Optimisation of the Facility Layout Proble mm

Universidade Técnica de Lisboa Instituto Superior Técnico

Optimisation of the Facility Layout Problem

Ricardo Jorge Gomes Mateus (Licenciado)

Dissertação para obtenção do Grau de Mestre em Investigação Operacional e Engenharia de Sistemas

Lisboa, Dezembro de 2000

ABSTRACT

The optimisation of industrial layout problems is addressed to take account for the economic importance and complexity inherent in such problems. A generic mathematical model for the Facility Layout Problem (FLP) is presented based on a rectilinear distance representation. An economic goal is considered as well as different real problem characteristics. Besides different equipment orientations, distance restrictions and nonoverlapping constraints, the model allows the definition of production sections along with safety and operability constraints in a non-rectangular available space. The previous state-of-the-art is expanded to include the existence of different input/output connectivity points within each equipment unit, irregular equipment units and 3D continuous space representation along with multifloor constraints. Also, an original formulation is addressed to model the simultaneous layout and design of FLP. In particular, the simultaneous layout, design and scheduling of multipurpose batch facilities is developed as a representative application. All optimisation models are formulated as Mixed Integer Linear Problem (MILP) problems, which are solved through the use of a standard branch and bound (B&B) optimisation package (CPLEX). The model applicability is explored through the presentation of various case studies. In general, good solution performances are presented.

Keywords: Optimal Layout, Industrial Facilities, Simultaneous Approach, Variable Layout, Design, Mathematical Programming

III

RESUMO

O problema do layout é de reconhecida complexidade e importância económica porquanto estas decisões influenciam significativamente outras a tomar no contexto da gestão do sistema produtivo. Um modelo matemático genérico para o problema do layout de instalações (FLP) é construído sobre uma representação baseada em distâncias rectilineares. Um objectivo económico é proposto juntamente com diversas características de problemas reais. Para além de diferentes orientações dos equipamentos, restrições de distância e de não-sobreposição, o modelo permite definir secções de produção e restrições de segurança e operacionalidade num espaço restrito não-rectangular. O vigente state-of-the-art é desenvolvido com a introdução de múltiplos pontos de ligação entre equipamentos, geometrias irregulares e a representação no espaço contínuo 3D com restrições multi-piso. Adicionalmente, uma formulação original é apresentada para a modelização simultânea do layout e projecto do FLP. Nesse contexto, é apresentada uma aplicação para a resolução do layout, projecto e escalonamento de instalações fabris multi-tarefa descontínuas. Todos os modelos são formulados através da Programação Linear Inteira Mista e resolvidos pelo método de branch and bound usando o software de optimização CPLEX®. A aplicabilidade do modelo é ilustrada através da resolução de vários casos de estudo. De uma forma geral, são apresentados bons resultados.

Palavras-Chave: Layout,

Instalações

Industriais,

Abordagem Simultânea,

Layout

Variável,

Projecto,

Programação Matemática

II

ACKNOWLEDGEMENTS

I would like to thank sincerely my supervisor Ana Paula Barbosa-Póvoa for the opportunity and constant support. She was a real supervisor in the full strength of the word, encouraging, elucidating, advising and setting free whenever necessary. I must also thank Tânia Pinto-Varela for helping me in various ways, friendship and wonderful working environment where a real team spirit was cultivated. This work wouldn’t have been possible without the resources and financial support offered by the Departamento de Engenharia de Sistemas at Instituto Nacional de Engenharia e Tecnologia Industrial (INETI). In particular, the guidance, understanding and positive attendance granted from Dr. Augusto Novais was of inestimable value. Also, I would like to thank many colleagues and friends at INETI, especially Pedro Carmona, Emídia Martins, Lígia Paula, Paulo Ferreira, Zé Barral, Rita Fuentes, Ana Margarida, Elsa Bernardo, Sérgio Estevão and Ricardo Estevão. Thanks to all my colleagues from MIOES for their valuable suggestions. In particular, Isabel Gomes-Salema, which supported, professionally and friendly, many stages of this work. Also, thank you to Pedro Matos, Vasco Ribeiro and Zé Coelho for your new friendship. Finally, I would like to thank my parents, Ana, Tininha, Chainho, Hugo, Mário and all “mabecos” for your real everlasting friendship.

This work has been supported by program PRAXIS XXI – grant PRAXIS/2/2.1/TPAR/453/95.

IV

CONTENTS

1

2

3

4

Introduction.............................................................................................................. 7 1.1

Motivation and Introduction ........................................................................................7

1.2

Problem Definition.......................................................................................................9

1.3

Thesis Outline ............................................................................................................10

Literature Review .................................................................................................. 13 2.1

Quadratic Assignment Problem .................................................................................14

2.2

Graph-Theoretic Approach ........................................................................................15

2.3

Heuristics ...................................................................................................................16

2.4

Mixed Integer Programming ......................................................................................17

2.5

Discussion and Objectives .........................................................................................18

The Layout Problem – Concepts and Assumptions............................................ 21 3.1

Types of Layouts in the Production Scope ................................................................21

3.2

The Facility Layout Problem .....................................................................................24

3.3

Block Layout versus Detailed Layout........................................................................24

3.4

Objective Functions ...................................................................................................25

3.5

Distance Assumptions................................................................................................28

Optimal 2D Layout of Industrial Facilities ......................................................... 31 4.1

Introduction................................................................................................................31

4.2

Layout Problem Statement and Characteristics .........................................................32

4.3

Layout Mathematical Formulations ...........................................................................33 4.3.1 Basic Detailed Layout Model ..........................................................................36 4.3.2 Layout with Production Sections .....................................................................52

4.4

Contents

Conclusions and Future Work....................................................................................57

V

5

Optimal 3D Layout of Industrial Facilities ......................................................... 59 5.1

Introduction................................................................................................................59

5.2

Layout Problem Statement and Characteristics .........................................................60

5.3

Layout Mathematical Formulations ...........................................................................62 5.3.1 Basic 3D Layout Model...................................................................................65 5.3.2 3D Layout with Multifloor ..............................................................................82 5.3.3 3D Layout with Production Sections ...............................................................89

5.4

6

Conclusions and Future Work....................................................................................94

Optimal Layout and Design of Industrial Facilities: A Simultaneous Approach .................................................................................... 97 6.1

Introduction................................................................................................................97

6.2

Design and Layout Problem Statement and Characteristics ......................................98

6.3

Design and Layout Mathematical Formulation..........................................................99 6.3.1 Basic 2D Layout Model with Variable Equipments and Connections ..........101

6.4

7

Conclusions and Future Work..................................................................................112

Simultaneous Layout and Design of Multipurpose Batch Processing Facilities: An Application ..................................................... 113 7.1

Introduction..............................................................................................................113

7.2

Problem Representation ...........................................................................................114

7.3

Problem Statement and Characteristics....................................................................115

7.4

Mathematical Formulation.......................................................................................117 7.4.1 Simultaneous 2D Layout and Design of Multipurpose Batch Processing Facilities......................................................120

7.5

8

Conclusions and Future Work..................................................................................135

Conclusions and Recommendations................................................................... 137

References............................................................................................................. 139

Contents

VI

Chapter 1 INTRODUCTION

1.1 - Motivation and Introduction Facility Layout Problems (FLP) can arise, at industrial level, due to a large number of new situations such as industry expansion, product addition or deletion, process changes, plant retrofit, adoption to new safety standards, amongst others. Its aims are similar independent of whether the organisations are manufacturing or services. Facilities are of crucial importance since they usually represent the largest and most expensive assets of the organisation (Canen and Williamson 1998). This economic weight along with the definitive nature of this kind of decisions presents a crucial key success factor in an organisation. Also, layout decisions often introduce significant effects in the production system and are often difficult to address. Therefore, a systematic and efficient way to deal with such problems is desirable. The FLP is a very active area and has been investigated by industrial engineers for several years. In fact, many papers have been published in the literature -see for comprehensive reviews Francis and White (1974), Mecklenburgh (1985), Kusiak and Heragu (1987) and Meller and Gau (1996). Meller and Gau (1996) identified nearly 100 papers published on FLP in the last 10 years. The Layout problem is usually formulated so as to guarantee the best equipment spatial arrangement in a restricted space accounting for the necessary connectivity while considering a set of plant goals. As mentioned by Francis and White (1974) and in detail later on (see chapter 3), some of these goals may include the minimisation of the plant investment, space efficient utilisation, operation safety, comfort, flexibility of arrangement and operation as well as minimisation of materials handling costs. The published literature concerning the FLP presents different ways of addressing the problem from where four main methodologies can be identified. These are respectively the Quadratic Assignment Problem (QAP), the Graph-Theoretic approach, Heuristics and finally the Mixed Integer Problem formulations (Meller and Gau 1996). Within these works the FLP has been essentially studied independently of any design consideration. Thus no systematic simultaneous approach to deal with the detailed design and layout of industrial facilities has been presented to the best of our knowledge. This is however

Introduction

7

an important point to be addressed since, as referred above, layout decisions often introduce significant effects in the production system and when addressing both problems simultaneously these interactions can be easily studied. Based on this, the first aim of this work was to fill in this gap. In order to achieve this goal the methodology developed along this thesis started by first addressing the FLP on its own. A more generic model than the ones presented in the literature and flexible enough to incorporate various types of layout (process, cellular or product layouts) was aimed at. The Mixed Integer Linear Programming (MILP) approach was the method chosen since apart from guaranteeing the optimal solution allows open models that can easily incorporate new problem characteristics. A generic MILP layout model has been developed considering a two dimensional (2D) continuous space. Different structural and operational characteristics, not yet addressed in the literature, have been studied. These are respectively, different equipment orientations, distance restrictions, non-overlapping constraints, different equipment connectivity inputs and outputs, irregular equipment shapes and space availability. Also, and in operational terms, production together with operational sections are modelled as well as safety and operability restrictions. This model is later extended to the three dimensional (3D) space where multifloor allocations are considered. Then, based on these 2D and 3D FLP generic models a variable layout model is proposed where variable allocation of equipment units and connections is studied. In this way the design and layout can be addressed simultaneously since the proposed model can be easily incorporated to generic design models. This model applicability is explored and the case of the detailed design and layout of a multipurpose batch plant is presented. Thus, scheduling, design and layout aspects of multipurpose batch industrial facilities are considered into a single model and promising results obtained.

Following on, the layout problem will be generically characterised and subsequently the thesis outline presented.

Introduction

8

1.2 - Problem Definition

The complete general facility layout problem addressed in this work can be stated formally as follows:

Given -

A set of equipment items and their geometrical shapes and sizes;

-

Input and output points locations inside each equipment space;

-

The connectivity structure;

-

Safety and operability minimum and maximum distances between equipment elements;

-

Production section characteristics and associated equipment units involved, if existent;

-

Multifloor characteristics and associated equipment involved (only in 3D layout case);

-

Space availability;

-

Cost of all connectivity structure;

-

Cost of all equipment units (only in the variable layout case);

Determine -

The optimal plant equipment arrangement – coordinates and orientation;

-

The optimal connectivity structure (inputs and outputs location);

-

Floor equipment allocation (only in 3D layout case).

So as to optimise a given quantitative objective function, generally the minimisation of the plant layout connectivity cost which models the materials handling cost, while fulfilling all the constraints defined.

This thesis assumes a continuous layout space where rectilinear distances between output and input points are used. This allows a more realistic estimation of the piping costs as opposed to direct connections. In fact, in real industrial environments the piping and instrumentation network is usually built in either aerial or underground right angle corridors (Bawa 1995). The equipment unit space considers, not only the central unit of operation, but also all the associated auxiliary equipment and instrumentation as well as the required space for maintenance and operability. Connection costs include capital and operational costs. In the 3D layout case, these are related with the direction of the fluids within the connection and the associated distance.

In the variable layout case, equipment items and connectivity structure can be defined as variable items whose allocation is defined after the model optimisation. If this is so, the

Introduction

9

objective function may include other goals namely the capital cost of equipments and/or connections.

The work presented in this thesis is focused mainly on the layout of industrial equipment units (i.e., machines, processing equipments, etc.). However this formulation can be easily used in service environments, for instance and applied to the more generic problem of workshop facilities where each equipment item represents a department with associated area and input/output aisles.

1.3 - Thesis Outline

This thesis is divided into eight chapters. In this chapter, Chapter 1, the motivation along with a short introduction to the layout problem is presented. The problem and the main objective are formally defined and some generic assumptions taken along this thesis adressed. In Chapter 2, the literature review of the field is discussed and important gaps identified. Based on this analysis, the main objectives of the thesis are expressed in order to build a more generic model for the layout problem. Chapter 3 addresses important background features, basic concepts and assumptions taken as basis along this thesis, such as traditional types of layout, objective functions and distance assumptions. In Chapter 4, an original 2D continuous mathematical model is presented for the simultaneous solution of the block and detailed layout problem, which considers different inputs and outputs within each equipment as well as irregular shapes. Production sections, safety and operability restrictions as well as space restrictions that lead to a non-rectangular available area can also be modeled. In Chapter 5, the 2D model, presented previously, is extended to account for the location of equipments inside a 3D multifloor continuos space. In the multifloor case, the equipments can be allocated within a pre-defined fixed number of floors with pre-defined height or over a variable number of floors each one with a variable height. Chapter 6 presents a variable mathematical formulation where the simultaneous approach to the layout and design of facilities is modelled as a single global generic problem. Based on the features presented in chapter 4, the model has the particularity of been easily adapted to any kind of design problem and is essentially focused on the layout characteristics while the design aspects are considered by the possible existence of a certain equipment or connection.

Introduction

10

Chapter 7 addresses an application of the model proposed in the previous chapter, the simultaneous layout and design of multipurpose batch processing facilities is studied. The resulting model determines simultaneously the optimal plant topology, layout and plant operation (schedule and resources consumption and production). Representative examples illustrating and demonstrating the applicability of the diverse models are addressed in each chapter. Finally, Chapter 8 resumes the work presented in this thesis. The areas of original features achieved are outlined. Also, work in process along with significant recommendations for future works are argued.

Introduction

11

Introduction

12

Chapter 2 LITERATURE REVIEW

The present work was motivated in the sequence of a PhD project (Barbosa-Póvoa 1994) named “detailed design and retrofit of multipurpose batch plants”. Based on this, the first aim of this work was to fill the existent gap in the literature to model simultaneously the design (or retrofit) and layout problems of multipurpose batch facilities. In order to achieve this goal a more generic Facility Layout Problem (FLP) model was developed where new characteristics not yet covered in the literature were incorporated. In this way, a model allowing the handling of various types of layout (i.e. process, cellular or product layouts) was obtained. The Mixed Integer Linear Programming (MILP) method was the approach chosen to model the problem. A number of reasons motivated this decision. First this thesis on the layout was meant to be a generalisation of the above mentioned work by Barbosa-Póvoa, which was built around a MILP formulation. In addition the MILP offers the benefit of guaranteeing, in theory, an optimal solution. Finally, the MILP formulation allows open models, which grants the necessary flexibility to introduce new problem characteristics resulting in generic frameworks.

This chapter covers the current state-of-the-art in solution methodologies for the layout problem, in particular for the FLP. Advantages of each approach are outlined, as well as their main limitations and gaps.

Layout algorithms guaranteeing an optimal solution are NP-complete (Sahni and Gonzalez 1976), that is, there isn’t any known algorithm that solves this problem in polynomial time, therefore, optimal procedures adopted are only solvable for dimension limited problems.

Obviously, first approaches to the layout problem relied on the creativity and experience of engineers. All alternatives were judged against a set of pre-defined characteristics where a compromise solution was chosen as the best (possible) solution. This simple straightforward method showed up to be slow and fastidious, neither efficient nor effective. Thus, new methods urged to be found. However, it must be said that interaction and human experience must never be apart at any point in the decision process.

Literature Review

13

Initially, as the empirical knowledge was being written new rules and patterns were concurrently being found contributing to the increase in the performance of the previous existent methods, originating the first group of heuristics.

In the middle 70’s, a disputed debate started in the scientific literature addressing this field. Computer algorithms performance versus visual inspection methods were in confront after the publication of a paper by Scribian and Vergin (1975) where better layout solutions were presented for the latter. Subsequently, this paper was criticised by Buffa (1976) and Coleman (1977) and extended by Block (1977).

Meanwhile, Muther (1955, 1973) presents a new revolutionary approach named Systematic Layout Planning (SLP). For each department pair, an initial preference rate chart is defined where the relative importance of each department being close to every other department is expressed. Next, a first schematic diagram is developed connecting the departments with different thickness lines evaluating closeness ratings. This initial diagram is progressively modified through a trial and error method until departments with high closeness ratings are adjacent and all constraints (e.g., area) satisfied.

Since then numerous authors have studied the layout problem and diverse methodologies have been published.

Francis and White (1974) and Mecklenburgh (1985) have accumulated detailed information on facility layout. Later, Kusiak and Heragu (1987) and Meller and Gau (1996) have reported a large number of research studies in review papers. These translate different types of applications to the main areas of workshops and manufacturing units. Four main methodologies types can be identified in the literature, respectively, the Quadratic Assignment Problem, the GraphTheoretic approach, Heuristics and more recently the Mixed Integer Programming formulation.

2.1 - Quadratic Assignment Problem (QAP) The Quadratic Assignment Problem (QAP) was the first mathematical model to address the layout problem. It is a special case of the facility layout problem since it assumes equal areas for each department or equipment items as well as fixed and known locations. This was first introduced by Koopmans and Beckman (1957) and later applied to a wide range of applications as reported by Meller and Gau (1996). QAP is NP-complete (Garey and Johnson 1979) Literature Review

14

therefore difficult arises in order to solve the problem. Several model alternatives have been proposed to bypass some of the referred restricted assumptions, namely by Kusiak and Heragu (1987), Rosenblatt and Golany (1991) and Liao (1993). Different types of solution methods were reported ranging from branch and bound algorithms, first introduced by Gilmore (1962) and Lawler (1963) or more recently by Kettani and Oral (1993), to recent general heuristics (Smith and MacLeod 1988, Kaku et al. 1991, Kaku and Rachamadugu 1991, Harmonosky and Tothero 1992), including simulated annealing (Wilhelm and Ward 1987), tabu search (Skorin-Kapov 1994) and genetic algorithms (Conway and Venkataramanan 1994, Tate and Smith 1995) as well as hybrid algorithms (Bland and Dawson 1994).

2.2 - Graph-Theoretic Approach Graph-Theoretic approaches optimise the layout taking into account the desirability of locating each pair of departments adjacent to each other (Foulds and Robinson 1978). Each facility is defined as a node within a graph–network where, initially, areas and shapes are ignored. Arcs represent satisfied pre-defined desirable adjacencies between pairs of facilities (nodes). The generic objective function (Meller and Gau 1996) to be optimised relies on the assumption that materials handling costs are reduced whenever two departments are adjacent. SLP differs from this approach (Gaither 1992) only for letting the objective function to include other goals (safety, supervision, etc.). Like in the pure QAP approaches, unequal area problems of even small size cannot be solved to optimality (Meller and Gau 1996). Several research papers have been published on this subject where different models and algorithms characteristics have been explored (Boswell 1992, Jayakumar and Reklaitis 1994, Kim and Kim 1995, Watson and Giffin 1997, Foulds and Partotiv 1998). For example, Jayakumar and Reklaitis (1996) considered the multifloor layout for batch plants where the main objective was not to find the optimal plant layout but the optimal allocation of units to floors, a particular case of the general layout problem. A graphical heuristic approach was developed to provide an upper bound to the true optimal value while a mathematical programming formulation was used to give the lower bound. Large-scale problems were solved using this approach.

Literature Review

15

2.3 - Heuristics Different types of Heuristics have been developed to address the layout problem. These appear not only as solution algorithms but also as model approaches, which try to exploit the layout problem characteristics. The main drawbacks are that generally no optimum can be guaranteed and the performance of a heuristic versus another method can not be established. However, these types of approaches appear as quite effective in solving several layout problems, mainly because they can solve higher dimension problems. The diversity of heuristic approaches creates difficulties in their identification. Some of the existent taxonomies classify them, just to name a few, as adjacency based algorithms vs. distance based algorithms, construction algorithms, improvement algorithms, n-way exchange algorithms, modified penalty algorithms as well as various classes of meta-heuristics. CRAFT developed by Armour and Buffa (1963) is an example of an improvement algorithm using 2 and 3-way exchange algorithms with a distance based objective function. SHAPE (Hassan et al. 1986) uses the same objective function type but explores a construction algorithm. The Deltahedron approach first published by Foulds and Robinson (1978) and later on extended by several authors (Al-Hakim 1991, Boswell 1992, Leung 1992) uses a construction adjacency based algorithm to determine the best layout. MULTIPLE is an improvement-type algorithm developed by Bozer et al. (1994) to address the multifloor problem using a discrete representation. Abdinnour-Helm and Hadley (1995) proposed an iterative layout heuristic for multifloor facilities. Aneke and Carrie (1986) studied the flow-line layout using a generic heuristic approach. Raoot and Rakshit (1991) developed a set of heuristics to model different placement procedures and Urban (1998) explored the dynamic layout problem using generic heuristics. Also, Gunn and Al-Alsadi (1987), Suzuki et al. (1991a), Amorese et al. (1991) and Meller and Bozer (1996) addressed the general layout problem for process plants using heuristic approaches. In particular, Meller and Bozer (1996) addressed the multifloor problem where a simulated annealing method was used to extend the previous MULTIPLE algorithm. Furthermore, meta-heuristics such as simulated annealing (Heragu and Alfa 1992, Tam 1992, Chwif et al. 1998), tabu search (Chiang and Kouvelis 1996) and genetic algorithms (Mak et al. 1998, Tavakkoli-Moghaddain and Shanyan 1998, Castell et al. 1998) have been extensively used. These have exploited different layout characteristics from the block to a more detailed layout problem. Hybrid algorithms, especially with simulated annealing, are presented (Heragu and Alfa 1992) as the most promising in terms of results achieved, i.e. solution quality and consumed time.

Literature Review

16

2.4 - Mixed Integer Programming Finally, and more recently, the Mixed Integer Programming has been receiving some attention in the scientific community as a way of modelling the layout problem. Montreuil (1990) was the first to formulate the FLP as a mixed integer program where a distance based objective was used in a continuous layout representation. This appears as an extension of the traditional discrete QAP formulation. Later, Heragu and Kusiak (1991) developed a specialised case of this model where the department orientation, length and width were given. Suzuki et al. (1991b) used integer programming to model the multifloor problem in batch plants, in a preference based objective function instead of an economic goal. Houshyar and White (1993) exploited the space discretization FLP with an adjacency based objective function. Lacksonen (1994) proposed a two-step algorithm for solving the layout problem while assuming that the departments can have varying areas. This algorithm is able to solve a general dynamic layout problem with varying departmental areas assuming that all are rectangular. The same author (Lacksonen 1997) extended the proposed model to deal with unequal areas and rearrangement costs. However, the model could only be solved to optimality for small problems. Penteado and Ciric (1996) proposed a mathematical programming model based method where particular attention was given to the safety aspects. The problem was solved as a relaxed MINLP and its application was shown through the solution of a 2D small case study. Jayakumar and Reklaitis (1996) studied the layout problem for multi-level plants where the minimization of the materials handling costs were taken as the main objective. An integer nonlinear model was developed which was linearized to a MILP problem providing a good approximation. More recently, Foulds et al. (1998) using the integer mathematical programming addressed the facilities layout problem with forbidden areas. Also, Georgiadis et al. (1997, 1999) used a space discretization technique to consider the allocation of equipment items to floors as well as the block layout of each floor. The main limitation of this formulation was the discretization of the available space that often could result in sub-optimal solutions and poor solution performances. Papageorgiou and Rotstein (1998) proposed a mathematical programming model for determining the optimal process plant layout. Their model employed a continuous space representation and took account of many important features of the plant layout problem. The complications arising from different equipment orientations, distance restrictions, nonoverlapping and space availability were taken into account. All the connectivity structures present in the plant were defined between the geometrical centres of each piece of equipment with only equipment rectangular shapes being considered. In operational terms, production or Literature Review

17

operational sections were modelled as well as safety and operability restrictions. However, some real case improvements allowing for a more detailed formulation can be introduced, as it will be discussed later. Meller et al. (1999) reformulated the Montreuil’s model tightening the feasible solution space by means of the acyclic subgraph structure underlying the model. Kim and Kim (1999) considered the problem of locating input and output (I/O) points of each department for a given block layout with the objective of minimising the total direct transportation distance. A new branch and bound algorithm was proposed which seems to perform efficiently even for large-size problems. However, the simultaneous solution of the block problem and the I/O points locations was not yet been solved. Finally, the simultaneous approach of the layout optimisation with other industrial decisions (design, scheduling, etc.) is a less studied field. Realff et al. (1996) presented a simultaneous approach to solve the design, scheduling and layout of pipeless batch plants, although assuming a discrete space layout with pre-defined positions. Thus, a generalised model allowing for the allocation of variable equipment units and connections is a promising and imperative field of study and will be next addressed.

2.5 – Discussion and Objectives This thesis, based on the reviewed work, in particular Montreuil (1990) and Papageorgiou and Rotstein (1998), develops a more generic mathematical programming approach for the generalised facilities detailed layout problem where some of the identified drawbacks pointed out above are addressed as well as important layout characteristics studied. It is clear that the layout problem has been extensively studied by the scientific community along with an increasing trend to address more realistic problems. However, some gaps within the existent work can still be identified. As already stated, traditionally all the connectivity structures present in a plant are defined between the geometrical centers of each piece of equipment. This is a tenable assumption, which can, nevertheless, lead to sub-optimal solutions. Therefore, a model that considers different inputs and outputs within a piece of equipment seems to be more realistic. Also, previous works allow usually only rectangular shapes to describe the equipment units. Such an assumption will be extended by enabling irregular shape representations, which are more realistic models. Finally, although the optimal layout of facilities within a 2D continuous space has been developed previously, 3D layout has not yet been comprehensibly studied. This is however often present at real industrial applications, thus

Literature Review

18

an extension to account for the location of the equipment units within a 3D continuous space is demanding where multifloor constraints may be applied. Based on these gaps, the present dissertation tries to develop a more generic model for the layout problem. This model uses the mathematical programming approach where the following points are originally studied: -

Existence of different input/output connectivity points inside each equipment unit;

-

Irregular shape equipment unit representations;

-

3D continuous space representation with multifloor constraints;

-

Simultaneous design and layout problems.

Literature Review

19

Literature Review

20

Chapter 3 THE LAYOUT PROBLEM - Concepts and Assumptions

This chapter focuses the background, basic concepts and assumptions taken as basis along this thesis.

3.1 - Types of Layouts in the Production Scope Production diversity and volume are key decision factors when dealing with production systems. The former concerns the mix of products produced by the organisation while the production volume is related with the amount of work needed to its production, throughout a specified time period. The volume/variety ratio can be seen as an indirect index of the degree of specialisation of the various production resources leading, in practice, to different types of production systems. These are usually divided in four main groups: - Project; - Job-Shop; - Batch Flow; - Continuous Flow.

Project applies frequently to a large and bulky product with unique and complex conception and execution. Typical examples are bridges and building construction or aircraft and ships manufacture. Therefore, it implies a huge allocation amount of resources, during a short period of time and a long time of delivery. Job-Shop can also be classified as Batch Flow, although some authors consider both separately. Within this clarification, Job-Shop concerns products with least conception and execution complexity against the previous, but larger and dissimilar individual demands. Usually are simultaneously manufactured short quantities of a great diversity of products, with various

The Layout Problem – Concepts and Assumptions

21

operational sequences and short execution times, sharing concurrently the same production resources. Common examples are small older metallomechanical plants. Batch Flow occurs when product demand has a considerable dimension and there are expressive setup times. Therefore the imposing size batch production in an extent larger than the orders, using the existent production resources concurrently. Furthermore, the product diversity is smaller and operational sequences more similar. Examples are typical textile and furniture factories. In Continuous Flow, production is associated with a small related set of products when not always unique. High volumes are processed, often, in more than one continuous production line. Commonly are implemented in large plants, such as, oil refinery, iron foundry or automobile factories. Some authors define further another category called Line Flow. In this way they differentiate clearly, the manufacture of a single product (continuous flow) and the manufacture of a related and limited set of products (line flow). Obviously, these two approaches can lead to different layouts (production lines).

Summarising, in a dissimilar production is essential the use of changeable and multi-task production resources. Those resources are shared by the different products within the production cells/sections according to the products operational similarities (job-shop, batch flow). As far as the production volume increases, a greater shared resource allocation per product becomes economically justifiable. At the extreme, when the production volume is huge with a restrict product diversity the continuous flow appears as the more profitable solution.

Thus, the production system types predisposes in a decisive way the types of layout to be implemented in the plant (see figure 3.1), traditionally named as: - Static product or fixed position layout; - Process layout; - Group or cellular manufacturing (CM) layout; - Product or production line layout.


22

Volume/ Diversity

Specialisation

+

Continuos Flow

Product Line Product Line

Batch Flow

Cellular Process

Job Shop Project -

Process Process Static

Production System

+

Layout

Figure 3.1 – Production Systems vs. Layouts

Static product plant layouts are normally used with Project production. The product is manufactured or assembled at a fixed location while machinery and workmanship move and work around it, according to the product requirements. Process layouts are commonly used in Job-Shop and Batch Flow productions. Production resources are static and grouped spatially by nature or type. A large amount of materials handling is present since spare parts need to be constantly moved between departments. The processes specialisation is therefore allowed. Group layout cluster production resources by product families. The equipments are grouped according to the product families, instead of their physical characteristics, forming operational sequences. Each group forms a cell. Each cell has its own materials handling system, such as a robot or a conveyor system. Batch Flow production is usually associated with this type of layout. Product layout is intrinsically related to Continuous Flow production. The production resources are frequently complex and automated and also spatially disposed according to the product operational sequence, typically in line.

In short, there is a parallelism among production volume and diversity product, production system and layout, as can be seen in figure 3.1. Nevertheless, it must be stated that this conformity is not at all rigid.


23

3.2 – The Facility Layout Problem It is important to note that in this dissertation the attention is focused on the layout of industrial equipment units (i.e. machines, processing equipment etc.) generally known as the Facility Layout Problem (FLP). Meller et al. (1999) presented a good definition to the FLP that was initially adopted literally in this dissertation: “in the FLP we are to find a non-overlapping planar orthogonal arrangement of n rectangular departments within a given rectangular facility of size Lx x Ly so as to minimise the distance based measure

∑f

ij

⋅ d ij where fij is the amount of flow between departments i

j >i

and j and variable dij is the rectilinear distance between their centroids”. However, the developed formulation can easily be applied to the more generic problem of workshops facilities where each equipment represents a department with area and related input/output aisles. Thus, it must be stated that, along this thesis, facilities, departments, blocks and equipments are used as synonymies to express the space used by the central unit of operation and all the associated auxiliary equipment and instrumentation as well as the required area for maintenance and operability. It is also assumed that this space is described by rectangular shapes. This assumption is based in current industrial practice, where the starting point for the layout design is often a set of rectangular modules comprising a central processing unit(s) along with the associated equipment and instrumentation (Bawa 1995).

3.3 – Block Layout versus Detailed Layout As stated by Meller and Gau (1996), due to its inherent complexity, the layout problem can be solved in two steps. In a first stage the block layout problem is solved then the detailed layout problem is obtained. The block layout problem (see figure 3.2) concerns the optimal arrangement of the relative location of each department, namely its dimensions and geometry, within a given facility. Subsequently, on the solution previously obtained, further work can be performed to obtain the detailed layout (see figure 3.3). In this case, different levels of generalisation can be addressed through the modelling of different layout characteristics such as I/O points locations, aisle structures and the exact layout within each department.


24

1

2

5

3

4

Figure 3.2 - Block Layout

Figure 3.3 - Detailed Layout

The work of this thesis aims to develop the two approaches simultaneously into a single one. The level of detail achieved addresses important real problems such as flow-line layout, production equipment layout (i.e. machines), cellular manufacturing design and group technology layout amongst others.

3.4 – Objective Functions The problem is often formulated with the aim of determining the most efficient physical arrangement of a production system accounting for a large set of layout decisions. These decisions must account for a whole range of factors such as maintenance, space availability, operability and process needs while considering a plant goal.


25

As stated before, FLP applies similarly in manufacturing and services. Accordingly to the nature of the space whose layout is to be optimised (Francis and White 1974, Gaither 1992, Meller and Gau 1996, Canen and Williamson 1998, Lin and Sharp 1999), typical objectives can be stated (see table 3.1):

Table 3.1 – Objectives for the Layout Problem Objectives for Manufacturing Operation Layouts: i. Achieve the strategic and operational goals with least capital investment ii. Reduce materials handling costs, products and information between departments iii. Provide enough production capacity iv. Allow space for production resources (machines, workers, administrative, leisure other personal-care areas, etc.) v. Increase productivity vi. Provide for production flexibility (product volume and diversity – product mix) vii. Secure employee safety viii. Environment quality (e.g. lighting, ergonomics, handicapped access) and health (e.g. pollution and noise level) ix. Allow ease of supervision and maintenance x. Conform to site and building constraints xi. Aesthetic factors xii. Surrounding (e.g. external access, site appearance, direction of sun and prevailing wind, and community environment) Additional Objectives for Service Operation Layouts: xiii. Provide for customer comfort and convenience xiv. Reduce travel of personnel and customers xv. Allow attractive display and setting for customers xvi. Provide for privacy in work areas Additional Objectives for Warehouse Operation Layouts: xvii. Allow ease of inventory record keeping (stock picking, order filling) and material accounting xviii. Promote efficient loading and unloading of shipping vehicles xix. Increase warehousing space Additional Objectives for Office Operation Layouts: xx. Reinforce organisation structure xxi. Promote communication between work areas xxii. Reduce travel of personnel and customers xxiii. Provide for privacy in work areas


26

The objectives presented can be categorised in two main groups: -

Qualitative (e.g., goals vi, vii, xi, xiii, xvi and xx traditionally valued through multi-criterion approaches);

-

Quantitative (e.g., goals ii, iii, v, xiv and xix).

Lin and Sharp (1999) present a set of new innovative indices for the flow issues of the layout. These are clearness; space sufficiency and utilisation; aisle; distance and volume density; and building expansion.

In this work, the aim is to optimise an economic goal, essentially based on the minimisation of materials handling costs. This goal has been traditionally adopted since 20% to 50% of operating expenses are due to materials handling and an efficient layout optimisation may imply a cost reduction about 10% to 30% (Tompkins and White 1984). This goal is often translated into the minimisation of the plant layout investment, more specifically the reduction of the materials handling cost, such that all the physical and operational plant restrictions are observed.

Nevertheless, the optimal solution found must be always seen as a relative optimal solution since it was build over a model, which is only an attempt to represent the complexity of real case problems. Thus the solutions obtained must be used having in mind this limitation. In short, the model must be seen as a Decision Support Tool that helps the decisor.

The generic Objective Function (OF) to be used in this thesis is based on the principle that materials handling costs increase with the distance that the unit load must travel between departments:

Min

∑∑(f i

ij

⋅ cij ) ⋅ d ij

(3.1)

j

where the variable d ij is the distance between departments i and j; f ij describes the flow (or number of travels) from department i to department j; and cij the cost to move one unit load, in one distance unit, from i to j. This OF can be divided in piping costs and pumping costs.


27

If the layout problem is defined in the 3D space, the vertical distance must be considered in addition to the horizontal distance. Also, if the OF to be optimised includes the design problem, a new term must be added to equation (3.1), therefore the following equation is obtained:

Min

∑∑ f i

ij

⋅ (cijH ⋅ dijH + cijV ⋅ dijV ) + ∑ Ci ⋅ Ei

j

(3.2)

i

where cijH ( cijV ) defines the horizontal (vertical) materials handling cost and d ijH ( dijV ) the horizontal (vertical) distance between departments i and j. Ci denotes the capital cost of the department i that must be summed whenever the equipment is present ( Ei ).

Thus, the general OF chosen embodies two main expressions: variable operational costs addressing horizontal and upward flows (for the 3D case); and fixed capital costs addressing connections and equipments costs (for the variable layout case).

3.5 – Distance Assumptions The distance ( d ij ) in equation (3.1) can be measured in a variety of ways (Francis and White 1974, Meller and Gau 1996). As for the metric used (Tompkins and White 1984) to measure the distance between the coordinates (x,y) of two department points i and j, this can be defined through:

•

Euclidean distance: appropriate when distances are measured along a straight-line path connection

(x •

between

department

points,

for

example,

conveyor

travel:

− x j ) + (yi − y j ) 2

i

two 2

Rectilinear distance: the most common used, where the travel is defined along parallel paths to the orthogonal axes: xi − x j + y i − y j


28

As for the location of the connection input/output (I/O) points coordinates between departments, three ways are traditionally used to measure the distance between them:

•

Centroid-to-centroid (CTC) distance: this distance is defined between the coordinates of the centroids of each department where each centroid represents simultaneously the input and output points of each department. The major drawbacks of this approach are: optimal layouts defined with concentric rectangles (Tompkins and White 1984); departments areas very long and narrow (Tate and Smith 1993) and the difficulty to model irregular shapes (e.g. L-shaped) that may have a centroid defined outside the department (Francis and White 1974);

•

Expected distance (EDIST): this distance measures the expected (rectilinear) between two departments when its I/O points are not known (Bozer and Meller 1997) addressing some of the shortcomings related with the CTC distance measure;

•

Distance between connection I/O points: this distance is defined between the coordinates of the output and input points defined within each department.

The distance metric adopted along this dissertation is the rectilinear distance since the connection network is usually built in either aerial or underground right angle corridors (Bawa 1995). Rectilinear distances are assumed, in this case, to give a more realistic estimation of the connection costs as opposed to direct connections. This distance is defined between the connection I/O points. A new formulation will be built to achieve this goal, based on the work of Papageorgiou and Rotstein (1998).

Based on the assumptions here presented, the mathematical models developed will be presented in the following chapters.


29


30

Chapter 4 OPTIMAL 2D LAYOUT OF INDUSTRIAL FACILITIES

4.1 - Introduction Layout is an important and complex aspect of the design of industrial plants. In this chapter a mathematical model is presented for the design of efficient and generic industrial layouts, where a simultaneous solution of the block and detailed layout problem is considered. An original model that considers different inputs and outputs inside each equipment as well as irregular shapes is developed. The optimal plant layout is obtained based on the minimisation of the connectivity cost, providing the optimal arrangement of all the equipment items as well as their interconnecting pipe-work over a two dimensional (2D) area. Different topological characteristics are considered such as different equipment orientations, distance restrictions, non-overlapping constraints, different equipment connectivity inputs and outputs, irregular equipment shapes and space availability over a 2D continuous area. In operational terms, production together with operational sections are modeled as well as safety and operability restrictions. A Mixed Integer Linear Problem (MILP) is developed where binary variables are introduced to characterize topological choices and continuous variables describe the distances and locations involved. To conclude, the applicability of the proposed formulation is illustrated via a set of representative examples.

This chapter is organised as follows. In the next section, the main features of the layout problem are stated. In section 4.3, a mathematical formulation for the industrial layout is developed as a MILP. The layout is first formulated in its more simple form and later on extended to address the existence of production sections. Some examples are solved to show the model applicability. Finally, section 4.4 presents some conclusions and directions for further analysis on the following chapters.

Optimal 2D Layout of Industrial Facilities

31

4.2 - Layout Problem Statement and Characteristics The layout of industrial plants as addressed in this chapter can be stated formally as follows:

Given •

A set of equipment items and their geometrical shapes and sizes over a 2D space;

•

Input/output points locations within each equipment area;

•


•

Space and equipment allocation limitations;

•

Safety and operability minimum and maximum distances between equipment items;

•

Production sections characteristics and related equipment unit involved, if present;

•

Space availability;

•

The capital and operational cost of all connectivity structures;

•

The optimal plant equipment arrangement – coordinates and orientation;

•

Connectivity structure (inputs and outputs coordinates).

Determine

So as to minimise the plant layout connectivity cost which models the materials handling cost, in terms of rectilinear distances.

The detailed layout problem is formulated assuming a set of equipment units and the associated connectivity. A 2D plant is considered over a continuous area where both rectangular and irregular shapes can be used to describe the equipment units. Multiple equipment connectivity inputs and outputs are considered as well as space limitations and safety and operability constraints.

Each equipment item is defined within the model as an equipment unit g, which can be described by an irregular shape formed by a set of equipment elements j (see figure 4.1). These equipment elements have a rectangular shape with given dimensions over the x and y–axis (αj,

βj) and possible multiple input and output points. Figure 4.1 shows an irregular equipment unit g formed by three rectangular equipment elements j1, j2 and j3 – in its original position defined with a 0° rotation over the x and y-axis. Out of


32

three possible combinations, the following two are assumed: j1 with j2 and j2 with j3. Each link is established by the definition of the fixed relative distance between the centroids of the two equipment elements. For the latter this is expressed as ∆xj2,j3 and ∆yj2,j3, respectively over the x and y-axis. The equipment unit depicted has one input point (oi1) and two output points (oi2 and oi3) defined respectively on equipment elements j3 and j1. For instance, the input point oi1 is fixed by the definition of the relative distance, in the x and y-axis (∆xj3,oi1, ∆yj3,oi1), between its location and the centroid of the equipment element j3 where it is circumscribed (see figure 4.1).

y Left oi2

Right

oi3

Back

∆xj2,j3

β j1

∆yj2,j3 j1

j2 αj

∆yj3,oi1

1

j3

oi1

Front

∆xj3,oi1

x

Figure 4.1 – Equipment Unit – Original Position

4.3 – Layout Mathematical Formulations Based on the above model characteristics the following indices, sets, parameters and associated variables are defined:

Global Indices: g – equipment unit j, j´– equipment element oi, oi’ – point (output or input) p, p´- production section


33

Sets:

J g = {j: set of equipment elements j that form unit g} G j = {g: equipment unit containing equipment element j} – Unique set Linksg = {(j,j´): set of linked equipment elements pairs (j,j´) ∈ Jg}

Zonx max = {(j,j´): set of equipment elements pairs (j,j´): Gj ≠ Gj´ with an upper limit distance between them along the x-axis}

Zony max = {(j,j´): set of equipment elements pairs (j,j´): Gj ≠ Gj´ with an upper limit distance between them along the y-axis} OI j = {oi: set of (output or input) points of equipment element j}

Pj = {p: production section of equipment element j} - Unique set J p ={j: set of equipment elements j within production section p} ZonX max = {(p,p´): set of production sections pairs with an upper limit distance between them over the x-axis}

ZonY max = {(p,p´): set of production sections pairs with an upper limit distance between them over the y-axis} Parameters:

C oi ,oi ' – cost per meter of the connection between the output point oi and the input point oi’

α j , β j – dimensions of equipment element j over the x and y-axis respectively x max , y max – maximum area over the x and y-axis ∆x j ,oi , ∆y j ,oi – relative distance between the point oi and the geometrical center of the

equipment element j respectively in the x and y-axis, as stated by the original equipment representation (0° rotation) min Zx min j , j ' , Zy j , j ' - minimum distance allowed between equipment elements j and j´ over the

x and y-axis respectively max Zx max j , j ' , Zy j , j ' - maximum distance allowed between equipment elements j and j´ over

the x and y-axis respectively ∆x j , j ' , ∆y j , j ' - relative distance between the geometrical centers of equipment elements

(j,j´) : Gj = Gj´ as defined by its original position, with a 0° rotation over both the x and y-axis, respectively


34

min ZX pmin , p´ , ZY p , p´ - minimum distance allowed between sections p and p´ in the x and y-

axis respectively max ZX pmax , p´ , ZY p , p´ - maximum distance allowed between sections p and p´ in the x and y-

axis respectively

Variables: Continuous Variables - all defined as positive variables: xj, yj – coordinates of the geometrical center of equipment element j xoi, yoi – coordinates of the point oi lj – length of equipment element j dj – depth of equipment element j Rioi,oi’ – relative distance in x coordinates between the output point oi and the input point oi’, if oi is to the right of oi’ Leoi,oi’ – relative distance in x coordinates between the output point oi and the input point oi’, if oi is to the left of oi’ Baoi,oi’ – relative distance in y coordinates between the output point oi and the input point oi’, if oi is in the back of oi’ Froi,oi’ – relative distance in y coordinates between the output point oi and the input point oi’, if oi is in front of oi’ Doi,oi’ – total rectilinear distance between the output point oi and the input point oi’ Xp, Yp - coordinates of the geometrical center of the production section p Lp, Dp – length and depth of the production section p Binary Variables: og - equipment unit g orientation: = 1 if the length of all the equipment elements j∈Jg (parallel to the x axis) is equal to α j ; 0 otherwise o1g, o2g, o3g and o4g – definition of the equipment unit g anti-clockwise rotation, expressed in multiples of 90° (respectively: 0°, 90°, 180° and 270°) from the original equipment representation

Two models are going to be developed. The first describes the generic detailed layout problem while the second is an extension of the former accounting for the existence of production sections.


35

4.3.1 – Basic Detailed Layout Model

The following objective function and constraints characterise the generic model for the industrial layout problem:

Objective Function:

The minimisation of the total connectivity cost is considered. This involves the cost of the physical connections and the operating costs caused by material transfers occurring within the connection.

Min

∑

C oi , oi ' ⋅ D oi , oi '

(4.1)

( oi , oi ' ) | C oi ,oi ' ≠ 0

All the possible outputs and inputs of equipment units g with a non-zero connection cost are considered.

Constraints:

A number of different types of constraints need to be introduced in order to model the detailed layout characteristics. These include equipment orientation, equipment irregular shapes, input/output locations, connectivity distances, equipment non-overlapping, safety and operability restrictions, and finally, area allocation constraints.

Equipment Orientation Constraints: for the equipment orientation it is assumed that the equipment unit rotation is allowed over the x and y-axis and is defined through the value of variable og (see figure 4.2). Based on this, the following equations are written:

l j = α j ⋅ ο g + β j ⋅ (1 − ο g )

d j =α j + β j −lj

∀g , j | j ∈ J g

∀j

(4.2)

(4.3)

where through the use of the equipment elements dimensions (αj, β j) and the value of the orientation binary variable (og), the equipment element length (lj) and depth (dj) are obtained.


36

Thus, from equation (4.2) if og =1 then the length of each equipment element j, belonging to unit g, is equal to αj, otherwise is given by βj. On the other hand, equation (4.3) states that the depth of equipment element j is equal to the remaining equipment dimension. That is, if lj is equal to αj then dj is equal to βj, otherwise dj is equal to αj.

Equipment Irregular Shapes Constraints: an equipment unit g may have an irregular form dictated from the set of its equipment elements j. Each equipment element j within the equipment unit set (Jg) is related to the remaining equipment elements through the definition of pre-defined distances between them (∆xj,j´, ∆yj,j´). Knowing the pre-defined set of linked equipment elements pairs (Linksg) and assuming that each equipment unit g is at one of the four orthogonal positions allowed – o1g, o2g, o3g and o4g – obtained by successive anti-clockwise rotations from the original position in relation to the x and y-axis, we can write the following equations, assuming two generic equipment units with geometric centers at (xj, yj) and (xj’, yj’):

x j ' = x j + o1g ⋅ ∆x j , j ' − o2 g ⋅ ∆y j , j ' − o3 g ⋅ ∆x j , j ' + o4 g ⋅ ∆y j , j ' ∀g, j, j' | ( j, j' ) ∈ Linksg

(4.4)

y j ' = y j + o1g ⋅ ∆y j , j ' + o 2 g ⋅ ∆x j , j ' − o3 g ⋅ ∆y j , j ' − o 4 g ⋅ ∆x j , j '

∀g , j , j ' | ( j , j ' ) ∈ Links g

(4.5)

And since each equipment unit can only exist in a single position:

ο1 g + ο 2 g + ο 3 g + ο 4 g = 1

∀g

(4.6)

Also, the equipment unit orientation is obtained from:

ο g = ο1 g + ο 3 g

∀g

(4.7)

The possible four ortoghonal positions of an equipment unit with two equipment elements j and j´ (1 and 2, respectively) are shown in figure 4.2. The user must settle the original representation (0° rotation) through the definition of parameters ∆x1,2 and ∆y1,2. Constraints (4.4) and (4.5) applied for each case are shown in subtitle.


37

0°°:

90°°: o2g = 1; og = 0

o1g = 1; og = 1

Original representation (user defined)

∆x1,2

(x2,y2) (x2,y2) ∆y1,2

(x1,y1)

(x1,y1)

−∆ y1,2

∆x1,2

 x2 − x1 = ∆x1, 2   y 2 − y1 = ∆y1, 2

 x2 − x1 = − ∆y1, 2   y 2 − y1 = ∆x1, 2

180°°: o3g = 1; og = 1

270°°: o4g = 1; og = 0

(x1,y1) (x2,y2)

−∆x1,2

−∆y1,2

(x1,y1)

(x2,y2)

∆ y1,2

−∆x1,2

 x 2 − x1 = − ∆x1, 2   y 2 − y1 = − ∆y1, 2

 x 2 − x1 = ∆y1, 2   y 2 − y1 = − ∆x1, 2

Figure 4.2 – Equipment Unit Orientation

Input/Output Constraints: different input and output point locations (xoi, yoi) are considered within the model which are defined within each equipment element j. As the previous constraints, its locations can easily be calculated using the orientation variables, as defined in figure 4.3, combined with the equipment element geometrical center coordinates (xj, yj). Therefore, along with constraints (4.6) and (4.7), we have:

x oi = x j + ο1 g ⋅ ∆x j ,oi − ο 2 g ⋅ ∆y j ,oi − ο 3 g ⋅ ∆x j ,oi + ο 4 g ⋅ ∆y j ,oi

∀g , j ∈ J g , oi ∈ OI j


(4.8)

38

y oi = y j + ο1 g ⋅ ∆y j ,oi + ο 2 g ⋅ ∆x j ,oi − ο 3 g ⋅ ∆y j ,oi − ο 4 g ⋅ ∆x j ,oi

∀g , j ∈ J g , oi ∈ OI j

(4.9)

where xoi and yoi characterise the different point coordinates within the equipment area used for the description of the connectivity existence. For model simplicity these points were defined as global in the model but it is easily understood that they are located within the different equipment elements present. This is guaranteed through the definition of the relative distance (∆xj,oi, ∆yj,oi) between the points and the geometrical center of the respective equipment element j.

0°°:

90°°: o2g = 1; og = 0

o1g = 1; og = 1

Original representation (user defined) (xoi6,yoi6)

(xoi6,yoi6)

−∆ y4,oi6

∆x4,oi6

∆y4,oi6

∆x4,oi6

(x4,y4)

(x4,y4)

∆x3,oi5

(x3,y3)

∆y3,oi5

(x3,y3) (xoi5,yoi5)

(xoi5,yoi5) −∆ y3,oi5

∆x3,oi5

 xoi 6 = x 4 + ∆x 4,oi 6   y oi 6 = y 4 + ∆y 4,oi 6

 xoi 6 = x 4 − ∆y 4,oi 6   y oi 6 = y 4 + ∆x 4,oi 6

180°°: o3g = 1; og = 1

270°°: o4g = 1; og = 0 ∆y3,oi5

(xoi5,yoi5)

(xoi5,yoi5) −∆x3,oi5

(x3,y3)

−∆y3,oi5

−∆ x3,oi5

(x3,y3) (x4,y4)

(xoi6,yoi6)

−∆x4,oi6

−∆y4,oi6

(x4,y4)

∆y4,oi6

−∆ x4,oi6

 xoi 6 = x 4 − ∆x 4,oi 6   y oi 6 = y 4 − ∆y 4,oi 6

(xoi6,yoi6)

 x oi 6 = x 4 + ∆y 4,oi 6   y oi 6 = y 4 − ∆x 4,oi 6

Figure 4.3 – Equipment Unit Input/Output Points


39

Figure 4.3 exhibits the possible four-ortoghonal positions of an equipment unit with two equipment elements j and j´, respectively equipment elements 3 and 4. Point oi6 (xoi6, yoi6) is within equipment element 4 and point oi5 (xoi5, yoi5) is within equipment element 3. For illustration, the relative constraints for the coordinates of point oi6 (xoi6, yoi6) are shown in subtitle.

Distance Constraints: rectilinear distances are assumed between each output and input point of each equipment element. Accounting for a 2D area the total distance for each non-zero cost connection (oi, oi’) is given by:

Doi ,oi ' = Rioi ,oi ' + Leoi ,oi ' + Baoi ,oi ' + Froi ,oi '

∀(oi, oi ' ) | Coi ,oi ' ≠ 0

(4.10)

where, the relative distances (Ri, Le, Ba and Fr) are obtained from:

Rioi ,oi ' − Leoi ,oi ' = xoi − xoi '

∀(oi, oi ' ) | Coi ,oi ' ≠ 0

(4.11)

Baoi ,oi ' − Froi ,oi ' = yoi − yoi '

∀(oi, oi ' ) | Coi ,oi ' ≠ 0

(4.12)

If the output point oi is to the right of the input point oi’, then xoi is greater than xoi’. Thus, from equation (4.11) we have Le (from Left) equal to zero and Ri (from Right) takes the positive distance difference between the connection points, since the problem being solved is a minimisation and, therefore, only one variable, at most, of the pair (Ri, Le) will be non-zero at the optimal solution. The same idea is translated into equation (4.12) in terms of the y-axis (Fr Front and Ba - Back).

Non-overlapping Constraints: two-equipment elements j and j´ cannot occupy the same physical location. Tsai et al. (1993) proposed an efficient way to model this problem that was later explored by Papageorgiou and Rotstein (1998). A set of disjunctive non-overlapping constraints is formulated to guarantee this condition which it was adapted to this model:

x j − x j ' + M x ⋅ ( E1 j , j ' + E 2 j , j ' ) ≥

l j + l j'

2


∀j , j ' | j ' > j ∧ G j ≠ G j´

(4.13)

40

x j ' − x j + M x ⋅ (1 − E1 j , j ' + E 2 j , j ' ) ≥

l j + l j'

y j − y j ' + M y ⋅ (1 + E1 j , j ' − E 2 j , j ' ) ≥

y j ' − y j + M y ⋅ (2 − E1 j , j ' − E 2 j , j ' ) ≥

∀j , j ' | j ' > j ∧ G j ≠ G j´

2 d j + d j'

2

(4.14)

∀j , j ' | j ' > j ∧ G j ≠ G j´

(4.15)

∀j , j ' | j ' > j ∧ G j ≠ G j´

(4.16)

d j + d j'

2

where only one of the above constraints can be active. This is obtained through the possible four combinations of the values of the two new auxiliary binary variables (E1j,j´ and E2j,j´). The values of Mx and My are taken as suitable upper bounds on the distance between any twoequipment elements j and j´ given by:   M x = min  x max , ∑ max(α j , β j ) j  

(4.17)

  M y = min  y max , ∑ max(α j , β j ) j  

(4.18)

Safety/Operability Constraints: often minimum and maximum distances between equipment items are defined due to safety and operability conditions. In order to guarantee a certain minimum possible distance between equipments, constraints (4.13) to (4.16) must be replaced by:

x j − x j ' + ( M x + Zx min j , j ' ) ⋅ ( E1 j , j ' + E 2 j , j ' ) ≥

l j + l j'

2

+ Zx min j, j' ∀j , j ' | j ' > j ∧ G j ≠ G j´

x j ' − x j + ( M x + Zx min j , j ' ) ⋅ (1 − E1 j , j ' + E 2 j , j ' ) ≥

l j + l j'

2

+ Zx min j, j' ∀j , j ' | j ' > j ∧ G j ≠ G j´


(4.19)

(4.20)

41

y j − y j ' + ( M y + Zy min j , j ' ) ⋅ (1 + E1 j , j ' − E 2 j , j ' ) ≥

d j + d j'

2

+ Zy min j, j'

∀j , j ' | j ' > j ∧ G j ≠ G j´

y j ' − y j + ( M y + Zy min j , j ' ) ⋅ ( 2 − E1 j , j ' − E 2 j , j ' ) ≥

d j + d j'

2

(4.21)

+ Zy min j, j'

∀j , j ' | j ' > j ∧ G j ≠ G j´

(4.22)

On the other hand, maximum possible distance restrictions can be modelled as follows:

x j − x j' ≤

x j' − x j ≤

y j − y j' ≤

y j' − y j ≤

l j + l j'

2 l j + l j'

2

+ Zx max j, j '

∀( j , j ' ) ∈ Zonx max

(4.23)

+ Zx max j, j '

∀( j , j ' ) ∈ Zonx max

(4.24)

d j + d j'

2 d j + d j'

2

+ Zy max j, j'

∀( j , j ' ) ∈ Zony max

(4.25)

+ Zy max j, j'

∀( j , j ' ) ∈ Zony max

(4.26)

Also, these constraints may be defined over input and/or output points due to operability conditions such as expedition. For instance, for two points oi and oi’:

xoi − xoi ' ≤ Zx oimax , oi '

∀(oi, oi ' ) ∈ Zoix max

(4.27)

xoi ' − xoi ≤ Zx oimax , oi '

∀(oi, oi ' ) ∈ Zoix max

(4.28)

y oi − y oi ' ≤ Zy oimax ,oi '

∀(oi, oi ' ) ∈ Zoiy max

(4.29)

y oi ' − y oi ≤ Zy oimax ,oi '

∀(oi, oi ' ) ∈ Zoiy max

(4.30)


42

max where Zxoimax ,oi´ and Zy oi ,oi´ are the maximum distance allowed between points oi and oi´

respectively in the x and y-axis. Zoixmax and Zoiymax are the set of point pairs (oi, oi´) with a distance upper limit between them respectively in the x and y-axis. A special care must be taken when defining the maximum distances since due to the equipment unit/elements dimensions the applicability of these restrictions may result in an infeasible solution. For instance, in a situation where two connection points are within its equipment elements and the sum of its margins is greater than the maximum allowed distance.

Allocation Constraints: additional constraints must be written whenever the available area is limited to the dimension of a given industrial plant. Thus, on the one hand, lower bounds on the equipment coordinates are defined so as to avoid intersection of equipment elements with the axes:

xj ≥

yj ≥

lj 2

dj 2

∀j

(4.31)

∀j

(4.32)

Similarly, upper bound constraints are written to force the equipment allocation within a predefined rectangular area confined by (0, 0) and (xmax, ymax):

xj +

yj +

lj 2

dj 2

≤ x max

≤ y max

∀j

∀j

(4.33)

(4.34)

For large plant areas, the limits xmax and ymax can be replaced by Mx and My, respectively.

Also, space restrictions that lead to a non-rectangular available area can be modelled by defining pseudo-equipment items with fixed sizes and locations that constrain the available area.


43

In conclusion, the objective function (4.1) along with an appropriate selection of constraints (4.2) to (4.34) define the MILP model for the layout of industrial plants without production sections. The introduction of production sections into the layout problem is addressed later as an extension of the basic layout problem defined in this section.

Examples The Generic Algebraic Modelling System (GAMS, Brook et al. 1988) was used coupled with the CPLEX optimisation package (version 6.5). All the problems were solved with a 5% margin of optimality on a Pentium II, 450 MHz.

Example 1 – Geometrical Centres versus Input/Output Points:

The example proposed by Jayakumar and Reklaitis (1996), studied by Papageorgiou and Rotstein (1996) and later considered by Barbosa-Póvoa et al. (2000) is presented to show the model applicability. The problem characteristics are summarised in table 4.1 and figure 4.4. All connections are assumed to have one currency unit (c.u.) per meter and, initially, no space restrictions are applied. An allocation pre-defined area of 20m x 20m is considered.

Table 4.1 – Unit Dimensions (m) Units Unit_1 Unit_2 Unit_3 Unit_4 Unit_5

α 13.8 2.7 14.4 5.8 9.0

β 3.0 2.6 2.2 5.7 2.3

Coffee

Hot Water

Percolator 1 Spray Drier 3 Cyclone 2

Press 4 Waste Solution

Dry Instant Coffee

Water

Drier 5

Wet Coffee Grounds

Water

Figure 4.4 – Equipment Flowsheet


44

The example studied, initially, is assumed to have all equipment inputs and outputs defined to/from the equipment geometrical centres (case 0) and subsequently the model presented in this chapter will be analysed (case 1). For case 1, equipment dimensions along with input and output points characteristics are shown in table 4.2. Connections established as well as their costs are presented in table 4.3.

Table 4.2 – Equipment and Input/Output Point Characteristics Units

α

β

Unit_1

13.8

3.0

Unit_2

2.7

2.6

Unit_3

14.4

2.2

Unit_4 Unit_5

5.8 9.0

5.7 2.3

Input

∆xin

∆yin

OI1

-1

0

OI2 OI3 OI4 OI5

-3 7 1 4

1 0 -2.5 0

Output OI6 OI7 OI8 OI9

∆xout 6 0 0 0

∆yout 0 -1 -1 1

OI10

2

2.5

Table 4.3 – Connections and Associated Costs Unit Output OI6 OI7 OI8 OI9 OI10

Unit Input OI1 OI2 OI3 OI4 OI5

Cost (cu/m) 1 1 1 1 1

The final results are shown in table 4.4 and the layout solutions depicted in figure 4.5 respectively for cases 0 and 1. Analysing the results it can be seen that the value of the objective function is smaller in case 1 (6.9 against 18.15) which is reflected in a different optimal plant layout (see figure 4.5). Therefore it can be stated that the assumption of geometrical centres (case 0) as focal points for all the connectivity structure may lead to sub-optimal layout solutions. Also, in computational terms (case 0 vs. case 1), the formulation presented, although characterised by a large number of variables and equations, presents a better solution performance (table 4.4). Thus, the proposed model seems to be promising in order to solve a more generic layout problem when compared to the model presented by Papageorgiou and Rotstein (1998).


45

Table 4.4 – Problem Statistics Statistics (case) Obj. Function (OF) CPU´seconds LP Nodes Iterations Nr Integer Var (NIV) Nr Variables (NV) Nr Constraints (NC)

Geometrical Centres (0) 18.15 0.77 838 1 406 25 71 86

Different Inputs/Outputs (1) 6.9 0.49 179 749 45 111 116

5

5

4

2

3

4

1

1

2 3

Case 0

Case 1 Figure 4.5 – Optimal Plant Layout

Example 2 – Equipment with irregular Shapes, Safety/Operability restrictions and Area restrictions:

Taking as basis the case 1 presented in Example 1, the same problem was studied assuming equipments with irregular forms (equipment units 2 and 5) along with the presence of safety and operational restrictions as well as area restrictions. For the case of equipments with irregular forms, unit 2 is formed by two elements 2a and 2b, while unit 5 is formed by three elements 5a, 5b and 5c – see table 4.5. Table 4.6 shows how the elements were linked. The equipment element dimensions along with the input/output data are shown in table 4.5. All connections are assumed to have one unitary c.u. per meter (see table 4.7) and an initial area of 20m x 20m is considered.


46

Table 4.5 – Equipment Characteristics α

β

Elem_1

13.8

3.0

Unit_2

Elem_2a Elem_2b

2.7 2.6

2.6 4.8

Unit_3

Elem_3

14.4

2.2

Unit_4

Elem_4 Elem_5a Elem_5b Elem_5c

5.8 9.0 2.3 2.3

5.7 2.3 6.0 6.0

Units

Elements

Unit_1

Unit_5

Input

∆xoi

∆yoi

OI1

-1

0

OI2 OI3 OI4 OI5

-3 7 1 0

1 0 -2.5 -1

Output OI6 OI7 OI8 OI9

∆xoi 6 0 0 -1.15

∆yoi 0 -1 -1 -0.4

OI10

2

2.5

Table 4.6 – Links for Equipments with Irregular Forms Element j Elem_2a Elem_5a Elem_5a

Element j' Elem_2b Elem_5b Elem_5c

∆x 2.60 5.65 -5.65

∆y -1.05 -1.85 -1.85

Table 4.7 – Connections and associated Costs oi (output) OI6 OI7 OI8 OI9 OI10

oi’ (input) OI1 OI2 OI3 OI4 OI5

Cost (cu/m) 1 1 1 1 1

Two different cases were solved. Case 1 does not consider any safety/operational or area restrictions while in case 2 an area restriction is present as well as safety/operational restrictions are imposed. The latter implies minimum and maximum distances that must be observed between some equipment elements. Thus for elements 1 and 4 a minimum distance of 2 is considered (Zx1,4min = Zy1,4min = 2) and for elements 3 and 5a a maximum distance of 3 is defined (Zx3,5amax = Zy3,5amax = 3). The final layout configurations are shown in figure 4.6. In case 2 the layout is altered due to the presence of a restricted area (10 m x 10m) in the lower left corner. Thus, the layout obtained in case 1 is no longer possible in the new available area. On the other hand, the maximum distance imposed between elements 3 and 5a places them closer compared with case 1. As a result the objective function for the new layout (case 2) is depreciated by one (1) c.u. (see table 4.8). From these results it can be seen that the existence of operational constraints do influence the final layout configurations as well as the presence of area limitations, as it could be expected.


47

1

2a 2b 3 5b

5a 5c

5b 4

4 5a 3 2b 2a

1

5c

Case 2 (area restriction in black)

Case 1

Figure 4.6 – Optimal Plant Layout The problem statistics and the value of the objective functions are shown in table 4.8 where it can be seen that the problem is solved quite rapidly in both cases. Table 4.8 – Problem Statistics Case 1 2

OF 11.3 12.3

CPU'sec 32.85 5.93

Nodes 18 975 2 478

Iterations 78 376 14 157

NIV 73 89

NV 151 171

NC 196 232

Example 3:

Based on Example 2 a new example was solved where seven equipment units are considered along with input and output points (oi) coincident. The plant flowsheet is shown in figure 4.7. The involved equipment characteristics (table 4.9), element links for irregular units (table 4.10) and connections involved and associated costs (table 4.11) are depicted. Coffee Percolator Hot Water

1 Spray Drier 3 Cyclone 2


Dry Instant Coffee 6

Water

Drier 5

Wet Coffee Grounds 7

Water

Figure 4.7 – Plant Flowsheet


48

Table 4.9 – Equipment Characteristics Units Unit_1 Unit_2 Unit_3 Unit_4 Unit_5 Unit_6 Unit_7

Elements Elem_1 Elem_2a Elem_2b Elem_3 Elem_4 Elem_5a Elem_5b Elem_5c Elem_6a Elem_6b Elem_7

α 13.8 2.7 2.6 14.4 5.8 9.0 2.3 2.3 6.0 2.0 4.0

β 3.0 2.6 4.8 2.2 5.7 2.3 6.0 6.0 2.0 4.0 7.0

Input

∆xoi

∆yoi

OI1

-1.3

0

OI2 OI3 OI4

7 1 0

OI5 OI6

Output OI7

∆xoi 6.5

∆yoi 0

OI8 OI9 OI3

-1.15 -7 1

-0.4 1 -2.5

OI10

1

2.8

0 -2.5 -1

0 2

2 3.5

Table 4.10 – Links for Equipments with Irregular Forms Element j Elem_2a Elem_5a Elem_5a Elem_6a

∆x 2.60 5.65 -5.65 0.00

Element j' Elem_2b Elem_5b Elem_5c Elem_6b

∆y -1.05 -1.85 -1.85 3.00

Table 4.11 – Connections and Associated Costs oi (output) OI7 OI7 OI8 OI8 OI9 OI3 OI10

oi’ (input) OI1 OI2 OI2 OI3 OI5 OI4 OI6

Cost (cu/m) 15 15 15 1 15 1 15

An available area of 25m x 25m was considered. No operational or other type of constraints were imposed. The optimal plant layout is shown in figure 4.8. 5a 5c

5b

7

2b 3

2a

4

6b 6a

1

Figure 4.8 – Optimal Plant Layout


49

The optimal computational results are presented in table 4.12:

Table 4.12 – Problem Statistics OF 93.05

CPU'sec 92.33

Nodes Iterations 33 631 199 584

NIV 135

NV 235

NC 330

Example 4:

The example proposed by Barbosa-Póvoa et al. (1994) for the design of batch plants where no layout aspects were considered was studied while introducing the necessary modifications to account for different input and output points. The unit characteristics are given in table 4.13 while the connection definition and associated costs are shown in table 4.14. The equivalent equipment flowsheet is shown in figure 4.9. Finally, an allocation pre-defined area of 25m x 25m was considered.

Table 4.13 – Unit Characteristics Units Unit_V1

α 5.0

β 3.0

Unit_V2

6.0

6.0

Unit_1a

6.0

6.0

OI1

-3

0

Unit_1b

5.0

5.0

Unit_2a

6.0

6.0

Unit_R2

4.5

4.5

Unit_R4

5.0

5.0

Unit_V5 Unit_V6 Unit_V5a Unit_V6a

5.0 6.0 2.0 3.0

3.0 6.0 1.0 2.0

OI2 OI3 OI4 OI5 OI6 OI7 OI8 OI9 OI10 OI11

0 -3 3 0 -2 0 -2.5 0 1 1.5

-2.5 -3 -3 -2 -2 -2 0 -3 0 0

Input


∆xoi

∆yoi

Output OI12 OI13 OI14 OI15 OI16 OI17 OI18 OI19 OI20 OI21 OI22

∆xoi 2.5 -2 2 3 3 0 2.5 -2.5 0 -2 2

∆yoi 0 3 3 1 -1 2.5 2.5 2.5 2 2 2

50

V5

V1

1a

2a

V6

1b

V5a

V2

R4 R2 V6a

Figure 4.9 – Plant Flowsheet Table 4.14 – Connections and Associated Costs oi (output) OI12 OI13 OI14 OI16 OI15 OI17 OI20 OI18 OI19 OI21 OI22

oi’ (input) OI1 OI2 OI5 OI3 OI6 OI4 OI7 OI8 OI9 OI10 OI11

Cost (cu/m) 1 20 5 10 1 20 5 10 10 1 1

The final results are shown in figure 4.10 and table 4.15.

V6

V6a

V5

2a R4

1a

V1

1b

R2

V2



51

Table 4.15 – Problem Statistics OF 32

CPU's 677.78

Nodes Iterations 216 734 1 436 004

NIV 165

NV 309

NC 386

Although this problem is of a significant greater magnitude - 11 equipment units and 22 points of input/output are used - it was solved in an acceptable computational time. Thus, the proposed model seems promising to solve a quite general facility layout problem.

4.3.2 – Layout with Production Sections

Within the layout problem, pre-defined production sections are often present (e.g. storage, cutting, evaporation, filling) due to operational requirements such as safety, materials handling and resources management amongst others. In order to extend the previous basic layout model to account for this characteristic the following constraints are added/modified to the above model:

Non-overlapping Constraints: when considering production sections, two types of these constraints are defined. These are respectively the ones applied to single equipment elements (i) and the ones applied to the production sections (ii):

(i) Non-overlapping Constraints for Equipment Elements: constraints (4.13) to (4.16) still hold but they are just applied within the different production sections (Pj = Pj’) to which the equipment elements belong;

(ii) Non-overlapping Constraints for Production Sections: as for the equipment elements also the non-overlapping of production sections must be considered. These are similar to the equivalent equipment constraints although small changes must be introduced, resulting in:

X p − X p ' + Mx'⋅( S1 p , p ' + S 2 p , p ' ) ≥

L p + L p'

X p ' − X p + Mx '⋅(1 − S1 p , p ' + S 2 p , p ' ) ≥

2 Lp + Lp'


2

∀p, p' | p' > p

∀p, p '| p ' > p

(4.35)

(4.36)

52

Yp − Yp ' + My '⋅(1 + S1p , p ' − S 2 p , p ' ) ≥

Yp ' − Yp + My '⋅(2 − S1p , p ' − S 2 p , p ' ) ≥

Dp + Dp '

2

∀p, p '| p ' > p

(4.37)

∀p, p '| p ' > p

(4.38)

Dp + Dp'

2

where, Mx’ and My’ are defined as suitable upper bounds on the distance between any two production sections p and p’, respectively on the x and y-axis. As before two new auxiliary binary variables are created (S1p,p’ and S2p,p’) and again only one of the above constraints can be active – disjunctive condition.

Safety/Operability Constraints: also between production sections, minimum and maximum possible distances can be defined in order to guarantee safety and operability conditions. Constraints (4.19) to (4.26) defined over equipment elements still hold because they can exist even for equipment elements belonging to different production sections. The same is applied to input and/or outputs points. Furthermore, as for the equipments also a certain minimum distance between production sections may be considered. These constraints are similar to the equipment constraints although small changes must be introduced. Thus, to model a certain minimum possible distance between production sections, inequations (4.35) to (4.38) must be replaced by:

X p − X p ' + ( Mx'+ ZX pmin , p ' ) ⋅ ( S1 p , p ' + S 2 p , p ' ) ≥

Lp + Lp '

2

+ ZX pmin , p' ∀p, p '| p ' > p

X p ' − X p + ( Mx'+ ZX pmin , p ' ) ⋅ (1 − S1 p , p ' + S 2 p , p ' ) ≥

Lp + Lp '

2

+ ZX pmin , p' ∀p, p '| p ' > p

Yp − Yp ' + ( My '+ ZYpmin , p ' ) ⋅ (1 + S1 p , p ' − S 2 p , p ' ) ≥

Dp + Dp '

2

Dp + Dp '

2

(4.41)

+ ZYpmin , p' ∀p, p '| p ' > p


(4.40)

+ ZYpmin , p' ∀p, p '| p ' > p

Yp ' − Yp + ( My '+ ZYpmin , p ' ) ⋅ ( 2 − S1 p , p ' − S 2 p , p ' ) ≥

(4.39)

(4.42) 53

Still addressing the problem of guaranteeing a certain number of operability/safety restrictions a maximum possible distance between production sections may also be imposed. For the plant production sections the following constraints are written:

X p − X p' ≤

X p' − X p ≤

Yp − Yp ' ≤

Yp ' − Y p ≤

Lp + L p '

2 Lp + L p '

2

Dp + Dp '

2 Dp + Dp '

2

+ ZX pmax , p'

∀( p, p' ) ∈ ZonX max

(4.43)

+ ZX pmax , p'

∀( p, p' ) ∈ ZonX max

(4.44)

+ ZYpmax , p'

∀( p, p' ) ∈ ZonY max

(4.45)

+ ZYpmax , p'

∀( p, p' ) ∈ ZonY max

(4.46)

Allocation Constraints: as for the non-overlapping constraints also in this case we have two types of constraints. Ones describing the equipment elements allocation within the respective production section area (i) and the others the allocation of the production sections within the available plant area (ii):

(i) Equipment Elements Allocation Constraints: the allocation of each equipment within its production section is modelled using the following constraints:

xj +

xj −

yj +

lj

2 lj

2 dj

2

≤ Xp +

≥ Xp −

≤ Yp +

Lp

2 Lp

2 Dp

2

∀p, j ∈ J p

(4.47)

∀p, j ∈ J p

(4.48)

∀p, j ∈ J p

(4.49)


54

yj −

dj

≥ Yp −

2

Dp

2

∀p, j ∈ J p

(4.50)

(ii) Production Sections Allocation Constraints: constraints (4.31) to (4.34) are modified resulting in:

Xp ≥

2 Dp

Yp ≥

2

Xp +

Yp +

Lp

Lp

2

Dp

2

∀p

(4.51)

∀p

(4.52)

≤ x max

∀p

(4.53)

≤ y max

∀p

(4.54)

In conclusion, the objective function (4.1) along with constraints (4.2) to (4.30) and constraints (4.35) to (4.54) applied to the production sections, define the MILP model for the layout with production sections.

Examples

Two examples presented in section 4.3.1 are now solved considering production sections. These are respectively examples 3 and 4 that constitute the new examples 1S and 2S.

Example 1S:

The example 3 presented in section 4.3.1 is solved considering three production sections. Production section S1 is formed by equipment units 1, 2 and 4, production section S2 by unit 3 and unit 5 and finally production section S3 by equipment units 6 and 7. The optimal plant layout is shown in figure 4.11 while the problem statistics appear in table 4.16.


55

S3

6a 7

6b

S1

5b

1

4

3

5a

5c

2b 2a

S2

Figure 4.11 – Optimal Plant Layout Table 4.16 – Problem Statistics OF 228.35

CPU's 21.42

Nodes 17 094

Iterations 72 617

NIV 61

NV 173

NC 194

Due to the existence of production sections, which impose an extra area restriction, the final objective function increases from the initial problem (228.35 against 93.05, see tables 4.16 and 4.12), as it could be expected.

In terms of problem statistics it is important to note that the model with production sections is of a lower magnitude due to the relations between the equipment and the existence of the production sections. Thus it exhibits a better solution performance (21.42 CPU against 92.33 CPU – see tables 4.16 and 4.12).

Example 2S:

The example 4 presented in section 4.3.1 is solved considering three productions sections, namely: section S1 – raw materials storage formed by storage vessels V1 and V2; section S2 – processing formed by units 1a, 1b, 2a, R2 and R4; and section S3 – products storage formed by storage vessels V5, V5a, V6 and V6a. The results are shown in figure 4.12 and table 4.17.


56

S3 V6 V6a

V5

R4

2a

R2 V2 1b

1a

S2

V1

S1

Figure 4.12 – Optimal Plant Layout with Production Sections Again, due to the existence of production sections, the final objective function differs from the initial problem (96 against 32) and also the model with production sections shows a much better solution performance (36.74 CPU against 677.78 CPU – see tables 4.17 and 4.15).

Table 4.17 – Computational Statistics OF 96

CPU's 36.74


NIV 95

NV 251

NC 258

4.4 - Conclusions and Future Work The layout problem of industrial plants over a 2D continuous space is addressed. A mixed integer formulation is proposed where the optimal plant layout is obtained based on a specified economic goal. This is defined as the minimisation of the connectivity structure cost involving not only all the possible physical costs for each connection, but also the operating costs related to the material transfers occurring within. Important topological aspects describing real issues are considered. These are different equipment orientation, different equipment inputs and outputs, space availability, rectangular and irregular shapes and safety and operability restrictions translated into distance restrictions (minimum and maximum). Also, the existence of production sections is addressed where all the mentioned characteristics still hold.


57

The problems studied were solved with a very satisfactory calculation performance and hence the proposed formulation seems promising in describing the layout problem. In conclusion it can be stated that a very generic formulation is presented and the obtained layout results describe closely real life situations. In the next chapters, it will be exploited the generalisation of the proposed model to a 3D space along with the simultaneous design and layout of industrial facilities.


58

Chapter 5 OPTIMAL 3D LAYOUT OF INDUSTRIAL FACILITIES

5.1 - Introduction

Facilities layout is concerned with the spatial arrangement of a set of departments or equipments. At the design level this problem constitutes an important stage that often results in a complex problem due to the high number of decisions involved. A Mixed Integer Linear Programming (MILP) formulation, for the optimal layout of facilities within a 2D continuous space has been developed in the previous chapter. However, at industrial level is often the case that a three dimensional (3D) space defines the layout problem that can impose additional restrictions to the equipment orientation (e.g. height restrictions). Also, other important aspects such as, the location of equipment units within different plant floors is usually considered when designing the layout of an industrial plant. In this chapter the 2D model is used as a basis and extended to account for the location of equipment units within a 3D multifloor continuous space. Different topological characteristics are considered such as equipment orientations, distance restrictions, non-overlapping constraints, equipment connectivity inputs and outputs, rectangular and irregular equipment shapes, space availability and multi-floor locations over a 3D space. In operational terms, production or operational sections are modelled as well as safety and operability restrictions. The final model is again described as a MILP where binary variables characterise topological choices and continuous variables describe the distances and locations involved. Finally, the applicability of the proposed formulation is shown through the resolution of a set of representative examples exploring the different model characteristics.

This chapter is structured as follows. In the next section, the layout problem statement and characteristics, as considered in this work, are described. Then, in section 5.3, the mathematical formulation is presented and developed as a MILP. The 3D layout is formulated taking as basis the 2D layout model, presented in the previous chapter. This involves the description of the basic 3D formulation followed by its extension to the treatment of multifloor allocation and the possible existence of production sections. Each one of these aspects is illustrated by the


59

resolution of a set of illustrative examples. Finally, in section 5.4, the chapter ends with some conclusions and the identification of possible areas for further analysis.

5.2 – Layout Problem Statement and Characteristics The layout of industrial plants as addressed in this chapter can be stated formally as follows:

Given •

A set of equipment items and their geometrical shapes and sizes over a 3D space;

•

Input and output point locations inside each equipment space;

•


•


•


•

Production section characteristics and associated equipment units involved, if present;

•

Multifloor characteristics and associated equipment restrictions, if present;

•

Space availability;

•

The capital and operational cost of all connectivity structures;

•

Optimal plant equipment arrangement – coordinates and orientation;

•

Optimal floor equipment allocation;

•

Optimal connectivity structure (inputs and outputs).

Determine

So as to minimise the plant layout cost including capital and operational connectivity cost.

The layout problem is formulated assuming a set of rectangular or irregular equipment units and the associated connectivity over a continuous 3D layout space where rectilinear distances are assumed. Capital costs model the connectivity costs in terms of rectilinear distances while operational costs relate the direction of the fluids with the associated distance. Multiple equipment connectivity inputs and outputs are considered as well as space limitations and safety and operability constraints.


60

Equipment units may have a rectangular or irregular shape. For the latter, a set of rectangular equipment elements j model the entire unit which have given dimensions over the x, y and z–axis (αj, β j, θj) and possible multiple input and output points. For example, as shown in figure 5.1, an irregular equipment unit g is formed by three rectangular equipment elements j1, j2 and j3 – in its original position defined with a 0° rotation over the x and y-axis. For combinations j1 with j2 and j2 with j3, each link is established by the definition of the fixed relative distance between the centroids of the two equipment elements. Thus, considering the link (j2;j3), this is expressed as: ∆xj2,j3; ∆yj2,j3; and ∆zj2,j3, respectively over the x, y and z-axis. In terms of input and output points, one input point (oi1) and one output point (oi2) are defined respectively over the equipment elements j1 and j3. For instance, the input point oi1 is fixed by the definition of the relative distance, in the x, y and z-axis between its location and the centroid of the equipment element j1 (∆xj1,oi1, ∆y j1,oi1, ∆z j1,oi1), while the same applies to the output point oi2 but in this case the centroid is the one of equipment element j3.

z Above ∆yj1,oi1 ∆xj1,oi1

oi1 ∆xj2,j3

oi2

∆zj1,oi1

∆zj2,j3

j1

θ j3

Below

j2 Back

y j3

β j3 α j3 Front

Left

Right

x

Figure 5.1 – Equipment Unit – Original Position

Using these problem characteristics the mathematical formulation for the generalised 3D layout problem can now be developed.


61

5.3 – Layout Mathematical Formulations The mathematical formulation for the layout problem is presented here. First, the global indices, sets, parameters and variables are defined. Then, the mathematical model for the 3D layout problem is developed. This considers the generalisation to 3D of the 2D model proposed in the previous chapter where not only the z coordinate is considered explicitly but also the possibility of having multifloor allocation is present. Two situations are modelled in this case. On the one hand a pre-defined fixed number of floors with fixed associated height is considered. On the other hand a variable number of floors with a variable height is also explored. Finally, production sections are addressed.

The indices, sets, parameters and associated variables used within the layout problem formulation are the following:

Global Indices: g – equipment unit j, j´– equipment element oi, oi’ – point (output or input) p, p´- production section n – floor Sets:

J g = {j: set of equipment elements j that form unit g} G j = {g: equipment unit containing equipment element j} – Unique set Linksg = {(j,j´): set of linked equipment elements pairs (j, j´) ∈ Jg}

Zonx max = {(j,j´): set of equipment elements pairs (j, j´): Gj ≠ Gj´ with an upper limit distance between them along the x-axis}

Zony max = {(j,j´): set of equipment elements pairs (j, j´): Gj ≠ Gj´ with an upper limit distance between them along the y-axis}

Zonz max = {(j,j´): set of equipment elements pairs (j, j´): Gj ≠ Gj´ with an upper limit distance between them along the z-axis} OI j = {oi: set of (output or input) points of equipment element j}

N j ={n: set of floors n where equipment element j must/can be allocated} J f ={j: set of equipment elements j with floor allocation restrictions}


62

Pj = {p: production section of equipment element j} - Unique set J p ={j: set of equipment elements j within production section p} ZonX max = {(p,p´): set of production sections (p, p´) pairs with an upper limit distance between them over the x-axis}

ZonY max = {(p,p´): set of production sections (p, p´) pairs with an upper limit distance between them over the y-axis}

ZonZ max = {(p,p´): set of production sections (p, p´) pairs with an upper limit distance between them over the z-axis} Parameters:

C oiH,oi ' , C oiAb,oi ' , C oiBe,oi ' – horizontal, downward and upward cost, respectively, of the connection defined between the output point oi and the input point oi’

C oi ,oi ' – cost of the connection defined between the output point oi and the input point oi’

α j , β j ,θ j – dimensions of equipment element j over the x, y and z-axis respectively x max , y max , z max – maximum space over the x, y and z-axis ∆x j ,oi , ∆y j ,oi , ∆z j ,oi – relative distance between the point oi and the geometrical center of

the equipment element j respectively in the x, y and z-axis, as stated by the original equipment representation (0° rotation) min min Zx min j , j ' , Zy j , j ' , Zz j , j ' - minimum distance allowed between equipment elements j and j´

over the x, y and z-axis, respectively max max Zx max j , j ' , Zy j , j ' , Zz j , j ' -maximum distance allowed between equipment elements j and j´

over the x, y, and z-axis, , respectively ∆x j , j ' , ∆y j , j ' , ∆z j , j ' -relative distance between the geometrical centers of equipment

elements (j∧j´) : Gj = Gj´ as defined by its original position, with a 0° rotation over the x, y and z-axis, respectively min min ZX pmin , p´ , ZY p , p´ , ZZ p , p´ -minimum distance allowed between sections p and p´ in the x, y

and z-axis, respectively max max ZX pmax , p´ , ZY p , p´ , ZZ p , p´ -maximum distance allowed between sections p and p´ in the x, y

and z-axis, respectively

h n - fixed height of floor n


63

n hf min - minimum height allowed for floor n n - maximum height allowed for floor n hf max

Variables: Continuous Variables – all defined as positive variables: xj, yj, zj – coordinates of the geometrical center of equipment element j xoi, yoi, zoi – coordinates of the point o. lj – length of equipment element j dj – depth of equipment element j Rioi,oi’ – relative distance in x coordinates between the output point oi and the input point oi’, if oi is to the right of oi’ Leoi,oi’ – relative distance in x coordinates between the output point oi and the input point oi’, if oi is to the left of oi’ Baoi,oi’ – relative distance in y coordinates between the output point oi and the input point oi’, if oi is in the back of oi’ Froi,oi’ – relative distance in y coordinates between the output point oi and the input point oi’, if oi is in front of oi’ Aboi,oi’ – relative distance in z coordinates between the output point oi and the input point oi’, if oi is above oi’ Beoi,oi’ – relative distance in z coordinates between the output point oi and the input point oi’, if oi is below oi’ DHoi,oi’

– horizontal rectilinear distance between the output point oi and the input point oi’

Doi,oi’ – total rectilinear distance between the output point oi and the input point oi’ Xp, Yp, Zp - coordinates of the geometrical center of the production section p Lp, Dp, Hp – length, depth and height of the production section p hfn - variable height of floor n Binary Variables og - equipment unit g orientation: = 1 if the length of all the equipment elements j∈Jg (parallel to the x axis) is equal to α j ; 0 otherwise o1g, o2g, o3g and o4g – definition of the equipment unit g anti-clockwise horizontal rotation, expressed in multiples of 90° (respectively: 0°, 90°, 180° and 270°) from the original equipment representation

f jn - equipment element j allocation floor: = 1 if equipment element j is allocated to floor n; 0 otherwise


64

Three models are going to be presented. The first describes the basic detailed 3D layout model, the second is an extension of the first considering simultaneously the multifloor presence, finally, the existence of production sections is introduced.

5.3.1 – Basic 3D Layout Model

The 2D model, presented in chapter 4, is now extended in order to build the generalised 3D layout model. The following objective function and constraints characterise the generic model for the basic 3D layout problem:

Objective Function:

The minimisation of the total connectivity cost is considered. This involves the cost of the physical connections and the operating costs caused by upward and horizontal material transfers occurring within the connection. The elevation of materials to higher floors is incorporated into the objective function with an additional cost when compared to the cost associated with horizontal transfers. The downward pumping cost is usually neglected, since the flow is often guaranteed by the gravity, although the connection cost still exists and has an associated cost. Thus, the objective function (4.1) is changed resulting in:

Min

∑

[C

H oi , oi '

⋅ D oiH, oi ' + C oiAb, oi ' ⋅ Ab oi , oi ' + C oiBe, oi ' ⋅ Be oi , oi '

]

(5.1)

( oi , oi ')

If there is no cost differences among the orientation of the flows, equation (4.1) is maintained where all possible outputs and inputs of equipment unit g with a non-zero connection cost are considered.

Constraints:

Different types of constraints are incorporated in order to model the 3D FLP, namely, equipment orientation, equipment irregular shapes, input/output locations, distance constraints, nonoverlapping, safety/operability restrictions and space allocation constraints.


65

Equipment Orientation Constraints: the equipment unit rotation is only allowed over the x and yaxis (through the value of variable og). This is considered as a good assumption since it is often the case that the equipment items within production systems are characterised by a single one side up, not allowing the rotation over the z-axis. Thus, there is no variable definition for the height. Instead, the height for each equipment element is fixed by the parameter θ j . Based on this, equations (4.2) and (4.3) still hold, without the need of any kind of modification, for the 3D model. 0°°: o1g = 1; og = 1

90°°: o2g = 1; og = 0


∆x

j1

,j2

∆xj1,j2

j2 ∆zj1,j2

j1

∆zj1,j2 j1

j2

 x j 2 − x j1 = − ∆y j1, j 2 = 0   y j 2 − y j1 = ∆x j1, j 2  z − z = ∆z j1 j1, j 2  j2

180°°: o3g = 1; og = 1

270°°: o4g = 1; og = 0 −∆ x

j1 , j2

 x j 2 − x j1 = ∆x j1, j 2   y j 2 − y j1 = ∆y j1, j 2 = 0  z − z = ∆z j1 j1, j 2  j2

−∆ xj1,j2

j1 ∆zj1,j2

∆zj1,j2 j2

j2

j1

 x j 2 − x j1 = − ∆x j1, j 2   y j 2 − y j1 = − ∆y j1, j 2 = 0  z − z = ∆z j1 j1, j 2  j2

 x j 2 − x j1 = ∆y j1, j 2 = 0   y j 2 − y j1 = − ∆x j1, j 2  z − z = ∆z j1 j1, j 2  j2

Figure 5.2 – Equipment Unit Orientation


66

Equipment Irregular Shapes Constraints: an equipment unit g may have an irregular form in the 3D space defined through the equipment elements j of its set Jg. A pre-defined set (Linksg) of equipment elements - linked in pairs (j,j´) - is defined with given distances between them over the 3D space (∆xj,j´, ∆yj,j´, ∆zj,j´ ). Thus, equations (4.4) and (4.5) still hold along with:

z j ' = z j + ∆z j , j '

∀g , j , j ' | ( j , j ' ) ∈ Links g

(5.2)

where, the coordinate over the z-axis is considered. Furthermore, since each equipment unit can only exist in a single position equations (4.6) and (4.7) are maintained. For illustration purposes, consider figure 5.2, where constraints (4.4), (4.5) and (5.2) are applied for each of the possible four orthogonal positions of an equipment unit g – see subtitle. The original representation is user defined by the definition of parameters ∆xj1,j2, ∆yj1,j2, ∆zj1,j2 and is assumed with a 0° orientation where og=1 and o1g=1.

Input/Output Constraints: within each equipment element j forming the equipment unit g, different input and output points locations (xoi,yoi,zoi) are allowed. These points are defined as global although they are located inside of the different equipment elements present in the plant. For the x and y-axis the points location can easily be calculated using the orientation variables, as defined in figure 5.3, combined with the coordinates of the geometrical centers of the equipment elements (xj,yj,zj) and the relative distance (∆xj,oi, ∆yj,oi, ∆zj,oi) between these geometrical centers and the points, therefore equations (4.6) to (4.9) must be present. For the z-axis, and since the orientation variable is not required, due to the assumption that no rotation over the z-axis is allowed, the following equation is added:

z oi = z j + ∆z j ,oi

∀g , j ∈ J g , oi ∈ OI j

(5.3)

Figure 5.3 depicts the possible four-ortoghonal positions of an equipment unit with two equipment elements, namely j1 and j2 along with the input/output point oi1. For illustration purposes, it also shows in subtitle the relative constraints (4.8), (4.9) and (5.3) for the coordinates of point oi1 (xoi1, yoi1, zoi1).


67

0°°: o1g = 1; og = 1

90°°: o2g = 1; og = 0 (xoi1,yoi1)


−∆ yj2,oi1 ∆zj2,oi1

∆xj2,oi1

∆xj2,oi1 (xj2,yj2) (xoi1,yoi1)

∆zj2,oi1 (xj2,yj2)

j1

(xoi1 ,yoi1 )

∆yj2,oi1

j2

j1

j2

 xoi1 = x j 2 + ∆x j 2,oi1   y oi1 = y j 2 + ∆y j1, j 2  z = z + ∆z j2 j 2 , oi1  oi1

 xoi1 = x j 2 − ∆y j 2,oi1   y oi1 = y j 2 + ∆x j1, j 2  z = z + ∆z j2 j 2 , oi1  oi1

180°°: o3g = 1; og = 1

270°°: o4g = 1; og = 0

−∆ xj2,oi1

∆zj2,oi1

(xoi1,yoi1)

j1 −∆ yj2,oi1

∆zj2,oi1

(xj2,yj2 )

−∆ xj2,oi1

(xj2,yj2)

j2 ∆yj2,oi1

j2

j1

 xoi1 = x j 2 − ∆x j 2,oi1   y oi1 = y j 2 − ∆y j1, j 2  z = z + ∆z j2 j 2 ,oi1  oi1

 xoi1 = x j 2 + ∆y j 2,oi1   y oi1 = y j 2 − ∆x j1, j 2  z = z + ∆z j2 j 2 ,oi1  oi1

Figure 5.3 – Equipment Unit Input/Output Points

Distance Constraints: rectilinear distances are assumed between each output and input point within the plant. If objective function (4.1) is used then equation (4.10) must be replaced by:

Doi ,oi ' = Rioi ,oi ' + Leoi ,oi ' + Froi ,oi ' + Baoi ,oi ' + Aboi ,oi ' + Beoi ,oi '

∀(oi, oi ' ) | Coi ,oi ' ≠ 0

(5.4)

where, the relative distances (Ri, Le, Fr, Ba) are obtained from equations (4.11) and (4.12).


68

Ab and Be are determined by:

Aboi ,oi ' − Beoi ,oi ' = zoi − zoi '

∀(oi, oi ' ) | Coi ,oi ' ≠ 0

(5.5)

If the output point oi is above the input point oi´ then zoi is greater than zoi’. Therefore Beoi,oi´ takes a null value while Aboi,oi´ takes a positive value, since this is a minimisation problem. The same applies to equations (4.11) and (4.12) under the x and y-axis.

On the other hand, if objective function (5.1) is used then the following equation must replace (4.10) instead of equation (5.4):

DoiH, oi ' = Ri oi , oi ' + Le oi ,oi ' + Ba oi , oi ' + Froi , oi '

∀(oi, oi ' ) | C oiH, oi ' ≠ 0

(5.6)

and the domain of equations (4.11), (4.12) and (5.5) must be changed accordingly:

Ri oi , oi ' − Le oi ,oi ' = x oi − x oi '

∀(oi , oi ' ) | C oiH,oi ' ≠ 0

(5.7)

Ba oi ,oi ' − Froi ,oi ' = y oi − y oi '

∀(oi, oi ' ) | C oiH,oi ' ≠ 0

(5.8)

Aboi , oi ' − Be oi ,oi ' = z oi − z oi '

∀(oi, oi ' ) | C oiAb, oi ' ≠ 0 ∨ C oiBe, oi ' ≠ 0

(5.9)

Non-overlapping Constraints: two-equipment elements j and j´ cannot occupy the same physical location. The formulation presented by Tsai et al. (1993) is adapted to model this restriction. Based on that, a set of disjunctive non-overlapping conditions is formulated to guarantee the non-overlapping. Thus, constraints (4.13) to (4.18) must no longer be present. Instead, new ones are added:

x j − x j ' + M x ⋅ ( E 2 j , j ' + E3 j, j ' ) ≥

x j ' − x j + M x ⋅ ( E1 j , j ' + E 3 j , j ' ) ≥

l j + l j' 2

l j + l j' 2


∀j , j '| j ' > j ∧ G j ≠ G j´

(5.10)

∀j , j '| j ' > j ∧ G j ≠ G j´

(5.11)

69

y j − y j ' + M y ⋅ ( E1 j , j ' + E 2 j , j ' ) ≥

d j + d j'

y j ' − y j + M y ⋅ (2 − E1 j , j ' − E 2 j , j ' ) ≥

z j − z j ' + M z ⋅ (2 − E 2 j , j ' − E 3 j , j ' ) ≥

z j ' − z j + M z ⋅ (2 − E1 j , j ' − E 3 j , j ' ) ≥

∀j , j '| j ' > j ∧ G j ≠ G j´

2

d j + d j' 2

θ j + θ j' 2

θ j + θ j' 2

∀j , j ' | j ' > j ∧ G j ≠ G j´

(5.12)

(5.13)

∀j , j '| j ' > j ∧ G j ≠ G j´

(5.14)

∀j , j '| j ' > j ∧ G j ≠ G j´

(5.15)

where E1j,j´, E2j,j´ and E3j,j´ are auxiliary binary variables. The unfolding of these variable values allows 8 possible combinations. In order to guarantee the presence of only 6 possible combinations (to 6 constraints) and such that that only one of the above constraints is active, the following conditions must be added:

E1 j , j ' + E 2 j , j ' + E 3 j , j ' ≥ 1

∀j , j ' | j ' > j ∧ G j ≠ G j´

(5.16)

E1 j , j ' + E 2 j , j ' + E 3 j , j ' ≤ 2

∀j , j ' | j ' > j ∧ G j ≠ G j´

(5.17)

As shown in table 5.1, with these constraints, solely six possible combinations are allowed from where only one constraint can be active:

Table 5.1 – Possible Combinations E1 1 0 0 1 0 1

E2 0 1 0 1 1 0

E3 0 0 1 0 1 1

Nr Active Constraint (5.10) (5.11) (5.12) (5.13) (5.14) (5.15)

The values of Mx, My and Mz are taken as suitable upper bounds on the distance between any two-equipment elements j and j´, given by:   M x = min  x max , ∑ max(α j , β j ,θ j ) j  


(5.18)

70

  M y = min  y max , ∑ max(α j , β j ,θ j ) j  

(5.19)

  M z = min  z max , ∑ max(α j , β j ,θ j ) j  

(5.20)

Safety/Operability Constraints: safety and/or operability restrictions can be modelled using minimum and maximum distances between equipment items. In order to guarantee a certain minimum possible distance between equipment elements, constraints (4.19) to (4.22) must no longer be present and constraints (5.10) to (5.15) must be replaced by:

x j − x j' + (M x + Zxmin j , j ' ) ⋅ (E2 j , j ' + E3 j , j ' ) ≥

l j + l j'

x j ' − x j + (M x + Zxmin j , j ' ) ⋅ (E1 j , j ' + E3 j , j ' ) ≥

y j − y j' + (M y + Zymin j , j ' ) ⋅ (E1 j , j ' + E2 j , j ' ) ≥

2

+ Zxmin j, j '

l j + l j' 2 d j + d j' 2

z j' − z j + (M z + Zzmin j , j ' ) ⋅ (2 − E1j , j ' − E3 j , j ' ) ≥

∀j, j'| j' > j ∧ G j ≠ G j´

(5.22)

+ Zymin j, j '

∀j, j'| j' > j ∧ G j ≠ Gj´

(5.23)

+ Zymin j, j'

∀j, j'| j' > j ∧ Gj ≠ Gj´

(5.24)

+ Zzmin j, j '

∀j, j'| j' > j ∧ Gj ≠ Gj´

(5.25)

+ Zzmin j, j '

∀j, j'| j' > j ∧ Gj ≠ Gj´

(5.26)

2

z j − z j' + (M z + Zzmin j , j ' ) ⋅ (2 − E2 j , j ' − E3 j , j ' ) ≥

θ j +θ j ' 2

θ j +θ j ' 2

(5.21)

+ Zxmin j, j '

d j + d j'

y j' − y j + (M y + Zymin j, j ' ) ⋅ (2 − E1j, j ' − E2 j, j ' ) ≥

∀j, j'| j' > j ∧ G j ≠ Gj´

Again, equations (5.16) and (5.17) must be present.

On the other hand, for maximum possible distance restrictions, constraints (4.23) to (4.26) are still used along with:

z j − z j' ≤

θ j + θ j' 2

+ Zz max j, j'

∀( j , j ' ) ∈ Zonz max


(5.27)

71

z j' − z j ≤

θ j + θ j' 2

+ Zz max j, j'

∀( j , j ' ) ∈ Zonz max

(5.28)

Allocation Constraints: when the available facility space is constrained to a limited value, additional constrains must be introduced into the model. On the one hand, in order to avoid intersection of equipment elements with the axes a lower bound to the equipment coordinates must be guaranteed. Thus, in the 3D space, additionally to constraints (4.31) and (4.32) the following constraint on the z-axis must be introduced:

zj ≥

θj 2

∀j

(5.29)

Similarly, upper bound constraints are written to force the equipment allocation within a predefined space confined by the points (0,0,0) and (xmax,ymax,zmax). Again, and for the 3D space, additionally to constraints (4.33) and (4.34) the next one must be defined:

zj +

θj 2

≤ z max

∀j

(5.30)

For large plant areas, the limits xmax, ymax and zmax can be replaced by Mx, My and Mz, respectively. Also, as shown in chapter 4, space restrictions that lead to a non-rectangular available space can be modelled by defining pseudo-equipment items with fixed sizes and locations that constrain the available space.

In conclusion, objective function (4.1) or (5.1) along with an appropriate selection of constraints (4.2) to (4.9), (4.11), (4.12), (4.23) to (4.26), (4.31) to (4.34) and (5.2) to (5.30) define the MILP model for the basic 3D layout of industrial plants. However, this does not account for the existence of multifloor and production sections characteristics which may be present in real plants. In order to overcome this, an extension of the model will be present later in this chapter.


72

Examples

In order to illustrate the applicability of the proposed model to the 3D layout of industrial plants a set of illustrative examples are solved. The Generic Algebraic Modelling System (GAMS, Brook et al. 1988) was used coupled with the CPLEX optimisation package (version 6.5). All the problems were solved with a 5% margin of optimality on a Pentium II, 450 MHz.

Example 1 – Equipments with irregular shapes in the 3D space

The example 2 presented in the previous chapter is solved considering the equipment unit allocations within a 3D space. The minimisation of the connectivity cost, equation (4.1), is used to find the optimal plant layout. The equipment flowsheet is shown in figure 5.4 where a set of equipment units with irregular forms is present. Equipment unit 2 is formed by elements 2a and 2b, while equipment unit 5 is formed by elements 5a, 5b and 5c (see table 5.2). Equipment and input/output characteristics are also shown in table 5.2. The links between these elements are shown in table 5.3. All connections are assumed to have a unitary c.u. per meter (see table 5.4) and an initial available space of 20m x 10m x 10m is considered. No operational or other type of constraints are imposed. Coffee

Percolator Hot Water

1 Spray Drier 3 Cyclone 2


Dry Instant Coffee

Water

Drier 5

Wet Coffee Grounds

Water

Figure 5.4 – Equipment Flowsheet


73

Table 5.2 – Equipment and Input/Output Point Characteristics Units

Elements

Unit_1

α

β

θ

Elem_1

13.8

3.0

3.0

Unit_2

Elem_2a Elem_2b

2.7 2.6

2.6 4.8

2.6 2.6

Unit_3

Elem_3

14.4

2.2

2.2

Unit_4

Elem_4 Elem_5a Elem_5b Elem_5c

5.8 9.0 2.3 2.3

5.7 2.3 6.0 6.0

3.0 2.3 2.3 2.3

Unit_5

In

∆xoi

∆yoi

∆zoi

OI1

-1

0

1.3

OI2 OI3 OI4 OI5

-3 7 1 0

1 0 -2.5 -1

1.1 -1.1 1.5 1.15

Out OI6 OI7 OI8 OI9

∆xoi 6 0 0 -1.15

∆yoi 0 -1 -1 -0.4

∆zoi 1.5 -1.5 1.3 1.3

OI10

2

2.5

1.5

Table 5.3 – Links for Equipment Units with Irregular Shapes Element j Elem_2a Elem_5a Elem_5a

Element j' Elem_2b Elem_5b Elem_5c

∆x 2.60 5.65 -5.65

∆y -1.05 -1.85 -1.85

∆z 0.0 0.0 0.0

Table 5.4 – Connections definition and associated Costs oi (output) OI6 OI7 OI8 OI9 OI10

oi’ (input) OI1 OI2 OI3 OI4 OI5

Cost (cu/m) 1 1 1 1 1

Exact optimal equipment coordinates and dimensions are shown in table 5.5. The optimal 3D plant layout configuration obtained is depicted in figure 5.5. The main window depicts a SW isometric perspective of the plant, the top left window depicts the front side perspective, middle left window depicts the top side perspective and, finally, below left window depicts the right side perspective. Future optimal layout figures will be shown with this structure.

Table 5.5 – Optimal Plant Layout Coordinates and Dimensions Units

Elements

xj

yj

zj

lj

dj

θj

Unit_1

Elem_1

6,9

8,1

3,7

13,8

3

3

Elem_2a

15,8

7,1

3,9

2,7

2,3

2,6

Elem_2b

18,4

6,05

3,9

2,6

4,8

2,6

Unit_3

Elem_3

9,9

6,1

1,1

14,4

2,2

2,2

Unit_4

Elem_4

14,25

2,9

3,7

5,7

5,8

3

Elem_5a

11,75

3,9

6,35

9

2,3

2,3

Elem_5b

6,1

5,75

6,35

2,3

6

2,3

Elem_5c

17,4

5,75

6,35

2,3

6

2,3

Unit_2

Unit_5


74

Figure 5.5 - Optimal Plant Layout The associated computational results are presented in table 5.6. The problem was solved in a reasonable time and from the final plant layout it can be seen how the equipment units are located so as to minimise the total connectivity cost.

Table 5.6 – Problem Statistics OF

CPU's

Nodes

Iterations

NIV

NV

NC

13.75

50.69

26 416

80 434

97

203

326

Example 2

The previous example is now optimised considering different costs associated with the upward, downward and horizontal transfers. Thus, objective function as expressed in equation (5.1) is used. The connection costs used are shown in table 5.7.

Table 5.7 – Connections and associated Costs (c.u./m) Output OI6 OI7 OI8 OI9 OI10

Input OI1 OI2 OI3 OI4 OI5


CH 1 1 1 1 1

CAb 0.5 0.5 0.5 0.5 0.5

CBe 1.5 1.5 1.5 1.5 1.5

75

The final optimal 3D plant layout configuration is depicted in figure 5.6, its coordinates and dimensions in table 5.8. The associated computational results are presented in table 5.9.

Figure 5.6 – Optimal Plant Layout Table 5.8 – Optimal Plant Layout Coordinates and Dimensions Units

Elements

xj

yj

zj

lj

dj

θj

Unit_1

Elem_1

13,1

2,75

8,5

13,8

3

3

Elem_2a

4,1

2,75

4,3

2,7

2,6

2,6

Elem_2b

1,5

3,8

4,3

2,6

4,8

2,6

Unit_3

Elem_3

11,1

5,15

5,9

14,4

2,2

2,2

Unit_4

Elem_4

5,15

5,2

1,5

5,7

5,8

3

Elem_5a

6,8

1,15

1,85

9

2,3

2,3

Elem_5b

1,15

3

1,85

2,3

6

2,3

Elem_5c

12,45

3

1,85

2,3

6

2,3

Unit_2

Unit_5


CPU's 18.07

Nodes 8 800

Iterations 30 828

NIV 97

NV 203

NC 326

When comparing the results of both examples it can be seen that a different objective function value (10.6 against 13.75) was obtained as well as a different plant layout (see figure 5.6 and table 5.8 versus figure 5.5 and table 5.5). The latter translates a configuration where all the output points are above, or at the same level, of all the input points – configuration characterised by a smaller objective function since the costs of upward and downward transfers are minimised


76

within the global fixed costs. In conclusion, direction flow costs should not be left aside if they may appear important at a cost level.

Example 3

Based on the previous example 1 (see also example 3 presented in chapter 4) a new example is solved where seven equipment units are considered within a 3D space allocation. Objective function (4.1) is used – minimisation of the connectivity cost. The plant flowsheet is drawn in figure 5.7 while the associated equipment characteristics are shown in tables 5.10 and 5.11. Connections and associated costs are exhibited in table 5.12. An initial available space of 15m x 12m x 12m is considered. No operational or other type of constraints were imposed. Coffee

Hot Water

Percolator 1 Spray Drier 3 Cyclone 2

Dry Instant Coffee 6

Water

Press 4

Wet Coffee Grounds 7

Drier 5

Waste Solution

Water


Table 5.10 – Equipment and Input/Output Point Characteristics Units Unit_1 Unit_2 Unit_3 Unit_4 Unit_5

Unit_6 Unit_7

Elements Elem_1 Elem_2a Elem_2b Elem_3 Elem_4 Elem_5a Elem_5b Elem_5c Elem_6a Elem_6b Elem_7

α 13.8 2.7 2.6 14.4 5.8 9.0 2.3 2.3 6.0 2.0 4.0

β 3.0 2.6 4.8 2.2 5.7 2.3 6.0 6.0 2.0 4.0 7.0

θ 2.0 2.6 2.6 4.0 2.5 2.3 2.3 2.3 2.0 2.0 3.0


In

∆xoi

∆yoi

∆zoi

OI1

-1.3

0

1.3

OI2 OI3 OI4

7 1 0

0 -2.5 -1

-2.0 1.25 -1.0

OI5 OI6

0 2

2 3.5

Out OI7

∆xoi 6.5

∆yoi 0

∆zoi -1.0

OI8 OI9 OI3

-1.15 -7 1

-0.4 1 -2.5

1.3 2.0 1.25

OI10

1

2.8

1.15

1.0 -1.5

77

Table 5.11 – Links for Equipment Units with Irregular Shapes Element j Elem_2a Elem_5a Elem_5a Elem_6a

Element j' Elem_2b Elem_5b Elem_5c Elem_6b

∆x 2.60 5.65 -5.65 0.00

∆y -1.05 -1.85 -1.85 3.00

∆z 0.0 0.0 0.0 0.0

Table 5.12 – Connections and associated Costs Output OI7 OI7 OI8 OI8 OI9 OI3 OI10

Input OI1 OI2 OI2 OI3 OI5 OI4 OI6

Cost (c.u./m) 15 15 15 1 15 1 15

The final optimal 3D plant layout configuration is shown in figure 5.8 while the associated coordinates and dimensions in table 5.13. Optimal computational results are presented in table 5.14.


Elements

xj

yj

zj

lj

dj

θj

Unit_1

Elem_1

7,7

9,8

3,6

13,8

3

2

Elem_2a

13,65

8,5

1,3

2,6

2,7

2,6

Elem_2b

12,6

5,9

1,3

4,8

2,6

2,6

Unit_3

Elem_3

7,2

7,2

4,6

14,4

2,2

4

Unit_4

Elem_4

9,7

2,9

3,85

5,7

5,8

2,5

Elem_5a

8,2

1,15

7,75

9

2,3

2,3

Elem_5b

2,55

3

7,75

2,3

6

2,3

Elem_5c

13,85

3

7,75

2,3

6

2,3

Elem_6a

5,2

8,2

7,6

2

6

2

Elem_6b

2,2

8,2

7,6

4

2

2

Elem_7

3,55

3,7

10,4

4

7

3

Unit_2

Unit_5

Unit_6 Unit_7


78



CPU's 125.12

Nodes 37 548

Iterations 121 130

NIV 185

NV 320

NC 573

When comparing the results obtained against the example 1 it can be seen a considerable increase in the value of the objective function (122.55 versus 13.75) due to the presence of two additional equipment units and related connections as well as the smaller available plant space. In this case since the example was augmented its resolution took a larger time (125.12 sec. against 50.69 sec.), nevertheless, a reasonable CPU time was still obtained.

Example 4

A more complex example is solved. This is based on the example 4 proposed in chapter 4, which is extended to the 3D case. The problem is characterised by eleven equipment units and several input/output points. The equipment flowsheet is present in figure 5.9. The minimisation of the connectivity costs - objective function (4.1)- without operability considerations is used. The unit characteristics are given in table 5.15 while the connection definition and associated costs are shown in table 5.16. An initial available space of 10m x 10m x 15m is considered.


79

V5

V1

1a

2a

V6

1b

V5a

V2

R4 R2 V6a

Figure 5.9 – Equipment Flowsheet Table 5.15 – Equipment and Input/Output Point Characteristics Units Unit_V1

Elements Elem_V1

α 5.0

β 3.0

θ 2.0

In

Unit_V2

Elem_V2

6.0

6.0

3.0

Unit_1a

Elem_1a

6.0

6.0

3.0

OI1

-3

3

1.5

Unit_1b

Elem_1b

5.0

5.0

2.5

Unit_2a

Elem_2a

6.0

6.0

3.0

Unit_R2

Elem_R2

4.5

4.5

2.0

Unit_R4

Elem_R4

5.0

5.0

2.5


Elem_V5 Elem_V6 Elem_V5a Elem_V6a

5.0 6.0 2.0 3.0

3.0 6.0 1.0 2.0

3.0 2.0 1.0 1.5


2.5 -3 3 2.25 -2.5 2.5 -2.5 3 1 1.5

-2.5 -3 -3 -2.25 -2.5 -2.5 1.5 -3 0.5 1

1.25 -1.5 -1.5 -1 1.25 -1.25 1.5 1 -0.5 0.75

∆xoi

∆yoi

∆zoi

Out ∆xoi OI12 2.5 OI13 -3 OI14 3 OI15 3 OI16 3 OI17 2.5 OI18 3 OI19 -3 OI20 2.25 OI21 -2.5 OI22 2.5

∆yoi 1.5 3 3 3 -3 2.5 3 3 2.25 2.5 2.5

∆zoi 1 1.5 1.5 -1.5 1.5 1.25 1.5 1.5 -1 1.25 1.25

Table 5.16 – Connections and associated Costs Output OI12 OI13 OI14 OI16 OI15 OI17 OI20 OI18 OI19 OI21 OI22


Input OI1 OI2 OI5 OI3 OI6 OI4 OI7 OI8 OI9 OI10 OI11

Cost (c.u./m) 1 20 5 10 1 20 5 10 10 1 1

80

The optimal 3D plant layout configuration obtained is drawn in figure 5.10 while its coordinates and dimensions are shown in table 5.17. The associated computational results are presented in table 5.18.


Elements

xj

yj

zj

lj

dj

Unit_V1

Elem_V1

7,5

3,5

9,5

3

5

2

Unit_V2

Elem_V2

4

3

1,5

6

6

3

Unit_1a

Elem_1a

3

3

7

6

6

3

Unit_1b

Elem_1b

3,5

2,5

4,25

5

5

2,5

Unit_2a

Elem_2a

3

3

10

6

6

3

Unit_R2

Elem_R2

2,75

7,25

4

4,5

4,5

2

Unit_R4

Elem_R4

7,5

7,5

4,25

5

5

2,5

Unit_V5

Elem_V5

3,5

7,5

10

5

3

3

Unit_V6

Elem_V6

3

3

12,5

6

6

2

Unit_V5a

Elem_V5a

9

9,5

6

2

1

1

Unit_V6a

Elem_V6a

9

7,5

6,25

2

3

1,5

NIV 220

NV 419

NC 661

θj


CPU's 191.2

Nodes 35 890

Iterations 299 738

As it can be seen through table 5.18, although a complicated example was considered, the model was still solved in a reasonable time. Therefore it can be concluded that the proposed model seems adequate to perform the layout of industrial plants within a 3D space.


81

5.3.2 – 3D Layout with Multifloor

The presence of floor restrictions is usually an issue at industrial facility problems. Some equipment items may present restrictions within the possible floors allocation (Jf) while others may be allocated freely within the 3D space. Two situations are considered in this paper: (1) a pre-defined fixed number of floors with fixed height and (2) a variable number of floors with variable height. In mathematical terms the following equations must be added to the model presented previously.

(1) pre-defined fixed number of floors with fixed height: considering the height of the equipment elements along with the set of floors (Nj) where they must/can be allocated as well as its associated heights (hn, where n is the number of the floor), the equipment floor allocation is obtained from:

zj −

θj 2

=

∑f

n j

⋅ hn

∀j ∈ J f

(5.31)

n∈N j

Since each equipment element can only be allocated to a single floor the following equation must be together considered:

∑f

n j

=1

∀j ∈ J f

(5.32)

n∈N j

Note that, the case where the equipment element j allocation is defined over one possible allocation floor with null value on the z-axis, the equality (=) in equation (5.32) can be replaced by the operator ≤ and the binary variable fjn omitted in constraints (5.31) and (5.32). Also, when equipment element j has a pre-defined unique floor allocation equation (5.32) can be omitted and in equation (5.31) the right hand side sum replaced by the fixed height of the floor. These changes allow the reduction in the number of equations as well as variables.

(2) Variable number of floors with variable height: in this case the number of floors will be defined based on the use of floors. Nevertheless, it must be set the maximum number of floors allowed. Thus, the equipment floor allocation is given by:

zj −

θj 2

=

∑f

n j

⋅ hf

n

∀j ∈ J f

(5.33)

n∈N j


82

where the product

f jn ⋅ hf

n

is non-linear. This however can be exactly linearized by

considering the auxiliary continuous variable FH nj , defined as FH nj = f jn ⋅ hf n , and therefore equation (5.33) replaced by:

zj −

θj 2

∑ FH

=

n j

∀j ∈ J f

(5.34)

n∈N j

Simultaneously, the following constraints must be added:

FH nj ≤ M '⋅ f jn

FH nj ≤ hf

n

∀j , n

(5.35)

∀j , n

(5.36)

FH nj ≥ hf n + M '⋅( f jn − 1)

∀j , n

(5.37)

with,

M ' = z max

(5.38)

On the other hand, in order to sort the floors by theirs heights and limit therefore the solution feasible space, it can be added:

hf n ≤ hf

n +1

hf n ≤ z max

∀n | 0 ≤ n < max{n} ∀n | n = max{n}

(5.39)

(5.40)

where max{n} defines the cardinal number of the greatest floor defined in the model.

Note that, it is also possible to define allocation intervals to the floor heights considering the following generic constraint:

n n hf min ≤ hf n ≤ hf max

∀n


(5.41)

83

n n where hf min and hf max represent respectively the minimum and maximum possible height of

floor n.

In conclusion, objective function (4.1) or (5.1) along with an appropriate selection of the constraints (4.2) to (4.9), (4.11), (4.12), (4.23) to (4.26), (4.31) to (4.34) and (5.2) to (5.30) along with the ones presented in this section namely constraints (5.31) to (5.33) and (5.35) to (5.41) define the MILP model for the 3D layout of industrial plants with multifloor allocation and without production sections. The latter characteristic will be explored later in the next section.

Examples

In order to illustrate the 3D model extension for multifloor allocation a set of examples are solved. Again, the GAMS/CPLEX package is used and a margin of 5% of optimality considered.

Example 5 – Equipments in the 3D space with fixed multifloor

The example 4 presented in section 5.3.1 is again studied where fixed multifloor constraints are addressed. Three floors define the plant layout: Floors 0, 1 and 2, with heights of 0m, 6m and 12,5m respectively. An allocation available space of 15m x 10m x 15m is considered. Again, objective function (4.1) is used to find the optimal plant layout. Considering the plant flowsheet, shown in figure 5.9, equipment element V2 must be allocated to floor 1, equipment element V6 to floor 2 and equipment element 2a must be allocated to floor 1 or 2. Also, equipment elements 1a, R4, V5 and V6a must be allocated to any floor (0, 1 or 2) in the plant while all other equipment elements (V1, 1b, R2 and V5a) can be allocated to any point within the available allocation space. No operational or other type of constraints were imposed. The results show the optimal plant layout as depicted in figure 5.11 and table 5.19. The problem statistics appear in table 5.20.


84

Figure 5.11 –Optimal Plant Layout Table 5.19 – Optimal Plant Layout Coordinates and Dimensions Units

Elements

xj

yj

zj

lj

dj

θj

Unit_V1

Elem_V1

5,5

8

2

5

3

2

Unit_V2

Elem_V2

12

4,5

7,5

6

6

3

Unit_1a

Elem_1a

6

3,5

1,5

6

6

3

Unit_1b

Elem_1b

6,5

4

4,75

5

5

2,5

Unit_2a

Elem_2a

6

3,5

7,5

6

6

3

Unit_R2

Elem_R2

11,25

5,25

11,5

4,5

4,5

2

Unit_R4

Elem_R4

11,5

5,5

13,75

5

5

2,5

Unit_V5

Elem_V5

1,5

4

7,5

3

5

3

Unit_V6

Elem_V6

6

3,5

13,5

6

6

2

Unit_V5a

Elem_V5a

14,5

9

14,5

1

2

1

Unit_V6a

Elem_V6a

13

1,5

13,25

2

3

1,5

NV 435

NC 675


CPU's 73.54

Nodes 7 091

Iterations 155 655

NIV 236

When comparing the results obtained against the one without floor considerations it can be observed that even considering a larger available space, the presence of additional multifloor constraints impose a penalty value in the optimal objective function value (185 against 153, see tables 5.20 and 5.18). This is explained by the extra costs associated with the final plant connectivity, imposed by the floor allocations.


85

Example 6 – Equipments in the 3D space with variable multifloor and zoning

Again example 4 from section 5.3.1 is considered but this time with variable multifloor constraints and minimum allowed distance constraints between equipment items. Three plant floors are proposed: 0, 1 and 2, without associated fixed heights. The minimisation of the capital connectivity cost -equation (4.1)- is considered over an available space of 12m x 12m x 15m. As before, equipment element V2 must be allocated to floor 1, equipment element V6 to floor 2 and equipment element 2a must be allocated to floors 1 or 2. Equipment elements 1a, R4, V5 and V6a must be allocated to any floor in the plant (0, 1 or 2) while all others equipment elements (V1, 1b, R2 and V5a) can be allocated to any point within the available allocation space. Safety/operational restrictions are also imposed translated into minimum possible distances between equipment elements. Thus, for equipment elements 1a-R2 a minimum distance of 5m must exist while for units 1b-R2, 2a-R2 as well as R4-R2 this minimum distance is equal to 3m. The associated optimal plant layout obtained is shown in figure 5.12 while optimal equipment coordinates and dimensions are exhibited in table 5.21. The problem statistics appear in table 5.22.



86


Elements

xj

yj

zj

lj

dj

θj

Unit_V1

Elem_V1

4,5

8,5

11

3

5

2

Unit_V2

Elem_V2

9

8

10,5

6

6

3

Unit_1a

Elem_1a

3

3

10,5

6

6

3

Unit_1b

Elem_1b

8,5

2,5

10,75

5

5

2,5

Unit_2a

Elem_2a

3

3

13,5

6

6

3

Unit_R2

Elem_R2

8,25

8,75

3

4,5

4,5

2

Unit_R4

Elem_R4

3,5

4

3,25

5

5

2,5

Unit_V5

Elem_V5

8,5

4,5

13,5

5

3

3

Unit_V6

Elem_V6

3

9

13

6

6

2

Unit_V5a

Elem_V5a

0,5

2,5

5

1

2

1

Unit_V6a

Elem_V6a

1,5

7,5

2,75

3

2

1,5

NV 454

NC 744


CPU's 220.64

Nodes 20 932

Iterations 451 496

NIV 220

Optimal floor heights were located respectively at 2m, 9m and 12m. All the equipment units with floor allocation took a different floor comparing with the previous example. Although the presence of additional multifloor and zoning constraints normally impose a penalty value in the objective function, in this case, the increment on the available plant space (12x12x15 against 10x10x15) led to a global decrease in the optimal value of the objective function (68 against 153, see tables 5.22 and 5.20). As expected the presence of additional constraints impose an extra computational time to the problem (220.64 sec. versus 73.54 sec., see tables 5.22 and 5.20).

Example 7 – Equipments in the 3D space with variable multifloor and floor height allocation intervals

The example 3 presented in section 5.3.1 is solved considering variable multifloor constraints and floor height allocation intervals. The same available space (15m x 12m x 12m) is considered. Two floors describe the plant (floors 0 and 1) and the minimisation of the connectivity cost - objective function (4.1) - is used Equipment element 1 must be allocated to floor 0; equipment element 4 to floor 1 and equipment elements 2a/2b and 7 must be allocated in floors 0 or 1. All others equipment units (3, 5, and 6) can be allocated to any point within the available allocation space.


87

Additional constraints are imposed. The height of floor 0 must be lower than the height of floor 1, and the latter must be lower than the maximum height space (zmax). Also, the height of floor 0 must be within the interval defined between 0m and 1m and the height of floor 1 must be within the interval defined between 6m and 10m (overwrites zmax). The optimal plant layout is shown in figure 5.13 and table 5.23 while the problem statistics appear in table 5.24.


Elements

xj

yj

zj

lj

dj

θj

Unit_1

Elem_1

7,7

8,2

1

13,8

3

2

Elem_2a

13,65

5,35

1,3

2,6

2,7

2,6

Elem_2b

12,6

2,75

1,3

4,8

2,6

2,6

Unit_3

Elem_3

7,2

8

4

14,4

2,2

4

Unit_4

Elem_4

9,7

2,9

9,55

5,7

5,8

2,5

Elem_5a

8,2

4,85

7,15

9

2,3

2,3

Elem_5b

13,85

3

7,15

2,3

6

2,3

Elem_5c

2,55

3

7,15

2,3

6

2,3

Elem_6a

5,2

9

7

2

6

2

Elem_6b

2,2

9

7

4

2

2

11,35

7,8

9,8

7

4

3

Unit_2

Unit_5

Unit_6 Unit_7

Elem_7


CPU's 11.97

Nodes 2 556


Iterations 13 134

NIV 185

NV 338

NC 617

88

Optimal floor 0 and 1 heights were located respectively at 0m and 8,3m. Equipment unit 2 took floor 0 position while equipment unit 7 took floor 1. The optimal objective function value increased from the initial problem (250.9 against 122.55, see tables 5.24 and 5.14), as it could be expected due to the presence of multifloor constraints leading to a more spread spatial arrangement than the example 3 layout.

5.3.3 – 3D Layout with Production Sections

In the FLP, pre-defined production sections are usually present (e.g. storage, cutting, evaporation, filling) due to operational requirements such as safety, materials handling and resources management amongst others. In order to extend the previous layout models –with or without multifloor conditions- so as to account for the production sections, the following constraints are added/modified to the 3D layout model:

Non-overlapping Constraints: when considering production sections, two types of these constraints are defined. These are respectively the ones applied to single equipment elements (i) and the ones applied to the production sections (ii):

(i) Non-overlapping Constraints for Equipment Elements: constraints (5.10) to (5.17) still hold but they are just applied within the different production sections (Pj=Pj’) to which the equipment elements belong;

(ii) Non-overlapping Constraints for Production Sections: as for the equipment elements also the non-overlapping of production sections must be considered. Constraints (4.35) to (4.38) are replaced by new ones. These are similar to the equivalent equipment constraints although small changes must be introduced:

X p − X p ' + Mx'⋅( S 2 p , p ' + S 3 p , p ' ) ≥

X p ' − X p + Mx'⋅( S1 p , p ' + S 3 p , p ' ) ≥

L p + L p' 2 L p + L p'


2

∀p, p' | p ' > p

(5.42)

∀p, p' | p' > p

(5.43)

89

Y p − Y p ' + My '⋅( S1 p , p ' + S 2 p , p ' ) ≥

D p + D p'

D p + D p'

Y p ' − Y p + My '⋅(2 − S1 p , p ' − S 2 p , p ' ) ≥

Z p − Z p ' + Mz '⋅(2 − S 2 p , p ' − S 3 p , p ' ) ≥

∀p, p' | p' > p

2

∀p, p' | p ' > p

2 H p + H p'

Z p ' − Z p + Mz '⋅(2 − S1 p , p ' − S 3 p , p ' ) ≥

(5.44)

(5.45)

∀p, p'| p' > p

2 H p + H p'

(5.46)

∀p, p' | p ' > p

2

(5.47)

with:

S1 p , p ' + S 2 p , p ' + S 3 p , p ' ≥ 1 ∀p, p ' | p ' > p

(5.48)

S1 p , p ' + S 2 p , p ' + S 3 p , p ' ≤ 2 ∀p, p ' | p ' > p

(5.49)

where, Mx’, My’ and Mz’ are defined as suitable upper bounds on the distance between any two production sections p and p’, respectively on the x, y and z-axis. As before, three new auxiliary binary variables (S1p,p’, S2p,p’ and S3p,p’) are created and again only one of the above constraints (5.42 to 5.47) can be active – disjunctive condition.

Safety/Operability Constraints: as before, minimum and maximum possible distances between production sections can be defined in order to guarantee safety and operability conditions. Constraints (4.23) to (4.26) and (5.21) to (5.28) defined over equipment elements still hold since they can exist even for equipment elements belonging to different production sections. Furthermore, as for the equipments also a certain minimum distance between production sections may be considered. These conditions are similar to the equipment constraints although small changes must be introduced. Constraints (4.39) to (4.42) must no longer exist. Thus, to model a certain minimum possible distance between production sections, inequations (5.42) to (5.47) must be replaced by:

X p − X p ' + ( Mx'+ ZX pmin , p' ) ⋅ (S 2 p, p' + S 3 p, p' ) ≥


L p + L p' 2

+ ZX pmin , p'

∀p, p'| p' > p

(5.50) 90

L p + L p'

X p ' − X p + ( Mx'+ ZX pmin , p ' ) ⋅ ( S1 p , p ' + S 3 p , p ' ) ≥

Y p − Y p ' + ( My '+ ZY pmin , p ' ) ⋅ ( S1 p , p ' + S 2 p , p ' ) ≥

2

D p + D p' 2

Y p ' − Y p + ( My '+ ZY pmin , p ' ) ⋅ ( 2 − S1 p , p ' − S 2 p , p ' ) ≥

Z p − Z p' + (Mz'+ZZ pmin , p ' ) ⋅ (2 − S 2 p , p ' − S 3 p , p ' ) ≥

Z p' − Z p + (Mz'+ZZ pmin , p ' ) ⋅ (2 − S1p , p ' − S 3 p, p ' ) ≥

+ ZX pmin , p'

+ ZY pmin , p'

D p + D p' 2

H p + H p' 2

H p + H p' 2

+ ZY pmin , p'

+ ZZ pmin , p'

+ ZZ pmin , p'

∀p, p'| p' > p

∀p, p'| p' > p

∀p, p ' | p ' > p

∀p, p'| p' > p

∀p, p'| p' > p

(5.51)

(5.52)

(5.53)

(5.54)

(5.55)

And again, constraints (5.48) and (5.49) must be present.

Still addressing the problem of guaranteeing a certain number of operability/safety restrictions a maximum possible distance between production sections may also be imposed. Along with constraints (4.43) to (4.46), for the 3D layout with plant production sections, the following constraints must be added:

Z p − Z p' ≤

Z p' − Z p ≤

H p + H p' 2 H p + H p' 2

+ ZZ pmax , p'

∀( p, p' ) ∈ ZonZ max

(5.56)

+ ZZ pmax , p'

∀( p, p' ) ∈ ZonZ max

(5.57)

Allocation Constraints: also in this case we have two types of constraints. Ones describing the equipment elements allocation within the respective production section space (i), and the others the allocation of the production sections within the available plant space (ii):


91

(i) Equipment Elements Allocation Constraints: Constraints (4.47) to (4.50) still hold. Along, the allocation of each equipment within its production section in the 3D layout problem is modelled using the additional constraints:

zj +

zj −

θj 2

θj 2

≤ Zp +

≥Zp −

Hp 2 Hp 2

∀p, j ∈ J p

(5.58)

∀p, j ∈ J p

(5.59)

(ii) Production Sections Allocation Constraints: constraints (4.31) to (4.34), (5.29) and (5.30) must be modified resulting in the previous defined constraints (4.51) to (4.54) along with:

Zp ≥

Zp +

Hp 2 Hp 2

∀p

≤ z max

(5.60)

∀p

(5.61)

In conclusion, objective function (4.1) or (5.1) with constraints (4.2) to (4.9), (4.11), (4.12), (4.23) to (4.26), (4.31) to (4.34), (4.43) to (4.54), (5.2) to (5.33) and (5.35) to (5.41) along with the ones presented in this section -(5.42) to (5.61)- applied to production sections define the MILP model for the 3D layout with multifloor and production sections.

Examples

A new example is now solved considering production sections.

Example 8 – Equipments in the 3D space with variable multifloor and production sections

The example 6 with variable multifloor as presented in section 5.3.2 is solved but this time without zoning constraints but with production section constraints. Objective function (4.1) is used to find the optimal plant layout. An allocation available space of 20m x 19m x 19m with


92

three production sections is considered. Production section S1 is formed by equipment units V1 and V2, production section S2 by units 1a, 1b, 2a, R2 and R4 and finally production section S3 by units V5, V5a, V6 and V6a (see figure 5.9) The resulting optimal plant layout is shown in figure 5.14 and table 5.25 and 5.26 while the problem statistics appear in table 5.27.

Figure 5.14 – Optimal Plant Layout Table 5.25 – Optimal Production Section Coordinates and Dimensions Production Section

Xp

Yp

Zp

Lp

Dp

Hp

SP_1

5,5

3

8,5

11

6

3

SP_2

5,5

10,333

7,5

11

8,667

11

SP_3

14

8,333

7,5

6

10,667

11


Elements

xj

yj

zj

lj

dj

θj

Unit_V1

Elem_V1

8,5

4,5

9

5

3

2

Unit_V2

Elem_V2

3

3

8,5

6

6

3

Unit_1a

Elem_1a

8

9

8,5

6

6

3

Unit_1b

Elem_1b

2,5

8,5

8,75

5

5

2,5

Unit_2a

Elem_2a

8

9

11,5

6

6

3

Unit_R2

Elem_R2

8,25

8,25

5,5

4,5

4,5

2

Unit_R4

Elem_R4

8

8,5

3,25

5

5

2,5

Unit_V5

Elem_V5

13,5

4,5

11,5

5

3

3

Unit_V6

Elem_V6

14

9

11

6

6

2

Unit_V5a

Elem_V5a

12

11,5

5

2

1

1

Unit_V6a

Elem_V6a

12

9,5

2,75

2

3

1,5


93


CPU's 243.49

Nodes 37 734

Iterations 647 316

NIV 115

NV 367

NC 482

Optimal floor heights were located respectively at 2m, 7m and 10m. When comparing the equipment floor allocation with example 6, only equipment unit 1a shifted its allocation from floor 2 to floor 1. Although the presence of production section constraints often impose a penalty value in the objective function, the increment on the available plant space (20x19x19 against 12x12x15) led to a global decrease in the optimal value of the objective function (60 against 68, see tables 5.27 and 5.22).

5.4 - Conclusions and Future Work The layout problem of industrial facilities over a 3D continuous space has been considered. A generic model was developed where important real aspects were considered. Different topological characteristics such as, different equipment orientations, distance restrictions, different equipment connectivity inputs and outputs, rectangular and irregular equipment shapes, space availability and multifloor allocations were modelled. For the case of multifloor allocations two cases were studied. These are the equipment allocation within a pre-defined fixed number of floors with fixed height and over a variable number of floors each floor with a variable height. Also, and looking into operational terms, the existence of production or operational sections as well as the presence of safety and operability restrictions were incorporated into the model. The final model is described trough the formulation of a MILP which guarantees the optimal plant layout considering not only the connectivity capital costs but also operational costs – these associated essentially with flow pumping costs. Thus a trade-off between capital and operational costs is obtained at the final plant layout. Several examples were solved and different layout characteristics illustrated. The solution of such problems was obtained in reasonable computational times. Although, it is a fact that for highly complicated examples the model may be quite time consuming. Thus, one of the points that may be considered for future work is the study of alternatives to reduce the computational time where the characteristics of the problem and/or the application of other solutions techniques should be explored.


94

In terms of model application, in the next chapter the proposed models will be generalised to deal with the simultaneous design and or retrofit of generic industrial facilities. In this case, not only layout aspects are considered but also design and operational characteristics are present and influence the final layout design.


95


96

Chapter 6 OPTIMAL LAYOUT AND DESIGN OF INDUSTRIAL FACILITIES: A SIMULTANEOUS APPROACH

6.1 - Introduction The equipment layout decision is an essential stage in the design of industrial plants. Traditionally, layout issues have been considered a posteriori once the main plant design stage has been completed. This can be explained due to the high level of complexity often associated with each one of the problems on its one. However, the interactions of layout with the rest of the design decisions are often quite strong and this renders a simultaneous approach more desirable. Thus a simultaneous approach to the design and layout of facilities appears as an important problem to be addressed. In the previous chapters a generic mathematical model for the optimal facility layout problem was presented. That formulation is extended in this chapter so as to consider simultaneously the design (choice of equipments and connections) and layout aspects (spatial arrangement of equipments and connections) accounting for possible strong interactions that may exist when designing industrial facilities. The model developed is essentially focused on the layout characteristics while the design aspects are addressed by considering the possible existence of a certain equipment item or connection. This existence is defined by adequate design models developed based on the design problem characteristics. A generic model is obtained where no assumptions were made for the equipment or connection choices. These are handled explicitly when defining the optimal layout of the plant, so as to leave open the possibility of incorporating the optimal layout with different design models. In this way, different operational and topological problem characteristics, often dictated by the type of plant being designed (ex: cellular plant, flow-shop or job-shop structures and operations) can be addressed. The optimal plant layout is obtained based on the minimisation of the rectilinear connectivity cost -where different topological and operational characteristics are considered- along with the equipment unit capital costs over a two dimensional (2D) continuous area. In topological terms different aspects are considered such as equipment orientations, distance restrictions, nonoverlapping constraints, equipment connectivity inputs and outputs, rectangular or irregular equipment shapes and space availability. On the other hand, in operational terms, production and operational sections are modelled as well as safety and operability restrictions.

Optimal Layout and Design of Industrial Facilities: A Simultaneous Approach

97

The final model is a Mixed Integer Linear Problem (MILP) where binary variables are introduced to characterise design and topological choices while continuous variables describe the distances and locations involved. To conclude, the applicability of the proposed formulation is illustrated via the solution of representative examples.

6.2 – Design and Layout Problem Statement and Characteristics The design and layout of industrial plants as addressed in this chapter can be stated formally as follows:

Given •

A possible set of equipment items and their geometrical shapes and sizes over a 2D area;

•

Input and output points locations inside each equipment space;

•

A possible connectivity structure;

•


•


•

Production section characteristics and associated equipment units involved, if present;

•

Space availability;

•

Capital and operational cost of all connectivity structures;

•

Cost of all equipment units;

•

Other design constraints;

•

Optimal plant equipment choices;

•

Optimal plant equipment arrangement – coordinates and orientation;

•

Associated connectivity structure (inputs and outputs).

Determine

So as to optimise a given quantitative objective function, generally the minimisation of the plant layout equipment and connectivity cost, while fulfilling all the constraints defined.


98

The design of industrial plants is on itself a complex problem whose options depend on the type of plant in study (e.g. flow-shop, job-shop, etc.). For instance if a multipurpose plant is to be designed the choices involved at the design stage are completely diversified from the ones that will be taken when designing a dedicated plant. Based on this, the design aspects introduced in this chapter are restricted to the possibility of having or not a certain equipment item (e.g. unit or connection) in the final plant. Thus, no assumption or decision is made on the choice of any type of equipment leaving this possibility to the use of appropriated models that will be easily incorporated in the present model. This choice is defined through the value of variables that model the equipment items presence (Eg for units and Ec for connections). As defined in chapter 4 the equipment units are assumed to have a rectangular or irregular shape. For the latter, a set of rectangular equipment elements j model the entire unit which have given dimensions over the x and y–axis (αj, βj) and possible multiple input and output points.

Using these problem characteristics the mathematical formulation for the simultaneous design and 2D layout problem can now be developed.

6.3 – Design and Layout Mathematical Formulation The following indices, sets, parameters and variables are defined:

Global Indices: g, g’ – equipment unit j, j´– equipment element oi, oi’ – point (output or input) c – connection Sets:

J g = {j: set of equipment elements j that form unit g} G j = {g: equipment unit containing equipment element j} – Unique set Linksg = {(j, j´): set of linked equipment elements pairs (j, j´) ∈ Jg}

Zonx max = {(j, j´): set of equipment elements pairs (j, j´): Gj ≠ Gj´ with an upper limit distance between them along the x-axis}


99

Zony max = {(j, j´): set of equipment elements pairs (j, j´): Gj ≠ Gj´ with an upper limit distance between them along the y-axis} OI j = {oi: set of (output or input) points of equipment element j} OI c = {(oi, oi’): set of output oi and input oi’ pairs defining connection c} Gc = {(g, g’): set of equipment units pairs (g, g’) connected by connection c}

Parameters:

C oi ,oi ' – cost per meter of the connection between the output point oi and the input point oi’

CC g0 – fixed capital cost of equipment unit g

α j , β j – dimensions of equipment element j over the x and y-axis respectively x max , y max – maximum area over the x and y-axis ∆x j ,oi , ∆y j ,oi – relative distance between the point oi and the geometrical center of the

equipment element j respectively in the x and y-axis, as stated by the original equipment representation (0° rotation) min Zx min j , j ' , Zy j , j ' - minimum distance allowed between equipment elements j and j´ over the

x and y-axis respectively max Zx max j , j ' , Zy j , j ' - maximum distance allowed between equipment elements j and j´ in the x

and y-axis respectively ∆x j , j ' , ∆y j , j ' - relative distance between the geometrical centers of equipment elements

(j, j´) : Gj = Gj´ as defined by its original position, with a 0° rotation over both the x and y-axis Variables: Continuous Variables - all defined as positive variables: xj, yj – coordinates of the geometrical center of equipment element j xoi, yoi – coordinates of the point oi lj – length of equipment element j dj – depth of equipment element j Rioi,oi’ – relative distance in x coordinates between the output point oi and the input point oi’, if oi is to the right of oi’ Leoi,oi’ – relative distance in x coordinates between the output point oi and the input point oi’, if oi is to the left of oi’


100

Baoi,oi’ – relative distance in y coordinates between the output point oi and the input point oi’, if oi is in the back of oi’ Froi,oi’ – relative distance in y coordinates between the output point oi and the input point oi’, if oi is in front of oi’ Doi,oi’ – total rectilinear distance between the output point oi and the input point oi’ Binary Variables: og - equipment unit g orientation: = 1 if the length of all the equipment elements j∈Jg (parallel to the x axis) is equal to α j ; 0 otherwise o1g, o2g, o3g and o4g – definition of the equipment unit g anti-clockwise rotation, expressed in multiples of 90° (respectively: 0°, 90°, 180° and 270°) from the original equipment representation Eg - equipment unit g existence: = 1 if the equipment unit g is present in the optimal solution problem; 0 otherwise Ec - connection c existence: = 1 if the connection c is present in the optimal solution problem; 0 otherwise.

Using the above definitions and variables the mathematical model will now be presented.

6.3.1 – Basic 2D Layout Model with Variable Equipments and Connections

The following objective function and constraints characterise the generic model for the 2D industrial design and layout problem with variable equipments and connections:

Objective Function:

The definition of the Objective Function (OF) depends on the concurrent design model chosen to model the simultaneous layout and design of the FLP. The OF here adopted was based on the minimisation of the total connectivity cost considered along with the capital cost of equipment units. The former involves the cost of the physical connections and the operating costs caused


101

by material transfers occurring within the connection where all the possible outputs and inputs of equipment units g with a non-zero connection cost are considered.

Min

∑

C oi ,oi ' ⋅ D oi , oi ' + ∑ CC g0 ⋅ E g

( oi , oi ')|C oi , oi ' ≠ 0

(6.1)

g

Constraints:

A number of different types of constraints need to be introduced in order to model the problem. These, include equipment orientation, equipment irregular shapes, input/output locations, connectivity distances, equipment non-overlapping, safety and operability restrictions, and finally, area allocation constraints.

Equipment Orientation Constraints: for the equipment orientation it is assumed that the equipment unit rotation is allowed over the x and y-axis and is defined through the value of the variable og (see figure 4.2) if the equipment unit (Eg) is present. Based on this, the following equations are written:

l j = α j ⋅ο g + β j ⋅ E g − β j ⋅ og

d j = (α j + β j ) ⋅ E g − l j

∀g , j | j ∈ J g

∀g , j | j ∈ J g

(6.2)

(6.3)

where through the use of the equipment element dimensions (αj,βj) and the value of the orientation binary variable (og), the equipment element length (lj) and depth (dj) are obtained. Two situations may occur. In the first one, if equipment unit g is present (Eg=1), in equation (6.2) for og =1 the length of each equipment element j, belonging to unit g, is equal to αj, otherwise is given by βj. On the other hand, equation (6.3) states that the depth (dj) of equipment element j is equal to the remaining equipment dimension. That is, if lj is equal to αj then dj is equal to βj, otherwise dj is equal to αj which defines og=0. In the second one, if equipment unit g is not present (Eg=0) then lj and dj take the value of zero.

Equipment Irregular Shapes Constraints: an equipment unit g may have an irregular form dictated from the set of its equipment elements j. Each equipment element j within the equipment unit set (Jg) is related to the remaining equipment elements through the definition of


102

pre-defined distances between them (∆xj,j´, ∆yj,j´). Knowing the pre-defined set of linked equipment element pairs (Linksg) and assuming that each equipment unit g is at one of the four orthogonal positions allowed – o1g, o2g, o3g and o4g - (see figure 4.2) the following equations for two generic equipment units with geometric centres at (xj, yj) and (xj’, yj’) are defined:

x j ' = x j + o1g ⋅ ∆x j , j ' − o2 g ⋅ ∆y j , j ' − o3g ⋅ ∆x j , j ' + o4 g ⋅ ∆y j , j '

∀g, j, j'| ( j, j' ) ∈ Linksg

(6.4)

y j ' = y j + o1g ⋅ ∆y j , j ' + o2 g ⋅ ∆x j , j ' − o3g ⋅ ∆y j , j ' − o4 g ⋅ ∆x j , j '

∀g , j, j'| ( j, j' ) ∈ Linksg

(6.5)

And since each equipment unit can only exist in a single position:

ο1g + ο 2 g + ο 3 g + ο 4 g = E g

∀g

(6.6)

If equipment unit g is present (Eg=1) then, one of the above orthogonal binary variables (o1g, o2g, o3g and o4g) must take obligatorily a unitary value (binary variable). Therefore the two equipment elements j and j’ are linked by the pre-defined distances defined in equations (6.4) and (6.5). If the equipment unit is not present (Eg=0) then its coordinates (x and y) are coincident and not relevant.

The equipment unit orientation is obtained from:

ο g = ο1 g + ο 3 g

∀g

(6.7)

As described in chapter 4 (Figure 4.2) four possible ortoghonal positions of an equipment unit are considered.

Input/Output Constraints: different input and output point locations (xoi, yoi) are considered which are defined inside each equipment element j. As the previous constraints, its locations can easily be calculated using the orientation variables, as defined in figure 4.3, combined with the equipment element geometrical center coordinates (xj, yj). Thus, along with constraints (6.6) and (6.7), the following equations are defined:

xoi = x j + ο1g ⋅ ∆x j ,oi − ο 2g ⋅ ∆y j ,oi − ο 3g ⋅ ∆x j ,oi + ο 4g ⋅ ∆y j ,oi

∀g, j ∈ J g , oi ∈ OI j


(6.8)

103

yoi = y j + ο1g ⋅ ∆y j ,oi + ο 2g ⋅ ∆x j ,oi − ο 3g ⋅ ∆y j ,oi − ο 4g ⋅ ∆x j ,oi

∀g, j ∈ J g , oi ∈ OI j

(6.9)

where xoi and yoi characterise the different point coordinates used for the description of the connectivity existence. For model simplicity these points were defined as global in the model but it is easily understood that they are located within the different equipment elements present (see figure 4.3). Again, if equipment unit g is present (Eg=1) then, one of the orthogonal binary variables (o1g, o2g, o3g and o4g) must take obligatorily a unitary value, and therefore the points coordinates are defined through the definition of the relative distance (∆xj,oi, ∆yj,oi) between the points and the geometrical centers of the respective equipment element j - equations (6.8) and (6.9). On the other hand, if the unit is not present (Eg=0) its coordinates (xoi with xj and yoi with yj) are coincident and not relevant.

Distance Constraints: rectilinear distances are assumed between each output and input point of equipment elements. Accounting for a 2D dimensional area the total distance for each non-zero cost connection (oi, oi’) is given by:

Doi , oi ' = Rioi , oi ' + Le oi ,oi ' + Ba oi ,oi ' + Froi , oi '

∀c, oi, oi ' | (oi, oi ' ) ∈ OI c

(6.10)

where the relative distances (Ri, Le, Ba and Fr) are obtained from:

(1 − E c ) ⋅ M "+ Rioi ,oi ' − Le oi , oi ' ≥ x oi − x oi '

∀c, oi, oi ' | (oi, oi ' ) ∈ OI c

(6.11)

Rioi , oi ' − Le oi , oi ' ≤ x oi − x oi ' + (1 − E c ) ⋅ M "

∀c, oi, oi' | (oi, oi ' ) ∈ OI c

(6.12)

(1 − E c ) ⋅ M "+ Ba oi , oi ' − Froi ,oi ' ≥ y oi − y oi '

∀c, oi, oi' | (oi, oi' ) ∈ OI c

(6.13)

Ba oi , oi ' − Froi , oi ' ≤ y oi − y oi ' + (1 − E c ) ⋅ M "

2 ⋅ Ec ≤ E g + E g '

∀c, oi, oi ' | (oi, oi ' ) ∈ OI c

∀c , g , g ' | ( g , g ' ) ∈ G c

(6.14)

(6.15)

where,

M " = x max + y max


(6.16) 104

Constraint (6.15) implies that a connection c can only exist (Ec=1) if the two equipment items (g and g’) that it connects exist (Eg=1 and Eg’=1). A connection c is defined between the output point oi of an equipment and the input point oi’ of another. If a connection is not present (Ec=0) inequations (6.11) to (6.14) are inactive due to the big value of M”. Otherwise, if a particular connection is active (Ec=1) inequations (6.11) and (6.12) model the equality: Ri oi ,oi ' − Le oi ,oi ' = x oi − x oi ' ; while inequations (6.13) and (6.14) represent a similar equation defined over the y-axis: Ba oi ,oi ' − Froi ,oi ' = y oi − y oi ' . Therefore, if the output point oi is to the right of the input point oi’, then xoi is greater than xoi’. Thus, constraints (6.11) and (6.12) imply that Le (from Left) equals to zero and Ri (from Right) takes the positive distance difference between the connection points, since this is a minimisation problem. The same idea is translated into constraints (6.13) and (6.14) in terms of the y-axis (Fr - Front and Ba - Back).

Non-overlapping Constraints: two-equipment elements j and j´ cannot occupy the same physical location as described in chapter 4. Based on that formulation, a set of disjunctive nonoverlapping constraints is formulated to guarantee this condition. However, this condition must be adapted to the case where a variable layout is modelled. Thus the following constraints are written:

M x ⋅ (2 − Eg − Eg ' ) + x j − x j ' + M x ⋅ (E1 j , j ' + E2 j , j ' ) ≥

l j + l j'

(6.17) 2 ∀g, g ' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ G j´

M x ⋅ (2 − Eg − Eg ' ) + x j ' − x j + M x ⋅ (1 − E1 j, j ' + E2 j , j ' ) ≥

l j + l j'

(6.18) 2 ∀g, g ' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ G j´

M y ⋅ (2 − Eg − Eg ' ) + y j − y j ' + M y ⋅ (1 + E1 j, j ' − E2 j, j' ) ≥

d j + d j'

2 ∀g, g' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ G j´

M y ⋅ (2 − Eg − Eg ' ) + y j' − y j + M y ⋅ (2 − E1j, j ' − E2 j, j ' ) ≥

(6.19)

d j + d j'

2 ∀g, g' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ G j´


(6.20)

105

Note that these constraints only have a meaning if the two equipment elements (j and j’) belong to two equipment units (g and g’ respectively) that exist in the problem. Thus, if any or both of the units existence binary variables is null (Eg=0 and/or Eg’=0) constraints (6.17) to (6.20) become inactive, due to the value of the upper bounds Mx and My. Otherwise, if both the equipment units are present (Eg=1 and Eg’=1), the first term becomes null and only one of the above constraints can be active – disjunctive condition. This is obtained through the possible four combinations of the values of the two auxiliary binary variables (E1j,j´ and E2j,j´). The values of Mx and My are taken as suitable upper bounds on the distance between any two-equipment elements j and j´ given by:   M x = min  x max , ∑ max(α j , β j ) j  

(6.21)

  M y = min  y max , ∑ max(α j , β j ) j  

(6.22)

Safety/Operability Constraints: often minimum and maximum distances between equipment items are defined due to safety and operability conditions. In order to guarantee a certain minimum distance between equipment items, constraints (6.17) to (6.20) must be replaced by:

M x ⋅ (2 − Eg − Eg ' ) + x j − x j' + (M x + Zxmin j , j ' ) ⋅ (E1 j , j ' + E2 j , j ' ) ≥

l j + l j'

+ Zxmin j, j '

2 ∀g, g' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ G j´

M x ⋅ (2 − Eg − Eg ' ) + x j ' − x j + (M x + Zxmin j , j ' ) ⋅ (1 − E1 j , j ' + E2 j , j ' ) ≥

l j + l j'

+ Zxmin j, j '

2 ∀g, g' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ G j´

M y ⋅ (2 − Eg − Eg' ) + y j − y j' + (M y + Zymin j , j ' ) ⋅ (1 + E1j , j ' − E2 j , j ' ) ≥

(6.24)

d j + d j'

+ Zymin j, j ' 2 ∀g, g' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ Gj´

M y ⋅ (2 − Eg − Eg' ) + y j' − y j + (M y + Zymin j , j ' ) ⋅ (2 − E1j , j ' − E2 j , j ' ) ≥

(6.23)

d j + d j'

+ Zymin j, j '

2 ∀g, g' , j, j'| j'∈ J g ' > j ∈ J g ∧ G j ≠ Gj´


(6.25)

(6.26)

106

On the other hand, maximum possible distance restrictions can be modelled as follows:

x j − x j' ≤

x j' − x j ≤

y j − y j' ≤

y j' − y j ≤

l j + l j' 2 l j + l j' 2

x + Zxmax j , j ' + M ⋅ (2 − Eg − Eg ' )

∀g, g' , j, j'| ( j ∈ J g , j'∈ J g' ) ∈ Zonxmax

(6.27)

x + Zxmax j , j ' + M ⋅ (2 − Eg − Eg ' )

∀g, g' , j, j'| ( j ∈ J g , j'∈ J g' ) ∈ Zonxmax

(6.28)

y + Zymax j , j ' + M ⋅ (2 − Eg − Eg ' )

∀g, g', j, j'| ( j ∈ J g , j'∈ J g' ) ∈ Zonymax

(6.29)

y + Zymax j , j ' + M ⋅ (2 − Eg − Eg ' )

∀g, g', j, j'| ( j ∈ J g , j'∈ J g' ) ∈ Zonymax

(6.30)

d j + d j' 2 d j + d j' 2

Again, these constraints (6.23) to (6.30) are only active if both the equipment units are present (Eg=1 and Eg’=1). Also, these constraints may be defined over specific input and/or output points due to operability conditions such as expedition (see chapter 4).

Allocation Constraints: additional constraints must be written whenever the available area is limited to the dimension of a given industrial facility. Thus, on the one hand, lower bounds on the equipment coordinates are defined so as to avoid intersection of equipment elements with the axes:

xj ≥

yj ≥

lj 2

dj 2

∀j

(6.31)

∀j

(6.32)

Similarly, upper bound constraints are written to force the equipment allocation within a predefined rectangular area confined by (0, 0) and (xmax, ymax):

xj +

lj 2

≤ E g ⋅ x max

∀g , j | j ∈ J g


(6.33)

107

yj +

dj 2

≤ E g ⋅ y max

∀g , j | j ∈ J g

(6.34)

If the unit g to which the equipment element j belongs does not exist or has not been chosen with the design decisions, both coordinates xj and yj as well as the element dimensions take the value of zero. For large plant areas, the limits xmax and ymax can be replaced by Mx and My, respectively. Again, space restrictions that lead to a non-rectangular available area can be modelled by defining pseudo-equipment items with fixed sizes and locations that constrain the available area (see chapter 4).

In conclusion, the objective function (6.1) along with an appropriate selection of constraints (6.2) to (6.34) define the MILP model for the layout of industrial plants with variable equipments and connections. This model can easily be extended to include the presence of possible production/operational sections as modelled in chapter 4. Also, and considering a possible layout definition over a three dimensional (3D) space, as presented in chapter 5, the above simultaneous design and layout model still applies where only the equations modelling the 3D space need to be introduced.

Example

One of the examples proposed in chapter 4 is studied again now accounting for the variable existence of equipment units and connections, along with previous additional characteristics there presented. The Generic Algebraic Modelling System (GAMS, Brook et al. 1988) was used coupled with the CPLEX optimisation package (version 6.5). The problem was solved with a 0% margin of optimality on a Pentium II, 450 MHz considering as objective function the minimisation of the total connectivity cost along with capital cost of equipment units – equation (6.1). Table 6.1 shows the equipment unit characteristics while the involved connections and associated costs are presented in table 6.2. The plant flowsheet is depicted in figure 6.1 that must be within a pre-defined rectangular area of (25x25) m2.


108

With the purpose of illustrating the model applicability a set of hypothetical constraints concerning the equipment and connection design were added. In terms of equipment units it is assumed that only one (disjunctive constraints) of the following equipment pairs could exist: Unit V1 or Unit V2; Unit V5 or Unit V6; Unit V5a or Unit V6a. On the other hand, all other equipment items must obligatory be present. Additionally, and concerning the connections it is assumed that a connection must exist if the sink and source units exist (e.g.: connections c2 and c3 only exist if Unit V2 is present in the final solution), and do not exist otherwise.

Table 6.1 – Unit Characteristics Units Unit_V1

Cost (cu) 500

α 5.0

β 3.0

Input

∆xoi

∆yoi

Unit_V2

450

6.0

6.0

Unit_1a

800

6.0

6.0

OI1

-3

0

Unit_1b

900

5.0

5.0

500

6.0

6.0

Unit_R2

700

4.5

4.5

Unit_R4

700

5.0

5.0


600 300 450 250

5.0 6.0 2.0 3.0

3.0 6.0 1.0 2.0


0 -3 3 0 -2 0 -2.5 0 1 1.5

-2.5 -3 -3 -2 -2 -2 0 -3 0 0

Unit_2a

Output OI12 OI13 OI14 OI15 OI16 OI17 OI18 OI19 OI20 OI21 OI22

c8

V5

c9

V6

c10

V5a

c11

V6a

∆xoi 2.5 -2 2 3 3 0 2.5 -2.5 0 -2 2

∆yoi 0 3 3 1 -1 2.5 2.5 2.5 2 2 2

c4

V1

c1

1a

2a

c6

1b

c2

c5

V2

R4 c3

R2

c7



109

Table 6.2 – Connections and Associated Costs Connections C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

oi (output) OI12 OI13 OI14 OI16 OI15 OI17 OI20 OI18 OI19 OI21 OI22

oi’ (input) OI1 OI2 OI5 OI3 OI6 OI4 OI7 OI8 OI9 OI10 OI11

Cost (cu/m) 1 20 5 10 1 20 5 10 10 1 1

The optimal plant flowsheet is shown in figure 6.2 and the correspondent layout is depicted in figure 6.3. Vessels V2, V6 and V6a were chosen since they lead to a minimum plant cost – objective function defined.

c4

1a

2a

c9

V6

R4

c11

V6a

c6

1b

c2

c5

V2 R2

c3

c7

V6a

Figure 6.2 – Optimal Plant Flowsheet

R4

R2 V2 1b

2a V6

1a

Figure 6.3 – Optimal Plant Layout (25m x 25m)


110

The problem statistics and the optimal value of the objective function are shown in table 6.3.

Table 6.3 – Computational Statistics OF 4626.5

CPU's 92.88


NIV 178

NV 331

NC 429

In order to expose the trade-off between cost minimisation and available allocation area the same problem was solved but considering a different plant area - 20x12 m2 - against the previous 25x25 m2. The results lead to the optimal plant flowsheet drawn in figure 6.4. The associated optimal layout configuration is depicted in figure 6.5. The computational statistics along with the objective function are shown in table 6.4.

c4

V1

c1

1a

2a

c9

V6

R4

c11

V6a

c6

1b c5

R2

c7

Figure 6.4 – Optimal Plant Flowsheet

V1

1a

1b

R2

R4 2a

V6 V6a

Figure 6.5 – Optimal Plant Layout (20m x 12m)


111

Table 6.4 – Computational Statistics OF 4700.5

CPU's 496.91


NIV 178

NV 331

NC 429

Due to the reduced available allocation area, vessel V2 was replaced by vessel V1 and the value of the objective function increased by 74 c.u. (50 c.u. from vessel V1 shift and 24 c.u. from additional connection costs).

6.4 – Conclusions and Future Work A variable layout formulation is presented. This models a simultaneous approach to the solution of design and layout problems as a single global problem. This formulation has the particularity of been easily adapted to any kind of design problem model where the characteristics of a given plant are explored. The proposed model considers the layout problem over a 2D continuous area and addresses important topological aspects such as different equipment orientations, space availability, different equipment inputs and outputs, rectangular and irregular shapes as well as safety and operability restrictions translated into distance restrictions (minimum and maximum). The model results in a mixed integer formulation that provides the optimal plant layout based on a specified economic goal. This goal is defined as the minimisation of the connectivity structure cost and equipment unit capital cost. In conclusion, it can be stated that a very generic formulation is presented and the obtained layout results describe closely real life situations. As future work it will be exploited the application of the proposed model to the simultaneous design and layout of industrial multipurpose batch facilities.


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Chapter 7 SIMULTANEOUS LAYOUT AND DESIGN OF MULTIPURPOSE BATCH PROCESSING FACILITIES - AN APPLICATION

7.1 – Introduction The design of multipurpose batch processing facilities involves a large number of interacting decisions. An important part of the latter is layout considerations concerning the spatial allocation of equipment items and their interconnections. Traditionally, such layout issues have been considered a posteriori once the main plant design task has been completed. However, the strong interactions of layout with the rest of the design decisions suggest a simultaneous approach solution. This chapter presents a mathematical formulation for the simultaneous design and layout problem of batch processing facilities. Based on the design model proposed by Barbosa-Póvoa and Macchietto (1994), detailed layout aspects are studied using the generalised approach of the work proposed in chapter 6. The resulting model determines simultaneously the optimal plant topology (i.e. the choice of the plant equipment and the associated connections), the optimal layout (i.e. the arrangement of processing equipments and storage vessels and their interconnecting pipe-work over a 2D space) as well as the optimal plant operation (schedule and resources consumption and production). The complications arising from operational conditions, suitability and availability of equipment, the various layout constraints and the presence of equipment items of various sizes are taken into account. The problem is formulated as a Mixed Integer Linear Problem (MILP) where binary variables are introduced to characterise operational and topological choices. The applicability of the proposed model is illustrated via a set of representative examples.

Simultaneous Layout and Design of Multipurpose Batch Processing Facilities: An Application

113

7.2 - Problem Representation The maximal State-Task Network (mSTN) as defined by Barbosa-Póvoa (1994) is used to represent the batch design problem. The process recipes are defined through a State-Task Network (STN) graph representation (Kondili et al. 1988) and the plant is characterised through a normal flowsheet, that is an equipment network of vessels, processing units and possible connections. The STN represents the precedence structure of the product networks where materials are defined as State nodes and the operations transforming input material states into output material states are described through Task nodes. Additionally, material proportions, processing times and utility requirements for each task must be defined. Having the process recipes along with the plant description (equipment characteristics and plant structure) the maximal STN (mSTN) automatically combines the operations and equipment network by performing the mapping between both. This mapping is defined by: - The suitability of each unit in the equipment network to carry out processing tasks and to store material states; - The suitability of connections to transport material states; - The resources - equipments, utilities, operators, etc. - required by each task.

The resulting mSTN is characterised by various types of nodes and arcs (see figure 7.1): eTask i/g nodes representing the processing tasks i which can be performed in unit g; eState s/g nodes representing materials in state s that can be stored in unit g; iState/oState nodes representing the origin/location of the material entering/leaving the eTask defined as zero capacity states and; finally, if exists a direct link between the input/output points defined through these last states, a transfer task (tTask arc) is introduced to represent this link.

iState

oState

eTask

tTask

iState

eTask

Figure 7.1 - mSTN Representation: Nodes and Arcs The main advantage of this representation is that it unambiguously and explicitly represents the location of all material states within the plant as well as the allocation of processing, storage and transfer tasks that are potentially necessary and structurally feasible within the problem in study.


114

7.3 – Problem Statement and Characteristics The simultaneous design and layout of batch processing facilities can be stated as follows:

Given •

The STN description of product recipes with associated parameters and resource requirements - equipment, utilities, etc.;

•

A plant superstructure - network of possible units and connections - with associated capacities and suitability;

•

Production demands, a time horizon (H), operation mode and availability profiles of all resources;

•

Equipment item geometrical shapes and sizes over a 2D area;

•

Input and output point locations within each equipment space;

•


•


•

Capital costs for units and connections;

•

Optimal operations schedule - sizes, allocation and timing of all batches, storage

Determine

and transfers; •

Optimal plant configuration - equipment network and sizes;

•

Optimal plant equipment arrangement - coordinates and orientation;

•

Optimal associated connectivity structure – inputs and outputs.

So as to optimise a given economic objective function, namely the minimisation of the capital cost of the plant, while fulfilling all the constraints defined.

The developed model allows the description of general processing networks described by multiproduct and multipurpose plants. The mathematical formulation depends on a discretization of time such that the planning horizon H is divided into a number of elementary steps of fixed length δ. Thus, all the task processing times and the planning horizon are defined as integer multiples of this elementary step δ. All the process events are allowed to occur only at the interval boundaries (the vertical arrows in figure 7.2) and not between them. For instance, the amount of material transferred by transfer task π at the beginning of period t, represented by the continuous variable BTπ ,t , can


115

only occur in any time boundary between 0 and H. On the other hand, the variables Wi , g ,t and

Bi , g ,t , describing respectively the occurrence starting time and the batch of task i in unit g at time t, are only defined for any time boundary between 0 and H-pi (where pi is the duration of task i). In this way no tasks are allowed to finish after the allowed time horizon H. For simplicity in figure 7.2 a processing time of one (pi=1) was assumed thus variables Wi , g ,t and

Bi , g ,t are defined until H-1.

Production Time

δ

t Ini

0

1

2

...

...

H-1

H

Wi , g ,t ; Bi , g ,t S sini

S s ,t ; BTπ ,t ; Ds ,t ; Rs ,t Figure 7.2 - Process Events and Planning Horizon

Design and operation decisions are represented by continuous variables (batch sizes, equipment capacities, amounts of materials, etc.) and discrete choices by binary variables (equipment existence, task allocations to equipment and time, etc.). Equipment items (units, dedicated storage) and connections are selected optimally from the defined plant superstructure while operation is optimised so as to satisfy all constraints. Product requirements are defined for each product as fixed or variable within ranges. Demands (and supplies) may be associated to specific equipment. Equipments in discrete size ranges, mixed storage policies, shared intermediate material, material merging, splitting and recycling, in-phase and out-of-phase operation in any combination as well as instantaneous transfers are allowed Also, a single campaign structure with fixed product slate is assumed within a non-periodic operation. The plant is defined within a 2D continuous space. Equipment items to be allocated in the available space are described by rectangular or irregular shapes. Rectilinear distances are assumed providing a more realistic estimate of the piping costs as opposed to direct connections. Multiple equipment connectivity inputs and outputs are considered as well as space limitations.


116

Finally, the objective function is defined in terms of the capital cost of the plant. This accounts for the unit and connection costs – a function of the selected equipment capacity, material and suitability. For the connections, the final piping length – rectilinear distance within the plant – is also considered.

7.4 – Mathematical Formulation The model for the simultaneous design and layout of batch processing facilities includes the mathematical formulation presented in chapter 6 along with the mathematical formulation that will be defined in this chapter. The following formulation is adapted from the design with non-periodic operation defined in Barbosa-Póvoa (1994) where, based on the above model characteristics, the following indices, sets, parameters and associated variables are defined:

Global Indices: besides the indices g [defined as j in Barbosa-Póvoa (1994)], c and oi defined in the previous layout chapter, it must be introduced: t, t’ – time i – processing and storage task (eTask)

π – transfer task (tTask) s – mSTN state (including iStates ∪ oStates ∪ eStates) sSTN – STN state sSTNProd – product STN state Sets: besides the set OIc defined in the previous layout chapter:

PU = {g: set of processing equipment units} DSV = {g: set of dedicated storage vessels} I g = {i: set of eTasks i which can be performed in equipment unit g}

e state = {s: set of eStates s} i state = {s: set of iStates s} o state = {s: set of oStates s} s STN = {s: set of oStates s} S s STN = {s: set of mSTN eStates s generated from STN state sSTN}


117

S s STNProd = {s: set of mSTN eStates s generated from STN product state sSTNProd}

K i = {g: set of processing equipment units g suitable for eTask i} K s = {g: set of dedicated storage vessels g suitable for storing eState s} I c = {π: set of tTasks π which can be performed in connection c}

∏ sink = {π: set of tTasks π to which equipment unit g is a sink} g Tsin = {i: set of eTasks i receiving material from state s} Tsout = {i: set of eTasks i producing material to state s}

∏ ins = {π: set of tTasks π transferring material from state s} ∏ out s = {π: set of tTasks π transferring material to state s} Parameters:

CC g0 – fixed capital cost of equipment unit g CC 1g – size dependent capital cost of equipment unit g CC c0 – fixed capital cost of connection c CC c2 – cost per meter of the connection c

pilag , s – lag time of the input state s entering eTask i relative to the start of the eTask i piproc , s – procesing time for eTask i to produce the output state s relative to the start of the eTask i

{

pi – duration of eTask i: max piproc ,s

}

ρ is – proportion of material from input state s entering eTask i

ρ is – proportion of material to output state s leaving eTask I φ imax ,g - maximum utilisation factor of eTask i in equipment unit g (it may be considered as a conversion factor between the maximum batch size units and the capacity units of the processing vessel (e.g. kg/l) or just a maximum fraction of the capacity that can be used for eTask i)

φ imin ,g - minimum utilisation factor of eTask i in equipment unit g (minimum capacity fraction that can be used)

φ s, g - size factor of eState s, conversion factor between the maximum amount stored and the capacity units of the storage vessel g dedicated to eState s (i.e. kg/l or just fraction used)


118

φπ ,c - size factor of tTask π in connection c, conversion factor between the maximum amount transferred and the capacity units of the connection (i.e. kg/l or just fraction used)

V gmin - minimum capacity of equipment unit g V gmax - maximum capacity of equipment unit g BTcmin - minimum capacity of connection c BTcmax - maximum capacity of connection c S sini - the initial amount of material in eState s at the beginning of the production (before t = 0)

S siniSTN Pr od - the initial amount of material in product STN state sSTNProd at the beginning of the production (before t = 0) STNProd Qsmin at the end of STN Prod - minimum production requirement of product STN state s

the horizon STNProd Qsmax at the end of STN Prod - maximum production requirement of product STN state s

the horizon

Dsmin - minimum amount of intermediate material delivery to the outside from STN STN ,t state sSTN at the beginning of period t

Dsmax - maximum amount of intermediate material delivery to the outside from STN STN ,t state sSTN at the beginning of period t

Rsmin - minimum amount of intermediate material received from the outside into STN STN ,t state sSTN at the beginning of period t

Rsmax - maximum amount of intermediate material received from the outside into STN STN ,t state sSTN at the beginning of period t Variables: Continuous Variables - all defined as positive variables:

Bi , g ,t – the amount of material that starts undergoing eTask i in unit g at the beginning of period t (t=0..H-pi)

S s ,t – the amount of material in eState s at the beginning of period t (t=0..H) BTπ ,t – the amount of material transferred by transfer task π at the beginning of period t (t=0..H)


119

Ds ,t – the amount of material delivered from state s to the outside at the beginning of period t (t=0..H)

Rs ,t – the amount of material received from the outside into state s at the beginning of period t (t=0..H)

V g – capacity of equipment unit g

BTc – capacity of connection c Binary Variables: besides the binary variables Eg and Ec defined in the previous layout chapter, representing respectively the installation of unit g and connection c, there is:

Wi , g ,t – processing eTask i in unit g allocation at t: =1 if unit g starts processing eTask i at the beginning of period t; 0 otherwise (t=0..H-pi) Using the above definitions and variables the mathematical model will now be presented.

7.4.1 – Simultaneous 2D Layout and Design of Multipurpose Batch Processing Facilities

The following objective function and constraints characterise the generic model for the simultaneous 2D industrial layout problem and design of batch processing facilities:

Objective Function:

Several objective functions (OF) can be considered, as defined in Barbosa-Póvoa (1994). The one chosen in here is the minimisation of the capital cost, including fixed and variable costs for equipment units and connections.

Min

∑ ( c , oi , oi ')|( oi , oi ')∈OI c

( CC c0 ⋅ E c + CC c2 ⋅ D oi ,oi ' ) +

∑ (CC

0 g

⋅ E g + CC 1g ⋅ V g )

(7.1)

g

Other OFs could be easily applied, namely the maximisation of the plant profit, given products demands and prices, as defined in Barbosa-Póvoa (1994).


120

Constraints:

A number of additional types of constraints need to be introduced in order to model the simultaneous layout and design characteristics along with the ones defined in the previous chapter for the detailed variable layout. These include processing unit existence, capacity and batch size, storage, connectivity, mass balances and production requirement constraints.

Processing Unit Existence Constraints: the assignment of units to processing tasks is established by two basic rules: at any time, each equipment unit is either idle or processing a single task; tasks cannot be pre-empted once they have been started.

t ≤ H − pi

∑ ∑

Wi,g,t' ≤ E g

∀g,t | g ∈ PU , t = 0,..., H − 1

(7.2)

i∈I g t' = t − pi +1≥ 0

If a suitable task Wi,g,t is processed in unit g then equipment g is installed (Eg=1). Otherwise, if the equipment unit is not installed no tasks can be transformed in this unit. The sum over the period t’ guarantees the continuity of task i. The sum over all tasks i excludes the chance to occur another potential tasks processing in unit g along that time period.

Capacity and Batch Size Constraints: the capacity of each equipment unit is directly related to the amount of the material (batch size) a task i must process in that unit. If there is no transformation, the batch size is zero. Otherwise, it must be within the maximum min ( φ imax , g ⋅ V g ⋅ Wi , g ,t ) and minimum ( φ i , g ⋅ V g ⋅ Wi , g ,t ) available unit capacity. These bilinear terms

can be reformulated into a linear form according to Shah (1992), using the following three constraints:

max φ imin ⋅ (1 − Wi , g ,t ) ≤ Bi , g ,t , g ⋅ Vg − Vg

Bi , g ,t ≤ φ imax ,g ⋅Vg

∀i, g , t | g ∈ K i , t = 0,..., H − pi

∀i, g , t | g ∈ K i , t = 0,..., H − pi

(7.3)

(7.4)

These constraints ensure that the batch size Bi , g ,t is defined between the available capacity, if equipment unit g is chosen to transform task i at any time t. Furthermore, if a suitable task Wi,g,t


121

is processed in unit g at any time t then the following constraint forces the batch size Bi , g ,t to be zero:

Bi , g ,t ≤ V gmax ⋅ Wi , g ,t

∀i, g , t | g ∈ K i , t = 0,..., H − pi

(7.5)

The capacity V g is defined within the range of the available capacities for unit g, according to these constraints:

Vgmin ⋅ E g ≤ Vg

∀g

(7.6)

V g ≤ V gmax ⋅ E g

∀g

(7.7)

Storage Constraints: for dedicated storage vessels, an optimal storage policy can be determined using the following constraints:

S ini s ≤ Vg ⋅ φ s, g

∀s, g | g ∈ K s

(7.8)

S s ,t ≤ V g ⋅ φ s , g

∀s, g , t | g ∈ K s , t = 0,..., H

(7.9)

If the amount S s ,t of material in eState s at the beginning of period t is greater than zero, then the associated capacity V g takes the correspondent positive value weighted by the size conversion factor φ s, g . If unlimited capacity is assumed, usually for the feed and product states, these constraints are omitted and only applied for the intermediate states. The case where multipurpose vessels are used to store material is modeled as a processing unit.

In order to avoid the use of a storage vessel as a simple passing through point, without having any material in storage during all the planning horizon, and therefore, in that cases, force the inclusion of such a vessel in the plant structure, the following constraint must be present:

H

∑ ∑ BTπ

,t

− V gmax ⋅ E g ≤ 0

∀g | g ∈ DSV

(7.10)

t π ∈Π sink g


122

Although this constraint must be present in the design problem on its own, it can be dropped when considering the simultaneous layout and design problem whenever the problem being solved is a connectivity minimisation.

Connectivity Constraints: Each transfer task π between two connection points has only one state of material associated. However, each connection might be suitable to transfer different kinds of materials. Although the plant superstructure may include many kind of possible transfer tasks, minimisation of the plant cost will typically result in the installation of a limited set of connections. Each connection c has one existence variable (Ec). The number of binary variables is therefore reduced. The amount of material BTπ ,t transported by transfer task π at the beginning of period t is constrained by the available capacity BTc of its associated connection c weighted by the size conversion factor φ π ,c :

BTπ ,t ≤ BTc ⋅ φ π ,c

∀c, π , t | π ∈ I c , t = 0,..., H

(7.11)

On the other hand, the capacity BTc of each connection is defined between pre-defined capacity lower and upper bounds, respectively:

BTcmin ⋅ Ec ≤ BTc

∀c

(7.12)

BTc ≤ BTcmax ⋅ Ec

∀c

(7.13)

Mass Balance Constraints: mass balance relates to the steady equilibrium that must always exist between materials delivery and receiving. Thus, the amount of material in a state at any time period must equal the same material being produced, consumed and transferred by all incident tasks along with the amount of material of the state existing in the previous time period. Thus: For the eStates:

∑ BTπ

S s , 0 = S sini +

,0

π ∈Π out s

S s ,t = S s ,t −1 +

∑ BTπ π ∈Π out s

∑ BTπ

−

,0

− Ds , 0 + Rs , 0

∀s | s ∈ e state

(7.14)

− D s ,t + R s , t

∀s, t | s ∈ e state , t = 1,..., H

(7.15)

π ∈Π ins

,t

−

∑ BTπ

,t

π ∈Π ins


123

For the iStates:

0 = − ∑ ∑ ρ iin, s ⋅ Bi , g ,t − plag ≤ H − p + i,s

i∈Tsin g∈K i

i

∑ BTπ

,t

∀s, t | s ∈ i state , t = 0,..., H

(7.16)

∀s, t | s ∈ o state , t = 0,..., H

(7.17)

π ∈Π out s

For the oStates:

0=

∑ ∑ρ i∈Tsout

out i,s

g∈K i

⋅ Bi , g ,t − p proc ≤ H − p − i ,s

i

∑ BTπ

,t

π ∈Π in s

Production Requirement Constraints: production requirements relate to the amount each product must be produced. For each product, the sum of all materials generated from the same STN product state ( s STN Prod ) stored or delivered in any unit in the plant at the end of the production horizon (H) can float within pre-defined minimum ( Qsmin STN Prod ) and maximum ( Qsmax STN Prod ) production requirements, respectively:

Qsmin STN Prod ≤

∑ (S

s,H

s∈s STN Prod

∑ (S

− S siniSTN Prod + Ds , H )

∀s STN Prod | Qsmax STN Prod > 0

(7.18)

− S siniSTN Prod + Ds , H ) ≤ Qsmax STN Prod

∀s STN Prod | Qsmax STN Prod > 0

(7.19)

s∈s STN Prod

s,H

For each intermediate delivery ( Ds ,t ), the same idea is translated, applied to all different time intervals t present in the planning horizon, resulting in the following lower and upper bound constraints, respectively:

Dsmin ≤ STN ,t

∑D s∈S

∑D s∈S

s ,t

s ,t

∀s STN , t | Dsmax > 0, t = 0,..., H STN ,t

(7.20)

∀s STN , t | t = 0,..., H

(7.21)

s STN

≤ Dsmax STN ,t

s STN


124

Finally, for each receipt of material ( Rs ,t ) at different time periods t again lower and upper bounds can be defined:

∑R

Rsmin ≤ STN ,t

s∈S

∑R s∈S

s ,t

s ,t

∀s STN , t | Rsmax > 0, t = 0,..., H STN ,t

(7.22)

∀s STN , t | t = 0,..., H

(7.23)

s STN

≤ Rsmax STN ,t

s STN

The combination of the presented production requirements constraints allows the definition of arbitrary time profiles of product deliveries and raw materials to/from individual locations as well as global production amounts.

In conclusion, the objective function (7.1) along with an appropriate selection of constraints (6.2) to (6.34) and constraints (7.2) to (7.32) applied to the design of batch processing facilities, define the MILP model for the simultaneous layout and design of multipurpose batch processing plants.

Examples The Generic Algebraic Modelling System (GAMS, Brook et al. 1988) was used coupled with the CPLEX optimisation package (version 6.5). All the problems were solved on a Pentium II, 450 MHz.

Example 1

A case study based on one of the examples proposed by Barbosa-Póvoa and Macchietto (1994) and also presented in Barbosa-Póvoa et al. (1999) is presented to illustrate the model applicability. A plant must be designed to produce two different products (P1 and P2) through the following process recipe (see figure 7.3): - Task T1: heat raw material S1 for 2 hours to produce the unstable intermediate S3; - Task T2: process raw material S2 for 2 hours to form the intermediate S4;


125

- Task T3: mix intermediate material S3 with material S4 in the ratio of 60:40 and let them react for 4 hours to form product P1; - Task T4: mix S3 with P1 in the ratio of 60:40 and let them react for 2 hours to form product P2.

S1

T1

2 hr. 60%

S3 60%

Heat 2 hr.

S2

T2 Process

4 hr.

S4

40%

T3

T4 40%

2 hr.

P2

≥ 80 ton.

Reaction

P1

≥ 80 ton.

Reaction

Figure 7.3 – Process Recipe

The production requirements of this plant are 80 tonne for each material P1 and P2 over a time horizon H of 8 hr. The example is solved considering two different cases: Case 1 – The stand-alone design problem, as proposed by Barbosa-Póvoa and Macchietto (1994), without layout considerations. The solution provides the final plant topology and associated schedule for an objective function defined as the sum of all units installed cost –considering only parameter CC g0 in equation (7.1), all other taken as null. Case 2 – The design and layout problems are solved simultaneously using the model proposed in this chapter. The solution provides the optimal plant topology, layout and schedule for an objective function defined as the total cost of units and connectivity –considering only parameters CC g0 and CC c2 in objective function (7.1), all other taken as null. A limited availability area of (21x6) m2 is considered.

The plant superstructure used is shown in figure 7.4 and the equipment characteristics are given in table 7.1 where units 1a, 1b, 1c, 2a and V4 have fixed capacity. Vessels V1 and V2 have initially 200 tonne and 100 tonne each of S1 and S2, respectively. Connections, assuming unlimited capacity and associated suitability are exhibited in table 7.2. All connection costs value 500 c.u./m.


126

Figure 7.4 – Plant Superstructure Topology Table 7.1 – Equipment Characteristics Unit

Suitab.

Capac. [tonne]

Costs [103 c.u.]

Dimens. α/β

1a

T1/T2

70

14

8/8

1b

T1/T2

70

15

6/4

1c

T1

70

18

6/2

2a

T3/T4

120

40

5/5

V1

Store S1

Unlim.

1

6/3

V2

Store S2

Unlim.

1

6/3

V4 V5 V6

Store S4 Store P1 Store P2

50 Unlim. Unlim.

3 1 1

5/1 4/3 4/3

In

∆xoi/∆ ∆yoi

Out

∆xoi/∆ ∆yoi


-3/-0.5 -3/-1.5 -3/1.5 -3/0 -3/0 -2.5/-1 0/-2.5 -2.5/2 2.5/0 -2.5/1.5

OI14 OI15 OI16 OI17 OI18

3/-0.5 3/-1.5 3/-1.5 3/0.5 3/0

OI19

2.5/1

OI20

2.5/-2

OI21 OI22 OI23 OI24 OI25 OI26 OI27

3/1 3/-1 3/0.5 3/1 3/0.5 0/0.5 -2/-1

OI11 OI12 OI13

-2.5/0 -2/0 -2/0

Table 7.2 – Connections and associated Suitability Connections C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14

Suitability S1 S2 S1 S2 S4 S3/S4 S4 S3/S4 S5 S6 S5 S4 S1 S3

Output OI21 OI24 OI22 OI25 OI15 OI14 OI16 OI17 OI19 OI20 OI27 OI26 OI23 OI18

Input OI1 OI2 OI3 OI4 OI11 OI8 OI11 OI6 OI12 OI13 OI9 OI7 OI5 OI10


127

Figure 7.5 shows the mSTN representation and depicts the relevant information concerning eStates, iStates, oStates, eTasks, tTasks (π) and associated connections c, as well as sets Tsin ,

Tsout , ∏ ins and ∏ out s . T1/1c

c14/π20

T1/1a

c6/π7

c14/π19

c13/π18

S1/V1

c1/π1

c6/π6

c3/π3 T1/1b

c8/π10 T4/2a

c10/π14

S6/V6

π17 c8/π11

c11/π15

T3/2a S2/V2

c2/π2

T2/1a

c9/π13

S5/V5

c6/π8 c5/π5 S4/V4

c4/π4

c12/π16

c7/π9 T2/1b

c8/π12

Figure 7.5 – mSTN Representation

max min Production requirements bounds Qsmin , Dsmax , R smin and R smax are all STN Prod , Q STN Prod , D STN STN STN STN s s ,t ,t ,t ,t

assumed to be null, with the exception of product STN states S5 (product P1) and S6 (product P2), as stated before, which have the following minimum and maximum production min max max max requirements: QSmin 5 = Q S 6 = Q S 5 = Q S 6 =80. All utilisation factors φ i ,g are equal to the unity

and φ imin ,g to zero. All size factors φ s, g equal the unity as well as φ π , c .

The results obtained showed that, for case 1, the final plant is characterised by the choice of processing units 1a, 1b and 2a along with vessels V1, V2, V5 and V6 (see figure 7.6). This corresponds to a capital cost of 73000 c.u.. These units and related capacities (Vg) along with the associated connections and respective capacities (BTc) are depicted in figure 7.6. The optimal schedule is similar to the one obtained in case 2 (see figure 7.8) only differing in the installation of unit 1a instead of unit 1c. In operational terms unit 1a is performing task T1, unit 1b task T2 and unit 2a is a multi-task unit performing tasks T3 and T4. No intermediate storage was chosen.


128

V1

1a

c1 (67.2)

V5

c6 (67.2) c9 (80)

2a c10 (80)

V2

c4 (44.8)

1b

V6

c8 (44.8)

Figure 7.6 – Optimal Plant Topology for Case 1

For case 2, where aspects of operation, design and layout are considered simultaneously, the results obtained are different. In this case the processing units chosen are respectively 1b, 1c and 2a apart from tanks V1, V2, V5 and V6 (see figure 7.7). The optimal OF corresponds to a capital cost of 77000 c.u.. The operational schedule and layout are respectively illustrated in figures 7.8 and 7.9.

V1

1c

c13 (67.2)

V5

c14 (67.2) c9 (80)

2a c10 (80)

V2

1b

c4 (44.8)

V6

c8 (44.8)

Figure 7.7 – Optimal Plant Topology for Case 2

1c

T1 (67.2)

1b

T2 (44.8)

T1 (48)

T3 (112)

2a V1

T4 (80) 84.8

132.8

V2

55.2

V5

80 80

V6 0

1

2

3

4

5

6

7

8

Figure 7.8 – Optimal Plant Scheduling for Case 2


129

Figure 7.9 – Optimal Plant Layout for Case 2

The different optimal plant topology (figures 7.6 and 7.7) is explained by the space availability restrictions. In this context unit 1a, which needs higher space requirements (8/8) when compared with unit 1c (6/2), can not be installed in the final plant due to the restricted total space availability (21/6). Indeed, when trying to optimise the layout of the plant topology obtained from case 1, with the space restrictions defined in case 2, an infeasible problem is obtained. This shows how it may be important to consider design and layout aspects simultaneously when designing the plant.

In terms of computer statistics the final results are shown in table 7.3 (both cases with 1% margin of optimality).

Table 7.3 – Problem Statistics Case 1 2

OF 73 000 77 000

CPU's 0.17 2.48

Nodes 6 326

Iterations 28 3 854

NIV 79 196

NV 481 758

NC 756 1 105

Both cases are solved quite efficiently, although it is worth noting that case 1 does not cover layout.

Example 2

Based on one of the examples proposed by Barbosa-Póvoa (1994), a new example is presented. A plant must be designed to produce two different products (P1 and P2) through the following process recipe (see figure 7.10): - Task TT1: heat feed F1 for 1 hour to produce intermediate I1; - Task TT2: mix feed F2 and feed F3 in the ratio of 50:50 and let them react for 2 hours to form 40% of intermediate I2;


130

- Task TT3: mix I1 (hot F1) and intermediate I2 in the ratio of 40:60 and let them react for 2 hours to form I3 and product 1 (state P1) in the ratio of 60:40; - Task TT4: mix feed F3 and intermediate I3 in the ratio of 20:80 and let them react for 1 hour to form the impure material represented by state I4; - Task TT5: distil impure material I4 to separate product 2 (state P2), after 1 hour, and recover pure intermediate I3, after 2 hours, which is recycled in the ratio of 90:10.

40%

TT1

I1

60%

Heat 2 hr.

F2

50%

TT2

TT3

40%

Mix

I2

60%

1 hr.

TT4

≥ 20 ton.

2 hr.

I4

Mix

TT5

90% 1 hr.

Distil

20%

Mix 50%

P2

I3 10%

F1

≥ 30 ton.

2 hr. 40%

80%

1 hr.

P1

F3

Figure 7.10 – Process Recipe

The design and layout problems are solved simultaneously. The solution provides the final plant topology, layout and schedule for an objective function defined as the sum of the cost of units (fixed and capacity dependent) and the cost of connections (fixed and distance dependent) objective function (7.1). The minimum production requirements of this plant are 20 and 10 tonne for materials P1 and P2, respectively, over a time horizon H of 12 hours. A limited availability area of (20x20) m2 is considered. The equipment characteristics are given in table 7.4 where capacities are given in tonne and costs in 103 currency units (c.u.). Connections, assuming unlimited capacity, associated suitability and costs are exhibited in table 7.5. Vessels VV1, VV2 and VV3 have initially 50 tonne of F1, 100 tonne of F2 and 100 tonne of F3, respectively.


131

Table 7.4 – Equipment Characteristics Capac. Min/Max

Costs Fix./Var.

Dim. α/β

In

∆xoi/∆ ∆yoi

Out

∆xoi/∆ ∆yoi

20 / 50

10 / 0.02

5/3

OI1

-2.5 / 0

OI11

2.5 / 0

50 / 70

15 / 0.05

7/4

OI2

-3.5 / -2

OI12

3.5 / 2

50 / 70

12 / 0

8/3

OI3

-4 / 1.5

OI13

4 / 1.5

50 / 80

15 / 0.03

6/4

OI4

0 / -2

OI14

-3 / 2

Store F1

Unlimit.

0

4/2

OI15

1/0

VV2

Store F2

Unlimit.

0

7/3

OI16

3.5 / 0

VV3

Store F3

Unlimit.

0

5/3

OI17

2.5 / 1.5

VV4

Store I1

10 / 30

3 / 0.01

4/2

OI5

-2 / -1

OI18

2/1

VV5

Store I3

10 / 70

1 / 0.01

4/2

OI6

-2 / 0

OI19

2/0

VV6

Store I2

10 / 60

1.5 / 0.01

6/3

OI7

-3 / 1.5

OI20

3 / 1.5

VV7

Store I4

50 / 100

2 / 0.02

5/4

OI8

-2.5 / 2

OI21

0 / -2

V8

Store P1

Unlimit.

0

8/4

OI9

4/1

V9

Store P2

Unlimit.

0

7/2

OI10

3.5 /0

Unit

Suitab.

H

St

TT1 TT2, TT3, TT4 TT2, TT3, TT4 TT5

VV1

R1 R2

Table 7.5 – Connections, Suitability and associated Costs (c.u./m) Connections

Cost

Suitability

Out

In

1 2 3 4 7 9 10 11 12 13 14 15 16 24 25 26 27 28 31 32 33 34 35 36 37 38 39 40 41 42 43

5 / 0.02 7 / 0.03 7 / 0.03 7 / 0.03 3 / 0.015 5 / 0.02 5 / 0.02 5 / 0.02 7 / 0.03 7 / 0.02 5 / 0.02 5 / 0.02 5 / 0.02 2 / 0.01 2 / 0.01 5 / 0.02 5 / 0.02 7 / 0.03 0 0 0 0 0 0 0 0 0 0 0 0 0

I1 I1 I1 I2, I3 I4 I3 I2 I4 I4 I2, I3 I3 I2 I4 I3 I3 I4 I3 I3 F1 F2 F2 F3 F3 I3 P2 P1 P1 I1 I1 I2 I2

OI11 OI11 OI11 OI12 OI12 OI12 OI12 OI12 OI13 OI13 OI13 OI13 OI13 OI19 OI19 OI21 OI14 OI14 OI15 OI16 OI16 OI17 OI17 OI14 OI14 OI12 OI13 OI18 OI18 OI20 OI20

OI5 OI2 OI3 OI3 OI4 OI6 OI7 OI8 OI4 OI2 OI6 OI7 OI8 OI2 OI3 OI4 OI6 OI3 OI1 OI2 OI3 OI2 OI3 OI2 OI10 OI9 OI9 OI2 OI3 OI2 OI3


132

Figure 7.11 shows the mSTN representation and depicts the relevant information concerning eStates, iStates, oStates, eTasks, tTasks (π) and associated connections c, as well as sets Tsin ,

Tsout , ∏ ins and ∏ out s . π8/c2

π25/c38 P1/V8 π29/c39 π35/c40

π1/c31

F1/VV1

TT1/H

π9/c1

TT3/R1

I1/VV4

π28/c9

π10/c3

I3/VV5

π36/c41 π30/c14

TT3/R2

π11 π12/c4

T2/R1

π2/c32

π37/c42 π13/c10

F2/VV2

I2/VV6 π14/c15

π32/c13

π38/c43

π3/c33 π22/c27 π4/c34

TT2/R2

π15/c13 π16 π27

π5/c35 π34/c25 π33/c24 π21/c36

F3/VV3 π6/c34

TT4/R1

π17/c7 π18/c11

π7/c35

TT5/St

π39/c26

I4/VV7 π19/c16

TT4/R2

π20/c12

π24/c37

P2/V9

π23/c28 π31 π26/c4

Figure 7.11 – mSTN Representation

max min Production requirements bounds Qsmin , Dsmax , R smin and R smax are all STN Prod , Q STN Prod , D STN STN STN STN s s ,t ,t ,t ,t

assumed to be null, with the exception of product STN states P1 and P2, which have the max min following minimum and maximum production requirements: QPmin 1 =20; Q P1 =300; Q P 2 =10; max min QPmax to 0. All size factors 2 =200. All utilisation factors φ i ,g are equal to the unity and φ i ,g

φ s, g equal 1 as well as φπ ,c . The optimal plant is characterised by the choice of processing units H, R2 and St along with vessels VV1, VV2, VV3, VV5, V8 and V9 (see figure 7.12). These units and related capacities (Vg) along with the connections installed and respective capacities (BTc) are depicted in figure 7.12. This corresponds to a capital cost of 40 956.395 c.u..


133

VV5

VV2

(58.5) c25 (31.5)

c27 (58.5)

c33 (35)

VV1

H

c31 (18.667)

R2

c3 (18.667)

(20)

c35 (35)

St

c12 (65)

(70)

(65)

c39 (18.667)

c37 (6.5)

V8

V9

VV3

Figure 7.12 – Optimal Plant Topology

The operational schedule is illustrated in figure 7.13. In operational terms unit H performs task TT1, unit R2 is a multitask unit performing TT2, TT3 and TT4, and finally, unit St performs task TT5. In terms of storage, the choice of VV5 indicates the need of storing intermediate material I3 while the remaining storage vessels are allocated to raw materials (VV1, VV2, VV3) and final products (V8 and V9). The optimal plant produces 32.3 tonne of P1, stored in vessel V8, and 10 tonne of P2, stored in vessel V9, meeting the required production requirements.

H R2

TT2

TT1

TT1

(18.667)

(13.667)

TT3

(70)

TT4

(46.667)

TT2

(35)

St

TT5 50

VV1

TT3

(51.25)

TT4

(34.167)

(65)

TT5

(35)

31.333

(65)

17.667

65

VV2

39.375

65

VV3

58 32.375

19.375 58.5

VV5

31.5

V8

32.333

18.667

V9

10

3.5

0

1

2

3

4

5

6

7

8

9

10

11

12

Figure 7.13 – Optimal Plant Scheduling


134

The optimal layout obtained is depicted in figure 7.14:

VV1

H R2 VV3 VV5

St VV2 V8

V9


In terms of computer statistics the final results are shown in table 7.6 (with 0.001% margin of optimality). The problem is solved quite efficiently.

Table 7.6 – Problem statistics OF 57400.59

CPU's 47.51

Nodes 2 462

Iterations 39 871

NIV 356

NV 1 599

NC 2 315

7.5 – Conclusions and Future Work The simultaneous approach to the solution of the design and layout problems was applied to real case problems within multipurpose batch processing facilities. This was based on the detailed design model developed by Barbosa-Póvoa and Macchietto (1994). The simultaneous layout and design model considered the layout problem over a 2D continuous area, as presented in chapter 6, along with design constraints such as processing unit existence, capacity and batch size, storage, connectivity, mass balances and production requirements. Again, the problem results in a MILP that provides the optimal plant layout, design and schedule based on a specified economic goal: minimisation of costs. Several examples were solved and good results obtained. In conclusion, it can be stated that the interaction of the variable layout formulation with a specific design case -multipurpose batch facilities- was achieved with success.


135

As future work it can be easily exploited the application of the proposed model considering other layout characteristics, as the ones defined in chapters 4 and 5, along with the more generic multipurpose batch processing characteristics as presented in Barbosa-Póvoa (1994).


136

Chapter 8 CONCLUSIONS AND RECOMMENDATIONS

The optimisation of industrial layout problems has been addressed by several works in the past decades, namely the Facility Layout Problem (FLP) as traditionally defined in the literature, in an attempt to take account for the importance and complexity inherent in such problems. This thesis has presented a generic mathematical model for the FLP, based on the optimisation of an economic goal considering different real problem characteristics. Besides different equipment orientations, distance restrictions, non-overlapping constraints and space availability, the model allows the definition of production sections along with safety and operability constraints. In particular, the previous state-of-the-art has been expanded to include: - The existence of different input/output connectivity points inside each equipment unit; - Irregular shape equipment unit representations; - 3D continuous space representation along with multifloor constraints. These features represent an important step towards a higher level of realism in FLPs. Furthermore, in order to give flexibility enough to the proposed models so as to incorporate the defined characteristics in concurrence with any generic design models, a variable layout model was proposed. In this way, the simultaneous layout and design of FLP can be, for the first time, addressed. In particular, the simultaneous layout, design and scheduling of multipurpose batch plants was developed as a representative application. The extension of the concurrent approach to consider also 3D and production sections constraints is postponed but identified as easily accomplishable. In all cases, the optimisation problems have been formulated as MILPs, which were solved through the use of a standard branch and bound (B&B) optimisation package: CPLEX. The model applicability is explored through the presentation of various case studies. In general, good solution performances were presented. Nevertheless, further improvements can be made towards the reduction of computational time, in particular, exploration of model efficiency and algorithm solutions. In the first case, new mathematical entities could be obtained in order to reduce the number of binary variables or constrain the space of feasible solutions. Some efforts have already been

Conclusions and Recommendations

137

made to achieve this goal, ones adopted others discarded. On the other hand, the intrinsic presence of degenerate solutions due to the presence of a relative distance objective function could be further studied in order to be reduced. Relevant possibilities are the introduction of the area cost minimisation, which however will lead to non-linear models but whose application would result in less degenerate solutions. Also, the presence of auxiliary constraints that would force the layout to be adjacent to the axes is an option. Although, this is an open field of study where significant improvements can be brought. In the second case, algorithm solutions can be explored via parameter optimisation in the B&B solver, for instance, adopting different horizontal vs. vertical direction branching strategies. On the other side, it can be explored other optimisation methods, such as outer approximation methods and benders decomposition – being presently in study (Gomes-Salema 2000) - and/or study problem intrinsic characteristics to overcome the computational complexity. The development of hybrid models (heuristics and exact algorithms) can be put into perspective to further explore the MILP model presented. A prototype application taking as basis the characteristics presented in this thesis and the simulation annealing approach presented by Chwif et al. (1998) was implemented and good results obtained (Mateus et al. 1999). Furthermore, the model developed can easily be migrated from manufacturing contexts to services organisations. The main change coming from a distance based objective function to an adjacency based one. Finally, the implementation of a Java application to fulfil all the model potentialities in an efficient and systematic way is already in development and should be pursued. Also, and in order to explore the applicability of the model, the study of real industrial case studies should be undertaken. In this context, a safeguard idea must be outlined that is: although the proposed models theoretically guarantee the optimum this is achieved based on a set of assumptions that must be taken into account. The model results can, in any case, be used as a good starting point to the reality and a prompt way to build if-then scenarios or as a good alternative generator in order to determine the real optimum or the solution to be implemented.

138

REFERENCES

Abdinnour-Helm, S.; and Hadley, S. W.; “An iterative layout heuristic for multi-floor facilities”, Technical Report (Dept. of Business Administration), Univ. of Wisconsin-River Falls, 1995 Al-Hakim, L. A.; “Two graph-theoretic procedures for an improved solution to the facilities layout problem”, International Journal of Production Research, 29(8), 1701-1718, 1991 Amorese, L.; Gena, V.; and Mustacchi, C.; “A heuristic for the compact location of process components”, Chemical Eng. Science, 32, 119-124, 1991 Aneke, N. A. G.; and Carrie, A. S.; “A design technique for the layout of multi-product flowlines”, International Journal of Production Research, 30(10), 2467-2476, 1986 Armour, G. C.; and Buffa, E. S.; “A heuristic algorithm and simulation approach to relative allocation of facilities”, Management Science, 9, 294-309, 1963 Barbosa-Póvoa, A. P. F. D.; “Detailed design and retrofit of multipurpose batch plants”, PhD thesis, Imperial College of Science, Technology and Medicine, University of London, 1994 Barbosa-Póvoa, A. P. F. D.; and Macchietto, S.; “Detailed design of multipurpose batch plants”, Computers Chemical Engineering, 18(11/12), 1013-1042, 1994 Barbosa-Póvoa, A. P. F. D.; Mateus, R.; and Novais, A. Q.; “Simultaneous design and layout of batch processing facilities”, in Proceedings of the Foundations of Computer-Aided Process Design, Colorado, U. S. A., 1999 Barbosa-Póvoa, A. P. F. D.; Mateus, R.; and Novais, A. Q.; “Optimal facility layout – A generic approach to process plants”, in Proceedings of the Manufacturing and Production & Control and Logistics, Grenoble, France, 2000 Bawa, A.; “Optimum routing of pipelines in predefined corridor network”, in Intergraph USGUG, USA, 1995

References

139

Bland, J. A.; and Dawson, G. P.; “Large-scale layout facilities using a heuristic hybrid approach algorithm”, Applied Mathematical Modelling, 18(9), 500-503, 1994 Block, T. E.; “A note on a ‘comparison of computers, algorithms and visual based methods for plant layout’ by Scriabin, M. and Vergin, R. C.”, Management Science, 24(2), 1977 Boswell, S. G.; “TESSA – A new greedy algorithm for facilities layout planning”, International Journal of Production Research, 30(8), 1957-1968, 1992 Bozer, Y. A.; Meller, R. D.; and Erlebacher, S. J.; “An improvement-type layout algorithm for single and multiple floor facilities”, Management Science, 40(7), 918-932, 1994 Bozer, Y. A.; and Meller, R. D.; “A reexamination of the distance-based facility layout problem”, IIE Transactions, 29, 549-560, 1997 Brook, A.; Kendrick, D.; and Meeraus, A.; “GAMS: A user’s guide”, San Francisco: The Scientific Press, 1988 Buffa, E. S.; “On a paper by Scribian and Vergin”, Management Science, 23(1), 1976 Canen, A. G.; and Williamson, G. H.; “Facility layout overview: towards competitive advantage”, Facilities, 16(7/8), 198-203, 1998 Castell, C. M. L.; Lakshmanan, R.; Skilling, J. M.; and Bañares-Alcántara, R.; “Optimisation of process plant layout using genetic algorithms”, Computers Chemical Engineering, 22S, S993S996, 1998 Chiang, W. C.; and Kouvelis, P.; “An improved tabu search heuristic for solving facility layout design problems”, International Journal of Production Research, 34(9), 2565-2586, 1996 Chwif, L.; Barretto, M. R. P.; and Moscato, L. A.; “A solution to the facility layout problem using simulated annealing”, Computers In Industry, 36, 125-132, 1998 Coleman, D. R.; “Plant layout: computer versus humans”, Management Science, 24(1), 1977 Conway, D. G.; and Venkataramanan, M. A.; “Genetic search and the dynamic facility layout problem”, Computers & Operations Research, 21(8), 955-960, 1994 Foulds, L.R.; and Robinson, D. F.; “Graph theoretic heuristics for the plant layout problem”, International Journal of Production Research, 16(1), 27-37, 1978

References

140

Foulds, L. R.; Hamacher, H. W.; and Wilson, J. M.; “Integer programming approaches to facilities layout models with forbidden areas”, Annals of Operations Research, 81, 405-417, 1998 Foulds, L. R.; and Partotiv, F. Y.; “Integrating the analytical hierarchy process and graph theory to model facilities layout”, Annals of Operations Research, 82, 435-451, 1998 Francis, R. L.; and White, J. A.; “Facility layout and location: An analytical approach”, Englewood Cliffs, NJ: Prentice-Hall, 1974 Gaither, N.; “Production and operations management”, 5th Edition, The Dryden Press International Edition, 1992 Garey, M. R.; and Johnson, D. S.; “Computers and intractability: A guide to the theory of NPcompleteness”, New York: W. H. Freeman and Co., 1979 Georgiadis, M. C.; Rotstein, G. E.; and Macchietto, S.; “Optimal layout design in multipurpose batch plants”, Ind. Eng. Chem. Res., 36(11), 4852-4863, 1997 Georgiadis, M. C.; Schilling, G.; and Rotstein, G. E.; “A general mathematical programming approach for process plant layout”, Computers Chemical Engineering, 23(7), 823-840, 1999 Gilmore, P. C.; “Optimal and suboptimal algorithms for the quadratic assignment problem”, Journal of the Society for Industrial and Applied Mathematics, 10, 305-313, 1962 Gomes-Salema, I.; “Aplicação de métodos de decomposição e heurísticos a layout industrial”, Tese de Mestrado, Instituto Superior Técnico, University of Lisbon, Portugal, 2000 Gunn, D. J.; and Al-Alsadi, A. D.; “Computer aided layout of chemical plants: A computational study method and case study”, Computer Aided Design, 19,131-140, 1987 Harmonosky, C. M.; and Tothero, G. K.; “A multi-factor plant layout methodology”, International Journal of Production Research, 30(8), 1773-1789, 1992 Hassan, M. M. D.; Hogg, G. L.; and Smith, D. R.; “SHAPE: A construction algorithm for area placement evaluation”, International Journal of Production Research, 24, 1283-1295, 1986 Heragu, S. S.; and Kusiak, A.; “Efficient models for the facility layout problem”, European Journal of Operational Research, 53, 1-13, 1991

References

141

Heragu, S. S.; and Alfa, A. S.; “Experimental analysis of simulated annealing based algorithms for the layout problem”, European Journal of Operational Research, 57(2), 1992 Houshyar, A.; and White, B.; “Exact optimal solution for facility layout: deciding which pairs of locations should be adjacent”, Computers and Industrial Engineering, 24(2), 177-187, 1993 Jayakumar, S.; and Reklaitis, G. V.; “Chemical plant layout via graph partitioning - I. Single level”, Computers Chemical Engineering, 14, 441-458, 1994 Jayakumar, S.; and Reklaitis, G. V.; “Chemical plant layout via graph partitioning – II. Multiple levels”, Computers Chemical Engineering, 20, 563-578, 1996 Kaku, B. K.; Thompson, G. L.; and Morton, T. E.; “A hybrid heuristic for the facilities layout problem”, Computers & OR, 18, 241-253, 1991 Kaku, K.; and Rachamadugu, R.; “Layout design for flexible manufacturing systems”, European Journal of Operational Research, 57, 224-230, 1992 Kettani, O.; and Oral, M.; “Reformulating quadratic assignment problems for efficient optimization”, IIE Transactions, 25, 97-107, 1993 Kim, J-Y; and Kim, Y-D; “Graph theoretic heuristics for unequal-sized facility layout problems”, Omega, 23(4), 391-401, 1995 Kim, J-G; and Kim, Y-D; “A branch and bound algorithm for locating input and output points of departments on the block layout”, Journal of the Operational Research Society, 50, 517-525, 1999 Kondili, E.; Pantelides, C. C.; and Sargent, R. W. H.; “A general algorithm for scheduling batch operations”, in International Symp. On Process Systems Engineering, 62-75, Sydney, Australia, 1988 Koopmans, T. C.; and Beckmann, M.; “Assignment problems and the location of economic activities”, Econometrica, 25(1), 53-76, 1957 Kusiak, A.; and Heragu, S. S.; “The facility layout problem”, European Journal of Operational Research, 29, 229-251, 1987 Lacksonen, T. A.; “Static and dynamic facility layout problems with varying areas”, Journal of Operational Research Society, 45(1), 59-69, 1994

References

142

Lacksonen, T. A.; “Pre-processing for static and dynamic facility layout problems”, International Journal of Production Research, 35(4), 1095-1106, 1997 Lawler, E. L.; “The quadratic assignment problem”, Management Science, 9, 586-599, 1963 Leung, J.; “A new graph-theoretic heuristic for facility layout”, Management Science, 38(4), 594-605, 1992 Liao, T. W.; “Design of line type cellular manufacturing systems for minimum operating and total material handling costs”, International Journal of Production Research, 32(2), 387-397, 1993 Lin, L. C.; and Sharp, G. P.; “Quantitative and qualitative indices for the plant layout evaluation problem”, European Journal of Operational Research, 116, 100-117, 1999 Mak, K. L.; Wong, Y. S.; and Chan, F. T. S.; “A genetic algorithm for facility layout problems”, Computer Integrated Manufacturing, 11(1-2), 113-127, 1998 Mateus, R.; Pinto-Varela, T.; and Matos, P.; “Aplicação de meta-heurísticas ao problema de layout de equipamentos”, doc. n.º 021/2000, DMS/INETI, Lisbon, Portugal, 2000 Mecklenburgh, J. C.; “Process plant layout”, UK: Institution of Chemical Engineers, 1985 Meller, R. D.; and Bozer, Y. A.; “A new simulated annealing algorithm for the facility layout problem”, International Journal of Production Research, 34(6), 1675-1692, 1996 Meller, R. D.; and Gau, K-Y; “Facility layout objective functions and robust layouts”, International Journal of Production Research, 34(10), 2727-2742, 1996 Meller, R. D.; Narayanan, V.; and Vance, P. H.; “Optimal facility layout design”, Operations Research Letters, 23(3), 117-127, 1999 Montreuil, B.; “A modeling framework for integrating layout design and flow network design”, in Proceedings from the Material Handling Research Colloquium, 43-58, Hebron, Kentucky, 1990 Muther, R.; “Practical plant layout”, McGraw-Hill, New York, NY, 1955 Muther, R.; “Systematic layout planning”, Cahners Books, Boston, MA, 1973 Papageorgiou, L. G.; and Rotstein, G. E.; “Continuous domain mathematical models for optimal process plant layout”, Industrial Eng. Chemical Res., 37, 3631-3639, 1998

References

143

Penteado, F. D.; and Ciric, A. R.; “An MILP approach for safe process plant layout”, Industrial Eng. Chemical Res., 35, 1354-1361, 1996 Raoot, A. D.; and Rakshit, A.; “A fuzzy approach to facilities layout planning”, International Journal of Production Research, 29(4), 835-857, 1991 Realff, M. J.; Shah, N.; and Pantelides, C. C.; “Simultaneous design, layout and scheduling of pipeless batch plants”, Computers Chemical Engineering, 20(6/7), 869-883, 1996 Rosenblatt, M. J.; and Golany, B.; “A distance assignment approach to the facility layout problem”, European Journal of Operational Research, 29(4), 835-857, 1991 Sahni, S.; and Gonzalez, T.; “P-complete approximation problems”, Journal of the Association for Computing Machinery, 23, 555-565, 1976 Scribian, M.; and Vergin, R. C.; “Comparison of computer algorithms and visual based methods for plant layout”, Management Science, 22(2), 1975 Shah, N.; “Efficient scheduling, planning and design of multipurpose batch plants”, PhD thesis, Imperial College of Science, Technology and Medicine, University of London, 1992 Skorin-Kapov, J.; “Extensions of the tabu search adaptation to the quadratic assignment problem”, Computers & OR, 21(8), 855-865, 1994 Smith, J. M.; and MacLeod, R.; “A relaxed assignment algorithm for the quadratic assignment problem”, INFOR, 26(3), 170-190, 1988 Suzuki, A.; Fuchino, T.; and Muraki, M.; “An evolutionary method for arranging the plot plant for process plant layout”, Journal of Chemical Eng. of Japan, 42, 226-231, 1991a Suzuki, A.; Fuchino, T.; and Muraki, M.; “Equipment arrangement for batch plants in multifloor buildings with integer programming”, Journal of Chemical Engineering of Japan, 24, 737-742, 1991b Tam, K. Y.; “A simulated annealing algorithm for allocating space to manufacturing cells”, International Journal of Production Research, 30(1), 63-87, 1992 Tate, D. M.; and Smith, A. E.; “A genetic approach to the quadratic assignment problem”, Computers and Operation Research, 22(1), 73-83, 1995

References

144

Tavakkoli-Moghaddain, R.; and Shanyan, E.; “Facilities layout design by genetic algorithms”, Computers and Industrial Engineering, 35(3/4), 527-530, 1998 Tompkins, J. A.; and White, J. A.; “Facilities planning”, John Wiley & Sons (New York), 1984 Tsai, R. D.; Malstrom, E. M.; and Kuo, W.; “Three dimensional palletization of mixed box sizes”, IIE Transactions, 25(4), 64-75, 1993 Urban, T. L.; “Solution procedures for dynamic facility layout problem”, Annals of Operations Research, 76, 323-342, 1998 Watson, K. K.; and Giffin, J. W.; “The vertex splitting algorithm for facilities layout”, International Journal of Production Research, 35(9), 2477-2492, 1997 Wilhelm, M. R.; and Ward, T. L.; “Solving quadratic assignment problems by ‘simulated annealing’”, IIE Transactions, 19, 107-119, 1987

References

145