the university of burdwan

THE UNIVERSITY OF BURDWAN

Doctoral Program: Business Administration

Doctoral Thesis

In Search of Optimum Assembly Line Balancing

DEBDIP KHAN

Thesis Advisor: Professor Dilip Roy

Department of Business Administration January 2011

Dedicated to My parents

This is to certify that the work reported in the thesis entitled “In Search of Optimum Assembly Line Balancing” submitted by Debdip Khan in partial fulfillment for the degree of the Doctor of Philosophy in Arts (Department of Business Administration), is a faithful and bona-fide research work carried out under my supervision and guidance. The results of the investigation reported in this thesis have not so far been submitted for any other Degree or Diploma. The assistance and help during the course of investigation have been duly acknowledged. However, some parts of this thesis have been published in various journals for purely academic purpose.

Date: (Dilip Roy)

I, Debdip Khan, do hereby declare that the work reported in the thesis entitled “In Search of Optimum Assembly Line Balancing” which is being submitted to The University of Burdwan, Burdwan, West Bengal, India in partial fulfillment for the degree of the Doctor of Philosophy in Arts (Department of Business Administration), is a faithful and bona-fide research work and has not been submitted earlier to any other University or Institution for any other degree.

Date: (Debdip Khan)

This thesis is based on the following published works and a few works submitted for publication:

Published: 1) Roy, D. and Khan, D. (2010). Assembly line balancing to minimize balancing loss and system loss. Journal of Industrial Engineering International, 6(11), 1-5. 2) Roy, D. and Khan, D. (2010). Integrated model for line balancing with workstation inventory. International Journal of Industrial Engineering Computation, 1, 139-146. 3) Roy, D. and Khan, D. (2011). Optimum assembly line balancing by minimizing balancing loss and a range based measure for system loss. Management Science Letters, 1, 13-22.

Accepted for publication: Roy, D. and Khan, D. (2010). Optimum assembly line balancing: A stochastic programming approach. International Journal of Industrial Engineering Computation, 2, Accepted for publication.

Submitted for publication: Roy, D. and Khan, D. (2010). Optimum assembly line balancing: A linear programming approach under dual objectives. Journal of Industrial Engineering International, Submitted for publication.

Acknowledgements First of all, I would like to express my deepest sense of gratitude to my guide Professor Dilip Roy, Head, Department of Business Administration, The University of Burdwan, for giving the opportunity of developing my Ph.D. thesis under his supervision. I am grateful for his scholarly guidance, inspiring and valuable suggestions, tirelessly monitoring of the work and for his timely corrections with positive criticism, without which this study would not have been completed or even started. I thank him for believing in me, sometimes more than what I believe myself. I am thankful to the authorities of The University of Burdwan for giving me permission to do this research work. I express my gratitude to all the members of the Centre for Management Studies for creating a pleasant place to work and for providing me with all the resources necessary to carry out this work with important suggestion. I would like to thank to all my teachers, colleagues, co-researcher and friends for their help and cooperation and a special thank goes to my friend Sumit for his continuous support and inspiration. Finally, I would like to thank my family members, specially my mother Suchitra and my father Janmejoy Khan for being my silent source of strength. They always supported me and motivated and encouraged me to continue towards the completion of this thesis. This achievement is theirs. Date: ( Debdip Khan )

i

Table of Content Page No.

Acknowledgements

i

Table of Content

ii

Index of Figures

v

Index of Tables

vii

Chapter 1. Introduction 1.1

Introduction

1

1.2

Structure of the Thesis

4

1.3

Assembly Line: Basic concepts and classification

6

1.4

State-of-the-Art

27

1.5

Simple Assembly Line balancing Problems (SALBP)

33

1.6

Generalized Assembly Line balancing Problems (GALBP)

35

Chapter 2. Existing Methodology 2.1

Introduction

38

2.1.1

Heuristic Approach

39

2.1.2

Simulation Approach

41

2.1.3

Programming Approach

42

2.2

Other Methodologies

43

2.3

Research Gap

51

2.4

Objectives

53

Chapter 3. Solutions to Assembly Line balancing Problem by Heuristic Methods 3.1

Introduction

55

3.2

Different Measures of Balancing loss and System loss

57

3.3

VBMS: The Variance Based Measure for System Loss for SALBP

58

3.3.1

Problem Description

58

3.3.2

Notation

59

3.3.3

Methodology

60 ii

Page No.

3.3.4

Algorithm

66

3.3.5

Worked Out Example

68

3.3.6

Sequential approach under heuristic method

76

3.3.7

Conclusion

81

RBMS: The Range Based Measure for System loss for SALBP

82

3.4.1

Problem Description

82

3.4.2

Notation

83

3.4.3

Methodology

84

3.4.4

Algorithm

88

3.4.5

Worked Out Example

90

3.4.6

Conclusion

97

Comparative study between VBMS and RBMS

98

3.4

3.5

Chapter 4. Assembly Line Balancing by Linear Programming Approach 4.1

Introduction

100

4.2

Notation

104

4.3

Methodology

105

4.4

Mathematical Formulation

107

4.5

Worked Out Example

109

4.6

Conclusion

112

Chapter 5. Assembly Line Balancing by Stochastic Programming Approach 5.1

Introduction

113

5.2

Notation

114

5.3

Methodology

116

5.4

Mathematical Formulation

117

5.5

The Algorithm

123

5.6

Worked Out Example

124

5.7

Conclusion

127

Chapter 6. Integrated model for Line Balancing with Workstation Inventory Management iii

Page No.

6.1

Introduction

129

6.2

Notation

133

6.3

Methodology and Mathematical Formulation

134

6.4

The Algorithm

140

6.5

Worked Out Example

141

6.6

Conclusion

144

Chapter 7. Designing of an Assembly Line based on Reliability 7.1

Introduction

146

7.2

Notation

147

7.3

Methodology and Mathematical Formulation

149

7.4

The Algorithm

154

7.5

Worked Out Example

155

7.6

Conclusion

159

Chapter 8. Conclusions 8.1

Conclusion

161

8.2

Scope

164

References

166

iv

Index of Figures Page No.

Figure

1.1

Ford assembly line in 1913

7

1.2

Ford motor company assembly line

8

1.3

The engine goes on the chassis

9

1.4

Radiator assembly

10

1.5

Grille assembly

10

1.6

Firewall preparation

11

1.7

Dashboard assembly

11

1.8

Under body protection

12

1.9

A complete body being readied

12

1.10

Body ready to be lowered onto the chassis

13

1.11

Three rows of completed cars

13

1.12

The car gets to know gasoline Under its own power

14

1.13

A car is driven to the delivery department

14

1.14

Single-model line

18

1.15

Mixed-model line

18

1.16

Multi-model line

19

1.17

Serial line

21

1.18

Two-sided line

22

1.19

Parallel workstation

23

1.20

Parallel lines

23

v

Page No.

Figure

1.21

U-shaped line

24

1.22

circular or closed line

24

1.23

Robotic line

26

1.24

Manual line

26

3.1

Precedence diagram of workstations

69

3.2


90

4.1

Precedence diagram of workstations along with the task times

110

5.1


125

6.1

Precedence diagram of workstations along with the task times

142

7.1


155

vi

Index of Tables Page No.

Table

Solutions under Variance Method 3.01

Workstation-wise line balancing configurations with trial cycle time 31

71

3.02


72

3.03


72

3.04


73

3.05


74

3.06

Workstation-wise line balancing final configurations

75

Solutions under Kilbridge and Wester Method 3.07


78

3.08


78

3.09


78

3.10


78

3.11


78

3.12


79

Solutions under Positional Weight (PW) Method 3.13


80

3.14


80

3.15


80

3.16


80

3.17


81

Solutions under Range Method 3.18


92

3.19


93

3.20


94

3.21


95

vii

Page No.

Table

3.22


96

3.23


96

Solutions under LP Approach 4.1

Precedence relation and task times of work elements

111

4.2

Final optimum configuration

112

Solutions under Stochastic Programming Approach 5.1


126

5.2


126

Solutions under workstation inventory Approach 6.1

Precedence relation, task times of work elements and cost parameters

143

6.2

Final optimum configuration and policy

144

Solutions under Reliability Approach 7.1


156

7.2

Trial configuration

157

7.3


159

viii

In Search of Optimum Assembly Line Balancing Chapter 1: Introduction

Chapter 1 Introduction

1.1 Introduction Assembly line production is one of the widely used basic principles in a production system. It is in use in many production and manufacturing systems, particularly those involving a large volume of a single product. The basic objective is to go for the division of total work into small sub works so as to maximize the system productivity (Amen, 2001). The configuration of the line and the distribution of work along the line are

1


fundamental to the system’s efficiency. A complex optimization problem arises when technological constraints and a given objective are also taken into consideration. The problem of Assembly Line Balancing deals with the distribution of activities among the workstations so that there will be maximum utilization of human resources and facilities without disturbing the work sequence. So, in an Assembly Line Balancing Problem (ALBP), a set of tasks have to be assigned to an ordered sequence of workstations in such a way that precedence constraints (i.e., predecessor task must be completed before successor tasks) are maintained and an efficiency measure is optimized, in respect of the number of workstations or the cycle time. The simplest case, referred to in the literature is SALBP: Simple Assembly Line Balancing Problem (Baybars, 1986; Scholl and Becker, 2006), where a serial line processes a single model of one product. Basically, the problem is restricted by technological precedence relations and the cycle time constrains. On the other hand, GALBP: Generalized Assembly Line Balancing Problems are considered to be those that take into account other attributes like parallel workstations, multiple process or multiple product and system restrictions. A great

2


diversity of GALBP has been considered in the literature, which include, for example, mixed-models, parallel workstations, U-Shaped lines, unequally equipped workstations and multiple objectives (Becker and Scholl, 2006). Regarding the conventional terminology (Baybars, 1986; Scholl, 1999), when the objective is to minimize the number of workstations given an upper bound on the cycle time, the problem is referred to as ALBP-1. If the objective is to minimize the cycle time given the number of workstations, the problem is called ALBP-2. The huge complexity of problems involving assembly alternatives has led to the use of a two-stage approach. The system designer selects one of the possible variants according to criteria such as total processing time, cost, resource allocation, and task parallelism (Lambert, 2006 and Senin et al., 2000). However, by following this two-stage procedure it cannot be guaranteed that an optimal solution of the global problem can be obtained, because the decisions taken by the system designer restrict the problem as a problem of balancing loss which occurs due to uneven allocation of tasks to workstations. Assembly lines involving human

3


elements have another pressing problem. “The losses resulting from workers’ variable operation times” is known as system loss (Wild, 2004) which is more important than balancing loss. But unfortunately very little attention has been given on system loss. Therefore, to increase the efficiency and to solve an ALBP that involves processing alternatives, all possibilities of the balancing process must be considered. For this purpose, in this thesis both the balancing loss problem and the system loss problem are jointly considered. The Optimum Assembly Line Balancing Problem (OALBP), the new problem firstly introduced, defined and studied in this doctoral thesis, considers the possibility of minimizing both the balancing loss and system loss.

1.2 Structure of the Thesis This thesis consists of eight chapters and is structured as follows. Chapter 1 introduces the problem addressed in this thesis and presents the State-of-the-art. It discusses the main concepts related to assembly

4


systems and gives an overview of the problems that have been addressed in the literature, including the proposed solution procedures. Chapter 2 presents existing methodologies, techniques and outlines the aims of this work. Chapter 3 introduces, defines and characterizes the Simple Assembly Line Balancing and solves the problem using Heuristic approach proposing two different Measures for System Loss. Chapter 4 presents a mathematical formulation of the Assembly Line balancing problem. Mathematical programming models are proposed and the performances are evaluated by optimization software. Chapter 5 deals with the stochastic approach to include stochastic time set up for Assembly Line and handling of complex Assembly Line balancing problem. Chapter 6 handles the problem of integrated model for Assembly Line Balancing with Work station Inventory Management Chapter 7 introduces the reliability concepts in Assembly Line. A new type of Assembly Line is proposed with solution to maintain a desired level of reliability for the setup. Chapter 8 presents the conclusions and direction for future research.

5


1.3 Assembly Line: Basic Concepts and Classification This section introduces the basic concepts and classifications of assembly lines. It describes classical assembly line balancing problems. For the purpose of problem identification it gives the overview of some classification schemes. It also gives an overview of the variety of problems and solution procedures for various research purposes. Optimization problems involving alternative configurations are presented in order to outline the problem under our study. Generally, an assembly line consists of a sequence of N, where N is a positive integer, workstations. These workstations are usually connected by a transportation mechanism such as a conveyor belt. Through this belt the product units flow. To produce or manufacture a specific product, each workstation repeatedly performs a set of tasks. Tasks require certain time to be processed. This time is known as task time or assembly time of that particular task. Tasks are related amongst one another according to the existing technological constraints or precedence constraints. By general assumption tasks are indivisible in nature, i.e. one task can be processed in only one workstation.

6


The famous example of an assembly line is Ford’s Assembly Line (see Figure 1.1) between 1908 and 1915. In Automobile industry, Henry Ford used that in his production plant for model-T (see Figure 1.2). Components were manufactured in the moving line and allow mass production at low cost.

Figure 1.1: Ford assembly line in 1913

7

In Searchh of Optimum m Assembly Line Balaancing uction Chapteer 1: Introdu

Figure 1.22: Ford mottor companny assemblly line The Venetian Arsennal, the firsst factory inn the worldd developeed the methodds of maass-produccing warshhips in the t early 16th cen ntury. Producction was much m fasteer and reqquired less wood. At the peak of its efficienncy, the Arsenal A was able to pproduce neearly one sship per daay. In 1799, Eli Whitnney introdduced the assemblyy lines in the Ameerican facturing system. s Inn 1901 Raansom Eli Olds pattented thee first manufa assembbly line cooncept. Hiss Olds Mootor Vehiclle Companny was thee first factoryy in Americca to mass--produce aautomobiles (Wikipeddia, 2010).

8


For example, simplest motor company assembly line is given below (Figure 1.3 – Figure 1.13)

Figure 1.3: (01) The engine goes on the chassis

9


Figure 1.4: (02) Radiator assembly

Figure 1.5: (03) Grille assembly

10


Figure 1.6: (04) Firewall preparation

Figure 1.7: (05) Dashboard assembly

11


Figure 1.8: (06) Under body protection

Figure 1.9: (07) A complete body being readied

12


Figure 1.10 : (08) Body ready to be lowered onto the chassis

Figure 1.11: (09) Three rows of completed cars

13


Figure 1.12:(10)The car gets to know gasoline Under its own power

Figure 1.13: (11) A car is driven to the delivery department

14


Though, Assembly lines are most commonly found in the automobile industry, many other sectors like daily life goods, are also organized in assembly lines. For example, the final assembly of products such as wristwatch, washing machines, refrigerators, radio, TV and personal computers (Amen, 2001). More recently, assembly lines have also gained importance in low volume production of customized products (Scholl et al., 2007).

Basic Concepts • Processing tasks or Work elements: A processing task (work element) i is an indivisible working unit. This processing task is generally known as task. The time taken by the task for being processed is known as processing time of that task and generally denoted by ti. The total work required to manufacture a product in an assembly line is divided into a set of K independent and identifiable tasks.

15


• Workstations: These are the line components where tasks are processed. Workstation can involve a human or robotic operator, certain equipment and some specialized processing mechanisms.

• Cycle time C: It is the time available in each workstation. This is the time to complete the tasks required to process a unit of product. The production rate is equal to 1/ C units of product per time unit. The cycle time is also defined as the time interval between the processing of two consecutive units (Peeters, 2006). • Precedence relations: These are defined by the technological precedence requirements. They determine the partial order in which tasks can be performed in the assembly line. Predecessor should be processed first, then the successor. A task cannot be processed until all its immediate predecessors have already been processed. Precedence diagram or precedence graph generally represents precedence relations. • Line balancing: It is the process of distributing the K tasks among the N workstations. The distribution will be in such a way that, precedence constraints and other constraints are satisfied. And at the same time efficiency measure is optimized. Classical objective

16


is to minimize N for a desired cycle time C, or to minimize C for given N. • Workstation load Wj: It is the total number of tasks assigned to workstation j. • Workstation time t(Wj): It is the sum of the task times ti of all tasks assigned to workstation j. • Workstation idle time Lj: It is the difference between the cycle time and the workstation time.

According to different criteria and need, there exists a great variety of assembly lines. The characteristics include the layout and the shape of the line. The number of products and models being processed in the line are also taken into consideration. Different types of workstation and the variability of the task processing times are also responsible for variety. Classification may be summarized for some most relevant attributes based on the research studies of Boysen et al. (2007a, 2007b), Becker and Scholl (2006), Hao (2005), Miralles (2004), Rekiek (2001) and Scholl (1999).

17


Classification of Assembly Lines According to the number of products or models • Single-model line: A single model of a unique product type is produced in this line configuration (see Figure 1.14). This type of model neither changes the product type nor changes its setup for production.

Figure 1.14: Single-model line • Mixed-model line: Different types of products are produced simultaneously in this line (see Figure 1.15). Here, units of different models are produced in a mixed sequence. All the models produced in this line require basically the same manufacturing tasks. As a result, no setup time is involved in this type of production process.

Figure 1.15: Mixed-model line

18


• Multi-model line: In this type of line, different models with significant differences amongst one another are produced. Similar kind of product are grouped together to form batch (see Figure 1.16). Then sequences of batches are processed, containing either the same model or a group of similar models. Here, intermediate setup tasks are involved.

Setup

Setup

*

*

Figure 1.16: Multi-model line

According to task durations • Deterministic: Certainty is there in the deterministic system because here all task processing times are fixed and known.

19


• Stochastic line: This type of setup is full of uncertainty and chance factor. Task processing times may be significantly affected from different causes like the ability or motivation of human operators. The processing time of one or more task is considered to be probabilistic due to the stochastic nature of human behavior.

• Dynamic line: In this type of line processing, times vary over time. It can be reduced by improving the assembly process.

According to the line shape or layout • Serial lines: Workstations are consecutively arranged in a straight line and products units are processed throughout the line (see Figure 1.17).

20


Figure 1.17: Seriaal line

• Two-sid ded lines: There aree two seriaal lines in pparallel. Pair of oppositee workstattions geneerally know wn as leftt-hand sidee and right-haand side simultaneeously proocesses thhe same task. Automo otive indusstry is a veery commoon examplee of this tyype of line (seee Figure 1.18).

211


Figure 1.118: Two-sidded line

• Parallel workstattions: Herre, two or more m workkstations arre put in paralllel. So thhe work element cann be distrributed bettween several workstatioons (see Figgure 1.19).

222


Fiigure 1.19: Parallel workstation w

• Parallel lines: In this configguration, prroduction system invvolves multiplee products.. Each linee is designeed for a paarticular prroduct or produuct family (see Figuree 1.20).

Figgure 1.20: P Parallel linnes

233


• U-Shap ped lines: The worksstations arre arrangedd in a U-shhaped line whhere both end e of thee line are closed to each otherr (see Figure 1.21). 1

Figure 1.221: U-shapped line • Circle/cclosed linees: Workstaations are arranged a a around a cirrcular conveyoor belt or something like l that (see Figure 11.22).

c or closed linee Figgure 1.22: circular

244


According to the work flow • Synchronous lines: In synchronous lines, all workstations have a common cycle time. Here, production rate is fixed. Start time of all workstations is same and advancement of the work element is constant.  • Asynchronous lines: All workstations can work at different speeds. After completion of task they are transferred to the next workstation. If the next workstation is busy in processing another task then the task is stored in the buffer until the next workstation is free. • Feeder lines: These are secondary subassemblies lines. It provides a main line with sub assemble components.

According to the level of automation • Robotic lines: These lines are fully automated and operated by robots generally used in automobile industry (see Figure 1.23).

25


Figure 1.223: Roboticc line • Manual liines: Tasks are performed by huuman (see Figure 1.2 24).

Figgure 1.24: Manual linne

266


To solve a problem of line balancing, first we need the characterization of the line which determines the type of balancing problem. The simplest balancing problem is single-model, serial, paced, deterministic line. Other line configurations come with additional decision problems. For example, mixed model line faces the problem of sequencing along with the balancing problem and lot sizing problem affects multi-model line. Parallel lines suffer from a decision of how many lines to be configured. Assignment of both tasks and robots to the workstations is the problem of Robotic lines. On the other hand buffers and their positioning are needed for Asynchronous lines where as the production rates should be synchronized whenever feeder lines are concerned.

1.4

State-of-the-Art

If we examine the history of industrial advancement we may note that the initial development of infrastructure led to increase in the market size which in turn resulted in improved technological and managerial skills for remaining afloat in the face of growing competition. Different types of production processes have come into existence, each meeting the special

27


requirement of the demand and or technology of the offer under consideration. According to Wild (2004), there are basically three types of production process namely mass production, batch production and jobbing. During the days of mass production era, when customization was not an important consideration and the products were more or less standardized, aim of the corporate planners was to enjoy maximum market share by producing the offer at minimum cost. In that context, mass production assumed high priority. In the recent past, balancing of assembly line has again assumed importance. Breaking down of the country boundaries and formation of a global village has increased the size of the market beyond imagination. As a result, competitors are many in number, each trying to get the maximum foot hold in the market. Throughout the globe, corporate houses have started thinking globally based upon global culture and global standard, except for food and other culturally sensitive items where multi-domestic form is the best choice (Keegan, 1995). Thus, in general there is emphasis on standardization and cost reduction for translation of core competence into competitive advantage. It may therefore be observed that in this age of globalization companies are

28


trying their best to increase their volume of production. As these products are more or less globally standardized companies have started re-thinking in terms of balanced flow lines to reduce the time and cost and increase the output. Though line balancing problem assumed importance during the mass production era (Roy, 2006), the first analytical treatment of assembly line balancing made an inroad in the literature during 1950s with the works of Bryton (1954) and Salveson (1955). Salveson (1955) and Bowman (1960) considered the linear programming approach to arrive at an optimum solution for this problem. However, in view of the complicated nature of the problem their optimum solutions turned out to be impractical and occasionally unusable.

These difficulties led to

introduction of heuristic methods by Kilbridge and Wester (1961), Assembly Line balancing using the ranked positional weight technique by Helgerson and Birnie (1961), Moodie and Young (1965) and Mansoor (1968). Later Dar-El (1973) suggested a heuristic technique, namely MALB, for balancing large but single-model assembly lines. Over the years interesting usage of simulation approach is also available for solving the line balancing problem. A general method for machine scheduling by Charlton and Death(1969),

29

assembly line


balancing using Best Bud search by Nevins(1972), and simulation to Generate the Data to Balance an Assembly Line by Grabau, Maurer, and Ott.(1997) are now available with the production planners for a close workstation management. Use of simulated annealing to solve a multi-objective assembly line balancing problem with parallel workstations has been studied by McMullen and Frazier(1998). During the last few decades several researchers have made serious attempts to handle this problem of line balancing by mainly using optimization techniques. Earlier references are those of Hoffman(1963), Mansoor and Yadin(1971) and Geoffrion(1976) where the purpose of mathematical programming was to give a clear insight into this complex but important problem. Buxey (1974) emphasized on the configuration of multiple workstations. Vrat and Virani (1976) presented a cost model for optimal mix of balanced stochastic assembly line and the modular assembly system for a customer oriented production system. Later, Van Assche and Herroelen (1979) have proposed an optimal procedure for the single-model deterministic assembly line balancing problem. Graves and Lamer (1983) used an integer programming procedure for designing an assembly system and Talbot and Patterson (1984) used the same for

30


solving the assembly line balancing problems. Infact in the mid 80’s some researchers gave emphasis on application part like application of operational research models and techniques in flexible manufacturing systems (Kusiak,1986) and application of a hierarchical approach for solving machine grouping and loading problems of flexible manufacturing systems (see Stecke, 1986). Sarin and Erel (1990) developed a cost model for the single-model stochastic assembly line balancing problem for the objective of minimizing the total labour cost and the expected incompletion cost arising from tasks not completed within the prescribed cycle time. For the multi-product assembly line balancing problem, Berger et al (1992) adopted Branch-and-bound algorithms and the problem of balancing assembly lines with stochastic task processing times using simulated annealing was addressed by Suresh and Sahu (1994). The study of Nkasu and Leung (1995) adopted the methodology of stochastic modeling, whereby various probability distributions are integrated within a modified COMSOAL algorithm, as a means of addressing the uncertainties associated with key assembly line balancing variables, such as cycle time and task times. In 1998, Pinnoi and Wilhelm dealt with the problem of system design using the branch and cut approach. In 2002, Nicosia et al introduced the concept

31


of cost and studied the problem of assigning operations to an ordered sequence of non-identical workstations, which also took into consideration the precedence relationships and cycle time restrictions. Their purpose was to minimize the cost of the assignment by using a dynamic programming algorithm. Erel et al (2005) presented a beam search-based method for the stochastic assembly line balancing problem in U-lines. In 2006, Zhao et al dealt with sequence-to-customer goal with stochastic demands for a mixed-model assembly line For minimizing the number of stations and task duplication costs in the mixed-model

assembly

line

balancing

problem,

Bukchin

and

Rabinowitch (2006) proposed a branch-and-bound based solution. Agarwal and Tiwari (2008) proposed a collaborative ant colony algorithm to stochastic mixed-model U-shaped disassembly line balancing and sequencing problem. Gamberini et al (2009) presented a multiple single-pass heuristic algorithm solving the stochastic assembly line rebalancing problem developed for the purpose of finding the most complete set of dominant solutions representing the Pareto front of the problem. Though the trend in heuristic solution has dampened down over years, yet a few new approaches have been suggested as may be seen

32


from the survey of algorithms for the simple assembly line balancing problems by Baybars (1986). Subsequently, Whitley (1989) has suggested the “GENITOR algorithm” and Scholl and Becker (2003) have suggested State-of-the-art to deal with the balancing problem. From the literature it is clear that for all those existing methods the main consideration was balancing loss and there were mainly three streams of attack such as heuristic approach, simulation approach and programming approach to handle balancing loss in assembly line balancing problem.

1.5

Simple Assembly Line Balancing Problems (SALBP)

SALBP is completely restricted by the technological precedence relations and the cycle time constraints. A huge amount of research work has been devoted to this type of problem (Baybars, 1986; Ghosh and Gagnon, 1989; Scholl, 1999 and Becker and Scholl, 2006). This type of assembly line processes is a unique model of a single product. All the input parameters are known. Task processing times are of deterministic type. These times are also independent of the workstations. Setup times are considered to be negligible. None of the task processing times is greater

33


than the cycle time. All workstations should be equally equipped so that any workstation can process any one of the tasks. Tasks must be processed only once. It cannot be split among workstations and should be completed in one workstation only. There is technological precedence. So, task cannot be processed in arbitrary sequences. All tasks must be processed. Precedence relations and cycle time constraints are the restrictions. Apart from these two, no other restriction is to be considered.

Versions of SALBP According to Scholl (1999), there are four versions of SALBP: • SALBP-1: minimizes the number of workstations N given a cycle time C. • SALBP-2: aims at minimizing the cycle time C given the number of workstations N. • SALBP-E: seeks to maximize the line efficiency E, where E = Tsum/(N.C) and Tsum is the summation of all task processing times. • SALBP-F: is a feasibility problem that tries to establish whether a feasible task assignment exists for a given cycle time C and a number of workstations N.

34


Most of the research works, done on SALBP, keep SALBP-1 in mind. But, SALBP-2 appears to be more relevant than SALBP-1(Miralles, 2004). It is because SALBP-1 is suitable only for designing an assembly line. But, every time for balancing and rebalancing of an existing line, SALBP-2 is required.

1.6 Generalized Assembly Line Balancing Problems (GALBP) In generalized assembly line balancing problems one or more assumptions of the simple case are relaxed (Baybars, 1986; Scholl and Becker, 2006). Some common GALBP are: U-Shaped Assembly Line Balancing Problem (UALBP) This is U-shaped lines. This configuration is considered to be more flexible. It allows more possibilities on how to assign tasks to workstations. The reason for this is that tasks can be assigned when either its predecessors have already been assigned, whereas with serial lines a task can be assigned only when its predecessors have been assigned. Its variants are: UALBP-1, UALBP-2 and UALBP-E respectively (Baybars, 1986).

35


Mixed-model Assembly Line Balancing Problem (MALBP) Kubiak and Suresh (1991), Bard et al. (1992), Bukchin (1998), Merengo et al. (1999), Bukchin et al. (2002), Karabati and Sayin (2003), Ponnambalam et al. (2003), Spina et al. (2003), Bukchin and Rabinowitch (2006) have addressed MALBP in their works. Different models of the same product are inter-mixed. On the same line these products are to be assembled. So, the sequence of different models has to be determined. MALPB-1, MALBP-2 and MALBP-E are the different types present here. Robotic Assembly Line Balancing Problem (RALBP) Robotic line is considered here. Problem considers the assignment of set of tasks and the set of robots to workstations (Rubinovitz and Bukchin, 1993; Tsai and Yao, 1993; Hong and Cho, 1999). Multi-objective Assembly Line Balancing Problem (MOALBP) Several optimization objectives are considered simultaneously. Agpak and Gokcen (2005) deal with a problem that seeks to minimize both the number of workstations and the total assembling cost or the amount of

36


resources. Most GALBP are multi-objective (Kim et al., 1996; Malakooti and Kumar, 1996; McMullen and Frazier, 1998; Bukchin and Masin, 2004). The characteristics of the line and the layout of the system Bukchin and Rubinovitz (2003) addressed a problem of parallel workstations. Multiple workstations are considered by Buxey (1974) and problem involving parallel tasks was addressed by Pinto et al. (1975). There are also many other problems including two-sided lines (Kim et al., 2000; Bartholdi, 1993), buffered or parallel lines commonplace in a multi-model context (Suer, 1998), multi-product lines (Pastor et al., 2002) and complex layouts involving lines with different shapes (Bukchin et al., 2006). Task durations Task durations are of different type. Processing times may be dependent on the sequence (Spina et al., 2003) or operator (Corominas et al., 2008). Given this backdrop in terms of classification and characterization of assembly lines we propose to review the existing methodologies for arriving at a balancing solution.

37

Chapter 2

Existing Methodology

2.1 Introduction Several techniques have been used so far for assembly line balancing. There are mainly three streams of attack to solve ALBP. All the existing techniques can be classified mainly into three categories such as heuristic approach, simulation approach and programming approach.

38

In Search of Optimum Assembly Line Balancing Chapter 2: Methodology

2.1.1 Heuristic Approach Heuristic is an adjective for experience-based techniques. It helps in problem solving, learning and discovering. A heuristic method is used to rapidly come to a solution that is hoped to be close with the best possible answer, or 'optimal solution'. A heuristic is a general way of solving a problem. Another name for heuristic methods is “Heuristics” when it is used as a noun. Generally, heuristics stand for strategies using readily accessible, though loosely applicable, information to control problem solving in human beings and machines. A heuristic is a technique designed to solve a problem. It ignores whether the solution can be proven to be correct, but usually produces a good solution.

Heuristics are intended to gain

computational performance or conceptual simplicity, potentially at the cost of accuracy. A heuristic method can accomplish its task by using search trees. Heuristic is an experience-based method that can be used as an aid to solve process design problems. It varies from size of equipment to operating conditions. Time for solving problems can be reduced by using heuristics. They are intended to be used as aids in order to make quick

39


estimates and preliminary process designs. Part of this method is using an educated guess, an intuitive judgment, or common sense. A heuristic is a general way of solving a problem. In more precise terms, heuristics stand for strategies using readily accessible, though loosely applicable, information to control problem solving in human beings and machines.

Directed trial and error is the most fundamental heuristic. This can be used in everything from very common work to higher one. For example, matching bolts to nuts or to finding the values of variables in algebra problems. So, this method has versatile application. Here are a few other commonly used heuristics (Polya, 1945) •

If you are having difficulty understanding a problem, try drawing a picture.

•

If you can't find a solution, try assuming that you have a solution and seeing what you can derive from that ("working backward").

•

If the problem is abstract, try examining a concrete example.

•

Try solving a more general problem first (the "inventor's paradox": the more ambitious plan may have more chances of success).

40


In Assembly line balancing problem, a heuristic is an experiencebased method that can be used as an aid to solve process design problems, varying from size of equipment to operating conditions. By using heuristics, time can be reduced when solving problems. Because heuristics are fallible, it is important to understand their limitations. They are intended to be used as aids in order to make quick estimates and preliminary process designs (Wikipedia, 2010).

2.1.2 Simulation Approach Simulation is the imitation of some real thing, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system. Key issues in simulation include acquisition of valid source information about the relevant selection of key characteristics and behaviours, the use of simplifying approximations and assumptions within the simulation, and fidelity and validity of the simulation outcomes (Wikipedia, 2010).

41


There are many uses of simulation techniques particularly those cases where actual experiment is not always possible. It may be used for the optimization of performance for any system or setup. It is also very popular testing and training for safety purpose, for e.g., flight simulators for training aircraft pilots. Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real system cannot be accessed.

2.1.3 Programming Approach Mathematical programming refers to choosing the best solution from some set of available alternative solutions. This is the case of solving problems in which one seeks to minimize or maximize a function by systematically choosing the values of real or integer variables from within an allowed set. This formulation, using a scalar, real-valued objective function, is probably the simplest example; the generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, it means finding "best available" values of some objective function given a defined domain,

42


including a variety of different types of objective functions and different types of domains (Wikipedia, 2010).

2.2 Other Methodologies Simulated annealing (SA) is a generic probabilistic metaheuristic for the global optimization problem of applied mathematics, namely locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more effective than exhaustive enumeration — provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution. The name and inspiration came from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. The heat causes the atoms to become unstuck from their initial positions (a local minimum of the

43


internal energy) and wander randomly through states of higher energy; the slow cooling gives them more chances of finding configurations with lower internal energy than the initial one. By analogy with this physical process, each step of the SA algorithm replaces the current solution by a random "nearby" solution, chosen with a probability that depends both on the difference between the corresponding function values and also on a global parameter T (called the temperature), that is gradually decreased during the process. The dependency is such that the current solution changes almost randomly when T is large, but increasingly "downhill" as T goes to zero. The allowance for "uphill" moves saves the method from becoming stuck at local optima—which are the bane of greedier methods (Wikipedia, 2010).

Branch and bound (BB or B&B) is a general algorithm for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization. It consists of a systematic enumeration of all candidate solutions, where large subsets of fruitless candidates are discarded by using upper and lower estimated bounds of the quantity

44


being optimized (Wikipedia, 2010). Branch and cut (sometimes written as branch-and-cut) is a method of combinatorial optimization for solving integer linear programs, that is, linear programming problems where some or all the unknowns are restricted to integer values. The method is a hybrid of branch and bound and cutting plane methods. This method solves the linear program without the integer constraint using the regular simplex algorithm. When an optimal solution is obtained, and this solution has a non-integer value for a variable that is supposed to be integer, a cutting plane algorithm is used to find further linear constraints which are satisfied by all feasible integer points but violated by the current fractional solution. If such an inequality is found, it is added to the linear program, such that resolving it will yield a different solution which is hopefully "less fractional". This process is repeated until either an integer solution is found (which is then known to be optimal) or until no more cutting planes are found. At this point, the branch and bound part of the algorithm is started. The problem is split into two versions, one with the additional constraint that

45


the variable is greater than or equal to the next integer greater than the intermediate result, and one where this variable is less than or equal to the next lesser integer. In this way new variables are introduced in the basis according to the number of basic variables that are non-integers in the intermediate solution but which are integers according to the original constraints. The new linear programs are then solved using the simplex method and the process repeats until a solution satisfying all the integer constraints is found. During the branch and bound process, further cutting planes can be separated, which may be either global cuts, i.e., valid for all feasible integer solutions, or local cuts, meaning that they are satisfied by all solutions fulfilling the side constraints from the currently considered branch and bound subtree (Wikipedia, 2010). Linear programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or minimum cost) in a given mathematical model for some list of requirements represented as linear equations. More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and inequality types constraints. Given a polytope and a real-valued affine function defined on this polytope, a linear

46

In Searchh of Optimum m Assembly Line Balaancing Chapter 22: Methodollogy

prograamming method m willl find a point on the polytoope wheree this functioon has thee smallest (or largeest) value if such ppoint existts, by searchiing throughh the polyttope verticees. Linearr programs are probleems that cann be expressed in cannonical form m:

where x represennts the vecttor of variaables (to bee determinned), c and b are wn) coefficiients and A is a (know wn) matrixx of coefficcients. vectorss of (know The exxpression to be maxximized orr minimizeed is calleed the objeective functioon (cTx in this t case). The T equatiions Ax ≤ b are the coonstraints which w specifyy a convex x polytopee over whhich the obbjective fuunction is to be optimized. (In thhis context,, two vectoors are com mparable w when every entry o in onee is less-thhan or equual-to the corresponnding entrry in the other. Otherw wise, they are a incompparable.) Linearr programm ming can be b applied to various fields of study. s It is used most eextensively y in busineess and ecoonomics, buut can alsoo be utilizeed for some engineerinng problem ms. Industtries that use linearr program mming

47


models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design (Wikipedia, 2010). Integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming, which is also known as mixed integer programming (Wikipedia, 2010). Stochastic programming is a framework for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters. When the parameters are known only within certain bounds, one approach to tackling such problems is called robust optimization. Here the goal is to find a solution which is feasible for all such data and optimal in some sense. Stochastic programming models are similar in style but take advantage of the fact that probability distributions governing the data are

48


known or can be estimated. The goal here is to find some policy that is feasible for all (or almost all) the possible data instances and maximizes the expectation of some function of the decisions and the random variables. More generally, such models are formulated, solved analytically or numerically, and analyzed in order to provide useful information to a decision-maker. As an example, consider two-stage linear programs. Here the decision maker takes some action in the first stage, after which a random event occurs affecting the outcome of the first-stage decision. A recourse decision can then be made in the second stage that compensates for any bad effects that might have been experienced as a result of the first-stage decision. The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions (a decision rule) defining which second-stage action should be taken in response to each random outcome (Wikipedia, 2010).

Ant Colony Optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. The algorithm was aiming to search for an optimal

49


path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants (Wikipedia, 2010).

Genetic algorithm (GA) is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems. Genetic algorithms belong to the larger class of evolutionary algorithms (EA), which generate solutions to optimization problems using techniques inspired by natural evolution, such as inheritance, mutation, selection, and crossover. In a genetic algorithm, a population of strings (called chromosomes or the genotype of the genome), which encode candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem, evolves toward better solutions. Traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possible. The evolution usually starts from a population of randomly generated individuals and happens in generations. In each generation, the

50


fitness of every individual in the population is evaluated, multiple individuals are stochastically selected from the current population (based on their fitness), and modified (recombined and possibly randomly mutated) to form a new population. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population. If the algorithm has terminated due to a maximum number of generations, a satisfactory solution may or may not have been reached (Wikipedia, 2010).

2.3Research Gap A comprehensive literature review has been carried out. After analyzing the previous research works reported in the literature concerning assembly line balancing problems, it is observed that: the balancing

51


problem studied in all the above mentioned methods are oriented towards balancing loss and can be best used in transfer lines where work elements are preferably performed by machines/ robots. Assembly lines involving human elements have a different pressing problem of system loss. Variations in the idle times in different work stations may lead to a behavioral problem and there may be system loss due to stochastic nature of the time elements. It has been stated in the literature (Wild, 2004) that losses resulting from the above mentioned causes are more important than balancing loss. This research work addresses a new generalized problem: the Optimum Assembly Line Balancing Problem (OALBP). Such a problem considers the possibility of assembly alternatives maintaining the precedence relations constraints. In this doctoral thesis this problem (OALBP) is defined and formalized

via

heuristic

methods,

programming

approach

and

mathematical models. Moreover, a significant number of constructive methods are proposed here for successfully solving the assembly line balancing problems. By suggesting a measure for system loss, this thesis

52


will design an assembly line where dual objectives of minimization of both balancing loss and system loss can be met and extend this minimization problem from deterministic domain to stochastic domain as well as increase the reliability of the setup.

2.4 Objectives This work addresses Optimum assembly line balancing problem that has not been previously considered in the literature. Therefore, the principal objective of this work is to find optimum solution for simple and complex Assembly Line Balancing problems. In order to accomplish the main objectives, the following specific objectives are considered. 1. To suggest a measure for system loss. 2. To design an assembly line where dual objectives of minimization of both balancing loss and system loss can be met.

53


3. To extend this minimization problem from deterministic domain to stochastic domain. 4. To increase the reliability of the setup.

54

Chapter 3 Solution of Assembly Line Balancing Problem by Heuristic Methods

3.1 Introduction Heuristic is an adjective for experience-based techniques. It helps in problem solving, learning and discovery. A heuristic method is used to rapidly come to a solution that is hoped to be close to the best possible

55

In Search of Optimum Assembly Line Balancing Chapter 3: Solution of ALBP by Heuristic Methods

answer, or 'optimal solution'. A heuristic is a general way of solving a problem and another name for heuristic methods is “Heuristics” when it is used as a noun. Generally, heuristics stand for strategies using readily accessible, though loosely applicable, information to control problem solving in human beings and machines. A heuristic is a technique designed to solve a problem. It ignores whether the solution can be proven to be correct, but usually produces a good solution. Heuristics are intended to gain computational performance or conceptual simplicity, potentially at the cost of accuracy.

56


3.2 Different Measures for System loss In this section we will concentrate on different types of losses occur due to balancing of assembly line and try to measure those losses. There are mainly two types of losses – Balancing loss and System loss. Balancing loss is the losses resulting from uneven allocation of work to workstations. It has been found that the differences in average operation times of workers on assembly lines are not primarily a result of differences in standard or work station times, but largely a result of differences in the speed at which workers work. In other words, System loss is the losses resulting from workers’ variable operation times (Wild, 2004). To examine the efficiency of an assembly line the said losses should be measured and compared with the ideal state. The measure for balancing loss, B, proposed in the literature is given by B = { ( NC - ∑ Ti ) / NC }. 100% where, N is number of workstations, Ti task time or assembly time of ith job and C is the cycle time.

57


But there is no such measure for system loss in the literature. We propose two different approaches for measuring system loss, the first one is based on the variance of idle time of tasks to workstations, i.e, Variance Based Measure for System loss (VBMS) and the second one is based on range, i.e., Range Based Measure for System loss (RBMS). Both these approaches are discussed in the following sections of this chapter.

3.3 VBMS: The Variance Based Measure for System loss for SALBP 3.3.1

Problem description

Unfortunately, the balancing problem studied in all the literature are oriented towards balancing loss and can be best used in transfer lines where work elements are preferably performed by machines/ robots. Assembly lines involving human elements have a different pressing problem of system loss. Variations in the idle times in different work stations may lead to a behavioral problem and there may be system loss due to stochastic nature of the time elements. It has been stated in the literature (see Ray Wild, 2004) that losses resulting from the above mentioned causes are more important than balancing loss.

58


Increase in the cycle time is a crude solution to system loss; a better solution can be obtained through pacing. By adjusting the interwork station distances and the speed of the conveyor belt, one can provide more time to workers in a work station. An alternative concept is to provide the work station with buffer stocks of semi-finished items. Optimum buffer stock capacity can be calculated using inventory cost, cost of idle facility, number of workstations, and extent of time variations in workstation. Our objective in this current work is to design an assembly line where dual objectives of minimization of balancing loss and of system loss can be met. For this we install our optimization method through multistage simulation approach. The procedure we propose herein will be generic in nature and can be used with different types of balancing methods so that the size of the balancing problem or the complexity of the same cannot act as hindrance.

3.3.2

Notation K

59

number of jobs.


N

3.3.3

number of workstations.

Ti

task time or assembly time of ith job.

Lj

idle time of jth work station.

Nmin

minimum number of workstation.

C

cycle time.

Ct

trial cycle time.

Cmax

maximum cycle time.

Cmin

minimum cycle time.

S

slackness.

St

slackness for trial cycle time Ct.

V

variance of idle time.

B

balancing loss.

Methodology

To examine the efficiency of an assembly line one uses the concept of balancing loss, which is defined as the loss resulting from uneven allocation of work elements to workstations. The measure of balancing loss, proposed in the literature is given by B = { ( NC - ∑ Ti ) / NC }. 100%

60


We have already pointed out that the balancing loss, B has so far remained the basic consideration for designing an assembly line. The underlying objective is to minimize B subject to precedence constraints. However, our proposed work is a multi-objective one where minimization of balancing loss is to be addressed along with system loss, another measure of efficiency of an assembly line. But no concrete measure of system loss has been suggested so far. Once a measure is identified, objective of the line planner will be to minimize the system loss too. In case the twin objectives of minimization of balancing loss and system loss are achieved the resultant line balancing solution will enjoy greater applicability and wider acceptability. So, we first make an attempt to measure the system loss and suggest a method for reducing the system loss along with the balancing loss. As system loss arises out of workers’ variable operation time any configuration where one workstation has no idle time and another workstation has high idle time may result in high disruption in the system. In contrast, if the workstations have idle time of nearly equal length the chance of disruption will be less. If we look at balancing loss only, idle time for each workstation may significantly vary. This is

61


because idle time for each workstation is the difference between the cycle time and the operation time of the station, where cycle time is externally fixed. We believe that system will be stable if the idle time of each workstation is more or less same. So, our target will be to reduce the variation in idle times of different workstations. As system loss arises out of workers’ variable operation time, any configuration where one workstation has no idle time and another workstation has high idle time may result in high disruption in the system. Therefore, a measure for system loss may be considered as the variance of idle time (V). The stability of the total system will be maximum when the variance will be minimum. Since, we are going by dual objective we need to simultaneously or sequentially minimize balancing loss and system loss. This in turn can be expressed as the dual objective of minimizing the number of workstations and variance of idle times. It is easy to observe that minimization of balancing loss does not lead to a unique solution but reduces the solution set to a manageable one. As a result, if we divide the multi-objective problem into two stages it will be easier to handle the balancing task. Given a choice of cycle time C one can arrive at the

62


minimum number of workstations. Given this minimum number of workstation we get a set of feasible solutions to line balancing problem each optimizing the balancing loss. Within the feasible set of solutions our objective is to allocate nearly equal amount of work to each station or to divide the total work content of the job as evenly as possible among the stations to achieve the selected objective of minimizing system loss. Since the length of work time, or operating time, for which a component is available at each workstation is known as cycle time (C), the solution to line balancing problem is dependent on the choice of C. Now, given a choice of C it may be noted that the theoretical minimum number of workstations, Nmin, must satisfy the following constraints K K ∑Ti/C ≤ Nmin ≤ ∑Ti/C +1 i=1 i=1 It may not be always possible to achieve this minimum number of workstations. Number of workstation may be sequentially increased to arrive at the first feasible solution giving thereby the optimum choice of N. Keeping in mind the fact that Nmin is an integer if we consider that Nmin is given we may reformulate the above inequations in terms of C and note that the minimum number of workstations remains the same for a range of C values. Writing Cmax as the maximum value of C and Cmin as the

63


minimum value of C, for which the minimum number of workstations remains the same, we have from the right hand inequality K Nmin ≤ ∑ Ti / C + 1 i=1 for which an equivalent condition on C is K C ≤ ∑ Ti / ( Nmin – 1) i=1 Thus, we arrive at a choice of maximum value of C as the highest integer contained in K ∑ Ti / ( Nmin – 1) i=1 Thus, Cmax

=

K ∑ Ti / ( Nmin – 1) i=1

Similarly, from the left hand inequality we have K ∑ Ti / C ≤ Nmin i=1

Thus, we get another condition on C as K C ≥ ∑ Ti / Nmin i=1

64


so that, the minimum value of C will be the highest integer contained in K ∑ Ti / Nmin + 1 i=1 Thus, Cmin =

K ∑ Ti / Nmin + 1 i=1

Thus, the choice of C may vary from Cmin to Cmax without disturbing the Nmin value or the optimality of the assembly line in respect of balancing loss. Now, given a cycle time, C, one may conceptually start from Cmin and move up to C to arrive at the set of feasible workstation configurations and maintain the same cycle time C by uniformly adding a slackness to each work station. In case a trial cycle time is denoted by Ct, then Ct must satisfy the condition Cmin ≤ Ct ≤ C with slack time St satisfying the condition St = C - Ct . In this way, the generation of alternative solutions can be increased by manifolds. For each solution, there will be different idle times in different work stations. We next calculate the variance of all idle times, Lj’s, of a particular solution of the solution set. Thus, for each configuration we get

65


a measure V. This V will vary from configuration to configuration according to the distribution of jobs to the workstations. The distribution which gives minimum V is the trial optimal solution for the balancing problem, reducing both the balancing loss and system loss. Final optimal solution will be the one that has least V value across trial cycle times Cmin ≤ Ct ≤ C.

Given the above mentioned line of attack for an assembly line balancing problem under the dual objective of minimizing both the balancing system losses we next develop an algorithm to make the suggested procedure computationally functional so as to deal with problems both large and small and simple and complex. To start with, we consider the simulation search method.

3.3.4

The Algorithm

Set cycle time C, determine the minimum number of workstations Nmin and calculate the Cmin value. 1. Set the cycle time at Cmin. 2. Prepare the list of all Unvisited tasks – call it List U.

66


3. Prepare List R from the tasks of List U with no immediate predecessor or whose immediate predecessors have been visited. These tasks are ready for selection. 4. Prepare List A from the tasks of List R having assembly time less than that of cycle time and is allowable for inclusion. 5. Randomly select a task from the List A and reset the cycle time as {Ct – assembly time}. 6. If cycle time is less than the assembly time, then open new workstation and reinitialize cycle time to its original value and repeat the above steps until all nodes are visited. 7. After getting the complete distribution of tasks to workstations calculate the variance of idle times after adding the slack time. 8. After each run the variance is compared with the previous least variance. If the new variance is less than the previous least variance the new solution is stored as the basis for next comparison. 9. Increase the cycle time by one unit until it crosses C value. If C value is crossed go to step 12. 10. Repeat step 2 to 10.

67


11. Check whether all the work elements have been assigned to specified number of workstations. If not, increase the value of Nmin by 1 and go to step 2. 12. Print the best solution in terms of overall minimum variance. Undertaking a large number of runs, one can get the optimum solution whose variance is minimum. Converting the proposed algorithm in C language in our numerical study, we have carried out for each trial a run of 2000 replications. However, depending on the speed and capacity of the computing facility one may change the number of replications.

3.3.5

Worked Out Example

To explain how the proposed algorithm works we consider a standard assembly line balancing problem as presented in Ray Wild (2004).

68


II

I

9

6 1

III

4

IV

V

VI

VII

IX

VIII

X XI

10 9 5

5

5 5

5 2

17

4 6

5 10

2 12

5 13

4 14

12 15

10 16

15 18

5

8

19

6 8

6 11

Fig 3.1: Precedence diagram of work stations.

In this example, assembly of a simple component requires the completion of 21 work elements, which are governed by certain precedence constraints, as shown in the above figure. This precedence diagram shows circles representing work elements placed as far to the left as possible with all the arrows joining circles sloping to the right. The figures above the diagram are column numbers. Elements appearing in column I can be started immediately, those in column II can be started only after one or more in column I have been completed, and so on. Our problem is to arrive at an optimum configuration of the workstations given a cycle time. This optimum solution will be dependent on C.

69

6 21

10

7

3

20


Let us consider for our trial run a cycle time of 35 time units. Then, for C=35, Nmin works out as Nmin = [ ∑ Ti / C + 1 ] = [ 143/35 +1] =5. Any configuration with N=5 will ensure the same B value and hence for all these configurations balancing loss will be least. To tackle the problem of system loss one can obtain Cmax and Cmin values in the following way: Cmax = [ ∑ Ti / (Nopt-1) ] = [ 143/(5-1) ] = [ 35.75 ] = 35. Cmin = [ ∑ Ti / Nopt +1 ] = [ 148/5 ] = [ 29.6 ] = 29. For minimizing system loss we consider a trial cycle time Ct. Here Ct should start from Cmin and proceed up to C = 35. To ensure C = 35 we add to each work station a corresponding slack value St = 35 - Ct. Thus, our iteration starts from Ct =29, with a slackness, S29 = 35 29 = 6. For Ct=29, no solution is available even after 2000 runs. This is consistent with our intuition that for a very tight situation like Ct=29 chance of generating a feasible solution is remote. We next consider Ct=30 with St=5. Here again, there does not exist any feasible solution. We next consider Ct=31 with St=4. For Ct=31 we get 10 solutions in 20000 runs. These solutions are described below.

70


01

Workstation 1 Elements Idle tim e 1,3,2,5,8 01

02

1,3,8,2,7

01

03

2,3,7,8,1

01

04

2,3,1,7,8,

01

05

2,3,7,8,11

01

06

3,1,7,8,11

00

07

3,7,8,1,2

01

08

2,3,1,5,8

01

09

3,7,8,11,2

01

10

1,2,3,8,11

00

Sol

n

Workstation 2 Elements Idle tim e 6,4,10,11, 00 7,12 6,11,5,10, 00 4,12 11,6,4,5, 00 10,12 5,11,6,10, 00 12,13,14 1,4,6,5,10, 00 12 2,6,5,10, 01 12,4 5,6,10,12, 00 13,14,11 11,6,10,4, 00 7,12 6,1,5,4,10, 00 12 4,6,7,5,10, 01 12

Workstation 3 Elements Idle tim e 13,9,14,15 00 13,9,14,15

00

9,13,14,15

00

Workstation 4 Elements Idle tim e 16,19,17, 01 20 16,17,19, 01 20 16,18,17 01

4,9,15

00

16,18,17

01

9,13,14,15

00

9,13,14,15

00

4,9,15

00

13,9,14,15

00

16,19,17, 20 16,19,17, 20 16,19,17, 20 16,18,17

13,14,9,15

00

13,14,9,15

00

Workstation 5 Element Idle s tim e 18,21 10 18,21

10 10

01

20,19, 21 20,19, 21 18,21

10

01

18,21

10

01

18,21

10

01

19,20 21

10

16,18,17

01

10

16,17,18

01

19,20, 21 20,19, 21

Tab 3.01: Workstation-wise line balancing configurations with trial cycle time 31.

The corresponding minimum variance works out as 14.639999. We next consider Ct=32 with St=3. For Ct=32 we get 6 solutions in 20000 runs.

71

10

10

Varia nce 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999


01


02

3,2,7,6,8

04

03

2,3,7,8,6

04

04

1,2,5,6,10

07

Workstation 2 Elements Idle tim e 15,10,8, 03 12,11,13 1,5,10,11, 03 12,13 1,11,5,10, 03 12,13 4,3,7,8,12 02

05

2,1,6,5,10

07

3,4,8,7,12

06

2,1,5,6,10

07

3,8,4,11

Soln

Workstation 3 Elements Idle tim e 4,14,9 09 4,9,14

09

4,9,14

09

9,11,13,14

07

02

11,9,13,14

07

03

9,7,12,13, 14

06

Workstation 4 Elements Idle tim e 15,16,17, 00 20 15,16,17, 00 20 15,16,19 00 15,16,17, 20 15,16,19 15,16,17, 20

00 00 00

Workstation 5 Element Idle s tim e 19,18, 01 21 18,19, 01 21 17,18, 01 20,21 18,19, 01 21 17,18, 01 20,21 19,18, 01 21

Varia nce 9.839 999 9.839 999 9.839 999 9.04 9.04 7.44


Here, we get the minimum variance as 7.44. We next consider Ct=33 with

01


Workstation 2 Elements Idle tim e 11,1,4,9 02

02

1,3,8,4

04

11,2,7,9,5

02

03

2,1,6,5,4

04

9,3,10,8

04

04

1,3,8,4

04

9,7,2,5,6

04

05

3,2,8,6,1

04

5,10,4,9

04

06

3,2,8,1,6

04

4,5,7,9

04

07

1,4,2,5,6,

04

9,10,3,8

04

08

3,1,4,8

04

2,5,9,7,6

04

09

1,2,6,4,5

04

9,3,8,7

04

Sol

n

Workstation 3 Elements Idle tim e 5,10,12,13 00 ,14,15 6,10,12,13 01 ,14,15 7,12,13,14 11 ,11 11,10,12, 11 13,14 11,7,12,13 11 ,14 11,10,12, 11 13,14 7,12,11,13 11 ,14 10,12,13, 11 14,11 10,12,11, 11 13,14

Workstation 4 Elements Idle tim e 16,19,17, 03 20 16,18,17 03 15,16,19

01

15,16,17, 20 15,16,19

01

15,16,19

01

15,16,17, 20 15,16,17, 20 15,16,19

01

01

01 01

Workstation 5 Element Idle s tim e 18,21 12

Varia nce

20,19, 21 18,17, 20,21 18,19, 21 18,17, 20,21 18,17, 20,21 19,18, 21 18,19, 21 17,18, 20,21

02

15.43 999 12.24

02

12.24

02

12.24

02

12.24

02

12.24

02

12.24

02

12.24


St=2. For Ct=33 we get 9 solutions in 20000 runs. These solutions are:

72

12

17.04


Here, the minimum variance is 12.24. We next consider Ct=34 with St=1. For Ct=34 we get 11 solutions in 2000 runs. These solutions are described below:

02

Workstation 1 Elements Idle tim e 2,1,4,6,5, 00 10 2,6,3,7,1,8 00

03

2,1,4,3,6

02

04

3,7,8,1,11

03

05

3,7,8,11,1

03

06

3,2,8,1,11

03

07

2,3,8,11,1

03

08

3,2,8,11,1

03

09

3,1,8,11,2

03

10

3,8,11,1,7

03

11

3,7,8,1,11

03

Sol

01

n

Workstation 2 Elements Idle tim e 3,7,8,12, 02 11,13 5,10,11,12 02 ,13,4 9,8,7,11,5 02 2,6,5,4,10, 12 2,5,6,10,4, 12 7,5,6,4,10, 12 6,5,7,10, 12,13,14 7,5,4,6,10, 12 6,7,5,10, 12,13,14 2,5,6,10, 12,13,14 2,6,5,10, 12,13,14

Workstation 3 Elements Idle tim e 14,9,15 08


9,14,15

08 06

04

10,12,13, 14,15 9,13,14,15

03

04

13,9,14,15

03

16,19,17, 20 16,19,17, 20 16,19,17, 20 16,18,17

04

13,9,14,15

03

16,18,17

04

04

4,9,15

03

04

13,9,14,15

03

04

4,9,15

03

16,17, 20, 04 19 16,19,17, 04 20 16,17,18 04

04

4,9,15

03

04

4,9,15

03

16,19,17, 20 16,18,17

04

Workstation 5 Element Idle s tim e 20,19, 13 21 18,21 13

04

18,21

13

04

18,21

13

04

19,20, 21 19,20, 21 18,21

13

13

18,21

13

20,19, 21 18,21

13

04 04

20,19, 21


Here, the minimum variance is 14.639998. We next consider Ct=35 with St=0. For Ct=35 we get 12 solutions in 2000 runs. These solutions are described below:

73

13

13 13

Varia nce 21.43 999 21.43 999 16.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999 14.63 999


03

Workstation 1 Elements Idle tim e 2,3,7,6,8, 01 11 3,8,7,11,2, 01 6 3,2,1,8,7,6 01

04

3,2,1,5,4

02

Workstation 2 Elements Idle tim e 1,5,10,12, 03 4,13 1,5,10,12, 08 13,14 4,5,9,10, 04 12 9,6,7,8,11 04

05

1,2,4,3,7

02

6,9,8,11,5

04

06

2,1,3,6,7,5

02

04

07

3,7,1,8,11

04

10,12,8,13 ,4,14 4,9,2,6,5

08

1,4,9,3

02

04

09

3,7,1,2,5,6

02

7,8,11,2,6, 5 8,11,4,9

10

02

4,7,8,12, 13,14 7,9,8,11,5

04

11

3,1,2,5,6, 10 2,1,6,4,3

12

2,1,6,3,4

03

8,11,5,9, 10

03

Sol

01 02

n

03

02

04

03

Workstation 3 Elements Idle tim e 14,9,15 09 4,9,15

04

11,13,14, 15 10,12,13, 14,15 10,12,13, 14,15 9,11,15

08

10,12,13, 14,15 10,12,13, 14,15 10,12,13, 14,15 9,11,15 10,12,13, 14,15 7,12,13, 14,15

07 07 07 07 07 07 07 07 07

Workstation 4 Elements Idle tim e 16,19,17, 05 20 16,19,17, 05 20 16,17,19, 05 20 16,19,17, 05 20 16,19,17, 05 20 16,17,19, 05 20 16,19,17, 05 20 16,19,17, 05 20 16,19,17, 05 20 16,19,17, 05 20 16,19,17, 05 20 16,17,19, 05 20

Workstation 5 Element Idle s tim e 18,21 14

Varia nce

18,21

14

19.44

18,21

14

19.44

18,21

14

17.04

18,21

14

17.04

18,21

14

17.04

18,21

14

17.04

18,21

14

17.04

18,21

14

17.04

18,21

14

17.04

18,21

14

16.64

18,21

14

16.64


Here, the minimum variance is 16.64. It may be observed from the above trial runs that the minimum variance is closely related to the number of solutions obtained for different Ct values. If the number of solutions decreases the minimum variance also decreases. It may also be noted that minimum value is obtained when flexibility in the system is moderate and slackness is also moderate. Further, the variance increases in the extreme situations.

74

21.44


As stated earlier our objective is to minimize the variance of idle times. So we will take that configuration whose minimum variance is the minimum of all the minimum variances. Let us say optimum variance Vopt. i.e., Vopt = Min (min (Vij )) , j i where Vij represents the variance of idle time in different workstations of ith solution for jth cycle time. Here Vopt = 7.44 corresponding to Ct = 32 and hence the optimum configuration comes out as Workstation 1 Elements Idle tim e 2,1,5,6,10 07


Workstation 3 Elements Idle tim e 9,7,12,13, 06 14

Workstation 4 Elements Idle tim e 15,16,17, 00 20

Workstation 5 Element Idle s tim e 19,18, 01 21

Tab 3.06: Workstation-wise line balancing final configurations.

It may be observed from the above example that the proposed approach, in spite of its sequential nature, has led to simultaneous minimization of balancing and system losses. The concept of varying cycle time and then adding slack time helps in generating more feasible solutions than the standard simulation approach proposed in COMSOAL and modified COMSOAL methods.

75

Varia nce 7.44


3.3.6

Sequential approach under heuristic method

To establish the strength of our proposed method and its generic nature we next apply the same concept on heuristic method suggested by Kilbridge and Wester (1961). In

Kilbridge and Wester Method one

generates a configuration by examining the precedence diagram which is constructed with the work elements being placed in different vertical columns. The work element of column I need not follow any work elements. The work elements which are immediately following these are shown under column II. Those work elements immediately following the work elements in column II are shown in column III and so on. So in the first instance we construct precedence diagram ‘column wise’. Then a table is constructed based on the precedence diagram and we group the work elements into a work station: moving from column I-to-column IIto-column III, and so on. We take care that the total station time does not exceed the cycle time. To balance the line, we can take advantage of the moveability of some of the work elements to the right-hand side so that if in the grouping of later work stations some gaps in the idle time are to be

76


filled, this could be done by moving the moveable work elements to the right hand side and assigning them to the desired workstations. We have modified this heuristic method with a sequential procedure and start from Cmin as the cycle time and end up to C, the given cycle time. In each case we add the slack time St= C – Ct if the trial cycle time is Ct. The same twin objectives of minimizing the balancing loss and system loss can be considered here again. For the example worked out in section 4 based on simulation technique we have applied a generic approach with heuristic solution. We present below the observed results which are encouraging when compared with original solution obtained by Kilbridge and Wester. The final solution is given by the following configuration with variance value as 14.64. It may be noted that system loss is more in case of this heuristic method than that in case of simulation method. This is because simulation method converges to the optimum solution. Heuristic method does not ensure optimality but provides reasonability in the solution in a single run. Sequential solutions under modified heuristic method are given below.

77

In Search of Optimum Assembly Line Balancing Chapter 3: Solution of ALBP by Heuristic Methods Workstation 1 Elements Idle tim e 1,2,3,5,8 01

Workstation 2 Elements Idle tim e 6,7,10,11, 00 12,13,14




Varia nce 14.64

Tab 3.07: Workstation-wise line balancing configurations with trial cycle time 31. Workstation 1 Elements Idle tim e 1,2,3,4,6 00





Varia nce 16.64


Workstation 2 Elements Idle tim e 6,7,8,10, 00 11,12,13,




Varia nce 21.04


Workstation 2 Elements Idle tim e 5,6,7,10, 02 11,12,13,




Vari ance 21.44

Tab 3.10: Workstation-wise line balancing configurations with trial cycle time 34. Workstation 1 Elements Idle tim e 1,2,3,5,7,8 00

Workstation 2 Elements Idle tim e 4,6,9,10, 00 12,13



Workstation 5 Element Idle s tim e 21 29


78

Vari ance 129.0 4


Thus, among these limited sets of solution the one generated by Ct = 31 provides the locally optimum solution that minimizes balancing loss and conditionally minimizes system loss. The final solution is Workstation 1 Elements Idle tim e 1,2,3,5,8 05






The same generic approach of ours can be applied on Ranked Positional Weight method, another heuristic method, to arrive at the solution of line balancing with minimum balancing loss and moderate system loss. In this method we assign elements to the workstations according to their precedence relationship and positional weight (PW). A task is prioritized based on the cumulative assembly time associated with itself and its successors. Tasks are assigned in this order to the lowest numbered feasible workstation. Cumulative remaining assembly time constrains the number of workstations required. The entire procedure requires computation of positional weight PW(i) of each task. By this method we get the following solutions:

79

Varia nce 14.64






Varia nce 16.64


Workstation 1 Elements Idle tim e 1,3,2,5,6,7 00





Varia nce 21.04

Tab 3.14: Workstation-wise line balancing configurations with trial cycle time 33. Workstation 1

Workstation 2

Workstation 3

Workstation 4

Workstation 5

Elements

Elements

Elements

Elements

Element s

1,3,2,5,6,7

Idle tim e 01

4,10,8,12, 13,11

Idle tim e 01

9,14,15

Idle tim e 08

16,18,17

Idle tim e 04

19,20, 21

Idle tim e 13

Varia nce

21.04

Tab 3.15: Workstation-wise line balancing configurations with trial cycle time 34. Workstation 1 Elements Idle tim e 1,3,2,5,6,7 02






Thus, among these limited sets of solution the one generated by Ct = 32 provides the locally optimum solution that minimizes balancing loss and conditionally minimizes system loss. The final solution is

80

Varia nce 21.04







3.3.7

Conclusion

We have presented a sequential approach for balancing an assembly line with twin objectives of minimization of balancing loss and system loss. Our approach is a generic one, which is capable of solving different line assembly problem with a reasonable computation time. From the solution set generated by simulation search final choice is made based on optimum number of workstations and minimum variance. In our algorithm first we have taken into consideration balancing loss issue and generate a set of solutions. Among those solutions the best solution is to be selected based on system loss criterion. One may think of considering System loss first and then Balancing loss. But in that case the number of solutions will be very large. As a result optimization will be very difficult to handle. The same

81

Varia nce 16.64


problem will arise if we want to simultaneously minimize System loss and Balancing loss. This approach according to numerical study is giving a better set of configurations because we are using some amount of slackness (St ) in each workstation with trial cycle time Ct varying from Cmin to C.

3.4 RBMS: The Range Based Measure for System loss for SALBP 3.4.1 Problem description Assembly lines involving human elements have a different pressing problem of system loss. It has been stated in the literature (see Ray Wild, 2004) that system loss is more important than balancing loss. Variations in the idle times in different work stations may lead to a behavioral problem. Also, there may be system loss due to stochastic nature of the time elements. Our objective in this current work is to design an assembly line where dual objectives of minimization of balancing loss and of system loss can be met. For this purpose, we first propose a range based measure

82


for system loss (RMS) and then install our optimization method through simulation approach. The procedure we propose here will be generic in nature and can be used with different types of balancing methods.

3.4.2

Notation K

number of jobs.

N

number of workstations.

Ti

task time or assembly time of ith job,

Lj

idle time of jth work station,

Nmin

minimum number of workstations.

C

cycle time.

Ct

trial cycle time.

Cmin

minimum cycle time.

S

slackness.

St

slackness for trial cycle time Ct.

R

range of idle times.

B M

i = 1, 2, …..K. j = 1, 2, …..N.

balancing loss. Range based Measure of System loss (RMS) = R/(minimum idle time)

83


3.4.3

Methodology We know, disruption in the system takes place in configurations

where one workstation has no idle time and another workstation has high idle time. In contrast, if the workstations have idle times of nearly equal length the chance of disruption will be less. We may then tend to consider the difference between maximum idle time and minimum idle time as a measure of the system loss with this belief that a system will be stable if the idle time of each workstation is more or less same. However, in the extreme case when there is no idle time for any of the work stations this difference will be zero but that situation will lead to high system loss. Thus, a minimum idle time is needed in the system. As the minimum idle time increases system loss decreases. Keeping these views in mind, system loss can be measured through range with lower value preferred over higher value and can also be measured through minimum idle time with higher value preferred over lower value. To make them unidirectional and suggest a unit free measure, we consider a combined range based measure of system loss as

84


Since cycle time is externally fixed for each workstation, idle time may significantly vary if we look at balancing loss only. So, our target will be to minimize the system loss due to uneven idle times for a particular set up that meets the balancing loss requirement. In place of sequentially minimizing balancing loss and system loss if one proposes to simultaneously minimize these dual objectives the solution set becomes unnecessarily large. It is easy to observe that minimization of balancing loss reduces the solution set to a manageable one. So, the problem will be easier to handle if we divide the dual objective problem into two stages, each stage having a single objective. Given a cycle, time one can arrive at the minimum number of workstations. Then, given this minimum number of workstation, we can generate a set of feasible solutions to line balancing problem each optimizing the balancing loss within a range of cycle time. Then, on each of these solutions, the condition of system loss can be applied and optimum solution to the overall problem can be obtained. By definition, the length of work time, or operating time, for which a component is available at each workstation is known as cycle

85


time ( C ). Now, given a choice of C, it may be noted that the theoretical minimum number of workstations, Nmin, must satisfy the following condition where Nmin is an integer : K K ∑Ti/C ≤ Nmin ≤ ∑Ti/C +1 i=1 i=1

(1)

It may not be always possible to attain Nmin. value due to precedence constraints. Then, to arrive at the first feasible solution number of workstation may be sequentially increased. The above inequations can be reformulated in terms of C if Nmin is given. From the left hand side of the inequality (1), we have, K ∑ Ti / C ≤ Nmin i=1 for which an equivalent condition on C is K C ≥ ∑ Ti / Nmin i=1

,

so that, the minimum value of C, Cmin, will be the highest integer contained in

86


K ∑ Ti / Nmin + 1 , i=1 i.e., Cmin =

K ∑ Ti / Nmin + 1 . i=1

The optimality of the assembly line in respect of balancing loss or the Nmin value will not be disturbed, if the choice of cycle time C varies from Cmin to C. Now, given a cycle time, C, one may conceptually start from Cmin and move up to C to arrive at the set of feasible workstation configurations with same balancing loss. In this process if Ct is the trial cycle time, one has to add a slackness St to each workstation to maintain the same cycle time C. This slackness St is calculated as St = C - Ct , where Ct is the trial cycle time satisfying the condition Cmin ≤ Ct ≤ C. This sequential approach can generate a reasonably good set of alternative solutions with the same balancing loss and forcefully induce similar idle times. From this solution set, we next calculate the range of idle time for each solution. Range is nothing but the difference between maximum and minimum idle time among workstations in a particular

87


configuration or set up. Using minimization of M = ( range / minimum idle time ) as a choice mechanism, the solution set can be reduced to an optimal one where both balancing and system losses will be least. We present an algorithm to make the suggested procedure computationally functional. This algorithm will meet the dual objective of minimizing both the balancing and range based measure of system losses for any assembly line balancing problem and will help us to deal with large and complex problems.

3.4.4 The Algorithm 1. Set cycle time C, determine the minimum number of workstations Nmin and

calculate the Cmin value.

2. Set the cycle time at Cmin. 3. Prepare List U which is the list of all uncovered tasks. From the tasks of List U with no immediate predecessor or whose immediate predecessors have been undertaken, prepare List R. These tasks are ready for selection and from the tasks of List R having task time / assembly time less than that of cycle time List A is prepared which are acceptable for immediate inclusion.

88


4. Select a task from the List A randomly and reset the cycle time as {Ct – assembly time}. 5. Repeat steps 3 – 4 if the cycle time is more than the elemental time. 6. If cycle time less than the elemental time then open a new work station and repeat steps 3 – 5. 7. Calculate the M value of idle times after getting the complete distribution of tasks to workstations. After each run, the current M value is compared with the previous least M value. If the new M value, is less than the previous least M value the new solution is stored as the basis for next comparison. Otherwise the new solution is discarded. 8. Increase the trial cycle time by one unit and repeat the entire exercise until it crosses C value. If C value is crossed go to step 10. 9. Repeat steps 3 to 8. Check whether all the work elements have been assigned to specified number of workstations. If not, increase the value of Nmin by 1, recalculate Cmin and go to step 2. 10. Print the solution. This is the best solution in terms of minimum balancing loss B, and minimum system loss, M.

89


We have converted the above algorithm into C language programme for our numerical study. It may be noted that by considering a large number of runs for each trial, one can reasonably arrive at the optimum solution whose M is minimum within a set of solutions with minimum balancing loss.

3.4.5 Worked Out Example For demonstration purpose, we consider a standard assembly line balancing problem, as presented in Ray Wild (2004). I

II 9

6 1

III

4

IV

V

VI

VII

VIII

IX

10

5

4 6

5

2 12

10

5 13

4 14

12 15

10 16

8

15

6 11

Figure 4.1 : Precedence diagram of work stations.

90

6 21

10 19

6 8

20

18

5 7

3

5

17

5

5

XI

9

5

2

X


There are 21 work elements in this example. These work elements are governed by certain precedence constraints, as shown in the above figure. In the precedence diagram circles shows work elements and figures against them show task times. The underlying task is to arrive at an optimum configuration of the workstations, given a cycle time of 35 time units. Then, for C=35, Nmin works out as Nmin = [ ∑ Ti / C + 1 ] = [ 143/35 +1] =5. For all these configurations, balancing loss will be least as any configuration with N=5 will ensure the same B value. One can obtain Cmin values in the following way to tackle the problem of system loss. Cmin = [ ∑ Ti / Nopt +1 ] = [ 148/5 ] = [ 29.6 ] = 29. We consider a trial cycle time Ct which should start from Cmin and proceed up to C = 35 for minimizing system loss. To ensure C = 35, we add to each work station a corresponding slack value St = 35 - Ct. Thus, our iteration starts from Ct =29, with a slackness, S29 = 35 29 = 6. For Ct=29, no solution is available even after 25,000 runs. We next consider Ct=30 with St=5. There also, no feasible solution exists.

91


This is consistent with our intuition that for very tight situations like Ct=29 and Ct=30, chance of generating a feasible solution is remote. We next consider Ct=31 with St=4. Here, we get 10 solutions in 25,000 runs. These solutions are described below.

01 02 03 04 05 06 07 08 09 10

n

Workstation 1 Elemen Idle ts time

Workstation 2 Elements Idle time

Workstation 3 Eleme Idle nts time

1,3,2,5, 8 1,3,8,2, 7 2,3,7,8, 1 2,3,1,7, 8, 2,3,7,8, 11 3,1,7,8, 11 3,7,8,1, 2 2,3,1,5, 8 3,7,8,1 1,2 1,2,3,8, 11

6,4,10,11 ,7,12 6,11,5,10 ,4,12 11,6,4,5, 10,12 5,11,6,10 ,12,13,14 1,4,6,5,1 0,12 2,6,5,10, 12,4 5,6,10,12 ,13,14,11 11,6,10,4 ,7,12 6,1,5,4,1 0,12 4,6,7,5,1 0,12

13,9,14 ,15 13,9,14 ,15 9,13,14 ,15 4,9,15

01 01 01 01 01 00 01 01 01 00

00 00 00 00 00 01 00 00 00 01

00 00 00 00

9,13,14 00 ,15 9,13,14 00 ,15 4,9,15 00 13,9,14 00 ,15 13,14,9 00 ,15 13,14,9 00 ,15

Workstation 4 Eleme- Idle nts tim e 16,19,1 01 7, 20 16,17,1 01 9, 20 16,18,1 01 7 16,18,1 01 7 16,19,1 01 7, 20 16,19,1 01 7, 20 16,19,1 01 7, 20 16,18,1 01 7 16,18,1 01 7 16,17,1 01 8

Workstation 5 ElemIdle ents tim e 18,21 10

10

2.50

18,21

10

10

2.50

20,19, 21 20,19, 21 18,21

10

10

2.50

10

10

2.50

10

10

2.50

18,21

10

10

2.50

18,21

10

10

2.50

19,20 21

10

10

2.50

19,20, 21 20,19, 21

10

10

2.50

10

10

2.50

Range

Sol

Table 3.18: Workstation-wise line balancing configurations with trial cycle time 31.

The minimum M value works out as 2.50. We next consider Ct=32 with St=3. For Ct=32 we get 7 solutions in 25,000 runs.

92

M


n

01 02 03 04 05 06 07

Workstation 2 Elemen Idle ts time


Workstation 4 Element Idle s time


1,2,6,5 ,10 2,6,1,5 ,10 2,6,1,5 ,10 1,2,6,5 ,10 2,1,5,6 ,10 2,6,3,7 ,1

07

07

15,16,1 7, 20 15,16,1 9 15,16,1 9 15,16,1 7, 20 15,16,1 7, 20 15,16,1 7, 20

00

19,18, 21

01

07

2.33

00

01

07

2.33

01

07

2.33

00

17,18, 20,21 18,17, 20,21 19,18, 21

01

07

2.33

00

18,19, 21

01

07

2.33

05

11,9,13 ,14 13,11,1 4,9 7,12,9, 13,14 9,7,12, 13,14 14,8,11 ,9 4,11,9

00

18,19, 21

01

07

2.33

2,1,6,3 ,5

04

3,7,12, 8,4 4,3,8,7, 1,2 3,8,11, 4 4,3,8,1 1 3,4,7,1 2, 13 5,10,12 ,13,8,1 4 7,10,12 ,13,14, 8

05

11,4,9

07

15,16,1 9

00

18,17, 20,21

01

07

2.33

07 07 07 07 04

02 02 03 03 03

07 06 06 06 07

00


Here, we get the minimum of M value as 2.33. We next consider Ct=33 with St=2. For Ct=33 we get 17 solutions as

93

Range

Sol


M


Workstation 2

Workstation 3

n

10 11 12 13 14 15 16 17

1,3,2,6 ,8 2,6,1,3 ,8 1,2,4,6 ,5 1,2,6,4 ,5 2,1,4,5 ,6 2,6,3,8 ,1 1,3,4,8 2,1,4,6 ,5

02

19,20, 21 18,21

05

1,4,9,5

03

04

04

04

9,2,5,6, 10 5,10,4,9

04

4,5,7,9

04

04

3,7,8,9

04

04

10,9,3,8

04

04

10,9,3,4

04

04

4,5,7,9

04

04

7,2,5,9,6

04

04

3,7,10, 12,13,14

04

16,18, 17 16,19, 17, 20 16,18, 17 16,17, 19, 20 16,19,1 7, 20 16,19,1 7, 20 15,16, 17, 20 15,16, 19 15,16, 19 15,16, 17, 20 15,16, 19 15,16, 17, 20 15,16, 19 15,16, 17, 20 15,16, 17, 20 15,16, 17, 20 15,16, 19

03

03

9,13,14, 15 13,14,9, 15 9,13,14, 15 9,13,14, 15 13,14,9, 15 9,13,14, 15 11,10,12, 13,14 11,10,12, 13,14 11,7,12,1 3,14 11,7,12, 13,14 11,10,12, 13,14 11,10,12, 13,14 11,7,12, 13,14 7,11,12, 13,14 10,11,12, 13,14 10,12,11, 13,14 9,8,11

02

05

6,11,4,5, 10,12 6,5,7,4, 10,12 6,1,5,10, 12,4 4,11,5,6, 10,12 6,4,7,11, 10,12 7,4,6,5, 10,12 1,5,4,9

03 02 03 03 03 02

03 02 02 02 03

04

02 02 02 02 02 11 11 11 11 11 11 11 11 11 11 11

03 03

M

12

10

2.50

12

10

2.50

Idle time

09

Elements

08

Idle time

07

Elements

06

Idle time

05

Elements

04

Idle time

03

Workstation 5

Elements

02

2,3,1,8 ,7 3,2,1,8 ,11 3,7,8,2 ,11 3,1,7,8 ,2 1,2,5,3 ,8 3,8,1,2 ,11 2,6,3,8 ,7 3,7,2,6 ,8 3,8,1,4

Idle time

01

Elements

Sol

Workstation 4

Range

Workstation 1

12

10

2.50

03

20,19, 21 18,21

12

10

2.50

03

18,21

12

10

2.50

03

18,21

12

10

2.50

01

18,19, 21 18,17, 20,21 18,17, 20,21 19,18, 21 18,17, 20,21 19,18, 21 18,17, 20,21 18,19, 21 19,18, 21 19,18, 21 18,17, 20,21

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

02

10

3.33

01 01 01 01 01 01 01 01 01 01


94


Here, minimum of M works out as 2.50. We next consider Ct=34 with St=1. For Ct=34 we get 8 solutions in 25,000 runs. These solutions are described below: Workstation 1 Sol

01 02 03 04 05 06 07 08

n

Elements

Workstation 2

Idle Elements time 3,7,8,1,11 03 2,6,5,4,10, 12 3,7,8,11,1 03 2,5,6,10,4, 12 3,2,8,1,11 03 7,5,6,4,10, 12 2,3,8,11,1 03 6,5,7,10, 12,13,14 3,2,8,11,1 03 7,5,4,6,10, 12 3,1,8,11,2 03 6,7,5,10, 12,13,14 3,8,11,1,7 03 2,5,6,10, 12,13,14 3,7,8,1,11 03 2,6,5,10, 12,13,14

Workstation 3

Workstation 4

Workstation 5

Idle Elements Idle Elements time time 04 9,13,14,15 03 16,19,17, 20 04 13,9,14,15 03 16,18,17

Idle Elements Idle time time 04 18,21 13 10

2.50

04

04

13,9,14,15 03

16,18,17

04

04

4,9,15

04

13,9,14,15 03

04

4,9,15

03

16,17, 04 20, 19 16,19,17, 04 20 16,17,18 04

04

4,9,15

03

04

4,9,15

03

03

16,19,17, 04 20 16,18,17 04

19,20, 21 19,20, 21 18,21

13

10

2.50

13

10

2.50

13

10

2.50

18,21

13

10

2.50

20,19, 21 18,21

13

10

2.50

13

10

2.50

20,19, 21

13

10

2.50


Here, the minimum value of M is 2.50. We next consider Ct=35 with St=0. For Ct=35 we get 2 solutions in 25,000 runs. These solutions are described below:

95

Range M

In Search of Optimum Assembly Line Balancing Chapter 3: Solution of ALBP by Heuristic Methods Workstation 1 Elements Idle time





Ran ge

M

01

2,1,6,4,3

03

7,9,8,11,5

03

18,21

14

11

3.33

2,1,6,3,4

03

8,11,5,9, 10

03

16,19,17, 20 16,17,19, 20

05

02

10,12,13, 14,15 7,12,13, 14,15

05

18,21

14

11

3.33

Sol

n

07 07


Here, the minimum value of M is 3.67. As stated earlier, our objective is to globally minimize the M value based on all such solutions. So the optimum configuration comes out as





01

1,2,6,5,10

07

3,7,12,8,4

02

11,9,13,14

07

02

2,6,1,5,10

07

4,3,8,7,12

02

13,11,14,9

07

15,16,17, 20 15,16,19

00

03

2,6,1,5,10

07

3,8,11,4

03

06

15,16,19

00

04

1,2,6,5,10

07

4,3,8,11

03

06

2,1,5,6,10

07

03

06

06

2,6,3,7,1

04

05

4,11,9

07

07

2,1,6,3,5

04

3,4,7,12, 13 5,10,12,13 ,8,14 7,10,12,13 ,14,8

05

11,4,9

07

15,16,17, 20 15,16,17, 20 15,16,17, 20 15,16,19

00

05

7,12,9,13, 14 9,7,12,13, 14 14,8,11,9

19,18, 21 17,18, 20,21 18,17, 20,21 19,18, 21 18,19, 21 18,19, 21 18,17, 20,21

00

00 00 00

Table 3.23: Workstation-wise line balancing final configurations.

for cycle time C = 35 and trial cycle time Ct=32 with slack St=3. Above example shows that the proposed approach has led to simultaneous minimization of balancing and system losses. It may be noted that global minimum value of M is obtained when flexibility in the system is moderate and slackness is also moderate. Further, the M value

96

Range

Soln


M

01

07

2.33

01

07

2.33

01

07

2.33

01

07

2.33

01

07

2.33

01

07

2.33

01

07

2.33


increases in the extreme situations. Compared to the standard simulation approach proposed in COMSOAL and modified COMSOAL methods, our proposed method is better because it generates more feasible solutions by using the concept of varying cycle time and then adding slack time. This forcefully induces more uniformity among the idle times because a fixed slack value is added to each workstation.

3.4.6 Conclusion The objective of this work was to develop an efficient solution for minimizing system loss and balancing loss of any Assembly Line Balancing Problem. For this purpose we have proposed a new measure for system loss, RMS, a Range based Measure of System loss. This solution aims to achieve a balanced distribution of work between different work stations. We have introduced the concept of trial cycle time, slack time in addition to cycle time so that more sets of feasible solutions can be obtained. Our approach is a generic one, which is capable of solving different line assembly problem with a reasonable computation time.

97


From the trial solution sets, final choice is made based on optimum number of workstations and RMS value. In our approach we first consider minimization of balancing loss and then minimization of system loss. In this process we have reduced the simultaneous optimization problem to a sequential optimization problem. This has been done to keep the number of solution to a manageable form. This approach according to numerical study is giving a better set of configurations because we are using some amount of slackness in each workstation with trial cycle time varying from Cmin to C.

3.5 Comparative study between VBMS and RBMS The two measures of system loss, we proposed herein, have different importance and applications.

Variance based measure examines the

overall state of control on distribution of tasks to work stations. On the other hand Range Based Measure for System loss has a complete control over the distribution of tasks to work stations, as we are emphasizing to restrict the distribution within a certain upper and lower

98


bound taking care of the scatter components and thus, gives a better solution. In variance based measure, we have to calculate variance for each setup. Complexity is high here and that increases computational time of the program. On the other side, range based measure is better to install as it takes relatively less computational time as well as gives better solution. If we examine the variance of the range based solutions we will get much better result of least variance. But if we calculate variance first and then go for range then we will not obtain as good result as the previous case. So range based measure is the better one and beat measure will be first generate the solution set by range based measure and then take the final configuration which has least variance value.

99

Chapter 4 Assembly Line Balancing by Linear Programming Approach

4.1 Introduction The need for high volume and low cost production has resulted in replacement of traditional production methods by Assembly Lines. The general line balancing problem is a difficult optimization problem where

100

In Search of Optimum Assembly Line Balancing Chapter 4: ALB by LP approach

constraints are not restricted to precedence constraints only. There may be problems of divisibility of a work element. Also, zonal constraints may be operative. To solve a line balancing problem, the approach followed in OR is to simplify the same by bringing down the level of complexity to a solvable state. The classic OR definition of the line balancing problem, given by Becker and Scholl (2006), is as follows: Given a set of tasks of various durations, a set of precedence constraints among the tasks, and a set of workstations, assign each task to exactly one workstation in such a way that no precedence constraint is violated and the assignment is optimal. This simplified problem of optimization can be classified mainly into two categories. (1)

The number of workstation is to be minimized for a given cycle

time. This cycle time can not be exceeded by the total task time of work elements assigned to any of the workstations. (2)

The cycle time is to be minimized given the number of

workstation. This cycle time is equal to the largest total task time of the work elements assigned to the workstations.

101


It is by far these two variants of line balancing that have been widely researched. Attempts to solve these optimization problems started during 1950s. That time, the focus of attention was on the core problem of configuration, which is the assignment of tasks to stations. Bowman (1960) was the first to consider the linear programming approach to arrive at an optimum solution to the line balancing problem. During the last few decades, several researchers handled this problem of line balancing by using different optimization techniques. Hoffman (1963), Mansoor and Yadin (1971) and Geoffrion (1976) used mathematical programming to present a clear formulation of the problem and solve the same. Later, Van Assche and Herroelen (1979) have proposed an optimal procedure for the single-model deterministic assembly line balancing problem.

Integer programming procedure was used by Graves and

Lamer (1983) for designing an assembly system. Infact, in the mid 80’s some researchers gave emphasis on application part like application of operational research models and techniques in flexible manufacturing systems (Kusiak,1986) and application of a hierarchical approach for solving

machine

grouping

and

loading

problems

of

flexible

manufacturing systems (see Stecke, 1986). Berger et al (1992) adopted

102


Branch-and-bound algorithms for the multi-product assembly line balancing problem. In 1998, Pinnoi and Wilhelm dealt with the problem of system design using the branch and cut approach. Nicosia et al (2002) introduced the concept of cost and studied the problem of assigning operations to an ordered sequence of non-identical workstations, which also took into consideration the precedence relationships and cycle time restrictions. The purpose was to minimize the cost of the assignment by using a dynamic programming algorithm. They also introduced several rules to reduce the number of states in the dynamic programming formulation. In 2006, Bukchin and Rabinowitch proposed a branch-andbound based solution for the mixed-model assembly line-balancing problem for minimizing stations and task duplication costs.

Problem description The balancing problems studied in all the above mentioned methods are oriented towards minimization of either balancing loss or cost of assignment. These methods can be best used in transfer lines where work elements are preferably performed by machines. Assembly lines involving human elements have another pressing problem. “The losses

103


resulting from workers’ variable operation times” is known as System loss (see Ray Wild, 2004) which is more important than balancing loss. Our objective in this current work is to design an assembly line where dual objectives of minimization of balancing loss and system loss can be met. For this purpose, we first propose a measure for system loss (MSL) and then install an optimization method through Linear Programming approach.

4.2 Notation

104

K

number of jobs

N

number of workstations

ti

task time or assembly time of ith job

Wj

jth workstation

a(i,j)

binary measure for assignment of task i to workstation j

Lj

idle time of jth work station

Nmin

minimum number of workstation for a given cycle time

C

cycle time

Ct

trial cycle time

Cmin

minimum cycle time for a given K


St

slackness for trial cycle time Ct, i.e., St = C - Ct

B

balancing loss, i.e., {(NC - ∑ Ti) / NC}*100%

R

range of idle times L1, L2, ….., LN

M

measure of system loss (MSL) = R / minimum idle time

4.3 Methodology Balancing loss occurs due to the uneven allocation of work to station. Mostly, one uses the concept of balancing loss, B to examine the efficiency of an assembly line. Our proposed work is a multi-objective one where minimization of balancing loss is to be addressed along with system loss. According to Ray Wild (2004) this System loss arises out of workers’ variable operation time. However, no standard measure has been proposed so far on the system loss. We propose to consider the difference between maximum idle time and minimum idle time as a measure of the system loss with this expectation that a system will be stable if the idle time of each workstation is more or less same. However, in the extreme case when there is no idle time for any of the work stations this difference will be zero. But that situation will lead to high system

105


loss. Thus, a minimum idle time is needed in the system and if the minimum idle time increases then the system loss decreases. Keeping these two issues in mind, system loss can be measured through range with lower value preferred over higher value and can also be measured through minimum idle time with higher value preferred over lower value. To make them unidirectional and suggest a unit free measure, we consider a combined range based measure of system loss as M = range / minimum idle time. Given a choice of C, it may be noted that the theoretical minimum number of workstations, Nmin, must satisfy the following constraints: K K ∑Ti/C ≤ Nmin ≤ ∑Ti/C +1, i=1 i=1 from where we arrive at Cmin, the minimum value of C with the same balancing loss, as K Cmin = ∑ Ti / Nmin + 1 . i=1 So, for a given a cycle time, C, one may conceptually start from a trial cycle time, Ct which satisfies the condition Cmin ≤ Ct ≤ C. In that case each workstation can be provided with at least a minimum slack time St. Then with Ct as the trial cycle time one can arrive at the set of

106


optimum workstation configurations and maintain the targeted cycle time C by uniformly adding to each workstation a slackness St to Ct. For optimum configuration of each trial cycle time, we get a value of the measure M. The configuration that gives minimum value of M is the final optimal solution. We next develop a mathematical formulation of the problem for arriving at the minimum M value.

4.4 Mathematical Formulation Since our aim is to minimize M, the ratio of range of idle times and minimum idle time, we propose to minimize {p – q}/{C – Ct + q} where p is the maximum idle time, q is the minimum idle time under trial cycle time Ct. Thus the objective function is z = {p – q}/{C – Ct + q} and our objective is to minimize z. Let us consider the binary variable a (i, j ) such that

a (i, j )

=

1

if i ∈ Wj

i th task is assigned to Wj,

0

if i ∉ Wj

i th task is assigned to Wj,

and is true for i = 1, 2, ….., K, j = 1, 2, ….., N.

107


Then, under the condition that the ith task can be assigned to only one workstation N

∑

a (i, j)

j =1

=1

must hold for all i = 1, 2, ….., K.

Also, according to precedence constraints if task i′ is to be assigned before assigning task i, that is i ′ < i , then j

a (i, j ) ≤

∑ a (i ′, r ) r =1

∀

i′ < i

Under the condition that p is the maximum idle time for trial cycle time Ct, we have, K ⎡ ⎤ p ≥ ⎢C t − ∑ t i a (i , j ) ⎥ i =1 ⎣ ⎦

j = 1, 2, ….., N.

Similarly, for q as the minimum idle time for trial cycle time Ct, K ⎡ ⎤ ⎢C t − ∑ t i a (i, j ) ⎥ ≥ q i =1 ⎣ ⎦

j = 1, 2, ….., N.

Thus, an integer programming formulation of the optimization problem can be written as: Minimize z = {p – q}/{C – Ct + q} Subject to constraints,

108


(i)

N

∑ a(i, j )

∀ i

= 1.

j =1

(ii)

a (i , j ) ≤

j

∑ a(i′, r )

∀ i′ < i

r =1

(iii)

⎡

K

⎤

⎣

i =1

⎦

p ≥ ⎢C t − ∑ t i a (i, j )⎥

(iv)

K ⎡ C − ⎢ t ∑ t i a(i, i =1 ⎣

(v)

p≤C

(vi) (vii) (viii)

⎤ j )⎥ ≥ q ⎦

q≥ 0 a (i, j ) = 0,1

∀ i, j

Ct = Cmin, Cmin +1, ……,C

4.5 Worked Out Example To explain how the proposed algorithm works, we consider in Figure 4.1 an assembly line balancing problem from Ray Wild (2004). A figure within a circle represents task number and that close to a circle represents corresponding task time. Precedence constraints are represented by arrows.

109


9

6 1

4

10 9 5

5

5 5

5 2

17

5

4 6

10

2 12

5 13

4 14

12 15

10 16

15 18

8

19

6 8

6 21

10

5 7

3

20

6 11

Figure 4.1: Precedence diagram of workstations along with the task times.

This problem can be summarized in a tabular form in terms of the binary variables a(i,j)s and is given in the Table 4.1.

110


Work Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Work Station Activity Immediate Time Predecessor 1 2 3 4 6 a(1,1) a(1,2) a(1,3) a(1,4) 5 a(2,1) a(2,2) a(2,3) a(2,4) 8 a(3,1) a(3,2) a(3,3) a(3,4) 9 1 a(4,1) a(4,2) a(4,3) a(4,4) 5 1, 2 a(5,1) a(5,2) a(5,3) a(5,4) 4 2 a(6,1) a(6,2) a(6,3) a(6,4) 5 3 a(7,1) a(7,2) a(7,3) a(7,4) 6 3 a(8,1) a(8,2) a(8,3) a(8,4) 10 4 a(9,1) a(9,2) a(9,3) a(9,4) 5 5, 6 a(10,1) a(10,2) a(10,3) a(10,4) 6 8 a(11,1) a(11,2) a(11,3) a(11,4) 2 10, 7 a(12,1) a(12,2) a(12,3) a(12,4) 5 12 a(13,1) a(13,2) a(13,3) a(13,4) 4 13 a(14,1) a(14,2) a(14,3) a(14,4) 12 9, 11, 14 a(15,1) a(15,2) a(15,3) a(15,4) 10 15 a(16,1) a(16,2) a(16,3) a(16,4) 5 16 a(17,1) a(17,2) a(17,3) a(17,4) 15 16 a(18,1) a(18,2) a(18,3) a(18,4) 10 16 a(19,1) a(19,2) a(19,3) a(19,4) 5 17 a(20,1) a(20,2) a(20,3) a(20,4) 6 18, 19, 20 a(21,1) a(21,2) a(21,3) a(21,4) Table 4.1: Precedence relation and task times of work elements.

Let us consider for our study a cycle time of 35(=C) time units. This results in Nmin = 5 and Cmin = 29 and 29 ≤ Ct ≤ 35 with St = 35 - Ct. The

111

5 a(1,5) a(2,5) a(3,5) a(4,5) a(5,5) a(6,5) a(7,5) a(8,5) a(9,5) a(10,5) a(11,5) a(12,5) a(13,5) a(14,5) a(15,5) a(16,5) a(17,5) a(18,5) a(19,5) a(20,5) a(21,5)


optimum configuration is presented in Table 5.2 where M value is 1.5 and balancing loss is 18.3%.

C

Work Station 1

Work Station 2

Work Station 3

Work Station 4

Work Station 5

M

31

1,2,3,5,8

6,7,10,11,12,13 ,14

4,9,15

16,17,19

18,20,21

1.5

Table 4.2: Final Optimum Configuration

4.6

Conclusion We have presented a mathematical programming approach for

balancing an assembly line with twin objectives of minimization of balancing loss and system loss. From the solution set generated by the optimization technique, final choice is made based on optimum number of workstations and minimum value of a measure of system loss, proposed herein. This study is giving an ideal set of configurations because we can ensure some amount of slackness (St) in each workstation when the trail cycle time is less than the targeted cycle time.

112

Chapter 5 Assembly Line Balancing by Stochastic Programming Approach

5.1 Introduction Minimization of balancing loss or cost of assignment was the only important consideration for the previous researchers. These methods can be best used in the case of transfer lines because in a transfer line

113

In Search of Optimum Assembly Line Balancing Chapter 5: ALB by Stochastic programming approach

elements are preferably performed by machines. Assembly lines involving human elements have another pressing problem. “The losses resulting from workers’ variable operation times” is known as System loss (see Ray Wild, 2004) and this loss is perhaps more important than the losses resulting from uneven allocation of work elements to workstations. Consequently, the problem of line design is not only the equal division of work among the stations or the adaptation of tasks to the speed of the workers but also to provide some amount of slackness in each workstation to take care of the variability of the elemental times. Our objective in this current work is to design an assembly line where dual objectives of minimization of balancing loss and system loss can be met by switching over from the domain of deterministic set-up to the domain of stochastic set-up. We propose an optimization method based on stochastic programming approach for that purpose.

5.2 Notation

114

E(.)

statistical expectation operator

K

number of jobs


N


N(µ, 2) normal distribution with mean µ and variance

2

expected task time of ith job variance task time of ith job the upper

point of N(0,1)

ti

random task time or assembly time of ith job

Wj

jth workstation

a(i,j)

binary measure taking value 1 for assignment of task i to workstation j

115

Lj

variable idle time of jth work station

Nmin


C

cycle time

Ct

trial cycle time

Cmin


St


B

balancing loss

V

variance of idle times, L1, L2,.., LN.


5.3 Methodology The main cause of balancing loss, as pointed out earlier, is the uneven allocation of work to different workstations. Generally, to examine the efficiency of an assembly line one uses the concept of balancing loss, B. We propose a completely different approach. Our proposed work is a multi-objective one. We have taken into consideration system loss as well as balancing loss. This System loss arises out of workers’ variable operation time (Ray Wild, 2004). But no standard measure has been proposed so far to examine the extent of system loss. We propose to consider a measure for system loss. As we know, system loss arises out of workers’ variable operation time, so any configuration where one workstation has high idle time and another workstation has no idle time will create high disruption. In the deterministic set up that will lead us to consider the variance of the idle times (V) as a measure for system loss for the system. The stability of the total system will be maximum when this variance will be minimum (Roy and Khan, 2010). Under stochastic task times the objective of our proposed method is minimization of

116


expected values of B and V, subject to precedence constraints and chance constraints in terms of cycle time.

5.4 Mathematical Formulation Let us consider the binary variable a (i, j ) such that a (i , j ) =

1

if i ∈ Wj

0

if i ∉ Wj

i th task is assigned to Wj, i

th

task

is

assigned to Wj, and is true for i = 1, 2, ….., K, j = 1, 2, ….., N. Then, under the condition that the ith task can be assigned to only one workstation, the following condition must hold for all i = 1, 2, ….., K. ∑

,

=1

(1)

according to precedence constraints if task i′ is to be

Further,

assigned before assigning task i, that is i ′ p i , then ,

≤ ∑

′

,

∀ i′ p i

(2)

Human beings are involved in completion of tasks involved in assembly line. So, depending upon variations in human skills and behavior, the task

117


time of each job will become a random variable. Therefore, we should consider both expected time for completion of each job and the extent of variability. Let µ be the expected time for completing ith job. Then, the expected balancing loss of the system should be ∑

100% ,

∑

100%. (3) Or, At the same time, the measure of system loss should be calculated taking the expectation of the variance of random idle times, i.e. E(V). By definition, ∑ ∑

∑ ∑

Or,

∑

, ,

Hence the expectation of V can be simplified as ∑ = ∑

∑ ,

118

∑ ∑′

, ′

′

,

′

,

∑

In Search of Optimum Assembly Line Balancing Chapter 5: ALB by Stochastic programming approach ∑

=

∑

∑′

′

′

,

′

,

∑

∑′

′

′

,

′

,

∑

, ∑

=

∑

∑

=

∑

,

,

∑

,

∑

,

,

,

Thus,

E(V)=

∑

∑

,

+

∑

∑

, (4)

Since the task times are random variables, the condition for completion of

tasks in a workstation within the assigned cycle time can be best described in terms of chance constraints . ∑

1

,

Equivalently it can be expressed as .

119

0

1

,

where 0

1,

j = 1,2,….,N.


.

or, 1

1-

Φ

Φ

or, Φ

Φ

So, (5) ∑

But,

,

∑

=

,

∑

=

,

(6)

and, Var(

∑

∑

, ∑

,

∑ =∑ =∑

120

,

, , ,

(7)


Now with the help of equation (6) and equation (7), equation (5) can be rewritten as,

Or,

∑

,

Or,

∑

,

Or,

∑

,

,

∑

(8)

Thus, chance constraints regarding cycle time can be reduced to the following deterministic constraints ∑

,

∑

,

(9)

Finally, combining (1), (2), (3), (4) and (9) a deterministic problem for stochastic model formulation of the optimization problem can be written as: Minimize E(B)

121

∑


Minimize E(V) = ∑ ∑

∑

∑

,

+

,

Subject to constraints, (i)

∑

,

1

(ii)

,

∑

′

(iii)

a (i, j ) = 0,1

(iv)

∑

,

∀ i ∀ i′ p i

,

∀ i, j

∑

+

,

∀ j

To assign equal importance to each workstation we consider

(10) =

∀ j.

Further

may be considered 0.05 for which

= 1.6449. One way of

dealing with dual objective is to combine them with weights or priorities. In that case the reduced objective can be written as: Minimize ∑

Z ∑

122

∑

∑ ,

,

∑

,


,

where 0 ≤

1,

1. However, we prefer to

sequentially undertake the task of minimization by generating in the first instant feasible solutions under the objective of minimization of E(B) 1

under

0and then obtaining the final solution by

imposing the second objective of minimization of E(V) with 1. To generate the set of feasible solutions we consider a

0

sequential approach of assigning trial cycle time and resulting in slack time, to be assigned to each workstation meeting the optimality condition arising out of the first objective of (10).

5.5 The Algorithm 1. Calculate the theoretical minimum number of workstations, Nmin, following the formula ∑

µ

∑

µ

1

2. Calculate minimum cycle time, Cmin, using the relation, ∑

µ

1

3. Set the cycle time at Cmin.

123


4. Make an attempt to get feasible solution following the algorithm of Roy and Khan (2010) with usual cycle time constraints replaced by (10)(iv). 5. If no feasible solution is obtained then increase Nmin by 1 and go to step 3.

6. Within the generated set calculate E(V) for each set and save the E(V) value. 7. Compare the E(V) with the previous value of E(V). If the current E(V) is lower than the previous one then save the current value of E(V). 8. When all the feasible sets are over we get the final solution to the optimization problem.

5.6 Worked Out Example We consider in Figure 5.1 an assembly line balancing problem from Ray Wild (2004) for the purpose of explaining how the proposed model works. Task number is represented by the figure within a circle.

124


1

4

9

5

2

17

6

10

12

13

14

15

7

3

16

18

20

21

19

8

11

Figure 5.1 : Precedence diagram of workstations.

This problem is summarized in a tabular form in terms of work elements, immediate predecessor(s), expected task durations and their variability is given in Table 5.1. For this particular problem, following the algorithm, we first get minimum number of workstation with cycle time C = 35. Nmin works out as 5. So, minimum trial cycle out as

∑

1 , i.e.

comes

29. Since Cycle time C is

35, the trial cycle time starts with 29 and goes upto 35. But among the initial trial cycle times as 29, 30, 31, 32, 33, 34, 35 we get the first feasible solution at Ct = 33.Thus, our trial cycle times of 33, 34 and 35. The final optimum configuration has been obtained from trial cycle time

125


as 34 with slackness

1. This optimum configuration is

presented in Table 5.2.

Expected Activity Variance of Time activity time 1 6 0.09 2 5 0.0625 3 8 0.16 4 1 9 0.2025 5 1, 2 5 0.0625 6 2 4 0.04 7 3 5 0.0625 8 3 6 0.09 9 4 10 0.25 10 5, 6 5 0.0625 11 8 6 0.09 12 10, 7 2 0.01 13 12 5 0.0625 14 13 4 0.04 15 9, 11, 14 12 0.36 16 15 10 0.25 17 16 5 0.0625 18 16 15 0.5625 19 16 10 0.25 20 17 5 0.0625 21 18, 19, 20 6 0.09 Table 5.1 : Precedence relation and task times of work elements. Work Element

C

Work Station 1

34 2, 3, 6, 7, 8

Immediate Predecessor

Work Station 2 1, 4, 5, 11

Work Work Station 3 Station 4 9, 10, 12, 15, 16, 19 13, 14

Table 5.2 : Final Optimum Configuration

126

Work E(V) Station 5 17, 18, 20, 6.70 21 7


This optimum configuration speaks of 5 workstations with work elements 2, 3, 6, 7, 8 assigned to workstation 1, work elements 1, 4, 5, 11 assigned to workstation 2, work elements 9, 10, 12, 13, 14 assigned to workstation 3, in workstation 4 work elements 15, 16, 19 are assigned and work elements 17, 18, 20, 21 assigned to workstation 5 for trial cycle time 34. Finally, the optimum value of E(V) comes out as 6.707.

5.7

Conclusion

A mathematical programming approach is presented here for balancing an assembly line with twin objectives of minimization of balancing loss and system loss. As system loss arises out of variations in human behavioral, stochastic setup is needed for describing the situation, representing of the problem and arriving at the optimum solution of the same. Reduction of the stochastic setup into deterministic constraints has been indicated under normality assumption. A sequential approach has been installed to arrive at the final solution. Our approach being a generic one, it is capable

127


of solving different line assembly problem with reasonable computation time. Final choice can be made based on optimum number of workstations and minimum value of expected variance of the idle times, proposed herein as a measure of system loss.

128

Chapter 6 Integrated model for Line Balancing with Workstation Inventory Management

6.1 Introduction Integrated line balancing involves a network of interconnected activities for the ultimate provision of products and service packages required by the end customers. The domain of integrated line balancing covers all movements and storage of raw materials, inventory of work-in-progress

129

In Search of Optimum Assembly Line Balancing Chapter 6: Integrated LB with WS Inventory.

and finished goods from point-of-origin to point-of-consumption. Basically, integrated line balancing is the planning, organizing and controlling of sourcing, procurement, conversion and logistic activities. It is used to characterize all the inter-related components and processes required to ensure that the right amount of product is in the right locations at the right time at the lowest possible cost. In the recent past, integrated line balancing has become a very important part of any business activity. This importance is going to increase further due to growing uncertainty in the global business environment and cost minimization has again become the bull’s eye. However, the form of integrated line balancing depends on the type of industry under study. Industries can be broadly divided into two types on the basis of their final offers. One is the manufacturing sector and other one is the service sector. So far as manufacturing sector is concerned, there exists a very close relationship between the increase in productivity and increase in resultant profit. Since integrated line balancing is a management technique through which the better quality products are delivered to the customers at lower costs by making a balance between the inbound and the outbound logistics, this increases productivity, improves quality and adds to profit. For most of the manufacturing sector,

130


a line of production is located between inbound and outbound logistics. Therefore, this line of production plays an important role in the overall integrated line balancing process. The details may differ, but the basic flow line principles remain the same. Items are processed as they pass through a series of workstations along a production line, i.e., different tasks are performed in different workstations. One or more tasks can be performed in a workstation but the activity time for each workstation should be near about equal so that the line will be balanced. In a nutshell, the problem of line balancing deals with the distribution of activities among the workstations so that there will be maximum utilization of human resources and facilities without disturbing the work sequence. But this problem should not be addressed in isolation. This is because in each workstation supply of materials and planning of inventory play important roles for ensuring smooth functioning of the assembly line. So, the excellence in integrated line balancing can only be achieved by undertaking a system approach and by simultaneously minimizing the total inventory cost and balancing the line of production. Better coordination of different links that connect source with the destination can create competitive advantage for a company. Infact, along with the company, its suppliers, its channel members and its customers can all

131


benefit from better coordination and usage of such linkages. For example, if we consider the case of modern rice milling facilities and if we examine the conversion area, we can classify the activities involved with the conversion process into four different groups such as steeping, parboil, dryer and hulling. Though, each group of activity may consist of one or more activities, those four groups are performed in four separate areas. For each separate area, we may consider sub-operative areas forming a network of workstations and a network of stock chambers. Therefore, to maximize the efficiency of the entire chain of operation, balancing of the assembly line is needed. And for each workstation, configuration of separate stock chamber is also needed to feed the respective workstation. For these stock chambers, the need for inventory planning, which is one of the major decision areas in integrated line balancing can be highly felt. Since we have to perform all these sub-activities under an overall system activity, the approach should be an integrated system approach where the inventory decision of integrated line balancing and balancing decision of assembly line will be considered jointly as one set of decisions. To study the integrated problem of supply chain, there is a need to link line balancing and customers’ rate of demand with the entire supply chain. This linking has an important role towards achieving excellence in

132


the complete task of moving from source to destination. Our objective in this current work is to design the integrated model for line balancing with workstation inventory where, given the rate of customers’ demand, the total of inventory cost and the cost of balancing loss of the assembly line will be jointly minimized so that the entire optimization approach can be a holistic one.

6.2 Notation K

number of jobs

N


ti

task time or assembly time of ith job

Wj

jth workstation

a(i, j )

assignment variable taking value 1 if task i is assigned to workstation j and taking value 0, otherwise

133

Lj

idle time of jth work station

C

cycle time

Ct

trial cycle time


Cmin

minimum cycle time

RL

lowest rate of market demand

RH

highest rate of market demand

Qj

ordered quantity for materials to be used in workstation j for workelement

Co

ordering cost of materials

CHi

holding cost of materials for ith job

WHj

total cost of holding of materials for jth workstation

M

constant multiplying factor, to be determined from average cost of one man-hour

6.3 Methodology and Mathematical Formulation To formulate the integrated problem of cost minimization, we like to split the same into three interrelated parts, viz. determination of customers’ rate of demand, deciding about the stock of materials for consumption during production activity and balancing of the workstations. At the end, we propose to rejoin their part-wise measures on a common scale and undertake the joint optimization task.

134


For us, on one hand the assembly line will be balanced, when the idle time in each work station is in minimum level and in turn, the cost of the production will be minimized when the balancing loss is minimized. The measure of balancing loss of an assembly line is defined as the loss resulting from allocation of work elements to workstations and is given by (see Ray Wild, 2004) K ⎫ ⎧ B = ⎨( NC − ∑ t i ) / NC ⎬.100% , i =1 ⎭ ⎩

where the numerator of which indicates the idle time per C unit time of work. The corresponding cost per unit time can be expressed as, K

( NC −

∑t i =1

C

i

) M

(1)

Inventory cost that mainly deals with ordering cost, holding cost and the shortage cost is minimized under the plan of optimum ordering quantity. If we do not permit any shortage that badly affects the total assembly line, the total cost of the supply chain for our purpose will be the sum total of ordering cost, average holding cost and the cost of balancing loss to be observed per unit time. To link the entire system with the rate of customers’ demand we consider the inverse of the same to arrive at the

135


cycle time of operation. Due to variations in the market demand it is preferable to express the same in terms of lower and upper bounds, RL and RH respectively. Then the cycle time must lie in the interval [1/RH, 1/RL]. Now, given a cycle time C, demand, Dj, for the materials may be expressed as Dj =

K

∑ a(i, j ) /

C under the assumption that each work

i =1

element has one unit of consumption. Holding cost for a particular K

workstation j will be WHj = ∑ (a (i, j ) * CH i ) , under a similar assumption i =1

and argument. With the help of our proposed notation, we can express the ordering cost for the jth workstation as, K

∑ a (i , j ) i =1

Qj

*

Co , C

Therefore, the total ordering cost for the N stock chambers of the Nworkstation line balancing system is

⎡K ⎤ ( , ) a i j ∑ ⎢ N Co ⎥ i =1 ⎢ * ⎥ ∑ Q C⎥ j =1 ⎢ j ⎢⎣ ⎥⎦

136

(2)


Average

holding

cost

for

the

jth workstation

will

be

K

Q j ∑ (a (i, j ) * CH i ) / 2 . Hence, the total average holding cost for the i =1

system is as follows, (3)

⎡ K ⎤ ∑ ⎢Q j ∑ (a(i, j ) * CH i )⎥ / 2 j =1 ⎣ i =1 ⎦ N

⎡

K

⎤

⎣

i =1

⎦

To this end we add the cost of balancing loss which is ⎢ NC − ∑ t i ⎥ * M / C. The cycle time C, being linked with the average rate of demand of the customers, may be determined in terms of an interval corresponding to interval estimator of average rate of demand of the customers. Thus, C varies between two points Cmin = 1/RH and Cmax = 1/RL and the optimum C along with optimum Qj values will be determined from the total cost of the plan. Thus, our objective is to minimize total cost of the supply chain including N stock chambers and N workstations where the integrated objective function, as obtained from (1), (2) and (3), is

⎡K ⎤ a ( i , j ) ∑ ⎢ N K Co ⎥ N ⎡1 K ⎤ ⎤ ⎡ i=1 Z = ∑⎢ * ⎥ + ∑⎢ Qj ∑(a(i, j) *CHi )⎥ + ⎢NC− ∑ti ⎥ * M / C Qj C ⎥ j=1 ⎣ 2 i=1 j =1 ⎢ i=1 ⎦ ⎦ ⎣ ⎢⎣ ⎥⎦

137


and our objective is to minimize Z, subject to (i) precedence constraints as given by the technology and (ii) cycle time constraints as determined from the market demand.

Under the condition that the ith task can be assigned to only one workstation, we must have, N

∑a(i, j) = 1 j =1

i = 1, 2, …, K.

Also, according to precedence constraints if task i′ is to be assigned before assigning task i, that is i ′ < i , then j

a (i, j )

≤

∑a(i′,r) . r=1

∀ i′ < i

Further, since each workstation can at the most be assigned C unit of time we have, K

∑a(i, j)t ≤ C i=1

i

∀ j=1, 2, … , N

Thus, a mathematical programming formulation of the integrated optimization problem can be written as,

138


minimize ⎤ ⎡K a(i, j ) N ⎢∑ K K C ⎥ N ⎡1 ⎤ ⎤ ⎡ Z = ∑ ⎢ i =1 * o ⎥ + ∑ ⎢ Q j ∑ (a(i, j ) * CH i )⎥ + ⎢ NC − ∑ t i ⎥ * M / C Qj C ⎥ j =1 ⎣ 2 i =1 j =1 ⎢ i =1 ⎦ ⎦ ⎣ ⎥⎦ ⎢⎣

subject to, N

∑ a(i, j )

=1

i= 1, 2, …. , K

j =1

j

a(i, j )

≤

∑ a(i′, r )

∀ i ′ < i and i ′, i = 1,2, …. , K

r =1

K

∑ a(i, j )t i =1

i

≤C

∀ j=1, 2, … , N

(4)

C min ≤ C ≤ C max a (i, j ) = 0,1

∀ i, j

i= 1, 2, …. , K.

j=1, 2, …, N.

Obviously, this is a mixed nonlinear programming problem. To arrive at the optimum solution we may take the course of iterative algorithm. For this, we formulate holding cost, ordering cost and cost of balancing loss and add them up to get the total cost of the supply chain and denote it by Z. Iteratively we minimize Z by optimization procedure to get the minimum cost of the chain. Inputs of the system include the values of

139


ordering cost, holding cost and average cost of one man hour. Highest and lowest rate of market demand is also fed into the program. After successful iterations we get the desired cycle time, assignments of work elements to workstations and order quantity for each stock chamber associated with each workstation. The next section includes an worked out example to demonstrate the functioning of the proposed method.

6.4 The Algorithm 1.

Formulate the objective function.

2.

Complete the model formulation by restricting the objective

function using precedence constraints, keeping in mind the zoning constrains and nonnegative constraints. 3.

Set cycle time C, determine the minimum number of workstations

Nmin and calculate the Cmin value. 4.

Set the trial cycle time Ct at Cmin.

5.

Solve the formulated problem for the particular Ct value.

6.

After getting the complete distribution of tasks to workstations, the

value of objective function is calculated.

140


7.

For each Ct value, the new value of the objective function is

compared with the previous value of the objective function. If the new value of the objective function is less than the previous least value of the objective function, the new solution is stored as the basis for next comparison. Otherwise keep the previous one as the basis for comparison. 8. Increase the trial cycle time Ct by one unit until it crosses C value. If C value is crossed, go to step 11. 9.

Repeat step 5 to 8.

10.

Check whether all the work elements have been assigned to

specified number of workstations. If not, increase the value of Nmin by 1 and go to step 4. 11.

Print the best solution in terms of overall minimum value of

objective function.

6.5 Worked Out Example To explain how the proposed method works, we consider in Figure 6.1 a conversion process where figure within a circle represents task number and that close to a circle represents corresponding task time. Precedence constraints are represented by the arrows.

141


9

6 1

4

10 9 5

5

5 5

5 2

17

4 6

5 10

2 12

5 13

4 14

12 15

10 16

15 18

5

8

19

6 8

6 21

10

7

3

20

6 11

Figure 6.1 : Precedence diagram of workstations along with the task times.

Work elements and precedence constraints as obtained from this conversion process are fed into the program developed for this purpose. In addition, we have given the values of ordering cost (Co), holding cost for each job (CHi), task time or assembly time of each job (ti), trial cycle time (Ct) and minimum cycle time (Cmin) and average cost of one manhour (M) as input data. Cycle time (C) and ordered quantity for each work station (Qj) are the decisions variables. We will determine the optimum values of those variables from the iterative run of our program. The conversion process can be summarized in a tabular form in terms of the binary variables

142


a(i,j)s and is given in Table 6.1 along with the choices of the cost parameters.

Work

Activity Time

Element 1 6 2 5 3 8 4 9 5 5 6 4 7 5 8 6 9 10 10 5 11 6 12 2 13 5 14 4 15 12 16 10 17 5 18 15 19 10 20 5 21 6 Co = 50

Immediate

Work Station CHi

Predecessor 15 10 10 1 20 1, 2 18 2 25 3 22 3 9 4 10 5, 6 12 8 10 10, 7 13 12 15 13 30 9, 11, 14 20 15 10 16 40 16 10 16 17 17 13 18, 19, 20 10 M = 25

1

2

3

4

5

a(1,1) a(2,1) a(3,1) a(4,1) a(5,1) a(6,1) a(7,1) a(8,1) a(9,1) a(10,1) a(11,1) a(12,1) a(13,1) a(14,1) a(15,1) a(16,1) a(17,1) a(18,1) a(19,1) a(20,1) a(21,1)





Table 6.1 : Precedence relation, task times of work elements and cost parameters

The optimum policy and configuration is presented in Table 6.2.

143


Work Station 1 2 3 4 5

Assigned tasks 1, 2, 3, 8, 11 5, 6, 7, 10, 12, 13, 14 4, 9, 15 16, 18 17, 19, 20, 21

Ordered quantity 17 13 14 18 12

Cycle time 31.00

Table 6.2 : Final Optimum Configuration and policy

Thus there will be 5 workstations with 5 stock chambers. Balancing loss of the system works out as 7.412%. The total cost of operation per unit time is 4667.7 unit.

6.6

Conclusion

We have presented a mathematical programming approach for solving an integrated model of line balancing with workstation inventory management problem with the objective of integrated cost minimization, resulting in optimum provision of materials in each stock chamber and determination of ideal cycle time so that the demand of the market can be met in time.

144


Till date, determination of cycle time was overlooked. Techniques were there for isolated minimization of balancing loss given a cycle time. Following our approach, we can easily calculate the ideal cycle time for our system according to the demand in the market. It will be possible to adjust the cycle time as and when needed. Our model will also minimize raw materials inventory as well as finished goods inventory and the total cost of the chain. We know that the cost of the supply chain is mainly the running cost of the chain. Our objective is to minimize this operational cost. More we can reduce the operational cost greater will be our flexibility in adding value to the chain. As a result of this point wise optimization and chronological improvement in the system, the quality of the supply chain will get increased providing distinctive competence to the company.

145

Chapter 7 Designing of an Assembly Line based on Reliability

7.1 Introduction Assembly line balancing problem is not only the problem of line design with the equal division of work among the stations or the adaptation of tasks to the speed of the workers but also to provide some amount of slackness in each workstation to take care of the stability of the system. And as we know, the success of an organization depends not only

146

In Search of Optimum Assembly Line Balancing Chapter 7: Assembly Line based on Reliability.

on quality and reliability of the final product, but also on the reliability of the production set up. Otherwise system failure may result in irregular supply of the item which will reduce the profit of the organization by increasing the cost of production or loss of customers or both. So, there should be both reliable products with reliable production set up for smooth and stable production. The objective of the current work is to design a stable assembly line where system failure will be minimum along with minimum number of workstations. For that purpose, expected balancing loss is minimized under the stochastic domain and then the reliability of the assembly line has been maximized. We propose a two-stage optimization method based on stochastic simulation approach to solve this problem.

7.2 Notation E(.)

147

statistical expectation operator

K

number of jobs

N



N(µ, 2) normal distribution with mean µ and variance

2

expected task time of ith job variance task time of ith job ti

random task time or assembly time of ith job

Wj

jth workstation

a(i,j)

binary measure taking value 1 for assignment of task i to workstation j

148

Lj

variable idle time of jth work station

Nmin


C

cycle time

Ct

trial cycle time

Cmin


St


B

balancing loss

Rj

reliability of the jth workstation

RAL

reliability measure for assembly line


7.3 Methodology and Mathematical Formulation Under the deterministic setup, the uneven allocation of work to different workstations results in loss of time. The efficiency of an assembly line is therefore measured in terms of this balancing loss, B. Under the stochastic setup this balancing loss itself becomes a random variable. One may like to minimize the expected value of the same. But this measure alone is not sufficient to ensure efficiency of the production system. There must be some consideration for ensuring high chance of meeting the cycle time requirements in each workstation. Drawing analogy with the concept of product reliability in terms of meeting the mission requirement, we may define the reliability of a workstation in terms of idle time meeting the non-negativity restriction. Thus, reliability of jth workstation, Rj can be defined as .

0 .

Then the assembly line can be viewed as an arrangement of N workstations in series in the sense if one workstation fails to meet the cycle time requirement the assembly line itself faces operational failure. We therefore propose to consider reliability of the assembly line along

149


with expected balancing loss as dual measures of system efficiency. Thus, the stability of the total system will be maximum when the expected balancing loss will be minimum and the reliability measure is maximum. Therefore, the objective of our proposed method is to minimize the expected values of B and maximize the measure of reliability, RAL, subject to precedence constraints. Let us consider the binary variable a (i, j ) such that 1 if i ∈ Wj i th task is assigned to Wj,

a (i, j ) =

0 if i ∉ Wj i th task is not assigned to Wj, and is true for i = 1, 2, ….., K, j = 1, 2, ….., N. The following condition must hold for each i = 1, 2, ….., K, under the restriction that the ith task can be assigned to only one workstation, ∑

,

=1

(1)

Further, according to precedence constraints if task i′ is to be assigned before assigning task i, that is i ′ < i , then ,

≤ ∑

′

,

∀ i′ < i

(2)

Now, the task time of each job will become a random variable depending upon variations in human skills and behavior. Therefore, we should consider both expected time and the extent of variability for completion

150


of each job as human beings are involved in completion of tasks in the assembly line. Let µ be the expected time for completing ith job. Then, the expected balancing loss of the system should be ∑ ∑

Or,

100% ,

100%.

(3)

Since the task times are random variables, the condition for completion of tasks in a workstation within the assigned cycle time can be best described in terms of chance measure, . ∑

,

,

j = 1,2,….,N.

Equivalently, it can be expressed as .

0

.

= 1

Φ

under normality of the each elemental times. Thus,

151


=Φ

The reliability of the assembly line, RAL, can be expressed as ∏ following the properties of the series system and the fact that workstations are arranged in series. Thus, the optimization framework of the line balancing problem can be expressed in terms of the following objectives ∑

Minimize ∏

Maximize

subject to the following constraints, : (i) (ii) (iii)

∑

, ,

1 ∑

a (i, j ) = 0,1

∀ i ′

,

∀ i′ < i ∀ i, j

We prefer to address the above optimization problem in two stages. First, we undertake the task of minimization of E(B) by generating in the first instant feasible solutions under the objective of minimization of E(B). Then we obtain the final solution of the problem by imposing the second objective of maximization of RAL. To generate the set of feasible

152


solutions we consider a sequential approach of assigning trial cycle time that results in slack time. This slack time is to be assigned to each workstation meeting the optimality condition arriving out of the first objective of the above formulation. The trial cycle time starts from some lowest value and get increased step by step so as to reach the maximum limit C. Determination of the lowest value depends on following consideration. Given a choice of C, it may be noted that the theoretical minimum number of workstations, Nmin, must satisfy the following constraints: K K ∑Ti/C ≤ Nmin ≤ ∑Ti/C +1, i=1 i=1 from where we arrive at Cmin, the minimum value of C, as Cmin =

K ∑ Ti / Nmin + 1 . i=1

Thus, given a cycle time, C, one may conceptually start from a trial cycle time, Ct, satisfying the condition Cmin ≤ Ct ≤ C, to arrive at the set of feasible workstation configurations and maintain the same cycle time C by uniformly adding to each workstation a slackness St to Ct, where St = C - Ct.

153


7.4 The Algorithm Our proposed two-stage procedure with sequential generation of feasible solution and selection of final solution is best described by the following algorithm. 1. Calculate the theoretical minimum number of workstations, Nmin, following the formula

∑

µ

∑

µ

1

2. Calculate the minimum cycle time, Cmin, using the relation, ∑

µ

1

3. Set the cycle time at Cmin. 4. Generate feasible solutions following the simulation algorithm of Roy and Khan (2010). 5. If no feasible solution is obtained then increase minimum cycle time Cmin by 1 and go to step 3. 6. Within the generated set of feasible solutions calculate RAL value for each solution of the set and save the RAL value that gives the maximum of the set. 7. Compare the RAL with the previous value of RAL. If the current RAL is greater than the previous one then save the current value of RAL.

154


8. When all the feasible sets are covered we get the final solution to the optimization problem, under simulation approach

7.5 Worked Out Example Figure 1 represents an assembly line balancing problem. This is a famous problem studied in Ray Wild. We have adopted it for the purpose of explaining how the proposed model works. The numerical figure within a circle represents the task number.

1

4

9

5

2

6

17

10

12

13

14

7

3

8

16

18

19

11

Figure 7.1 : Precedence diagram of workstations.

155

15

20

21


In Table 7.1, the above mentioned problem is summarized in terms of work elements, immediate predecessor(s), expected task durations and their variances.

Work Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Immediate Predecessor 1 1, 2 2 3 3 4 5, 6 8 10, 7 12 13 9, 11, 14 15 16 16 16 17 18, 19, 20

Expected Activity Time 6 5 8 9 5 4 5 6 10 5 6 2 5 4 12 10 5 15 10 5 6

Variance of activity time 0.09 0.0625 0.16 0.2025 0.0625 0.04 0.0625 0.09 0.25 0.0625 0.09 0.01 0.0625 0.04 0.36 0.25 0.0625 0.5625 0.25 0.0625 0.09

Table 7.1 : Precedence relation and task times of work elements.

Using the above table, we can easily get the minimum number of workstation, Nmin as 5. So, minimum trial cycle

156

comes out as


∑

µ

1 , i.e.

29 time unit. Now, we consider in

the final solution the Cycle time C as 35 time unit. So, the trial cycle Trial Cycl e Time

S l

Workstation 1

Workstation 2

Workstation 3

Workstation 4

Workstation 5

1

1,3,2,5,8

6,4,10,11,7,12

13,9,14,15

16,19,17, 20

18,21

2

1,3,8,2,7

6,11,5,10,4,12

13,9,14,15

16,17,19, 20

18,21

3

2,3,7,8,1

11,6,4,5, 10,12

9,13,14,15

16,18,17

20,19, 21

4

2,3,1,7,8,

5,11,6,10,12,13, 14

4,9,15

16,18,17

20,19, 21

5

2,3,7,8,11

1,4,6,5,10,12

9,13,14,15

16,19,17, 20

18,21

6

3,1,7,8,11

2,6,5,10, 12,4

9,13,14,15

16,19,17, 20

18,21

7

3,7,8,1,2

5,6,10,12,13,14, 11

4,9,15

16,19,17, 20

18,21

8

2,3,1,5,8

11,6,10,4,7,12

13,9,14,15

16,18,17

19,20 21

9

3,7,8,11,2

6,1,5,4,10,12

13,14,9,15

16,18,17

19,20, 21

1,2,3,8,11

4,6,7,5,10,12

13,14,9,15

16,17,18

20,19, 21

1,2,6,5,10

3,7,12,8,4

11,9,13,14

15,16,17, 20

19,18, 21

2

2,6,1,5,10

4,3,8,7,12

13,11,14,9

15,16,19

17,18, 20,21

3

2,6,1,5,10

3,8,11,4

7,12,9,13,14

15,16,19

18,17, 20,21

4

1,2,6,5,10

4,3,8,11

9,7,12,13,14

15,16,17, 20

19,18, 21

5

2,1,5,6,10

3,4,7,12, 13

14,8,11,9

15,16,17, 20

18,19, 21

6

2,6,3,7,1

5,10,12,13,8,14

4,11,9

15,16,17, 20

18,19, 21

7

2,1,6,3,5

7,10,12,13,14,8

11,4,9

15,16,19

18,17, 20,21

1

2,3,1,8,7

6,11,4,5, 10,12

9,13,14,15

16,18,17

19,20, 21

2

3,2,1,8,11

6,5,7,4,10,12

13,14,9,15

16,19,17, 20

18,21

3

3,7,8,2,11

6,1,5,10, 12,4

9,13,14,15

16,18,17

20,19, 21

4

3,1,7,8,2

4,11,5,6, 10,12

9,13,14,15

16,17,19, 20

18,21

5

1,2,5,3,8

6,4,7,11, 10,12

13,14,9,15

16,19,17, 20

18,21

31

1 0 1 32

33

157

RAL 0.8734504 76 0.8734504 76 0.8531308 99 0.8528978 41 0.8734504 76 0.8716554 55 0.8732118 67 0.8531308 99 0.8531308 99 0.8513776 36 0.7750863 0.7713410 63 0.7788128 26 0.7825943 43 0.7837068 58 0.7862868 27 0.7824874 68 0.8531308 99 0.8716554 55 0.8531308 99 0.8734504 76 0.8734504

In Search of Optimum Assembly Line Balancing Chapter 7: Assembly Line based on Reliability. 6

3,8,1,2,11

7,4,6,5,10,12

7

2,6,3,8,7

1,5,4,9

8

3,7,2,6,8

1,4,9,5

9

3,8,1,4

9,2,5,6,10

16,19,17, 20

18,21

15,16,17, 20

18,19, 21

15,16,19

18,17, 20,21

15,16,19

18,17, 20,21

1,3,2,6,8

5,10,4,9

15,16,17, 20

19,18, 21

2,6,1,3,8

4,5,7,9

15,16,19

18,17, 20,21

1,2,4,6,5

3,7,8,9

15,16,17, 20

19,18, 21

1,2,6,4,5

10,9,3,8

15,16,19

18,17, 20,21

2,1,4,5,6

10,9,3,8

15,16,17, 20

18,19, 21

2,6,3,8,1

4,5,7,9

15,16,17, 20

19,18, 21

1,3,4,8

7,2,5,9,6

15,16,17, 20

19,18, 21

2,1,4,6,5

3,7,10,12,13,14

9,8,11

15,16,19

18,17, 20,21

1

3,7,8,1,11

2,6,5,4,10, 12

9,13,14,15

16,19,17, 20

18,21

2

3,7,8,11,1

2,5,6,10,4, 12

13,9,14,15

16,18,17

19,20, 21

3

3,2,8,1,11

7,5,6,4,10, 12

13,9,14,15

16,18,17

19,20, 21

4

2,3,8,11,1

6,5,7,10, 12,13,14

4,9,15

16,17, 20, 19

18,21

5

3,2,8,11,1

7,5,4,6,10, 12

13,9,14,15

16,19,17, 20

18,21

6

3,1,8,11,2

4,9,15

16,17,18

20,19, 21

7

3,8,11,1,7

4,9,15

16,19,17, 20

18,21

8

3,7,8,1,11

4,9,15

16,18,17

20,19, 21

1

2,1,6,4,3

7,9,8,11,5

16,19,17, 20

18,21

2

2,1,6,3,4

8,11,5,9, 10

16,17,19, 20

18,21

1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7

9,13,14,15 11,10,12, 13,14 11,10,12, 13,14 11,7,12,13,1 4 11,7,12,13,1 4 11,10,12, 13,14 11,10,12,13, 14 11,7,12,13,1 4 7,11,12,13,1 4 10,11,12, 13,14 10,12,11, 13,14

34

6,7,5,10, 12,13,14 2,5,6,10, 12,13,14 2,6,5,10, 12,13,14

35

10,12,13, 14,15 7,12,13, 14,15

76 0.8716554 55 0.7697563 35 0.7660368 52 0.7764669 63 0.7799043 32 0.7761358 13 0.7800751 1 0.7763057 66 0.7800751 1 0.7799043 32 0.7802370 89 0.7797389 9 0.8716554 55 0.8513776 36 0.8513776 36 0.8637867 8 0.8716554 55 0.8436920 15 0.8637867 8 0.8436920 15 0.7880773 09 0.7880773 09

Table 7.2 : Trial Configurations

time starts with 29 time unit and goes upto 35 time unit. For the given problem we get no feasible solution for the trial cycle times as 29 and 30.

158


For the rest of the cycle times we get feasible solutions. These trial configurations are presented in Table 7.2.

The final solution is presented in Table 7.3 for trial cycle time as 31 time unit.

C

Workstation 1

31

2, 3, 7, 8, 11

Workstation 2 1, 4, 6, 5, 10, 12

Workstation 3 9, 13, 14,15

Workstation 4 16, 19, 17, 20

Workstation 5

RAL

18, 21

0.873450476

Table 7.3 : Final Optimum Configuration

In the optimum configuration it is found that the optimum value of RAL comes out as 0.873450476 with the configuration of 5 workstations with work elements 2, 3, 7, 8, 11 assigned to workstation 1, work elements 1, 4, 5, 6, 10, 12 are in workstation 2, work elements 9, 13, 14, 15 assigned to workstation 3, in workstation 4 work elements 16, 19, 17, 20 are assigned and work elements 18, 21 assigned to workstation 5.

7.6

Conclusion

Reliability of a production system is as important as the product reliability and we have considered reliability optimization problem for an

159


assembly line. For that purpose a mathematical programming approach is followed. Since, system failure is due to variations in human behavior a stochastic setup has been considered for describing the situation. A twostage approach has been installed to arrive at the final solution to the dual objective problem. Reliability of an assembly line has been evaluated under normality assumption. The proposed approach is a generic one and capable of solving different large or small, simple or complex assembly problems under different distributional assumptions. Final system can be designed based on minimum number of workstations and maximum value of the system reliability.

160

Chapter 8 Conclusion

8.1 Conclusion The Optimum Assembly Line Balancing Problem (OALBP) has been formulated and addressed in this doctoral thesis. This problem of assembly line balancing deals with the distribution of activities among the workstations so that there will be maximum utilization of human resources and facilities without disturbing the work sequence.

161

In Search of Optimum Assembly Line Balancing Chapter 8: Conclusion

The objective of this work was to develop an efficient solution for minimizing system loss and balancing loss of any Assembly Line Balancing Problem. As there was no measure for system loss in the literature, we first proposed two measures for system loss. We introduced the concept VBMS, a variance based measure of system loss and RMS, range based measure of System loss. Those measures aim to achieve a balanced distribution of work elements among different work stations. By comparing and integrating these two measures, one may get better efficiency. For the purpose of arriving at an optimum or near optimum solution, we have introduced the concept of trial cycle time and slack time in addition to cycle time so that more sets of feasible solutions can be generated. This approach of ours is a generic one, which is capable of solving different line assembly problem with a reasonable computation time. In our proposed approach we first consider minimization of balancing loss and then minimization of system loss. In this process, we have reduced the simultaneous optimization problem to a sequential optimization problem. This has been done to keep the number of solution to a manageable form. This approach according to numerical study is giving a better set of configurations because we are providing with some

162


amount of slackness in each workstation with trial cycle time varying from Cmin to C. We have also presented a mathematical programming approach for balancing an assembly line under similar objective. From the solution set generated by the optimization technique, final choice is made based on optimum number of workstations and minimum value of a measure of system loss, proposed herein. Truly speaking, system loss arises out of variations in elemental times due to variations in human behavior. So, stochastic setup is more appropriate to describe the situation or to represent the problem. Stochastic programming results the optimum solution of the same. Reduction of the stochastic setup into deterministic constraints has been indicated under normality assumption. Thereafter, a sequential approach has been installed to arrive at the final solution. Reliability of a production system is as important as the product reliability and we have considered reliability optimization problem for an assembly line. For that purpose a mathematical programming approach is followed. The basic aim is to build a stabilized system with maximum reliability.

163


We have presented a mathematical programming approach for solving an integrated model of line balancing with workstation inventory management problem with the objective of integrated cost minimization, resulting in optimum provision of materials in each stock chamber and determination of ideal cycle time so that the demand of the market can be met in time. Following our approach, we can easily calculate the ideal cycle time for our system according to the demand in the market. Our model will not only minimize raw materials inventory and finished goods inventory but also minimize the total cost of the production and increase the profit for the company.

8.2 Scope Major contribution of this thesis is introduction of a new simple assembly line balancing problem. This Optimum Assembly Line Balancing Problem is introduced and defined with examples. The next aim of this thesis is to bridge the gap between research works and real life applications. Future research work may mainly involve in exploring the

164


other methods to solve this new problem efficiently. In order to increase the practicality of the proposed problem, its definition can be extended by including a new variant of SALBP which can be defined as Type-3 simple assembly line balancing problem or SALBP-3. Here main focus will be on the stability of the system. The objective of SALBP-3 may be defined as to maximize production rate maintaining a desired level of reliability of the setup which is to be predefined, i.e., minimize cycle time C with a given level of reliability. It may be recalled that the existing variants SALBP-1 minimizes the number of workstations N given a cycle time C and SALBP-2 aims at minimizing the cycle time C given the number of workstations N. We are of the opinion that this work under the stochastic setup with reliability consideration for the setup will give the future direction of work in assembly line. Ensuring desired level of reliability will be the main task of this type of stable assembly line under stochastic variation in elemental times.

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