Optimisation of facility layout design problem with ...

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Optimisation of facility layout design problem with safety and environmental factors by stochastic DEA and simulation approach a

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A. Azadeh , T. Nazari & H. Charkhand

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School of Industrial and Systems Engineering and Center of Excellence for Intelligent Based Experimental Mechanics, University College of Engineering, University of Tehran, Tehran, Iran b

HSE Department, National Petrochemical Company of Iran, Asaluyeh, Iran Published online: 17 Dec 2014.

Click for updates To cite this article: A. Azadeh, T. Nazari & H. Charkhand (2014): Optimisation of facility layout design problem with safety and environmental factors by stochastic DEA and simulation approach, International Journal of Production Research, DOI: 10.1080/00207543.2014.986294 To link to this article: http://dx.doi.org/10.1080/00207543.2014.986294

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International Journal of Production Research, 2014 http://dx.doi.org/10.1080/00207543.2014.986294

Optimisation of facility layout design problem with safety and environmental factors by stochastic DEA and simulation approach A. Azadeha*, T. Nazaria and H. Charkhandb a School of Industrial and Systems Engineering and Center of Excellence for Intelligent Based Experimental Mechanics, University College of Engineering, University of Tehran, Tehran, Iran; bHSE Department, National Petrochemical Company of Iran, Asaluyeh, Iran

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(Received 8 January 2013; accepted 27 October 2014) This article presents an integrated computer simulation–stochastic data envelopment analysis (SDEA) approach to deal with the job shop facility layout design problem (JSFLD) with stochastic outputs and safety and environmental factors. Stochastic outputs are defined as non-crisp operational and deterministic inputs. At first, feasible layout alternatives are generated under expert decision. Then, computer simulation network is used for performance modelling of each layout design. The outputs of simulation are average time-in-system, average queue length and average machine utilisation. Finally, SDEA is used with Lingo software for finding the optimum layout alternative amongst all feasible generated alternatives with respect to stochastic, safety and environmental indicators. The integrated approach of this study was more precise and efficient than previous studies with the stated outputs. The results have been verified and validated by principal component analysis. The unique features of this study are the ability of dealing with multiple inputs (including safety) and stochastic (including environmental) outputs. It also uses mathematical programming for optimum layout alternatives. Moreover, it is a practical tool and may be applied in real cases by considering safety and environmental aspects of the manufacturing process within JSFLD problems. Keywords: job shop facility layout design (JSFLD); job shop; simulation; stochastic data envelopment analysis (SDEA)

1. Introduction Facility layout design (FLD) is a crucial task in redesigning, expanding or designing the manufacturing systems, for example job shop systems. The job shop facility layout design problem (JSFLD) problem involves determining the arrangement and the location of equipment, workstations, offices, etc. within a job shop system by considering the interconnections through sequential facilities as well as walks and vehicle transportations. The most common objectives of layout problems in the literature are minimisation of the transportation costs of raw material, parts, tools, work-in-process (WIP), and finished products amongst the facilities (McKendall, Shang, and Kuppusamy 2006; Önüt, Tuzkaya, and Doğaç 2008), facilitate the traffic flow and minimisation the costs of it (Balakrishnan 2003), maximisation of the layout performance (Zhang, Teng, and Shi 2008), minimisation of the dimensional and form errors of products depending on the fixture layout (Prabhaharan, Padmanaban, and Krishnakumar 2006; Chen, Ni, and Xue 2007), minimisation of the total number of loop traversals for a family of products (Satheesh Kumar, Asokan, and Kumanan 2007) increasing the employee morale, minimisation of the risk of injury of personnel and damage to property, providing supervision and face-to-face communication (Heragu 1997). Algorithmic approaches usually simplify both design constraints and objectives in reaching a total objective to obtain the solution of the problems. These approaches lead to generation of efficient layout alternatives base on expert decision. Because, the success of this process strongly depends on the generation of quality design alternatives provided by an expert designer. Considering material handling costs as the main objective, several heuristic and meta-heuristic approaches have been presented in the literature for various facility layout problems (Ye and Zhou 2007; Diego-Mas et al. 2009; Ku, Hu, and Wang 2010). Layout generation and evaluation is often a challenging and time-consuming task due to its inherent multiple objective natures and its difficult data collection process (Lin and Sharp 1999). Different methodologies have been presented in the literature to deal with such problems. The algorithmic approaches have mainly focused on minimising flow distance in order to minimise material handling costs, and the procedural approaches have heavily relied on the experience and judgment of expert designers. In this regard, Yang, Su, and Hsu (2000) showed that neither algorithmic nor

*Corresponding author. Emails: [email protected], [email protected] © 2014 Taylor & Francis

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procedural FLD methodology is necessarily effective in solving FLD problems. Following this idea, different studies have been conducted to cover the existent gap in the FLD problems (Cambron and Evans 1991; Foulds and Partovi 1998; Yang, Su, and Hsu 2000). Yang et al. (2012) proposed a hybrid approach which integrates analytic hierarchy process (AHP) and preference ranking organisation methods for enrichment evaluations with the purpose of solving a facility layout problem. On the other hand, several studies have attempted to determine the efficiency of different layout alternatives and rank these decision-making units (DMUs) in a better fashion. In order to rank the DMUs, Yang and Kuo (2003) and Azadeh and Izadbakhsh (2008) considered three quantitative and three quantitative performance indicators in a flow SFLD problem. Nevertheless, neither Yang nor Kuo (2003) nor Azadeh and Izadbakhsh (2008) provided a comprehensive decision aiding tool for FLD problems. Therefore, a more comprehensive approach should be developed to incorporate all required features of manufacturing system to the ranking models and so provide a thorough and more real decision aiding tool for decision-making processes. Simulation is a tool with the ability to use data to evaluate a current facility layout, show potential improvement areas and objectively evaluate various alternatives, and it is used widely in the literature (Savsar 1991; Azadivar and Wang 2000; Pagell and Melnyk 2004; Zhou, AbouRizk, and AL-Battaineh 2009; Hsieh et al. 2012). Zhou, AbouRizk, and AL-Battaineh (2009) integrated general-purpose simulation to model the space, logistics and resource dynamics with genetic approaches (GAs) for optimising the layout based on various constraints and rules and implementing a site layout optimisation system within a simulation environment. Jithavech and Krishnan (2010) presented a simulation-based method for predicting the uncertainty associated with the layout and validated their simulation approach against analytical procedures. DEA was developed by Charnes, Cooper, and Rhodes (1978). It was extended by Banker, Charnes, and Cooper (1984) with presentation of BCC model and Charnes and Cooper (1985) with presentation of additive model; this model is nonparametric and linear programming method for estimating technical efficiency of each DMU with multiple input and output. Bretholt and Pan (2013) tests several nonparametric DEA models for their ability to accurately decompose CO2 emissions change. This article has presented several models that reduce undesirable outputs. Vaninsky (2010) proposes the environmental performance of regions and largest economies of the world; actually, the efficiency of their energy sectors is estimated. Two essentially different methodologies, DEA and stochastic frontier analysis, are used to obtain upper and lower boundaries of the environmental efficiency index. Regions and national economies having low level or negative dynamics of environmental efficiency are determined. Staub, Souza, and Tabak (2010) use DEA to compute efficiency scores in Brazilian banks. Yang et al. (2012) used DEA for ranking DMUs. Friesner, Mittelhammer, and Rosenman (2013) DEA is amongst the most popular empirical tools for measuring cost and productive efficiency within an industry. However, there is a weakness in conventional DEA model. Conventional DEA does not allow to use stochastic variable as an input and output data, and the result is sensitive to such variables and none of the previous studies considered and presented a unique methodology for JSFLD problems with stochastic outputs. Azadeh et al. (2011) proposed an integrated fuzzy simulation–fuzzy data envelopment analysis (FSFDEA) algorithm to cope with a special case of single-row facility layout problem (SRFLP). The proposed FSFDEA algorithm is capable of modelling and optimising small-sized SRFLP’s in stochastic, uncertain and nonlinear environments. Qin and Liu (2010) develop a new class of chance model about DEA in fuzzy random environments, in which the inputs and outputs are assumed to be characterised by fuzzy random variables with known possibility and probability distributions. Moreover, there are usually stochastic data with respect to layout problems in general and JSFLD problems in particular. This means data could not be collected and analysed by deterministic models, and new approaches for tackling such problems are required. This gap motivated the authors to develop a unique approach to handle such gaps in JSFLD problems. In order to incorporate stochastic variable into the DEA, we use CCR model that objective function as the maximum of the expected ratio of the weighted output to weighted inputs and reliability function subject to several chance constraints and Li (1998) presents stochastic data envelopment analysis (SDEA) developed by tacking random disturbances into account for the possibility of variation in inputs and outputs with other DMU. A major contribution on the stochastic DEA may be found in the work of Sengupta (1982, 1987, 1988, 1989, 1995) who has extensively studied the research topic, using the chance constraint programming (CCP) proposed by Charnes and Cooper (1963). An important feature of such studies is that stochastic variables are incorporated into DEA, and then, the stochastic DEA is reformulated into a deterministic equivalent. As a result of such reformulation, the stochastic DEA can be solved by any commercial computer software including linear programming and quadratic programming. Besides Sengupta’s studies, Cooper, Huang, and Li (1996) and Cooper et al. (1998) have proposed other innovative approaches which incorporate the concept of satisfying into stochastic DEA formulations under CCP. Udhayakumar, Charles, and Kumar (2011) consider inputs and outputs are stochastic and proposed the stochastic objective function, and chance constraints are directly used within the genetic process. Wu et al. (2012) proposes a stochastic DEA model considering

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undesirable outputs. This model introduces the concept of risk to define the efficiency of DMUs. In conclusion, the model with broad applicability has a more superior analysis capacity than the existing model. A shortcoming of the above research efforts is that they did not document how to use the stochastic DEA (SDEA) for strategic planning and other decisional issues. Hence, Sueyoshi (2000) apply the stochastic approach to plan the restructure strategy of a Japanese petroleum company and so compare the result with real form with considered three categories large, medium and small. SDEA is the most important category of the literature related to this work, which has been widely used in different research works in the literature for operations evaluation and ranking of DMUs. Tavana et al. (2012) proposed three fuzzy DEA models with respect to probability–possibility, probability–necessity and probability–credibility constraints. In addition to addressing the possibility, necessity and credibility constraints in the proposed SDEA model, we also consider the probability constraints. Ertay and Ruan (2005) presents a decision-making methodology based on data envelopment analysis (DEA), which uses both quantitative and qualitative criteria, for evaluating FLD. AHP is then applied to collect qualitative data related to quality and flexibility. Yang and Kuo (2003) combined AHP and DEA approach to solve layout problems. They also used computer-aided layout-planning tool to generate a considerable numbers of layout alternatives. Azadeh et al. (2011) integrated FSFDEA algorithm to cope with a special case of SRFLP. Ertay, Ruan, and Tuzkaya (2006) presented the evaluation of the FLD by developed a robust layout framework based on DEA/AHP methodologies with VisFactory tool. They also consider the fuzzy sets when collecting and analysing data. Jahanshahloo et al. (2013) proposed a new approach to determine the optimal distribution of process facilities based on the common set of weights DEA model. Maniya and Bhatt (2011) proposed an alternative decision-making methodology with preference selection index method to the choice of optimal FLD alternative. The proposed methodology is user-friendly, easy to understand, less calculation and systematic. Hadi-Vencheh and Mohamadghasemi (2013) incorporate qualitative criteria and quantitative criteria for evaluating facility layout patterns (FLPs). They present a decision-making methodology based on a simple nonlinear programming model and AHP. Based on this motivation, an integrated simulation–stochastic DEA approach is presented in this article to locate the optimum layout through a set of feasible solution. First, different layout alternatives are generated by decision experts. Then, discrete-event simulation, as a robust performance evaluation, and modelling tool is used to model the generated layout alternatives, with respect to a set of operational data. Simulation is a flexible and powerful tool for visualising and manipulating the system under study and can be used in different situations to make the company agile in implementing changes in a swift and effective manner based on a confident analysis. The results of the simulation model include the average utilisation of machines (i.e. stage), average time-in-system for a given number of products, average available and average length in queue. On the other hand, we consider safety indicators which are emergency equipment layouts, safety rout and brightness deign as inputs and air pollution and tangible pollution as outputs. The nine performance indicators are then imported to SDEA model in order to determine the technical efficiency and rank of each layout alternative (DMU). The results show that the proposed integrated computer simulation–SDEA approach yields a more comprehensive and applicable framework for JSFLD in comparison with the previous studies. To the best of our knowledge, this is the first study in the literature that presents such integrated approach based on computer simulation and DEA for JSFLD problems.

2. Motivation and significance There are usually stochastic outputs with respect to layout problems in general and JSFLD problems in particular. This means data could not be collected and analyzed by deterministic models and new approaches for tackling such problems are required. Also, almost all previous studies do not consider the impacts of safety and environmental issues on JSFLD problems. This gap motivated the authors to develop a unique approach to handle such gaps in JSFLD problems. The integrated simulation-stochastic DEA presents improved solution to the JSFLD problems with random variables whereas previous studies present incomplete and non exact alternatives. Also, it provides a comprehensive analysis on the JSFLD problems with stochastic outputs. The superiority and effectiveness of the proposed integrated approach is compared with deterministic DEA and principal component analysis (PCA) methodologies through an actual case study. The unique features of the proposed integrated approach are the ability of dealing with multiple inputs and random outputs and optimization through SDEA and applicability in real cases due to considering operational aspects of the manufacturing process within JSFLD problems.

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3. Method 3.1 System Description A practical case, which is in regard to Arya Sasol Polymer Company, is the producer of polymer products in Pars Special Economic & Energy Zone in Bushehr Province in Iran. This company is a joint venture between National Petrochemical Company of Iran and Sasol Polymer of Germany which is also a subsidiary of Sasol limited. In 1998, Pars Special Economic and Energy Zone was established to utilise gas from a South Pars field in a bid to develop oil, gas, petrochemical and their related downstream industries. South Pars area contains one of the world`s largest independent gas fields. Holding more than 14 trillion cubic metres of gas and 18 billion barrel of condensates, the field contains 8% of world`s gas and over 48% of Iran`s total known gas reserves. The actual case of this study illustrates the efficiency and effectiveness of the proposed approach. The maintenance section of Arya Sasol Company consists of ten stages. Figure 1 presents the existing layout of the ten stages. The manager of the plant would like to assure their future plant layout is efficient in supporting production activities. If the current layout is not efficient, the company would like to know what layout alternatives are efficient. The experience learnt from this study will provide the guidelines for future FSFLD optimisation and planning. The following assumptions are considered in the proposed approach. Considering the stated assumptions, the simulation–stochastic DEA approach is explained next.
Due to the low inventory cost of Arya Sasol process, the most desirable layout is the one that produces the most quantity of products within a given period of time;
The manufacturing system is job shop which consists of ten sequential stages (i.e. machines);
The material flow is initiated from each stage;
The processing times are modelled based on probability distribution, and the nature of the manufacturing system is such that the processing times can be obtained by PERT/CPM method. PERT/CPM is a management science technique for planning activity times and scheduling, whilst this study uses the technique to estimate the expected value and variance of each output. Using the proposed approach, we can incorporate stochastic DEA for all outputs. Moreover, it is assumed that HSE data are obtained from answer sheet and used as input and output indicators in SDEA model;
Layout alternatives consist of stochastic indicators. 3.2 The integrated approach This study presents an integrated simulation–stochastic DEA–multi-attribute approach to deal with the JSFLD problems with stochastic outputs. Figure 2 presents a schematic view of the proposed approach. In the following section, the steps of the proposed approach have been applied to the Arya Sasol process. In summary, the proposed approach is achieved as follows:
Step 1: Collect the required data for designing the layout of the manufacturing plant such as the total space of the plant, space of each machine, etc.
Step 2: Generate different layout alternatives with respect to decision expert.
Step 3: Collect the required data for the manufacturing process such as, processing times, travelling times between sequential machines, etc., which can be obtained from expert judgments and history of the manufacturing plant. Moreover, data are presented by probability distributions.

Figure 1. The current plant layout of maintenance section of Arya Sasol Company.

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Figure 2. Schematic view of SDEA–computer simulation approach.


Step 4: Develop the simulation network model of each layout alternative using some additional information such as machines’ processing times, which can be obtained from historical data and the experts of manufacturing plant by probability distribution.
Step 5: Analyse and retrieve three operational indicators from simulation model to be used for further analysis in stochastic DEA.

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Step 7: Incorporate layout dependent indicators (safety and environmental indicators) and operational indicators (average length, average machine utilisation (AMU) and average time-in-system) to the SDEA models to rank the generated layout alternatives and identify the optimum alternative. The safety indicators are considered as inputs, operational and environment indicators are considered as outputs of SDEA models.
Step 8: Compare SDEA rankings with principal component analysis (PCA) to see if there is a significant difference between results.

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4. Experiments 4.1 Data collection for JSFLD Data collection should include characteristics of products, quantities, routing, support and time considerations in order to assure the validity of the input data at the design stage. The outputs of this step are used in generating different layout alternatives. Table 1 presents the facility sizes of the ten stages. Also, the available width and length of the plant are 55 and 40 m, respectively. As mentioned, operational indicators are defined as average queue length, AMU and average time-in-system. They are retrieved as outputs of simulation models. Furthermore, minimising average time-in-system essentially would guarantee to produce the most quantity of products within a given period of time. In addition, safety and environment indicators are obtained from answer sheet. Layout dependent indicators include flow distance (the sum of the products flow volume and rectilinear distance between the centre of two facilities), safety and environmental factors. 4.2 Data collection for the manufacturing process To illustrate the efficiency of the proposed approach in evaluating the generated layout alternatives from operational viewpoints, a set of operational data from a case study in Azadeh et al. (2010, 2011) is applied to the IC packaging process with deterministic and fuzzy data. The required data for modelling Arya Sasol process are shown in Table 2. The historical production data are collected from the company’s shop floor control system. The set-up times, mean time between failures (MTBF) and mean time to repair (MTTR) are stochastic data analysed by commercial curve fitting software, Expert Fit. The resulting distributions for each machine type are validated by both χ2 and Kolmogorov–Smirnov tests for their goodness of fit. There are neither historical data nor robust time studies on processing times. Moreover, expert judgment is used to derive to processing times. This is because of in real word deterministic data dose not exit. Furthermore, expert judgment is used to establish stochastic numbers. Thus, all processing times are stochastic as shown in Table 2. The processing times for the stages in Table 3 are dependent on material type, regardless of product size whilst, the processing times for the stages in Table 3 are independent of product type. There are 7 product types. It is assumed that the flow of WIP between sequential stages has approximately 30 metre per hour velocity (considering all waste times). Thus, the time taken to transfer WIP between each two sequential stages can be calculated by dividing the distance into the flow velocity as shown in Table 4.

Table 1. Facilities (stages) sizes. No.

Name

1 2 3 4 5 6 7 8 9 10

Trash Welding Boring Balance Assemble Inspection Asset Electrical shop Valve shop Pump shop

Size (m2) 420 225 81.0 220 200 100 154 250 360 300

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4.3 Simulation network modelling

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Six product types and ten resources are considered as entities and servers in this model, respectively. Each product type has two attributes identifying its type and starting time and is emanated in network by a CREATE node, and also, the type and starting time of it are determined by an ASSIGN node positioned after the CREATE node. It is assumed that all product types have equal per cents of the overall demand. Hence, the simulation network is modelled with 40 entities for each product type. A product (entity) is sent to original network. If the specific machine processing this product is available, then it is assigned to the machine for during the processing time. Otherwise, this material must be awaited in the file number of the stage. This process is done with ten AWAIT nodes. Processing time of each product on each machine in each step is defined during the activity. After entering all 280 entities to the TERMINATE node, the simulation will be completed. After entering all entities to the TERMINATE node, the simulation will be terminated. In this paper, we have seven material type and 40 entities for each type, and therefore after entering 280 entities to the TERMINATE node, the simulation will be terminated. The stochastic operational indicators are AMU in each stage, average length and average time-in-system. Simulation is performed by considering uncertain values. There are four operational stochastic indicators obtained from simulation model. The reader should note that the simulation was replicated 100 times and the average of 100 runs was used for each DEA model. Stochastic shape is used to decrease the relative error in comparison with the deterministic models. 4.4 Applied stochastic DEA model It is interested to investigate the efficiency of different layouts. The stochastic outputs and deterministic inputs are inputted to SDEA model to obtain the ranking results. In most general decision-making cases, the decisions are based on concurrent inputs and outputs data. Stochastic DEA seems to be convenient for problems associated with stochastic data sets. This is because most indicators for layout alternatives are judgmental and stochastic nature. Sueyoshi (2000) proposed Pn a new method for ranking the efficient units based on CCR model. This was achieved by omitting the constraint j¼1 sj ¼ 1 to the BCC model and obtaining the results for CCR model. The stochastic input-oriented CCR model for ranking the layout alternatives is as follows: Stochastic DEA model:  s  P max E ur ^yrk m P

r¼1

s:t : vi xik ¼ 1 i¼1 0P 1 s B r¼1 Pr@ P m i¼1

ur ^yrj

(1.1)

C  bj A  1  a j

vi xij

ur  0; vi  0

The constraints of Equation (1.1), including the stochastic process, can be rewritten as follows: ! s m X X ur ^yrj  bj vi xij  1  aj Pr r¼1

(1.2)

i¼1

Equation (1.2) is equivalent to: 0P s

1 m s P P ur yrj ur ð^yrj  yrj Þ bj vi xij  Br¼1 C C  1  aj PrB  i¼1 pffiffiffiffi r¼1 pffiffiffiffi @ A vj vj where y is the expected value of ^y and: 0

vð^y1j Þ covð^y1j ; ^y2j Þ vð^y2j Þ vj ¼ ðu1 ; . . .; us Þ  @ covð^y2j ; ^y1j Þ covð^ysj ; ^y1j Þ ...

... ... ...

0 1 1 u1 covð^y1j ; ^ysj Þ B  C C covð^y2j ; ^ysj Þ A  B @  A vð^ysj Þ us

(1.3)

(1.4)

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Here, Vj indicates the variance–covariance matrix of the jth DMU in which the symbol ‘v’ stands for a variance and the symbol ‘COV’ refers to a covariance operator. To reformulate Equation (1.3) by CCP, this study introduces the following new variable (^zj ): s P

^zj ¼ r¼1

ur ð^yrj  yrj Þ ; j ¼ 1; . . .; n: pffiffiffiffi vj

(1.5)

which follows the standard Normal distribution with zero mean and unit variance. Substitution of Equation (1.5) in Equation (1.3) produces: 0 1 m s P P bj vi xij  ur yrj B C i¼1 r¼1 C  1  aj ; j ¼ 1; . . .; n: PrB (1.6) pffiffiffiffi @^zj  A vj

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Since ^zj follows the standard normal distribution, the invertibility of Equation (1.6) is executed as follows: bj

s P vi xij  ur yrj i¼1 r¼1  F 1 ð1  aj Þ; pffiffiffiffi vj m P

j ¼ 1; . . .; n:

(1.7)

Here, F stands for a cumulative distribution function of the normal distribution and F−1indicates its inverse function. The DEA stochastic model (1.8) is obtained by replacing Equation (1.2) by Equation (1.7) and its resulting formulation becomes:  s  P Max E ur ^yrj ; s:t : bj

m P i¼1

vi xij 

s P

m P

r¼1

vi xik ¼ 1;

(1.8)

i¼1

pffiffiffiffi ur yrj  vj F 1 ð1  aj Þ; j ¼ 1; . . .; n;

r¼1

ur  0; r ¼ 1; . . .; s&vi  0; i ¼ 1; . . .; m:

To obtain a linear programming equivalent to Equation (1.8), this research assumes that a stochastic variable (^yrj ) of each output is expressed by ^yrj ¼ yrk þ brj n (r = 1, …, s, and j = 1, …, n), where yrk is an expected value of ^yrj , and brj is its standard deviation. Section 5 of this article describes how to determine the average and the standard deviation from prediction of a decision-maker(s). It is also assumed that a single random variable (ξ) follows a Normal distribution N(0, δ2). Under such an assumption, Vj becomes: 0 1 0 1 b21j b22j d2 . . . b21j b2sj d2 b21j d2 u1 !2 s B 2 2 2 C B C X 2 2 2 2 2  B b2j b1j d C b d . . . b b d C¼ 2j 2j sj vj ¼ ðu1 ; . . .; us Þ  B ur brj d (1.9) CB @ A @  A  r¼1 us b2sj b21j d2 ... ... b2sj d2 Paying attention to ^yrj ¼ yrk þ brj n, we reformulate the objective of Equation (1.8) as follows: ! m s s X X X vi xij  ur yrj  ur brj n F 1 ð1  aj Þ; j ¼ 1; . . .; n: bj i¼1

E

s X r¼1

! ur by rj

¼E

r¼1 s X r¼1

r¼1

! ur ðyrk þ brj nÞ

¼E

s X r¼1

! ur yrk

þE

s X r¼1

! ur brj n

¼

s X

ur yrk

r¼1

In the above model, indices i, r and j represent the inputs, outputs and layout alternatives, respectively. The stochastic outputs are availability, AMU in each stage, average time-in-system and environmental indicators (air and tangible pollutions). The deterministic input indicators are the three safety indicators (emergency equipment layout, safety rout and brightness deign). This is because inputs should be reduced whilst outputs should be increased in optimisation problems. xij and ^yrj are respectively input and output variables of SDEA model. This research assumes that a stochastic variable

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(^yrj ) of each output is expressed by ^yrj ¼ yrk þ brj d (r = 1,…, s and j = 1,…, n), where yrk is an expected value of by rj , andbrj is its standard deviation. It is also assumed that a single random variable (δ) follows a Normal distributiond  N ð0; 1Þ. The following linear programming model is therefore developed (1.10) by referring to Sueyoshi (2000), Charnes and Cooper (1963) and Cooper et al. (1996, 1998). max

s P

ur yrk

r¼1

s:t : bj

m P

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i¼1

vi xij 

s P

m P

vi xik ¼ 1

i¼1

ur yrj 

m P

ur brj d/ i¼1 ur  0; vi  0

(1.10) 1

ð1aj Þ

r¼1

In the above developed model (1.10) by this article, a is a parameter belonging to the interval [0 1]. Model (1.10) is a parametric linear programming model which can be used for obtaining the optimum solution for each given value of a (Sueyoshi 2000; Wu and Lee 2010). Since the objective of this study was to analyse the efficiency of layouts based on output indicators, the input-oriented CCR model has been utilised as a base model and the efficiency and rank of each layout is determined based on model (1.10) for different a and β values. Also, since a represents the certainty of the given indicators, as a gets near 0.5 and β gets near 1, the two DEA approaches (conventional DEA and SDEA) exhibit very similar results on these efficiency scores. 5. Computational results In this article, an integrated computer simulation–SDEA approach is presented to deal with the JSFLD problem in Arya Sasol process. As mentioned previously, Yang and Kuo (2003) and Azadeh and Izadbakhsh (2008) considered three quantitative and three qualitative indicators for evaluating the performance of feasible layout alternatives provided by a computer-aided layout-planning tool. Consequently, 45 layout alternatives have been generated. The simulations results are obtained as stochastic shape parameters as shown in Table 4. 5.1 Simulations results The operational data for the Arya Sasol process have been presented in Table 5, in which seven material and ten resources have been considered in order to test the model. Machines priorities and processing times for each stage are modelled and analysed by simulation. In this article, Visual SLAM is the simulation language used for solving the layout optimisation problem. The structure of this language is based on network modelling. Therefore, adding or removing some elements in different sections of the model is an easy task. In this study, Visual SLAM language is used to build the model and simulate the system. Computer simulation is used for solving this problem, since it provides a systematic for evaluating different layout scenarios based on generated data and the objective data to assist the decision-maker. In this model, a product is sent into the original network by a CREATE node. If the required resources (i.e. VALVE SHOP) are available they will be assigned to the product. Otherwise, this product must be awaited in the AWAIT node labelled ‘await’. In this model, ATRIB [2] is used to determine the TNOW; thus, changing this parameter results in analysis of different layouts. Since we have seven queues (product models) and ten resources that can be assigned to each of them; therefore, 45 layout alternatives are available for this problem. After performing each activity, the value of ATRIB [2] is determined in an ASSIGN node. Then, if the required resource (i.e. electrical) is available, it is assigned to the product. Otherwise, the product must be awaited in its specified AWAIT nodes. A FREE node, as a node for freeing the allocated resources within AWAIT node, is used to free the head; thus, it is returned to the network. COLCT nodes are then used to collect the information regarding time-in-system and layouts and print the report of it to an output file in a pre-defined format. After entering all entities to the TERMINATE node, the simulation will be terminated. At the end, control statements are used to equalise the variables in network of Visual SLAM. Moreover, the array statements make a table with seven rows. The required information about the product type depends on processing times and time taken to transfer WIP between each two sequential stages (to be read from Tables 2 and 3). The processing times of products on stages are determined for the model by array statements. Columns number is the same as the product number. The output of simulation data contains pessimistic estimate, mode and optimistic estimate for TIS, AMU, AL and AL. The main structure of the simulation model is composed of CREATE, ASSIGN, ACTIVITY, AWAIT, GOON, FREE, RESOURCE BLOCK, COLCT, and TERMINATE nodes.

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Table 2. Set-up times, MTBF and MTTR for machines. Number of machines

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Stage 1 2

5 6

3

2

4 5

5 2

6 7 8 9 10

1 8 1 12 10

MTBF (hour)

MTTR (hour)

(0.55, 75)c (0.5,

Exp (5.00) 0

Exp (2.1) 0

(0.75,

Exp (3.50)

Exp (1.4)

(0.25,

0 0

0 0

Set-up time (hour) Uniform Uniform 0.65)b Uniform 0.9)d 0 Uniform 0.5)a 0 0 0 0 0

Exp Exp Exp Exp Exp

(9.00) (4.20) (5.60) (12.3) (3.50)

Exp Exp Exp Exp Exp

(2.4) (5.0) (7.6) (11) (1.0)

Set-up needed for the material and fixture change. Set-up needed for material change. c Set-up needed for the size and material change. d Set-up needed for material change. a

b

Table 3. Processing time of each product type in each stage. Product type Stage 1 2 3 4 5 6 7 8 9 10 a

1

2

3

4

5

6

7

Ua (5,15) 0 0 U(2,6) U(4,8) 0 U(2,6) U(18,38) 0 0

0 U (1,3) U(3,5) 0 U(2,5) U(2,5) 0 0 U(8,16) 0

0 U(1,3) U(2,6) 0 U(2,5) U(2,5) 0 0 U(7,17) 0

U(3,7) U(2,6) 0 U(2,6) 0 0 U(7,13) 0 0 U(10,16)

U(3,7) U(2,6) 0 U(2,6) 0 0 U(5,15) 0 0 U (8,16)

U(3,7) U(2,6) 0 U(2,6) 0 0 U(7,13) 0 0 U(8,16)

U(3,7) U(2,6) 0 U(2,6) 0 0 U(2,8) 0 0 U(8,16)

U: Uniform distribution.

After defining the control statements, the simulation model will be ready to run. The simulation network has been modified based on the flow distances between each two sequential stages obtained by a computer-aided layout-planning tool for all 45 layout alternatives. The simulation output shows information given in the nodes of the model, such as the average utilisation of resources for each machine and the average time-in-system. By analysing the output of simulation, it is widely known that the results do not present same effects. Therefore, we change the undesirable (including ATIS and AL) to desirable outputs by reversion. Moreover, the two types of efficiencies are unified and referred to as ‘unified efficiency’ in this study (Sueyoshi and Goto (2011). Table 5 presents the average machine utilisation (AMU), average available (AA) inverse of average time-in-system (ATIS) and inverse of average length (AL), respectively, for all 45 layout alternatives. 5.2 Results of questionnaire yrk and brj of ^yrj may be estimated by PERT/CPM (programme evaluation and review technique/critical path method). PERT/CPM is a management science technique for planning activity times and scheduling, whilst this study uses the technique to estimate the expected value and variance of each output. Using the proposed approach, we can incorporate stochastic DEA regarding each output.

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Table 4. FROM-T0 chart matrix with velocity (distance/velocity). N

1–4

1–7

2–3

2–6

2–7

3–5

4–5

4–10

6–9

7–8

7–10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1.33 1.33 0.85 0.85 0.67 0.58 0.82 0.67 1.17 0.57 0.33 0.67 0.52 0.85 0.83 0.83 1.20 0.50 1.45 1.78 0.40 0.83 1.27 1.17 0.87 0.83 1.62 0.67 1.33 0.67 0.25 0.42 0.25 0.50 1.00 1.00 0.50 0.50 1.00 0.50 0.67 0.88 1.50 1.50 0.50

0.58 0.75 1.42 1.42 0.58 1.47 1.32 0.92 0.58 0.48 20.0 17.0 43.0 24.0 10.0 47.5 43.0 23.0 32.5 28.0 62.5 16.5 21.5 26.0 34.5 22.5 39.5 55.0 37.5 32.5 12.5 22.5 22.5 52.5 32.5 32.5 25.0 17.5 17.5 17.5 22.5 7.50 17.5 62.5 27.5

0.57 0.57 0.57 0.57 17.5 17.5 0.88 0.67 1.57 0.33 0.58 0.42 0.35 1.07 0.40 0.57 0.30 1.07 0.50 0.50 1.43 0.92 1.08 0.42 0.42 0.92 0.58 0.83 0.90 0.75 1.45 0.37 0.58 0.75 0.58 0.58 0.75 0.42 0.75 0.33 0.58 0.58 0.83 1.12 0.58

1.33 1.33 1.33 1.58 0.83 0.58 0.58 0.33 0.58 0.33 0.75 0.62 1.25 1.75 1.60 1.75 1.37 0.92 1.53 1.43 0.38 1.08 0.92 0.58 1.65 0.58 0.92 2.02 0.37 0.33 1.00 0.65 0.77 1.33 0.67 0.58 0.75 0.33 0.42 0.75 0.50 0.42 0.42 1.42 1.75

0.82 0.52 0.82 0.52 1.17 0.83 0.67 0.33 0.83 0.33 1.42 1.33 1.00 1.07 0.25 0.50 0.90 1.78 0.32 1.08 0.42 2.10 0.58 0.83 0.83 0.50 0.83 0.67 0.58 0.33 0.37 1.02 0.33 0.83 0.67 0.67 0.33 0.50 0.67 0.67 0.42 0.67 0.92 0.75 0.50

1.40 1.48 1.48 1.40 0.58 0.58 0.50 1.42 0.75 0.50 0.50 0.92 1.00 0.37 1.00 0.82 0.62 1.48 0.55 0.63 1.83 0.67 0.75 0.50 0.50 0.25 0.50 0.55 0.45 0.58 0.58 1.08 1.08 0.50 0.83 0.58 0.50 0.50 0.83 1.67 0.67 0.33 0.42 1.17 2.00

1.08 0.83 0.67 0.67 1.45 0.67 0.33 0.92 1.00 0.33 0.33 1.17 1.25 0.40 0.67 0.67 0.60 1.22 0.33 0.70 1.07 0.83 0.43 0.67 0.50 0.57 0.67 1.33 0.87 0.33 0.17 0.17 0.17 1.00 1.17 0.58 0.67 1.00 0.33 0.33 0.50 0.77 0.58 0.67 0.33

0.47 1.85 1.42 1.42 0.75 0.43 0.75 1.58 0.58 1.42 1.08 1.42 0.82 1.58 0.48 1.42 0.40 0.57 0.92 0.64 0.75 0.64 0.72 1.75 0.55 1.25 0.75 0.42 0.63 1.90 0.88 0.98 1.65 1.08 0.92 1.92 0.75 0.42 0.75 0.67 1.05 0.92 0.75 0.92 1.75

1.53 1.53 1.53 1.53 0.33 0.42 0.42 1.17 0.67 0.67 0.67 0.75 0.93 1.73 0.98 1.33 0.73 1.33 0.83 1.38 1.17 0.67 0.42 0.58 1.12 0.67 0.67 0.63 0.33 0.50 0.33 0.83 0.58 0.42 0.42 0.67 0.33 0.50 0.50 0.83 0.50 0.50 0.83 0.50 0.83

1.27 0.47 1.27 1.27 0.83 0.75 0.40 1.45 0.50 1.67 0.67 0.58 0.83 2.00 0.30 1.17 0.57 2.12 0.98 0.92 0.67 0.50 0.48 0.50 1.25 0.92 0.50 0.67 0.67 0.45 0.62 1.00 0.33 0.50 0.42 0.75 0.33 0.83 0.50 0.67 1.17 0.42 0.83 0.67 0.75

0.93 1.10 0.72 0.67 0.50 1.22 1.10 0.67 0.67 0.50 1.33 0.88 0.58 1.60 0.63 0.83 0.53 0.97 1.43 1.65 1.27 1.17 0.97 0.50 0.65 1.17 0.83 0.83 0.67 0.77 0.45 0.67 0.67 0.83 0.67 0.33 1.17 1.50 0.42 1.72 0.48 0.42 0.33 1.17 0.75

At first, we consider the maximum and minimum of each output. Decision-makers who are asked to predict the following three estimates on each output of the jth DMU: (a) the most likely estimate (MLrj ), (b) the optimistic estimate (OPrj ) and (c) the pessimistic estimate (EPrj ). The MLrj is the most realistic estimate of ^yrj . Statistically speaking, it is considered as the mode (the highest point) of the probability distribution for each output. The OPrj is intended to be the unlikely but possible output quantity if everything goes well. It can be considered as an estimate of the upper bound of the probability distribution. The EPrj is intended to be the unlikely but possible output quantity if everything goes badly. It is an estimate of the lower bound of the probability distribution.   yrk ¼ OPrj þ 4MLrj þ PErj =6 (2.1)

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Table 5. Simulation results.

Downloaded by [University of Tehran] at 11:54 30 December 2014

TISa

AMUb

AAc

ALd

N

PE

ML

OP

PE

ML

OP

PE

ML

OP

PE

ML

OP

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

0.0036 0.0038 0.0036 0.0032 0.0030 0.0029 0.0036 0.0029 0.0040 0.0038 0.0036 0.0036 0.0029 0.0045 0.0038 0.0041 0.0039 0.0037 0.0036 0.0034 0.0044 0.0035 0.0037 0.0041 0.0037 0.0030 0.0037 0.0036 0.0029 0.0036 0.0041 0.0036 0.0037 0.0045 0.0032 0.0029 0.0036 0.0036 0.0037 0.0036 0.0032 0.0033 0.0036 0.0032 0.0036

0.0143 0.0145 0.0141 0.0167 0.015 0.0151 0.0146 0.0138 0.0158 0.0139 0.0142 0.0136 0.0158 0.0149 0.0145 0.0146 0.0142 0.0126 0.0143 0.0131 0.0149 0.0147 0.0149 0.0150 0.0148 0.0142 0.0152 0.0140 0.0162 0.0144 0.0153 0.0140 0.0133 0.0141 0.0139 0.0141 0.0141 0.0151 0.0141 0.0143 0.0138 0.0135 0.0133 0.0138 0.0139

0.0330 0.0332 0.0344 0.0354 0.0349 0.0368 0.0368 0.0327 0.0347 0.0374 0.034 0.0331 0.0344 0.0317 0.0356 0.0326 0.0342 0.0332 0.0347 0.0335 0.0324 0.0329 0.0372 0.0337 0.0324 0.0340 0.0358 0.0353 0.0350 0.0360 0.0353 0.0343 0.0366 0.0351 0.0353 0.0339 0.0363 0.0363 0.0353 0.0351 0.0363 0.0357 0.0347 0.0314 0.0328

0.1870 0.1854 0.2110 0.1936 0.2036 0.2089 0.1950 0.2037 0.2025 0.1792 0.1852 0.1844 0.1959 0.2043 0.1829 0.1786 0.1922 0.1994 0.1974 0.1898 0.2002 0.1958 0.1992 0.1893 0.1965 0.1971 0.2066 0.1899 0.1997 0.2047 0.1629 0.1781 0.1821 0.1979 0.2050 0.1836 0.1881 0.1864 0.1928 0.2014 0.1846 0.1892 0.1976 0.2002 0.2056

0.2544 0.2556 0.2547 0.2535 0.2600 0.2581 0.2522 0.2580 0.2532 0.2516 0.2514 0.2496 0.2579 0.2529 0.2516 0.2544 0.2497 0.2543 0.2543 0.2544 0.2532 0.2546 0.2546 0.2518 0.2521 0.2523 0.2557 0.2496 0.2536 0.2571 0.2560 0.2526 0.2518 0.2549 0.2565 0.2541 0.2587 0.2535 0.2515 0.2546 0.2555 0.2560 0.2490 0.2560 0.2560

0.3240 0.3263 0.3260 0.3140 0.3383 0.3268 0.3239 0.3163 0.3097 0.3343 0.3021 0.3144 0.3469 0.3272 0.3338 0.3425 0.3289 0.3385 0.3354 0.3282 0.3067 0.3509 0.3044 0.3109 0.3143 0.3179 0.3149 0.3285 0.3121 0.3207 0.3281 0.3123 0.3171 0.3221 0.3094 0.3175 0.3297 0.3535 0.3100 0.3118 0.3342 0.3451 0.3192 0.3349 0.3200

4.8760 4.8737 4.8740 4.8860 4.8618 4.8732 4.8762 4.8837 4.8903 4.8657 4.8978 4.6992 4.8531 4.8728 4.8637 4.8574 4.8711 4.8615 4.8645 4.9014 4.8932 4.8491 4.8968 4.8908 4.8857 4.8821 4.8851 4.8715 4.8879 4.8793 4.8717 4.8877 4.8829 4.8777 4.8906 4.8789 4.8702 4.8465 4.8900 4.8882 4.8658 4.8566 4.8808 4.8651 4.8800

5.0456 5.0444 5.0457 5.0466 4.9400 4.9419 4.9478 4.9420 5.0468 4.9484 4.9486 4.9504 4.9421 4.9471 4.9484 4.9479 4.9503 5.0457 4.9479 5.0456 5.0468 5.0454 5.0454 4.9482 4.9477 4.9477 4.9443 4.9504 5.0464 4.9121 4.9440 4.9473 4.9482 5.0451 4.9435 5.0459 4.9413 5.0465 4.9485 5.0454 5.0446 4.9440 4.9510 4.9440 4.9440

5.3154 5.8520 6.0320 6.3720 5.8620 6.0032 6.4109 5.9371 6.4702 6.5032 5.9021 6.0931 6.1239 6.3782 5.9064 6.0981 6.2743 6.7329 5.9902 5.8998 5.9087 5.6982 5.9801 6.0961 6.2153 6.3721 6.4961 6.7251 6.0921 6.2135 6.6127 6.8126 6.7231 6.8809 6.9087 6.8765 6.7542 6.3092 7.0021 6.8765 6.9082 6.7092 6.9871 7.0987 7.0090

0.2503 0.2659 0.2595 0.2963 0.2377 0.2221 0.2751 0.2095 0.3462 0.2557 0.2623 0.2618 0.1923 0.3374 0.3168 0.2500 0.2458 0.3039 0.3032 0.2442 0.3235 0.2680 0.2783 0.3585 0.2916 0.2194 0.2789 0.2732 0.2488 0.2763 0.2980 0.2931 0.2606 0.3002 0.2504 0.2339 0.2371 0.2755 0.2956 0.2698 0.2419 0.2038 0.2472 0.2462 0.2584

1.9818 2.0602 1.9402 2.5974 2.0429 2.0458 2.0072 1.8073 2.292 1.8372 1.9467 1.8536 2.2589 2.1935 1.9904 2.0747 1.9406 1.6423 1.9924 1.7413 2.1191 2.0820 2.0768 2.1455 2.0790 1.9433 2.1547 1.9369 2.3574 1.9414 2.1000 1.9051 1.7301 1.9033 1.8406 1.9073 1.8406 2.1026 1.9011 1.9478 1.8136 1.7400 1.7621 1.9139 1.8646

3.2394 3.5063 2.8977 3.5971 3.6496 3.2175 2.8555 3.3356 2.6582 3.3167 2.5589 2.9343 2.5119 2.7241 3.3910 3.9984 3.3512 3.0769 3.3613 3.1162 3.6179 4.1736 4.5600 5.0025 4.7733 4.7916 4.6232 4.7962 5.2576 4.1511 3.9047 3.7693 3.4447 3.1990 4.1135 3.6470 3.3102 3.4783 4.7664 4.3085 3.6166 3.5600 4.6751 3.2873 4.2342

a

TIS: time-in-system. AMU: average machine utilisation. c AA: average available. d AL: average length. b

  brj ¼ OPrj  PErj =6

(2.2)

Results of the stated questionnaires are shown in Table 6. Moreover, mean and standard deviation of each output is obtained based on the Equations (2.1) and (2.2) as shown in Table 7.

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Table 6. Results of questionnaire (safety and environment indicators). Safety factor

Environment output

Downloaded by [University of Tehran] at 11:54 30 December 2014

Deterministic inputs a

b

N

EE

SR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

7.7 7.8 8.1 7.5 8.2 8.3 7.8 7.9 7.9 8.1 8.5 8.4 8.2 7.9 7.8 7.2 7.8 7.4 7.0 7.2 8.9 8.3 8.1 8.4 7.4 8.2 8.2 8.0 8.0 7.0 8.0 8.9 8.2 7.8 7.4 7.5 7.7 7.8 7.9 8.1 8.5 8.3 8.0 7.0 7.4

0.000 7.560 6.450 7.010 6.901 6.910 5.430 5.990 7.000 6.040 6.060 7.030 7.800 7.000 7.000 7.000 7.000 7.000 7.000 6.700 7.000 7.600 7.900 7.700 7.000 6.600 6.600 6.300 7.000 7.100 6.500 7.400 7.300 7.100 7.000 6.300 6.900 7.000 6.100 5.901 6.030 6.660 7.880 8.000 7.891

Air pollution

Tangible pollution

c

N

PE

ML

OP

PE

ML

OP

9.00 9.50 8.00 8.95 8.65 8.77 8.88 8.90 7.89 8.02 7.94 7.68 7.67 9.07 9.04 9.77 9.11 9.25 9.45 9.01 9.00 9.00 9.00 9.00 9.54 8.66 7.00 8.87 8.68 9.70 8.90 8.78 8.56 8.45 7.99 8.05 8.05 7.98 7.56 8.04 9.00 9.33 9.00 9.00 9.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

4 4 3 4 3 4 3 3 3 3 3 3 4 4 4 4 4 4 3 3 3 4 4 3 4 3 3 4 3 4 4 3 4 3 3 3 4 4 3 3 3 3 4 4 3

4.6 5.3 4.9 5.4 4.5 6.3 4.5 4.2 3.8 4.0 5.5 5.0 4.4 4.8 5.7 6.0 4.6 5.3 5.0 5.0 4.4 4.3 5.1 4.5 5.2 5.0 5.0 5.0 5.0 4.7 5.1 4.1 5.6 5.3 5.3 6.0 5.5 5.7 5.4 5.0 5.0 4.0 5.7 5.0 4.8

7 7 7 8 6 8 7 6 5 6 7 6 5 6 7 7 6 7 6 7 6 5 6 7 6 7 7 7 6 7 7 6 7 6 6 7 7 7 6 7 6 7 7 7 6

5 5 5 5 5 4 4 4 5 5 5 4 4 4 4 5 5 5 5 5 5 4 4 4 4 4 5 4 5 4 5 5 4 4 5 5 4 4 4 4 4 4 5 5 5

6.0 6.0 6.0 6.7 6.3 6.0 6.5 5.9 6.0 6.4 5.5 5.5 5.0 5.8 6.0 6.1 6.5 6.9 7.0 5.5 5.4 6.1 5.8 5.5 5.6 6.0 6.0 5.9 6.0 6.2 6.5 6.5 6.0 5.8 6.0 6.0 5.9 6.0 6.0 6.3 5.0 5.6 6.0 6.5 6.7

8 8 8 7 8 7 7 7 7 7 7 6 6 8 8 7 8 8 8 8 7 7 7 7 8 8 8 7 7 7 8 7 7 7 8 7 7 8 7 7 7 7 8 8 8

BD

a

EE: emergency equipment layout. SR: safety rout. c BD: brightness deign. b

5.3 Stochastic DEA results SDEA model discussed in previous section is used to evaluate the efficiency of each layout alternative and optimise the JSFLD problem with stochastic outputs. As mentioned, 45 layout alternatives and 9 performance indicators are considered and analysed by the stochastic DEA model. The SDEA model performs a full ranking on all 45 DMUs. Thus, optimal layout alternative could be obtained. Moreover, the input-oriented CCR model has been consider as the base model of SDEA. The linear programming SDEA model (1.2) has been utilised, and the efficiency and rank of each layout are

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Table 7. Mean and standard deviation of simulation and questionnaire. Output estimate Standard deviation Result of questionnaire

Simulation result

Downloaded by [University of Tehran] at 11:54 30 December 2014

Mean Result of questionnaire

Simulation result

N

TIS

AMU

AL

AA

APa

TPb

N

TIS

AMU

AA

AL

AP

TP

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

48.01 51.47 47.41 40.12 51.03 43.86 49.30 55.80 37.46 53.03 52.17 49.25 44.43 41.80 47.32 43.17 48.12 50.44 48.30 51.73 42.26 45.65 50.86 39.02 49.25 56.23 46.64 48.02 45.54 53.71 44.06 46.26 55.14 45.45 45.65 54.64 48.30 49.85 48.86 50.01 56.17 57.13 53.31 57.05 57.82

0.029 0.029 0.028 0.027 0.026 0.025 0.027 0.023 0.025 0.030 0.023 0.026 0.027 0.024 0.029 0.028 0.032 0.025 0.026 0.026 0.024 0.030 0.023 0.027 0.253 0.253 0.269 0.294 0.027 0.026 0.031 0.026 0.028 0.026 0.024 0.026 0.029 0.029 0.023 0.026 0.028 0.032 0.025 0.028 0.025

0.003 0.003 0.003 0.003 0.003 0.002 0.003 0.002 0.003 0.003 0.002 0.003 0.003 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.002 0.003 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.002 0.003 0.003 0.003 0.002 0.003 0.003 0.003 0.003 0.003 0.003

0.617 0.675 0.632 0.520 0.671 0.600 0.622 0.754 0.687 0.705 0.672 0.642 0.608 0.548 0.409 0.595 0.649 0.662 0.427 0.677 0.555 0.613 0.641 0.509 0.622 0.733 0.611 0.608 0.572 0.678 0.587 0.604 0.719 0.594 0.608 0.704 0.650 0.644 0.614 0.513 0.725 0.777 0.700 0.714 0.536

6.80 5.65 5.65 5.65 5.65 5.65 6.80 4.50 6.50 6.80 6.80 7.20 6.80 8.10 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 6.80 7.20 9.10 5.30 3.50 8.10 8.10 5.30 5.30 5.30 8.10 6.30 6.30 6.30 6.30 6.30

6.9 5.6 5.6 8.4 5.6 5.4 5.6 9.3 7.4 5.6 8.1 5.6 5.6 6.8 6.8 6.9 6.8 6.8 6.9 6.9 6.9 6.9 6.9 5.4 5.4 5.4 3.5 3.5 3.5 9.1 9.1 9.1 9.1 5.5 6.8 6.8 6.8 6.3 8.1 4.5 7.3 8.1 9.1 7.4 8.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

0.016 0.016 0.052 59.95 66.58 66.44 68.35 72.68 63.39 72.16 70.39 73.49 63.18 66.98 68.99 68.5 70.49 79.29 69.73 76.47 67.01 67.88 67.08 66.56 67.57 70.44 65.76 71.28 61.89 69.34 65.17 71.20 75.35 70.86 71.79 70.76 70.85 66.13 70.9 69.8 72.39 73.88 75.42 72.67 71.99

0.255 0.256 0.254 0.254 0.264 0.261 0.255 0.259 0.254 0.253 0.249 0.250 0.262 0.257 0.254 0.256 0.253 0.259 0.258 0.256 0.253 0.261 0.254 0.251 0.253 0.254 0.257 0.253 0.254 0.259 0.253 0.250 0.251 0.257 0.257 0.253 0.259 0.259 0.251 0.255 0.257 0.260 0.252 0.260 0.258

5.062 5.151 5.181 5.241 5.081 5.107 5.180 5.098 5.258 5.194 5.099 5.099 5.124 4.110 5.094 5.125 5.158 5.296 5.108 5.164 5.165 5.121 5.176 5.130 5.149 5.174 5.193 5.233 5.194 5.124 5.210 5.248 5.233 5.323 5.262 5.323 5.232 5.224 5.281 5.324 5.325 5.224 5.279 5.290 5.278

0.505 0.485 0.515 0.385 0.490 0.489 0.498 0.553 0.436 0.544 0.514 0.540 0.443 0.456 0.502 0.482 0.515 0.609 0.502 0.574 0.472 0.480 0.482 0.466 0.481 0.515 0.464 0.516 0.424 0.515 0.476 0.525 0.578 0.525 0.543 0.524 0.543 0.476 0.526 0.513 0.551 0.575 0.568 0.523 0.536

5.6 5.6 3.2 5.6 4.1 5.6 5.6 9.1 5.6 7.2 9.2 7.8 8.1 5.4 8.1 7.8 7.8 7.8 7.8 7.8 7.8 5.4 5.4 5.4 5.4 5.4 6.1 8.1 8.1 8.1 8.1 7.1 7.2 6.3 6.8 3.5 3.5 3.5 3.5 8.1 8.1 8.1 8.1 3.5 3.5

6.8 5.6 5.6 5.6 5.6 5.6 6.8 4.5 6.5 6.8 6.8 7.2 6.8 8.1 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 7.2 9.1 5.3 3.5 8.1 8.1 5.3 5.3 5.3 8.1 6.3 6.3 6.3 6.3 6.3

a

AP: air pollution. TP: tangible pollution.

b

determined based on the stated mathematical programme. As mentioned before, this study, identified 99.9 per cent level of uncertainty for the layout systems by expert judgments due to severe uncertainly that exists in various activities. The results of SDEA which are computed for various risk level (α) by using Lingo® are shown in Tables 8 and 9. The stated tables show that different risk and inspection levels would result into different decision-making process for the layout problems.

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Table 8. Technical efficiency of each layout with different level of risk and specification.

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β = 1.0 N

α = 0.05

α = 0.1

α = 0.2

α = 0.3

α = 0.5

α = 0.7

α = 0.8

α = 0.95

DEA

SDEA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

0.336 0.840 0.814 0.299 0.877 0.905 0.348 0.758 0.900 0.395 0.803 0.902 0.342 0.846 0.929 0.241 0.871 0.911 0.288 0.853 0.894 0.298 0.860 0.926 0.296 0.854 0.920 0.273 0.860 0.914 0.297 0.911 0.891 0.278 0.850 0.937 0.343 0.890 0.934 0.301 0.815 0.860 0.336 0.873 0.870

0.346 0.857 0.833 0.308 0.900 0.923 0.358 0.774 0.920 0.315 0.830 0.921 0.352 0.870 0.943 0.250 0.896 0.929 0.295 0.880 0.913 0.308 0.886 0.940 0.304 0.882 0.956 0.281 0.886 0.931 0.305 0.931 0.907 0.287 0.878 0.950 0.353 0.912 0.948 0.313 0.843 0.876 0.345 0.897 0.890

0.360 0.879 0.858 0.320 0.933 0.949 0.373 0.795 0.947 0.327 0.868 0.947 0.366 0.905 0.963 0.262 0.930 0.954 0.304 0.919 0.939 0.323 0.923 0.961 0.317 0.920 0.958 0.293 0.923 0.954 0.316 0.957 0.931 0.299 0.918 0.967 0.368 0.942 0.967 0.327 0.883 0.897 0.358 0.931 0.918

0.370 0.901 0.876 0.329 0.957 0.967 0.383 0.816 0.967 0.336 0.895 0.967 0.376 0.931 0.976 0.270 0.955 0.971 0.312 0.947 0.957 0.333 0.950 0.975 0.326 0.948 0.973 0.301 0.950 0.971 0.323 0.973 0.948 0.307 0.946 0.979 0.378 0.963 0.979 0.338 0.912 0.914 0.366 0.955 0.939

0.389 0.943 0.910 0.345 0.990 1.000 0.402 0.854 1.000 0.352 0.944 1.000 0.396 0.979 0.998 0.285 0.999 1.000 0.328 0.997 0.989 0.352 0.998 1.000 0.343 1.000 1.000 0.316 0.999 1.000 0.336 1.000 0.981 0.322 1.000 1.000 0.398 0.999 1.000 0.355 0.966 0.946 0.382 1.000 0.977

0.410 0.989 0.948 0.363 1.051 1.042 0.423 0.898 1.035 0.369 0.989 1.035 0.417 1.032 1.048 0.301 1.062 1.044 0.346 1.076 1.028 0.372 1.067 1.043 0.361 1.065 1.047 0.333 1.076 1.046 0.353 1.054 1.020 0.337 1.067 1.047 0.419 1.054 1.058 0.370 1.021 0.979 0.398 1.053 1.018

0.421 1.019 0.972 0.375 1.083 1.064 0.436 0.927 1.058 0.379 1.018 1.057 0.429 1.060 1.077 0.311 1.101 1.072 0.357 1.126 1.052 0.385 1.111 1.071 0.373 1.097 1.076 0.344 1.126 1.075 0.362 1.089 1.043 0.345 1.110 1.076 0.431 1.086 1.093 0.380 1.045 1.001 0.408 1.085 1.045

0.445 1.104 1.043 0.403 1.174 1.124 0.461 1.012 1.127 0.406 1.103 1.125 0.457 1.134 1.159 0.342 1.214 1.149 0.388 1.281 1.120 0.421 1.244 1.149 0.404 1.147 1.160 0.374 1.246 1.158 0.388 1.189 1.111 0.370 1.238 1.162 0.461 1.168 1.189 0.407 1.113 1.062 0.433 1.177 1.119

34 28 29 39 20 1 31 30 1 38 27 1 33 23 18 45 14 1 42 19 21 37 17 12 40 1 1 44 15 12 41 1 22 43 11 1 32 15 1 36 25 26 35 1 24

34 27 29 39 14 9 31 30 10 37 28 11 33 24 3 45 17 7 42 21 13 38 18 4 40 20 5 44 18 8 41 6 15 43 23 1 32 12 2 36 26 25 35 16 22

In order to incorporate stochastic variables into the DEA, we use CCR model with objective function as the maximum of the expected ratio of the weighted output to weighted inputs and reliability function subject to several chance constraints. Therefore, the constraints and objective were reformulated by CCP proposed by Charnes and Cooper. In the stochastic DEA models of these papers, outputs are stochastic variables. Hence, our formulation presented in this study can be considered as stochastic DEA. All inputs and outputs are based on Tables 5–7, and model 1.10 is utilised. The technical efficiency levels are shown in Tables 8 and 9 based on different α and β. The rank of each DMU based can then be determined. Noise was used for selection of β and α parameters. At first, one cell of data was selected by chance, and data of this cell were changed drastically (multiplied by 10), and then, the efficiency of DMUs was computed. Then, the average efficiency levels (α) of 45 DMUs for the two cases were compared (with and without noise) by correlation

16

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Table 9. Technical efficiency of each layout with different levels of risk and specification.

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β = 0.7

β = 0.8

Rank

N

α = 0.05

α = 0.1

α = 0.2

α = 0.3

α = 0.05

α = 0.1

α = 0.2

α = 0.3

SDEA

PCA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

0.2352 0.5882 0.5700 0.2093 0.6136 0.6333 0.2436 0.5304 0.6299 0.2137 0.5621 0.6314 0.2394 0.5922 0.6504 0.1690 0.6100 0.6373 0.2016 0.5973 0.6256 0.2084 0.6019 0.6481 0.2069 0.5975 0.6437 0.1911 0.6018 0.6395 0.2082 0.6378 0.6236 0.1946 0.5953 0.6559 0.2401 0.6230 0.6535 0.2110 0.5705 0.6021 0.2353 0.6109 0.6088

0.2425 0.6005 0.5835 0.2157 0.6309 0.647 0.2512 0.5424 0.6447 0.2206 0.5821 0.6454 0.2467 0.6103 0.6607 0.1751 0.6279 0.6509 0.2067 0.6172 0.6399 0.2159 0.6212 0.6589 0.2134 0.6183 0.6557 0.1972 0.6211 0.6521 0.2138 0.6523 0.6358 0.2009 0.6157 0.6654 0.2476 0.6390 0.6642 0.2192 0.5911 0.6135 0.2419 0.6287 0.6235

0.2517 0.6155 0.6006 0.2238 0.6530 0.6643 0.2610 0.5568 0.6632 0.2288 0.6077 0.6630 0.2561 0.6338 0.6738 0.1831 0.6509 0.6680 0.2129 0.6431 0.6574 0.2258 0.6462 0.6725 0.2218 0.6441 0.6706 0.2047 0.6462 0.6679 0.2209 0.6695 0.6517 0.2089 0.6423 0.6771 0.2573 0.6592 0.6770 0.2292 0.6177 0.6280 0.2503 0.6515 0.6423

0.2588 0.6304 0.6134 0.2299 0.6696 0.6770 0.2683 0.5711 0.6768 0.2350 0.6267 0.6769 0.2635 0.6518 0.6833 0.1892 0.6682 0.6800 0.2183 0.6629 0.6697 0.2332 0.6650 0.6824 0.2282 0.6636 0.6813 0.2104 0.6650 0.6794 0.2261 0.6814 0.6637 0.2148 0.6624 0.6854 0.2647 0.6739 0.6856 0.2366 0.6380 0.6401 0.2564 0.6686 0.6570

0.2688 0.6722 0.6515 0.2392 0.7012 0.7238 0.2784 0.6062 0.7120 0.2443 0.6424 0.7215 0.2736 0.6767 0.7433 0.1932 0.6971 0.7284 0.2304 0.6826 0.7150 0.2382 0.6879 0.7407 0.2365 0.6828 0.7356 0.2184 0.6878 0.7308 0.2379 0.7290 0.7127 0.2224 0.6803 0.7496 0.2744 0.7120 0.7468 0.2411 0.6520 0.6881 0.2690 0.6982 0.6958

0.2771 0.6863 0.6668 0.2465 0.7210 0.7394 0.2871 0.6199 0.7368 0.2521 0.6653 0.7376 0.2820 0.6975 0.7552 0.2001 0.7176 0.7439 0.2363 0.7053 0.7313 0.2468 0.7099 0.7530 0.2439 0.7066 0.7494 0.2253 0.7099 0.7452 0.2444 0.7454 0.7266 0.2296 0.7037 0.7604 0.2829 0.7303 0.7591 0.2506 0.6756 0.7011 0.2765 0.7185 0.7126

0.2876 0.7034 0.6864 0.2558 0.7463 0.7591 0.2983 0.6363 0.7580 0.2615 0.6946 0.7578 0.2927 0.7243 0.7701 0.2093 0.7439 0.7634 0.2433 0.7350 0.7513 0.2581 0.7385 0.7685 0.2535 0.7361 0.7664 0.2340 0.7385 0.7633 0.2525 0.7652 0.7448 0.2388 0.7340 0.7738 0.2941 0.7533 0.7737 0.2619 0.7060 0.7177 0.2860 0.7446 0.7341

0.2958 0.7205 0.7011 0.2628 0.7653 0.7737 0.3066 0.6527 0.7735 0.2685 0.7162 0.7736 0.3011 0.7449 0.7809 0.2163 0.7637 0.7772 0.2495 0.7576 0.7653 0.2666 0.7601 0.7799 0.2608 0.7584 0.7786 0.2405 0.7600 0.7765 0.2584 0.7787 0.7585 0.2455 0.7571 0.7834 0.3025 0.7701 0.7835 0.2704 0.7292 0.7315 0.2930 0.7641 0.7509

34 27 29 39 14 9 31 30 10 37 28 11 33 24 3 45 17 7 42 21 13 38 18 4 40 20 5 44 18 8 41 6 15 43 23 1 32 12 2 36 26 25 35 16 22

34 28 29 39 14 12 31 30 4 37 26 5 33 23 11 45 16 3 42 20 21 38 17 10 40 13 6 44 18 9 41 1 22 43 19 8 32 7 2 36 25 27 35 15 24

experiments. The larger value was selected as a risk level, and this method was repeated ten times. Therefore, risk level 0.05 with highest value amongst the other risk levels was selected for finding efficiency of each DMU. It is assumed that β equals 1 because the result of SDEA must be compared with DEA. Conventional DEA belongs to the risk-natural category (α = 0.5) under β = 1. Therefore, results of SDEA are compared with conventional DEA under aspiration level of 1 and risk level of 0.05 and the ranking results are shown in Tables 8 and 9. SDEA have different ranking in each risk and aspiration levels, and therefore, the results are integrated with PCA. The results of different risk levels under aspiration level 0.7 are considered as PCA indicators (Table 8). It is observed that by incorporating the stochastic indicators to the JSFLD problem, the ranking results have been considerably changed. For instance, layout alternative 36 is recognised as the most efficient layout by the proposed integrated approach, whilst the obtained ranks by PCA methods were 32, respectively. On the other hand, layout alternative 39 that

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Figure 3. Ranks of SDEA vs. DEA.

Table 10. The features of the SDEA–simulation approach vs. other methods.

Feature Method SDEA– Simulation approach DEA Approach Conventional Simulation Azadeh et al. (2010) Sueyoshi (2000) Bretholt and Pan (2013) Wu and Lee (2010)

Optimisation

Stochastic data modelling

High precision and reliability

Flexibility: Crisp simulation

Multi-criteria decision-making through DEA

Practicability in real cases

√

√

√

√

√

√

√

√ √

√

√

√

√

√

√

√

√

√

√

√

√

√

Multiple inputs and outputs

√

√

√ √

√

√

√ √

√

took the best rank amongst all alternatives using SDEA and PCA methods has been recognised as the 2nd ranked alternative by the proposed integrated approach. It is concluded that the existent difference between the results of the integrated simulation–stochastic DEA approach and the previous studies is due to its comprehensive standpoint in stochastic modelling the JSFLD problems. The ranking results show that considering stochastic operational indicators (simulation results), and HSE indicators provide more comprehensive insight to the decision-making process in JSFLD problems with stochastic outputs. Moreover, it should be noted that performing exact ranking amongst all layout alternatives could help policy makers and top managers to have precise understanding and improve existing systems with respect to facility layout performance. It must be stated that Vip plan opt Software was used for layout generations. The results of SDEA vs. DEA are shown in Figure 3. This means incorrect managerial decisions may occur if DEA is used as decision-making criteria.

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6. Conclusion There are usually stochastic data with respect to layout problems. This means data could not be collected and analysed by deterministic models, and therefore, new approaches for tackling such problems are required. This gap motivated the authors to develop a unique approach to handle such gaps in JSFLD problems. This study presented a unique approach based on computer simulation and SDEA to tackle the JSFLD problems with stochastic data in manufacturing systems. Moreover, this study considered operational and HSE indicators for evaluating the generated layout alternatives. An integrated simulation approach was then used to model the process with respect to the operational data (average time-in-system and, average length and AMU). Finally, the CCR input-oriented SDEA model was used to find the optimal layout design. In the proposed stochastic DEA, each layout alternative has been considered as a DMU. The results show that the proposed integrated approach provides an efficient approach in solving JSFLD problems with stochastic by incorporating a set of operational and dependent indicators (HSE). Moreover, the integrated approach presented in this study yields exact rankings whereas previous studies present incomplete and non-exact plant layout alternatives. The superiority and effectiveness of the proposed approach was quantitatively compared with PCA studies through a case study. The proposed approach would help policy makers and top managers to have a more comprehensive and thorough understanding the layout design aspects with respect to the operational features of the manufacturing processes. Although the proposed approach might be relatively time-consuming, it could be applied in real world problem as we mentioned in this paper due to its aforementioned advantages. Furthermore, benefits of the FLD optimisation will justify time and work which is used to implement the proposed approach. The integrated simulation–stochastic DEA approach is also compared with some of the relevant studies and methodologies in the literature. The approach is capable of dealing with operational indicators as well as uncertain indicators. It can handle complex layout problems in manufacturing systems due to utilisation of discrete-event simulation. Also, it has the ability of optimisation layout problems due to utilisation of SDEA which is able to find the optimal layout solution through ranking DMUs (i.e. layout alternatives) based on stochastic outputs. In addition, it provides a comprehensive and robust approach in solving real world JSFLD problems (Table 10). Acknowledgements The authors are grateful for the valuable comments and suggestions by the respected reviewers, which have enhanced the strength and significance of this work. This study was supported by a grant from the Iran National Science Foundation [grant number 93010029]. The authors are grateful for the financial support provided by the Iran National Science Foundation.

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