A novel algorithm for layout optimization of injection ...

Journal of Manufacturing Systems 33 (2014) 287–302

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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys

A novel algorithm for layout optimization of injection process with random demands and sequence dependent setup times A. Azadeh ∗ , S. Motevali Haghighi, S.M. Asadzadeh School of Industrial Engineering and Center of Excellence for Intelligent Based Experimental Mechanic, College of Engineering, University of Tehran, Iran

a r t i c l e

i n f o

Article history: Received 23 February 2013 Received in revised form 13 October 2013 Accepted 31 December 2013 Available online 11 February 2014 Keywords: Layout optimization Discrete event simulation Stochastic data envelopment analysis Stochastic demand Dependent set-up times Injection molding

a b s t r a c t Injection molding is an ideal manufacturing process for producing high volumes of products from both thermoplastic and thermo setting materials. Nevertheless, in some cases, this type of manufacturing process decelerates the production rate as a bottleneck. Thus, layout optimization plays a crucial role in this type of problem in terms of increasing the efficiency of the production line. In this regard, a novel computer simulation–stochastic data envelopment analysis (CS-SDEA) algorithm is proposed in this paper to deal with a single row job-shop layout problem in an injection molding process. First, the system is modeled with discrete-event-simulation as a powerful tool for analyzing complex stochastic systems. Then, due to lack of information about some operational parameters, theory of uncertainty is imported to the simulation model. Finally, an output-oriented stochastic DEA model is used for ranking the outputs of simulation model. The proposed CS-SDEA algorithm is capable of modeling and optimizing non-linear, stochastic, and uncertain injection process problems. The solution quality is illustrated by an actual case study in a refrigerator manufacturing company. © 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

Motivation and significance

1. Introduction

There are usually incomplete and stochastic data or lack of data with respect to layout problems. This means data could not be collected and analyzed by deterministic models. So, new approaches for tackling such problems are required. This gap motivated the authors to develop a unique approach to handle such gaps in layout problems. One assumption made in the vast majority of research on the job shop problem is that setup times are considered as a component of processing times. However, in several actual scheduling problems, it may not be realistic to assume this hypothesis. On the other hand, setup times for each job depend on type of the job and previous job that has been processed on the current machine. Consequently, it is more realistic to consider setup times separated from processing times. This gap motivated the authors to considered sequence dependent setup times for each machines. This is the first study that introduces an integrated CS-SDEA as an optimization approach for handling random demands and sequence dependent set up times with respect to layout problems in a special case of job shop manufacturing system.

Layout design in manufacturing systems is a crucial task in redesigning, expanding, or designing the system for the first time. Major considerations in designing a manufacturing layout can be minimizing material handling costs, frequency of products and employees among workstations, smoothing production, and providing a safe workplace for employees. The layout problem in manufacturing systems involves determining the location of machines, workstations, rest areas, inspection rooms, clean rooms, heat treatment stations, offices, and tool cribs to achieve the following objectives: minimization of the transportation costs of raw material, parts, tools, work-in-process, and finished products among the facilities [1,2], facilitate the traffic flow and minimization the costs of it [3], maximization of the layout performance [4], minimization of the dimensional and form errors of products depending on the fixture layout [5,6], minimization of the total number of loop traversals for a family of products [7] increasing the employee morale, minimization of the risk of injury of personnel and damage to property, providing supervision and face-to-face communication [8]. In this paper, a refrigerator manufacturing company is studied. The company in which the study took place is a major manufacturer of home appliances, especially refrigerators. A main process in manufacturing refrigerators is the injection in which the foam is injected between the metal body and plastic tub. The molded part

∗ Corresponding author at: Department of Industrial & Systems Engineering, University of Tehran, P.O. Box 14178-43111, Iran. Tel.: +98 21 88967810; fax: +98 21 66461680. E-mail address: [email protected] (A. Azadeh).

0278-6125/$ – see front matter © 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmsy.2013.12.008

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is then cooled and forms the final product. The case of injection molding process under study is used for producing four dissimilar types of refrigerator with different technical specifications in a feeder-line before transporting them to the assembly line. The injection molding process is composed of a sequence of manual and automated operations. This process comprises five stages including mold closing, filling, packing–holding, cooling and mold opening are preceded repeatedly for each product model (see [9]). Each type of refrigerators including 15, 16, 18 and 20 feet, has a specific fixture. The 15 feet refrigerators needs one fixture and other refrigerators need two fixtures. At first, product is placed in the fixture(s) and a conveyor transports the non-injected subassemblies (product in fixture(s)) to the hot room for preparing. There are four especially stations in this assembly line for each type of refrigerator; in the hot room, the non-injected subassemblies with fixture transport to station and wait for injection. In the next step the injecting machine charged to injected foam between the metal body and plastic tub (product in fixture(s)). After injection and cooling, fixtures are opened and injected subassemblies are transported to the next stages of manufacturing process. To remain competitive in the market, the company under study is preparing to improve the efficiency of its production line. Meanwhile, injection molding process, as a bottleneck, considerably decelerates the production rate and decreases the production line efficiency. Thus, the problem is to find and implement the optimal layout of stations so that the overall processing time is minimalized. In order to cope with aforementioned single-row facility layout problem (SRFLP), this paper presents a novel algorithm based on discrete-event-simulation and stochastic data envelopment analysis (SDEA). The proposed integrated algorithm consists of two main steps. First, simulation is used to model the process of foam injection. Discrete-event-simulation is known as a powerful and flexible tool for modeling, visualizing, and manipulating complex systems. With the aid of the proposed discrete-event-simulation model, key performance indicators of the system can be simply evaluated. In the second step, SDEA-output oriented model is utilized to rank different layout formations with respect to a set of key performance indicators obtained from the simulation models in order to determine optimum solutions. In this SRFLP, each layout is considered as a decision-making unit (DMU). Queue length (QL), machine utilization (MU), and time in system (TIS) are defined by the decision-makers of the company as primary evaluation measures. These indicators are considered as outputs of the SDEA model. The proposed SDEA approach specifies the strength and weakness of each layout formation in terms of technical efficiency. This in turn, helps the decision-makers to make right decisions regarding to various layouts and find the optimal one. The rest of this paper is organized as follows; Section 2 provides a review of the past literature in the area of this work. The proposed CS-SDEA algorithm is presented in Section 3. In Section 4, the implementation procedure of the simulation in the case of injection process has been investigated. A new stochastic DEA model is presented in Section 5. Section 6 provides the computational results of this study. Performance comparison is shown in Section 7 and the conclusions are presented in Section 8.

2. Literature review Optimization of facility layout has attracted many researchers and practitioners. Various methodologies have been presented in literature to deal with such problems. The related literature to layout optimization problem can be divided into fifth main categories. The first category is mathematical programming which is employed in several studies. In this regard, a mathematical programming model was presented by Wang et al. [10] to minimize

the total material handling distance on the shop floor. Their model could be applied in both inter-cell and intra-cell material handling distances for cellular manufacturing systems (CMSs). Mak et al. [11] developed a mathematical programming model to examine the machines’ for typical job-shop and flow-shop manufacturing environments. In their model pattern of material flow were considered. Georgiadis et al. [12] presented a general mathematical programming approach to solve the equipment allocation problem in a given two and three dimensional spaces. Their proposed problem was formulated using a Mixed-Integer Linear Programming (MILP) model so that equipment with various sizes and geometries were considered with the objective of minimizing total transportation, connection, land and floor construction costs. This model can be used in two and three dimensional spaces. Patsiatzis and Papageorgiou [13] presented a general mathematical programming formulation for the multi-floor process plant layout problem considering a number of costs, management, and engineering drivers within the same framework. They did not consider time as an important factor in their model. Solimanpur et al. [14] presented a mathematical programming formulation as a Quadratic Assignment Problem (QAP) for material flow among the cells in order to tackle the inter-cell layout problem. But intra-cell problem did not consider. A new mathematical programming model was developed by Castillo and Peters [15] in which the layout design and production planning are integrated to prescribe efficient multi-bay manufacturing facilities. This model is more useful for industrial manufacturing. Abboud et al. [16] surveyed mathematical programming models and methods used for the physical layout of printed circuit boards, in particular component placement and wire routing. Tavakkoli-Moghaddam et al. [17] presented a mathematical model to solve a facility layout problem in CMSs with stochastic demands in order to minimize total costs of inter and intra-cell movements in both machine and cell layout problems, simultaneously. Their study can be applied in uncertainty environment. A Mixed-Integer Nonlinear Mathematical Programming (MINLP) model was proposed by Solimanpur and Jafari [18] for determining the optimum layout of machines in a two-dimensional area considering four parameters including (a) production capacity of machines, (b) multiple machines of each type (machine redundancy), (c) processing route of parts, and (d) dimensions of machines. This MINLP model cannot be applied in stochastic environment. Sanjeevi and Kianfar [19] presented a polyhedral model of the triplet formulation of the SRFLP introduced by Amaral [20]. A multi-objective job shop problem was introduced in stochastic environment by Lei [21]. He proposed a multi-objective algorithm with lowest makespan and total tardiness ratio, but he did not consider stochastic factor for layout optimization. An entropy function was introduced by González-Cruz and Gómez-Senent Martínez [22], this function analyzed relationship between elements. This algorithm generates possible layout and select optimal layout with lowest entropy. Their model can be used in crisp environment. Computer simulation modeling is another widely used approach to deal with layout optimization problems. A systematic method based on computer simulation was presented by Savsar [23] to solve flexible facility layout problems, in which objectives such as minimization of total material handling costs, maximization of total closeness ratings between departments, minimization of expected future re-layout costs, and minimization of expected total material handling costs. In This model, possible future re-layouts were considered. Morris and Tersine [24] employed simulation to compare the process and cellular layouts in a dual resource (labor and equipment) constrained environment and examined the impact of the dual resource constrained shop on the relative performance of cell layouts vis-a-vis process layouts. They did not consider important key performance such as setup time in their model. Koh et al. [25,26] described a direct database-simulation which used the database


data model as the simulation model for solving a simulation-based JSSP. A facility layout optimization technique was presented by Azadivar and Wang [27] that considered the dynamic characteristics and operational constraints of the system as a whole. Their proposed approach was able to solve the facility layout design problem based on system’s performance measures, such as the cycle time and productivity. An optimization-oriented simulation-based approach for JSSP was proposed by Arakawa et al. [28] which incorporated capacity adjustment function with the aim of eliminating tardy jobs. The operational factors were not considered in their model. Pagell and Melnyk [29] studied the improvement of the overall operation of a service process, as found in one regional blood center, in which computer simulation was used to stimulate a critical analysis of the process. They investigated three layouts including the existing worker paced assembly line, a modified assembly line, and service cells. Tavakkoli-Moghaddam and Daneshmand-Mehr [30] presented a simulation model for solving JSSPs with the objective of makespan minimization. In the case of scheduling problems, using computer simulation models makes it possible to compare different rules and procedures, and so develop greater insight into the shop operations. Vinod and Sridharan [31] employed simulation to study scheduling rules for JSSP with sequence dependent setup times. Jithavech and Krishnan [32] presented a simulation-based method for predicting the uncertainty associated with the layout and validated their simulation approach against analytical procedures. In another study by Vinod and Sridharan [97], a discrete-event simulation model was developed to deal with a JSSP considering five new setup-oriented scheduling rules. Hence, simulation has attracted researchers to use it for solving different complex scheduling problems. The third category in literature is related to heuristic and metaheuristic and their applications in layout problem. Sim et al. [33] applied neural network (NN) approach to the dynamic JSSP. An NN was developed by Willems and Rooda [34] to solve deterministic JSSPs. Sabuncuoglu and Gurgun [35] proposed an NN approach to minimize mean tardiness and makespan in JSSP. Yang and Wang [36] presented an adaptive NN and heuristics hybrid approach for solving JSSP. Kumar et al. [98] presented a constructive heuristic to solve the single row facility layout problem (SRFLP) with the objective of minimizing the materials handling cost. In their proposed approach, the facilities with the highest frequency of parts between them and their adjacent locations were prior in adding to the solution sequence. These studies cannot be applied in uncertainty and stochastic environment. Ho and Moodie [37] proposed a two-phase heuristic procedure based on simulated annealing technique for solving a SRFLP within an automated manufacturing system. They took into consideration different evaluation criteria such as minimization of total flow distance and maximization number of in-sequence movements for finding the optimal layout formation. Genetic algorithm (GA) is a heuristic search method that is broadly used in various NP-hard combinatorial optimization problems. It has been shown in literature that various facilities layout optimization could make use of GAs so as to find high quality solutions. Krishnakumar and Melkote [38] introduced a layout optimization technique based on GA to minimize the deformation of the machined surface. Vallapuzha et al. [39] presented a GA-based optimization method that uses spatial coordinates to represent the locations of machines. Solimanpur et al. [40] proposed a binary programming approach to model the single row facility layout problem (SRFLP) in which the distances taken between machines were considered to be sequence dependent. They also applied an ant algorithm to solve the SRFLP. Chan et al. [41] studied multifactory and multi-product environment. They wanted to reduce production cost and management risk. For this aim, they used GA algorithm and proposed a new crossover mechanism that improved performance of GA. In another study, Chan et al. [42–44] introduced

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a flexible job-shop-scheduling problem. They considered resource constraints for this problem. To do this, GA algorithm was applied to solve assignment and scheduling problem. Their study can be used in flexible shop floor environment. Chan et al. [42–44] proposed a new GA algorithm for solving flexible manufacturing system (FMS) scheduling. In this study, maintenance factor was considered as important index. They considered two subjects: allocation of jobs to suitable factories and determination of the corresponding production scheduling in each factory. Maximizing of the system efficiency was the objective of this study. Their model is more realistic than classical production scheduling problem. Samarghandi et al. [95] proposed a particle swarm optimization algorithm to solve the SRFLP’s. Their proposed algorithm was able to map discrete feasible space of SRFLP’s to a continuous space through a novel encoding and decoding technique. In another study, Samarghandi and Eshghi [45] defined a theorem for finding the optimal solution of SRFLP’s in a particular case. They also proposed a new tabu search (TS) algorithm for solving the SRFLP and reducing the computational time in comparison with formerly used algorithms as well. Datta et al. [46] propose a new genetic algorithm that improved several previously algorithm. Aiello et al. [47] proposed a new multi-objective genetic algorithm. They considered Material handling costs, aspect ratio, closeness and distance requests as objective. A new neighborhood search heuristic (LK-INSERT) based on minimizing weighted sum of the distance between facilities proposed by Kothari and Ghosh [48]. Heuristic and meta-heuristic model are used for large size problem, in these studies, authors considered layout problem in deterministic environment. But in the real world, there are usually incomplete and stochastic data or lack of data with respect to layout problems. Multi-criteria decision making is the fourth and absolutely the most important category of literature related to this work, which has been widely used in different research works in the literature for operations evaluation and ranking of decision making units (DMUs). Andersen and Petersen [49] proposed a procedure called the super-efficiency method for ranking DEA efficient units. Superefficiency models are used to determine critical outputs. Different super-efficiency DEA models are introduced by Seiford and Zhu [50]. A complete list of super-efficiency DEA models is provided, in which the necessary and sufficient conditions are developed for the infeasibility of various super-efficiency DEA models. Superefficiency models have been deeply researched in the DEA literature [51–57]. Sueyoshi [58], introduce a stochastic DEA, the output of DEA is stochastic. He applied Chance Constraint Programming (CCP) and the estimation technique of PERT/CPM for stochastic mathematical programming. Tone [59] employed the efficiency Slacks-based Measure (SBM) to address super-efficiency models, which was different from Andersen and Petersen model. The proposed model of Andersen and Petersen [49] directly considered the slacks in inputs/outputs, while Tone [59] did not take into account the existence of slacks. Fuzzy DEA (FDEA) has been widely used in different research works in the literature for operations evaluation and ranking of DMUs [60,61]. Li et al. [57] theoretically provided a super-efficiency model to overcome some deficiencies in the previous models. Khodabakhshi [62] studied super-efficiency models on the basis of improved outputs. His proposed model was different from the proposed model of Andersen and Petersen [49], because it was not limited to use the input data for evaluating DMUs and could be used to obtain a total ordering of DMUs. Khodabakhshi et al. [63] developed an input-oriented super-efficiency measure in stochastic DEA by providing a deterministic equivalent of the stochastic super-efficiency model. They showed that their proposed deterministic model can be converted to a quadratic program and also presented a sensitivity analysis of their model. But in the real world, there are usually incomplete and stochastic data or lack of data. So, stochastic model can be showed more realistic rank of DMUs in stochastic and uncertainty environment.

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The hybrid and integrated model for solving facility layout problem is the another category. Gen et al. [64] provided a fuzzy multi-row machine layout formulation, in which the clearance between any two adjacent machines was given as a fuzzy set. Their proposed objective function was to maximize the minimum grade of satisfaction over machines and meanwhile, to minimize the total travel cost among machines, to do this, GA was investigated as a heuristic technique to solve the problem and computer simulation was used to demonstrate the performance of the proposed algorithm. Their integrated model can be applied in fuzzy system. Deb and Bhattacharyya [96] presented a distinct decision support system based on multifactor Fuzzy Inference System (FIS) for the development of facility layout with fixed pickup/drop-off points. Enea et al. [65] discussed the facility layout design problem taking into account the uncertainty of production scenarios using fuzzy numbers and the finite production capacity of the departments. Chan et al. [42–44] introduced machine-part grouping and cell layout problem. They considered a two-stage approach for this problem. In their model, machine cells and part families was first stage and machining sequence was considered in second stage. The objective was minimizing intercellular and intracellular part movements. In this study, linear mathematical and QAP model were applied. Also, GA algorithm was considered for this NP-hard problem. So, this model can be used for complex model. Yang and Feng [66] presented a multi-level warehouse layout problem under fuzzy environment. In their model, the monthly demand of each product type and the horizontal distance traveled by clamp track were considered to be fuzzy variables. A chance-constrained programming model was employed to minimize the total transportation costs as the objective function of the problem based on the credibility measure. Then, tabu search (TS) algorithm based on the fuzzy simulation was designed to solve the problem. Lian et al. [67] proposed a particle swarm optimization (PSO) algorithm to solve JSSP with the objective of minimizing makespan. They also used new valid algorithm operators and computer simulation to find the effectiveness of them in finding the optimal solution in medium sized JSSPs. Ertay et al. [61] considered quantitative and qualitative criteria for layout alternative. Analytic hierarchy process (AHP) method is applied to transform quantitative criteria to qualitative. After this processing DEA is applied to calculated efficiency score for each layout, then the best layout was selected, according to these criteria. Jeong et al. [68] presented a hybrid GA-simulation approach in which the GA was used for optimization of schedules, and the simulation was used to minimize the maximum completion time for the last job with fixed schedules from the GA. Zhang et al. [69] proposed a neighborhood structure to solve the JSSP by tabu search (TS) approach. They used neighborhood structure combined with the appropriate move evaluation strategy, and tested it with TS to find better upper bounds among some unsolved instances. A TS approach combined with simulated annealing (SA) and was proposed by Zhang et al. [4] to provide a robust and efficient methodology for the JSSPs. In their proposed approach, SA was used to find the best solutions inside big valley in which TS was enabled to re-intensify search from the promising solutions. A hybrid GA was presented by Gao et al. [70] for the Fuzzy Job Shop Scheduling Problem (FJSSP) with three objectives: makespan minimization, maximal machine workload minimization, and total workload minimization. Weckman et al. [71] developed an NN scheduler in which genetic algorithms (GAs) to generate optimal schedules to a known benchmark problem. An integrated multivariate and multi-attribute analysis approach based on AHP and PCA was proposed by Azadeh and Izadbakhsh [72] for solving plant facility layout design (FLD) problems. Yang and Kuo [60] and Azadeh and Izadbakhsh [72] considered three quantitative performance indicators in a Flow Shop Facility Layout Design (FSFLD) problem including distance, adjacency, and

shape ratio, and three qualitative performance indicators including flexibility, accessibility, and maintenance. Chang et al. [73] applied the fuzzy set theory to deal with the ambiguity and vagueness of linguistic expression in the office layout problem using an improved and Efficient Fuzzy Weighted Average (EFWA) algorithm. A modified shifting bottleneck heuristic (MSBH) was proposed by Topaloglu and Kilincli [74] in order to minimize makespan in a reentrant JSSP. They proposed a new sequencing heuristic for the single machine problem with maximum lateness sub-problem to handle large-size problems. Zhou et al. [75] introduced an integrated general simulation to model the space, logistics and resource dynamics with genetic approaches (GAs) for optimizing the layout based on various constraints and rules, and implementing a site layout optimization system within a simulation environment. An integrated Fuzzy Data Envelopment Analysis Fuzzy Simulation (FDEAFS) in cellular manufacturing was presented by Azadeh et al. [76]. They compared U shaped layout with other types, namely spiral, W, Z, zigzag and L for optimization of operator allocation in cellular manufacturing systems. A novel competitive co-evolutionary quantum genetic algorithm (CCQGA) was proposed by Gu et al. [77] to deal with a Stochastic Job Shop Scheduling Problem (SJSSP) for minimizing the expected value of makespan. Azadeh et al. [9] proposed an integrated FSFDEA algorithm to cope with a special case of single-row facility layout problem. The proposed FSFDEA algorithm is capable of modeling and optimizing small-sized SRFLP’s in stochastic, uncertain, and non-linear environments. An integrated nonlinear programming model (NLP) and AHP was considered for facility layout design. Hadi-Vencheh and Mohamadghasemi [78] used computer-aided layout-planning to generated layout alternative, the quantitative criteria were introduced for each layout and AHP was applied to determine weights of criteria. Then NLP model was proposed to solve facility layout by considering these criteria. An integrated analytic hierarchy process (AHP) and preference ranking organization methods (PROMETHEE) were used for evaluation facility layout problem by Yang and Deuse [79], in this approach; AHP applied for determining of the weights for each feature and PROMETHEE method ranked the layout by using these criteria. Another integrated methodology can be applied in facility layout design is synthetic value of fuzzy judgments and nonlinear programming (SVFJ–NLP). Mohamadghasemi and HadiVencheh [80] used SVFJ to collect performance measures related to qualitative criteria and then applied NLP to solve facility layout design. A stochastic job shop problem with normal processing time is introduced by Lei [81]. An efficient genetic algorithm (GA) is applied for this problem and the algorithm is compared with simulated annealing (SA) and a particle swarm optimization (PSO). Results of the comparisons show that GA has a better performance in stochastic job shop problem. In the most integrated method that introduced, authors did not considered stochastic keys performance index for job shop problem. There are cases where data could not be collected and analyzed by deterministic models. Therefore, new approaches for tackling such problems are required. This gap motivated the authors to develop a unique approach to handle such gaps in layout problems. One assumption made in the vast majority of research on the job shop problem is that setup times are considered as a component of processing times. Also, setup times for each job depend on type of the job and previous job that has been processed on the current machine. Thus, it is more realistic to consider setup times separated from processing time. This gap motivated the authors to considered sequence dependent setup times for each machines. This is the first study that introduces an integrated computer simulation–stochastic DEA (CS-SDEA) as an optimization approach for handling imprecision and non-linearity of layout problems in a special case of job shop manufacturing system. A novel stochastic output-oriented DEA model has been proposed to provide an optimal solution for the layout problem.


This approach specifies the strength and weakness of each layout within the manufacturing systems, in terms of technical efficiency. This in turn, helps decision-makers to make appropriate decisions about different layout alternatives to find the optimal one. The applicability and robustness of the proposed approach have been demonstrated in a refrigerator manufacturing company. To the best of our knowledge, this is the first study that presents an integrated CS-SDEA approach for optimizing the layout of a typical manufacturing process with random demands and sequence dependent set up times. 3. The integrated CS-SDEA algorithm This paper presents an integrated simulation–stochastic DEAmulti-attribute approach to deal with the SRFLP problems with stochastic outputs. First, simulation is used to model the process of foam injection. Discrete-event-simulation is known as a powerful and flexible tool for modeling, visualizing, and manipulating complex systems. With the aid of the proposed discrete-eventsimulation model, key performance indicators of the system can be simply evaluated. In the second step, SDEA-output oriented model is utilized to rank different layout formations with respect to a set of key performance indicators obtained from the simulation models in order to determine optimum solutions. 3.1. General framework An integrated simulation–stochastic DEA-multi-attribute approach is defined for SRFLP problem. Fig. 1 presents a schematic view of the proposed approach. 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 and space of each machine. Step 2: generate all possible layout alternatives. Step 3: collect the required data for the manufacturing process, such as processing times and traverse times between sequential machines, which can be obtained from expert judgments, documents and history of the manufacturing plant. Step 4: develop 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 plants. Step 5: analyze and retrieve optimistic (OP), most likely (ML) and pessimistic (PE) of three operational indicators (TIS, QL, and MU) from simulation model to be used for further analysis in stochastic DEA. Step 6: define mean and standard deviation of each operational indicators as outputs of SDEA. Step 7: compare stochastic DEA rankings with deterministic DEA to see if there is a significant difference between results. Step 8: verify and validate the stochastic DEA. Step 9: select the optimum layout design through noise analysis. 4. Injection molding process simulation Different simulation languages are available in the literature and market. In this paper, Visual SLAM is the simulation language used for solving the layout optimization problem. The structure of this language is based on network modeling. Therefore, adding or removing some elements in different sections of the model is an easy task [82]. In the presented problem, based on the network model, the products and machines are considered as entities and servers, respectively. Descriptions of network components

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are derived from Pritsker and O’Reilly [82]. Drake and Smith [83] introduced a framework for online simulation systems in operational planning, scheduling, and control of manufacturing systems. They identified five basic concepts for software design of an online simulation system and solved an example simulator. In this procedure, there is a budgetary limitation in the shop. Azadeh et al. [84] used a computer simulation model for a car industry using a just-in-time production system. Azadeh et al. [93] used a computer simulation for finding optimal solution of an assembly line in a shop floor. Grangeon et al. [94] proposed a generic simulation model for a hybrid flow-shop, where the job priorities at each machine are established dynamically. They used GPSSS (GPSS under simulation) language. Hence, the main goal of simulator is to facilitate the performance evaluation of different priority rules for job dispatching, concerning the mean flow time and make span as well as other performance criteria like average resource utilization, average queue length, etc. In real world problems, simulation plays a critical role in helping management to take the optimum decision throughout the sensitivity analysis, especially in problems with layout-dependent capacity and productivity constraints. In this section, before initializing a model of the system including its constraints, first the following assumptions should be considered: 1. Since the cycle time of the conveyor is fixed, the input rate of injection system is considered to be fixed and equal to 54 s. 2. Demands for 15, 16, 18 and 20 feet refrigerators are 20%, 15%, 30% and 35% of total demand. 3. The 15 feet model requires only one fixture; while other three types (i.e. 16, 18, and 20 feet) require two fixtures. 4. The capacity of the hot-room is two times more than the number of fixtures. Hence, there is sufficient space only for 14 noninjected refrigerators in the hot-room. 5. Since specifications of new products are vague and the types of foams are different, the time in which the foam is injected to the fixtures is variable and depends on the specification of both the foam and the refrigerator. It is assumed that this parameter is fixed and equal to Exp(120) seconds, as a statistical distribution for this item. This value has been obtained by statistical analysis on previous data. 6. For different refrigerator types, the times in which a refrigerator remains in fixture are statistical distributions obtained by statistical sampling methods. This time is considered as normal distribution (604, 68.03), normal distribution (747, 79.05), triangular distribution (602, 757, 913), and triangular distribution (483, 742, 853) for 15, 16, 18, and 20 feet refrigerators, respectively. 7. Sequence-dependent setup times are considered for each product placed in a fixture. Therefore, setup times for a specific product are variable and depend on the type of previous products in the fixture. Setup times have statistical distributions (Table 1) Fig. 2 illustrates the network-based modeling for the layout problem with four product types and two resources. This model includes an original network which represents the relationships among servers, entities, and activities in the system. In order to model the described system in Visual SLAM, the head (a tool used in the foam injection process) and the fixtures are considered as resources. In this process, the head is transported between stations and should be recharged after each injection took place. Therefore, this is a layout-dependent process. In Fig. 2, the head (a tool designed for foam injection) and the fixtures are taken up as resources. The products (i.e. 15, 16, 18, or 20 feet refrigerator) are sent into the original network by a CREATE node. These products have stochastic demand. Demand for 15, 16,

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Figures &Illustrations Collect the required data for manufacturing process

Collect the data required for layout design

Develop simulation models

Generate layout alternatives

Evaluate the performance of each layout alterative using provided simulation model

Define the obtained indicators as outputs of stochastic DEA

Apply the Developed Stochastic DEA for ranking and optimization of layout alternatives. Perform validation, verification and noise analysis

Fig. 1. Schematic view of the integrated simulation–stochastic DEA with random demands and sequence dependent set up times.

Table 1 Sequence dependent setup times. Product

15 feet 16 feet 18 feet 20 feet

Type of previous product 15 feet

16 feet

18 feet

20 feet

Exp(30) Exp(45) Exp(40) Exp(60)

Exp(45) Exp(30) Exp(50) Exp(55)

Exp(40) Exp(50) Exp(30) Exp(35)

Exp(60) Exp(55) Exp(35) Exp(30)

Fig. 2. Visual SLAM network of the injection process.


18 and 20 feet refrigerator is 15%, 30%, 15% and 15%, respectively. To do this, in ASSIGN node, we assign demand for each type. If the required resources (i.e. fixtures) are available they will be assigned to the product. Otherwise, this product must be awaited in the AWAIT node labeled “Inject Await”. Products must be remained in fixture. If the required head is available they will be assigned to the products. Otherwise, products must be awaited in the AWAIT node. In the next step, we consider the variable LL [1] with initial value of one. This variable shows the previous type of products. Sequence-dependent setup times are considered for each product placed in a fixture. After setup times, LTRIB variable is considered for each type of products. This variable is number of fixtures for each product. Then, traverse times for heads are calculated. In this step, ARRAY [8] is used to determine the layout design; this ARRAY shows the different layout of station. Therefore, 24 layout alternatives are available for this problem. In the next ASSIGN node, type of the previous assign is set to LL [1]. The foam is injected in the next step. Then, the head and fixtures are freed. A FREE node, as a node for freeing the allocated resources within AWAIT node is used to free the head and fixture. Thus, head and fixture return to the network. The number of returned fixtures is equal to LTIRB. COLCT node is then used to collect the information regarding time in system. After entering all entities to the TERMINATE node, the simulation will be terminated. 5. Stochastic DEA Consistent with DEA terminology, the term “DMU” refers to the individuals in the evaluation group or decision making units. DEA generates a surface called the frontier that follows the peak performers and envelops the remainder [85]. Fig. 3 illustrates the concepts of the empirical and theoretical production frontiers in a two-dimensional surface to generalize the case of a multi-dimensional surface. The theoretical frontier represents the absolute maximum possible production that a DMU can achieve in any level of inputs. However, the theoretical relationships between input and output parameters of a system are generally difficult to identify and to express mathematically. For this reason, the theoretical frontier is usually unknown. Therefore, the relative or empirical frontier based upon real DMU is used. The empirical frontier connects all the relatively best DMUs in the observed population. Note that if the performance of all observed DMUs is generally poor, then your empirical frontier gives you only the best of a bad lot. The theoretical frontier would clearly indicate that the poor DMUs were indeed poor. By providing the observed efficiencies of individual DMUs, DEA may help to identify possible benchmarks toward which performance can be targeted. The weighted combinations of peers and the peers themselves may provide benchmarks for relatively less

efficient organizations. The actual levels of input use or output production of efficient organizations (or a combination of efficient organizations) can serve as specific targets for less efficient organizations, while the processes of benchmark organizations can be promulgated for the information of managers of organizations aiming to improve performance. The ability of DEA to identify possible peers or role models as well as simple efficiency scores gives it an edge over the other measures. Sueyoshi [58] introduced a CCR stochastic models follows:

Max

Infeasible production areaa

Empirical fronttier

Regression linee

DMU UX

m

vi xik = 1

i=1 m

vi (ˇj xij ) −

Fig. 3. Empirical and theoretical frontier.

(1) s

ur {y¯ rj + br,j F −1 (1 − ˛j )≥0

r=1

i=1

ur , vr ≥0 The stochastic CCR model (1) evaluates the relative efficiencies of n DMUs (j = 1, n), each with m inputs and s outputs denoted by xij and yrj respectively. In model (1), ur and vi are the factor weights. br,j is SD of yrk and y¯ rk is the mean of yrk . He considered ˛j as a risk criterion representing of decision makers and ˇj as an expected efficiency level of the jth DMU (see [58]). In contrary to the above model, this paper presents an output-oriented CCR with stochastic outputs. This model can be used for evaluating DMUs in stochastic environment. The CCR output-oriented can be formulated as follows [86]:

Min

s

vi xik

r=1

s.t. m

ur yrk = 1

i=1 m

vi xij −

s

(2)

ur yrj ≥0

r=1

i=1

ur , vr ≥0 In above model (2), the probability of the first constraint is equal zero. Therefore this model must be reformulated as follows:

Min

s

vi xik

r=1

s.t. 1−ε≤

m

ur yrk ≤ 1 + ε

(3)

i=1

vi xij −

i=1

Inputs

ur y¯ rk

s.t.

m

Possible n production area

s r=1

Outputss ontier Theoretical fro

293

s

ur yrj ≥0

r=1

ur , vr ≥0 Thus, the probability of this constraint can be calculated. Because we consider stochastic outputs for this model, Sueyoshi’s

294


[58] method for reformulation of model (3) is used and formulated as follows: s

Min E(

vi xik )

We used this method for the first and second constraints to obtain the linear model. The proposed stochastic DEA model is formulated as follows: Min

m

r=1

s.t. Pr (1 − ε ≤

Pr

m

m i=1

vi xij −

s.t. ur yˆ rk ≤ 1 + ε)≥1 − ˛k s

(4)

ur yˆ rj ≥0

≥1 − ˛j

1−ε−

ur , vr ≥0

m

The above model needs to be reformulated. To do this, we applied Sueyoshi’s method. Input of this model is not stochastic, so the objective of models (3) and (4) are equal. The second constraint consists of two constraints, yˆ rj is stochastic output. Therefore, model (4) is rewritten as follows: s

vi xik

m

ur yˆ rk ≤ 1 + ε

Pr (1 − ε ≤

m

m i=1

vi xij −

≥1 −

˛k 2 (5)

s

ur yˆ rj ≥0

≥1 − ˛j

r=1

ur y¯ rk + brk F

r=1

vi (xij ) −

s

(˛k ) ≤ 0

ur y¯ rj + br,j F −1 (1 − ˛j )≥0

r=1

i=1

ur , vr ≥0 This model (9) is infeasible when the value of ε is relatively low. To alleviate this issue, trial and error are applied to detect the minimum value of ε.

z= ≥1 − ˛j ,

s u (ˆy r=1 r rj

− y¯ rj )

Vj

m ≤

vx − i=1 i ij

s

u y¯ r=1 r rj

⎡

(6)

v(ˆy1j )

...

⎢ Cov(ˆy , yˆ ) v(ˆy ) ⎢ 2j 1j 2j ⎣ ... ...

Cov(ˆy1j , yˆ sj )

Vj = (u1 , u2 , . . ., us ) × ⎢

Cov(ˆysj , yˆ 1j ) × (u1 , u2 , ..., us )T =

s

2

ur brj

⎤ ⎥ ⎥ ⎥ ⎦

v(ˆysj ) (7)

Incorporation models (6) and (7), we can have the following inequality model:

r=1

ur {y¯ rj + br,j F −1 (1 − ˛j )≥0

i∈I

Ln = total queue length for entity n

,

MUi

I

,

(10)

MUi

QL and TIS are the undesirable and MU is desirable outputs. This is because MU is desired to be increased whereas QL and TIS are undesirable and are required to be reduced to improve the performance. To do this we transform (1/yrj ) undesirable output, so that all of our outputs become desirable. Mean (12) and standard deviation (13) are calculated for each output by using estimation technique of PERT/CPM: y¯ rj = brj =

r=1

s

N

= utilization of machine i; I = total number of machines (11)

Vj

j = 1, . . ., n

L n∈N n

QL =

MU =

In model (6), Vj is the variance–covariance matrix of jth DMU. This matrix is calculated as follows:

vi (xij ) −

6.1. SDEA for optimization of the SRFLP

To obtain a linear model, we consider that yˆ rj equal to y¯ rj + brj ς. Where ς is single random variable with normal distribution (0, 2 ). Thus, we rewrite third stochastic constraint as follows:

i=1

(9) −1

We applied model (9) for our problem (SRFLP). One dummy input equal to one is used. Also, we have 3 outputs for each DMU including the queue length (QL), machine utilization (MU) and time in system (TIS) for each layout alternative. We calculate QL (10) and MU (11) as follows:

ur , vr ≥0

m

s

The major purpose of this study is to investigate the effects of uncertainty on the outputs of discrete-event-simulation and to find the best alternatives by utilizing the proposed SDEA.

˛ ur yˆ rk )≥1 − k 2

i=1

pr

ur y¯ rk + brk F −1 (1 − ˛k )≥0

6. Computational results

i=1

Pr

s

r=1

s.t. Pr

1+ε−

r=1

r=1

i=1

Min

vi xik

i=1

(8)

2MLrj + (OPrj + PErj )/2 3 OPrj − PErj 6

(12) (13)

Table 2 shows the outputs of the simulation model for the layout problem. Table 3 shows mean and standard deviation of each output. The simulation model is run 200 times with 200 entities. As stated before, we have 24 permutations for a single row layout problem with four facilities. Hence, all 24 layout alternatives have been studied. For selection the best ˛j for this problem, ˛j = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 were considered. The results of SDEA are shown in Tables 5 and 6 after applying SDEA with different ˛j . To select the best ˛j noise analysis is performed for each


295

Table 2 Value of indicators (outputs of simulation). Layout alternative

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

TIS

QL

MU

PE

OP

ML

PE

OP

ML

OP

PE

ML

5.18427E−05 5.37249E−05 5.08397E−05 5.48625E−05 5.48774E−05 5.45752E−05 5.28016E−05 5.40727E−05 5.45354E−05 4.99428E−05 5.53079E−05 5.33144E−05 5.45693E−05 5.45852E−05 5.21829E−05 5.26989E−05 5.47728E−05 5.21655E−05 5.51795E−05 5.47762E−05 5.20662E−05 5.41267E−05 5.40426E−05 5.58565E−05



0.137646 0.149368 0.143112 0.151572 0.150718 0.14301 0.142809 0.149401 0.148608 0.146246 0.153945 0.147532 0.14971 0.143827 0.143052 0.151034 0.148036 0.147663 0.152429 0.14455 0.145679 0.147468 0.146518 0.152695

0.203944 0.218312 0.199729 0.205804 0.22302 0.22154 0.204653 0.203449 0.209831 0.199848 0.209504 0.206274 0.217398 0.213066 0.199651 0.212865 0.209168 0.205154 0.220811 0.207168 0.208877 0.212303 0.199725 0.199779

0.171721 0.172381 0.172858 0.172293 0.17388 0.174161 0.173616 0.172741 0.171682 0.172446 0.1738 0.173042 0.173678 0.17296 0.171056 0.17197 0.174615 0.173094 0.174808 0.174686 0.171116 0.172711 0.173431 0.173092

7.2525 7.249 7.2295 7.24 7.265 7.238 7.284 7.2505 7.2525 7.2685 7.242 7.2495 7.246 7.2375 7.249 7.2375 7.233 7.2505 7.245 7.2485 7.245 7.246 7.2315 7.272

7.029 7.0315 7.0125 7.0155 6.989 7.0385 6.9995 7.014 7.008 7.0075 7.013 6.99 7.0035 7.016 6.9965 7.0105 6.9975 7.0515 6.9875 6.9575 7.021 6.9455 6.9645 6.9695

7.149 7.139 7.143 7.146 7.1435 7.143 7.1435 7.146 7.1455 7.1435 7.143 7.1455 7.149 7.1465 7.148 7.141 7.142 7.149 7.139 7.1395 7.1455 7.144 7.1425 7.1435

PE, pessimistic estimate; OP, optimistic estimate;

˛j . The results of the SDEA for ˛ = 0.1 and ˛ = 0.2 are shown in Tables 7 and 8, respectively. The results of other noise analysis are shown in Tables 14 to 29 of Appendix A. The correlation between the result of SDEA with primary and noisy data is calculated for each ˛j . Table 9 shows the results of correlation. According to Table 9, the SDEA model for this problem is more stable at ˛j = 0.2. Therefore, this level is the best value for ˛j . 6.2. Validation and verification If brj = 0 and the value of output equals to mean the result of SDEA and deterministic DEA must be similar. Table 10 compares

models (2) and (9) under this condition. The result shows that these two models have same results. Correlation analysis was performed for the result of these models by MINITAB® . Moreover, Pearson correlation of models (2) and (3) is equal to one. Thus, the proposed SDEA is validated and verified by deterministic DEA when brj = 0 and the value of output equals to mean. 7. Performance comparison In this section, the proposed CS-SDEA is compared against two other traditional algorithms namely as artificial neural network (ANN), and genetic algorithm (GA) to truly demonstrate its

Table 3 Mean and standard deviation for each indicator. Layout alternative

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

TIS

QL

MU

Mean

SD

Mean

SD

Mean

SD



0.171413 0.1762 0.172379 0.174425 0.17821 0.176866 0.173654 0.173969 0.174194 0.172647 0.176441 0.174329 0.17697 0.174789 0.171154 0.175297 0.175944 0.174199 0.178746 0.175077 0.17317 0.175103 0.173328 0.17414

0.01105 0.011491 0.009436 0.009039 0.01205 0.013088 0.010307 0.009008 0.010204 0.008934 0.00926 0.00979 0.011281 0.01154 0.009433 0.010305 0.010189 0.009582 0.011397 0.010436 0.010533 0.010806 0.008868 0.007847

7.14625 7.139417 7.135667 7.139917 7.138 7.141417 7.142917 7.141417 7.140417 7.141667 7.137833 7.136917 7.140917 7.139917 7.139583 7.135333 7.133083 7.149667 7.131417 7.127333 7.141333 7.127917 7.127667 7.135917

0.03725 0.03625 0.036167 0.037417 0.046 0.03325 0.047417 0.039417 0.04075 0.0435 0.038167 0.04325 0.040417 0.036917 0.042083 0.037833 0.03925 0.033167 0.042917 0.0485 0.037333 0.050083 0.0445 0.050417

The results of SDEA are shown in Table 4. The proposed SDEA model (9) is infeasible for some value of ε. Moreover, SDEA model is run for different value of ε, and when ε is equal to 0.014 the SDAE is feasible for all of DMUs. In addition, ˛j as a risk criterion equals to 0.05.

296


Table 4 Optimum value of ε and layout rankings. Layout alternative

ε

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

Rank

0

0.01

0.011

0.013

0.014

Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible

Infeasible 1.011366 1.01188 Infeasible Infeasible 1.01024 Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible 1.00904 Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible

1.009649 1.010345 1.010858 1.0106 Infeasible 1.00922 Infeasible 1.010946 Infeasible Infeasible 1.011109 1.01046 Infeasible 1.01046 Infeasible 1.011373 1.012094 1.008021 1.010374 Infeasible Infeasible Infeasible Infeasible Infeasible

1.007607 1.008302 1.008813 1.008556 1.011238 1.007179 Infeasible 1.008902 1.009418 1.010011 1.009064 1.010621 1.009253 1.008417 1.009911 1.009328 1.010047 1.005983 1.011316 Infeasible 1.008331 Infeasible 1.0123 Infeasible

1.006586 1.00728 1.007791 1.007535 1.010214 1.006158 1.009907 1.00788 1.008396 1.008987 1.008042 1.009597 1.008231 1.007395 1.008888 1.008305 1.009023 1.004963 1.010291 1.01245 1.007309 1.012813 1.011275 1.011755

performance. Before comparison, to synchronize the application of ANN and GA with our problem, we start with the definition of a criterion for goodness of a solution. For conformity with DEA terminology, we call this goodness as “efficiency”. The efficiency of a layout alternative (solution) can be calculated as a function of its performance indicators, here in terms of TIS, QL, and MU. We can write: Effi = f (TISi , QLi , MUi ) + εi

(14)

In which Effi is the efficiency score for solution i, and εi is the noise term representing the stochastic behavior in the modeling environment. The synchronized application of SDEA, ANN, and GA in our problem is to estimate the appropriate form of the f(.) function. DEA and SDEA estimate this function as the frontier

3 4 8 7 19 2 18 9 13 15 10 17 11 6 14 12 16 1 20 23 5 24 21 22

performance function according to their identified inputs and outputs. Non-parametric approaches such as DEA and SDEA can estimate the performance function in 14 according to the data available for performance indicators (TIS, QL, and MU). However, ANN and GA are parametric approaches where they need data on Effi to estimate f(.) in 14 but there is no data on Effi as this variable is the solution variable. Here the standard procedure is to move the efficiency term into the variable noise term and estimate one of performance indicators as a stochastic function of other performance indicators. (15)MUi = h(TISi , QLi ) + Effi In (15), the term h(TISi ,QLi ) is an expected parametric estimation of MUi namely as . Hence the efficiency of a solution is the surplus of the actual MUi from expected . When searching in the solution

Table 5 Efficiency results of SDEA. ˛ ε

0.1 0.013

0.2 0.010

0.3 0.010

0.4 0.010

0.5 0.010

0.6 0.010

0.7 0.010

0.8 0.010

0.9 0.010

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

1.00 1.00 1.01 1.00 1.01 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.00 1.01 1.01 1.01 1.00 1.01 1.01 1.00 1.01 1.01 1.01

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

0.996 0.997 0.998 0.997 0.998 0.997 0.998 0.997 0.998 0.998 0.998 0.998 0.998 0.997 0.998 0.998 0.999 0.996 0.999 1 0.997 1 1 0.999

0.996 0.997 0.997 0.997 0.998 0.996 0.998 0.997 0.997 0.997 0.997 0.998 0.997 0.997 0.998 0.998 0.998 0.995 0.999 1 0.997 1 0.999 0.999

0.994 0.995 0.995 0.995 0.996 0.994 0.995 0.995 0.995 0.995 0.995 0.996 0.995 0.995 0.995 0.996 0.996 0.993 0.997 0.998 0.995 0.998 0.997 0.997

0.992 0.993 0.993 0.993 0.993 0.992 0.993 0.993 0.993 0.993 0.993 0.994 0.993 0.993 0.993 0.994 0.993 0.991 0.994 0.995 0.993 0.995 0.995 0.994

0.992 0.992 0.993 0.992 0.992 0.991 0.993 0.992 0.993 0.993 0.992 0.993 0.992 0.992 0.993 0.993 0.992 0.991 0.992 0.994 0.992 0.995 0.994 0.993

0.986 0.97 0.982 0.976 0.967 0.969 0.978 0.975 0.978 0.983 0.968 0.976 0.972 0.976 0.983 0.976 0.967 0.977 0.967 0.967 0.983 0.972 0.973 0.967

0.957 0.927 0.949 0.94 0.922 0.928 0.943 0.94 0.942 0.95 0.928 0.939 0.928 0.939 0.95 0.936 0.919 0.94 0.919 0.918 0.947 0.93 0.936 0.929


297

Table 6 Rank results for SDEA. ˛ ε

0.1 0.013

0.2 0.010

0.3 0.010

0.4 0.010

0.5 0.010

0.6 0.010

0.7 0.010

0.8 0.010

0.9 0.010

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

3 3 8 3 19 2 14 8 8 14 8 14 8 3 14 8 14 1 19 23 3 23 21 21

2 5 5 5 19 2 14 5 5 14 5 14 5 5 14 5 14 1 19 21 2 24 21 21

1 3 9 3 9 3 9 3 9 9 9 9 9 3 9 9 19 1 19 22 3 22 22 19

2 4 4 4 14 2 14 4 4 4 4 14 4 4 14 14 14 1 20 23 4 23 20 20

2 4 4 4 16 2 4 4 4 4 4 16 4 4 4 16 16 1 20 23 4 23 20 20

2 4 4 4 4 2 4 4 4 4 4 18 4 4 4 18 4 1 18 22 4 22 22 18

3 3 14 3 3 1 14 3 14 14 3 14 3 3 14 14 3 1 3 22 3 24 22 14

24 8 20 13 1 7 18 12 18 21 6 13 9 13 21 13 1 17 1 1 21 9 11 1

24 5 21 15 4 6 19 15 18 22 6 13 6 13 22 11 2 15 2 1 20 10 11 9

environment, high surplus is an indication of the goodness of the solution. The problem of estimating function h(.) can be solved by ANN and GA. In ANN no explicit form of h(.) function is defined and the non-linearity and complexity of this function can be handled, appropriately, if the necessary data is available. In GA, we assume the translog functional form for g(.) function. GA will find the best parameters of this translog function. Following Fare et al. [87] and Coelli and Perelman [88], translog form is generally applied for distance function. Translog form of output distance function is shown in Eq. (16). ln(MUi ) = ˛0 + ˛1 ln(TISi ) + ˛2 ln(QLi ) + ˛3 ln(TISi ) ln(QLi ) + Effi (16)

7.1. ANN structure In this paper the effort is made to identify the best fitted network for the desired model according to the characteristics of the problem and ANN features. Fig. 4 shows the two layer feed forward model used for estimating MU. Each solution has three performance indicators. The input nodes are TIS and QL. Hidden neurons with different nonlinear transfer functions are used to process the information received by the input nodes. The best transfer function is discovered to be sigmoid transfer function. A linear transfer function is employed in the output node as desired for function estimation problems. The MLP’s most popular learning rule is the error back propagation algorithm [89,90]. The training and test data for finding the weights of architecture in Fig. 4 are generated randomly from a

Table 7 Results for SDEA with noisy data (˛ = 0.1). ˛ = 0.1

1

2

3

4

5

6

7

8

9

10

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

0.987 1.004 1.005 1.004 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 1.006 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.004 1.005 1.004 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 1.006 1.005 0.987 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.004 1.005 0.987 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 1.006 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.002 1.005 1.004 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 1.006 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.004 1.005 1.004 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.002 1.004 1.006 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.004 1.005 1.004 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 0.987 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.004 1.005 1.004 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 1.006 1.005 1.006 1.002 1.007 1.009 0.987 1.009 1.008 1

1.004 1.004 1.005 1.004 1.007 1.003 1.006 1.005 1.005 0.987 1.005 1.006 1.005 1.004 1.006 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.004 1.005 1.004 1.007 1.003 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 1.004 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

1.004 1.004 1.005 1.004 1.007 1.001 1.006 1.005 1.005 1.006 1.005 1.006 1.005 1.004 1.006 1.005 1.006 1.002 1.007 1.009 1.004 1.009 1.008 1.008

298


Table 8 Results for SDEA with noisy data (˛ = 0.2). ˛ = 0.2

1

2

3

4

5

6

7

8

9

10

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

1.001 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.001 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.002 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.003 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.002 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.002 1.001 1.005 1.006 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.004 1.002 1.007 1.006 1.006

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.004 1.006 1.006

Table 9 Correlation results. ˛

1

2

3

4

5

6

7

8

9

10

Average

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.996 0.999 1 1 0.994 0.935 0.935 0.617 0.657

0.923 0.99 0.966 0.994 0.999 0.933 0.935 0.692 0.742

0.996 0.999 0.981 0.994 0.994 0.931 0.935 0.862 0.858

0.997 0.994 0.981 0.999 0.994 0.936 0.868 0.802 0.892

0.975 0.968 0.966 0.927 0.994 0.931 0.935 0.676 0.797

0.923 0.99 0.966 0.798 0.994 0.81 0.718 0.842 0.896

0.828 0.94 0.981 0.798 0.912 0.81 0.977 0.676 0.891

0.923 0.94 0.882 0.927 0.813 0.931 0.935 0.803 0.727

0.948 0.98 0.822 0.927 0.86 0.931 0.881 0.874 0.829

0.999 0.959 0.924 1 0.86 0.81 0.868 0.828 0.925

0.9508 0.9759 0.9469 0.9364 0.9414 0.8958 0.8987 0.7672 0.8214

Table 10 Comparison of deterministic and stochastic DEA (when brj = 0). Layout alternative

Deterministic output-oriented CCR

Stochastic output-oriented CCR

Rank

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

1.000478 1 1.001962 1.001189 1 1 1.000945 1.00088 1.001211 1.00112 1.000606 1.001501 1.00009 1.00105 1.001412 1.001457 1 1 1 1 1.001167 1.001717 1.002279 1.000677

0.986471 0.986 0.987935 0.987172 0.986 0.986 0.986932 0.986867 0.987194 0.987105 0.986597 0.98748 0.986089 0.987036 0.987393 0.987437 0.986 0.986 0.986 0.986 0.987151 0.987693 0.988247 0.986667

9 1 23 17 1 1 13 12 18 15 10 21 8 14 19 20 1 1 1 1 16 22 24 11

normal distribution function. In our case, data on TIS, QL, and MU are assumed to be normally distributed with known mean and variance. Here, for each row of these performance indicators, we generated 300 random observations. Hence totally we have

24 × 30 = 7200 randomly generated observations from which 70% are used for training neural network and the rest is used to test ANN. We use ANN toolbox of MATLAB© for applying the training and test process.

A. Azadeh et al. / Journal of Manufacturing Systems 33 (2014) 287–302 Table 11 Possible values of ANN control parameters.

MU

Output Neuron

Output Layer

+1 HN 1

HN 2

Parameter

Options

Number of neurons in the hidden layer Transfer functions of hidden layer Transfer functions of output layer Learning algorithm

10, 20, 30, 40, 50 Logsig, Tansig, Purelin Logsig, Tansig, Purelin Trainlm, Traingd, Traingdm,

HN 10

…

Hidden Layer

299

for test data. The best parameter combination with minimum MAPE is highlighted in Table 11 with bold, underlined options. It should be noted that the application of ANN in this study is a function estimation application so as ANN finds a complex function that could be used for efficiency measurement of the possible problem solutions. The fact that we have proposed to use ANN and GA for problem solving makes it reasonable to compare the results of SDEA with ANN and GA.

HN: Hidden Neuron +1

Input Layer TIS

7.2. Specific GA-coding

QL

Fig. 4. The ANN architecture for estimating MU.

In this study we select the MAPE (Mean Absolute Percentage Error) as the fitness function to optimize ANN parameters. The fitness function is shown in Eq. (17). Note that MUactual and MUestimated are the actual and estimated machine utilizations, respectively and n is the number of observations in the test data set.

n

MAPE =

j=1

[(MUjactual − MUjestimated )/MUjactual ]

(17)

n

For a feed-forward two-layer neural network with backpropagation learning algorithm, the parameters that need to be tuned are the number of neurons in the hidden layer, the transfer functions of hidden and output layer, and the learning algorithm. In our ANN, the options for the parameters are presented in Table 11. For details please see the Help documentation of MATLAB© . A flexible code is developed in MATLAB© which enables us to test all combinations of the parameters for minimum MAPE of estimation

Genetic algorithm is similar to the natural evolution process where a population of a specific species adapts to the natural environment under consideration, a population of designs is created and then allowed to evolve in order to adapt to the design environment under consideration. These algorithms were directly described by Goldberg [91] and have taken attention to solve optimizing problems. The fundamental principal of genetic algorithms first was introduced by Holland [92]. Genetic algorithm encompasses three main operators: selection, crossover and mutation and which are described below briefly. Here GA is used to find the model coefficients in Eq. (16). Hence a chromosome in GA consists of four binary coding each corresponding to a single coefficient in Eq. (16), as follows: ˛0

˛1

˛2

˛3

Binary coding

Binary coding

Binary coding

Binary coding

The fitness function in GA is previously defined in Eq. (17) as the MAPE error for the test data. In other words, GA will search for the best model parameters according to the minimum test MAPE.

Table 12 Performance comparison with GA and ANN. Layout alternative

#01 (1234) #02 (1243) #03 (1342) #04 (1324) #05 (1423) #06 (1432) #07 (2134) #08 (2143) #09 (2314) #10 (2341) #11 (2413) #12 (2431) #13 (3124) #14 (3142) #15 (3241) #16 (3214) #17 (3412) #18 (3421) #19 (4123) #20 (4132) #21 (4213) #22 (4231) #23 (4312) #24 (4321)

The proposed CS-SDEA

ANN

GA

Efficiency

Rank

Efficiency

Rank

Efficiency

Rank

1.002 1.003 1.003 1.003 1.005 1.002 1.004 1.003 1.003 1.004 1.003 1.004 1.003 1.003 1.004 1.003 1.004 1.001 1.005 1.006 1.002 1.007 1.006 1.006

2 5 5 5 19 2 14 5 5 14 5 14 5 5 14 5 14 1 19 21 2 24 21 21

1.00588 1.00443 1.00301 1.00400 1.00360 1.00476 1.00506 1.00472 1.00415 1.00438 1.00370 1.00334 1.00416 1.00406 1.00425 1.00239 1.00291 1.00700 1.00163 1.00138 1.00447 1.00112 1.00100 1.00359

2 13 19 10 14 7 3 6 9 4 15 16 8 11 12 18 20 1 21 24 5 22 23 17

1.00604 1.00427 1.00318 1.00433 1.00387 1.00480 1.00516 1.00476 1.00447 1.00478 1.00382 1.00353 1.00461 1.00434 1.00424 1.00305 1.00257 1.00700 1.00208 1.00100 1.00471 1.00112 1.00103 1.00330

2 12 18 11 14 4 3 6 9 5 15 16 8 10 13 19 20 1 21 24 7 22 23 17

300


Table 13 The features of the simulation–stochastic DEA algorithm versus other methods. Method

Feature

Simulation–stochastic DEA algorithm Azadeh et al. [9] Genetic algorithm Neural network model Tabu search algorithm Particle swarm optimization Algorithm Azadeh et al. [76] Solimanpur and Jafari [18] Jithavech and Krishnan [32] Zhou et al. [75] Chang et al. [73]

Multiple outputs √ √ √ √ √ √ √ √ √ √

Stochastic outputs √

High precision and reliability √ √ √ √ √ √ √ √ √ √ √

√ √ √

√ √

In our GA, we use Roulette Wheel procedure for selection of chromosomes to form the next generation. We use a two-point crossover with a specified crossover probability. The crossover operation is applied with a probability of pc which takes the probabilistic values from 0.7 to 0.98. Mutation operator is another essential operator in genetic algorithm process and it acts on each chromosome after crossover operator in this way that a random number is produced for each bite of a chromosome, if this number is smaller than pm mutation will occur in that bite and otherwise it is not happened. If mutation is not applied, after crossover the offspring will enter the new generation. According to the researches, pm shows best while varying between 1% and 5%. The control parameters of GA are tuned by a random search procedure in which for every parameter a range has been specified. Crossover probability ranges between 0.7 and 0.98 with step 0.02; mutation probability ranges between 0.01 and 0.05 with step 0.01; Number of populations, as stopping criteria, ranges from 30 and 70 with step 10. For all possible combinations of these three parameters, GA has been run and the best combination of parameters is tuned according to the minimum MAPE error for test data. After running GA with the above parameter setting, the following final values are derived for the coefficients in model (16): ˛0

˛1

˛2

˛3

2.2488

0.0298

0.1572

0.0169

7.3. Comparison results The efficiency scores results from ANN and GA are mapped in the same range of SDEA, i.e. between 1.001 and 1.007, based on which layout alternatives are ranked. The results are presented in Table 12. The Spearman correlation between rank results of CS-SDEA and GA and ANN both are 67% and is significant at 99% confidence level. This high correlation confirms the significance and reliability of the CS-SDEA results. 8. Conclusion In this study, we propose a novel algorithm based on computer simulation and stochastic DEA to tackle the single row layout optimization problem in an injection molding job-shop system with random demand and sequence dependent set up times. In this problem, we considered stochastic demand and sequence dependent setup time. A discrete-event-simulation is used to model the injection process. Using the simulation models, all possible layout alternatives have been modeled and their performance measures including average queue length, average machine utilization and time in system have been obtained. A proposed SDEA model is developed and used to find the optimal layout

Multi-variate decision-making through new output-oriented Stochastic DEA √

Practicability in real world cases √ √ √ √ √ √ √ √ √ √ √

solutions. In the proposed Stochastic DEA, each layout alternative has been considered as a DMU. The results show that there are seven optimum layout alternatives for this problem. The CS-SDEA algorithm is also compared with some of the current studies and method such as GA, PSO, NN, TS algorithm in manufacturing systems. Its features are compared with previous models to show its advantages over previous models (Table 13). Acknowledgements The authors are grateful for the valuable comments and suggestion from the respected reviewers. Their valuable comments and suggestions have enhanced the strength and significance of our paper. This study was supported by a grant from University of Tehran (Grant No. 8106013/1/14). The authors are grateful for the support provided by the College of Engineering, University of Tehran, Iran. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jmsy. 2013.12.008. References [1] Wu Y, Appleton E. The optimization of block layout and aisle structure by a genetic algorithm. Computers and Industrial Engineering 2002;41(4):371–87. [2] Önüt S, Tuzkaya UR, Do˘gaç B. A particle swarm optimization algorithm for the multiple-level warehouse layout design problem. Computers and Industrial Engineering 2008;54(4):783–99. [3] Balakrishnan J. FACOPT: a user friendly facility layout optimization system. Computers and Operations Research 2003;30(11):1625–41. [4] Zhang B, Teng HF, Shi YJ. Layout optimization of satellite module using soft computing techniques. Applied Soft Computing 2008;8(1):507–21. [5] Prabhaharan G, Padmanaban KP, Krishnakumar R. Machining fixture layout optimization using FEM and evolutionary techniques. International Journal of Advanced Manufacturing Technology 2006;32(11/12):1090–103. [6] Chen W, Ni L, Xue J. Deformation control through fixture layout design and clamping force optimization. International Journal of Advanced Manufacturing Technology 2007;38(9/10):860–7. [7] Satheesh Kumar RM, Asokan P, Kumanan S. Design of loop layout in flexible manufacturing system using non-traditional optimization technique. International Journal of Advanced Manufacturing Technology 2007;38(5/6):594–9. [8] Heragu SS. Facilities design. Boston, MA: PWS Publishing; 1997. [9] Azadeh A, Moghaddam M, Asadzadeh M, Negahban A. An integrated fuzzy simulation-fuzzy data envelopment analysis algorithm for job-shop layout optimization: the case of injection process with ambiguous data. European Journal of Operational Research 2011;214(3):768–79. [10] Wang TY, Lin HC, Wu KB. An improved simulated annealing for facility layout problems in cellular manufacturing systems. Computers and Industrial Engineering 1998;34(2):309–19. [11] Mak KL, Wong YS, Chan FTS. A genetic algorithm for facility layout problems. Computer Integrated Manufacturing Systems 1998;11(1–2):113–27. [12] Georgiadis MC, Schilling G, Rotstein GE, Macchietto S. A general mathematical programming approach for process plant layout. Computers and Chemical Engineering 1999;23(7):823–40.

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