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CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND By Jeffrey Schaller A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 ACKNOWLEDGMENTS I would like to thank Dr. Selcuk Erenguc, my supervisory committee chairman, for his invaluable guidance and patient encouragement throughout my years in the program. My supervisory committee cochairman Dr. Asoo J. Vakharia has also provided invaluable guidance during the past year and I am thankful that he brought his expertise in cellular manufacturing to our department. I also would like to thank Dr. Harold Benson, Dr. Patrick Thomson, and Dr. ChungYee Lee for serving on my supervisory committee and for the fine instruction they provided in the classes I attended. I would also like to thank Dr. Antal Majthay for the direction he provided while supervising my assistantship assignments. Finally, and most importantly, I want to thank my mother and father for the love and support they have provided my entire life. Anything good that I have ever accomplished is a result of having the best parents. TABLE OF CONTENTS ACKNOWLEDGMENTS. . ii LIST OF TABLES vii LIST OF FIGURES. ix Abstract x Chapters 1 INTRODUCTION 1 A Brief Description of Cellular Layouts 1 Product Line Layout 1 Process Layout. 3 Cellular Layouts 4 Cell Formation 7 Early Cell Formation Techniques 8 Later Cell Formation Techniques 9 The Impact of Dynamic Demand on Cell Formation Decisions 11 Dynamic Demand's Impact on Cellular Layouts 11 Strategies to Handle Dynamic Demand while using a Cellular layout 16 Current Practices 18 Automotive Parts Manufacturer. 18 Enclosure Manufacturer 20 The Problem and its Terminology. 23 The Organization of this Proposal 24 2 REVIEW OF CELL FORMATION LITERATURE 26 Introduction 26 Procedures Based solely on a PartMachine Incidence Matrix 30 Objectives and Constraints. 31 Frequently used objectives 31 iii Frequently used constraints. 34 Other objectives . 35 Methods 37 Manual Identification. .. 38 Matrix Manipulation 39 Clustering using Similarity Coefficients 40 Graph Theory. 42 Mathematical Models 43 Other Methods 44 Evaluation of Procedures 44 Procedures Which use a PartMachine Incidence Matrix and Incorporate Data to Provide Weights for Parts or Machines. 46 Objectives and Constraints. 47 Methods 48 Routing Based Procedures Without Machine Capacities 49 Operations Sequences. . 50 Alternative Routings. . 51 Procedures Which Incorporate Setup Information. 52 Procedures Which Incorporate Machine Capacities 55 Model 1 55 Model 2 57 Methods 59 Procedures That Consider Multiple Scenarios for Part Volumes. 60 Other Procedures 63 Summary 64 3 MODELS 66 Introduction 66 Model I. 67 Description of Model I . 67 Model I Formulation . 69 Description of Model I Equations. 71 An Example of Model I . 74 Model II 77 Description of Model II. 77 Model II Formulation. . 79 Description of Model II Equations 81 An Example of Model II . 83 4 SOLUTION PROCEDURES. Introduction . Model I Heuristic . Lower Bounds for Model II. . Linear Programming Relaxation. Lagrangean Relaxation . Linear Programming Relaxation Tightened Adding Valid Inequalities Valid Inequality Set 1 . Valid Inequality Set 2 . Valid Inequality Set 3 . Heuristics for Model II . Heuristic 1. . Description of Heuristic 1 . Formal Procedure for Heuristic 1 Discussion of Heuristic 1 . Heuristic 2. . Heuristic 3A . Description of Heuristic 3A. Formal Procedure . Discussion of Heuristic 3A . Heuristic 3B . 5 DESCRIPTION OF EXPERIMENTS AND DATA S 87 88 91 91 92 by 94 94 94 96 S 97 98 98 S 98 100 101 S 101 S 105 .105 .107 S 110 111 114 Introduction 114 Experiments 115 Experiment 1 Data . 116 PartMachine Incidence Matrices 117 Processing Times . 119 Part Demand. 120 Inventory and Backorder Costs. 120 Machine Capacities and Number of Machines of Each Type 121 List of Potential Cells. 122 Cell Production and Intercell Production Costs 123 Summary 124 Experiment 2 Data . 125 6 RESULTS. 127 Results of Experiment 1 127 Lower Bounds for Model II Heuristic Results for Model Discussion of Results Lower Bounds. . Heuristics . Results of Experiment 2 . 7 CONCLUSIONS AND FUTURE RESEARCH. Conclusions . Future Research . APPENDICES A FORMULATION OF MODEL I EXAMPLE B FORMULATION OF MODEL II EXAMPLE C PROBLEM 1 DATA AND SOLUTION Problem 1 Data . Problem 1 Solution . LIST OF REFERENCES. . BIOGRAPHICAL SKETCH . S 127 I 130 . 131 131 S 136 . 138 S 142 S 142 . 144 147 S 149 S 151 . 151 S 171 S 178 S 200 LIST OF TABLES TABLE pace 1. Part Routing. 5 2. Cellular Layout. 5 3. An Example of a PartMachine Incidence Matrix. 8 4. Capacity Consideration . 10 5. Part Family Machine Requirements.. 12 6. Load on Machine Type by Period. 13 7. Average Load. 13 8. PartMachine Incidence Matrix for Model I Example. 75 9. By Part Demand.. 75 10. Machine Data. 75 11. Potential Cells. 84 12. Production Costs 85 13. PartMachine Data Sets . 118 14. Summary of Experimental Problems 124 15. L.P. and Lagrangean Relaxations 128 16. Valid Inequality Relaxations 129 17. Heuristics 1 and 2. 132 18. Heuristics 3A and 3B 133 19. Problem 1 Results . 140 20. Problem 2 Results 140 21. Problem 1 Selected Cells . 172 22. Part/Cell Processing . 173 LIST OF FIGURES FIGURE page 1. Product Line Layout . 2 2. Process Layout 3 3. A Process Layout. 6 4. A Cellular Layout.. . 6 5. Family 1 Load by Period.. .. 14 6. Family 2 Load By Period.. 14 7. Total Load by Period.. 15 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND By Jeffrey Schaller May 1996 Chairperson: Dr. S. Selcuk Erenguc Major Department: Decision and Information Sciences This research addresses the problem of configuring cell systems for environments that experience variable demand. Past cell formation models consider only average product demand to determine configurations of cells. Short term fluctuations in demand may cause a cell configuration that would appear to be good, when considering only average demand, to perform poorly because during some periods machines of a certain type may be under utilized in one cell while machines in another cell may not be able to produce the required products. For these reasons, a methodology for configuring cell systems in environments that experience variable demand is needed. In this dissertation, two mathematical models are formulated. The first model is used to develop a list of potential cells, and the second model is used to select cells to include in a shop. Since both models are computationally complex, this research develops heuristic procedures to solve the models. Given that other approaches can also be used to develop a list of potential cells, the primary focus of this research is on the second cell selection model. Lower bounding procedures are proposed for this model. In order to test the effectiveness of the lower bounding and heuristic procedures for the second model, fifteen problems from the literature were modified and solved optimally. The lower bounding and heuristic procedures were then used on these problems, and the lower bounds and objective function values are compared to the optimal objective function values. A second experiment was conducted, in order to test the effectiveness of the lower bounding and heuristic procedures on larger problems. Using two larger published data sets the results are compared to the results obtained using the lower bounding and heuristic procedures. CHAPTER 1 INTRODUCTION 1.1 A Brief Description of Cellular Layouts Cellular Manufacturing is a manufacturing application of the group technology philosophy where groups of processes are dedicated to sets of products or parts. The associated layout (referred to as cellular layout) is a form of production layout that combines features of two traditional forms of plant layout: product line layout and process layout. 1.1.1 Product Line Layout In a plant with a product line layout, machinery and equipment are dedicated to a specific product, and thus, machines needed to manufacture each product are laid out in a way that facilitates efficient material flow. The advantages of product line layout are fast throughput times, efficient material handling, simplified scheduling, and control and low workinprocess inventories. The disadvantages of product line layout are that it requires a high level of investment in specialized machinery and equipment, and it lacks flexibility because only one type of product can be manufactured without incurring a high changeover cost. This type of layout is typically used for high volume, standardized products. Examples are automobiles, television sets and appliances. An example of a product line layout is shown in Figure 1. , Product Flow [] Machines SB6 > Product 1 )I.> Product 2 J A N .> Product3 Figure 1. Product Line Layout. 1.1.2 Process Lavout A process layout groups machines which perform similar functions into departments. Many different types of products may use a specific machine type and each product may visit different machine types in various sequences. An example of a process layout is shown in Figure 2. + Product Flow l Machines Product 3 SProduct 2 Product 2 Figure 2. Process Layout. The products in this example have the same routings as the products in Figure 1 but notice that the material flows are much more jumbled than in Figure 1. The advantage of a process layout is that many different types of products can be manufactured on the same set of machines which leads to flexibility in the utilization of machinery. The disadvantages of a process layout are that material handling costs are high, material flows are complex which causes planning and control to be difficult, flow time is long, setup costs are high and workinprocess inventory is high. Typically low volume or nonstandardized products are produced using a process layout. Examples include (Hyer 1984): aerospace, agricultural machinery, business machines, control devices, diesel engine assemblies, machine tools, and machined parts. 1.1.3 Cellular Layouts Cellular layouts are hybrid layouts which combine features of both product and process layout. The objective of implementing a cellular layout is to enable a manufacturer of low volume specialized or custom products to obtain some of the benefits of product line layout while maintaining some of the flexibility of a process layout. This is done by forming product families which consist of products that require the same types of machines (preferably in the same sequence). A machine of each machine type (or more than one if required) required to produce the products in a family is put into a cell and is dedicated to producing a family of products. As an example consider the products shown in Table 1. Table 1. Part Routing. Product 1 2 3 4 5 6 Table 2. Machine Sequence 1 4 5 5 4 1 2 3 5 2 3 4 5 1 2 3 4 1 3 2 4 Cellular Layout. Machines 1,4,5 2,3,4,5 1,2,3,4 Products 1,2 3,4 5,6 Figure 3 (on the following page) shows the process layout for producing these products. Figure 4 shows the Cell 1 2 3 resulting product flows if the three cell layout in table 2 is used. Process Layout p Product Flow 1 Machines Product 2  Product 3  Product 4 > Product 5 Product 6 Figure 3. A Process Layout. Cellular Layout *0 Product Flow ] Machines Figure 4. A Cellular Layout. 7 Notice the simpler product flows in the cellular layout vs the process layout. Creating cells in this manner enables material handling costs to be reduced and materials handling flows to be simplified. This in turn can lead to reduced flow time, reduced workinprocess inventories and simplified planning and control. Setup time and cost for parts manufactured in a cell can typically be reduced by developing specialized procedures for setups required in the cell. Quality improvement is also usually a benefit because cell operators have more experience manufacturing a specific set of products. 1.2 Cell Formation In order to implement cellular manufacturing cells must be formed. The cell formation problem includes the following decisions. How many cells should there be? Which machine types should be included in a cell? How many machines of each type should be included in a cell? Which parts should a cell produce? 1.2.1 Early Cell Formation Techniques Most of the early cell formation algorithms used some form of partmachine incidence matrix. In a partmachine incidence matrix machines are represented as rows and parts are represented as columns in the matrix (or vice versa). Each element in the matrix is '1' if part j requires machine i for processing and '0' otherwise. Table 3. An example Machine Type A B C D a partmachine incidence matrix Table 3 shows an example of a partmachine incidence matrix. In this example part 1 requires machine types A and C but not B and D, and part 5 requires machine types A, B, and D but not C. The algorithms use the partmachine incidence matrix as input and then use various techniques to break up the parts and machines into cells. These techniques include manual identification (Burbidge 1963); clustering using measures of similarity (McAuley 1972, Carrie 1973); graph theory using measures of similarity (Rajagopalan and Batra 1975, Chandrasekharan and Rajagopalan 1986, Vannelli and Kumar 1986, and Kumar, Kusiak, and Vannelli 1986); and matrix manipulation (King 1980, King and Nakornchai 1982, Chan and Milner 1981, and Chandrasekharan and Rajagopalan 1986). 1.2.2 Later Cell Formation Techniques Many of the algorithms that were developed later tend to be more sophisticated and are based on mathematical models that include operational considerations in the objectives and constraints. These models also usually include some sort of integrality constraint. Operational considerations that have been incorporated into algorithms include operations sequences, machine setup similarities between parts, part volumes, and machine capacities. A typical integrality constraint is that the number of machines of a machine type that are assigned to a cell is required to be integer. Table 4. Capacity Consideration Part Hours required Family on machine type 1 1 2700 2 2500 Total 5200 # of type 1 machines available 3 Capacity per type 1 machine 2000 hrs Total capacity 6000 hrs For an example of why capacity, volume, and machine integrality are important considerations when making cell formation decisions, consider Table 4. In this example two part families have been formed and both families have parts which require processing on machine type 1. The table shows the hours required on machine type 1 for each family, the number of type 1 machines available, and the capacity per type 1 machine. If we could assign 1.5 type 1 machines to each family, we would have ample capacity to produce both families and everything would be fine. However, we cannot dedicate fractions of machines to families. If we assign two machines to family 1 and one machine to family 2, then there 11 will be idle capacity dedicated to family 1 while we do not have enough capacity to produce family 2. If we assign two machines to family 2 and one machine to family 1, a similar situation arises. One alternative to this problem is to combine both families and create a larger cell by assigning all the machines to this single cell. Using this alternative there is adequate capacity to produce both part families but, because the cell is larger and the parts that are produced in the cell are more diverse, some of the benefits of cellular manufacturing are likely to be lost. 1.3 The Impact of Dynamic Demand on Cell Formation Decisions 1.3.1 Dynamic Demand's Impact on Cellular Layouts Demand for a product is said to be dynamic if it varies from period to period. Past cell formation models consider only average product demand to determine the configurations of cells and which products to produce in a cell. Short term fluctuations in demand may cause a cell configuration that would appear to be good, when considering only average demand, to perform poorly. During some periods machines of a certain type may be underutilized in one cell while machines in another cell may not be able to produce the required Table 5. Part Family Machine Requirements. Part Family Required Machines 1 2 6 7 2 1 5 6 quantities of products. For an example consider the part families in Table 5. Suppose there are 2 machines of type 6 available and the capacity of a type 6 machine is 2500 units per period. The load on machine type 6 by period for each family is shown in Table 6. Table 7 (on the following page) shows the average load on machine type 6 for each period. If one machine of type 6 was dedicated to producing family 1 and the other was dedicated to producing family 2 there would be enough capacity in each cell to meet the average load. If you look at the byperiod requirements in Table 6, however, you can see that insufficient capacity would exist to produce family 2 while capacity would be underutilized in family l's cell during the first three periods. During the last three periods the situation is reversed. When looking at the load of both families in aggregate there is enough Table 6. Load on Machine Type by Period. Load on machine type 6 by period Period 1 2 3 4 5 6 Family 1 1480 2068 1256 2676 2831 3360 Family 2 2856 2914 3144 2154 738 674 Total 4336 4982 4400 4830 3569 4034 Table 7. Average Load Average Family 1 2278.5 Family 2 2080.5 Total 4358.5 Capacity 2500 2500 5000 capacity to produce both families during every period. Figures 5, 6, and 7 show the load variations graphically. Figure 5 (on the following page) shows the load caused by family 1, Figure 6 the load caused by family 2, and Figure 7 shows the load in aggregate. The aggregate load is much more stable than the individual family loads. This illustrates an advantage of a process layout when there is dynamic demand. By combining all the machines of a given type into a 3500 Family 1 Load by period ...ii I . 500 1 2 3 4 5 e PklWN Figure 5. Family 1 Load by Period. Family 2 Load by period Figure 6. Family 2 Load by Period. Y PloM Total Load by period WM 1000/ 1 2 3 4 5 6 Figure 7. Total Load by Period. department and allowing any product that requires that machine type to use any machine of that type flexibility is provided so that the peaks and valleys of a product's demand can be offset by other products. Studies have shown the adverse effects of variable demand on a cellular layout. Flynn and Jacobs (1986, 1987) conducted simulation studies which show that when demand variability is high a process layout is favored over a cellular layout in terms of mean flowtime. Garza and Smunt (1991) showed that demand variability can cause intercell flows in a cellular layout and this results in less stable flow time performance as compared to a process layout. 1.3.2 Strategies to Handle Dynamic Demand While Using a Cellular Layout There are strategies which can be used to offset the effects of demand variability in a cellular manufacturing system. These include combining cells to smooth out load variations, routing parts through alternative cells, let parts move between cells (intercell movement), hold inventories of parts, and incur backorders for parts. Combining cells to form larger cells allows the large cell to produce a greater variety of parts than the individual small cells. Since there are more parts it is likely that machine loads will be more stable in the large cell. The large number of parts produced by the cell will also cause more complex materials flows and higher material handling cost. Therefore, there is a tradeoff that must be considered when choosing a cell's size; stable loads in large cells vs lower material handling cost in small cells. Cells can also be formed so that there is more than one cell that can produce a part. Hence, if there is insufficient capacity to produce the part in its preferred cell, it may be possible to produce the part in a different cell. By having alternative cells, load variations can be 17 smoothed out while meeting product demand. The disadvantage of this approach is that it may be more costly to produce parts in their alternative cells and/or there may be a need to duplicate more machines to develop these types of configurations. If there is insufficient capacity for a machine type to produce all the parts required, then some parts can be moved to another cell which has that machine type for processing. This is called an intercell transfer. Ang and Willey (1984) showed that low levels of intercell transfer can greatly improve the flow time performance of a cellular layout. The disadvantages of intercell transfer are that this requires materials handling capability between cells, there is increased paper work required to track part movement, and quality could be adversely affected because of the loss of responsibility by a single cell for the part's quality. A final strategy that could be used is that during periods of low demand, inventories of parts can be produced which can be used to meet demands for parts during periods when insufficient capacity exists in the cells to produce all the parts required. Alternatively parts could be backordered when capacity is not available in a cell and then produced during a later period when there is adequate capacity in the cell. Using inventories and backorders will level out the load requirements in the cells, but the cost of holding inventories or backordering parts must be incurred. 1.4 Current Practices In order to find out how cellular layout is used in industry, we visited two plants that utilize a cellular layout. The first plant produces parts for automotive manufacturers. The second plant produces steel enclosures that are used for industrial applications to hold electrical circuitry. A brief description of how each plant uses a cellular layout and how they deal with demand variability presented next. 1.4.1 Automotive Parts Manufacturer This plant produces parts that are major components in the assembly of automobiles and some service parts. It uses product line, process, and cellular layout. Major factors in the choice of layout are maintaining machine utilization and efficient use of space. The plant has over 20 cells that are 19 used for production. In some cases the cells are production lines that produce one part. One cell is a small job shop that produces approximately 150 parts. The other cells are of varying sizes in terms of the number of machines they contain and the number of products they produce. There are certain processes in the plant which are strictly process departments. An example of this is the paint department. All parts that require painting go to the paint department (there are no paint facilities in the cells). The reason why this department was chosen to be a process layout is that one oven was sufficient to handle the entire plant's requirements. The plant also has 20 high speed roll mills. Fourteen of the mills are grouped together in a department and six have been moved to cells. The roll mills which have not been moved are large mills which perform complicated processing. It was decided that it would be too expensive and require too much space to duplicate these machines in multiple cells. The mills that were moved to cells are smaller and the capacity of the mills matches the requirements demanded by their cells. Some of the cells were created by combining production lines. This enabled the plant to better utilize equipment and save space. The plant experiences demand for products that is variable. This is particularly true with service demand which is very lumpy. A variety of strategies are used to handle this variability. Many of the low volume parts, including service parts, are produced in the large job shop cell. The varying demands of the individual parts tend to offset each other and help stabilize the load in the cell. Sometimes a large order is produced partially in advance of its due date and parts are inventoried. There are instances when orders are rescheduled to a later date. There are also times when production capacities in other cells are used to perform some operations and then intercell transfers of parts occur. When this happens, it is a disruption in the plant because all the presses for the job must be setup at the same time to maintain quality. The primary benefits obtained by using a cellular layout in this plant are reduced equipment and space, reduced material handling cost, and reduced setups. 1.4.2 Enclosure Manufacturer Approximately 2800 standard end items are produced in this plant. The plant also produces some custom jobs. The standard end items are produced and then sent to regional warehouses. Roughly 200 items account for 90% of the units produced by the plant. This plant originally used a process layout. The current layout slowly evolved to a hybrid cellular process layout. Currently there are three cells and plans for more cells are being formulated. The cells are designed to produce high volume products very efficiently and are setup as flow lines. Currently 75% of the unit volume and 40% of the dollar volume is produced using cells. High volume production in cells is undertaken to justify the investment in dedicated equipment and tooling. The demand for standard end items is variable and seasonal while demand for custom items tends to be lumpy. The plant responds to demand variability using a variety of methods. A portion of the plant is still using a process layout. This portion of the plant is generally used to produce low volume standard items and custom items. The demand for these items tend to be more erratic, but by producing many items on the same equipment, the load generated is fairly stable. High volume items are produced in the cells and the demand for these items tends to be fairly stable. There are 22 times, however, when some facilities in the plant are idle while other facilities become bottlenecks. The plant also had to invest in additional equipment to build cells that could meet the seasonality of demand. Sometimes custom jobs will be processed in cells if the equipment in the cell is underutilized. At other times, a product normally produced in a cell will be produced in the process portion of the plant if the cell capacity is exceeded. Inventory is also held and customer orders are backordered to help deal with demand variability. The following benefits have been attributed to the use of cellular layout in the plant. 1) Material handling cost has been significantly reduced because conveyors are used to move materials in the cells. 2) Setups have been significantly reduced because tools have been dedicated to products and designed for quick setup. 3) Productivity has increased. Labor costs have been cut in half. 4) Leadtimes have been reduced from 4 weeks to 2 weeks. 5) WIP inventory has been dramatically reduced. 6) Quality has been improved due to the fast feedback provided by the cellular layout. 7) The plant has become a safer place to work because there is 23 less material handling and simpler setups. 8) The fill rate on customer orders has increased from 50% to 80%. 1.5 The Problem and its Terminology Based on our experience and industry visits, the problem we address is as follows. Given a set of parts, a finite planning horizon with T periods, and a set of machine types, the problem is to specify a set of cells to be included in the shop and a production and inventory plan that will minimize total cost. For each cell the number of machines of each type must be specified. The set of cells to be included in the shop can contain cells of varying sizes in terms of the total number of machines included in each cell. For example the shop could include several small cells and a large "remainder cell." A remainder cell is a portion of the shop that typically is organized as a process layout. The production and inventory plan must specify the quantity produced of each part in each cell during each period, and the quantities of each part produced using intercell movement during each period. Each part has associated with it the following: (a) a set of machine types that are required to produce the part and the processing time required by the part on each machine type; (b) a demand deterministicc) for each period during the planning horizon; (c) a cost to hold one unit of inventory for one period (unit inventory holding cost); (d) a cost to hold one unit on backorder for one period (unit backorder cost); (e) costs to produce a unit of the part in various size cells (in terms of the number of machines contained in the cell). (These costs are referred to as cell production costs. It is assumed that larger cells have a higher cost); (f) a cost to produce a unit of the part in two or more cells (intercell movement cost). Each machine type has associated with it the following: (a) the number of machines of that type that are available and (b) the capacity of a single machine of that type. 1.6 The Organization of this Dissertation The remainder of this dissertation is organized as follows. Chapter 2 contains a survey of cell formation literature. Chapter 3 introduces the two models that can be 25 used to develop cellular configurations, and Chapter 4 describes solution procedures for these models. Chapter 5 describes the experimental data used to test the solution procedures described in Chapter 4, and Chapter 6 presents computational results of these tests. Finally Chapter 7 proposes future research directions. CHAPTER 2 REVIEW OF CELL FORMATION LITERATURE 2.1 Introduction The cell formation problem has attracted more academic interest than any other aspect of cellular manufacturing. The reason research has focused so heavily on cell formation problem is due to the complex nature of the cell problem. In order to design a cellular system decisions must be made as to how many cells there should be, which machine types should be included in a cell, and which parts should a cell produce. Arriving at these decisions constitutes a solution to the cell formation problem. Many aspects of the effectiveness and efficiency of an operation will be affected by the cell formation decisions. Examples are labor costs, investment, leadtimes for products, supervision of personnel, quality, and amount of paperwork needed to control the operation. This leads to the consideration of many possible objectives in arriving at a solution. The 26 problem is also complex because an entire machine usually must be assigned to a cell. Because of this most mathematical models of the problem are integer programming models which are difficult to solve. The procedures that have been developed to solve the cell formation problem are diverse with respect to 1) the objectives and constraints considered by the procedure, 2) the data required as input for the procedure, and 3) the techniques used by the procedure to solve the problem. The first two items are somewhat related. Usually, to incorporate an objective or a constraint into a problem, data are required to support that objective or constraint. For example, if a constraint of the problem is to use only existing equipment to configure cells, then data by machine type of the number of machines available are required. Many procedures use objectives that are surrogates for other objectives which if incorporated would require additional data and cause the problem to become more complex. For example, a possible objective when designing a cellular system is to maintain a minimum level of machine utilization for each machine. This objective would require data on part volumes, part processing times, and machine capacities. Many cell formation procedures use a surrogate measure for machine utilization based on the percentage of parts to be produced by the cell that uses a machine. This measure is used in the hope that insight into machine utilization will be gained without explicitly considering the data required to determine machine utilization. Of course, surrogate measures can sometimes turn out to be a poor representation for the desired objective or constraint. The diversity of the cell formation procedures with respect to the items mentioned above causes the development of a framework for reviewing the procedures to be difficult. The most comprehensive review of cell formation procedures to date was done by Wemmerl6v and Hyer (1986). Over 70 procedures are categorized in Wemmerl6v and Hyer's paper. The authors classify cell formation procedures based on which of four general approaches was used in the procedure. In all four approaches parts and machines must be selected for possible inclusion in the cellular manufacturing system. An evaluation step of the proposed cells is also required. The four approaches are 1) identify part families without the help of machine routings (including procedures that use group technology codes and those that do not); 2) identify 29 machining groups; 3) identify part families using routings; 4) identify part families and machine groups simultaneously. The advantages of this method of classification are that it is inclusive and provides classifications for procedures which perform similar functions. Within each of the last three approaches, however, the procedures can vary greatly with respect to the objectives and constraints considered explicitly by the procedure as well as the data required for the procedure. In this literature review we propose six categories which are based on the data used, by a procedure, to classify procedures. There are two reasons for classifying procedures this way. 1) Identifying data used by a procedure provides insight into factors that are explicitly considered by the procedure and helps determine the strengths and limitations of the procedure, and 2) procedures that use similar data can be compared with one another. The six categories that were chosen are 1) procedures that are based solely on data that can be represented by a partmachine incidence matrix, 2) procedures that use a partmachine incidence matrix and incorporate data which can be used to provide weights for parts or machines, 3) procedures that incorporate operations sequences, 4) procedures that consider part setup times or costs, 5) procedures that include part volumes, part processing times, and machine capacities, and 6) procedures that consider multiple scenarios for part volumes. Some procedures may fall into two or more categories. There, also, are a few procedures which use data that do not fall into any of the categories, therefore, a seventh category called "other" is included. For each category discussions of the objectives and constraints considered by procedures, and methods used by procedures to develop a solution are included. 2.2 Procedures Based solely on a PartMachine Incidence Matrix These are the most common of the cell formation procedures. The concept of a partmachine incidence matrix was described in chapter 1. As a brief refresher, in a part machine incidence matrix machines are represented as rows and parts are represented as columns in the matrix (or vice versa). An element in the matrix is '1' if part j requires machine i for processing and '0' otherwise. The advantages of these procedures are that they have low data requirements (all the data can be obtained from part routings) and they tend to be relatively simple and efficient. The disadvantage of these procedures is they do not consider factors such as operations sequences, part volumes, machine capacities, and demand variability. When a solution is developed using one of these procedures the cell designer must somehow factor in other important considerations to arrive at a final solution. There are several objectives and constraints considered by procedures in this category and many techniques have been used to develop solutions. 2.2.1 Obiectives and Constraints The most common objectives used in these procedures are to minimize exceptional elements, minimize duplicate machines, and maximize a measure of similarity between parts or machines assigned to a cell. The two most common constraints are maximum cell size and measure of similarity threshold. 2.2.1.1 Frequently used objectives Exceptional elements. Many procedures (Burbidge 1963, Vannelli and Kumar 1986, King 1980, Chan and Milner 1982, 32 King and Nakornchai 1982, Waghodekar and Sahu 1984, Boctor 1989, Chow and Hawaleshka 1993, Khator and Irani 1987, and Boctor 1991) have as an objective the minimization of exceptional elements. An exceptional element occurs when a part requires two or more cells for processing. This objective is important because exceptional elements would require increased material handling cost and paperwork cost, or subcontracting cost to remove the exceptional element from the plant. If many parts are exceptional elements when a cellular system is designed the benefits of the system will be greatly diminished. Machine duplication. Minimization of machine duplication is also an objective used by many procedures (Burbidge 1963, Vannelli and Kumar 1986, King 1980, Chan and Milner 1982, King and Nakornchai 1982, Khator and Irani 1987, and AlQattan 1990). Machine duplication (also sometimes referred to as a bottleneck machine) occurs when a machine type is required in two or more cells. When a machine type is required in two or more cells additional machines may be required which increases machine investment and may cause machine utilization to be lower vs a process layout. This objective is usually used in conjunction with the exceptional element constraint. Most procedures trade off exceptional elements against machine duplication. Vannelli and Kumar's (1986) procedure attempts to minimize the number of machine types that must be duplicated while allowing no exceptional elements. Measures of similarity. Several procedures attempt to maximize measures of similarity between pairs of parts or pairs of machines (Rajagopalan and Batra 1975, Carrie 1973, Mcauley 1972, Kusiak 1987, De Witte 1980, Chandrasekharan and Rajagopalan 1986a, Luong 1993, Askin et al. 1991, Srinivasan et al. 1990, Gunasingh and Lashkari 1989b, and Wei and Kern 1991). Measures of similarity between pairs of parts are based on how similar the processing requirements are for the parts. Measures of similarity between pairs of machines are based on which parts require the machines. Creating cells that produce parts that have similar processing requirements and machines that process similar parts is one of the main objectives of cellular manufacturing, therefore, this is an important objective. Most of the measures of similarity that are used in cell formation are based on some sort of similarity coefficient (the concept of a similarity coefficient will be described 34 in the next section) which can take on a value between 0 and 1. 2.2.1.2 Frequently used constraints Cell size. The most popular constraint used by these procedures is a cell size constraint (Chandrasekharan and Rajagopalan 1986, Boctor 1989, Luong 1993, Stanfel 1985, Boctor 1991, Wei and Kern 1989, AlQattan 1990, and Vannelli and Kumar 1986). The cell size constraint limits the number of machine types allowed in a cell. The constraint is imposed to limit the physical size of cells. It is generally assumed that intracell material handling cost will be insignificant but this assumption loses validity as cell size grows. This constraint is used with each of the objectives described above. Without this constraint, a cell consisting of all machine types could be created which would have no exceptional elements and would not require any duplicate machines. Similarity threshold. A second constraint which is sometimes imposed is a measure of similarity threshold (Carrie 1973, McAuley 1972, and Chandrasekharan and Rajagopalan). This constraint is usually used in conjunction with the objective that maximizes some similarity measure. 35 The similarity threshold constraint does not allow a machine or a part into a cell unless the cell's similarity measure is above the threshold. The threshold can be applied to a single pair of items (single linkage) in a cell, any pair of items (complete linkage) in the cell, or the average similarity between items (average linkage) in the cell. If the threshold constraint is not applied then a single large cell would result and the shop would be a process layout. If a high threshold is applied to a measure of similarity between machines the resulting layout would have small cells and possibly many exceptional elements. If a high threshold is applied to a measure of similarity between parts then the result would be many part families and to create cells for each family could require many machine types to be duplicated. Picking a similarity threshold requires judgement by the cell designer. Usually the procedures would be run with several similarity thresholds and then a part machine structure would be selected. 2.2.1.3 Other objectives Chandrasekharan and Rajagopalan (1986) and Kumar and Chandrasekharan (1990), and Stanfel (1985) have developed three measures which are slightly more sophisticated and use these measures as objectives. Grouping efficiency. The first measure is called grouping efficiency (Chandrasekharan and Rajagopalan 1986). This measure is a convex combination of withingroup utilization and intercell efficiency. Withingroup utilization is a measure of how heavily parts visit machines in a cell (summation of the number of machine types used by each part in each cell divided by the summation of the number of machine types times the number of parts in each cell). Intercell efficiency equals the fraction of exceptional elements divided by the number of non exceptional elements. This measure serves as a surrogate for a tradeoff between machine investment and materials handling cost. It is speculated that cell layouts with a high within group utilization will result in a shop in which machine types have high utilization because most of the parts in the cell use each machine type in the cell. This should result in a lower required machine investment than a shop that has a low withingroup utilization. A cell layout with a high intercell efficiency should have a low material handling cost because there are relatively few exceptional elements. Grouping efficacy. The second measure is called grouping efficacy (Kumar and Chandrasekharan 1990). Grouping efficacy is similar to grouping efficiency but is adjusted to take into consideration the density of the partmachine incidence matrix (the density of the partmachine incidence matrix is equal to the 1 entries in the matrix divided by total entries). Weighted transfers. Stanfel (1985) uses an objective that considers both intercell and intracell transfers. Intracell transfers for a part are based on the number of machine types that are in a cell that are not required to process the part. The objective weights both intercell and intracell transfers (intracell transfers usually have a lower weight). This objective serves as a surrogate for material handling cost and also considers cell size. Cells with more machine types are likely to be larger and require a higher intracell movement cost. 2.2.2 Methods A wide variety of methods have been used to solve the cell formation problem with only a partmachine incidence matrix as input. The most popular methods have been manual identification, matrix manipulation, clustering using similarity coefficients, graph theory, and solving mathematical models. 2.2.2.1 Manual identification Probably the earliest procedure to be used for cell formation is a manual procedure called production flow analysis developed by Burbidge (1963). This procedure uses part routings to analyze inter and intra departmental flows. There are two stages in production flow analysis: 1) factory flow analysis and 2) departmental flow analysis. Factory Flow analysis studies the basic routes in the factory and then eliminates exceptions. Departmental analysis is broken into group analysis and line analysis. In group analysis parts are divided into families so groups of machines can be put together so that the parts in a family can be completely manufactured by the group. The main criterion is that the division into groups should not necessitate an increase in the number of machines required. ElEssawy and Torrance (1972) also develop a manual procedure called component flow analysis. This procedure is similar to production flow analysis. Since production flow analysis and component flow analysis are manual procedures they become cumbersome to apply as the problem size becomes large. 2.2.2.2 Matrix manipulation This method attempts to create a block diagonal structure, in which almost all l's occupy the diagonal submatices, and almost all zeros occupy the offdiagonal submatrices, by manipulating the data in the partmachine incidence matrix. King (1980) was the first to use this approach. His procedure is called rank order clustering (ROC). Each row and column are considered as binary words, and rows and columns are alternately sorted until a block diagonal structure is obtained. King and Nakornchai (1982) revised the procedure so it would be more efficient to run on a computer (if there are many rows or columns in the partmachine incidence matrix the binary word approach cannot be used) and used a relaxation if there are bottleneck machines (bottleneck machines are machines which process many parts). Chan and Milner (1981) developed a procedure called the direct clustering algorithm (DCA). The advantage of this procedure is that it can start with any matrix and arrive at the same result (if the order of parts or machines is changed the ROC procedure may obtain a different result). The authors also show that DCA will converge to a solution in a limited number of iterations. 40 Chandrasekharan and Rajagopalan (1986) developed a procedure called modified rank ordering clustering algorithm (MODROC). This procedure builds on the ROC procedure by taking ROC output and using clustering to create cells. Khator and Irani (1987) developed a procedure called the occupancy value method (OV). The OV method eliminates limitations of other methods (ROC, DCA) by building up clusters along the diagonal using small selected sections of the larger matrix. 2.2.2.3 Clustering using similarity coefficients Clustering methods using similarity coefficients have been used by many researchers to develop procedures to solve the cell formation problem (Carrie 1973, McAuley 1972, Chandrasekharan and Rajagopalan 1987, Waghodekar and Sahu 1984, DeWitte 1980, Chow and Hawaleshka 1993, Luong 1993, Shafer and Rogers 1993, Askin et al. 1991, Srinivasan and Narendran 1991, and Wei and Kern 1989). These procedures use some measure of similarity between pairs or groups of parts (or pairs or groups of machines) based on the partmachine incidence matrix. A clustering procedure is then used to group similar items into clusters which become the basis for forming cells. The earliest clustering procedure for cell formation was developed by McAuley (1972). He used the Jaccard similarity coefficient. The Jaccard similarity coefficient for a pair of machines is defined as the number of parts processed by both machines divided by the number of parts processed by either machine (the similarity coefficient ranges between 0 and 1). McAuley then used single linkage cluster analysis to form groups of machines. Single linkage cluster analysis uses the most similar pairs of items between two groups to define the similarity between two groups. Other clustering methods have also been used. Two common methods are complete linkage cluster analysis and average linkage cluster analysis. Complete linkage cluster analysis defines the similarity between two groups by the least similar pair of items between the groups and average linkage cluster analysis uses the average similarity between the items in the groups. Seifoddini (1988, 1989a) compares single linkage clustering and average linkage clustering for cell formation and outlines the advantages and disadvantages of each method. Many similarity coefficients and distance measures have been used in clustering procedures for cell formation. Shafer and Rogers (1993a) did a comprehensive survey of these measures. 2.2.2.4 Graph theory Several procedures use graph theory to solve the cell formation problem. Rajagopalan and Batra (1975) were the first to use a graphtheoretic approach. In their procedure machines are considered vertices, and edges are relationships between the machines. The Jaccard similarity coefficient is computed for each pair of machines and an edge is included if the similarity coefficient is above a threshold value. The weight of an edge is equal to the similarity coefficient. A graph partitioning approach is then used to create machine groups and parts are allocated to machine groups to create manufacturing cells. Chandrasekharan and Rajagopalan (1986a) show that the problem can be represented as a bipartite graph in which one set of vertices represents machines and the other set of vertices represents parts. An edge exists between a machine vertex and a part vertex if the part uses the machine. Vannelli and Kumar (1986), and Kumar, Kusiak and Vannelli (1986) also use this approach and propose heuristics to find the minimal cut nodes needed to create the required number of cells. A cut represents intercell movement. 2.2.2.5 Mathematical models Several researchers have formulated models that require only the information provided by a partmachine incidence matrix. These models can then be solved optimally; or if the problem is too large a lower bound can be obtained, and heuristic procedures can be evaluated against the lower bound. Kusiak (1987) formulated the problem using the pmedian model. The objective is to maximize the similarities between parts that are assigned to the same part family (similarities refer to a similarity coefficient). The constraints in this model assign each part to exactly one family, specifies the number of part families, ensure that part i belongs to family j only if family j is formed, and ensure integrality. Shtub (1989) shows that Kusiak's model can be formulated as a generalized assignment problem. Srinivasan, Narendran, and Mahaderan (1990) formulate the problem as an assignment model which differs from Kusiak's model in that the number of groups to be created is not required as an input for the model. Boctor (1989, 1991) formulated a 01 integer program that has as its objective the minimization of intercell transfers (a machine type is assigned to a cell that is different from that of a part that requires processing on it). Constraint sets that assign each part and each machine type to only one cell are included and a constraint set for cell size is included. The author proves that the integrality constraints for part assignment can be dropped and an integer solution will still be obtained. 2.2.2.6 Other methods Many other methods have been used to develop procedures for cell formation. Our review of the literature found the following set theoreticPurcheck (1975), Sundaram and Fu (1987), and Purcheck (1985); string manipulationWu, Venugopal, and Barash (1986); polyhedral dynamicsRobinson and Duckstein (1986); neural networkChu (1993), and Karparthi and Surresh (1993); Hamiltonian pathAskin et al. (1991); branch and boundAlQattan (1990); and simulated annealingBoctor (1991). 2.2.3 Evaluation of Procedures Considering the amount of research that has been done to develop cell formation procedures, based on partmachine incidence matrices, there has been comparatively little research that compares the effectiveness of various procedures. The majority of research that has been done in comparing cell formation procedures is a comparison of various similarity measures and clustering procedures. Harrigan and Mosier (1988) tested four similarity coefficients used in clustering procedures. The objective of their study was to asses the appropriateness of cellular manufacturing within a particular manufacturing environment. Mosier (1989) tested similarity coefficients and clustering procedures using 30 generated problems which varied by cluster definition and block diagonal density. Shafer and Rogers (1993b) tested 16 similarity measures and four clustering procedures and compared the procedures based on various performance measures. Miltenburg and Zhang (1991) tested 9 procedures that used a variety of methods on 8 well known problems from the literature and 60 generated problems. Chu and Tsai (1990) tested three procedures that used matrix manipulation on 11 problems from the literature. Shafer and Meredith (1990) conducted a study that used actual data from three companies. Cell layouts were developed using 7 procedures. Two of the procedures used matrix manipulation, four procedures used clustering and one procedure was based on operations sequences. A simulation model was then built and the performance of each of the cell layouts generated by the procedures was tested. The performance measures used were average distance traveled, number of extracellular operations, average WIP, maximum WIP, and longest average queue. 2.3 Procedures Which Use a PartMachine Incidence Matrix and Incorporate Data to Provide Weights for Parts or Machines These procedures are similar to the procedures of section 2.2 with the exception that these procedures recognize that some parts are more important than others, and some machine types are more important than others. Data is incorporated into these procedures which allows parts and machine types to have various weights of importance. Part weights are usually based on the volume of units required for a part times a cost per unit. The cost per unit is usually based on the cost of intercell movement or an incremental subcontracting cost to remove the part from the system. Machine type weights are usually based on the cost of a single machine of a given type. 2.3.1 Objectives and Constraints The objectives used by these procedures are 1) minimize the cost of exceptional elements (Kusiak and Chow 1987, Seifoddini 1989c, Seifoddini and Wolfe 1987, and Kumar and Vannelli 1987); 2) minimize the cost to duplicate machines (Seifoddini 1989c, and Sundaran and Fu 1987); 3) maximize a measure of similarity (Steudel and Ballakur 1987, Balasubramanian and Panneerselvam 1993, Okogbaa et al. 1992, and Gupta and Seifoddini 1990). The two most popular constraints are cell size and measure of similarity threshold. The first objective is sometimes used by itself subject to the constraint that a machine type can only appear in one cell and a cell size constraint (Kusiak and Chow 1987, Kumar and Vannelli 1987); or in conjunction with the second objective (Seifoddini 1989). Some of the similarity measures used by these procedures incorporate weights. Steudel and Ballakur (1987) use a similarity measure called cell bond strength (CBS) which is based on the processing times of parts. Balasubramanian and Panneerselvam's (1993) measure of similarity is based on the number of excess moves a part requires in each cell. Okogbaa et al. (1992) base their measure on flows between machine types. Gupta and Seifoddini (1990) incorporate part volumes and processing times into a similarity coefficient. 2.3.2 Methods Several methods are used by these procedures. Seifoddini and Wolfe (1987) cluster using several similarity thresholds and then check the intercell movement cost for each threshold and choose the best to use as a layout. Gupta and Seifoddini (1990) cluster using complete linkage with a threshold constraint. Steudel and Ballakur (1987) use a dynamic programming algorithm to maximize their similarity coefficient. Balasubramanian and Panneerselvam (1993) first create a set of potential cells by creating a cell for each part and then creating additional potential cells using the rank ordering clustering procedure. They then select cells based on a warehouse covering algorithm. Kumar and Vannelli (1987) use a procedure which starts with part seeds or machine seeds. Part seeds are parts that do not use any common machine types and machine seeds are machine types which do not process any common parts. The procedure then uses objective function criteria to add parts and machines to cells. 2.4 Routing Based Procedures Without Machine Capacities These procedures require part routings to be used as opposed to a partmachine incidence matrix. The part routings provide additional information about a part's sequence of operations and possible alternative processing plans. The procedures in this category also can incorporate cost and volume information, but do not incorporate any information about machine capacities. The procedures are further categorized into two groups: 1) procedures which use operations sequences, and 2) procedures which use alternative routings. Procedures which consider operations sequences focus on efficient materials flows when forming cells. Procedures which consider alternate routings focus on equipment utilization and flexibility when forming cells. 2.4.1 Operations Sequences Vakharia and Wemmerl6v (1990), Selvam and Balasubramanian (1985), Tam (1990), and Logendran (1991) have developed procedures which incorporate operations sequences. The first three procedures use similarity coefficients. Vakharia and Wemmerl6v (1990) attempt to create cells that do not require parts to backtrack (An attempt is being made to create flowline cells in which there is a unidirectional flow thru the machines in the cell. If a part must move in the opposite direction of the flow then a backtrack has occurred). A similarity coefficient, that compares the number of machines in two groups that are used in the same sequence to the total number of machines in the groups, is used to cluster groups. Selvam and Balasubramanian (1985) develop a set of potential cells using a similarity coefficient based on operations sequences, and then select the desired number of cells by using a covering algorithm. The objective is to minimize material handling cost. Tam (1990) also develops a similarity coefficient based on operations sequences, and then uses a clustering method called knearestneighbor (KNN) to form cells. Logendran aggregates machine types in an attempt to minimize weighted intracell and intercell moves based on operations sequences. 2.4.2 Alternative Routings Kusiak (1987), Shtub (1989), and Sankaran and Kasilingam (1990) have developed models which incorporate the possibility of alternate routings. Kusiak (1987) formulates the problem as a pmedian model. The objective is to maximize the similarities between part routings assigned to the same family (similarities are based on a comparison of process plans for pairs of parts). The constraints in this model are only one routing for a part is selected, each part is assigned to one family, only N part families are formed where N is an input parameter, part i belongs to family j only if family j is formed, and integrality. Shtub (1989) shows that Kusiak's (1987) formulation is equivalent to the generalized assignment problem (GAP). Sankaran and Kasilingam (1990) formulate a model that maximizes the number of routings that can be completed in a cell. The purpose of this model is to design a cellular system that is flexible (parts will have alternate cells that can be used for processing). The constraints of the model are at least one routing is chosen for each part, cell size limit, a budget limit that can be spent on machines, and if a routing is assigned to a cell then all the machine types required by the routing are assigned to the cell. 2.5 Procedures Which Incorporate Setup Information Several procedures incorporate setup objectives into the cell formation problem. Setup is an important consideration in cell formation because by combining items with similar setups and dedicating equipment to producing them setup time can be substantially reduced which reduces labor costs and increases plant capacity. Shafer and Rogers (1991) formulated models which include minimizing setup time as an objective (minimizing intercellular movements and investment in machines are also objectives). Shafer and Roger's models also consider sequence dependent setups in the cell formation problem. The models combine a pmedian formulation with the travelling salesman problem (TSP). A two stage heuristic is used to solve the problem. In stage I a pmedian model is solved to form part families and cells. Stage II then solves a travelling salesman problem to determine sequences for each family. Chakravarty and Shtub (1984) developed a procedure to form cells that considers inventory and setup costs associated with batch sizes as well as materials handling costs. The authors develop a cost function for the total setup and inventory holding cost for all components at a machine and then derive an optimal production cycle time for a machine (lotsize). The authors then show how to assign a machine, that could be assigned to more than one group (based on processing sequences), to one of the groups based on the cost function. Askin and Subramanian (1987) constructed an economic model of manufacturing costs as a function of cell configuration. The model includes the following costs: setup, variable production, production cycle inventory, WIP, material handling, and fixed machine. Setup costs include both product and family setup costs. A heuristic, that is an extension of the approach of Boucher (1984) is used to 54 develop product and family cycle times and considers family setup costs, WIP costs and finished goods inventory costs. The procedure used to create cells consists of three sequential stages. ROC is used in the first stage to cluster the partmachine incidence matrix. Stage two attempts to combine adjacent groups based on an economic evaluation. Stage three aggregates groups based on machine capacities and further economic evaluation. Rajamani, Singh, and Aneja (1992) formulate a mathematical model for cell formation in environments where sequence dependent setups exist. The model considers the tradeoff between setup costs and machine investment (by dedicating machines to produce certain parts setup cost is reduced but machine investment increases). The model's objective function is to minimize the sum of total discounted cost of machines assigned to all the cells, and setup costs incurred due to sequence dependence of parts in each cell. The constraints in the model ensure each part is produced in a cell, each part has a place in a sequence, a sequence is defined, relationships between pairs of parts, and capacity constraints (these include processing time and setup time). 2.6 Procedures Which Incorporate Machine Capacities These procedures recognize that it may not be possible to process all parts in a desired cell due to limited machine capacity. These procedures require for each part a demand requirement and the processing time on each machine type, and for each machine type the time available for processing. The procedures in this category are based on mathematical models which require an integer number of machines of each type assigned to a cell. Two common models can be identified, although, there are several variations to each of these models. 2.6.1 Model 1 Shafer, Kern and Wei (1992), Shafer and Rogers (1991), Askin and Chin (1990), Choobineh (1988), Askin and Subramanian (1987), Vakharia, Chang and Selim (1993), Rajamani, Singh and Aneja (1990), Sankaran (1990), Sule (1991), and Kamrani and Pansaei (1993) use models which are variations of this basic model. These models include at least two items in the objective function. The first item deals with the cost of parts which cannot be produced in a single cell. Shafer, Kern and Wei (1992) consider a subcontracting cost to remove the part, the other models consider the cost to move parts between cells. The second item is the cost to use additional machines. If there is an infinite number of machines available and there is no cost to use machines a cell could be created for each part and no intercell transfers or subcontracting would be required. The models recognize that there is a tradeoff between the two objectives that were described. The cost of machines can be decreased at the expense of intercell material handling cost or subcontracting, and the reverse is true. These models contain constraints which do not allow the processing time of a machine type in a cell exceed the time available of a machine type in a cell. The models, with one exception, also contain constraints which restrict the size of a cell to k machine types where k is a required input parameter, or require cells defined by machine types as input (Shafer, Kern and Wei 1992, Askin and Subramanian 1987, Sule 1991, and Choobineh 1988). The exception is Sankaran (1990) who uses a constraint which requires a minimum level of similarity (based on parts which are processed) between machines in a cell. These models, with the exception of Sule (1991), require that a part is assigned to a single cell (a part's production requirement cannot be split among cells). There are several variations to this model. Askin and Chin (1990), Choobineh (1988), Vakharia, Chang and Selim (1993), and Sule (1991) incorporate operations sequences into the model. Choobineh (1988) and Rajamani, Singh and Aneja (1990) consider multiple process plans or alternate routings in their formulation. Shafer and Rogers (1991) include setup time in the objective function; and Askin and Subramanian (1987) include setup costs and inventory costs in the objective function. Kamrani and Pansaei (1993) do not include a cost for parts that cannot be produced in a cell in the objective function but include constraints which do not allow intercell transfers. 2.6.2 Model 2 Wei and Gaither (1990a, 1990b), Dahel and Smith (1991, 1993), Nagi, Harhalakis, and Proth (1990), and Logendran (1990, 1993) use models which are variations of this basic model. 58 These models include the cost of parts which cannot be produced in a single cell. Wei and Gaither (1990a, 1990b) consider a subcontracting cost to remove the part, Dahel and Smith (1991, 1993), Nagi, Harhalakis, and Proth (1990), and Logendran (1990, 1993) consider the cost to move parts between cells. These models also contain constraints which do not allow the processing time on a machine type in a cell exceed the time available of the machine type in the cell. These models contain two other sets of constraints: one that limits the number of machines that can be assigned to a cell to a prespecified number; and one that limits the number of machines that are available of each machine type. The second constraint is important because if an unlimited number of machines are available for each type then a cell could be created for each part and there would be no need for subcontracting or intercell transfers. Dahel and Smith (1991, 1993) incorporate part operations detail into their model. Nagi, Harhalakis, and Proth (1990) consider alternate routings in their formulation and allow a part's production requirement to be spread over several routings so machines can be better utilized. Wei and Gaither (1990b) include, along with the cost of bottleneck parts, average cell utilization, intracell load imbalances, and intercell load imbalances in the objective function. Each of the items is given a weight and the objective is to minimize the weighted sum of the items. Logendran (1990, 1993) includes both intracell and intercell moves in the objective. Weights are given to both types of moves (moves are based on operations sequences and a lower weight is usually given to intracell moves) and the objective is to minimize weighted moves. Logendran (1990) includes constraints that do not allow a machine's utilization to fall below a specified level; and Logendran (1993) includes machine utilization in the objective function. 2.6.3 Methods Since the models contain integer variables, reasonable size problems can be computationally difficult to solve, therefore, many of the procedures used are heuristics which may not solve the problem optimally. As in other categories a variety of methods are used in the procedures. Several procedures (Shafer and Rogers 1991, Askin and Chin 1990, Askin and Subramanian 1987, Logendran 1990, Nagi, Harhalakis, and Proth 1990, and Sule 1991) break the problem into stages and solve each stage sequentially. Some of the procedures solve the subproblems in each stage optimally. Shafer and Rogers (1991) break their formulation into two subproblems, a pmedian problem and a travelling salesman problem, and solve the subproblems optimally. Nagi, Harhalakis, and Proth (1990) create two subproblems and then iterate between the two subproblems until convergence to a local minimum is achieved. Other methods are also used. Wei and Gaither (1990) fix seed machines and then form cells by solving the integer program. Vakharia, Chang and Selim (1993) use simulated annealing and tabu search. Dahel and Smith (1991, 1993) use the constraint method to solve their multi objective model. 2.7 Procedures that Consider Multiple Scenarios for Part Volumes Procedures in this category recognize that demand for parts may not remain the same over time, or that demand for parts may be stochastic. Previous procedures, which considered part volumes, were based on annual or average demand and were considered to be deterministic. If demand varies from period to period or estimates for demand turn out to be inaccurate a cell layout may not perform as well as expected. Cell layouts created by procedures in this category should have the flexibility to perform well even if there are expected volume swings or volume differs from what was estimated. Little work has been done in this area. Vakharia and Kaku (1988, 1993) developed a procedure to handle longterm demand changes. Seifoddini (1990) developed a procedure for probabilistic demand estimates. Vakharia and Kaku (1988, 1993) focus on a strategy of part reallocation to respond to demand changes as opposed to partial cell system redesign or complete cell system redesign. This strategy is chosen because it has the lowest cost of change (if cell design is robust changes can be handled by this strategy). Vakharia and Kaku proposed a model. The objective is to minimize material handling cost and additional machine investment costs. The constraints of the model are each operation for each part is assigned to a cell, determination of the number of new machines required based on machine capacity, a cell is visited if an operation is performed in a cell, integrality, and nonnegativity. Additional constraints can be added to limit cell size to a specified number of machines. A heuristic is used to solve the model. Seifoddini's (1990) procedure incorporates the uncertainty of product mix into the final solution. The procedure has three major steps: 1) the product mix and the associated partmachine incidence matrix are expressed in the form of a probability function, 2) a set of alternate solutions are developed based on all possible product mixes, 3) a measure of effectiveness is developed and used as a criterion for evaluating different solutions for the purpose of choosing the best solution. This procedure assumes there is a discrete distribution of product mixes and partmachine incidence matrices. The procedure uses similarity coefficients and a clustering algorithm to develop a cell layout for each product mix that minimizes intercell material handling cost. The intercell material handling cost is then calculated for each product mix and each layout, and an expected intercell material handling cost is calculated 63 for each layout. The layout with the lowest expected cost is chosen. 2.8 Other Procedures Procedures in this category incorporate data that is not included in the previous categories. Gunasingh and Lashkari (1989b) formulate an integer programming model that groups machines based on machine tooling availability and parts tooling requirements. The authors define a compatibility index between a machine and a part. The index is based on the number of common tools between part i and machine j divided by the minimum of the number of tools part i requires or machine j has available. The objective of the model is to maximize the compatibility index. Constraints of the model include a cell size restriction, allocation of each machine to at least one cell, and integrality. Irani, Cavalier and Cohen (1993) consider both the layout of cells and machines in order to minimize the effect of intercell movement. The authors use a fromto travel 64 chart to layout machines within a cell and layout cells with respect to each other. The procedure also considers options such as placing a machine that is common to two cells, that are adjacent to each other, between the two cells so that greater flexibility of machine utilization can be achieved. Min and Shin (1991) formulated a model that incorporates human factors into cell formation. The authors believe that, in addition to the normal functions, cell formation should also allocate operators to cells, assign machines to each operator, and determine the job scope for each operator. The model includes many sets of constraints and is a multiobjective goal program. Some of the constraints and objectives that stem from incorporating human factors are maximum number of operators allowed in a cell, maximum number of parts that operator p can handle, assignment of parts to operators, wage rates of operators, and skill matching of an operator with a part. 2.9 Summary The review of procedures in the previous sections illustrates how diverse cell formation procedures are with 65 respect to factors considered and data required, as well as the methods used. A single procedure probably cannot be developed which considers all the possible factors that could impact on the performance of a cell layout. When choosing a cell formation procedure to apply to a particular situation selection should depend on the factors that appear to be most important. In many instances it may be appropriate to use several procedures and then compare the layouts that are generated. The review of the procedures in section seven indicates that more work could be done to develop procedures that create good cell layouts in situations that experience demand variability or demand uncertainty. Cellular layouts usually tradeoff strengths and weaknesses of product line layouts and process layouts. A key strength of a process layout is its flexibility to handle volume changes or uncertainty, product line layouts tend to be less flexible in these environments. Procedures which can evaluate whether a shop should be organized with small efficient cells dedicated to a few parts or larger less efficient cells which can be used to produce many parts, or a mixture of large and small cells should be of aid to the shops which exist in the environment described above. CHAPTER 3 MODELS 3.1 Introduction This chapter presents two models which can be used for identifying cellular configurations in the presence of dynamic demand. Model I attempts to form cells that do not exceed a predetermined cell size in terms of the number of machines contained in each cell. This model considers materials handling costs as well as inventory and backordering costs to select a solution. Model II develops a cell configuration for a shop by selecting cells from a set of potential cells, which could be of varying sizes. To develop a cell configuration, first several runs of model I could be made with varying cell size parameters as input. The cells that were formed in each of the runs can be included in the set of potential cells. Model II would then be run using this set of potential cells to create a cell configuration. 3.2 Model I 3.2.1 Description of Model I This model attempts to form cells that do not exceed a cell size constraint which is specified in terms of the number of machines a cell can contain. The model can also form a remainder cell which has no cell size constraint. The model uses a planning horizon which consists of T periods. The parts to be produced during the planning horizon are known. The demand for each part is known and can vary from period to period. Each part requires a set of machine types for production, and the processing time per unit for each part on each machine type, that is required by the part, is a known parameter. There are a limited number of machines of each type that can be used to produce parts. Each machine has a limited capacity available during each period in the planning horizon. The capacity of the machine types within each cell is an important constraint in this model. The total processing time required of a machine type within a cell during any period cannot exceed the total capacity of that machine type within that cell. In order to effectively utilize machine capacity the model allows for inventory planning decisions. The number of units produced of a part during a period can exceed the demand requirement for the period. The excess production can be stored in inventory and used to satisfy demand requirements in later periods. A part's demand requirement, for a period, may also be partially or completely backordered and excess production of the part during a later period may be used to fill backorders. If an inventory of a part is held then there is a cost associated with the inventory. If a backorder is incurred there is a cost associated with the backorder. Each part has a unit inventory holding cost and a unit backorder cost associated with it. These costs are on a per period basis. The unit cost of producing a part depends on where in the shop the unit is produced. If the unit is produced entirely in a cell that meets the cell size constraint then the production cost is zero. If the unit is produced entirely in the remainder cell there is a cost associated (by part) with producing the unit. If the unit is produced 69 using machines that reside in two or more cells there is an intercellular movement cost associated (by part) with producing the unit. Each part has a unit remainder cell production cost and a unit intercellular movement production cost associated with it. 3.2.2 Model I Formulation The formulation of model I is shown below. I T Min Z= E (MMRiXitr+MMIiXiti+hilit+biBit) iI tI Subject to K IitlBit1+1: Xitk+XitiIit+Bit=dit kI PijXitk CjYjk i1 for i= .,I; (2) t=l,...,T. for for for I K 1 jE PijXitk+ Pij (Xiti+Xitr) < CjAj for  ki 1 for k=l, ...,K & r; j=l, ... ,J; t=l,...,T. j=l, .. .,J; t=l, .,T for j=l,...,J. K E YiA3 Yk J Yjk for k=l,...,K Yjk is integer for all j & k. (7) All variables are a 0 (8) where I=The total number of parts. T=The total number of periods. J=The total number of machine types. K=The total number of cells not including a remainder cell. i=part index (i=1,...,I). t=period index (t=l,...,T). j=machine type index (j=l,...,J). k=cell index (k=l,...,K & r). MMRi=The cost to produce 1 unit of part i in the remainder cell. MMIi=The cost to produce 1 unit of part i using two or more cells (uses intercell movement). hi=The cost to hold 1 unit of part i for 1 period. bi=The cost to backorder 1 unit of part i for 1 period. dit=The demand for part i in period t. Pij=The processing time for part i on machine type j. Cj=The capacity in hours of 1 machine of type j. Aj=The number of machines available of type j. CS=The maximum number of machines allowed in a cell excluding the remainder cell. Xik=The number of units of part i produced in cell k during period t. Xi,,=The number of units of part i produced using intercell movement during period t. Iit=The inventory of part i at the end of period t. Bit=The number of units on backorder of part i at the end of period t. Yjk=The number of machines of type j assigned to cell k. 3.2.3 Description of Model I Equations (1) Objective function: The objective function is the sum over all parts and all periods of four terms for each part during each period. The first term is the cost of producing a unit of part i in the remainder cell times the number of units produced of part i in the remainder cell during period t. The second term is the cost of producing a unit of part i using intercell movement times the number of units produced of part i using intercell movement during period t. The third term is the cost to hold one unit of part i in inventory for one period times the number of units of part i that are in inventory at the end of period t. The last term is the cost to hold one unit of part i on backorder for one period times the number of units on backorder at the end of period t. (2) This set of equations represents material balance constraints. The first two terms on the left side of the equality (Iit_Bit,) represent the net inventory position of part i at the beginning of period t (or the end of period tl) and the last two terms on the left side of the equality (Iit+Bi,) represent the negative of the net inventory position of part i at the end of period t. The middle two K terms on the left side of the equality ( Xltk+Xiti) equals production in units of part i during period t. Therefore, the net inventory position of part i at the beginning of period t plus production of part i during period t minus the net inventory position of part i at the end of period t must equal the demand for part i during period t (the right side of the equality). There is a constraint for each part during each period. 73 (3) These constraints ensure that the total processing time required by parts produced in cell k during period t does not violate the capacity of machine type j in cell k. There is a constraint for each machine type, in each cell during each period. (4) These constraints ensure that the total processing time required by parts produced during period t does not violate the capacity of machine type j. There is a constraint for each machine type during each period. (5) These constraints ensure that the total number of machines of type j that are assigned does not exceed the number of machines of type j that are available. There is a constraint for each machine type. (6) These constraints require that cell k not contain more than a total of CS (the cell size parameter) machines. There is a constraint for every cell with the exception of the remainder cell. (7) These constraints require that the number of type j machines assigned to cell k be integer. There is a constraint for each machine type in each cell. (8) These constraints require all the production, inventory, and backorder variables to be nonnegative. 3.2.4 An Example of Model I The following example shows how model I is applied to a problem with 2 parts, 3 machine types, and two periods of demand. The partmachine incidence matrix for this example is shown in table 8. Part 1 requires machine types 2 and 7, and part 2 requires machine types 1 and 2. The cell size limit used in this example is two machines. A remainder cell can also be formed. Table 9 shows the demand by period, the unit inventory cost, and the unit backorder cost for each part. Table 10 shows the number of machines of each type that are available and the capacity in units per machine of each type. The cost to manufacture either part in a cell that meets the cell size restriction is 0, the cost to manufacture either part in the remainder cell is 1, and the cost to manufacture either part using intercell movement is 3. In the formulation of this problem we allow for two cells that meet the cell size constraint and a remainder cell. The objective for this problem is Min Z=Xllr+Xi2r+X21r+X22r+3Xii+3X12i+3X21i+3X22I+4B11+4B12+4B21 +4B22+I +112+I21 +I22 Table 8 PartMachine Incidence Matrix for Model I Example. Machine Type Part 1 2 7 1 1 1 2 1 1 Table 9 By Part Demand. Period Unit Unit Part 1 2 Inv. Cost B/o Cost 1 450 0 1 4 2 0 450 1 4 Table 10 Machine Data. Machine Type # Available Capacity/Machine 1 1 450 2 2 225 7 1 450 The full formulation for this problem is shown in appendix A. Examples of each of the different types of constraints will be shown below. The demand constraint for part 1, period 1 is Xiii+Xi2+Xinr+XIII,+B,,=450 The capacity constraint for machine type 2 in cell 1 during period 1 is X,,,+X211225Y21 The shop capacity constraint for machine type 2 during period 2 is X121+Xi22+X12r+X121+X221+X222+X22r+X22,i450 The machine availability constraint for machine type 7 is Y,7+Y72+Y7r:1 The cell size constraint for cell 1 is Y11+Y21+Y,712 The solution to this model is to form one cell with all the machines in it (a process layout), produce 450 units of part 1 during period 1, and produce 450 units of part 2 during period 2. With this solution no inventory or backorders would be required, and the objective function value equals 900. It is interesting to see how the solution would change if we had assumed a constant demand of 225 units for each part during each period. With this assumption, the model forms two cells. The first cell would have a type 2 machine and a type 7 machine. This cell would produce 225 units of part 1 in each period. The second cell would have a type 1 machine and a type 2 machine. This cell would produce 225 units of part 2 in each period. The objective function value would equal 0. If this plan were implemented and the demand turned out to be the same as in table 9 the cost would be 1125. 3.3 Model II 3.3.1 Description of Model This model selects cells, which will be formed to produce parts, from a set of potential cells. The objective of the model is to minimize production, inventory, and backordering costs. The model uses a planning horizon which consists of T periods. The parts to be produced during the planning horizon are known. The demand for each part can vary from period to period. Each part requires a set of machine types for production, and the processing time per unit for each part on each machine type, that is required by the part, is a known parameter. There are several potential cells that are being considered for inclusion in the shop. All the cells cannot be included because there are not enough machines available. There are several different types of machines in the shop and the number of machines available of each type must be specified as an input parameter for the model. The configuration of each potential cell in terms of the number of machines of each type required by the cell is known. Each machine has a limited capacity available during each period in the planning horizon. The capacity of the machine types within each cell is an important constraint in this model. The total processing time required of a machine type within a cell during any period cannot exceed the total capacity of that machine type within that cell. In order to effectively utilize machine capacity the model allows for inventory planning decisions. The number of units produced of a part during a period can exceed the demand requirement for the period. The excess production can be stored in inventory and used to satisfy demand requirements in later periods. A part's demand requirement, for a period, may also be partially or completely backordered and excess production of the part during a later period may be used to fill backorders. If an inventory of a part is held then there is a cost associated with the inventory. If a backorder is incurred there is a cost associated with the backorder. Each part has a unit inventory holding cost and a unit backorder cost associated with it. These costs are on a per period basis. The unit cost of producing a part varies depending on which cell the unit is produced in. Each part has a cost 79 assigned for each cell. These costs can reflect the material handling and other costs associated with producing a unit of the part in a particular cell. A unit of a part can also be produced by using machines that reside in two or more cells. If this occurs the unit will incur costs associated with intercell material handling and other costs, such as the paperwork required for intercell movement and tracking. 3.3.2 Model II Formulation The formulation of model II is shown below. 1 T K I T Min Z= (CiiXit,+hiit+biBiJt + (CikXik) iI t1 k1 iI tI Subject to K Iit1Biti+ E Xitk+XitI Ii+Bit=dit I PijXitk< CjZkMjk i1 SPijXitk+ pij (Xitz) CA iI kI iI K E MjZk Aj kI for i=l, ..,I; (10) t=1,...,T. for k=l,...,K & r; (11) for j=1, ...,J; for t=l,...,T. for j=l, ...,J; for t=l, ...,T (12) for j=1,...,J. (13) 80 Zk, {0,1} for k=l,...,K. (14) All other variables a 0 (15) where I=The total number of parts. T=The total number of periods. J=The total number of machine types. K=The total number of cells. i=part index (i=l,...,I). t=period index (t=l,...,T). j=machine type index (j=1,...,J). k=cell index (kl,...,K). Cik=The cost to produce 1 unit of part i in cell k. Cii=The cost to produce 1 unit of part i using two or more cells (uses intercell movement). hi=The cost to hold 1 unit of part i for 1 period. bi=The cost to backorder 1 unit of part i for 1 period. dit=The demand for part i in period t. Pij=The processing time for part i on machine type j. Cj=The capacity in hours of 1 machine of type j. Aj=The number of machines available of type j. Mjk=The number of machines of type j required to form cell k. Xilk=The number of units of part i produced in cell k during period t. Xi,,=The number of units of part i produced using intercell movement during period t. Iit=The inventory of part i at the end of period t. Bit=The number of units on backorder of part i at the end of period t. Zk=1 if cell k is formed; 0 otherwise. 3.3.3 Description of Model II Equations (9) Objective function: The objective function sums production, inventory, and backorder costs for all parts over all periods. There are four terms in the objective function. The first term sums across all parts, cells, and periods the cost to produce a unit of part i in cell k times the number of units produced of part i produced in cell k during period t. The second term sums the cost to produce a unit of part i using intercell movement times the number of units produced of part i using intercell movement during period t across all parts and all periods. The third term is the cost to hold one unit of part i in inventory for one period times the number of units of part i that are in inventory at the end of period t summed across all parts and all periods. The last term is the cost to hold one unit of part i on backorder for one period times the number of units on backorder at the end of period t summed across all parts and all periods. (10) This set of equations represents material balance constraints. The first two terms on the left side of the equality (Iit_Bit_1) represent the net inventory position of part i at the beginning of period t (or the end of period tl) and the last two terms on the left side of the equality (Iit+Bit) represent the negative of the net inventory position of part i at the end of period t. The middle two terms on the left side of the equality (F Xit+Xiti) equals ki production in units of part i during period t. Therefore, the net inventory position of part i at the beginning of period t plus production of part i during period t minus the net inventory position of part i at the end of period t must equal the demand for part i during period t (the right side of the equality). There is a constraint for each part during each period. 83 (11) These constraints ensure that the total processing time required by parts produced in cell k during period t does not violate the capacity of machine type j in cell k. There is a constraint for each machine type, in each cell during each period. (12) These constraints ensure that the total processing time required by parts produced during period t does not violate the capacity of machine type j. There is a constraint for each machine type during each period. (13) These constraints ensure that the cells selected for inclusion in the shop do not require more type j machines than are available. There is a constraint for each machine type. (14) These constraints require that cell k is either selected (Zk=l) or not selected (Zk=0) for inclusion in the shop. (15) These constraints require all the production, inventory, and backorder variables to be nonnegative. 3.3.4 An Example of Model II This example uses the data presented in tables 8 thru 10 of section 3.2.4. Table 8 shows the partmachine incidence matrix; table 9 shows the demand by period, the unit inventory cost, and the unit backorder cost for each part; and table 10 shows the number of machines of each type that are available and the capacity in units per machine of each type. Three potential cells are considered for inclusion in the shop. Table 11 shows the configuration of each cell in terms of the number of machines of each type that would be included in the cell if the cell is selected. Table 11 Potential Cells Machine Type Cell 1 2 7 1 0 1 1 2 1 1 0 3 1 2 1 Table 12 shows unit production costs for each part in each cell that the part can be completely manufactured in. Table 12 also shows unit production costs using intercell movement. The objective for this problem is Min Z=X13+X+X12 +X213+X223+3X1I+3X12I+3X21I+3X22I+4B11+4B12+4B21 +4B22+I11+I1+I1+I22 Table 12: Production Costs Part cell 1 cell 2 cell 3 Intercell 1 0 n/a 1 3 2 n/a 0 1 3 The full formulation for this problem is shown in appendix B. Examples of each of the different types of constraints will be shown below. The demand constraint for part 1, period 1 is Xiii+X13,+XI In+B11=450 The capacity constraint for machine type 2 in cell 1 during period 1 is X,111225Z, The shop capacity constraint for machine type 2 during period 2 is X121+X222+X123 +X12+X223+X22I 450 The machine availability constraint for machine type 2 is: Zi+Z2+2Z32 The solution to this model is to select only cell 3. Cell 3 contains all the machines in the shop so this solution is a Process layout. 450 units of part 1 should be produced in period 1, and 450 units of part 2 should be 86 produced in period 2. This solution requires no inventory or backorders, and the objective function value equals 900. It is interesting to see how the solution would change if we had assumed a constant demand of 225 units for each part during each period. With this assumption, the model selects cells 1 and 2 for inclusion in the shop. 225 units of part 1 would be produced in cell 1 during each period, and 225 units of part 2 would be produced in cell 2 during each period. The objective function value would equal 0. If this plan were implemented and the demand turned out to be the same as table 9 the cost would be 1125. CHAPTER 4 SOLUTION PROCEDURES 4.1 Introduction In this chapter, solution procedures for the models are proposed. Since model II develops a shop configuration from a list of potential cells, that may have been created by solving model I several times with different cell size parameters, the focus of the procedures proposed is on model II. A heuristic for solving model I is proposed in section 4.2. This heuristic can then be run, using several cell size parameters, to create a list of potential cells. Section 4.3 proposes lower bounding schemes for model II. Heuristics for solving model II are proposed in section 4.4 The lower bounding schemes in section 4.3 can be used, for large problems, to evaluate the heuristics of section 4.4. 4.2 Model I Heuristic This heuristic requires several iterations to develop a solution for model I. During each iteration an attempt is made to form one cell that meets the cell size constraint. If a cell is formed (that meets the cell size constraint) then the integer variables for that cell are fixed, and then the next iteration of the heuristic attempts to form another cell. If a cell is not formed (that meets the cell size constraint) the heuristic stops and a solution has been developed. The heuristic also stops if all the machines have been used to form cells that meet the cell size constraint. In addition to the notation of section 3.2.2, the following notation is required. Let n=The iteration index for the heuristic. Let Mjk=The number of type j machines required by the cell formed in iteration n of the heuristic. The following steps are required: Step 0. Initialization. set n=l; Step 1. Solve problem P1 (described below). Step 2. a. set a=1; for j=l to J do if Yjk>0 then a=0; b. set b=l; for j=l to J do K if Aj Mjk>0 then b=O; ki c. if a+b>0 then stop else n=n+l and repeat step 1. Problem P1 is as follows: I T Min Z= E (MMRiXitr+MMIiXiti+hilit+biBit) i1 t1 Subject to IitlBit1+ Xitk +Xitr+Xit Iit+Bit=dit k1 I PijXitk CjMjk Pi XinC 7 i~tn j~ Ep=!.Y (16) for i=l, ...,I; (17) t=1,...,T. for k=l, ...,nl; for j=l, ...,J; for t=l,...,T. for j=l, ...,J; for t=l, ...,T. (18) (19) 
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