A two-stage model for cost effective part family and machine cell formation

PII:

Computers ind. Engng Vol. 34, No. 4, pp. 759±776, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0360-8352(98)00103-X 0360-8352/98 $ - see front matter

A TWO-STAGE MODEL FOR COST EFFECTIVE PART FAMILY AND MACHINE CELL FORMATION S. M. TABOUN,1* N. S. MERCHAWI2 and T. ULGER1 Department of Industrial and Manufacturing Systems Engineering, University of Windsor, Windsor, Ontario, N9B 3P4 Canada 2 Department of Mechanical Engineering, Concordia University, Montreal, Quebec, H3G 1M8 Canada 1

AbstractÐThis research presents, implements and tests a two-stage procedure for cost e€ective part family and machine cell formation. First, the problem is formulated as a mixed integer mathematical model for simultaneous machine grouping and part family assignment. This model, which we refer to as the single-stage model, considers the cost trade-o€s of cell con®guration, machine procurement and salvage, subcontracting, inter-cell movement, and capital investment, all of which re¯ect the signi®cance of real life planning aspects. To alleviate the computational burden of this single-stage model, we decompose it into two stages: the ®rst stage is a heuristic for machine cell and part family formations; the second stage integrates the heuristic method with a mathematical program to optimize the various cost aspects. The ecacy of the proposed models is shown through a number of example problems. The results show that the two-stage procedure is powerful in the planing stages of large-size problems where the cost aspects are crucial. # 1998 Elsevier Science Ltd. All rights reserved. KeywordsÐcellular manufacturing systems, part grouping, optimization, integer programming, heuristics, inter-cell movement, subcontracting, investment costs

1. INTRODUCTION

Cellular Manufacturing Systems (CMS) are the result of direct application of Group Technology (GT) philosophy. Parts that have similar processing requirements, such as machines, tools, routes and/or geometrical shapes are classi®ed into part families. Machine cells contain groups of functionally dissimilar machine types; each machine cell processes one or more part families. Reorganization of the functional shop into manufacturing cells leads to several bene®ts, such as reductions in production costs, set-up and throughput times, work-in-process inventories, and material handling, as well as improvements in machine utilization. Part family and machine cell formations are the backbone of the structural design phase of manufacturing cells. Major steps of this phase include the identi®cation of parts and their process populations, machines, tools, ®xtures, pallets and material handling equipment. Other key variables may include the number of cells, machine types, di€erent part types, number of operations per part, alternative process plans and routings for each part, and part characteristics. In many situations, it is necessary to trade-o€ certain objectives related to structural design parameters and system performance variables [56]. Successful implementation of cellular manufacturing is heavily dependent on the determination of part families and machine cells, such that one or more families can be fully processed within a single cell. This problem is designated as ``machine-part grouping''. A signi®cant amount of e€ort has been directed toward part family and cell formation problems, and several approaches were utilized. Part families and machine cells were formed either sequentially or simultaneously, taking into consideration system constraints and various direct or indirect manufacturing costs. Burbidge [4±7] developed Production Flow Analysis (PFA) to form the production cells in such a way that each part is completely processed within a production cell. Production ¯ow analysis has been successfully used to develop models and algorithms using information from the machine-part incidence matrix. It was extended to consider cellular subsystem formation for *Corresponding author. 759

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assembly processes [14]. Various similarity measures were used in the development of models to form part families and machine cells. One of the early algorithms used to group machines is the single linkage cluster analysis (SLCA) introduced by McAuley [36], which is based on Jaccard's similarity coecient. The algorithm uses pairwise comparisons to calculate similarities between machines. A number of similarity measures based on machine-part matrix information were later introduced. Examples include the mutual similarity coecient, the absolute similarity coecient [15], the processing sequence similarity coecient [56], the pairwise proximity measure based on manufacturing operations [11], and the production data based similarity coecient [20]. Shafer and Rogers [47] presented a survey on similarity measures used in CMS and suggest a new similarity measure which eliminates certain biases. Various research e€orts present comparative studies of similarity coecients and clustering algorithms [49, 50]. Gu [17] proposed a process-based machine grouping method for cellular manufacturing systems. A clustering algorithm based on the process similarities is used for part family grouping. Cell formation is based on the similarities of the machines' operational functionalities as well as the formed part families. Many researchers have formulated the part family and machine cell formation problem as a 0±1 integer mathematical program. Literature reveals that various objective functions are formulated based on similarity measures, cost functions, and other performance measures. Although there are some limitations in solving large scale problems, mathematical programming techniques provide a useful tool in obtaining optimal solutions. Kusiak [25] considered an integer programming formulation of the clustering problem known as the p-median model. The number of part families (clusters) are speci®ed in p-median formulation with the objective of maximizing total part similarities. The p-median model was extended to deal with alternative process plans for each part type [26]. Choobineh [11] proposed a twostage procedure. The ®rst stage forms the part families by using clustering techniques. The second stage forms the machine cells with an integer programming model. The objective function is the sum of production costs of acquiring and maintaining the machine tools. Gunasingh and Lashkari [18] presented two 0±1 integer programs with a sequential modelling approach to the cell formation problem. The machines are grouped into cells based on their similarity in parts processing. The parts are then allocated to the appropriate machine groups based on processing requirements. Wei and Gaither [57] develop and optimally solve the cell formation problem using an integer programming model that minimizes the cost of manufacturing exceptional parts outside the system (subcontracting cost). No inter-cell movements are allowed in their model. Askin and Chiu [2] present an integer program for cell formation that minimizes machine and group overhead costs, as well as the costs of tooling and material handling. The model does not consider subcontracting of exceptional parts. Due to the complexity of the mathematical model, a heuristic grouping procedure, based on graph partitioning, is suggested and tested. Shafer and Rogers [45] present an integer programming formulation that forms part families and sequences the parts within a family such that set up times are minimized, along with intercellular moves. The utilization of cells and the cost of equipment are kept within certain allowable limits. The authors distinguish between 3 scenarios: (1) purchase all new equipment; (2) use only existing equipment; and (3) possibly add new equipment to existing equipment. A twostage heuristic procedure is presented for solving scenario 1, where stage 1 ®nds part families and machine cells such that machine cost is minimized. Stage 2 sequences the parts within each family as to minimize set up times. Logendran [34] observed the e€ect of the identi®cation of key machines in the cell formation problem of cellular manufacturing systems, and proposed one mathematical model to minimize the total number of inter- and intra-cell movements. Jain et al. [22] proposed a 0±1 integer programming model for the combined problem of forming cells and providing tools in a ¯exible manufacturing system with the objective of minimizing the overall system cost, which is de®ned as the sum of the annual part processing costs, tooling costs and annualized cost of machines. Taboun et al. [54] introduced a weighted similarity index and incorporated a mixed (0±1) non-linear mathematical programming model to form part families and machine cells. Liang and Taboun [31] presented a model to address the part selection and assignment problems sim-

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ultaneously for existing layouts, considering exceptional parts and inter-cell movements. The model also considers part demands and machine capacities. Shafer et al. [48] presented a mathematical programming model that deals with bottleneck machines and exceptional parts. The model considers inter-cellular transfer, machine duplication, and subcontracting costs. Chow and Hawaleshka [10] proposed a machine grouping algorithm that considers a new machine unit concept to minimize the inter-cellular part movements. Adil et al. [1] developed a mathematical model to form the machine cells by considering trade-o€s between the investment cost and the operational cost. Xie [60] presents a cell formation program that minimizes intercell movement under machine capacity constraints. Liao [32] developed a three-stage procedure for designing a line-type CMS minimizing operating and material handling costs. As illustrated by the above literature overview, the trend in cellular manufacturing and group technology research is to move towards part family and cell formation techniques that consider the various costs involved in setting up and operating the systems, as well as take into account the various real-life constraints. Some of the costs considered include: machine purchase cost; machine operating costs; inter-cell movement costs; cell overhead costs; tooling costs; set up time costs; and subcontracting costs. Some of the constraints considered include: cell size limits; utilization levels; machine capacity; funds available, etc. Previously published research basically di€ers in (1) which costs are considered, (2) which cost function is optimized, and (3) which constraints are taken into account. They also di€er in the heuristic procedure developed, if any, to solve the resulting complex model. Though the published literature considers various combinations of the costs and constraints mentioned above, none of them considers all of: (1) machine investment and operating costs; (2) cell con®guration costs; (3) idle time costs; (4) inter-cell movement costs; and (5) subcontracting costs, while taking into account constraints on the number of part families/cells formed, part family /cell size, and machine capacity. We feel that taking into account machine, inter-cell movement, as well as subcontracting costs is necessary to obtain better utilized cells. When these costs are not taken into account, part families/cells containing a few, though highly similar parts, may be formed. This leads to a greater imbalance in the machines and machine cell workloads, reducing their utilizations and increasing manufacturing costs. Taking into account the above manufacturing costs avoids the above situation by allowing some parts to be subcontracted or grouped with other families (including the possibility of being moved between cells), if this is more cost ecient than forming their own family/cell. In addition, the models in the literature assume that either a cellular manufacturing system is to be designed from scratch, or that the existing system is to be recon®gured (with or without buying new equipment). None of the models published is general enough to represent and solve all of the above scenarios simultaneously. In this paper, we present a new methods for developing part family and machine cell con®guration that takes into account the various manufacturing system costs and the constraints mentioned above and is general enough to handle existing systems to be recon®gured as well as design of new systems. First, the problem is formulated as a mathematical model which simultaneously forms the optimal part family and machine cell con®gurations for the planning horizon. This is referred to as the single-stage model. To alleviate the computational requirements of this model, we develop a two-stage model where stage 1 consists of a heuristic to form the machine cells and part families and stage 2 integrates the results of the heuristic with a mathematical model to optimize the various cost aspects. This paper is divided into ®ve sections. Section 2 discusses the single-stage mathematical model which simultaneously develops the optimal part family and machine cell con®gurations. Section 3 presents the two-stage model. Section 4 describes the numerical applications and results of the models and Section 5 presents our conclusions.

2. THE SINGLE-STAGE MODEL

In this section, we present the mathematical model to simultaneously form machine groups and part families in a cellular manufacturing system, considering all major costs associated with

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design and implementation of the system. To facilitate the presentation of the proposed model, we ®rst state the following assumptions. 1. Production information for all parts such as part demand, processing times, required machines, available machining capacities and various costs are known. 2. Inter-cell movements are allowed. 3. Subcontracting of any part type is possible but the whole demand of the particular part must be subcontracted. 4. There are no alternative process plans. 5. Machine setup times are assumed negligible, since this is one of the major advantages of adopting a CMS. The following notation is used throughout: i j k AICk AIC APTk CC Di ENMk Fi Mj Pk ITCk MAXC MINC MAXP MINP MCkj NPMk DNC PCk TPTik SCk SIMi1i2 SL SUBCi TMCkj Z Z1 Z2 Z3 Z4 Z5 Z6

part type index, i = 1, 2, . . . , N part family/cell index, j = 1, 2, . . . , N machine type index, k = 1, 2, . . . , K average inter-cell movement cost per unit processing time on type k machine average inter-cell movement cost (over all parts and all cells) average processing time per part on type k machine cell con®guration cost including the cost of lost production incurred during cell con®guration annual demand for part type i existing number of type k machine(s) in the whole system set of families that part i may belong to set of machines that part family j may require set of parts that require type k machine idle time cost for machine type k per unit time maximum number of part families allowed in the system minimum number of part families/cells allowed in the system maximum number of parts that can be assigned to a part family minimum number of parts that can be assigned to a part family annual capital cost for machine type k in cell j average number of parts per movement, when moving parts from machine type k desired number of cells in the system procurement cost associated with buying one type k machine, excluding the purchase price (i.e. costs of handling, loading, installation etc.) total processing time required for type k machine for type i part cost associated with selling machine type k, excluding the selling price (i.e. costs of advertizing, human resources in charge of the sale, etc.) similarity coecient between parts i1 and i2 a measure of overall system similarity level required cost of subcontracting part i total machining capacity for machine of type k in cell j total cost total cell con®guration cost total annual machine capital investment cost total cost associated with procurement and sales of machines total idle time cost total inter-cell movement cost total subcontracting cost

The decision variables are: Xij



1 if part i belongs to part family j, 0 otherwise: 

Si NC NUMk PURk SELk Ykj AMCkj ITk Mkj

1 0

if part i is subcontracted, otherwise:

number of formed part families/cells number of type k machine(s) that will exist number of new (purchased) type k machine(s) number of type k machine(s) to sell number of type k machine(s) required in cell j available machining capacity of type k machine in cell j, in excess of the processing capacity needed for parts in family j total idle time for type k machine(s) process time allocated out of machine type k in cell/part family j for processing parts in other families

The objective function of the model minimizes the total sum of six cost components considered in the design of the manufacturing cells for the planning horizon. The model does not aim to minimize a speci®c cost, but rather the total cost arising from trade-o€s between cost

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functions. Min

6 X Zi

…1†

iˆ1

Z1 ˆ CC  NC Z2 ˆ

K X N X

MCkj  Ykj

…2† …3†

kˆ1 jˆ1

Z3 ˆ

K X ‰…PCk  PURk † ‡ …SCk  SELk †Š

…4†

kˆ1

Z4 ˆ

K X

ITCk  ITk

…5†

kˆ1

Z5 ˆ

K X N X AICk  Mkj

…6†

kˆ1 jˆ1

AICk ˆ

Z6 ˆ

AIC NPMk  APTk

N X SUBCi  Si

…7†

…8†

iˆ1

Subject to the following constraints: N X

Xij ‡ Si ˆ 1, 8i ˆ 1,2, . . . ,N

…9†

jˆ1

MINC  NC  MAXC

…10†

N X Xjj ÿ NC ˆ 0

…11†

jˆ1

MINP  Xjj 

N X Xjj  MAXP  Xjj , 8j ˆ 1,2, . . . ,N

…12†

iˆ1 N X Xij ÿ N  Xjj  0,

8j ˆ 1,2, . . . ,N

…13†

iˆ1 N X …AMCkj ÿ Mkj † ÿ ITk ˆ 0, 8k ˆ 1,2, . . . ,K

…14†

jˆ1 N X Ykj ÿ ENMk ÿ PURk ‡ SELk ˆ 0, 8k ˆ 1,2, . . . ,K jˆ1

…15†

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…16†

iˆ1 j>i N X

…Di  TPTik †  Xij ÿ TMCkj  Ykj ‡ AMCkj ÿ Mkj ˆ 0

8j ˆ 1,2, . . . ,N, 8k ˆ 1, . . . ,K …17†

iˆ1

Xij …0,1†, Si …0,1† 8i ˆ 1,2, . . . ,N 8j ˆ 1,2, . . . ,N integer Ykj , NC, NUMk , PURk , SELk

8k ˆ 1,2, . . . ,K 8j ˆ 1,2, . . . ,N

…18† …19†

Equations (1)±(8) are the components of the objective function. Equation (2) speci®es the total cell con®guration cost, including the cost of installing or reallocating machines. Equation (3) determines the cost of capital investment associated with various types of machines, and Equation (4) determines the total procurement and salvage costs for the machines. Equation (5) speci®es the cost related with the idle time of the machines. Equations (6) and (7) determine the cost of inter-cell movement. The approach used here is based on a new type of cost measure, AlCk, de®ned as the average inter-cell movement cost per unit processing time on machine type k, and given mathematically by Equation (7). The rationale for this cost measure is that the more parts per movement and the longer the processing time required for each part at the destination machine, the more ecient the transfer is. AlCk takes into account the average inter-cell movement cost (over all parts) as well as the number of parts being moved together in one move and the process time requirements at the destination machine k. Using this approach, the inter-cell movement cost is AlCk multiplied by the processing time (Mkj) allocated out of machine type k in cell/part family j. This is mathematically expressed in Equation (6). Finally Equation (8) speci®es the costs related to subcontracting of parts. Constraint Equation (9) ensures that each part either belongs to exactly one part family or is subcontracted. Constraint Equation (10) speci®es the required number of cells/part families. Constraint Equation (11) ensures that the number of cells formed is NC. The cell numbering scheme used is such that for every part i, if a new cell is to be formed for that part, then that cell is numbered i. This scheme ensures that xjj=1 if cell j is formed and 0 otherwise. With this numbering scheme, a part index (i) can also be used as a cell index (j) and vice versa (see for example constraint Equation (16) explained later). Constraint Equation (12) ensures that part i belongs to part family j only when this part family is formed, and it brings an upper and a lower bound for the number of parts that can be assigned to a part family to balance the utilization of the cells. Constraint Equation (13) is a more general form of constraint Equation (12), with no lower bound and with an upper bound of N. N is the maximum possible number of parts in a family, since it is the total number of parts in the system. Constraint Equation (13) is used to complement constraint Equation (12) when MINP and MAXP are not speci®ed or are speci®ed with unrealistic values. Constraint Equation (14) speci®es that the idle time of all machines type k is the sum over all machines of that type in all cells j, of the available excess machining capacity of the machine minus the processing time allocated from that machine to parts in other families. Constraint Equation (15) determines the number of machines to purchase and to sell for each type for the planning horizon. Constraint Equation (16) ensures that the sum of similarity levels among all part families is higher than a certain level. Constraint Equation (17) sets a balance among total processing time, available machining time, idle time and required overtime for each type of machine. Constraints Equations (18) and (19) ensure the integrality of the model. For small size problems (see Tables 1 and 2), this model can be ecient. However, for medium- and large-size problems, the number of integer variables would increase the computational time dramatically. A two-stage model is developed to reduce the number of integer variables and corresponding constraints.

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Table 1a. Model sizes Number of 0±1 integer variables Description

min

max

2

Without MS/MM With MS/MM

Number of general integer variables

2

N N

N N

Number of constraints

min

max

min

max

NK K

NK NC  K

2N + K  (N + 2) + 1 N + 3K

2N + K  (N + 2) + 1 N + K  (NC + 2)

N: number of part types in the system; K: number of machine types in the system; NC: number of cells in the system.

Table 1b. Numerical example for model sizes Number of 0±1 integer variables

Number of general integer variables

Number of constraints

Description

min

max

min

max

min

max

Without MS/MM With MS/MM

729 27

729 27

324 12

324 36

403 63

403 87

Number of part types in the system = 27; number of machine types in the system = 12; number of cells in the system = 3.

3. THE TWO-STAGE MODEL

As indicated in the previous section, large-size problems require a signi®cant amount of computational times to obtain optimal solutions due to the existence of several integer variables, which increase exponentially with the increase in the number of parts and machines. In this section, a two-stage model is developed. In the ®rst stage, a heuristic, which we refer to as Maximum Similarity/Minimum Machines (MS/MM), forms part families and generates possible machines that may be included in the cell. The second stage of the model integrates the results of the heuristic with a mathematical model to optimize the total cost associated with machine cell con®guration. The basic ¯ow-chart of the two-stage algorithm is shown in Fig. 1. The MS/ MM heuristic is based on a measure of system similarity level. We discuss the system similarity measure in Section 3.1. Details of each of the two stages are described in Section 3.2 and Section 3.3, respectively. 3.1. Development of system similarity measure In this section, we derive a measure of the similarity level of a manufacturing system as a function of the similarities between the parts in the system. First, note that there are a number of types of similarity between two parts in the system. Typical types of similarity include machine similarity, tool similarity, and process sequence similarity. Let 1 denote the type of similarity used (1 = machine, 2 = tool, 3 = sequence). Then the similarity level in a created cell, for that speci®c type 1 would be calculated by using the following equation: CSIMlj ˆ 1

N

N

j X j …Nj ÿ 1†  Nj X SIMi1 i2 2 i ˆ1i2 >i1

…20†

1

where CSIM1j is the average type 1 similarity level in cell j, Nj the total number of part(s) in family j. To calculate the overall similarity in the system, each cell contributes with its similarity level, weighted by a factor wc given below. The approach taken here is that the weight factor wc only Table 2. Part demands and processing requirements for Example 1 Machine type Part No.

01

02

03

04

05

06

01 02 03 04 05 06 07

1.25 0 0.60 0 0 0 1.00

1.50 0 1.00 0 1.10 0 0

0.75 0 0 1.20 1.20 0 1.30

0 0.80 1.30 0 1.00 1.75 0.60

0 0.90 0 1.25 0 0 0

0 1.20 0 0.70 0 1.25 0

Part demands 200 175 100 250 280 150 120

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Fig. 1. Flow-chart of two-stage algorithm.

depends on the number of parts in that cell. The overall similarity level in the system is then given by: OSIMl ˆ

NC X wj  CSIMlj

…21†

jˆ1

where OSIMl is the overall type 1 similarity level in the system. wj ˆ

Nj N

The general similarity level of the system could be as follows: X Wl  OSIMl GSIM ˆ

…23†

l2SS

where GSIM is the general similarity level in the system, SS the set of similarity types {1 (machine), 2 (tool), 3 (sequence), . . . } and W1 the weight of similarity of type 1. This is determined based on the scarcity or importance of the corresponding resource (machines for machine similarity, tools for tool similarity, or the material handling system for processing sequence similarity) and must satisfy the following: X Wl ˆ 1 …24† l2SS

Equation (23) is the generalized case for considering di€erent similarity levels in the system. But for simplicity of analyzing the models, only machine similarity is considered in the following two-stage model. 3.2. Stage 1 The proposed algorithm is based on the criteria of Maximum Similarity/Minimum Machines (MS/MM). The MS/MM algorithm may be considered as Max/Min, which reduces the number of variables and minimizes the range between the maximum and minimum number of variables. The heuristic is divided into one main algorithm and two subroutines. In the ®rst subroutine (Procedure A), part families are created by using a similarity level without considering the minimum number of parts in a cell. In the second subroutine (Procedure B), the members of the part families whose sizes are less than a prestated minimum level are placed in the families with which they have the highest similarity. The ¯ow-chart of the main algorithm is shown in Fig. 2. The main algorithm inputs the desired number of cells, the minimum part family size, and a range of similarity levels to search. One similarity level is considered at each iteration of the algorithm. Procedure A is ®rst called to form initial part families, using this similarity level. Since

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Fig. 2. The main body of the MS/MM method.

Procedure A does not take into account the minimum part family size, this results in part families whose sizes can be greater than MINP or less than MINP. If the number of part families which pass the size constraint (i.e. whose size is greater than MINP) is equal to the desired number of cells, then Procedure B is called. This assigns the parts in the families which did not pass the test to the family, among those that passed the test, with which they have the highest similarity. Then the system similarity level and the required number of machines for this con®guration are calculated. If the system similarity level obtained in this iteration is higher than the one obtained in the previous iteration (the initial one being set to 0), or if the required number of machines found in this iteration is less than the one found in the previous iteration (the initial one being set to a large number) then the current con®guration will be used, otherwise the previous con®guration will be used, and the algorithm starts again with a new similarity level. When all similarity levels have been searched, the con®guration found is the one with

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Fig. 3. The ¯ow-chart of Procedure A.

Maximum Similarity and Minimum Number of Machines (MS/MM), given the initial search range. The ¯ow-chart of Procedure A is shown in Fig. 3. In this procedure, the similarity coecients between each pair of parts is ®rst calculated. A set is created with all parts in it (PART = {Pi; i = 1, 2, . . ., N}), and a family set is created (FAMILY). The ®rst part from set PART is removed and placed in set FAMILY. The procedure then loops through all remaining parts in set PART to ®nd parts that have a similarity level greater than or equal to the target level. These parts are removed from the set PART and placed in set FAMILY. Then the procedure considers each of these new parts placed in set FAMILY and similarly ®nds all parts in set PART with which they have target similarity level. These parts are removed from PART and placed in FAMILY. When each part in set FAMILY has been processed as above, the family is complete. If any parts are left in set PART, a new family is created containing the next part in set PART. If no parts are left in PART, the procedure terminates. The ¯ow-chart of Procedure B is shown in Fig. 4. This procedure ®rst calculates the size of the part families created by Procedure A. Then the families which satisfy the size requirements (i.e. whose sizes are greater than or equal to the minimum part family size) are identi®ed and their machine requirements are calculated. Then

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Fig. 4. The ¯ow-chart of Procedure B.

the procedure loops through the parts in the families which did not pass the size requirement. For each of these parts, the machine similarity level with each of the established families is calculated, and each part is placed with the family with which it has the highest machine similarity level. The machine requirements for that family are then updated. 3.3. Stage 2: model development In this section, we develop a mathematical model which is a modi®cation of the model of Section 2, such that no part family forming constraints are used since they are already formed in stage 1. The Objective function and constraints are as follows: MIN

6 X Zi

…25†

iˆ1

Z1 ˆ CC  NC Z2 ˆ

NC X X

MCk  Ykj

…26† …27†

jˆ1 k2Mj

Z3 ˆ

K X ‰…PCk  PURk † ‡ …SCk  SELk †Š

…28†

kˆ1

Z4 ˆ

K X

ITCk  ITk

…29†

kˆ1

Z5 ˆ

NC X X jˆ1 k2Mj

AICk  Mkj

…30†

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Z6 ˆ

N X SUBCi  Si

…31†

iˆ1

Subject to following constraints: Xij ‡ Si ˆ 1 8i ˆ 1,2, . . . ,N j ˆ 1,2, . . . ,NC X

…Di  PTik †  Xij ÿ TMCkj  Ykj ‡ AMCkj ÿ Mkj ˆ 0

8j ˆ 1,2, . . . ,NC, 8k 2 Mj

…32† …33†

i2Pk NC X

…AMCkj ÿ Mkj † ÿ ITk ˆ 0

8k ˆ 1,2, . . . ,K

…34†

jˆ1 NC X

Ykj ÿ PURk ‡ SELk ˆ ENMk

8k ˆ 1,2, . . . ,K

…35†

jˆ1

Xij ,…0,1† 8i ˆ 1,2, . . . ,N, j ˆ 1,2, . . . ,NC

…36†

integer Ykj , 8j ˆ 1,2, . . . ,NC 8k 2 Mj

…37†

The objective function is similar to the objective function of single-stage model, except for the cell con®guration cost function, since the number of cells is determined in stage 1. The twostage model does not contain part family forming constraints, since they have already been formed in stage 1. It also includes less constraints since the possible machine types that will exist in the cells has already been determined in stage 1. Constraint Equation (32) ensures that each part is assigned to a pre-determined cell or is subcontracted. Constraint Equation (33) sets a balance among total processing time, available machining capacity, idle time and required overtime for each type of machine. Constmint Equation (34) speci®es the idle time of the machines and constraint Equation (35) determines the number of machines to purchase and to sell for each type of machine. Constraints Equations (36) and (37) ensure the integrality of the model. 3.4. Complexity considerations Table 1 shows the numbers of 0±1 integer variables, general integer variables, and constraints, for both models developed in this research: the original mathematical model without MS/MM and the model developed with the MS/MM heuristic. Table 2 illustrates these results with a numerical example of 27 part types, 12 machine types, and 3 cells to be formed in the system. It is obvious that the MS/MM method reduces the models to much more manageable and easily solvable sizes.

4. COMPUTATIONAL RESULTS

The proposed MS/MM method is tested by using the examples described in Burbidge [5] and King [23] and comparing the results. We also compare the proposed two-stage model with the single-stage mathematical model. Large-size problems are solved using the proposed two-stage models. 4.1. Capability of MS/MM method Burbidge [5] presented a group technology problem with 14 machines and 43 parts, which was solved by manual trial-and-error method. The same initial machine-part matrix was also used in a study by King [23]. The example data is used in this study to test the proposed heuristic, and to compare its results with those of Burbidge [5] and King [23].

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Table 3. Cost parameters and planning horizon for Example 1 Description

Unit values

Planning horizon Unit cell con®guration cost Unit machine procurement cost Unit capital investment cost Unit idle time cost Unit inter-cell movement cost Unit part subcontracting cost

700 time units 100 cost units 10 cost units 15 cost units 40 cost units 10 cost units 80 cost units

The Burbidge solution resulted in ®ve cells, with three exceptional parts that cannot be completely assigned to a cell. In order to produce all of the parts within machine cells, the system must have 25 machines. When the proposed MM/MS heuristic was applied to this example, with the number of desired machine cells speci®ed as 5, the results obtained are similar to Burbidge's manual solution of 25 machines and 4 part families. In addition, similar results of the weighted system similarities are obtained using both methods. When the desired number of cells was speci®ed as four, the proposed algorithm resulted in a total of 22 machines to process all parts within the system, and 4 part families were formed. The weighted system similarity was 0.539. When the ROC algorithm [23] was applied to the same example data and four desired machine cells, results revealed that there were 2 exceptional parts, and a total of 23 machines were required to process all the parts. The weighted system similarity was 0.523. Comparing both results of the proposed MM/MS and ROC algorithms, it can be stated that the proposed MM/MS gives a better solution than ROC. It is clear that the total number of machines required to process all parts of the MS/MM solution are less than the ROC solution. Although a similar number of part families is obtained from both methods, the MS/MM procedure gives higher weighted system similarity. 4.2. Comparison of single-stage and two-stage models To test the validity of the two-stage model, hypothetical data of 7 parts and 6 machines is used. The part demands and processing requirements of this hypothetical example, referred to as Example 1, are given in Table 3. Various cost parameters and the planning horizon are given in Table 4. The data is used as an input to both models: single-stage and two-stage. Results of the single-stage model for two machine cells indicated that the ®rst machine cell consisted of machines 1, 2, 3 and 4, while the second machine cell consisted of machines 3, 5 and 6. On the other hand, their corresponding part families consisted of parts 1, 3, 5 and 7, and parts 2, 4 and 6, respectively. These results are identical to those obtained from the two-stage model with 2 as the desired number of cells. The results of both cases are summarized in Table 5. The various cost results of the optimal solution are given in Table 6. Comparison of the model constraints and decision variables of both models and the computational times associated with each one of them are given in Table 7. It is clear that, even though the proposed twostage model is not an optimization algorithm, it did result in the optimal solution for this example in a fraction of the CPU time of the optimizing single-stage model (2 s vs 41 min). 4.3. Testing and evaluation of two-stage models A detailed analysis considering system utilization and within-cell utilization is also done for two other larger hypothetical example applications, referred to as Example 2 and Example 3. Table 4. Con®guration results of both single- and two-stage models for Example 1 Members of part family Cell number 1 2

Members of machine group

part number

total

machine number

total

1±3±5±7 2±4±6

4 3

1±2±3±4 3±5±6

4 3

772

S. M. Taboun et al. Table 5. Cost results of both single- and two-stage models for Example 1

Description

Cost values

Cell con®guration cost Machine purchasing cost Capital investment cost Idle time cost Inter-cell movement cost Subcontracting cost

200 units 70 units 105 units 58 units 41 units 0 units

Total cost

474 units

Table 6. Comparison of time complexity of single-stage and two-stage models Models Comparison parameters

two-stage model

single-stage model

7 8 21 319 2s

49 42 63 54253 41 min

Number of 0±1 integer variable Number of general integer variable Number of constraints Number of iterations to reach optimum Required computer time to reach optimum

Machine similarity between parts was also considered for the example applications since the proposed models are based on machine allocation in Cellular Manufacturing Systems. The LINDO (Hyper version) software run on IBM Compatible 486 SX Personal Computer was used for all of the applications. The unit values of cost types that were used in the applications are summarized in Table 8. In both example applications, 13 part types and 12 machine types were considered. Part demands, machine requirements of parts, and processing times on these machines are given for Example 2 in Table 9 and for Example 3 in Table 10. Results of the ®rst stage are listed for Example 2 in Table 11 and for Example 3 in Table 12. The resulting con®gurations for the two

Table 7. Cost factors used in Examples 2 and 3

Cell set-up cost Machine purchasing cost Machine selling cost Capital investment cost Idle time cost Inter-cell movement cost Subcontracting cost

per unit cell per unit machine per unit machine per unit machine per unit time per unit time per unit part

Example 2

Example 3

100 10 10 25 1/40 1/5 75

100 10 10 15 1/40 1/10 80

Table 8. Machine requirements and processing times of parts for Example 2 Machine type Part No.

01

02

03

04

05

06

07

08

09

10

11

12

01 02 03 04 05 06 07 08 09 10 11 12 13

0.4 0.5 0 0.75 0 0.4 0 0 0.9 0 0 0 0

0 0 0.75 0 0.75 0.6 0 0.6 0 0.75 0 0.6 0.75

0 0.4 0.6 0 0.6 0.75 0 0 0.75 0.7 0.8 0.4 0

0.5 0 0 1.2 0 0 0.5 0 0 0 0 0 0

0.6 0 0.5 0 0 0 0.4 0.5 0 0.6 0 0 0

0 0.9 0 0 0 0 0 0 0.6 0 0 0.5 0.8

0 0 0 0 0.7 0.8 0 0.4 0.4 0 1 0.9 0.4

0.7 0 0 0.8 0 0 0.7 0 0 0 0.2 0 0

0 0 0.4 0.5 0.9 0 0.8 1 0 1 0.6 0 0.8

0 0 0.7 0 0.5 0 0 0.6 0 0.4 0 0 0.5

0 0.6 0 0.4 0 0 0.6 0 0 0 1.25 0 0

0 0.6 0 0 0 0.75 0 0 0.5 0 0 0.7 0

Demand 300 160 280 180 250 320 340 200 290 140 200 240 180

Part family and machine cell formation

773

Table 9. Results of MS/MM algorithm for Example 2 Members of part family

Members of machine group

Cell number

part number

total

machine number

total

1 2 3

1±4±7±11 2±6±9±12 3±5±8±10±13

4 4 5

1±4±5±8±9±11 1±2±3±6±7±8±10±11±12 2±3±4±5±6±7±9±10

6 9 8

Total

13

23

Table 10. Machine requirements and processing times of parts for Example 3 Machine type Part No.

01

02

03

04

05

06

07

08

09

10

11

12

Demand

01 02 03 04 05 06 07 08 09 10 11 12 13

0 0.4 0 0 0.9 0 0.5 0 0 0 0 0 0

0.75 0.6 0 0.6 0 0.75 0 0.6 0.75 0.6 0 0.4 0.5

0.6 0.75 0 0 0.75 0.7 0 0.4 0 0 0 0.75 0.8

0 0 0.5 0 0 0 0.8 0 0 0.75 0.7 0 0

0 0 0.4 0.5 0 0.6 0 0 0 1 0.9 0 0

0 0 0 0 0 0 0.6 0 0 0.5 0.8 0 0

0.7 0.8 0 0.4 0.4 0 0 0.9 0.4 0 0 1 0

0 0 0.7 0 0 0 1.2 0 0 0 0.5 0 0.7

0.9 0 0.8 1 0 1 0.6 0 0.8 0.4 0 0 0

0.5 0 0 0.6 0 0.4 0 0 0.5 0.6 0 0.6 0

0 0 0.6 0 0 0 1.25 0 0 0 1.2 0 0

0 0.75 0 0 0.5 0 0 0.7 0 0 0 0.75 0.6

Table 11. Results of MS/MM algorithm for Example 3 Members of part family

Members of machine group

Cell number

part number

total

machine number

total

1 2 3

3±7±11 2±5±8±12±13 1±4±6±9±10

3 5 5

1±4±5±8±9±11 1±2±3±6±7±8±10±11±12 2±3±4±5±6±7±9±10

6 9 8

Total

13

23

Table 12. Con®guration results of Examples 2 and 3 Members of part family Cell No. 1 2 3

Members of machine group

Example No.

part number

subcontracted parts

machine number

total

2 3 2 3 2 3

ÿ 3±7±11 2±6±9±12 2±5±8±12±13 3±5±8±10±13±14 1±4±6±9±10

1±4±7±11 ÿ ÿ ÿ ÿ ÿ

ÿ 4±8±9±11 ÿ 1±2±3±6±7±12 2±3±5±9±10 2±3±5±7±9±10

0 4 5 6 5 6

Table 13. Cost results of Examples 2 and 3 Cost type Cell con®guration cost 1 Machine procurement cost Capital investment cost

Example No.

Unit cost

2 3 2 3 2 3

200 300 100 160 250

300 240 440 340 440 320 350 340 260 160 80 180 170

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S. M. Taboun et al.

examples are listed in Table 13, and the arising costs for the examples are listed in Table 14. All the costs are unit values, and the real dollar values can be found by multiplying with an appropriate constant. It is clear that the proposed two-stage model gives good solutions in a reasonable time, and is therefore e€ective in handling large-size problems. 5. CONCLUDING REMARKS

In this study, problems arising during the implementation stages of Cellular Manufacturing Systems, such as the con®guration and recon®guration problems that exist in many manufacturing environments, are discussed. The paper presented a new two-stage model to address these problems. The approach is based on a single-stage mathematical program developed for simultaneous part and machine grouping and for seeking trade-o€s between cell con®guration, machine procurement and sale, capital investment, inter-cell movement and part subcontracting costs. The proposed two-stage model decomposes the problem in two stages: ®rst a heuristic is developed to reduce the number of variables and constraints in the models. The heuristic is based on the criteria of Maximum Similarity Minimum Machine number in the system and forms machine cells and part families. The results of the heuristic are then integrated with a mathematical model, to optimize the results with respect to the cost factors. The proposed MS/MM method was tested by using the example data of Burbidge [5] and King [23]. The proposed heuristic was also tested with the above example data while considering four machine cells. Results of the heuristic were superior to the ROC algorithm [23] in terms of number of machines, part families and weighted similarity of the system. The two models (single-stage as well as two-stage) were also tested and compared using hypothetical examples. It has been shown that the models gave reliable results and can be used to handle large-size problems e€ectively, as shown by the signi®cantly reduced CPU time requirements. Future research will extend the proposed approach to multi-period planning horizons to optimize the trade-o€s between system recon®guration costs when demands for the products change across periods. REFERENCES 1. Adil, GK, Rajmani, D and Strong, D A mathematical model for cell formation considering investment and operational costs. European Journal of Operations Research, 69, 1993, 330±341. 2. Askin RG, Chiu KS. A graph partitioning procedure for machine assignment and cell formation in group technology. International Journal of Production Research 28:1555±1572. 3. Bhat, MV and Haupt, A An ecient clustering algorithm. IEEE Transaction on Systems, Man, and Cybernetics, SMC6, 1976, 61±64. 4. Burbidge JL. Production ¯ow analysis. The Production Engineer 1971;April:139±152. 5. Burbidge JL. The Introduction of Group Technology. New York: Wiley, 1975. 6. Burbidge JL. Production Flow Analysis for Planning Group Technology. Reading: Clarendon, 1989. 7. Burbidge, JL Change to group technology: process organization is obsolete. International Journal of Production Research, 30, 1992, 1209±1219. 8. Carrie, A Numerical taxamony applied to group technology and plant layout. International Journal of Production Research, 13, 1975, 541±557. 9. Chan, HM and Milner, DA Direct clustering algorithm for group formation in cellular manufacture. Journal of Manufacturing Systems, 1, 1982, 65±74. 10. Chow, WS and Hawaleshka, O Minimizing intercellular part movements in manufacturing cell formation. International Journal of Production Research, 31, 1993, 2161±2170. 11. Choobineh, F A framework for the design of cellular manufacturing systems. International Journal of Production Research, 26, 1988, 1161±1172. 12. Chu, CH and Tsai, M A comparison of three array-based clustering techniques for manufacturing cell formation. International Journal of Production Research, 28, 1990, 1417±1433. 13. Co, HC and Arar, A Con®guring cellular manufacturing systems. International Journal of Production Research, 26, 1988, 1511±1522. 14. De Beer, C and De Witte, J Production ¯ow synthesis. CIRP Annals, 27, 1978, 389±392. 15. De Witte, H The use of similarity coecients in production ¯ow analysis. International Journal of Production Research, 18, 1980, 503±514. 16. El-Assawy, IFK and Torrance, J Component ¯ow analysis: an e€ective approach to production systems design. The Production Engineer, 51, 1972, 165±170. 17. Gu, P Process-based machine grouping for cellular manufacturing systems. Computers in Industry, 17, 1991, 9±17. 18. Gunasingh, KR and Lashkari, RS Machine grouping in cellular manufacturing systems ± an integer programming approach. International Journal of Production Research, 27, 1989, 1465±1473. 19. Gunasingh, KR and Lashkari, RS Simultaneous grouping of parts and machines in cellular manufacturing systems ± an integer programming approach. Computers and Industrial Engineering, 20, 1991, 111±117.

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