A methodology for designing flexible cellular manufacturing systems.

1. Introduction

Cellular manufacturing (CM), an application of group technology, entails the creation and operation of manufacturing cells. Each cell is dedicated to processing a specific set of part families. A manufacturing cell typically consists of several functionally dissimilar machines, whereas

Although the operational benefits of CM have been well-documented in the literature [1], it has also been argued that the implementation of cells could lead to a decrease in manufacturing flexibility [2]. The major difficulty with cells stems from potentially unstable machine utilizations due to dynamic and random variations in part demands [3]. This has led to some confusion as to the appropriateness of CM by industry users. On the one hand, companies would like to achieve the operational efficiencies through implementing CM systems, but, on the other hand, companies do not want to lose the strategic benefits of flexible operations. Further, as pointed out by Craig, et al. [4], flexibility is one of the critical dimensions of enhancing the competitiveness of organizations and hence the design of 'flexible' cells is an important issue [5].

This paper proposes a CF method that incorporates several flexibility criteria to guide the creation of 'flexible' cells. This approach is unique in several aspects. First, the cell system design generated is a function of the user priorities in terms of flexibility dimensions. This not only allows the user to incorporate preset user priorities but also allows the investigation of trade-offs between conflicting flexibility criteria. Secondly, although there is some prior research that has incorporated alternative process plans when identifying cell configurations (e.g, [6-11]), this is one of the first methods that focuses on part-operation requirements in creating cells. Most of the cell formation research to date (e.g., [12-20]) assumes that parts are processed on specific machine types and the assignment of operations to machines is determined a priori. However, to allow for flexibility in operation-machine assignments, we explicitly incorporate this decision into our procedure. Thirdly, the method proposed in this research includes an explicit improvement stage where the user can attempt to modify the candidate design to increase alternative (or all) types of system flexibility.

The remainder of this paper is organized as follows. In Section 2 the relevant literature is reviewed and the flexibility criteria of interest for CM are introduced. Section 3 describes the proposed CF procedure, and Section 4 presents an illustration of the methodology. Section 5 describes the experiment conducted to validate the proposed methodology and provides user guidelines for parameter settings depending on the flexibility criteria. Finally, the implications and conclusions of the research are discussed in Section 6.

2. Relevant literature

In recent years there has been a tremendous growth in the number of CF methods. The surge of interest in the area has been fueled not only by surveys that have shown the benefits of CM systems [1] but also because there is substantial industry interest in implementing CM systems. Comprehensive reviews of CF can be found in [5, 21-24].

In the context of this paper, two papers on CF are the most relevant. Tilsley and Lewis [25] were the first to propose a CF method where routing flexibility was a primary consideration. They essentially propose a system of 'cascade' cells that are created such that the more critical part families can be processed in more than one cell. Thus, machines required to process critical part families are allocated to more than one cell. Although the algorithmic details of the procedure are not provided, they do point out the importance of building in routing flexibility when machines within cells are subject to breakdowns. Machine downtime in a cell could be handled by having multiple machines of the same type in a cell or by routing operations performed on one machine in a cell to other machines in the same cell; however, these factors are not considered in their procedure.

Dahel and Smith [26] propose a procedure to create cells such that routing flexibility and cell independence could be simultaneously considered. They essentially formulate the CF problem as a multi-objective mathematical model that simultaneously attempts to create independent cells (by minimizing intercellular materials flows) and flexible cells (i.e., cells containing the largest variety of machine types). Their logic is that routing flexibility of the system is maximized when we can create such flexible cells.

In terms of flexibility dimesions, there has been a remarkable lack of interest in designing cells that can respond quickly to changes in the part demands (in terms of new part introduction and in terms of changes in volumes of current part). To address this issue, Vakharia, et al. [27] develop a framework and measures for different flexibility types relevant in the context of CM systems. These types are:

* machine type flexibility: the ability of the machines grouped into cells to process a large number of distinct operation types;

* routing flexibility: the ability of the cell system to process parts completely in multiple cells (referred to as process flexibility in [28]);

* part volume flexibility: the ability of the cell system to deal with volume changes in the current part mix; and

* part mix flexibility: the ability of the cell system to handle different product mixes with minimum disruption.

Of these flexibility types, routing, part volume, and art mix flexibilities are determined by the cell system design generated, whereas machine type flexibility is also a function of the technological constraints on the machines. Our primary objective in this paper is to consider all four types of flexibility in developing a CF method. This procedure is described in the next section.

3. Flexible cell formation (FCF) method

An overview of the proposed FCF method is shown in Fig. 1. Phase I identifies the most economical set of machine types to process the required operations of the entire part set based on machine costs, capabilities, and capacities. Phase II assigns individual part-operations o individual machines with an objective of providing n assignment that will lead to minimum material handling cost in final system design. Balancing material handling costs, current processing requirements, and flexibility to adapt to changes, Phase III forms candidate cells. The flexibility of this cellular configuration is then evaluated and improved in Phase IV on the basis of the current set of part types and demands. Each phase of the procedure allows user interaction in terms of parameter settings. Further, once the cell design (with a fixed set of parameters) is generated, measures to evaluate the design in several dimensions are provided. On the basis of such an evaluation, the user has the option of either completely regenerating the system design (i.e., start again at Phase I) or simply changing parameter settings at some intermediate level to modify the cell system design generated. Details of each phase are described next; the notation used for the complete method is shown in Table 1.

3.1. Phase I: assign operations to machine types

Given a machine type population with different processing capabilities (in terms of operation types), Phase I is concerned with assigning operation types to machine types. Each operation type represents a set of identical operations required by one or more parts. An operation type could be a generic process such as drilling, but would in most instances be more specific. For example, drilling with a specific tool and power requirement might be an operation type. Although Phase I would ordinarily be used to assign operation types, the methodology could also be used to assign individual operations. A comprehensive mathematical model for carrying out this assignment is as follows:

[Mathematical Expression Omitted] (1)

subject to

[summation of] [x.sub.mj] where m = 1 to M = 1 [for every]j; (2)

[summation of] [u.sub.mj][x.sub.mj] where j = 1 to J [less than or equal to] [U.sub.m][N.sub.m] [for every]m; (3)

[x.sub.mj] [element of] {0, 1} [for every]m [for every]j; (4)

[N.sub.m] [greater than or equal to] 0 and integer [for every] m. (5)

In this model, the objective function minimizes the total annual operating cost of the operation-machine assignments and the total annualized procurement cost of machines. The first constraint ensures that each operation is assigned to a single machine type. The second constraint ensures that operation - machine type assignments are feasible at the system level by ensuring that we acquire adequate capacity. In this case [u.sub.mj] is the fraction of machine type m required, assuming that all parts requiring operation j will be processed on machine type m. Because demand for each part type is assumed to be known, then if batch sizes are also known, the time needed for the required number of setups of each part can be included in [TABULAR DATA FOR TABLE 1 OMITTED] [u.sub.mj]. [U.sub.m](0 [less than or equal to] [U.sub.m] [less than or equal to] 1) is a user-specified parameter that indicates the amount of slack capacity that should be built into the system. The user parameter [U.sub.m] controls the volume flexibility of a cell system design. By setting lower values of [U.sub.m], we build more slack capacity into the system and hence, allow the system to handle part volume changes without disruption. Finally, the technological constraints on the decision variables [x.sub.mj] and [N.sub.m] are enforced in the last two sets of constraints. Note that if we do not require that a part operation always be performed on the same machine type, we can relax the integrality restrictions on the [x.sub.mj].

For a known set of [N.sub.m] values, this model reduces to the generalized assignment problem (GAP). In the context of our problem, we would know [N.sub.m] if we were reconfiguring an existing system into a cellular system. In contrast, when designing a completely new system, we propose that initial estimates of [N.sub.m] be obtained by linearizing the fixed machine cost and then selecting the least expensive machine type for each operation. With this assignment, estimate the number of machines of each type by computing [([[Sigma].sub.j] [u.sub.mj][x.sub.mj])/[U.sub.m]]. Once these estimates of [N.sub.m] have been obtained, we use the Martello and Toth [29] enumerative algorithm for obtaining good solutions to the GAP model. Note that the final [N.sub.m] values are based on the operation-machine type assignments resulting from applying this algorithm. A branch and bound strategy can be added to search for a better solution if desired. The output of this stage is the [x.sub.mj] and [N.sub.m] (if required) decision variables.

3.2. Phase II: assign part-operations to specific machines

Given operation assignments to machine types, Phase II assigns each part-operation to a specific machine of each type such that the similarity between part-operations assigned to a machine is maximized and the user-specified maximum machine usage ([U.sub.m]) is not exceeded. Knowledge of these operation to machine assignments is needed in Phase III to include minimization of material handling as a criterion in the machine grouping decision.

The goal in Phase II is to limit the number of inter-machine and intercell transfers that will result from the final cellular system design. We accomplish this by assigning sets of operations to the same machine if they satisfy either of two criteria. First, if multiple operations from the same part type are assigned to the same machine type in Phase I, these operations are now assigned to the same machine. Secondly, operations from parts that have similar machine type sequences, and therefore should be in the same cell, are assigned to the same machine. The technical procedure for this assignment is described below. The complete Phase II problem is separable by machine type and we solve this problem independently for each machine type that requires more than one machine.

As in [13] we use a graph partitioning model for selecting machine assignments. However, instead of creating duplicate nodes based on discrete approximations of processing time requirements for each part, we create a node for each operation and use actual time requirements. Thus, each node in the graph represents a specific part-operation assigned to this machine type in Phase I. The arc weight between each pair of nodes is 1 if both part-operations are required by the same part type. However, if a pair (say i and j) of nodes (part-operations) correspond to different parts (say [p.sub.i] and [p.sub.j], respectively), then the arc weight is defined as:

[S.sub.ij] = max{[N.sub.[p.sub.i][p.sub.j]]/[O.sub.[p.sub.i]], [N.sub.[p.sub.i][p.sub.j]]/[O.sub.[p.sub.j]]}, (6)

where [N.sub.[p.sub.i][p.sub.j]] is the number of common operation types for parts [p.sub.i] and [p.sub.j]. Thus, [S.sub.ij] measures the similarity between the processing requirements for the part types to which part-operations i and j belong. Accordingly, [S.sub.ij] represents the relative desirability for the two operations to be assigned to the same machine. Note that 0 [less than or equal to] [S.sub.ij] [less than or equal to] 1.

In the second stage, an initial partition for this graph is obtained by using the following heuristic. A part-operation pair with the lowest similarity is identified and each part-operation is assigned to two separate machines. This process is repeated iteratively until each machine has been assigned one part-operation. Then we sequentially assign the remaining part-operations to machines so as to maximize the internal similarity of all part-operations assigned to the same machine subject to [U.sub.m]. In the third and final stage, we improve on this initial partition of operations to machines by using a capacity-constrained variant of the Kernighan and Lin [30] graph partitioning procedure. The output of this phase is the specification of the [X.sub.[k.sub.m][j.sub.p]] decision variables.

3.3. Phase III: identify candidate manufacturing cells

Phase III of the FCF method involves clustering individual machines to identify candidate manufacturing cells. The individual part-operations performed in each cell are a function of the individual machines assigned to that cell (Phase II specifies the assignment of part-operations to individual machines). Our objective is twofold. We wish to form cells that are comprehensive for the current parts so as to minimize intercell material flows, but also to ensure flexibility to react to changes in processing requirements. The algorithm used to create the candidate manufacturing cells is similar to that of Phase II, except that we now consider partitioning the individual machine set into the number of desired cells. A major difference is in the manner by which the arc weights between individual machines are computed. The arc weight or desirability index [d.sub.n1,n2] between nodes n1 and n2 (each representing an individual machine of type m1 and m2, respectively) is defined as a convex combination of two parts:

[Mathematical Expression Omitted], (7)

where

[Mathematical Expression Omitted],

[W.sub.n1,n2] = proportion of machine n1 workload associated with parts that also use machine n2

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[f.sub.j1j2] = proportion of current parts that require operations j1 and j2

[Mathematical Expression Omitted].

[Mathematical Expression Omitted] is computed as the maximum proportion of machine n1 or n2's work that involves parts that also visit the other machine given the current part-operation assignments. [Mathematical Expression Omitted] is a function of the combined processing capability of machine pair n1 (of type m1) and n2 (of type m2). The motivation for [Mathematical Expression Omitted] rests in the desirability, in a dynamic environment, to create cells that have the ability to process future parts that exhibit minor technological modifications or to handle minor changes in part mix and volume. [Mathematical Expression Omitted] is the product of two terms. The first term is the proportion of part operations that can be performed by the combined set of machines n1 and n2. The second term in [Mathematical Expression Omitted] takes a weighted average over those operation pairs that require both machines of the likelihood that an arbitrary future part will require both operations. [Mathematical Expression Omitted] takes on the value 1 only if neither machine type m1 nor m2 can perform both operation j1 and j2, but this pair of operations can be completed by the union of the two machines. The weights [f.sub.j1j2] then indicate the probability that a future part will require this pair of operations. Together these indicate whether the synergism resulting from combining these two machines in the same cell is likely to be useful.

The parameter [Alpha] is defined by the user. A high value of [Alpha] indicates a user preference for designing cells based on the current part set and demand requirements. In contrast, by specifying a lower value of [Alpha], the user can instead opt to focus on the possible set of operations that could be performed by both machines n1 and n2 rather than the current set that have been assigned. If the firm has additional technological forecasting information beyond that contained in the current part set, this information can be easily entered into the decision process by modifying the [f.sub.j1j2] parameters accordingly.

Once the arc weights between all nodes have been defined, the user is required to specify the number of cells to be identified (i.e., C) and the variability to be permitted in cell sizes (i.e., V). Cell size is assessed in terms of the number of machines. C controls the number of partitions that will be created, whereas V controls the allowable range of the number of machines placed within each partition (cell). In practice, C would be set by organizational parameters such as size of worker teams, span of supervisory authority, and group dynamics. Given a range of cell sizes, the procedure can be applied for feasible values of C and the resultant solutions compared.

On the basis of these inputs, the procedure for initial partitioning and improvement described in Phase II is used to partition the set of nodes, each representing a machine, into machine groups (cells). The output of Phase III is the [Z.sub.[k.sub.m]c] decision variables and indicate the assignment of individual machines to cells.

The three user-defined parameters control the flexibility of cell system design in different ways. [Alpha] allows the incorporation of machine-level flexibility (in terms of possible operations that can be performed) during the cell design process. Thus, by varying [Alpha] the user can investigate how responsive the current part-operation to machine assignments are in the context of the set of operations which can be performed by each machine. The number of cells (C) can be hypothesized to impact the flexibility of the cell design. By setting a low value of C, the size of each cell in terms of machines will be larger. Hence, we can argue that there will be more routing flexibility built into the system as machine variety in each cell is likely to increase. Regarding the variability in cell size parameter (V), it can be hypothesized that by allowing cell sizes to be more variable, there is the likelihood that a 'master' cell consisting of a large number of machine types will be identified and thus routing flexibility will once again be higher. Likewise a few highly specialized cells can be formed for current highly similar part families when appropriate. Note that both cases (i.e., small C and high V) will probably increase the machine type flexibility of the cell system configuration. This is primarily the result of there being a single cell that will contain multiple machine types and hence new parts can be processed in such a cell.

3.4. Phase IV: improvement and evaluation

In Phase IV of the method we first attempt to improve upon the preliminary cellular configuration. Depending upon the user input, we can improve the routing flexibility and/or the volume flexibility of this preliminary system design. Given a cell system design (indicating the assignment of specific machines to cells and also the assignment of individual part-operations to machines in each cell), the routing flexibility for a part p in a cell c is computed as follows:

[Mathematical Expression Omitted], (8)

where

[Mathematical Expression Omitted]

and [K.sub.mc] is the number of machines of type m assigned to cell c. [R.sub.[cj.sub.p]] provides the proportion of time that cell c has at least one machine available that can perform operation [j.sub.p]. R[F.sub.cp] computes the proportion of time that cell c has a set of machines operational that are capable of completely processing part p. Note that if none of the machines in cell c can process an operation [j.sub.p] on part p, then R[F.sub.cp] = 0. Define [F.sub.cp] = 1 if R[F.sub.cp] [greater than] 0 and 0 otherwise. Then the routing flexibility of a given cell system design associated with an individual part p ([F.sub.p]) is simply the total number of cells that have the capability to process the part completely and is computed as

[F.sub.p] = [summation of] [F.sub.cp] where c = 1 to C (9)

and the aggregate routing flexibility of the system design is simply given by F = [summation of] [F.sub.p] where p = 1 to P. Given that this measure is influenced by the number of cells identified and does not indicate the balance of alternative cells across part types, an alternative aggregate measure of routing flexibility can be computed as the percentage of parts that can be routed through more than one cell for processing. Other possible alternatives include F[prime] = [summation of] log [F.sub.p] where p = 1 to P, which assigns an infinite penalty for creating a system with one or more parts that do not have a comprehensive cell, or F[double prime] = [summation of] log (1 + [F.sub.p]) where p = 1 to P.

To improve upon the routing flexibility of a given cell system design, we attempt to reassign parts, individual machines and/or part-operations in that order (without changing the total number of machines in the complete system). The general procedure is as follows. Each part is considered for reassignment to another cell that contains all the necessary equipment to process the part provided that machine capacity is available in that cell. After we perform this individually for every part, machine reassignment is considered if it will result in an increase in the number of cells to process one or more part types. Finally, the reassignment of part-operations is performed for those part types that cannot be completely processed in one cell. The objective of such a reassignment is to check whether we can identify another cell to completely process each part.

Another improvement algorithm available to the user at this Phase of the FCF method focuses on volume flexibility of the system design. Given a cell system design, the volume flexibility is computed as the maximum equal percentage (defined as [[Delta].sup.*]) increase in volume for all parts that can be handled without changing the system configuration. The improvement in volume flexibility is performed as follows. First we identify the bottleneck machine(s) in the system (i.e., the machine(s) that restrict the volume flexibility of the current design). Next, we attempt to reroute some load from this machine to another machine in the same cell that can perform the same operation. If this is not possible, we attempt to reroute the workload to a machine of the same type (or a machine that can perform the same operation) in another cell. The procedure is repeated iteratively until no further rerouting of load is possible.

Once the cell design has been improved upon, we evaluate it not only in terms of routing and volume flexibility (as defined earlier) but also in terms of mix flexibility. The flexibility of the cell system to respond to a change in part mix can be assessed depending upon the type of such change. A mix change occurs either when the relative volume requirements of the current part mix change, or when a new part with an associated operation set and volume requirement is introduced in the system. To assess the flexibility of a cell system to the first type of change (i.e., relative volume change), we compute the maximum percentage of demand for each current individual part (defined as [Mathematical Expression Omitted]) which can be accommodated within the current cellular configuration. Demand for part types other than p are held constant for this calculation.

To assess the flexibility of the cell system to respond to the introduction of a new part, we need to consider two aspects: (i) the amount of slack capacity in the cell system to completely process all the demand requirements for the new part, and (ii) whether there is a set of machines within any one cell with adequate slack capacity to completely process all the demand requirements for the new part. Obviously (ii) is preferred because the new part will be completely processed in a cell (leading to no increase in intercellular flows). However, if this is not possible, it may be possible to process the new part through multiple cells as long as adequate capacity is available in the system to do so. To assess these two aspects, we compute three measures.

The first measure is denoted by [[Gamma].sub.1] and is computed as the percentage of new parts that can be accommodated in the current configuration (even if these parts need to be processed in multiple cells). Hence, [[Gamma].sub.1] evaluates the availability of capacity in the aggregate cell system with reference to new parts. The second measure, [[Gamma].sub.2], is computed as the percentage of new parts that can be completely processed within an existing cell without regard to capacity constraints (i.e., there exists at least one primary cell to process the new part). Hence, this measure attempts to capture the part-mix flexibility of the cell design by restricting the intercellular flows of materials. The final measure ([[Gamma].sub.3]) integrates the first two measures and is computed as the percentage of new parts that can be completely processed within a single existing cell by explicitly considering the availability of capacity in the cell. These measures will be illustrated in the next section.

4. Illustration

The hypothetical example used to illustrate the procedure is as follows. There are 10 different machine types in the system. The system produces 19 parts requiring 12 different operation types. For each machine type, the procurement and operating costs ([F.sub.m] and [c.sub.m]) and the availability of each machine ([A.sub.m]) are shown in Table 2. The part demand in batches per year ([D.sub.p]), the batch size ([Q.sub.p]) and part processing requirements matrix are shown in Table 3. The run and setup times are shown in Table 4. For this example we assume that the operation characteristics are not part dependent. (The FCF method does not require this assumption because each part-operation is treated separately in Phase Il, and optionally in Phase I as well, but it does reduce the amount of data we must include in the table.) Thus, regardless of which part-operation is performed on a machine type, run and setup times are identical. We assume that the shop in question operates 8 hours per day, 250 days a year.

At Phase I, assume that [U.sub.m], the maximum allowable utilization of any machine, is 0.90 [for every] m. From Table 4 we can see that only operations 7, 9, and 10 can be processed on more than one machine type. We assume that the number of machines is not fixed; thus each operation is assigned to the machine type with minimum relaxed cost [u.sub.mj]([c.sub.m] + [F.sub.m]/[U.sub.m]). On the basis of the solution to Phase I, operation 7 is assigned to machine type 9; operation 9 to machine type 1; and operation 10 to machine type 10. The number of machines ([N.sub.m]) required of each type are (2,2,1,2,2,2,1,1,2,2).

Phase II is concerned with assigning each part-operation to a machine. Note that on the basis of the [N.sub.m] requirements from Phase I, this is only relevant for machine types 1, 2, 4, 5, 6, 9, and 10. Based on these part-operation-machine assignments, in Phase III we generate a candidate cell design with [Alpha] = 1 (to emphasize current part-operation to machine assignments), C = 4 (four-cell design) and V = 0 (i.e., cell size variability is not considered). Finally, in Phase IV, an improvement of the Phase III cell design is performed. We search for improved solutions in terms of volume and routing flexibility (in that order). The Phase III and IV part-operation assignments to individual machines and the cells to which these machines are allocated are shown in Table 5. Table 6 shows the final individual cell compositions (i.e., after Phase IV) in terms of machines and the average usage of each machine in each cell.

Table 2. Machine type data

Machine     Procurement        Operating          Availability
Type (m)    cost ([F.sub.m])   cost ([c.sub.m])   ([A.sub.m])

1             1.70                   1.70              0.85
2             1.80                   1.80              0.90
3             1.20                   1.20              0.60
4             1.80                   1.80              0.90
5             1.60                   1.60              0.80
6             1.60                   1.60              0.80
7             1.40                   1.40              0.70
8             1.80                   1.80              0.90
9             1.90                   1.90              0.95
10            1.60                   1.60              0.80

[TABULAR DATA FOR TABLE 3 OMITTED]

[TABULAR DATA FOR TABLE 4 OMITTED]

[TABULAR DATA FOR TABLE 5 OMITTED]

Table 6. Cell compositions after Phase IV

Cell      Machine type     Number          Av. usage
          assigned         of machines     per machine(*)

1             1                 1               0.576
              5                 1               0.583
              6                 1               0.532
              9                 1               0.652
             10                 1               0.485

2             2                 1               0.756
              4                 1               0.489
              6                 1               0.331
              8                 1               0.696
              9                 1               0.465

3             1                 1               0.470
              3                 1               0.234
              4                 1               0.639
              5                 1               0.421
              7                 1               0.447
4             2                 1               0.621
             10                 1               0.461

* This is computed as the average utilization per machine divided by
the average availability of the machine of that type (i.e.,
[A.sub.m]).

The Phase III and IV assignments of individual part-operations to machines and the cells to which individual machines are allocated are shown in Table 5. For example, part 2 requires operations (2,3,4,8). At Phase III, these operations are assigned to machine types (7,1,4,5), respectively as shown in column 3; and individual machines to which the load for these part-operations is allocated are placed in cells (3,1,3,3), respectively. Thus, for this part, one operation is performed in cell 1 while the other three operations are performed in cell 3. After implementing Phase IV, the load on machine type 1 in cell 1 (for the second operation on part 2) was rerouted to another machine of the same type in cell 3 and thus part 2 is completely processed in cell 3. As can be seen from Table 5, the Phase IV improvement procedure was able to reroute exceptional operations of five parts (nos 1, 2, 4, 5, and 11) such that all of them could be completely processed within a cell. However, parts 3, 10, and 13 are still processed in multiple cells. The cell compositions shown in Table 6 are the individual cell compositions after implementing Phase IV of the FCF method. Although cell size variability was set at 0%, cell 4 (two machines) is smaller than the remaining cells (each with five machines). This is because, to split up the 17 machines into four cells, the maximum size of a cell is set to [(17/4)] = 5.

Phase IV of the procedure also focuses on evaluating the generated cell design. In terms of routing flexibility of the current design, we can see that all part types except 3, 10, and 13 have all operations assigned to machines included in a single cell. Thus, there is at least one cell to completely process these part types. To assess volume flexibility of the cellular system shown in Table 5 (Phase IV assignment) and Table 6, we increased the demand (in batches per year) for every part type by the same percentage until the utilization of one or more machines exceeded the availability (i.e., [A.sub.m]; see Table 2). For this cell system, this occurred when the number of batches for all parts was increased beyond 145% of their current demand in Table 3. Hence, the volume flexibility (3) is 45% for the given cell system. In this case, the critical machine was machine type 2 in cell 2.

Regarding mix flexibility, the maximum percentage change in volume for an individual part ([[Delta].sub.p]) that can be handled by the current system (keeping all other part volumes fixed) is shown in Table 7. Further, for each individual part, we identify the related bottleneck machine type. Thus, Table 7 shows that for part type 3, we can accommodate at most a 740% increase in batch volume before machine type 7 in cell 3 becomes overloaded.

To assess the flexibility of the system to respond to the introduction of new parts, we proceed as follows. First, we generate the demand (in batches) and batch size (in units) for a potential new part by using a discrete uniform distribution with parameters (4,11) and (100,961), respectively. Secondly, for the part, we randomly generate the number of operations required for processing by using a discrete uniform distribution with parameters (2,6). These parameter settings are chosen on the basis of the minimum and maximum demand, batch sizes and number of operations required for all parts shown in Table 3. Thirdly, we generate the individual operations required to process the part randomly from the current set of operations as follows. The probability ([p.sub.j]) that a particular operation is selected first is based on the frequency ([f.sub.j]) that that operation is used for processing parts in the current mix and is computed as [f.sub.j]/([[Sigma].sub.j] [f.sub.j]). Once the first operation has been selected, the second operation for the new part is selected conditioned on the first. Thus, the probability that operation j1 is selected as the second operation given that operation j is the first operation is computed as f1[j.sub.1]/([[Sigma].sub.j][f1.sub.j]). In this case [f1.sub.1] is defined as the frequency that operation j1 appears along with operation j in the operation sequences for all parts. This sequential procedure is repeated until all the operations in the operation set for a new part have been generated.

Table 7. Mix flexibility of the cell design

Part (p)     Mix flex. (%) ([[Delta].sub.p])       Critical mach.

1             930                                    2
2             260                                    7
3             740                                    7
4             350                                    2
5             270                                    7
6             270                                    9
7             300                                    9
8            1960                                    7
9             370                                    4
10            220                                    9
11            390                                    1,5
12            280                                    2
13            790                                    2
14            380                                    2
15            340                                    2
16            290                                    2
17            260                                    2
18            360                                    2
19            560                                    2

Using the procedure described, we assessed the flexibility of the current system to respond to new part introduction. For the designed system, 68 % of 40 randomly generated new parts could be processed in the current system if we allowed intercellular flows of batches (i.e., [[Gamma].sub.1] = 68%); 48% of the new parts could possibly be processed within a single cell (without considering machine availability, i.e., [[Gamma].sub.2] = 48%); and 40% of the new parts could be processed within a single cell (i.e., without intercellular materials flows and explicitly considering machine availability, i.e., [[Gamma].sub.3] = 40%).

This concludes our illustration of the FCF method. We now turn to a discussion of the experiment performed to validate the usefulness of the routing and volume flexibility improvement algorithms (Phase IV of the method) and to provide guidelines as to how the parameter settings impact routing, volume and mix flexibility.

5. Experimental analysis

In order to evaluate the FCF method two experiments were performed. The first experiment used a designed experiment to test the value of the various steps and parameter settings in the FCF method to obtain flexible cell designs. The second experiment compared the FCF method to alternative methods for finding dense cells.

5.1. Evaluation of algorithmic steps and parameter settings

For the first experiment, the factors investigated and their treatment levels are:

(1) maximum allowable machine usage U (set at Phase I): 85% and 100%;

(2) parameter [Alpha] (set at Phase III): 0, 0.5, 1;

(3) number of cells C (set at Phase III): low, medium, high;

(4) cell size variability V (set at Phase III): 0% and 20%;

(5) improvement algorithm G (Phase IV): none (N), routing (R), volume (V), routing and volume (RV).

To carry out the experimental analysis, we used eight data sets (seven published and the hypothetical data set used in Section 4). The seven data set sizes were:

* P = 15, M = 10, J = 10 (see [31]);

* P = 19, M = 12, J = 12 (see [32]);

* P = 20, M = 8, J = 8 (see [33]);

* P = 24, M = 14, J = 14 (see [34]);

* P = 43, M = 16, J = 16 (see [7]);

* P = 43, M = 14, J = 14 (see [7]); and

* P = 15, M = 95, J = 10 (see [16]).

For each data set and factor/treatment level, we generated five different cell designs by randomly varying the operation-machine type input data. (The procedure used is as follows. First, using a discrete uniform distribution, we generate the number of operations that can be performed by a machine type. The parameters for the distribution are data set dependent. Let this be equal to [B.sub.m]. Secondly, we randomly pick [B.sub.m] operations from the total number of possible operations (with an equal probability of picking any operation) and assume that any machine of type m can perform all these operations. Note that we always ensure that there is at least one machine type that can carry out each generic operation.) ANOVA was used to analyze the results. (Owing to space limitations, we do not present the ANOVA results in the paper. The interested reader is referred to [35].) A summary of the results reveals that:

1. Statistically significant increases in routing flexibility were observed when the routing flexibility improvement algorithm was implemented in Phase IV. Similarly, when the volume flexibility improvement algorithm was implemented, statistically significant increases in volume flexibility were noted. These results point to the usefulness of implementing the improvement algorithms in Phase IV of the FCF method.

2. To maximize routing flexibility, the parameter [Alpha] should be set to focus on current part-operation assignments, and fewer cells with highly variable cell sizes should be created.

3. To maximize volume and mix flexibility (assessed by using [[Gamma].sub.3]), larger slack capacity should be built into the system. Further, a larger number of cells is preferred for maximizing volume flexibility, whereas more variable cell sizes are preferred for maximizing mix flexibility.

Summary recommendations for users of the FCF method based on these observations are shown in Table 8.

5.2. Comparison with other cell design approaches

Co and Araar [16] proposed a procedure for assigning operations to machines and machines to cells for the case of duplicate machines. These steps correspond to Phases II and III of our method with the exception that those authors concentrated on designing cells for the current static part mix. They demonstrate their method on a 15-part, 63-machine problem. Their method produces a grouping efficiency [7] of 0.68 for this problem. Setting U = 100%, C = 6 and [Alpha] = 1.0, and executing Phases II and III we obtained an improved grouping efficiency of 0.72. Cell density was improved and the number of exceptional part operations was reduced, indicating improvement in both aspects of the efficiency measure. This was further improved in Phase IV to obtain an efficiency of 0.73. By reassigning part operations, the FCF method was able to increase the comprehensiveness of the cell system.

Table 8. Experimentally determined preferred parameter settings(a)

Factor/                     Performance measure
user input
              Routing flex.   Volume flex.   Mix flex.
              (F)             ([Delta])      ([[Gamma].sub.3])

U             -               LOW            LOW
[Alpha]       HIGH            -              -
C             LOW             HIGH           -
V             HIGH            -              HIGH
G             R               RV             RV

a A "-" entry in the table indicates that the Factor/user input did
not significantly impact the flexibility measure.

6. Conclusions and implications

In this paper we have proposed a cell formation method that is unique in many respects. First, this method allows the user to design cellular systems that are flexible (in terms of responsiveness to part demand and part mix changes as well as in terms of routing flexibility). The method also allows the user to modify the preliminary cell design to increase the routing and/or part volume flexibility. Secondly, the decision hierarchy provides a computationally feasible approach for solving problems of realistic size while including consideration of important factors such as alternate machine types and costs. Thirdly, the FCF method can generate multiple cellular configurations depending upon user inputs. The user can trace out multiple solution alternatives by varying parameters on cell size, emphasis on current or future operations, and type of flexibility preferred. Additionally, the user can modify individual phases to achieve desired objectives or test varying strategies. Fourthly, through an experimental analysis, we have also been able to provide guidelines on the use of the procedure.

The method balances costs and flexibility. Phase I minimizes fixed machine cost plus direct processing costs. Phase II assigns operations to machines to take advantage of part similarities and operation flexibility of machines. In turn, this serves to limit final material handling costs due to intermachine and intercell transfers. Phase III groups machines by balancing actual material handling with system flexibility. Finally, Phase IV can be used to increase flexibility of the final system design. The proposed methodology provides one approach for dealing with the complexity of the overall cellular manufacturing configuration problem. Although we have provided evidence that the individual phases of the procedure are computationally efficient (polynomially bounded heuristics) and highly competitive with previous procedures for each subproblem, we believe the major contribution lies in the integrated methodology for designing cells with consideration of flexibility for a dynamic environment. Still, there is clearly potential for further research into exploring the cost-flexibility tradeoff.

Acknowledgements

This paper is based on work supported in part by the National Science Foundation under grant no. DDM-92-15432.

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Biographies

Ronald G. Askin is a Professor and Acting Department Head of Systems and Industrial Engineering at the University of Arizona. Dr Askin received a B.S. in Industrial Engineering from Lehigh University, and an M.S. in Operations Research and Ph.D. in Industrial and Systems Engineering from Georgia Institute of Technology. He is a Fellow of the IIE, and an active member of INFORMS, ASQC, and SME, having previously served as Program Co-Chair for the 1993 Phoenix ORSA/TIMS meeting, Chair of the ORSA Technical Section on Manufacturing Management, Chair of the Statistics Division of ASQC, and Program Chair for the 1996 IE Research Conference. He is the former editor of the IIE Transactions on Design and Manufacturing. He has published in various professional journals, predominantly in the areas of design and operational analysis of manufacturing systems. His current research is in the areas of cellular manufacturing and Just-in-time production. Dr. Askin co-authored the text Modeling and Analysis of Manufacturing Systems, which was awarded the 1994 lie Joint Publishers Book of the Year Award. Other awards he has received include the Shingo Award for Excellence in Manufacturing Research, IIE Transactions Development and Applications Award (coauthor), the ASEE/IEE Eugene L. Grant Award (coauthor), and a National Science Foundation Presidential Young Investigator Award. In addition, he has been the academic advisor for several award-winning student research projects. Dr. Askin has consulted with a variety of industrial companies in the general areas of facility design, process capability analysis and improvement, integrated product and process design, material flow control, and performance measurement of manufacturing systems.

Hassan M. Selim is a member of the Faculty of Engineering and Technology at Helwan University in Cairo, Egypt. He received his Ph.D. from the College of Business and Public Administration at The University of Arizona. His research interests are in the area of flexible manufacturing and cellular manufacturing systems.

Asoo. J. Vakharia (a senior member of IIE) is an Associate Professor in the Department of Decision and Information Sciences at the University of Florida. He received his Ph.D. degree in Operations Management from the University of Wisconsin-Madison. Asoo's research focuses. on the design and control of Cellular Manufacturing Systems as well as on the integration of the marketing and operations functions in service firms. His prior research has been published in Annals of OR, Decision Sciences, Discrete Applied Mathematics, IIE Transactions, Journal of Operations Management, Naval Research Logistics, and International Journal of Production Research. He is currently an Associate Editor for the International Journal of Flexible Manufacturing Systems and also serves on the Editorial Review Board of the Journal of Operations Management.