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Application of fuzzy decision making in part-machine grouping
Int. J. Production Economics 63 (2000) 181}193
Application of fuzzy decision making in part-machine grouping ZuK lal GuK ngoK r*, Feyzan Armkan Industrial Engineering Department, Gazi University, Maltepe 06570, Ankara, Turkey Received 9 October 1997; accepted 25 January 1999
Abstract In this study, fuzzy set theory (FST) is used to set out the cell layout. A new algorithm which will consider both design and manufacturing attributes and operation sequences as factors, is proposed to formulate the problem. The structure of the algorithm is based on fuzzy decision making system (FDMS). Hence three factors mentioned above are determined as input variables and fuzzi"ed using membership function concept. Then the pairwise comparison of the analytical hierarchy process (AHP), which ensures the consistency of the designer&s decisions when assigning the importance of one factor over another, is used to "nd the weights of these factors. Applying IF}THEN decision rules, parts relationship chart (PRC) is generated. After these steps, the traditional cell formation procedure is applied. Finally the proposed method is scored by performance measures such as machine investment, the amount of work load deviations within cell and between cells and the number of skippings. Also the comparison with AktuK rk's study (International Journal of Production Research 34 (8) (1996) 2299}2315) in respect to these performance measures is presented. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy decision making system; Cellular manufacturing
1. Introduction During the last ten years, there has been great interest in the area of group technology (GT). Cellular manufacturing (CM) is an application of GT which allows for small batch production to gain economical advantage over the traditional manufacturing systems. The identi"cation of part families and machine groups in design of cellular manufacturing system is commonly referred to as a cell. In CM, parts with design and manufacturing similarities are grouped into part families and the
* Corresponding author. E-mail address: [email protected] (Z. GuK ngoK r)
associated machines are grouped to form the cells. Such part families are assigned to cells composed by machines and set up in the order of best routine and process sequences of part families. Numerous papers can be found in the literature for cell formation. In this area, the following major approaches to model the problem have been employed: clustering analysis as matrix formation which uses some similarity coe$cient to obtain part family formations [2}5], production #ow analysis [6], mathematical programming [7}9], graphical analysis [10], neural network [11] and simulating annealing [12]. Coding systems like OPITZ [13] and MICLASS [14] are also used to group the parts according to their design attributes. Comprehensive reviews of the cell formation
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 1 0 - 9
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approaches are provided by WemmerloK v and Hyer [2,15}17]. Most of the above-mentioned approaches adopt either a sequential or a simultaneous procedure to group the parts and machines. The sequential procedure used in some of these studies determines the part families "rst, followed by machines assignment. On the other hand the simultaneous procedure determines the part families and machine groups concurrently. In this approach a part/machine matrix in a block diagonal form is obtained. Most of these applications deal only with cell formations. A common measure of performance is the number of exceptional elements. Additional measures, such as minimizing total material-handling costs, duplication costs or maximization of within cell utilization are considered only to avoid exceptional elements. Flow of the materials also is related to operation sequences. The spatial arrangement of machines inside cells and cell layout formations in#uence the e!ectiveness of cells. However there are only a few studies dealing with the spatial arrangements of machines inside the cells. The study of Greene and Sadowski [18] can be given as an example. During the cell formation consideration of only one factor such as similarity coe$cient, as used in many traditional studies, is not enough. All possible variables (factors) that can a!ect the design must be considered in cell design. Because the number, complexity and unclear nature of the variables may in#uence the decision maker's decision. There are many factors to be considered such as similarity of operation sequences, design similarities of the parts, the number of parts in each cell, machines required in each cell, additional machine investment for each cell. O!odile [19] also has addressed the problem of how to form part families using part classi"cation and coding analysis. One common weakness with conventional methods is that they implicitly assume that only disjoint part families exist in the data. That is, a part can only belong to one part family. In practice, however, there may exist some parts whose lineages are much less evident according to their technological and/or geometrical features. Therefore the FDMS provides a more accurate presentation of the problem in the domain of uncertain or
inexact information, and FST is used to express the linguistic variables which describe the quantitative and qualitative factors a!ecting the design of cell formation. The remainder of this paper is organized as follows. In Section 2, a brief introduction to FST is presented. Also a detailed discussion on FDMS is given. In Section 3, we discuss the fuzzy decision making in cell formation in general format. In Section 3.1, the proposed algorithm is explained in a wide perspective. In Section 4, the application of the algorithm to AktuK rk's [1] example is given. Finally, concluding remarks and the comparison to AktuK rk's [1] results are discussed in Section 5.
2. Introduction to fuzzy set theory The FST, introduced by Zadeh [20] to deal with vague, imprecise and uncertain problems, has been used as a modeling tool for complex systems, that can be controlled by humans but are hard to de"ne exactly. A collection of objects (universe of discourse) ; has a fuzzy set A described by a membership functions k that takes the values in the A interval (0, 1), k : ;(0, 1). Thus A can be represented A as A"Mk (u)/u; where u3;N. The degree that u A belongs to A is the membership function k (u). A More detailed discussion of FST can be found in [21}23]. FST has been applied to many production areas. Some of these can be found in [24}27]. In this paper the focus is on fuzzy decision making in the design of cellular manufacturing systems. It di!ers from [28,29] in that they use fuzzy clustering mathematics in manufacturing the part families formation rather than using fuzzy decision making. 2.1. Fuzzy decision making system Fuzzy logic control (FLC), initiated by the pioneering work of Mamdani and Assilian [30], is one of the most active areas in the application of FST. The basic idea is to incorporate the expert knowledge of a human operator in the design of the controller in controlling a process whose input output relationship is described by a collection of fuzzy control rules (e.g., IF}THEN rules) involving
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Fig. 1. Fuzzy decision making system (FDMS).
linguistic variables rather than a complicated dynamic model. FDMS consists of four principal components as shown Fig. 1. These four parts are;
value is calculated as
1. The fuzzi"cation interface: Measures the values of the input and output variables, transfers the range of these values into a corresponding universe of discourse and converts them into natural language such as high, low and very low. 2. The knowledge base: A data base that consists of the expert's knowledge of the application and the control rules of the process. The membership functions are determined and fuzzi"cation of the variables is achieved. 3. The decision making logic: It has the capability of simulating human decision making by performing approximate reasoning to achieve a desired control strategy. The rules are mostly in the form of IF}THEN form. The membership value of the control action of each rule is the minimum value of the input variables of membership values [30].
where R is the "nal crisp rating of activity, g the 0 fuzzy rating of the activity for the rule in consideration, i the rules that are used in the activity, R the numerical rating of the activity of the rule and k the membership value of the activity for the rule.
Label l #0/530- !#5*0/ "MinMLabel l l N. */1651(7!-6%),2, */165/(7!-6%)
(1)
4. The defuzzi"cation interface: Converts the fuzzy output into a crisp (nonfuzzy) value. One of the most common methods is the Centre of Area (COA) method [22]. In this method "nal crisp
+ kg .R R " i R , 0 +k R
(2)
An example of application of this type of FDMS is the facilities layout problem given in [31].
3. Fuzzy decision making in cell formation The fuzzy approach to the cell layout problem involves the identi"cation of linguistic variables that describe the quantitative and qualitative factors a!ecting the cell design and the selection and determination of these values and the membership functions of the linguistic variables and the development of heuristic procedures for the selection and placement of machine and parts for each cell. At the beginning of cell formation all related input variables are de"ned, then the values of these variables are determined and the membership functions are developed using expert's knowledge. Weights of factors of these variables are determined using analytical hierarchy process (AHP) which
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is based on pairwise comparisons of the input variables [32]. The basic AHP starts by dividing the decision problem into a hierarchy. Most systems can be modeled in a hierarchical fashion; that is, the model is constructed in several di!erent levels of increasing aggregation or priority, until a top level or objective is reached. Each level contains components with similar properties that allow comparison. The procedure of making pairwise comparisons greatly reduces the conceptual complexity of decision making. The scale comparison is usually taken to be in the range 1}9. These are, basically of equal importance (1), between equal an weak importance of one over another (2), weak importance of one over another (3), between weak and strong importance of one over another (4), strong importance (5), between strong and demonstrated importance of one over another (6), demonstrated importance (7), between demonstrated and absolute importance of one over another (8), absolute importance (9). Using this scale the participating managers assess the dominance of each element over the others with respect to each element of immediate higher level of hierarchy. This process of comparisons yields a relative scale of measurement of priorities or weights (w) of elements. That is, the scale measures the relative standing of the elements with respect to a criterion independently of any other criterion or element that may be considered for comparison. The comparisons are performed for the elements in a level with respect to all the elements in the level above. The "nal or global weights of the elements at the bottom level of the hierarchy are obtained by adding all the contributions of the elements in a level with respect to all the elements in a level above. Once the comparisons matrix is formed, the largest eigenvalue j should be approximately .!9 equal to n (the number of factors). The deviation of j from n provides a measure of consistency .!9 which measures the deviation of the judgements from consistent approximation. The `consistency indexa CI is given by j !n CI" .!9 . n!1
(3)
Consistency ratio, CR is obtained from the ratio of CI to RI (random index). Random index is the
index of the same order matrix, the elements of which are randomly generated [33]. Generally if the CR is less than 0.1, the consistency of the decision maker is considered satisfactory. The vector w is then presumed to be the decision makers optimal weights or priorities. In this study AHP only uses as pairwise comparisons of attributes to determine its factors' weight. These variables are fuzzi"ed using a set of membership functions such as `very lowa, `mediuma, `higha, and `very higha. The other common input factor is closeness rating (R), the rating given to the parts by designer is based on input variables. The rating ranges from `absolutely necessarya to `undesirablea. A closeness rating is assigned to represent the priority of parts collected in the same cell. Parts with similar rating have higher priorities in forming a cell collection. Designer assigns closeness rating (cost) (A"6, E"5, I"4, O"3, ;"2, X"1) of the parts to show the degree of relationship of the parts. The closeness rating is fuzzi"ed by the set of membership functions as in Fig. 2. After the fuzzi"cation processes of all input and output variables are completed, the next step is the establishment of the decision-making logic (decision rules). These rules which initiate the designers' decisions, usually take the form of the IF}THEN rules. Such rules are conveniently tabulated in look-up tables as shown in Tables 1}3. Using these tabulated rules, the minimum operator from Eq. (1), and the defuzzi"cation COA method in Eq. (2), the designer can "nd the crisp (exact) values of the parts relationship chart (PRC). PRC is the similarity coe$cient matrix which is used to develop the physical layout of cells in the plant, in traditional procedures. FST is used in generating the PRC which is the basis of cell layout. 3.1. The proposed algorithm In the process of cell layout design, variables (factors) that a!ect the design are mostly design and manufacturing features, operation sequences and operation times. The design similarities are important in the part design, manufacturing and purchasing phases, whereas the operation sequences are important factors in determining the spatial arrangement of machines inside the cells. The design
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Fig. 2. Membership functions of closeness rating.
Table 1 IF}THEN rules of factor 1 (F1) and its weight factor (WF)
Table 3 IF}THEN rules of factor 3 (F3) and its weight factor (WF)
F1/WF
VL
¸
M
H
VH
F3/WF
VL
¸
M
H
VH
VL ¸ M H VH
; ; O ; I
; ; O ; E
O O I O A
O O E E A
I E E A A
VL ¸ M H VH
; ; O I I
; O I I E
O I I E E
I I E E A
I E E A A
Table 2 IF}THEN rules of factor 2 (F2) and its weight factor (WF) F2/WF
VL
¸
M
H
VH
VL ¸ M H VH
; ; O I I
; ; I I E
; O I E E
O O E E A
O I E E A
and manufacturing similarities and operating sequences and operating times of parts in each cell are represented by such vague issues as the rating qualitative or quantitative relationships in each cell ranges from `absolutely necessarya to `undesirablea. After deciding on the variables that can a!ect the design, fuzzy set theory is used as the suitable theory to this process due to the unclear nature of the factor similarities of parts that are placed in the
same cell. Then, values of these variables are determined, the universe of discourse and the membership functions are de"ned. Finally, the following algorithm as shown below is developed. Step 1: Find the factors that a!ect most the design of the cells. Tabulate all the necessary information for the factors. De"ne the number of identi"cations similarities of these factors between each pair of parts. Using AHP, the weight of each factor is found. If the result of the AHP is consistent and acceptable (consistency ratio and maximum eigen value are smaller than 0.1 and about n, respectively), the weight of factors is considered one of the input factors that will in#uence the layout (Eq. (3)). The weights of the factors are fuzzi"ed. This process is repeated for all pairs of parts. Step 2: The next step is to fuzzify the input and output factors that a!ect the cell formations. Membership functions are developed with interviews of related people. The maximum rule of fuzzi"cation
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process, which can be explained by using the maximum membership value of a factor to "nd the fuzzy set (label), is employed for all parts. Step 3: The third step is to apply the IF}THEN decision rules developed by a designer as the result of the fuzzi"cation interface of the fuzzy decision making process. As shown in Tables 1}3 all input variables are tabulated. Step 4: Using the minimum operator rule (Eq.(1)) "nd the related rating and weight of each pair of parts. Step 5: The "nal step of the fuzzy decision making process is the defuzzi"cation interface. This step uses COA Method to calculate the "nal crisp value of PRC (Eq. (2)). This process is repeated for each pairs of parts. After obtaining PRC the following steps are employed. Step 6: Normalize the PRC which is in a matrix form, to have the value of unit measures. N "A /(A !A )2, (4) ij ij kj lj where A represents the greatest value of the kj matrix, A represents the lowest value of matrices lj and i, j show the part number. Step 7: Apply one of the clustering algorithms such as Single Linkage Clustering (SLC) as used by Singh and Rajamani [34]. This algorithm is summarized as follows. Find the similarity coe$cient of the parts (in this study PRC is used as the similarity coe$cient matrix as mentioned before) and determine the maximum value of the matrix. Two parts which correspond to the maximum value of the matrix are joined to obtain a new group. Similarity between the new group and remaining groups is computed by taking the maximum value of pairs. At the next step, again join the parts which have the maximum value. Repeat the same procedure to "nd the group for all parts. Step 8: De"ne the threshold value to obtain the necessary number of cells. Step 9: Spatial arrangement of machines in each cell for the remaining candidate alternatives are found by using the Computerized Relative Allocation of Facilities Technique (CRAFT) which is the most widely applied computerized layout program. Step 10: Calculate the requirement of each machine type in each cell by considering the demand requirements and capacity restrictions.
Step 11: The total machine investment, the amount of workload deviations within and between cells, and the number of skippings are calculated using the following equations: I"+ + X .C , (5) kj k j k|M where I is the total machine investment, C is the k "xed cost of machine type k, X is the number of kj machines of type k assigned to cell j. D"+ + (¹¸ !A¸ )2/n , (6) j j j j k|M where D is the workload deviation within a cell, ¸ the expected load of machine type k in cell j, A kj lj the average load in cell j, n the total number of j machines in cell j and M the set of all machine types. B"+ (¹¸ !E¸)2/N, (7) j j where B is the workload deviation between cells, ¹¸ the total load in cell j, EL the average load of j cells and N the number of cells. Step 12: Repeat all steps 9}11 for all de"ned cells determined in step 8. Select the best e!ective cell formation.
4. The problem de5nition An example problem is taken from AktuK rk's study [1] to illustrate the proposed fuzzy cell formation heuristic. The example considers 20 parts and each part requires at most 5 and at least 3 machine types selected from 10 machine types. In this study there are 9 design and manufacturing attributes coded in a polycode structure as summarized in Table 4 along with the operation sequences, processing times on each machine and the demand forecasts for each part. Each symbol in an operation sequence corresponds to a machine type that will perform the given operation. Three of the design and manufacturing attributes are selected with respect to the shape of the parts, and others are design properties of parts such as proportion of diameter to lengths, measure of the cogwheel, length and width, tolerance and seconder hole. The
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Table 4 Part, design and manufacturing attributes, routings and demand data Parts no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Operation sequences
B A F F A A F K A A A B M A F A A A F G
F B K G M B G A K K B F R R H B B B G R
G F M H T L R R M R R R T T R M F F R T
L R R M T L T T R T T T L
L T T G T
Design and man. attri.
Ope. times sequence num.
1 2 3 4 5 6 7 8 9
1 2 3 4 5
0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0
2 2 2 3 2 2 3 2 3 3 2 3 3 1 1 2 3 5 4 1
1 3 2 5 0 1 4 1 5 1 2 1 4 1 3 1 2 0 3 3
2 5 1 2 0 1 3 0 3 0 0 3 3 1 2 0 3 0 2 3
6 1 3 6 3 6 0 8 7 7 5 5 6 8 5 3 3 0 5 6
4 0 8 8 4 3 4 5 3 5 5 4 4 4 7 2 0 5 1 1
3 3 3 2 3 1 3 3 2 3 3 2 2 0 2 3 3 2 4 2
Table 5 Machine investment costs B
2 5 4 2 0 1 0 0 3 1 2 3 5 0 2 2 4 0 2 2
3 5 1 6 6 2 2 4 3 4 4 3 2 5 6 5 2 0 3 4
1 3 0 2 0 2 3 2 0 2 1 1 2 2 4 3 1 1 2 1
3 3 3 2 3 3 2 3 1 2 3 2 2 3 2 3 2 4 1 1
2 4 4 4 4 2 3 1 4 2 4 1 4 2 1 2 3 2 1 2
Average demand
150 226 335 446 274 171 218 273 307 414 223 378 328 280 270 182 244 152 366 226
3 2 2 1 5 1 2 2 2 3 1 1
1 3 4 2 1
Table 6 Similarities of design and manufacturing
Type of machine
A
F
G
H
K
L
M R
T
Cost of machine
106 136 65 14 103 61 126 93 94 70
investment costs of related machines are given in Table 5. Before the illustration of the algorithm, it is worthwhile to explain the factors that a!ect the cell design: the similarities between parts depend on qualitative and quantitative factors. Some factors may have greater e!ect on the designer's decisions. Hence three factors are selected. These are similarities of design and manufacturing (F1), similarities of operation times (F2) and similarities of operation sequences (F3). 1. Similarities of design and manufacturing (F1): The design similarities are important in the part design, manufacturing and purchasing phase. The design di!erences between parts of production
1 2 3 4 5 . . 20
1
2
3
4
1 1 4 2
2 0 2
1 4
1
3
0
0
4
5 222222 20
02
4
must be minimum in order to create cell design with similar parts. When di!erences of parts are reduced in the same cell, machine set up times are reduced and many other factors will be better than before. The rating of these relationships ranges from absolutely necessary to undesirable. Experts can give this variable some linguistic values such as high, low, etc. The number of identi"cation similarities between each pair of parts (Table 6) are fuzzi"ed as in Fig. 3. For
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Fig. 3. Membership functions of factor 1. Table 8 Similarities of operation sequences of parts
Table 7 Similarities of operation times of parts 1 1 2 3 4 5 . . 20
2
3
4
2 2 0 2
3 1 3
2 3
1
1
0
0
0
5 22222220
0 2222221
1 2 3 4 5 . . 20
1
2
3
4
2 1 2 0
1 1 1
2 1
0
1
0
1
2
5 22222220
1 2222223
Fig. 4. Membership functions of factor 2.
instance, parts 1 and 2 have only one common character at identi"cation number 6 and, parts 3 and 2 have only two common characters at identi"cation numbers 1 and 6. This process is repeated for all pairs of parts. 2. Similarities of operation times (F2): In order to organize the machine utilization and cell utilization in each cell and between cells, similarities of operation times are also important. Imbal-
ance of workload in each cell will cause a high level of work in process inventories. Machine utilization and cell utilization are measures of load imbalance within a cell and the amount of workload deviation within and between cells will show the cell utilization. Similarities of operation times are given in Table 7. Experts also fuzzi"ed this variable as high, low, etc. as shown in Fig. 4.
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3. Similarities of operation sequences (F3): Parts used in the same machine must be replaced in the same cell. Di!erences in the operation sequences will be minimized and spatial arrangement of machines inside the cells are performed. This factor is de"ned by the number of similarities of operation sequences of parts. For instance (Table 8), parts 1 and 2 have only 2 similar processes and this process is repeated for all pairs of parts. If there are inverse operation sequences, it must be taken into consideration. These are fuzzi"ed as in Fig. 5.
4.1. Methodology of solution
1. After selection of the most important factors, we assume that the designer collects all the information tabulated in Tables 4 and 5. Using the AHP process, the weight of each factor is found. For example, if the designer is assigning a value of 3 for the importance of factor 1 over factor 2. This means that there is a range between an equal and a weak importance of factor 1 over factor 2. A value of 5 assigned for the importance of factor 1 over factor 3 indicates a strong importance of factor 1 over factor 3. The result of weight factors calculation using AHP are given in Table 9. The result of AHP is acceptable (CR"0.0025 and maximum eigenvalue" 3.001). The weight of the factors is considered to be one of the input factors that in#uence the cell formations and is fuzzi"ed as in Fig. 6.
189
2. All three factors that a!ect the cell design are fuzzi"ed. Maximum rule of the fuzzi"cation process and Figs. 4}7 are used. Results of the information about the relation between parts 1 and 2 are shown below. Factor 1 has a value of 1 (Table 6); it belongs to the fuzzy subset low (L) with a membership value of 1 (Fig. 3). Its weight factor (WF) has a value 0.640 (Table 9) which belongs to the fuzzy subset Very High (VH) with a membership value of 1 (Fig. 6). Factor 2 has a value of 2 (Table 7); it belongs to the fuzzy subset Medium (M) with a membership value of 1 (Fig. 4). Its weight factor (WF) has a value 0.148 (Table 9) belongs to the fuzzy subset Low (¸) with a membership value 0.5 (Fig. 6). Factor 3 has a value of 2 (Table 8); it belongs to the fuzzy subset Medium (M) with a membership value of 0.87 (Fig. 5). Its weight factor 0.212 (Table 9) belongs to the fuzzy subset Low (¸) with a membership value 0.75 (Fig. 6).
Table 9 Results of AHP
F1 F2 F3
F1
F2
F3
W
1 1/3 1/5
5 1 2
5 1/2 1
0.640 0.148 0.212
Note: CR"0.0025, CI"0.001, maximum eigenvalue (j )" .!9 3.001.
Fig. 5. Membership functions of factor 3.
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3. This process is completed for all pairs of parts. Then IF}THEN decision rules are applied to the results. These rules are: (a) IF (F1) is (¸) and its (WF) is (VH) THEN the rating (R) is (E) (Table 1). (b) IF (F2) is (M) and its (WF) is (¸) THEN the rating (R) is (I) (Table 2). (c) IF (F3) is (M) and its (WF) is (¸) THEN the rating (R) is (I) (Table 3). 4. Using the minimum operator (Eq. (1)) rule 1, rating from Fig. 2 (E"5) has a membership value of the minimum of 1 and 1. Thus, rule 1 results in a rating of (I"4) with a membership value of 1, rule 2 results in a rating of (I"4) and with a membership value of 0.75, and rule 3 results in rating (I"4) with a membership value 0.5. The "nal step of fuzzy decision-making process is the defuzzi"cation interface. This step uses COA method to calculate the "nal crisp value (Eq. (2)). The relationship between parts 1 and 2 is shown below: PRC"((5)(1)#(4)(0.75)#(4)(0.5))/ (1#0.75#0.5)"4.44.
Fig. 6. Membership functions of the weight factor (WF) of the inputs.
Again this process is repeated for all parts. The new PRC as a result of FST is shown below in Table 10. Cells formation is developed using the PRC. The values of only the "rst "ve rows and columns and the 20th row are presented in Tables 7 and 8 due to space limitations. In, Table 10, the values of the "rst 10 rows and the 20th row are presented for similar reasons. The second part of the algorithm is applied as a traditional cell formation procedure. 5. Normalize the PRC matrix by using Eq. (4). Apply Single Linkage Clustering algorithm, "nd the number of cells. If the threshold value is taken as 0.88, part family formation is not valuable since two cells are formed and the remaining parts do not belong to any cell. These parts of each cell are (1, 13, 4, 6, 19, 12), (17, 16, 2, 11, 3), (5), (7), (8), (9), (10), (14), (15), (18), (20). If the threshold value is taken as 0.86 two cells are formed as shown Table 11. 6. The spatial arrangement of machines are found by using the CRAFT algorithm for the initial part families shown in Table 11. 7. Compute the required number of each machine type in each cell by considering the demand requirements and capacity restrictions. Initially, parts 18 and 20 are assigned to cell number 1, but they use machine type G. In order to minimize the cost of total machine investment they are moved to cell number 2, since it does not require any additional machine investment of G. Parts 10 and 8 have reverse operations in the same cell. It requires additional machine investment of machine type A. But the requirement of duplication of machine investment costs in the same cell
Fig. 7. Membership functions of deviation within cell.
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Table 10 Parts relationship chart (PRC)
1 2 3 4 5 6 7 8 9 10 . . 20
1
2
3
4
5
6
7
8
4.44 4.44 5.20 4.53 4.75 5.20 4.53 5.00 4.00
4.64 2.80 4.16 3.60 4.63 4.40 3.78 4.67
4.44 4.75 4.44 4.67 4.40 4.64 4.40
3.71 6.00 4.64 3.78 4.40 4.63
4.00 5.20 4.75 3.93 4.75
3.69 4.40 5.00 4.40
4.63 3.78 5.67
4.00 4.00
3.63
0
4.00
4.00
5.20
4.00
3.69
5.00
9
1022222 20
4.6722222222 4.25
Table 11 Part families and cells layout Cells
Cell number
Part family
Cell layout
2
1 2
8, 10, 9, 17, 16, 11, 3 5, 7, 1, 13, 4, 6, 19, 12, 14, 15, 20, 2, 18
A, B, F, K, A, M, R, T, T A, B, F, F, G, H, R, L, T, M, R, T
Table 12 Comparison of the proposed method and AktuK rk's algorithm
M/C investment Within cell deviation Between cell deviation Total number of skippings
Fuzzy decision making layout
AktuK rk's algorithm
1963 495 53 5879
2150 602 38 9894
is already acceptable instead of replacing these two parts in a separate cell. Another reason is that these two parts are processed by machine type K which is placed only in cell number 1. 8. The total machine investment (Eq. (5)), the amount of work load deviations within cells (Eq. (6)) and between cells (Eq. (7)), and the number of skippings are calculated as shown in Table 12.
5. Conclusion In this study, the cell formation problem is studied by using FDMS in the mean time consider-
ing the design and manufacturing attributes and operation sequences. FST comes in handy because of the imprecise structure of qualitative and quantitative factors which a!ect the cell design directly. Herein a new alternative fuzzy approach to design cells is proposed. After some numerical explanations, the advantages of this FDM approach over conventional clustering will be summarized. After forming cells, a score must be given to decide that it has an acceptable value. In the evaluation, two input factors which are PRCs and the workload deviations within cells, and one output factor as score of the output are used. The scoring process follows the FDMS. PRC is fuzzi"ed as in
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192
Fig. 3 and deviations within cells is fuzzi"ed as in Fig. 7. Overall parts relationship is 4.39. The rating is considered an input variable for the scoring process and 4.39 belong to the rating `Ia with the membership value of 0.87 which is found by using the maximum operator rule. The workload deviation within cell belongs to ¸ with a membership value of 0.95. IF}THEN procedure is established as in Table 13. The output variable (score) must also be fuzzi"ed such as Good, Bad, etc., using a set of membership functions (Fig. 8). The "nal crisp score is then transferred into a linguistic variable using the maximum operator rule. The "nal output crisp score is 3.28 which belongs to the linguistic variable FAIR with a membership value of 0.58 (Fig. 8). It is recommended to accept this score as the preferable cells layout. Besides these results mentioned above, the summary of the advantages of the FDM approach are given below: f It allows the designer to use all possible factors a!ecting the layout regardless of their qualitative or quantitative nature.
Table 13 IF}THEN rules of the score (S) for the rating (R) and deviation (D) R/D
¸
M
H
VH
A E I O ; X
F F F P B B
F F F P P P
E E E G G G
E E E G G G
f The designer can use all these factors in a scienti"c way instead of relying on pure judgement. f It not only reveals the speci"c part family that part belongs to, but also provides the degree of relationship of a part associated with each part family. This information would allow users a #exibility in determining to which part family a part should be assigned so that the workload balance among machine cells can be taken into consideration. f The use of the AHP gives a consistent method of "nding the weights of all factors. f This approach determines also the number of cells together with the spatial arrangement of machines to meet the required demand. f This method is independent of the number of parts and the size of the cell and can be adopted in a large scale of the design. Hence FDMS can be easily adapted to the cell formation process as in many other application areas. Statistical analysis (standard deviation) also shows the groupability of data. The "nal grouping e$ciency is strongly related to the standard deviations for matrices encountered in cell formation. It can be concluded that the working range must be between 0.2 and 0.35 [35]. The standard deviation of the similarity matrices in this study is 0.715 which is greater than this range. The reason for this result is that in cell formation only cell number 2 is available and other alternative number of cells formation are not considered. The results of the proposed method are compared to AktuK rk's [1] results for four performance measures which are the machine investment, workload deviation within and between cells and the number of skippings (Table 12). It can be easily seen that the proposed method is better than AktuK rk's while considering machine investment and deviation within cells, spatial arrangement is given as a result of skipping, which clearly indicates the advantage of the proposed method. References
Fig. 8. Membership functions for the score (S).
[1] M.S. AktuK rk, H.O. BalkoK se, Part-machine grouping using a multi-objective cluster analysis, International Journal of Production Research 34 (8) (1996) 2299}2315.
Z. Gu( ngo( r, F. Arıkan / Int. J. Production Economics 63 (2000) 181}193 [2] H.C. Chu, Clustering analysis in manufacturing cell formation, OMEGA 17 (2) (1989) 289}295. [3] C. Moiser, L. Taube, Weighted similarity measure heuristics for the group technology machine clustering problem, OMEGA 13 (6) (1989) 577}583. [4] H. Seifoddini, P.M. Wolfe, Selection of threshold value based on material handling cost in machine-component grouping, IEEE Transactions 19 (3) (1987) 266}270. [5] H. Seifoddini, Duplication process in machine cells formation in group technology, IEEE Transactions 21 (4) (1989) 382}388. [6] J.L. Burbidge, A manual method of production #ow analysis, Production Engineering 56 (1) (1977) 34}38. [7] A. Kusiak, The part family problem in #exible manufacturing systems, Annals of Operations Research 3 (1985) 279}300. [8] A. Kusiak, The generalized group technology concept, International Journal of Production Research 25 (4) (1987) 561}569. [9] D. Rajamani, N. Sing, Y.P. Aneja, Design of cellular manufacturing systems, International Journal of Production Research 34 (7) (1996) 1917}1928. [10] A. Vanelli, K.R. Kumar, A method "nding minimal bottleneck cells for grouping part-machines families, International Journal of Production Research 24 (2) (1986) 387}400. [11] S. Kaparthi, N.C. Suresh, A neural network system for shape-based classi"cation and coding of rotational parts, International Journal of Production Research 29 (9) (1991) 1771}1784. [12] F.F. Boctor, The minimum-cost machine-part cell formation problem, International Journal of Production Research 34 (4) (1996) 1045}1063. [13] H. Opitz, A Classi"cation System to Describe Workpieces, Parts 1 and 2, Pergamon Press, New York, 1972. [14] A. Houtzeel, An introduction to the MICLASS system, Proceedings of Computer Aided Manufacturing International Seminar of CAPP Applications, CAM.1, 1995, pp. 159}178. [15] U. WemmerloK v, N.L. Hyer, Procedures for the part family/ machine group identi"cation problem in cellular manufacturing, Journal of Operations Management 6 (1996) 125}147. [16] N. Singh, Design of cellular manufacturing systems: An invited review, European Journal of Operation Research 69 (3) (1993) 284}291. [17] O.F. O!odile, A. Mehrez, J. Grznar, Cellular manufacturing: A taxonomic review framework, Journal of Manufacturing Systems 13 (3) (1994) 196}220. [18] T.J. Greene, R.P. Sadowski, Cellular manufacturing control, Journal of Manufacturing Systems 2 (2) (1982) 137}144.
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[19] F.O. O!odile, Application of similarity coe$cients method to parts coding and classi"cation analysis in group technology, Journal of Manufacturing Systems 10 (6) (1991) 442}448. [20] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338}353. [21] H.J. Zimmerman, Fuzzy Sets, Decision Making, and Expert Systems, Kluwer Academic Publishers, Boston, MA, 1987. [22] C.C. Lee, Fuzzy logic in control systems: Fuzzy logic control}Parts I and II, IEEE Transactions on Systems, Man and Cybernetics 20 (1990) 404}435. [23] T.J. Ross, Fuzzy Logic with Engineering Applications, McGraw-Hill, New York, 1995. [24] H. Nojiri, A model of executive's decision process in new product development, Fuzzy Sets and Systems 7 (1982) 227}241. [25] J. Grobelny, The fuzzy approach to facilities layout problems, Fuzzy Sets and Systems 23 (1987) 175}190. [26] R.O. Duda, P.E. Hart, Pattern Classi"cation and Scene Analysis, Wiley, New York, 1973. [27] G.W. Evans, W. Karwowski, M.R. Wilhelm (Eds.), Applications of Fuzzy Set Methodologies in Industrial Engineering, Elsevier, Amsterdam, 1989. [28] H.C. Chu, J.C. Hayya, A fuzzy clustering approach to manufacturing cell formation, International Journal of Production Research 29 (7) (1991) 1475}1487. [29] A. Masnata, L. Settineri, An application of fuzzy clustering to cellular manufacturing, International Journal of Production Research 35 (4) (1997) 1077}1094. [30] E.H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man}Machine Studies 7 (1975) 1}13. [31] F. Dweiri, A.F. Meier, Application of fuzzy decision making in facilities layout planning, International Journal of Production Research 34 (11) (1996) 307}325. [32] T.L. Saaty, The Analytical Hierarchy Process, McGrawHill, New York, 1980. [33] T.L. Saaty, L.G. Vergas, The logic of priorities: Applications in business, energy, health, and transportation, Kluwer-Nijho!, Boston, 1982. [34] N. Singh, D. Rajamani, Cellular Manufacturing Systems Design, Planning and Control, Chapman and Hall, London, 1996. [35] M.P. Chandrasekaran, R. Rajagopalan, Groupability: An analysis of the properties of binary data matrices for group technology, International Journal of Production Research 27 (7) (1989) 1035}1052.
Application of fuzzy decision making in part-machine grouping ZuK lal GuK ngoK r*, Feyzan Armkan Industrial Engineering Department, Gazi University, Maltepe 06570, Ankara, Turkey Received 9 October 1997; accepted 25 January 1999
Abstract In this study, fuzzy set theory (FST) is used to set out the cell layout. A new algorithm which will consider both design and manufacturing attributes and operation sequences as factors, is proposed to formulate the problem. The structure of the algorithm is based on fuzzy decision making system (FDMS). Hence three factors mentioned above are determined as input variables and fuzzi"ed using membership function concept. Then the pairwise comparison of the analytical hierarchy process (AHP), which ensures the consistency of the designer&s decisions when assigning the importance of one factor over another, is used to "nd the weights of these factors. Applying IF}THEN decision rules, parts relationship chart (PRC) is generated. After these steps, the traditional cell formation procedure is applied. Finally the proposed method is scored by performance measures such as machine investment, the amount of work load deviations within cell and between cells and the number of skippings. Also the comparison with AktuK rk's study (International Journal of Production Research 34 (8) (1996) 2299}2315) in respect to these performance measures is presented. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy decision making system; Cellular manufacturing
1. Introduction During the last ten years, there has been great interest in the area of group technology (GT). Cellular manufacturing (CM) is an application of GT which allows for small batch production to gain economical advantage over the traditional manufacturing systems. The identi"cation of part families and machine groups in design of cellular manufacturing system is commonly referred to as a cell. In CM, parts with design and manufacturing similarities are grouped into part families and the
* Corresponding author. E-mail address: [email protected] (Z. GuK ngoK r)
associated machines are grouped to form the cells. Such part families are assigned to cells composed by machines and set up in the order of best routine and process sequences of part families. Numerous papers can be found in the literature for cell formation. In this area, the following major approaches to model the problem have been employed: clustering analysis as matrix formation which uses some similarity coe$cient to obtain part family formations [2}5], production #ow analysis [6], mathematical programming [7}9], graphical analysis [10], neural network [11] and simulating annealing [12]. Coding systems like OPITZ [13] and MICLASS [14] are also used to group the parts according to their design attributes. Comprehensive reviews of the cell formation
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 1 0 - 9
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approaches are provided by WemmerloK v and Hyer [2,15}17]. Most of the above-mentioned approaches adopt either a sequential or a simultaneous procedure to group the parts and machines. The sequential procedure used in some of these studies determines the part families "rst, followed by machines assignment. On the other hand the simultaneous procedure determines the part families and machine groups concurrently. In this approach a part/machine matrix in a block diagonal form is obtained. Most of these applications deal only with cell formations. A common measure of performance is the number of exceptional elements. Additional measures, such as minimizing total material-handling costs, duplication costs or maximization of within cell utilization are considered only to avoid exceptional elements. Flow of the materials also is related to operation sequences. The spatial arrangement of machines inside cells and cell layout formations in#uence the e!ectiveness of cells. However there are only a few studies dealing with the spatial arrangements of machines inside the cells. The study of Greene and Sadowski [18] can be given as an example. During the cell formation consideration of only one factor such as similarity coe$cient, as used in many traditional studies, is not enough. All possible variables (factors) that can a!ect the design must be considered in cell design. Because the number, complexity and unclear nature of the variables may in#uence the decision maker's decision. There are many factors to be considered such as similarity of operation sequences, design similarities of the parts, the number of parts in each cell, machines required in each cell, additional machine investment for each cell. O!odile [19] also has addressed the problem of how to form part families using part classi"cation and coding analysis. One common weakness with conventional methods is that they implicitly assume that only disjoint part families exist in the data. That is, a part can only belong to one part family. In practice, however, there may exist some parts whose lineages are much less evident according to their technological and/or geometrical features. Therefore the FDMS provides a more accurate presentation of the problem in the domain of uncertain or
inexact information, and FST is used to express the linguistic variables which describe the quantitative and qualitative factors a!ecting the design of cell formation. The remainder of this paper is organized as follows. In Section 2, a brief introduction to FST is presented. Also a detailed discussion on FDMS is given. In Section 3, we discuss the fuzzy decision making in cell formation in general format. In Section 3.1, the proposed algorithm is explained in a wide perspective. In Section 4, the application of the algorithm to AktuK rk's [1] example is given. Finally, concluding remarks and the comparison to AktuK rk's [1] results are discussed in Section 5.
2. Introduction to fuzzy set theory The FST, introduced by Zadeh [20] to deal with vague, imprecise and uncertain problems, has been used as a modeling tool for complex systems, that can be controlled by humans but are hard to de"ne exactly. A collection of objects (universe of discourse) ; has a fuzzy set A described by a membership functions k that takes the values in the A interval (0, 1), k : ;(0, 1). Thus A can be represented A as A"Mk (u)/u; where u3;N. The degree that u A belongs to A is the membership function k (u). A More detailed discussion of FST can be found in [21}23]. FST has been applied to many production areas. Some of these can be found in [24}27]. In this paper the focus is on fuzzy decision making in the design of cellular manufacturing systems. It di!ers from [28,29] in that they use fuzzy clustering mathematics in manufacturing the part families formation rather than using fuzzy decision making. 2.1. Fuzzy decision making system Fuzzy logic control (FLC), initiated by the pioneering work of Mamdani and Assilian [30], is one of the most active areas in the application of FST. The basic idea is to incorporate the expert knowledge of a human operator in the design of the controller in controlling a process whose input output relationship is described by a collection of fuzzy control rules (e.g., IF}THEN rules) involving
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183
Fig. 1. Fuzzy decision making system (FDMS).
linguistic variables rather than a complicated dynamic model. FDMS consists of four principal components as shown Fig. 1. These four parts are;
value is calculated as
1. The fuzzi"cation interface: Measures the values of the input and output variables, transfers the range of these values into a corresponding universe of discourse and converts them into natural language such as high, low and very low. 2. The knowledge base: A data base that consists of the expert's knowledge of the application and the control rules of the process. The membership functions are determined and fuzzi"cation of the variables is achieved. 3. The decision making logic: It has the capability of simulating human decision making by performing approximate reasoning to achieve a desired control strategy. The rules are mostly in the form of IF}THEN form. The membership value of the control action of each rule is the minimum value of the input variables of membership values [30].
where R is the "nal crisp rating of activity, g the 0 fuzzy rating of the activity for the rule in consideration, i the rules that are used in the activity, R the numerical rating of the activity of the rule and k the membership value of the activity for the rule.
Label l #0/530- !#5*0/ "MinMLabel l l N. */1651(7!-6%),2, */165/(7!-6%)
(1)
4. The defuzzi"cation interface: Converts the fuzzy output into a crisp (nonfuzzy) value. One of the most common methods is the Centre of Area (COA) method [22]. In this method "nal crisp
+ kg .R R " i R , 0 +k R
(2)
An example of application of this type of FDMS is the facilities layout problem given in [31].
3. Fuzzy decision making in cell formation The fuzzy approach to the cell layout problem involves the identi"cation of linguistic variables that describe the quantitative and qualitative factors a!ecting the cell design and the selection and determination of these values and the membership functions of the linguistic variables and the development of heuristic procedures for the selection and placement of machine and parts for each cell. At the beginning of cell formation all related input variables are de"ned, then the values of these variables are determined and the membership functions are developed using expert's knowledge. Weights of factors of these variables are determined using analytical hierarchy process (AHP) which
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is based on pairwise comparisons of the input variables [32]. The basic AHP starts by dividing the decision problem into a hierarchy. Most systems can be modeled in a hierarchical fashion; that is, the model is constructed in several di!erent levels of increasing aggregation or priority, until a top level or objective is reached. Each level contains components with similar properties that allow comparison. The procedure of making pairwise comparisons greatly reduces the conceptual complexity of decision making. The scale comparison is usually taken to be in the range 1}9. These are, basically of equal importance (1), between equal an weak importance of one over another (2), weak importance of one over another (3), between weak and strong importance of one over another (4), strong importance (5), between strong and demonstrated importance of one over another (6), demonstrated importance (7), between demonstrated and absolute importance of one over another (8), absolute importance (9). Using this scale the participating managers assess the dominance of each element over the others with respect to each element of immediate higher level of hierarchy. This process of comparisons yields a relative scale of measurement of priorities or weights (w) of elements. That is, the scale measures the relative standing of the elements with respect to a criterion independently of any other criterion or element that may be considered for comparison. The comparisons are performed for the elements in a level with respect to all the elements in the level above. The "nal or global weights of the elements at the bottom level of the hierarchy are obtained by adding all the contributions of the elements in a level with respect to all the elements in a level above. Once the comparisons matrix is formed, the largest eigenvalue j should be approximately .!9 equal to n (the number of factors). The deviation of j from n provides a measure of consistency .!9 which measures the deviation of the judgements from consistent approximation. The `consistency indexa CI is given by j !n CI" .!9 . n!1
(3)
Consistency ratio, CR is obtained from the ratio of CI to RI (random index). Random index is the
index of the same order matrix, the elements of which are randomly generated [33]. Generally if the CR is less than 0.1, the consistency of the decision maker is considered satisfactory. The vector w is then presumed to be the decision makers optimal weights or priorities. In this study AHP only uses as pairwise comparisons of attributes to determine its factors' weight. These variables are fuzzi"ed using a set of membership functions such as `very lowa, `mediuma, `higha, and `very higha. The other common input factor is closeness rating (R), the rating given to the parts by designer is based on input variables. The rating ranges from `absolutely necessarya to `undesirablea. A closeness rating is assigned to represent the priority of parts collected in the same cell. Parts with similar rating have higher priorities in forming a cell collection. Designer assigns closeness rating (cost) (A"6, E"5, I"4, O"3, ;"2, X"1) of the parts to show the degree of relationship of the parts. The closeness rating is fuzzi"ed by the set of membership functions as in Fig. 2. After the fuzzi"cation processes of all input and output variables are completed, the next step is the establishment of the decision-making logic (decision rules). These rules which initiate the designers' decisions, usually take the form of the IF}THEN rules. Such rules are conveniently tabulated in look-up tables as shown in Tables 1}3. Using these tabulated rules, the minimum operator from Eq. (1), and the defuzzi"cation COA method in Eq. (2), the designer can "nd the crisp (exact) values of the parts relationship chart (PRC). PRC is the similarity coe$cient matrix which is used to develop the physical layout of cells in the plant, in traditional procedures. FST is used in generating the PRC which is the basis of cell layout. 3.1. The proposed algorithm In the process of cell layout design, variables (factors) that a!ect the design are mostly design and manufacturing features, operation sequences and operation times. The design similarities are important in the part design, manufacturing and purchasing phases, whereas the operation sequences are important factors in determining the spatial arrangement of machines inside the cells. The design
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185
Fig. 2. Membership functions of closeness rating.
Table 1 IF}THEN rules of factor 1 (F1) and its weight factor (WF)
Table 3 IF}THEN rules of factor 3 (F3) and its weight factor (WF)
F1/WF
VL
¸
M
H
VH
F3/WF
VL
¸
M
H
VH
VL ¸ M H VH
; ; O ; I
; ; O ; E
O O I O A
O O E E A
I E E A A
VL ¸ M H VH
; ; O I I
; O I I E
O I I E E
I I E E A
I E E A A
Table 2 IF}THEN rules of factor 2 (F2) and its weight factor (WF) F2/WF
VL
¸
M
H
VH
VL ¸ M H VH
; ; O I I
; ; I I E
; O I E E
O O E E A
O I E E A
and manufacturing similarities and operating sequences and operating times of parts in each cell are represented by such vague issues as the rating qualitative or quantitative relationships in each cell ranges from `absolutely necessarya to `undesirablea. After deciding on the variables that can a!ect the design, fuzzy set theory is used as the suitable theory to this process due to the unclear nature of the factor similarities of parts that are placed in the
same cell. Then, values of these variables are determined, the universe of discourse and the membership functions are de"ned. Finally, the following algorithm as shown below is developed. Step 1: Find the factors that a!ect most the design of the cells. Tabulate all the necessary information for the factors. De"ne the number of identi"cations similarities of these factors between each pair of parts. Using AHP, the weight of each factor is found. If the result of the AHP is consistent and acceptable (consistency ratio and maximum eigen value are smaller than 0.1 and about n, respectively), the weight of factors is considered one of the input factors that will in#uence the layout (Eq. (3)). The weights of the factors are fuzzi"ed. This process is repeated for all pairs of parts. Step 2: The next step is to fuzzify the input and output factors that a!ect the cell formations. Membership functions are developed with interviews of related people. The maximum rule of fuzzi"cation
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process, which can be explained by using the maximum membership value of a factor to "nd the fuzzy set (label), is employed for all parts. Step 3: The third step is to apply the IF}THEN decision rules developed by a designer as the result of the fuzzi"cation interface of the fuzzy decision making process. As shown in Tables 1}3 all input variables are tabulated. Step 4: Using the minimum operator rule (Eq.(1)) "nd the related rating and weight of each pair of parts. Step 5: The "nal step of the fuzzy decision making process is the defuzzi"cation interface. This step uses COA Method to calculate the "nal crisp value of PRC (Eq. (2)). This process is repeated for each pairs of parts. After obtaining PRC the following steps are employed. Step 6: Normalize the PRC which is in a matrix form, to have the value of unit measures. N "A /(A !A )2, (4) ij ij kj lj where A represents the greatest value of the kj matrix, A represents the lowest value of matrices lj and i, j show the part number. Step 7: Apply one of the clustering algorithms such as Single Linkage Clustering (SLC) as used by Singh and Rajamani [34]. This algorithm is summarized as follows. Find the similarity coe$cient of the parts (in this study PRC is used as the similarity coe$cient matrix as mentioned before) and determine the maximum value of the matrix. Two parts which correspond to the maximum value of the matrix are joined to obtain a new group. Similarity between the new group and remaining groups is computed by taking the maximum value of pairs. At the next step, again join the parts which have the maximum value. Repeat the same procedure to "nd the group for all parts. Step 8: De"ne the threshold value to obtain the necessary number of cells. Step 9: Spatial arrangement of machines in each cell for the remaining candidate alternatives are found by using the Computerized Relative Allocation of Facilities Technique (CRAFT) which is the most widely applied computerized layout program. Step 10: Calculate the requirement of each machine type in each cell by considering the demand requirements and capacity restrictions.
Step 11: The total machine investment, the amount of workload deviations within and between cells, and the number of skippings are calculated using the following equations: I"+ + X .C , (5) kj k j k|M where I is the total machine investment, C is the k "xed cost of machine type k, X is the number of kj machines of type k assigned to cell j. D"+ + (¹¸ !A¸ )2/n , (6) j j j j k|M where D is the workload deviation within a cell, ¸ the expected load of machine type k in cell j, A kj lj the average load in cell j, n the total number of j machines in cell j and M the set of all machine types. B"+ (¹¸ !E¸)2/N, (7) j j where B is the workload deviation between cells, ¹¸ the total load in cell j, EL the average load of j cells and N the number of cells. Step 12: Repeat all steps 9}11 for all de"ned cells determined in step 8. Select the best e!ective cell formation.
4. The problem de5nition An example problem is taken from AktuK rk's study [1] to illustrate the proposed fuzzy cell formation heuristic. The example considers 20 parts and each part requires at most 5 and at least 3 machine types selected from 10 machine types. In this study there are 9 design and manufacturing attributes coded in a polycode structure as summarized in Table 4 along with the operation sequences, processing times on each machine and the demand forecasts for each part. Each symbol in an operation sequence corresponds to a machine type that will perform the given operation. Three of the design and manufacturing attributes are selected with respect to the shape of the parts, and others are design properties of parts such as proportion of diameter to lengths, measure of the cogwheel, length and width, tolerance and seconder hole. The
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187
Table 4 Part, design and manufacturing attributes, routings and demand data Parts no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Operation sequences
B A F F A A F K A A A B M A F A A A F G
F B K G M B G A K K B F R R H B B B G R
G F M H T L R R M R R R T T R M F F R T
L R R M T L T T R T T T L
L T T G T
Design and man. attri.
Ope. times sequence num.
1 2 3 4 5 6 7 8 9
1 2 3 4 5
0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0
2 2 2 3 2 2 3 2 3 3 2 3 3 1 1 2 3 5 4 1
1 3 2 5 0 1 4 1 5 1 2 1 4 1 3 1 2 0 3 3
2 5 1 2 0 1 3 0 3 0 0 3 3 1 2 0 3 0 2 3
6 1 3 6 3 6 0 8 7 7 5 5 6 8 5 3 3 0 5 6
4 0 8 8 4 3 4 5 3 5 5 4 4 4 7 2 0 5 1 1
3 3 3 2 3 1 3 3 2 3 3 2 2 0 2 3 3 2 4 2
Table 5 Machine investment costs B
2 5 4 2 0 1 0 0 3 1 2 3 5 0 2 2 4 0 2 2
3 5 1 6 6 2 2 4 3 4 4 3 2 5 6 5 2 0 3 4
1 3 0 2 0 2 3 2 0 2 1 1 2 2 4 3 1 1 2 1
3 3 3 2 3 3 2 3 1 2 3 2 2 3 2 3 2 4 1 1
2 4 4 4 4 2 3 1 4 2 4 1 4 2 1 2 3 2 1 2
Average demand
150 226 335 446 274 171 218 273 307 414 223 378 328 280 270 182 244 152 366 226
3 2 2 1 5 1 2 2 2 3 1 1
1 3 4 2 1
Table 6 Similarities of design and manufacturing
Type of machine
A
F
G
H
K
L
M R
T
Cost of machine
106 136 65 14 103 61 126 93 94 70
investment costs of related machines are given in Table 5. Before the illustration of the algorithm, it is worthwhile to explain the factors that a!ect the cell design: the similarities between parts depend on qualitative and quantitative factors. Some factors may have greater e!ect on the designer's decisions. Hence three factors are selected. These are similarities of design and manufacturing (F1), similarities of operation times (F2) and similarities of operation sequences (F3). 1. Similarities of design and manufacturing (F1): The design similarities are important in the part design, manufacturing and purchasing phase. The design di!erences between parts of production
1 2 3 4 5 . . 20
1
2
3
4
1 1 4 2
2 0 2
1 4
1
3
0
0
4
5 222222 20
02
4
must be minimum in order to create cell design with similar parts. When di!erences of parts are reduced in the same cell, machine set up times are reduced and many other factors will be better than before. The rating of these relationships ranges from absolutely necessary to undesirable. Experts can give this variable some linguistic values such as high, low, etc. The number of identi"cation similarities between each pair of parts (Table 6) are fuzzi"ed as in Fig. 3. For
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188
Fig. 3. Membership functions of factor 1. Table 8 Similarities of operation sequences of parts
Table 7 Similarities of operation times of parts 1 1 2 3 4 5 . . 20
2
3
4
2 2 0 2
3 1 3
2 3
1
1
0
0
0
5 22222220
0 2222221
1 2 3 4 5 . . 20
1
2
3
4
2 1 2 0
1 1 1
2 1
0
1
0
1
2
5 22222220
1 2222223
Fig. 4. Membership functions of factor 2.
instance, parts 1 and 2 have only one common character at identi"cation number 6 and, parts 3 and 2 have only two common characters at identi"cation numbers 1 and 6. This process is repeated for all pairs of parts. 2. Similarities of operation times (F2): In order to organize the machine utilization and cell utilization in each cell and between cells, similarities of operation times are also important. Imbal-
ance of workload in each cell will cause a high level of work in process inventories. Machine utilization and cell utilization are measures of load imbalance within a cell and the amount of workload deviation within and between cells will show the cell utilization. Similarities of operation times are given in Table 7. Experts also fuzzi"ed this variable as high, low, etc. as shown in Fig. 4.
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3. Similarities of operation sequences (F3): Parts used in the same machine must be replaced in the same cell. Di!erences in the operation sequences will be minimized and spatial arrangement of machines inside the cells are performed. This factor is de"ned by the number of similarities of operation sequences of parts. For instance (Table 8), parts 1 and 2 have only 2 similar processes and this process is repeated for all pairs of parts. If there are inverse operation sequences, it must be taken into consideration. These are fuzzi"ed as in Fig. 5.
4.1. Methodology of solution
1. After selection of the most important factors, we assume that the designer collects all the information tabulated in Tables 4 and 5. Using the AHP process, the weight of each factor is found. For example, if the designer is assigning a value of 3 for the importance of factor 1 over factor 2. This means that there is a range between an equal and a weak importance of factor 1 over factor 2. A value of 5 assigned for the importance of factor 1 over factor 3 indicates a strong importance of factor 1 over factor 3. The result of weight factors calculation using AHP are given in Table 9. The result of AHP is acceptable (CR"0.0025 and maximum eigenvalue" 3.001). The weight of the factors is considered to be one of the input factors that in#uence the cell formations and is fuzzi"ed as in Fig. 6.
189
2. All three factors that a!ect the cell design are fuzzi"ed. Maximum rule of the fuzzi"cation process and Figs. 4}7 are used. Results of the information about the relation between parts 1 and 2 are shown below. Factor 1 has a value of 1 (Table 6); it belongs to the fuzzy subset low (L) with a membership value of 1 (Fig. 3). Its weight factor (WF) has a value 0.640 (Table 9) which belongs to the fuzzy subset Very High (VH) with a membership value of 1 (Fig. 6). Factor 2 has a value of 2 (Table 7); it belongs to the fuzzy subset Medium (M) with a membership value of 1 (Fig. 4). Its weight factor (WF) has a value 0.148 (Table 9) belongs to the fuzzy subset Low (¸) with a membership value 0.5 (Fig. 6). Factor 3 has a value of 2 (Table 8); it belongs to the fuzzy subset Medium (M) with a membership value of 0.87 (Fig. 5). Its weight factor 0.212 (Table 9) belongs to the fuzzy subset Low (¸) with a membership value 0.75 (Fig. 6).
Table 9 Results of AHP
F1 F2 F3
F1
F2
F3
W
1 1/3 1/5
5 1 2
5 1/2 1
0.640 0.148 0.212
Note: CR"0.0025, CI"0.001, maximum eigenvalue (j )" .!9 3.001.
Fig. 5. Membership functions of factor 3.
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3. This process is completed for all pairs of parts. Then IF}THEN decision rules are applied to the results. These rules are: (a) IF (F1) is (¸) and its (WF) is (VH) THEN the rating (R) is (E) (Table 1). (b) IF (F2) is (M) and its (WF) is (¸) THEN the rating (R) is (I) (Table 2). (c) IF (F3) is (M) and its (WF) is (¸) THEN the rating (R) is (I) (Table 3). 4. Using the minimum operator (Eq. (1)) rule 1, rating from Fig. 2 (E"5) has a membership value of the minimum of 1 and 1. Thus, rule 1 results in a rating of (I"4) with a membership value of 1, rule 2 results in a rating of (I"4) and with a membership value of 0.75, and rule 3 results in rating (I"4) with a membership value 0.5. The "nal step of fuzzy decision-making process is the defuzzi"cation interface. This step uses COA method to calculate the "nal crisp value (Eq. (2)). The relationship between parts 1 and 2 is shown below: PRC"((5)(1)#(4)(0.75)#(4)(0.5))/ (1#0.75#0.5)"4.44.
Fig. 6. Membership functions of the weight factor (WF) of the inputs.
Again this process is repeated for all parts. The new PRC as a result of FST is shown below in Table 10. Cells formation is developed using the PRC. The values of only the "rst "ve rows and columns and the 20th row are presented in Tables 7 and 8 due to space limitations. In, Table 10, the values of the "rst 10 rows and the 20th row are presented for similar reasons. The second part of the algorithm is applied as a traditional cell formation procedure. 5. Normalize the PRC matrix by using Eq. (4). Apply Single Linkage Clustering algorithm, "nd the number of cells. If the threshold value is taken as 0.88, part family formation is not valuable since two cells are formed and the remaining parts do not belong to any cell. These parts of each cell are (1, 13, 4, 6, 19, 12), (17, 16, 2, 11, 3), (5), (7), (8), (9), (10), (14), (15), (18), (20). If the threshold value is taken as 0.86 two cells are formed as shown Table 11. 6. The spatial arrangement of machines are found by using the CRAFT algorithm for the initial part families shown in Table 11. 7. Compute the required number of each machine type in each cell by considering the demand requirements and capacity restrictions. Initially, parts 18 and 20 are assigned to cell number 1, but they use machine type G. In order to minimize the cost of total machine investment they are moved to cell number 2, since it does not require any additional machine investment of G. Parts 10 and 8 have reverse operations in the same cell. It requires additional machine investment of machine type A. But the requirement of duplication of machine investment costs in the same cell
Fig. 7. Membership functions of deviation within cell.
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Table 10 Parts relationship chart (PRC)
1 2 3 4 5 6 7 8 9 10 . . 20
1
2
3
4
5
6
7
8
4.44 4.44 5.20 4.53 4.75 5.20 4.53 5.00 4.00
4.64 2.80 4.16 3.60 4.63 4.40 3.78 4.67
4.44 4.75 4.44 4.67 4.40 4.64 4.40
3.71 6.00 4.64 3.78 4.40 4.63
4.00 5.20 4.75 3.93 4.75
3.69 4.40 5.00 4.40
4.63 3.78 5.67
4.00 4.00
3.63
0
4.00
4.00
5.20
4.00
3.69
5.00
9
1022222 20
4.6722222222 4.25
Table 11 Part families and cells layout Cells
Cell number
Part family
Cell layout
2
1 2
8, 10, 9, 17, 16, 11, 3 5, 7, 1, 13, 4, 6, 19, 12, 14, 15, 20, 2, 18
A, B, F, K, A, M, R, T, T A, B, F, F, G, H, R, L, T, M, R, T
Table 12 Comparison of the proposed method and AktuK rk's algorithm
M/C investment Within cell deviation Between cell deviation Total number of skippings
Fuzzy decision making layout
AktuK rk's algorithm
1963 495 53 5879
2150 602 38 9894
is already acceptable instead of replacing these two parts in a separate cell. Another reason is that these two parts are processed by machine type K which is placed only in cell number 1. 8. The total machine investment (Eq. (5)), the amount of work load deviations within cells (Eq. (6)) and between cells (Eq. (7)), and the number of skippings are calculated as shown in Table 12.
5. Conclusion In this study, the cell formation problem is studied by using FDMS in the mean time consider-
ing the design and manufacturing attributes and operation sequences. FST comes in handy because of the imprecise structure of qualitative and quantitative factors which a!ect the cell design directly. Herein a new alternative fuzzy approach to design cells is proposed. After some numerical explanations, the advantages of this FDM approach over conventional clustering will be summarized. After forming cells, a score must be given to decide that it has an acceptable value. In the evaluation, two input factors which are PRCs and the workload deviations within cells, and one output factor as score of the output are used. The scoring process follows the FDMS. PRC is fuzzi"ed as in
Z. Gu( ngo( r, F. Arıkan / Int. J. Production Economics 63 (2000) 181}193
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Fig. 3 and deviations within cells is fuzzi"ed as in Fig. 7. Overall parts relationship is 4.39. The rating is considered an input variable for the scoring process and 4.39 belong to the rating `Ia with the membership value of 0.87 which is found by using the maximum operator rule. The workload deviation within cell belongs to ¸ with a membership value of 0.95. IF}THEN procedure is established as in Table 13. The output variable (score) must also be fuzzi"ed such as Good, Bad, etc., using a set of membership functions (Fig. 8). The "nal crisp score is then transferred into a linguistic variable using the maximum operator rule. The "nal output crisp score is 3.28 which belongs to the linguistic variable FAIR with a membership value of 0.58 (Fig. 8). It is recommended to accept this score as the preferable cells layout. Besides these results mentioned above, the summary of the advantages of the FDM approach are given below: f It allows the designer to use all possible factors a!ecting the layout regardless of their qualitative or quantitative nature.
Table 13 IF}THEN rules of the score (S) for the rating (R) and deviation (D) R/D
¸
M
H
VH
A E I O ; X
F F F P B B
F F F P P P
E E E G G G
E E E G G G
f The designer can use all these factors in a scienti"c way instead of relying on pure judgement. f It not only reveals the speci"c part family that part belongs to, but also provides the degree of relationship of a part associated with each part family. This information would allow users a #exibility in determining to which part family a part should be assigned so that the workload balance among machine cells can be taken into consideration. f The use of the AHP gives a consistent method of "nding the weights of all factors. f This approach determines also the number of cells together with the spatial arrangement of machines to meet the required demand. f This method is independent of the number of parts and the size of the cell and can be adopted in a large scale of the design. Hence FDMS can be easily adapted to the cell formation process as in many other application areas. Statistical analysis (standard deviation) also shows the groupability of data. The "nal grouping e$ciency is strongly related to the standard deviations for matrices encountered in cell formation. It can be concluded that the working range must be between 0.2 and 0.35 [35]. The standard deviation of the similarity matrices in this study is 0.715 which is greater than this range. The reason for this result is that in cell formation only cell number 2 is available and other alternative number of cells formation are not considered. The results of the proposed method are compared to AktuK rk's [1] results for four performance measures which are the machine investment, workload deviation within and between cells and the number of skippings (Table 12). It can be easily seen that the proposed method is better than AktuK rk's while considering machine investment and deviation within cells, spatial arrangement is given as a result of skipping, which clearly indicates the advantage of the proposed method. References
Fig. 8. Membership functions for the score (S).
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