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Fuzzy decision support system for manufacturing
facilities layout planning
S.K. Deb
a,*, B. Bhattacharyya
b
a (^) Jorhat Engineering College, Jorhat 785007, India b (^) Production Engineering Department, Jadavpur University, Kolkata 700032, India
Received 1 November 2002; accepted 1 December 2003 Available online 28 March 2004
Abstract
Manufacturing facility layout problem is an unstructured decision-making problem due to natural vagueness associated with the inputs to the models. Arbitrary numerical ratings are assigned for relationship chart to determine facility selection routine. This paper presents a distinct decision support system based on multifactor fuzzy inference system (FIS) for the development of facility layout with fixed pickup/drop-off points. The algorithm searches several candidate points with different orientation of incoming machine blocks in order to minimize flow cost, dead space and area required for the development of layout. The proposed methodology is coded in C +^ language and implemented in a Pentium III, 550-MHz machine. The experimental results with a test problem are illustrated with encouraging result with its advanced soft computational effectiveness. D 2004 Elsevier B.V. All rights reserved.
Keywords: Facility layout; Fuzzy decision; Flow cost; Dead space; Minimum required area
- Introduction
The most significant objective of any enterprise has been the maximum utilization of facilities available to achieve desired goal of productivity and profitability. Two-dimensional facility layout deals with the selec- tion of most appropriate and effective arrangements of departments in the continuous plane to allow greater working efficiency [2,3]. Owing to the complex and unstructured nature of facility layout, many research-
ers have proposed various approaches, which had varying degrees of success in dealing with the com- plexities associated with the problem. Regardless of the type of data, there is an element of vagueness or fuzziness in it [6]. Traditional layout method treats these data as exact and cannot satisfy the desire of managers in handling real problems [12]. Kawasaki and Evans [9] illustrated the potential application of fuzzy set theory to various areas of production management. One of the prominent areas identified by the authors was facility planning which includes such problem as facilities layout design. Raoot and Rakshit [10] have also presented a frame- work of an algorithm for the development and eval- uation of a layout based on fuzzy linguistic variables
0167-9236/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2003.12.
- Corresponding author. c/o Dr. K. Patra, Department of Mathematics, Guahati University, Guwahati 781014, India. E-mail address: [email protected] (S.K. Deb).
www.elsevier.com/locate/dsw
Decision Support Systems 40 (2005) 305 – 314
and their fuzzy relation. The integrated expert system approach of Abdou and Dutta [1] determines the movement of material-handling equipments first to determine its effect on the layout. Recently, Dweiri [6] proposed a fuzzy decision- making application for developing relationship charts and comparing the layouts generated with them. The concept has been applied to develop layout in the line of computer-relative layout programme that was not efficient as compared to conventional layout proce- dure. He suggested further research work to improve the procedure or developing a new algorithm for the facility layout design. The facility layout problems fall in to the class of NP-complete solutions, and heuristic approaches are usually adopted to develop the layout [8]. Most of the models and algorithms available in the literature are based on the quadratic assignment problem with an objective to minimize transportation costs or maxi- mize total closeness rating. The facility selection routine required for the development of layout was solved by considering a single quantitative factor as flow chart. Moreover, the move (distance) traversed is considered from center to center of the departments without considering the practical issue of entry and exit of the departments. Most of the existing methods have been devel- oped on the grid-based system without considering actual dimensions of departmental block and entry/ exit locations, thus resulting in irregular shapes. Deb et al. [3] have already developed a hybrid modeling for the management of material-handling equipment selection planning while generating a manufacturing facility layout. The authors herein have also proposed different projects of integrating facility layout and material-handling equipment se- lection by using a knowledge base and optimization approach [5]. They utilized the material flow inter- action matrix in finding the facility placement sequence. The authors have already developed a decision model and algorithm for material-handling equipment selection routine under facility layout planning by using fuzzy multi-criteria decision-mak- ing methods [4]. Therefore, the present research work follows in the same direction of author’s previous work to integrate various linguistic assessments to evaluate facility selection routine and its impact on the
development of facility layout. The applicability of the suggested methodology is demonstrated with a six-machine layout problem considering subjective factors such as supervision, information and envi- ronmental condition. The material flows between the different departments are assumed as the objective factor for the development of selection routine. The heuristic search algorithms proposed in this paper take care of optimal placement of incoming facilities based on a multi-criteria optimization function. The performance of the proposed multifactor fuzzy facil- ity selection routine is compared with the multifactor normalized facility selection routine for the develop- ment of facility layout under the auspices of a manufacturing environment.
- Fuzzy set and decision-making system
A fuzzy set can be thought of a class of concepts/ objects in which no well-defined boundary exists between the concepts/objects that belong to the class and those which do not belong. Formally, if B={x (^) i | i qN} is a set of objects, then the fuzzy set C on B is defined by its membership function fC (x) that assigns to each element x qB a real number in the interval [0,1] which represents the grade of membership of x in C or the degree to which x belongs to C. Thus, C can be written as
C fC x x AxeB ; B 0 1
Linguistic variables are words in natural language, while numerical variables use numbers as values. Since words are usually less precise than numbers, linguistic variables provide a method to characterize complex systems that are ill structured to be de- scribed in traditional quantitative terms. A linguistic variable is defined by the name of the variable x, and the set term S(x) of the linguistic values of x with each value being a fuzzy number defined on U. For example, if material flow (MF) is a linguistic vari- able, its term set S(MF)={Very High (VH), High (H), Medium (M), Low (L), Very Low (VL)}, where each term is characterized by a fuzzy set in a universe of discourse U.
etc.). The values associated with different linguistic variables used in the formulation of proposed FIS are
(i) material flow (very high, high, medium, low, very low), (ii) supervision link (negligible, considerable, mod- erate, essential, very essential), (iii) environmental link (very safe, safe, unsafe, hazardous, very hazardous), and (iv) information link (very strong, strong, medium, weak, very weak).
The crisp output of the FIS measures the rating with standard associated values usually in practice [2,12] under facility layout planning X, U, O, I, E and A within the universe of discourse [0,6]. The universe of dis- course and set of the grades of membership were developed within the existing knowledge and experi- ence of facility layout designers using the subjective approach, which is in line with the view of Zadeh [13], who indicated that the grade of membership are sub- jective, in the sense that their specification is a matter of definition rather than experimentation. The shape of the membership function reflects the expert’s knowledge, experience and preference regarding the importance of different relationships (‘sharp’ slope for important relationship and ‘flat’ slope for less important relation- ships). The membership function of each linguistic value in the crisp output rating set R=[X,U,O,I,E,A] is shown in the following expressions:
X : 0 0 0 0 1 0
f x 1 x 0 VxV 1 0
U: 0 1 0 1 0 2 0
f x
x 0 VxV 1 0
2 x 1 0 VxV 2 0
O: 1 0 2 0 2 0 3 0
f x
x 1 1 0 VxV 2 0
3 x 2 0 VxV 3 0
I: 2 0 3 0 3 0 4 0
f x
x 2 2 0 VxV 3 0
4 x 3 0 VxV 4 0
E: 3 0 4 0 4 0 5 0
f x
x 3 3 0 VxV 4 0
5 x 4 0 VxV 5 0
A: 4 0 5 0 5 0 6 0
f x
x 4 4 0 VxV 5 0
6 x 5 0 VxV 6 0
In decision rules module, the expert’s decision- making ability is simulated based on a fuzzy concept. The entire knowledge of the decision maker is stored
Fig. 2. Multifactor fuzzy inference system for facility layout.
as rules in the knowledge base of the FIS. The development of rules may be time consuming. An intuitively developed strategy for finding the rating of each move based on the values of their relationships can be summarized as follows:
(i) If the material flow (MF) relationship between two facilities is very high, then they should be located very close to each other, i.e., rating given is ‘A’. (ii) If the supervision link (SL) between two facilities is very high, then they should be located very close to each other, i.e., rating given is ‘A’. (iii) If the environmental link (EL) between two departments is very hazardous, then they should be located very far to each other, i.e., rating given is ‘U’. (iv) If the information flow (IF) between two facilities is very strong, then they should be located very close to each other, i.e., rating given is ‘A’.
The mapping of the inputs to the outputs for a fuzzy system is in part characterized by a set of condition action rules in the form of IF– THEN. The connective ‘and’ is implemented as a fuzzy conjunction in a Cartesian product space in which the input variables take on their respective universe of discourses. For this study, the [minimum] operator will be used. The membership value of the control action of each rule is the minimum value of the input variables’ membership values. In this paper, multi- input, single-output (MISO) is considered under heu- ristic design rules in the following form:
IF MF is VH and IF is VH
THEN rating is A
IF MF is H and EL is hazardous
THEN rating is The values of linguistic variables are considered within a designed weighing scale [0, 10] with levels [VL, L, M, H, VH], and the values of the output rating are designed within a weighing scale [0,6] with generally accepted levels [X,U,O,I,E,A]. The triangular membership function is considered for objective variable material flow and rating score. The subjective variables supervision link, informa-
tion link and environmental link are considered as trapezoidal membership functions. The number of rules (N) used in controlling the system using fuzzy control is represented by:
N
^ m
j 1
j
n i 1 L (^) i
where, m = number of set of rules, L (^) i = number of membership functions or levels, N = number of input variables used in one set of rules. When m = 1, n = 4 and L (^) i = 5, then number of rules (N) becomes: 5 5 5 5 = 625. The following steps are established to find the selection routine of facilities in an open field:
- Prepare the input values for all moves (activities). The total activities are n (n 1), where n is number of departments.
- Find the minimum values of the input variables’ membership values using minimum operator for every activity.
- Scan the heuristic design rules for all the moves, and find the crisp output of the rating (R (^) ij^ c) by using center-of-area (COA) method.
- Determine the fuzzy rating matrix as R (^) n*n = [Rij^ c^ ]b i,j = 1,2 n and Rij^ c^ = 0 for i = j.
- Calculate the total fuzzy rating of the ith department with the other departments and find the maximum value ( Fk) to select the first facility as department ‘k’.
F (^) i
j
Rc ij Rc ji b i j 1 2... n
F (^) k max Fi b i 1 2... n
- Next, find the department that has maximum fuzzy rating value with the facility already included in the selection routine.
- Repeat step 6 until all facilities are included in the selection routine.
3.2. Facility placement routine
Heuristic is deterministic and hence suboptimal. Not that this is a bad thing. The generalized QAP facility layout formulation is NP complete. Indeed, the heuristic is novel and provides an interesting approach
unit flow cost coefficient (i.e., c (^) ij = 1) and unit penalty cost coefficient for the dead space (i.e., Pc = 1). The penalty cost coefficient is defined as the cost of dead space per unit area that is considered as a parameter, which varies from place to place.
3.3. Steps of algorithm for facility placement routine
- Find the facility selection routine applying multifactor FIS.
- Locate the first block at the center of the open plane horizontally.
- Select the next block for placement according to placement routine of facilities.
- Select the candidate point, and check the feasible quarter. If not feasible, go to step 7, else go to next step.
- Locate the block according to placement possibil- ities, and check for nonoverlapping. If not satisfied, repeat next possible placement as explained in Section 4, else go to step 6.
- Calculate the value of objective function, if it is better than previous update configuration and objective function value. Go to step 5 for searching other possibilities at the candidate point.
- Select next candidate point. If all candidate points of the selected block are already considered, go to step 8; otherwise, go to step 4.
- Select the next block. If all blocks are selected, go to step 9, else go to step 3.
Table 2 Fuzzy system input data and output rating
Move number
MF SL EL IL R^ c (^) ij^ R^ n (^) ij
1-2 1 5 9 10 3.00 0. 2-1 5 8 2 2 3.00 0. 1-3 2 3 8 5 1.00 0. 3-1 2 2 7 6 1.99 0. 1-4 1 1 6 8 1.00 0. 4-1 4 6 1 9 3.00 0. 1-5 2 8 5 2 3.00 0. 5-1 1 3 4 5 1.99 0. 1-6 3 5 2 8 3.00 0. 6-1 0 7 3 6 3.00 0. 2-3 1 2 5 7 1.99 0. 3-2 3 2 6 8 1.99 0. 2-4 2 9 9 5 3.00 0. 4-2 0 1 0 0 1.04 0. 2-5 1 2 1 2 1.09 0. 5-2 2 3 4 3 1.99 0. 2-6 2 4 6 9 3.00 0. 6-2 2 6 5 4 3.00 0. 3-4 3 8 3 3 3.00 0. 4-3 0 0 0 0 1.04 00 3-5 2 1 5 6 1.99 0. 5-3 0 0 0 0 1.04 00 3-6 1 1 2 2 1.99 0. 6-3 0 2 0 0 3.00 0. 4-5 1 1 1 5 1.00 0. 5-4 5 8 9 9 3.00 0. 4-6 2 5 6 1 3.00 0. 6-4 2 8 8 6 3.00 0. 5-6 1 3 5 4 1.99 0. 6-5 10 9 7 3 1.99 0. MF = material flow, SL = supervision link, EL = environmental link, IL = information link.
Table 3 Fuzzy crisp activity relation matrix Machines M1 M2 M3 M4 M5 M
M1 – 3.00 1.00 1.00 3.00 3. M2 3.00 – 1.99 3.00 1.09 3. M3 1.99 1.99 – 3.00 1.99 1. M4 3.00 1.04 1.04 – 1.00 3. M5 1.99 1.99 1.04 3.00 – 1. M6 3.00 3.00 3.00 3.00 1.99 –
Table 4 Multifactor-normalized relation matrix Machines M1 M2 M3 M4 M5 M M1 – 0.20 0.16 0.13 0.16 0. M2 0.18 – 0.17 0.01 0.11 0. M3 0.16 0.21 – 0.15 0.12 0. M4 0.19 0.05 00 – 0.38 0. M5 0.11 0.17 00 0.13 – 0. M6 0.12 0.16 0.07 0.20 0.16 –
Fig. 4. Membership function for material flow.
- Locate the block, which provides the best value of objective function.
- Experimentation and results
The experimentation was carried out in order to investigate the applicability and effectiveness of the proposed methodology based on multifactor fuzzy inference system, and the results have been com- pared with the facility selection routine developed as an extension of the multifactor plant layout method- ology by Harmonosky and Tothero [7]. The algo- rithm was coded in C +^ language, and the problem was run on an IBM Pentium III, 550-MHz machine. The data table related to machine dimensions, P/D locations and move characteristics used for computer simulation having six machines, 30 moves under the consideration of four influencing factors are taken from earlier research work [3– 5] and listed in Tables 1 and 2. The values of subjective variables supervi- sion link (SL), information link (IL), environment link (EL) and objective variable material flow (MF) are taken arbitrarily within designed weighing scale [0,10]. The values of crisp output rating (R (^) ij^ c^ )
obtained by applying the proposed methodology and normalized rating obtained by applying the methodology of Harmonosky and Tothero [7] for each move are presented in Table 2. The fuzzy activity relation matrix and normalized activity rela- tion matrix are shown in Tables 3 and 4 respectively. Membership functions of material flow, supervision link and score are shown in Figs. 3– 5. A triangular membership function is considered for the material flow and score (Fig. 6). For other variables, trape- zoidal membership functions are considered. The values of material flow cost, minimum required area and dead space for different selection routine are presented in Table 5. Figs. 7 and 8 show the layouts developed by using the proposed algorithm under multifactor fuzzy inference system and multifactor
Fig. 5. Membership function for supervision link.
Fig. 6. Membership function for output variable score.
Table 5 Experimental results using fuzzy and normalized approach Facility layout selection routine
Methods applied
Flow cost (T-m)
Minimum required area of layout
Dead space (m 2 )
Value of fuzzy score 1-2-6-4-5-3 (A) Multifactor normalized
2998 17280 7580 7.
6-4-2-1-5-3 (B) Multifactor fuzzy method
3015 17199 5499 7.
Fig. 7. Layout based on multifactor fuzzy selection routine.
References
[1] G. Abdou, S.P. Dutta, An integrated approach to facilities layout design using expert system, International Journal of Production Research 28 (1990) 685 – 708. [2] J.M. Apple, Plant Layout and Material Handling, Wiley, New York, 1977. [3] S.K. Deb, B. Bhattacharyya, S.K. Sorkhel, Management of machine layout and material handling system selection using hybrid approach, 1st International Conference on Logistic and Supply Chain Management, PSG Tech, India, 2001. [4] S.K. Deb, B. Bhattacharyya, S.K. Sorkhel, Material Han- dling Equipment Selection by Fuzzy Multi-Criteria Decision Making Methods. Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin, 2002. [5] S.K. Deb, B. Bhattacharyya, S.K. Sorkhel, Facility layout and material handling equipment selection planning using hybrid methodology, International Journal of Industrial Engineering 10 (3) (2003 September) 436 – 443. [6] F. Dweiri, Fuzzy development of crisp activity relationship charts for facilities layout, Computer and Industrial Engineer- ing 36 (1999) 1 – 16. [7] C.M. Harmonosky, K. Tothero, Multi factors plant layout methodology, International Journal of Production Research 30 (1992) 1773 – 1789. [8] S.S. Heragu, A. Kusiak, Machine layout: an optimization and knowledge-based approach, International Journal of Produc- tion Research 28 (1990) 615 – 635. [9] W. Kawasaki, G.W. Evans, A layout design heuristic employ- ing theory of fuzzy set, International Journal of Production Research 25 (1987) 1431 – 1450. [10] A. Raoot, A. Rakshit, A linguistic pattern approach for mul- tiple criteria facility layout problems, International Journal of Production Research 31 (1993) 203 – 222. [11] T.L. Saaty, The Analytical Hierarchy Process, McGraw-Hill, New York, 1980.
[12] J.A. Tompkins, J.A. White, Facilities Planning, Wiley, New York, 1984. [13] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338 – 553. [14] H.J. Zimmermann, Fuzzy Sets. Decision Making and Expert Systems, Kluwer Academic Publishing, Boston, 1987.
S.K. Deb is assistant professor of the Mechanical Engineering Department, Jorhat Engineering College, Jorhat-7, In- dia. He secured his MTech and MBA degree from IIT, Kharagpur and Gauhati University, respectively. Recently, he has obtained his PhD (Engg) from Jadavpur University, Kolkata, India. The author has teaching and research experience of about 18 years. His area of specialization is Facility Layout Planning and Operations Management. He has published several research papers in national and international journals.
B. Bhattacharyya is professor and for- mer head of the Production Engineering Department, Jadavpur University, Kol- kata, India. He did his MProd (Engg) and PhD (Engg) from Jadavpur Univer- sity, Kolkata. At present, he is the coor- dinator of Center of Advanced Studies (CAS) and Quality Improvement Pro- gram (QIP) of Jadavpur University. His area of specialization is non-traditional manufacturing and production manage- ment. He has published about 50 research papers in international and national journals.
