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Computers Ops Res. Vol. 22, No. 8, pp. 771-778, 1995 Copyright © 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0305-0548/95 $9.50+ 0.00

A MULTI-OBJECTIVE PROGRAMMING PROBLEM A N D ITS EQUIVALENT GOAL PROGRAMMING PROBLEM WITH APPROPRIATE PRIORITIES A N D ASPIRATION LEVELS: A FUZZY APPROACH B. K. M o h a n t y l t §

a n d T. A. S. V i j a y a r a g h a v a n 2 ~

1Tata Research Development and Design Centre, 1 Mangaldas Road, Prune 411 050, India and ZCentral Institute of Road Transport, Pune 411 026, India (Received July 1993; in revised.[orm July 1994)

Scope and Purpose In real life decision problems, a decision or a policy maker is fraught with difficulties to arrive at a decision, especially when multiple objectives are present in the decision and they are to be simultaneously achieved. This problem becomes more complicated when the objectives are conflicting, non commensurable and imprecise in nature. Goal programming is a suitable method to solve these types of problems. However in the methods based on goal programming mostly the goals and priorities are fixed a priori using the experienced judgments of the decision makers. A procedure in arriving at prioritizing the objectives and fixing the aspiration levels with out taking into account the subjective opinion of the decision maker is of immense use in many managerial and policy problem situations. A method based on the concepts of fuzzy sets is introduced here in this direction. An application in the area of urban vehicle scheduling is described to highlight the procedure. Abstract--A method has been introduced here to convert a multi-objective programming problem into an equivalent goal programming problem by appropriately fixing priorities and goals. The conversion method introduced in this paper, uses the concepts of conflict among objectives and the theory of fuzzy sets. The method is applied in a real life urban scheduling problem of a State Transport Undertaking in India to highlight the method proposed here.

1. I N T R O D U C T I O N A m o n g v a r i o u s m e t h o d s to solve m u l t i - o b j e c t i v e p r o g r a m m i n g p r o b l e m , the m e t h o d s b a s e d o n g o a l p r o g r a m m i n g is f o u n d to be useful in m a n y real-life s i t u a t i o n s [6]. H o w e v e r , the m a i n p r o b l e m in u s i n g t h e s e m e t h o d s is in fixing the g o a l s a n d p r i o r i t i e s a priori w i t h c e r t a i n d e g r e e of subjectivity. T h i s is b e c a u s e v e r y often a d e c i s i o n m a k e r has n o k n o w l e d g e a b o u t fixing t h e p r i o r i t i e s a n d the g o a l s to the objectives. S i n c e the o b j e c t i v e s are usually c o n f l i c t i n g a n d in the a b s e n c e o f k n o w l e d g e a b o u t t h e i r t r a d e - o f f r e l a t i o n s , v e r y often the priorities a n d g o a l s are fixed arbitrarily. T h i s m a y l e a d to w r o n g results. T h e r e f o r e , it is n e c e s s a r y to fix the g o a l s a n d the p r i o r i t i e s in a m u l t i - o b j e c t i v e p r o g r a m m i n g p r o b l e m ( M O P ) by k e e p i n g a view t o w a r d s the d e g r e e of conflict a m o n g the objectives. lBhaba Krishna Mohanty is presently a Member of Technical Staff at the Tata Research Development and Design Centre, Pune, India. He has done his M.Sc. (Mathematics} in Berhampur University, Orissa, India and his PG Diploma in Mathematics and Doctorate in Operations Research at the Indian Institute of Technology, Kharagpur, India. He has published several papers in the journals of Fuzzy Sets and Systems, hlternational Journal of Systems Scienee, hlternational Journal 0[" Production Research and Journal of hltelligent and Fuzzy Systems. He worked as an Associate Professor in the Xavier Institute of Management, Bhubaneswar, India and as a Scientist at CFTRI Mysore, India. His area of interest include MCDM, Fuzzy Logic, Software Reliability and Fuzzy Game Theory. ++T. A. S. Vijayaraghavan is presently on the faculty at the Central Institute of Road Transport, Pune, India. He has done his M.Stat. and Postgraduate Diploma in SQC & OR in the Indian Statistical Institute, New Dehli, India and his doctoral program at the Indian Institute of Management, Bangalore, India. Prior to joining here he worked as an Associate Professor at the Xavier Institute of Management, Bhubaneswar, India in the area of Systems and Operations Management. He has published a number of articles and technical papers in National and International Journals. His area of interest include Operations Research, General Management and Transportation Planning. § Indian Institute of Management, Prabandh Nagar, Off. Sitapur Road, Lucknow 226 013, India 771

B. K. Mohanty and T. A. S. Vijayaraghavan

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In this paper, a method has been introduced to fix the priorities and goals using the concepts of fuzzy sets along with the notion of degree of conflict among objectives. Numerous literature is available for solving M O P problems I-3, 4, 5, 12]. Among the methods as claimed by many researchers the techniques based on fuzzy sets reflects the true realities I-9, 11]. Fuzzy-based techniques for solving the M O P problem are found in [1, 2, 8-11-1. Even though the literature is rich in this area, we have not come across any work which objectively derives an equivalent goal programming from a multi-objective programming problem. As the techniques of solving the goal programming (GP) and M O P aim at obtaining a single solution in many operational settings, it is necessary to have some sort of equivalent or transformation methods between them. That is, each objective in a M O P should have appropriate aspiration levels for a M O P equivalent GP problem. In real life situations generally the objectives are conflicting, non-commensurable and imprecise or fuzzy in nature. To deal quantitatively with these types of qualitative situations and hence to arrive at a satisfiable efficient solution very often it requires an appropriate compromise among the objectives. That is, trade-off relations among the fuzzily defined objectives are viable for the one purpose. As the objectives are fuzzy it will be more realistic if we can establish this trade-off relation in the form of fuzzy relations or fuzzy sets. We have used the concept of Zimmermann's [11] membership function approach to achieve this trade-off relation in the form of fuzzy sets. This membership function helps in linearly expressing the degrees of conflict among the objectives. It is also numerically explicit in the implicitly defined conflicts between the objectives. This approach is more compatible with Cohort's [5] gradient method used in the paper. The construction of membership function for the objectives in a M O P problem is constructed by using the method given in [11]. The membership functions give an insight with regard to fixing appropriate priorities and goals of the objectives. The angle 0 between the gradients of a pair of objectives characterizes the degree of conflict between the two objectives. The degree of conflict is zero when 0---0 and maximum when 0 = ~. As a result of this approach, a function of conflict between pairs of objectives which uses the theory of fuzzy sets can be constructed. These functions always incorporate a pair of objectives at a time and the conflict functions are always defined for the objectives pair-wise. A symmetric matrix is then constructed where the entries of the matrix indicate a numerical measure of the degrees of non-conflict among the objectives and the extent to which an objective is not conflicting with the other objectives. Besides, the matrix enables to arrive at a preference structure among the objectives and thereby the priorities of the objectives. The methods so developed have been applied to an urban vehicle scheduling problem for illustrative purpose. In Section 2, the formulation of membership functions to the objectives is given. Section 3 presents the theory underlying conflict among objectives and construction of function of conflict by using fuzzy sets. In Section 4, a method for constructing the matrix of conflict among the objectives is described. The priority structure of objectives is also derived in this section. In Section 5, an urban vehicle scheduling problem is illustrated to demonstrate the method. 2. FORMULATION OF MEMBERSHIP FUNCTIONS Let Zl(x), Zz(x) . . . . . ZR(X) be the k number of objectives which are to be optimized, subject to a given set of linear constraints. Using the concept of fuzzy sets, the membership functions can be defined based on the following steps given by Zimmermann [11]. Step 1: Max Zi(x) st: x e X

i= 1, 2 . . . . . k

where X denotes the decision space and x e R", i.e. x is a n dimensional decision vector. let x* (i = 1, 2. . . . . k) be the optimal solution to the objective Zi(x). Take Zi(x*) = Z maX. Step 2." Find min s Zi(x*)= Z mi"

V i.

A multi-objective programming problem: a fuzzy aproach

773

Step 3: Define the membership function of the objective Zi(x) as 1

Z~(x) >1Z ? aX

I

Z i ( x ) - Z mi"

#z,(x)= ~ ~

z~'i"~