Expert Systems With Applications 64 (2016) 228–238
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Expert Systems With Applications journal homepage: www.elsevier.com/locate/eswa
Multi-objective non-linear programming problem in intuitionistic fuzzy environment: Optimistic and pessimistic view point Deepika Rani a, T.R. Gulati a, Harish Garg b,∗ a b
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247 667, INDIA School of Mathematics, Thapar University, Patiala-147 004, INDIA
a r t i c l e
i n f o
Article history: Received 26 January 2016 Revised 31 May 2016 Accepted 22 July 2016 Available online 28 July 2016 Keywords: Multi-objective Preference function Intuitionistic optimization technique Optimistic view Pessimistic view
a b s t r a c t The objective of this manuscript is to present an algorithm for solving multi-objective optimization problem under the optimistic and pessimistic view point. The conflicting natures of the different objective have been handled by defining the membership functions corresponding to it in parabolic intuitionistic fuzzy set environment and thus the problem becomes parabolic multi-objective non-linear optimization programming problem (PMONLOPP). A linear and non-linear membership functions corresponding to each objective has been taken in account. An illustrative examples from transportation as well as in manufacturing systems are reported and compared with the typical approaches exist in the literature. As shown, the solutions obtained by the proposed approach are superior to those of existing best solutions reported in the literature. Further-more, experimental results indicate that the proposed approach may yield better solutions to these types of problems than those obtained by using current algorithms. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction In real life decision making process, the systems and their corresponding decisions are becoming complicated day by day and hence it will be difficult for the decision maker’s for getting an accurate decision within a reasonable time. Moreover, the system depends on so many factors that contains a lot of uncertainties during the collection of data phase. Thus, the analysis corresponding to these data are also uncertain, vague and imprecise. In order to handle these uncertainties and impreciseness in the data, Attanassov (1986) presented intuitionistic fuzzy sets (IFSs), which is an extension of fuzzy set (Zadeh, 1965), and are characterized by a membership degree, a non-membership degree and a hesitancy degree. Angelov (1997) gave an application of the IFSs to optimization problems. His technique is based on maximizing the degree of membership (satisfaction), minimizing the degree of non-membership (dissatisfaction) and the crisp model is formulated using the IF aggregation operator (Bellman & Zadeh, 1970). While addressing real world problems, a multiobjective model with fuzzy parameters is more realistic than the one with deterministic. A solution that optimizes all the objectives
∗
Corresponding author. E-mail addresses:
[email protected] (D. Rani),
[email protected] (T.R. Gulati),
[email protected] (H. Garg). URL: http://sites.google.com/site/harishg58iitr/ (H. Garg) http://dx.doi.org/10.1016/j.eswa.2016.07.034 0957-4174/© 2016 Elsevier Ltd. All rights reserved.
simultaneously is rarely possible. Hence, an aspiration level for each objective is decided depending upon the decision maker’s choice. Obtaining the exact aspiration level is not necessary always and one tries to find a solution as close as possible to the decided aspiration level. In most of the fuzzy multi-objective optimization literature, this is done by maximizing the degree of membership function for each of the objective. Furthermore, IFSs has been proven to be highly useful to deal with uncertainty and vagueness, and hence by applying this concept, it is possible to reformulate the optimization problem by using degree of rejection of the constraints and the value of the objective which are non-admissible. Pramanik and Roy (2005) solved a vector optimization problem using an intuitionistic fuzzy goal programming. Garg (2013) proposed a technique for analyzing the behavior of industrial systems in terms of various reliability parameters using vague set theory. Chakrabortty, Pal, and Nayak (2013) proposed a method to solve the multi-objective EPQ inventory model with fuzzy inventory costs and fuzzy demand rate and the problem is solved with IFO technique. Garg and Rani (2013) presented an efficient technique for computing the membership functions of various reliability parameters using PSO and IFS theory. Garg and Sharma (2013) presented a method for solving multiobjective optimization problem using fuzzy and particle swarm optimization techniques. Also, Garg (2015) presented a hybrid GAGSA algorithm for finding the optimal solution by utilizing the uncertain and vague information. Apart from that a lot of work has been done to develop and enrich the IFS theory given in
D. Rani et al. / Expert Systems With Applications 64 (2016) 228–238
Bit, Biswal, and Alam (1993); Chakraborty, Jana, and Roy (2015a); 2015b); Chandra and Aggarwal (2014); Dey and Roy (2015); Dubey, Chandra, and Mehra (2012); Gani and Abbas (2014); Garg (2016a); 2016b); Garg and Rani (2013); Garg, Rani, and Sharma (2013); 2014); Garg, Rani, Sharma, and Vishwakarma (2014a); 2014b); Hadi-Vencheh, Hejazi, and Eslaminasab (2012); Huang, Gu, and Du (2006); Hussain and Kumar (2012); Islam and Roy (2006); Jana and Roy (2007); Kumar and Hussain (2014); 2015); Kundu, Kar, and Maiti (2013); Nehi and Hajmohamadi (2012); Pandian (2014); Singh, Abdullah, Mohamed, and Noorani (2015); Singh and Yadav (2014); 2015); Stanciulescu, Fortemps, Install, and Wertz (2003). In particular, most of the research on transportation problems is limited to the single or multi-objective fuzzy linear programming. But, in today, most of the real-world decision-making problems in economic, technical and environmental ones are multi-dimensional and multi-objective. It is significant to realize that multiple objectives are often non-commensurable and conflict with each other in the optimization problem. An objective within exact target value is termed as fuzzy goal. So a multi-objective model with fuzzy objectives is more realistic than the one with deterministic. However, if the decision maker has some preference or biasness towards a particular objective, then a linear membership function may not serve the purpose. Thus, a conflicting nature between the objectives is resolved with the help of defining their non-linear membership functions. Also, there may exist situations when the problems can be modeled only as non-linear fuzzy optimization problems which may not be modeled and solved efficiently by using the traditional techniques. Thus, in such circumstances, there is a need to modify the approach by taking into the account of decision maker preferences towards the objective in both optimistic and pessimistic view. Motivated by this idea, in this study an algorithm has been proposed for solving multi-objective problems in fuzzy environment. IFS theory has been used for resolving the conflict nature between the objectives where the degree of attainability and non-attainability of objectives are represented by nonlinear functions. Based on their corresponding membership functions, in the present work, the problem has been analyzed in two different ways - the optimistic and the pessimistic, and is explained in the subsequent sections. The model so obtained is solved to obtain the pareto optimal solution. The proposed technique has been illustrated through manufacturing and transportation problems. To the best of our knowledge, no one has applied the concept of considering membership as well as non-membership function for these problems. Moreover, the extension of this work over the others is to consider the input parameters as non-linear numbers instead of the fixed or linear ones. Rest of the paper is organized as follows: The next section contains some preliminaries to be used later in the paper. Parabolic fuzzy number has been defined and their ordering is suggested using the preference function. The model for the crisp as well as fuzzy multi-objective non-linear programming problem (MONLPP) has been presented in Section 3. Section 4 explains the different views of the decision maker and summarizes the solution algorithm. The numerical illustration of the proposed technique is given in Section 5 by solving two different MONLPPs. The paper ends with conclusions and future scope in Section 6. 2. Preliminaries Let X be the classical set of objects. Definition 1 (Kaufmann & Gupta, 1991; Zadeh, 1965). The set of ordered pairs A˜ = {(x, μA˜ (x )) : x ∈ X } is said to be a fuzzy set (FS), where the evaluation function μA˜ : X → [0, 1] is called the membership function.
229
Fig. 1. Parabolic fuzzy number.
Definition 2 (Attanassov, 1986). An intuitionistic fuzzy (IF) set A˜ I ∈ X is defined as an ordered triplet {x, μA˜ I (x ), νA˜ I (x ), x ∈ X }, where μA˜ I (x ) (the degree of membership) and νA˜ I (x ) (the degree of non-membership) are the functions from X to [0,1], i.e., μA˜I (x ), νA˜I (x ) : X → [0, 1] such that 0 ≤ μA˜I (x ) + νA˜I (x ) ≤ 1 for all x ∈ X. 1 − μA˜ I (x ) − νA˜ I (x ) represents the degree of hesitation or indeterminacy of x being in A˜ I ∈ X. Definition 3 (Kaufmann & Gupta, 1991). The α -cut of a fuzzy set A˜ is the crisp set Aα with members of universal set X such that the membership degree is at least α , i.e., Aα = {x ∈ X : μA˜ (x ) ≥ α}, 0 ≤ α ≤ 1. Definition 4 (Kaufmann & Gupta, 1991). The triplet (a, b, c) denoting the lower, modal and upper value of a membership function, is said to be triangular fuzzy number if its membership function is given by
⎧ x − a ⎪ , a≤x≤b ⎪ ⎪ ⎪ ⎨ b−a 1, x=b μA(x ) = c − x ⎪ , b≤x≤c ⎪ ⎪ ⎪ ⎩ c−b 0,
otherwise
Definition 5. A fuzzy number denoted by A˜ p (a, b, c ) is said to be parabolic fuzzy number, if its membership function is given by
⎧ x−a 2 ⎪ ⎪ , a≤x≤b ⎪ ⎪ ⎪ ⎨ b−a 1, x=b μAp (x ) = 2 c − x ⎪ ⎪ , b≤x≤c ⎪ ⎪ ⎪ ⎩ c−b 0,
otherwise
Its general shape is shown in Fig. 1. Definition 6. A parabolic fuzzy number A˜ p = (a, b, c ) is said to be non-negative if a ≥ 0, while zero fuzzy number if a = 0, b = 0 and c = 0. Definition 7 (Defuzzified value or the preference index:). Defuzzification results in deterministic values of a fuzzy number. It becomes important in some applications like hardware where the system’s operations depend upon the crisp data exchange. For defuzzification, a number of methods have been suggested in the literature (Ross, 2004). The centroid and maxima methods are most widely used among them. The centroid method finds a balancing point of a property that can be the area of highest intersection, the area of largest fuzzy set or the weight of each fuzzy set etc. Whereas, the maxima methods searches for the highest peak. In this paper, we will use the centroid of area for finding the defuzzified value or the preference index (f) of the parabolic fuzzy number which is given by:
xμAp (x )dx f = x , p (x )dx x μA
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where x is the output variable and μAp (x ) is the membership function.
x
μAp (x )dx = =
b
x − a 2 b−a
a
1 ( b − a )2
+
c b
( x − a )3
c − x 2 c−b
b
+
3
a
1 ( c − b )2
( c − x )3 −3
Definition 9 (Complete optimal solution). Let S denote the set of all feasible solutions of Problem (P1 ). x∗ ∈ S is said to be its complete optimal solution, if and only if for all x ∈ S, φ i (x∗ ) ≤ φ i (x); 1 ≤ i ≤ l and φ i (x∗ ) ≥ φ i (x); l + 1 ≤ i ≤ l.
c b
However, when there are more than one objective in the problem then complete optimal solution is rarely possible. An alter to this, a concept of pareto optimality or compromise solution arises which are defined as below.
1 1 = [ ( b − a ) + ( c − b )] = ( c − a ) 3 3
b c c − x 2 x−a 2 xμAp (x )dx = x + x b−a c−b x a b =
1
( b − a )2
x (x − a ) (x − a ) − 3 12 3
4
=
b( b − a ) ( b − a )2 − 3 12
+
Definition 10 (Pareto-optimal solution). For a multi-objective programming problem (P1 ), a feasible solution x∗ ∈ S is said to be pareto-optimal or optimal compromise solution if there does not exist any other feasible solution x such that φ i (x) ≤ φ i (x∗ ) for 1 ≤ i ≤ l and φ i (x) ≥ φ i (x∗ ) for l + 1 ≤ i ≤ l with at least one inequality holding as strict inequality.
b
1 x ( c − x )3 ( c − x )4 + + 2 −3 −12 (c − b )
a
c b
b( c − b ) ( c − b )2 − 3 12
(c − b ) ( 3b + a ) + ( 3b + c ) 12 12 (c − a ) = ( a + 2b + c ) 12 =
(b − a )
Hence the preference index f for the parabolic fuzzy number A˜ p = (a, b, c ) is
f = ( a + 2b + c )/4. Theorem 1. The preference index f is a linear function. Proof. Let A˜ p = (a, b, c ) and B˜ p = (a1 , b1 , c1 ) be two parabolic fuzzy numbers. For λ > 0 ∈ R, λA˜ p B˜ p = (λa + a1 , λb + b1 , λc + c1 ).
f (λA˜ p B˜ p ) = =
•
•
4
λ ( a + 2 b + c ) + ( a1 + 2 b1 + c1 ) 4
= λ f (A˜ p ) + f (B˜ p )
λc + a1 + 2(λb + b1 ) + λa + c1
4 λ ( c + 2 b + a ) + ( a1 + 2 b1 + c1 ) = = λ f (A˜ p ) + f (B˜ p ) 4
Hence, f is a linear function.
Definition 8. For any two parabolic fuzzy numbers A˜ p and B˜ p , the ordering can be defined as: (i) (ii) (iii)
Based on these pareto solution, a decision maker may choose according to his or her preference. Also, in most of the practical situations due to complexity of the real world, decision makers often come to face uncertainties in the determination of parameters. This imprecision/uncertainty is inevitable and hence the data is not precise all the time and is given as estimates. These ambiguities arise mainly because of the imprecision in judgement, lack of evidence, insufficient information, environmental conditions, insufficient involvement of all the concerned persons, short of alertness of the present market, unawareness of customers etc. Some of the real time situations, when such kind of imprecision may exist in various transportation systems, are as follows:
λa + a1 + 2(λb + b1 ) + λc + c1
For λ < 0 ∈ R, λA˜ p B˜ p = (λc + a1 , λb + b1 , λa + c1 ).
f (λA˜ p B˜ p ) =
where φ i (x), 1 ≤ i ≤ l and ψ j (x), 1 ≤ j ≤ r are the real valued linear or non-linear functions and x is an n tuple.
A˜ p ≥ B˜ p if f (A˜ p ) ≥ f (B˜ p ) A˜ p > B˜ p if f (A˜ p ) > f (B˜ p ) A˜ p = B˜ p if f (A˜ p ) = f (B˜ p )
3. Modeling
•
•
• • • • • •
A product is transported for the very first time to a consumer point and the decision maker is not very much sure about the transportation cost. Sudden change in environmental conditions like rain, storm or the bad road conditions, traffic jams, road works etc. more time can be taken for the transportation causing an unexpected change in the transportation cost as well as time. Unsureness of the decision maker about the stocks available because of estimation error or insufficient information etc. Different routes or modes of transportation causes change in cost and time. Variation in the load of transportation mode used. The shortage of a high demand product. An unexpected change in the price of fuel or raw material etc. Uncertainty about the demand of a newly launched product. The demand of seasonal products keeps on varying each season. Production varies with availability of raw materials and labour, proper working of machines etc.
An optimization model can be made more realistic, adoptable by human decision process if its parameters are assumed to be flexible in nature. Hence, the fuzzy logic optimization model has been introduced and correspondingly to problem (P1 ), the fuzzy optimization problem becomes.
(P2 )
A deterministic multi-objective non-linear programming problem can be modeled as:
(P1 ) Min φi (x ), 1 ≤ i ≤ l Max φi (x ), l + 1 ≤ i ≤ l subject to ψ j (x ) ≤ c j , 1 ≤ j ≤ r ψ (x ) ≥ c , r + 1 ≤ j ≤ r j
j
ψ j (x ) = c j , r + 1 ≤ j ≤ r x ≥ 0,
Min φ˜ i (x ), 1 ≤ i ≤ l Max φ˜ i (x ), l + 1 ≤ i ≤ l subject to ψ˜ j (x ) ≤ c˜ j , 1 ≤ j ≤ r
ψ˜ j (x ) ≥ c˜ j , r + 1 ≤ j ≤ r ψ˜ j (x ) = c˜ j , r + 1 ≤ j ≤ r x ≥ 0, where φ˜ i (x ), 1 ≤ i ≤ l and ψ˜ j (x ), c˜ j ; 1 ≤ j ≤ r are taken to be fuzzy numbers.
D. Rani et al. / Expert Systems With Applications 64 (2016) 228–238
In the present study, we use parabolic fuzzy numbers to re Ki β flect the uncertainty. Let φi (x ) = k=1 aik nm=1 xmm , 1 ≤ i ≤ l
P j γ and ψ j (x ) = p=1 b j p nm=1 xmm , 1 ≤ j ≤ r. We call the model as parabolic multi-objective non linear programming problem (PMONLPP) which can be written as:
Min φ˜ ip (x ) =
(P3 )
Ki
n β p a˜ik xmm , 1 ≤ i ≤ l
Max φ˜ ip (x ) =
Ki
b˜ pjs
s=1 Sj s=1
Sj
(3.1)
n γ b˜ pjs xmm = c˜pj , r + 1 ≤ j ≤ r
s=1
(3.2)
(3.3)
m=1
4. Solution algorithm After using the preference function, the reformulated crisp model of Problem (P3 ) can be written as:
Min
φip (x ) =
Ki
aˆik
m=1
k=1
Max
φip (x ) =
Ki k=1
n β xmm , 1 ≤ i ≤ l
aˆik
n β xmm , l + 1 ≤ i ≤ l m=1
subject to Sj
bˆ js
s=1 Sj
m=1
bˆ js
s=1 Sj
n γ xmm ≤ cˆ j , 1 ≤ j ≤ r n γ xmm ≥ cˆ j , r + 1 ≤ j ≤ r m=1
bˆ js
s=1
n γ xmm = cˆ j , r + 1 ≤ j ≤ r m=1
xi ≥ 0, 1 ≤ i ≤ n, p p p where aˆik = f (a˜ik ), bˆ js = f (b˜ js ) and cˆ j = c˜ j .
Theorem 2. An pareto-optimal solution X ∗ = (x∗1 , x∗2 , ..., x∗n ) of Problem (P4 ) is also a pareto-optimal solution for the Problem (P3 ). Proof. Since X∗ is the pareto-optimal solution of the problem (P4 ), so X∗ is also its feasible solution and the following holds: Sj s=1 Sj
s=1
∗γ
xm m ≤ cˆ j , 1 ≤ j ≤ r
m=1 n
bˆ js
s=1 Sj
n
bˆ js
∗γ
xm m ≥ cˆ j , r + 1 ≤ j ≤ r
m=1 n
bˆ js
∗γ
xm m ≤ c˜pj , 1 ≤ j ≤ r
b˜ pjs
n
∗γ
xm m ≥ c˜pj , r + 1 ≤ j ≤ r
m=1
b˜ pjs
n
∗γ
xm m = c˜pj , r + 1 ≤ j ≤ r
m=1
x = (x1 , x2 , ..., xn ) ≥ 0. Hence X∗ is also a feasible solution of Problem (P3 ). As X∗ is a pareto-optimal solution of (P4 ), it means there does not exist any other feasible solution X = (x1 , x2 , ..., xn ) such that φ i (X) ≤ φ i (X∗ ) for 1 ≤ i ≤ l and φ i (X) ≥ φ i (X∗ ) for l + 1 ≤ i ≤ l with at least one inequality holding as strict inequality.
Ki n βm That is, we have no X such that aˆ m=1 xm ≤ k=1 ik n
Ki Ki ∗βm βm n aˆ for 1 ≤ i ≤ l and aˆ m=1 xm m=1 xm ≥ k=1 ik k=1 ik n
Ki ∗βm aˆ for l + 1 ≤ i ≤ l with at least one inequality m=1 xm k=1 ik holding as strict inequality. Since aˆik , bˆ js and cˆ j are preference function values, i.e., aˆik = f (a˜ik ), bˆ js = f (b˜ js ) and cˆ j = f (bˆ js ) and f is linear, so we have no
Ki
Ki β ∗β p p X such that k=1 a˜ik nm=1 xmm ≤ k=1 a˜ik nm=1 xm m for 1 ≤ i ≤ l
Ki
Ki βm ∗βm p n p n and a˜ a˜ for l + 1 ≤ i ≤ l with m=1 xm ≥ m=1 xm k=1 ik k=1 ik at least one inequality holding as strict inequality. Hence proved. In order to get the optimal solution corresponding to these programming problem, different preferences corresponding to objective functions have been taken. Then, based on the views of decision makers in terms of either optimistic or pessimistic, a membership and non-membership functions corresponding to each objective has been constructed and are described briefly as below. Let Li and Ui respectively be the lower and upper bounds for the objective i and α i be the respective tolerance. As l objectives φ i ; i = 1, 2, ..., l are to be minimized, the degree of decision maker’s satisfaction increases as each objective value approaches its respective lower bound Li and he is fully satisfied if all the objectives reach their lower bounds. But it is quite common that, in practically, attaining of these lower bound are not to be exact. For this, based on the decision and judgement of the decision maker, the degree of attainability (μLi (φi (x ))) and non-attainability (νLi (φi (x ))) of Li , respectively have been interpreted in two different ways- the optimistic view and the pessimistic view. The optimistic view: We define the membership (μLi (φi )) and non-membership (νLi (φi )) functions for the ith objective as follows:
∗γ
xm m = cˆ j , r + 1 ≤ j ≤ r
m=1
xi ≥ 0, 1 ≤ i ≤ n, That is
s=1
n m=1
s=1 Sj
x = (x1 , x2 , ..., xn ) ≥ 0.
(P4 )
b˜ pjs
s=1
m=1
Sj
∗γ
xm m = f (c˜pj ), r + 1 ≤ j ≤ r
m=1
Sj
n γ xmm ≥ c˜pj , r + 1 ≤ j ≤ r
n
f (b˜ pjs )
Since the preference function f is linear, so
m=1
b˜ pjs
∗γ
xm m ≥ f (c˜pj ), r + 1 ≤ j ≤ r
m=1
s=1
m=1
n γ xmm ≤ c˜pj , 1 ≤ j ≤ r
n
f (b˜ pjs )
x = (x1 , x2 , ..., xn ) ≥ 0.
subject to Sj
∗γ
xm m ≤ f (c˜pj ), 1 ≤ j ≤ r
m=1
s=1
n β p a˜ik xmm , l + 1 ≤ i ≤ l
k=1
s=1 Sj
n
f (b˜ pjs )
Sj
m=1
k=1
Sj
231
μLi (φi (x )) =
⎧ 1, ⎪ ⎪ ⎨ t
Ui − φit (x )
U t − Lti ⎪ ⎪ ⎩ i 0,
φi ( x ) ≤ L i , Li ≤ φi (x ) ≤ Ui ,
φi (x ) ≥ Ui
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D. Rani et al. / Expert Systems With Applications 64 (2016) 228–238
Fig. 2. Membership functions for minimizing objective in optimistic view.
Fig. 3. Membership functions for minimizing objective in pessimistic view.
νLi (φi (x )) =
⎧ 0, ⎪ ⎪ ⎨
φi ( x ) ≤ L i
φ (x ) − , Li ≤ φit (x ) ≤ Ui + αi t t ( U ⎪ i + αi ) − Li ⎪ ⎩ 1, φi (x ) ≥ Ui + αi t i
Its possible shape has been shown in Fig. 2. From it, it has been observed that in the interval [Ui , Ui + αi ], the membership degree of achieving the goal is zero but the other is not, i.e., the decision maker is not interested to accept the values more than Ui but at the same time not strictly rejecting the values from Ui to Ui + αi . That is why such an approach is called optimistic. The pessimistic view: We define μLi (φi ) and νLi (φi ) as follows:
μLi (φi (x )) =
⎧ 1, ⎪ ⎪ ⎨ t
The optimistic view: We define the functions as follows:
Lti
Ui − φit (x )
φi ( x ) ≤ L i
, Li ≤ φi (x ) ≤ Ui , U t − Lti ⎪ ⎪ ⎩ i 0, φi (x ) ≥ Ui ⎧ 0 , φi (x ) ≤ Ui − αi ⎪ ⎪ ⎨ φ t (x ) − (U − α )t i i i νLi (φi (x )) = , Ui − αi ≤ φi (x ) ≤ Ui ⎪ Uit − (Ui − αi )t ⎪ ⎩ 1, φi (x ) ≥ Ui
Its possible general shape has been shown in Fig. 3. From it, it has been observed that in the interval [Li , Ui − αi ], the membership degree of achieving the aspired goal is not one but the nonmembership degree is zero, i.e., the decision maker is not interested to reject the values from Li to Ui − αi and at the same time not accepting them completely. For this reason, such an approach is called pessimistic. As the objectives φ i (i = l + 1, l + 2, ..., l) are to be maximized, the degree of satisfaction of the decision maker increases as each objective value approaches its respective upper bound Ui and is fully satisfied if all the objectives reach their upper bounds. Similar to the case of minimizing objectives, here also the situation has been summarized in two different ways- the optimistic and pessimistic view.
⎧ 0, ⎪ ⎨ φ t (x ) − Lt
φi ( x ) ≤ L i
, Li ≤ φi (x ) ≤ Ui , U t − Lti ⎪ ⎩ i 1, φi (x ) ≥ Ui ⎧1, φi (x ) ≤ Li − αi ⎪ ⎨ t Ui − φit (x ) νUi (φi (x )) = , Li − αi ≤ φi (x ) ≤ Ui t t ⎪ ⎩ U − (Li − αi ) 0, φi (x ) ≥ Ui
μUi (φi (x )) =
i
i
and their corresponding shapes have been represented in Fig. 4. In the pessimistic view these functions are defined as
⎧ 0, φi ( x ) ≤ L i ⎪ ⎪ ⎨ φ t (x ) − Lt i i μUi (φi (x )) = , Li ≤ φi (x ) ≤ Ui , t t U − L ⎪ i i ⎪ ⎩ 1, φi (x ) ≥ Ui ⎧ 1 , φi ( x ) ≤ L i ⎪ ⎨ t t νUi (φi (x )) = (Li + αi ) − φi (x ) , Li ≤ φi (x ) ≤ Li + αi ⎪ ⎩ (Li + αi )t − Lt 0, φi (x ) ≥ Li + αi and their corresponding shapes are shown in Fig. 5 when the aim is to maximize the objective. Here, in the interval [Li + αi , Ui ), the membership degree of achieving the aspired goal is not one but the other degree is zero, i.e., the decision maker is not rejecting the values between Li + αi to Ui but at the same time not accepting them completely too. Now, our main objective is to increase the level of satisfaction of the decision maker and decrease the level of dissatisfaction, i.e, to increase the degree of attainability and decrease the degree of non-attainability. This can be done as follows: Let λ = min{μLi (φi (x )), μUi (φi (x ))}; and λ = max{νLi (φi (x )), νUi (φi (x ))}; for i = 1, 2, . . . , l , l + 1, . . . , l then the problem can be modeled as:
D. Rani et al. / Expert Systems With Applications 64 (2016) 228–238
233
Fig. 4. Membership functions for maximizing objective in optimistic view.
Fig. 5. Membership functions for maximizing objective in pessimistic view.
(P5 ) Max λ, Min λ , subject to μLi (φi (x )) ≥ λ; 1 ≤ i ≤ l , μUi (φi (x )) ≥ λ; l + 1 ≤ i ≤ l νLi (φi (x )) ≤ λ ; 1 ≤ i ≤ l , νUi (φi (x )) ≤ λ ; l + 1 ≤ i ≤ l xm ≥ 0; 1 ≤ m ≤ n, λ, λ ≥ 0; λ + λ ≤ 1; and constraints 3.1 − 3.3 This model is reduced to the deterministic single objective programming problem as follows. In optimistic decision maker’s point of view, the model becomes:
Max (λ − λ ), subject to
Uit − φit (x ) ≥ λ(Uit − Lti ); 1 ≤ i ≤ l ,
φit (x ) − Lti ≥ λ(Uit − Lti ); l + 1 ≤ i ≤ l φit (x ) − Lti ≤ λ [(Ui + αi )t − Lti ]; 1 ≤ i ≤ l , Uit − φit (x ) ≤ λ [Uit − (Li − αi )t ]; l + 1 ≤ i ≤ l xm ≥ 0; 1 ≤ m ≤ n, λ, λ ≥ 0; λ + λ ≤ 1; and constraints 3.1 − 3.3 For the pessimistic decision maker, the model takes the following form:
the membership of x being much more than the membership of y, y will still be the optimal choice of the decision maker. So preserving the same order of comparison, Yager (2009) substituted the concept of μ(x ) − ν (x ) by a more generalized function say f(x), where f (x ) = μ(x ) + kπ (x ); k ∈ (0, 1] and π (x ) = 1 − μ(x ) − ν (x ). k can take any value in the interval (0,1]. The larger value represents that the indeterminacy gets resolved more in favor of membership functions whereas the smaller value indicates it in favor of non-membership. Hence, for the optimistic and pessimistic decision maker’s, the function f takes the form as given below. Define the function f(φ i (x)) for all i; 1 ≤ i ≤ l. Let us say fLi (φi (x )) for i; 1 ≤ i ≤ l and fUi (φi (x )) for i; l + 1 ≤ i ≤ l. So, Problem (P5 ) can be re-casted as:
(P6 ) Max χ , subject to fLi (φi (x )) ≥ χ ; 1 ≤ i ≤ l , fUi (φi (x )) ≥ χ ; l + 1 ≤ i ≤ l xm ≥ 0; 1 ≤ m ≤ n and constraints 3.1 − 3.3
χ = min{ fLi (φi (x )), fUi (φi (x ))}, χ ∈ [0, 1]. Theorem 3. The optimal solution x∗ ∈ S of Problem (P6 ) is a pareto optimal solution of the multi-objective Problem (P4 ).
Max (λ − λ ) subject to
Proof. Let us assume that x∗ is not a pareto optimal solution of (P4 ). Therefore, there exist x ∈ S such that
φit (x ) − Lti ≥ λ(Uit − Lti ); l + 1 ≤ i ≤ l φit (x ) − (Ui − αi )t ≤ λ [Uit − (Ui − αi )t ]; 1 ≤ i ≤ l , (Li + αi )t − φit (x ) ≤ λ [(Li + αi )t − Lti ]; l + 1 ≤ i ≤ l xm ≥ 0; 1 ≤ m ≤ n, λ, λ ≥ 0; λ + λ ≤ 1;
φi (x ) ≤ φi (x∗ ) for 1 ≤ i ≤ l and φi (x ) ≥ φi (x∗ ) for l + 1 ≤ i ≤ l
Uit − φit (x ) ≥ λ(Uit − Lti ); 1 ≤ i ≤ l ,
and constraints 3.1 − 3.3 But there are certain demerits of the above technique which were pointed out by Yager (2009). For example, let the membership and non-membership for two alternatives x and y be as μ(x ) = 0.67, μ(y ) = 0.2 and ν (x ) = 0.82, ν (y ) = 0. So, μ(x ) − ν (x ) = −0.15 and μ(y ) − ν (y ) = 0.2. Then according to the above technique, the calculated values shows that inspite of the fact that
with at least one of the inequality holding as strict inequality. Since the membership function fLi (φi (x )) is strictly decreasing with increasing value of the corresponding objective φ i (x) and fUi (φi (x )) is strictly increasing with increasing value of corresponding φ i (x), so we have
fLi (φi (x )) ≥ fLi (φi (x∗ )) for 1 ≤ i ≤ l
and fUi (φi (x )) ≥ fUi (φi (x∗ )) for l + 1 ≤ i ≤ l with at least one of them holds as strict inequality.
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φit (x ) − Lti
Hence χ = min{ fLi (φi (x )), fUi (φi (x ))} ≥ min{ fLi (φi (x∗ )), fUi (φi (x∗ ))} = χ ∗ (say ) which is contradiction to the fact that x∗ is an
(1 − k )
optimal solution of Problem (P6 ). Hence proved.
xm ≥ 0; 1 ≤ m ≤ n,
fLi (φi (x )) =
φi ( x ) ≤ L i
φit (x )−Lti −k (U + t t + k, i αi ) −L
and
fUi (φi (x )) =
Li ≤ φi (x ) ≤ Ui
⎪ ⎪ ⎪ φ t (x )−Lt ⎪ ⎪k − k (Ui +i αi )t −Li t , ⎪ ⎪ ⎩
Ui ≤ φi (x ) ≤ Ui + αi
⎧ 0, ⎪ ⎪ ⎪ Uit −φit (x) ⎪ k + k, ⎪ ⎪ ⎨ (Li −αi )t −Uit
φi (x ) ≤ Li − αi Li − αi ≤ φi (x ) ≤ Li
φi (x ) ≥ Ui + αi
0,
+ k ≥ χ ; l + 1 ≤ i ≤ l
χ ∈ [0, 1]
and constraints 3.1 − 3.3
Optimistic view
⎧ 1, ⎪ ⎪ ⎪ U t −φ t ( x ) ⎪ ⎪ (1 − k ) iU t −Li t ⎪ ⎪ i i ⎨
Uit − Lti
φit (x )−Lti
(1 − k ) U t −Lt ⎪ i i ⎪ ⎪ Uit −φit (x ) ⎪ ⎪ ⎪ ⎩ −k U t −(Li −αi )t + k, Li ≤ φi (x ) ≤ Ui 1, φi (x ) ≥ Ui
For optimistic decision maker, Problem (P6 ) takes the following form:
χ , subject to U t − φ t (x ) φit (x ) − Lti (1 − k ) i t i t − k + k ≥ χ ; 1 ≤ i ≤ l (Ui + αi )t − Lt Ui − Li φit (x ) − Lti k−k ≥ χ ; 1 ≤ i ≤ l (Ui + αi )t − Lt Uit − φit (x ) k + k ≥ χ ; l + 1 ≤ i ≤ l (Li − αi )t − Uit φ t (x ) − Lti U t − φit (x ) (1 − k ) i t −k t i + k ≥ χ ; l + 1 ≤ i ≤ l t U − (Li − αi )t Ui − Li xm ≥ 0; 1 ≤ m ≤ n, χ ∈ [0, 1] and constraints 3.1 − 3.3 Max
Thus, the overall solution algorithm can be summarized as follows: 1. Model the problem as multi-objective non-linear programming problem. 2. Get the crisp equivalent of the model by using the preference function. 3. Obtain the ideal solution, say X1 , X2 , \ldots, Xl , ..., Xl corresponding to each objective by solving individually and construct the pay-off matrix as follows.
X2 . . . Xl
φ1 ( x )
φ2 ( x )
···
φl ( x )
···
φl ( x )
φ1 ( x ) φ1 (X2 )
φ2 ( x ) φ2 (X2 )
φl ( x ) φl (X2 )
. . .
··· ··· . . . ···
φl ( x ) φl (X2 )
. . .
··· ··· . . . ···
φ1 (Xl )
φ2 (Xl )
. . .
φl (Xl )
. . .
φl (Xl )
4. Based on this pay-off matrix, ideal and anti-ideal values corresponding to each objective has been constructed as Li = min{φi (X1 ), φi (X2 ), . . . , φi (Xl )} and Ui = max{φi (X1 ), φi (X2 ), . . . , φi (Xl )} for all i. 5. Define the membership and non-membership functions for each of the objective and use the technique explained above to get the solution. 6. The algorithm stops if the decision maker is satisfied with the obtained solution. Otherwise, the key parameters, i.e., preferences of each objective function, the tolerance for each objective etc. can be altered to meet the choice. The process is repeated until the decision maker is satisfied with the obtained solutions. We assume that the decision maker is satisfied with the obtained results so we will show just one run of the algorithm.
Pessimistic view
5. Numerical illustration
and
Example 1 (In manufacturing). Suppose a production company has to produce three kinds of products P1 , P2 and P3 in a specified period of time (say 1 month). The production of each of the product needs three different kinds of raw materials R1 , R2 and R3 . To produce a single unit of P1 , the use of R1 , R2 and R3 is about 2, 4 and 3 units, respectively. The requirement of raw materials to produce each unit of P2 is around 3, 2, 2 units and for that of product P3 is units 4, 2 and 3 approximately. The planned available resource R1 is about 360 units with an additional 10 units in safety store for the emergency purpose which are administrated by the manager. The estimated available amount of R2 is 350 units with an error of about 10 units. For better quality of the manufactured product about 325 units of R3 must be utilized with some allowed tolerance by the managerial board. To reach the goals, let the planned production of P1 , P2 and P3 be x1 , x2 and x3 , respectively. Further assume that the units cost (UCi ) and unit sale price (USi ) p p p p 1/a of P1 , P2 and P3 are UC1 = c˜1 , UC2 = c˜2 , UC3 = c˜3 , US1 = s˜1 /x1 1 ,
⎧ 1, φi ( x ) ≤ L i ⎪ ⎪ ⎪ Uit −φit (x ) ⎪ ( 1 − k ) + k, Li ≤ φi (x ) ≤ Ui − αi ⎪ Uit −Lti ⎪ ⎨ t t fLi (φi (x )) = (1 − k ) Ui −t φi (tx ) Ui −Li ⎪ ⎪ ⎪ φit (x )−(Ui −αi )t ⎪ ⎪ −k U t −(U −α )t + k, Ui − αi ≤ φi (x ) ≤ Ui ⎪ i i ⎩ i 0, φi (x ) ≥ Ui
fUi (φi (x )) =
⎧ 0, ⎪ ⎪ ⎪ φ t (x )−Lt ⎪ (1 − k ) Ui t −Lt i ⎪ ⎪ i i ⎨ −k
(Li +αi )t −φit (x )
φi ( x ) ≤ L i + k, Li ≤ φi (x ) ≤ Li + αi
(Li +αi )t −Lt ⎪ ⎪ ⎪ φ t (x )−Lt ⎪ ⎪ (1 − k ) Ui t −Lt i + k, ⎪ ⎩ i i
1,
Li + αi ≤ φi (x ) ≤ Ui
φi (x ) ≥ Ui
The obtained crisp non-linear model is as given below:
χ , subject to U t − φ t (x ) (1 − k ) i t i t + k ≥ χ ; 1 ≤ i ≤ l
Max
Ui − Li
φit (x ) − (Ui − αi )t + k ≥ χ ; 1 ≤ i ≤ l Uit − Lti Uit − (Ui − αi )t φ t (x ) − Lti (Li + αi )t − φit (x ) (1 − k ) i t −k + k ≥ χ ; l + 1 ≤ i ≤ l t (Li + αi )t − Lt Ui − Li (1 − k )
Uit − φit (x )
−k
p
1/a
p
1/a
U S2 = s˜2 /x2 2 and U S3 = s˜3 /x3 3 , where a1 , a2 , a3 are positive real numbers. The estimated time of production for each unit of P1 , P2 and P3 is around 5, 7, 6 h, respectively. The manger wishes a production schedule that maximizes his profit and minimize the total time. The problem can be modeled as follows:
Min φ˜ 1p (x ) = 5˜ p x1 7˜ p x2 6˜ p x3 Max φ˜ 2p (x ) = s˜1p x11−1/a1 c˜1p x1 s˜2p x12−1/a2
D. Rani et al. / Expert Systems With Applications 64 (2016) 228–238 p
p 1−1/a3
c˜2 x2 s˜3 x3
s. t.
p
s.t
c˜3 x3
˜ p 2˜ p x1 3˜ p x2 4˜ p x3 ≤ 360 ˜ 4˜ p x1 2˜ p x2 2˜ p x3 = 350
(5.1)
p
˜ p 3˜ p x1 2˜ p x2 3˜ p x3 ≤ 360 x = ( x1 , x2 , x3 ) ≥ 0. We assume that all the uncertain parameters estimated by the decision maker to be parabolic fuzzy numbers. Let 2˜ p = (1.5, 2, 4 ), ˜ p 3˜ p = (2, 3, 4.5 ), 5˜ p = (3, 5, 6 ), 6˜ p = (3, 6, 7 ), 7˜ p = (5, 7, 10 ), 325 p p ˜ = (340, 350, 353 ), 360 ˜ = (355, 360, 370 ), = (310, 325, 325 ), 350 p p p p p c˜1 = (6, 8, 9 ), c˜2 = (7, 10, 12 ), c˜3 = (7, 7, 8 ), s˜1 = (48, 50, 53 ), s˜2 = p (42, 45, 46 ), s˜3 = (60, 60, 65 ), a1 = 2, a2 = a3 = 3. Using the preference function, the equivalent crisp model for Eq. (5.1) can be written as:
235
2.375x1 + 3.125x2 + 3.5x3 ≤ 361.25
(5.2)
3.5x1 + 2.375x2 + 2.375x3 = 348.25 3.125x1 + 2.375x2 + 3.125x3 ≥ 321.25 x1 , x2 , x3 ≥ 0. Solving φ 1 and φ 2 separately under the given set of constraints, we get the solutions as X1 = (92.53, 0, 10.27 ) and X2 = (52.58, 15.05, 54.09 ). The ideal and anti-ideal values for each of the objective are found to be L1 = 496, U1 = 656.32, L2 = 18.81 and U2 = 565.32. Let the allowed tolerances defined by the decision maker be α1 = 65 and α2 = 70. As the φ 1 is to be minimized and φ 2 is to be maximized, the degree of attainability and nonattainability of respective lower and upper bounds can be defined as follows: Optimistic view: Constructing and using the functions f L1 and fL2 corresponding to their respective objectives, the crisp model can be written as:
χ (565.32 )t − (50.25x11/2 − 7.75x1 + 44.5x22/3 − 9.75x2 + 61.25x23/3 − 7.25x3 )t k + k ≥ χ; (−88.81 )t − (565.32 )t (50.25x11/2 − 7.75x1 + 44.5x22/3 − 9.75x2 + 61.25x23/3 − 7.25x3 )t − (−18.81 )t (1 − k ) − (565.32 )t − (−18.81 )t (565.32 )t − (50.25x11/2 − 7.75x1 + 44.5x22/3 − 9.75x2 + 61.25x23/3 − 7.25x3 )t +k≥χ k (565.32 )t − (−88.81 )t (4.75x1 + 7.25x2 + 5.5x3 )t − (496 )t k−k ≥χ (721.36 )t − (496 )t (656.36 )t − (4.75x1 + 7.25x2 + 5.5x3 )t (4.75x1 + 7.25x2 + 5.5x3 )t − (496 )t (1 − k ) −k +k≥χ t t (656.36 ) − (496 ) (721.36 )t − (496 )t Max
s.t.
(5.3)
2.375x1 + 3.125x2 + 3.5x3 ≤ 361.25 3.5x1 + 2.375x2 + 2.375x3 = 348.25 3.125x1 + 2.375x2 + 3.125x3 ≥ 321.25 x1 , x2 , x3 ≥ 0. t > 0 is defined by the decision maker. Pessimistic view:
χ (50.25x11/2 − 7.75x1 + 44.5x22/3 − 9.75x2 + 61.25x23/3 − 7.25x3 )t − (−18.81 )t (1 − k ) +k≥χ (565.32 )t − (−18.81 )t (50.25x11/2 − 7.75x1 + 44.5x22/3 − 9.75x2 + 61.25x23/3 − 7.25x3 )t − (−18.81 )t (1 − k ) − (565.32 )t − (−18.81 )t (51.19 )t − (50.25x11/2 − 7.75x1 + 44.5x22/3 − 9.75x2 + 61.25x23/3 − 7.25x3 )t k +k≥χ (51.19 )t − (−18.81 )t (656.36 )t − (4.75x1 + 7.25x2 + 5.5x3 )t (1 − k ) +k≥χ (656.36 )t − (496 )t (656.36 )t − (4.75x1 + 7.25x2 + 5.5x3 )t (4.75x1 + 7.25x2 + 5.5x3 )t − (591.36 )t (1 − k ) −k +k≥χ t t (656.36 ) − (496 ) (656.36 )t − (591.36 )t
Max s.t.
(5.4)
2.375x1 + 3.125x2 + 3.5x3 ≤ 361.25 3.5x1 + 2.375x2 + 2.375x3 = 348.25 3.125x1 + 2.375x2 + 3.125x3 ≥ 321.25 x1 , x2 , x3 ≥ 0. t > 0 is defined by the decision maker.
φ1 (x ) = 4.75x1 + 7.25x2 + 5.5x3 Max φ2 (x ) = 50.25x11/2 − 7.75x1 + 44.5x22/3 Min
− 9.75x2 + 61.25x23/3 − 7.25x3
Both the optimistic and pessimistic models are solved for different values of k and t = 2 with the help of Lingo 15.0 software. For optimistic decision maker, solution in tabular as well as graphical form are given in Table 1 and Fig. 6, respectively. On
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Table 1 Solutions for optimistic view point when t = 2. k
x1
x2
x3
φ 1 (x)
φ 2 (x)
χ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
53.17 53.74 55.13 56.93 58.49 59.88 61.11 62.20 63.19 64.08
10.63 6.44 4.79 4.57 4.38 4.22 4.07 3.94 3.83 3.72
57.64 60.99 60.59 58.16 56.04 54.17 52.51 51.02 49.69 48.48
646.65 637.40 629.84 623.43 617.80 612.96 608.59 604.63 601.22 597.99
439.07 448.91 459.51 471.16 483.99 498.09 513.71 531.19 549.98 561.89
0.09 0.19 0.26 0.33 0.39 0.44 0.48 0.52 0.56 0.59
Container volume = x1 x2 x3 p Number of trips = 70˜0 0 /x1 x2 x3 ˜ p (70˜0 0 p /x1 x2 x3 ) Shipment cost of the product = 275 p p ˜ x1 x2 + 60 ˜ x1 x3 + 40 ˜ p x2 x3 Container cost = 50 The PMONLPP can be modeled as:
Minimize
p
˜ φ˜ 1 (x ) p = (70˜0 0 /x1 x2 x3 )275 p
p
p p
˜ x1 x2 60 ˜ x1 x3 40 ˜ x2 x3 50 Minimize
p
φ˜ 2 (x ) p = 70˜0 0 /x1 x2 x3
subject to x1 x2 x3 ≥ 0˜ p ˜ x1 x2 x3 ≤ 600
(5.5) p
x = ( x1 , x2 , x3 ) ≥ 0. p Let the parameters estimated by the decision maker be 70˜0 0 = p p ˜ = (264, 275, (6965, 70 0 0, 7035 ), 60˜ 0 = (590, 60 0, 610 ), 275 ˜ p = (22, 25, 30 ), 30 ˜ p = (25, 30, 38 ), 20 ˜ p = (17, 20, 23 ). 280 ), 25 Using the preference index for the parabolic fuzzy numbers, the deterministic model for problem (5.5) can be written as:
Minimize
φ1 (x ) = 1910375.01/x1 x2 x3 + 51.5x1 x2
+ 62.26x1 x3 + 40x2 x3 Minimize
φ2 (x ) = 70 0 0/x1 x2 x3
subject to
(5.6)
x1 x2 x3 ≥ 0 x1 x2 x3 ≤ 600 x1 , x2 , x3 ≥ 0.
Fig. 6. Graphical representation of the solutions for optimistic view point.
the other hand, for the pessimistic decision maker, solution is x1 = 68.01, x2 = 3.28, x3 = 43.12, φ1 = 583.99, φ2 = 394.13 and χ increases from 0.54 to 0.96 as k increased from 0.1 to 1. We find that the obtained solutions as well as level of satisfaction goes on improving with increasing value of k. Larger value of χ favors the alternatives with memberships more and lower value favors the alternatives towards the non-memberships. Example 2 (In transportation). About 70 0 0 m3 of a granular product is to be transported across a river using a boat. Irrespective of the amount to be shipped, the transportation cost per trip is about Rs. 275. The cost of the container (rectangular parallelepiped) that is to be used for shipment depends upon its dimensions and is as: Cost of the base and top of the container is about Rs. 25 per square meter Front and back costs about Rs. 30 per square meter Cost of ends is about Rs 20 per square meter
On solving the each objective under the given set of constraints by using LINGO, the following solutions and pay-off matrix are obtained: X1 = (5.69, 8.85, 7.32 ) and X2 = (12.74, 7.72, 6.10 ) and
X1 X2
φ1 ( x )
φ2 ( x )
12960.47 14971.54
11.67 19.06
let the tolerances defined by the decision maker for φ 1 (x) and φ 2 (x) be α1 = 10 0 0 and α2 = 5. Using these values, the optimistic and pessimistic models are obtained by defining the membership and non-membership as below: The optimistic view: χ (1910375.01/x1 x2 x3 + 51.5x1 x2 + 62.26x1 x3 + 40x2 x3 )t − (12960.47 )t χ s.t. ≤1− (15971.54 )t − (12960.47 )t k (70 0 0/x1 x2 x3 )t − (11.67 )t χ ≤ 1 − (24.06 )t − (11.67 )t k (14971.54 )t − (1910375.01/x1 x2 x3 + 51.5x1 x2 + 62.26x1 x3 + 40x2 x3 )t (1 − k ) (14971.54 )t − (12960.47 )t ) (1910375.01/x1 x2 x3 + 51.5x1 x2 + 62.26x1 x3 + 40x2 x3 )t − (12960.47 )t −k +k≥χ (15971.54 )t − (12960.47 )t (19.06 )t − (70 0 0/x1 x2 x3 )t (70 0 0/x1 x2 x3 )t − (11.67 )t (1 − k ) −k +k≥χ (19.06 )t − (11.67 )t (24.06 )t − (11.67 )t Max
x1 x2 x3 ≥ 0, x1 x2 x3 ≤ 600 x1 , x2 , x3 ≥ 0.
Due to the space and weight restrictions on the boat only one container can be taken during a trip and the volume of that container should not be more than about 600 m3 . The decision maker’s objective is to find the dimensions of the container so that entire product is transported from one bank to another at a minimum cost and in minimum number of trips possible. To reflect the uncertainties in judging the exact values of the parameters, let us assume that they are represented with the help of parabolic fuzzy numbers. Let x1 , x2 and x3 represent the length, breadth and height of the container. Then, the
t > 0 is defined by the decision maker.
The pessimistic view: Max χ , subject to
(14971.54 )t − (1910375.01/x1 x2 x3 + 51.5x1 x2 + 62.26x1 x3 + 40x2 x3 )t +k≥χ (14971.54 )t − (12960.47 )t t t (19.06 ) − (70 0 0/x1 x2 x3 ) (1 − k ) +k≥χ (19.06 )t − (11.67 )t (14971.54 )t − (1910375.01/x1 x2 x3 + 51.5x1 x2 + 62.26x1 x3 + 40x2 x3 )t (1 − k ) (14971.54 )t − (12960.47 )t ) (1 − k )
D. Rani et al. / Expert Systems With Applications 64 (2016) 228–238
237
Table 2 Results comparison. Example
Zimmermann’s approach
Maximum additive operator
Maximum product operator
Example 1 Example 2
φ1 = 583.92, φ2 = 394.12 φ1 = 16427.23, φ2 = 11.67
φ1 = 637.05, φ2 = 549.29 φ1 = 16423.93, φ2 = 11.68
φ1 = 593.20, φ2 = 422.33 Unbounded solution
(1910375.01/x1 x2 x3 + 51.5x1 x2 + 62.26x1 x3 + 40x2 x3 )t − (13971.54 )t +k≥χ (14971.54 )t − (13971.54 )t t t t t (19.06 ) − (70 0 0/x1 x2 x3 ) (70 0 0/x1 x2 x3 ) − (14.06 ) (1 − k ) −k +k≥χ (19.06 )t − (11.67 )t (19.06 )t − (14.06 )t −k
x1 x2 x3 ≥ 0, x1 x2 x3 ≤ 600 x1 , x2 , x3 ≥ 0. t > 0 is defined by the decision maker.
For t =2, k = 1/2, the obtained solution for the optimistic decision maker is λ = 0.5, x1 = 11.21, x2 = 7.63, x3 = 7.02, φ1 = 14628.56, φ2 = 11.66 and for the pessimistic decision maker, solution is λ = 1, x1 = 9.75, x2 = 6.57, x3 = 9.36, φ1 = 14629.81, φ2 = 11.68 5.1. Comparative study The above examples are also solved with some other techniques existing in the literature, like Zimmermann approach, maximum additive operator, maximum product operator (Zimmermann, 1978) by considering the non-linear linear membership function (t = 2) for each of the objective. The non-membership function is not taken into account in these methods. The general models for these techniques are as given below and a comparative study of the obtained results is given in Table 2. Zimmemann’s approach
χ , subject to μLi (φi (x )) ≥ χ , 1 ≤ i ≤ l μUi (φi (x )) ≥ χ , l + 1 ≤ i ≤ l Maximize
and constraints 3.1 − 3.3
χ = min{μLi (φi (x )), μUi (φi (x ))}; 1 ≤ i ≤ l, χ ∈ [0, 1], xi ≥ 0.
where
6. Conclusion and future scope In this paper, an attempt has been made to solve the multiobjective non linear programming problems in fuzzy environment and has been justified by solving two numerical problems, one occurring in the manufacturing system and another in the transportation. Membership function plays a major key role while designing a model in fuzzy sense. Most of the techniques in the literature are based on constructing only the linear membership functions for the fuzzy objective or constraints. But, we have resolved the mutual conflicting nature of the objectives by constructing the region of satisfaction by taking membership as well as nonmembership functions. Moreover, the linear membership functions do not always do justice while modeling a real life decision model. So, general non-linear membership functions have been considered in optimistic as well as pessimistic view. The obtained results are found to be better than those obtained by considering the membership functions only. While constructing the optimistic and pessimistic models, we have chosen the same value of t for the membership and non-membership functions. To get the results of his/her interest, the decision maker can alter this value along with the allowed tolerances depending upon the choice. We have assumed that the decision maker is satisfied with the results obtained in one run of the algorithm. In case he is not satisfied, these tolerance values can be altered and sensitivity analysis can be applied to see the change in results with change in these parameters. The technique effectively deals with vagueness and subjectivity of the decision maker and may be helpful in solving decision making problems in the field of production, planning, manufacturing and transportation. In future, this approach can be studied with some other type of non-linear membership and non-membership functions like exponential, hyperbolic type etc., instead of triangular or parabolic.
Maximum additive operator
Maximize
μLi (φi (x )) + μU j (φ j (x )); 1 ≤ i ≤ l , l + 1 ≤ j ≤ l
subject to
μLi (φi (x )), μU j (φ j (x )) ∈ [0, 1], xi ≥ 0 and constraints 3.1 − 3.3 Maximum product operator
Maximize
μLi (φi (x )) ∗ μU j (φ j (x )); 1 ≤ i ≤ l , l + 1 ≤ j ≤ l
subject to
μLi (φi (x )), μU j (φ j (x )) ∈ [0, 1], xi ≥ 0. and constraints 3.1 − 3.3 From these computational results, it has been concluded that, in case of Example 1, for increasing the values of k, the obtained values of φ 1 and φ 2 improves from 646.65 to 597.99 and 439.07 to 561.89, respectively and hence their corresponding level of satisfaction also increases. Further, these objective values are better than that obtained by any of the existing techniques as given in Table 2. Based on the proposed approach, the decision makers’ may choose their desired level of results according to their satisfaction levels and hence it can be more beneficial and fruitful for the real world optimization problems. Similarly, the results for the considered transportation problem can be compared.
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