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Journal of Intelligent & Fuzzy Systems 28 (2015) 843–865 DOI:10.3233/IFS-141366 IOS Press
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A new approach to solve multi-objective multi-choice multi-item Atanassov’s intuitionistic fuzzy transportation problem using chance operator Dipankar Chakrabortya , Dipak Kumar Janab,∗ and Tapan Kumar Royc
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a Department of Mathematics, Heritage Institute of Technology, East Kolkata Township, Chowbaga Road, Anandapur,
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Kolkata, West Bengal, India b Department of Applied Science, Haldia Institute of Technology, Haldia, Purba Midnapur, West Bengal, India c Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, India
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Abstract. In some practical situations the decision maker is interested in setting multi aspiration levels for objectives that may not be expressed in a specific manner. So in this paper, Atanassov’s intuitionistic fuzzy transportation problem with multi-item, multi-objective function assuming multiple choices is considered. We have modeled multi-objective multi-choice multi-item Atanassov’s intuitionistic fuzzy transportation problem (MMMIFTP), and its several special cases. Possibility, necessity and credibility measures for Atanassov’s intuitionistic fuzzy numbers for the first time have been developed here. Solution methodology of those models using chance operator has been discussed. A real life example is presented to illustrate proposed models numerically and the results are compared. The optimal results are obtained by using three different soft computing techniques (i) Interactive satisfied method, (ii) Global criteria method and (iii) Goal programming method.
1. Introduction
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Keywords: Atanassov’s intuitionistic fuzzy, chance operator, possibility, necessity, creditability, multi-objective, multi-choice, multi-item, transportation problem
Transportation problem is a well known problem of operations research that can be formulated and solved as a linear programme. The classical transportation problem (TP) refers to a special class of linear programming problems. Transportation problem was first introduced by Hitchcock [27]. Several researchers have shown their ∗ Corresponding author. Dipak Kumar Jana, Department of Applied Science, Haldia Institute of Technology, Haldia, Purba Midnapur-721657, West Bengal, India. Tel.: +91 9474056163; Fax: 03224 252800; E-mails:
[email protected] (D. Chakraborty);
[email protected] (Dipak Kumar Jana); roy t
[email protected] (Tapan Kumar Roy).
interest in transportation problem in fuzzy environment (cf. Pramanik et al. [41, 42]). Chanas et al. [20] analysed the transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers. Ojha et al. [15] proposed transportation problem with fuzzy-stochastic cost. Tang and Gong [17] focused on hybrid two-stage transportation and batch scheduling problem. Chanas and Kuchta [18] has developed the the concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Recently multi objective, multi-choice and multiitem optimization have gained more importance for decision making problems in many areas like industry, health care, transportation etc. Biswal and Acharya [5]
1064-1246/15/$27.50 © 2015 – IOS Press and the authors. All rights reserved
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
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istic fuzzy optimization. Xu and Zhou [25] discussed possibility, necessity and credibility measures for fuzzy optimization. Maity and Maiti [32] presented the possibility and necessity constraints for multiitem production-inventory scenario via optimal control theory. Two-warehouse supply-chain model under possibility/necessity/credibility measures have been developed by Das et al. [35]. Panda et al. [34] proposed imprecise constraints single period inventory model with imperfect production and stochastic demand. Measure theory has been used by Ban [30] to developed Atanassov’s intuitionistic fuzzy-valued possibility and necessity measures. Table-1 represents the summary of related literature for transportation problem. With our best knowledge none of them introduced multi objective multi choice multi item transportation problem in Atanassov’s intuitionistic fuzzy environment and its solution methodology using chance operator. The rest of this paper is organized as follows. In section 2, we recall some preliminary knowledge about Atanassov’s intuitionistic fuzzy. Section 3 provided possibility, necessity and credibility measures in Atanassov’s intuitionistic fuzzy number. In section 4, we have proposed MMMIFTP and its several special cases have been discussed in this section. The solution methodology of the proposed models using chance operator has been discussed in section 5. In section 6, a real life example is solved using three different soft computing techniques (i) Interactive satisfied method, (ii) Global criteria method and (iii) Goal programming method. Results of different models and methods are also discussed and compared in this section. Section 7 summarizes the paper as well as some directions for future research.
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presented the transformation of a multi-choice linear programming problem. Mahapatra et al. [33] developed multi-choice stochastic transportation problem involving extreme value distribution. Goal programming approach for a fuzzy multi-choice transportation problem has been presented by Dutta and Murthy [13]. El-Wahed [16] presented a multi-objective transportation problem under fuzziness. Kapur [11] proposed mathematical methods of optimization for multi-objective transportation systems. Lau et al. [8] discussed fuzzy guided multi-objective evolutionary algorithm model for solving transportation problem. Interactive fuzzy goal programming for multi-objective transportation problems has been discussed by ElWaheda and Lee [9]. Gupta and Kumar [38] proposed a new method for solving linear multi-objective transportation problems with fuzzy parameters. Islam and Roy [39] developed entropy based geometric programming and its application of transportation problems under fuzzy environment. Ojha et al. [10] presented a multi-item transportation problem with fuzzy tolerance. Sancak and Salman [12] proposed multi-item dynamic lot-sizing with delayed transportation policy. Biswal and Acharya [40] has done a case study on Multi-choice multi-objective mathematical programming model for integrated production planning. Bankian-Tabrizi et al. [6] solved fuzzy multi-choice goal programming. Atanassov’s intuitionistic fuzzy set(A-IFS) is one of generalization of fuzzy set theory. A-IFS was first introduced by Atanassov [21]. Fuzzy sets is totally characterized by the membership function but A-IFS is described by not only membership function but also a non-membership function so that the sum of both values lies between zero and one [22]. Many researchers have shown their interest in the study of Atanassov’s intuitionistic fuzzy sets/numbers and its application in optimization problem(c.f. [1], [2], [3], [4], [14], [24], [26], [31], [37]). Mahapatra and Roy [19] have discussed about Atanassov’s intuitionistic Fuzzy Number and its arithmetic operation with application on system failure. Solution methodology of linear programming with triangular Atanassov’s intuitionistic fuzzy number has been presented by Dubey and Mehra [23]. Simplex Method in Atanassov’s intuitionistic fuzzy environment has been presented by Parvathi et al. [28]. Hussain et al. [7], Gani [29] discussed method for solving Atanassov’s intuitionistic fuzzy transportation problem. Possibility, necessity and credibility measures play a significant role in fuzzy and Atanassov’s intuition-
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2. Preliminaries Definition 2.1. Atanassov’s Intuitionistic Fuzzy Set (A-IFS)( [21], [22]): Let E be a given set and let A ⊂ E be a set. An A-IFS A∗ in E is given by A∗ = {< x, µA (x), νA (x) >; x ∈ E} where µA : E → [0, 1] and νA : E → [0, 1] define the degree of membership and the degree of non-membership of the element x ∈ E to A ⊂ E satisfy the condition 0 ≤ µA (x) + νA (x) ≤ 1. Definition 2.2. Atanassov’s Intuitionistic Fuzzy Number (A-IFN) [7]: An Atanassov’s intuitionistic I = {< x, µ I , ν I >} of the real line fuzzy subset A A A is called A-IFN, if the followings holds (i) there exist m ∈ , µ (m) = 1 , ν (m) = 0. AI AI
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Table 1 Summary of related literature for transportation problem Objective
Choice
Item
Environments
Multi Single Single Single Single Multi Single Single Multi Multi Multi Single Multi
Single Single Single Single Single Single Single Single Single Single Single Single Multi
Single Multi Single Single Multi Single Single Single Single Single Single Single Multi
Fuzzy Crisp, Atanassov’s intuitionistic Fuzzy Atanassov’s intuitionistic Fuzzy Fuzzy Crisp Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy Atanassov’s intuitionistic Fuzzy
1
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(ii) µ is continuous mapping from to the closed AI interval [0, 1] and x ∈ , the relation 0 ≤ µ (x) + AI ν (x) ≤ 1 holds. I A The membership function and non-membership I is of the following form function of A ⎧ f1 (x), m − α ≤ x ≤ m; ⎪ ⎪ ⎪ ⎨ 1, x = m; µ (x) = AI ⎪ h1 (x), m ≤ x ≤ m + β; ⎪ ⎪ ⎩ 0, otherwise.
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Non-membership function ⎧ ⎪ f2 (x), m − α ≤ x ≤ m, ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ (f1 (x) + f2 (x)) ≤ 1; ⎪ ⎪ ⎪ ⎨ 0, x = m; ν (x) = AI ⎪ h2 (x), m ≤ x ≤ m + β , ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ (h1 (x) + h2 (x)) ≤ 1; ⎪ ⎪ ⎪ ⎩ 1, otherwise.
Here f1 (x) and h1 (x) are strictly increasing and decreasing function in [m − α, m] and [m, m + β] respectively and f2 (x) and h2 (x) are strictly decreasing and increasing function in [m − α , m] and [m, m + β ] I . α are respectively, where m is the mean value of A β are called the left and right spreads of membership function µ (x) respectively. α are β are called the left A (x) and right spreads of non-membership function ν A respectively. Definition 2.3. Trapezoidal Atanassov’s Intuition istic Fuzzy Number(TIFN): Let a1 ≤ a1 ≤ I in written as a2 ≤ a3 ≤ a4 ≤ a4 . A TIFN A (a1 , a2 , a3 , a4 )(a1 , a2 , a3 , a4 ) has membership function (c.f. fig.-1)
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Gupta and Kumar 2012 [38] Mahapatra et al. 2012 [33] Hussain and Kumar 2012 [7] Nagoorgani and Ponnalagu 2012 [14] Ojha et al. 2013 [15] Kapur 1970 [11] Chanas and Kuchta 1996 [18] Chanas et al. 1984 [20] Lau et al. 2009 [8] El-Wahed 2001 [16] Islam and Roy 2006 [39] Ojha et al. 2013 [15] Propsed
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Authors and years
0
a′1
a1
a2
a3
a4
a′4
Fig. 1. Membership and non-membership function of TIFN.
⎧ x−a 1 ⎪ ⎪ a2 −a1 , a1 ≤ x ≤ a2 ; ⎪ ⎪ ⎨ 1, a ≤ x ≤ a ; 2 3 µ (x) = I a4 −x A ⎪ , a ≤ x ≤ a 3 4; ⎪ a4 −a3 ⎪ ⎪ ⎩ 0, otherwise. and non-membership function ⎧ a2 −x ⎪ , a ≤ x ≤ a2 ; ⎪ ⎪ a2 −a1 1 ⎪ ⎪ ⎨ 0, a ≤ x ≤ a ; 2 3 ν (x) = I x−a3 A ⎪ , a3 ≤ x ≤ a4 ; ⎪ ⎪ a −a3 ⎪ ⎪ ⎩ 4 1, otherwise. Definition 2.4. Triangular Atanassov’s Intuitionistic Fuzzy Number(TrIFN): Let a1 ≤ a1 ≤ a2 ≤ a3 ≤ a3 . I in written as (a1 , a2 , a3 )(a , a2 , a ) has A TrIFN A 1 3 membership function (c.f. fig.-2)
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3. Possibility, Necessity and Credibility Measures of Atanassov’s Intuitionistic Fuzzy Number
1
I and B I be two A-IFN with Definition 3.1. Let A , µ and non-membership membership function µ BI AI function ν , ν respectively and R is the set of real I I B A numbers. Then I ∗B I ) = sup min(µ I , µ I ), x, y ∈ R, x ∗ y Posµ (A B A
a′ 1
a1
a2
a′ 3
a3
and
Fig. 2. Membership and non-membership function of TrIFN.
⎧ a2 −x ⎪ , a1 ≤ x ≤ a2 ; ⎪ a2 −a1 ⎪ ⎪ ⎪ ⎨ 0, x = a ; 2
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ν (x) = x−a2 AI ⎪ , a2 ≤ x ≤ a3 ; ⎪ ⎪ a3 −a2 ⎪ ⎪ ⎩ 1, otherwise.
I = I is denoted as A Definition 2.5. A positive TrIFN A (a1 , a2 , a3 )(a1 , a2 , a3 ) where a1 > 0. I = (a1 , a2 , a3 )(a , a2 , a ) Definition 2.6. [36] Let A 1 3 I = (b1 , b2 , b3 )(b , b2 , b ) be two TrIFN then and B 1 3 (i) (ii) (iii) (iv) (v)
where the abbreviation Posµ , Posν represents possibility of membership and non-membership function, and Nesµ , Nesν represents necessity of membership and non-membership function. * is any of the relations , ≤, ≥, =. The dual relationship of possibility and necessity gives that
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and non-membership function
I I , µ ), x, y ∈ R, x ∗ y Nesµ (A ∗ B ) = inf max(µ BI AI I I Nesν (A ∗ B ) = inf max(ν , νBI ), x, y ∈ R, x ∗ y AI
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⎧ x−a 1 ⎪ ⎪ a2 −a1 , a1 ≤ x ≤ a2 ; ⎪ ⎪ ⎨ 1, x = a ; 2 µ (x) = a3 −x AI ⎪ , a ≤ x ≤ a3 ; 2 ⎪ a3 −a2 ⎪ ⎪ ⎩ 0, otherwise.
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0
I I Posν (A ∗ B ) = sup min(ν , νBI ), x, y ∈ R, x ∗ y AI
I ⊕ B I = (a1 + b1 , a2 + b2 , a3 + b3 )(a + A 1 b1 , a2 + b2 , a3 + b3 ). I = (ka1 , ka2 , ka3 )(ka , ka2 , ka ) if k ≥ 0. kA 1 3 I = (ka3 , ka2 , ka1 )(ka , ka2 , ka ) if k < 0. kA 3 1 I B I = (a1 − b3 , a2 − b2 , a3 − b1 )(a − A 1 b3 , a2 − b2 , a3 − b1 ) I ⊗ B I = (a1 b1 , a2 b2 , a3 b3 )(a b , a2 b2 , a b ) A 3 3 1 1
I ∗ B I ∗ B I ) = 1 − Posµ (A I ) Nesµ (A I ∗ B I ∗ B I ) = 1 − Posν (A I ) Nesν (A I ∗ B I ∗ I represents complement of the event A where A I B . I be a A-IFN. Definition 3.2. Measure of AI Let A I Then the fuzzy measure of A for membership and non-membership function are I = λPosµ A I + (1 − λ)Necµ A I Meµ A I = λPosν A I + (1 − λ)Necν A I Meν A where λ(0 ≤ λ ≤ 1) is the optimistic-pessimistic parameter to determine the combined attitude of a decision maker. If λ = 1 then Meµ = Posµ , Meν = Posν . If λ = 0 then Meµ = Nesµ , Meν = Nesν . If λ = 0.5 then Meµ = Crµ , Meν = Crν , where Cr is the credibility measure.
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
b1
a1
b2
a2
b3
0
a3
I and I Fig. 3. Membership function of TrIFN A BI and Posµ (A ≤ BI ).
3.1. Measures of triangular Atanassov’s intuitionistic fuzzy number
a′1
b2
a2
b′3
a′3
I and Fig. 4. Non-membership function of TrIFN A BI and I ≤ Posν (A BI ).
1
0,
a2 −b2 , I ) = B ⎪a2 −a1 +b3 −b2
⎪ ⎩
1,
a2 ≤ b2 ;
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I ≤ Posν (A
⎧ ⎪ ⎪ ⎨
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OR
I = (a1 , a2 , a3 )(a , a2 , a ) and B I = (b1 , b2 , Let A 1 3 b3 )(b1 , b2 , b3 ) are two TrIFNs. From the definition-3 I ≤ B I ) for membership and nonpossibility of (A membership function (c.f. fig.-3 and 4) are as follows ⎧ 1, a2 ≤ b2 ; ⎪ ⎨ b3 −a1 I I Posµ (A ≤ B ) = b3 −b2 +a2 −a1 , a2 > b2 , a1 < b3 ; (1) ⎪ ⎩ 0, a1 ≥ b3 .
b′1
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a2 > b2 , b3 > a1 ; (2)
b3 ≤ a1 .
I ≥ B I ) for From the definition-3.1 possibility of (A membership and non-membership function (c.f. fig.-5 and 6) are as follows ⎧ a2 ≥ b2 ; ⎪ ⎨1, a −b I I 3 1 ≥B )= Posµ (A a −a +b −b , a2 < b2 , a3 > b1 ; ⎪ ⎩ 3 2 2 1 0, a3 ≤ b1 .
I
I
Posν (A ≥ B
⎧ 0, ⎪ ⎪ ⎨
b2 −a2 ) = b2 −b +a −a2 , 1 3 ⎪
⎪ ⎩ 1,
a2 ≥ b2 ;
a2 < b2 , a3 > b1 ;
a3 ≤ b1 .
0.8 0.6 0.4 0.2 0
b1
a1
a2
b2
b3
a3
I and I Fig. 5. Membership function of TrIFN A BI and Posµ (A I ≥ B ).
I ≤ B I ) are as Now by definition-3.1 necessity of (A follows I ≤ B I ) Nesµ (A I > B I ) = 1 − Posµ (A ⎧ 0, b1 ≤ a2 ; ⎪ ⎨ b1 −a2 = a3 −a2 −b2 +b1 , b1 > a2 , b2 < a3 ; ⎪ ⎩ 1, b2 ≥ a3 .
(3)
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I ≤ B I ) Meµ (A I ≤ B I ≤ B I ) + (1 − λ)Nesµ (A I ) = λPosµ (A ⎧ 0, b3 ≤ a1 ; ⎪ ⎪ ⎪ ⎪ b3 −a1 ⎨ λ b3 −b2 +a2 −a1 , a2 > b2 , a1 < b3 ; = b −a 1 2 ⎪ λ + (1 − λ) a3 −a2 −b2 +b1 , b1 > a2 , b2 < a3 ; ⎪ ⎪ ⎪ ⎩ 1, b2 ≥ a3 .
1 0.8 0.6 0.4
I ≤ B I ) Meν (A
b′1
a′1
a2
b2
b′3
a′3
I and Fig. 6. Non-membership function of TrIFN A BI and I I Posν (A ≥ B ).
I ≥ B I ) are as Now by definition-3.2 measure of (A follows
I ≤ B I ) Nesν (A
OR
I ≥ B I ) Meµ (A
(4)
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I > B I ) = 1 − Posν (A ⎧ 0, b2 ≥ a3 ; ⎪ ⎪ ⎪ ⎨ a3 −b2 = , b2 < a3 , b1 > a2 ; a −a −b +b ⎪ 2 2 1 ⎪ ⎪ 3 ⎩ 1, a2 ≤ b1 .
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I ≤ B I ≤ B I ) I ) + (1 − λ)Nesν (A = λPosν (A ⎧ 1, b3 ≤ a1 ; ⎪ ⎪ ⎪ ⎪ a2 −b2 ⎪ ⎪ ⎨ λ a2 −a +b −b2 + (1 − λ), a2 ≥ b2 , b3 > a1 ; 1 3 = a3 −b2 ⎪ ⎪ b2 < a3 , b1 a2 ; (1 − λ) , ⎪ ⎪ a3 −a2 −b2 +b1 ⎪ ⎪ ⎩ 0, b2 ≥ a3 .
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0.2
I ≥ B I ) are as Now by definition-3.1 necessity of (A follows
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I ≥ B I < B I ) = 1 − Posµ (A I ) Nesµ (A ⎧ 0, b3 ≥ a2 ; ⎪ ⎨ a2 −b3 = a2 −a1 −b3 +b2 , a2 > b3 , b2 > a1 ; ⎪ ⎩ 1, b2 ≤ a1 .
I ≥ B I < B I ) = 1 − Posν (A I ) Nesν (A ⎧ 0, b2 ≤ a1 ; ⎪ ⎪ ⎪ ⎨ b2 −a1 = , b2 > a1 , a2 > b3 ; a −a −b +b ⎪ 2 2 1 3 ⎪ ⎪ ⎩ 1, a2 ≤ b3 . I ≤ B I ) are as Now by definition-3.2 measure of (A follows
I ≥ B I ≥ B I ) + (1 − λ)Nesµ (A I ) = λPosµ (A ⎧ 0, a3 ≤ b1 ; ⎪ ⎪ ⎪ ⎪ a −b ⎨ λ a3 −a23 +b21 −b1 , a2 < b2 , a3 > b1 ; = −b3 ⎪ , b2 > a1 , b3 < a2 ; λ + (1 − λ) a2 −aa21 −b ⎪ ⎪ 3 +b2 ⎪ ⎩ 1, b2 ≤ a1 .
I ≥ B I ) Meν (A I ≥ B I ≥ B I ) + (1 − λ)Nesν (A I ) = λPosν (A ⎧ 1, b1 ≥ a3 ; ⎪ ⎪ ⎪ ⎪ b2 −a2 ⎪ ⎪ ⎨ λ b2 −b +a −a2 + (1 − λ), a2 < b2 , a3 > b1 ; 1 3 = b2 −a1 ⎪ ⎪ , b2 > a1 , a2 > b3 ; (1 − λ) ⎪ ⎪ a −a −b +b 2 2 ⎪ 1 3 ⎪ ⎩ 0, b2 ≤ a1 . For λ = 0.5 I ≤ B I ) Crµ (A ⎧ 0, b3 ⎪ ⎪ ⎪ ⎪ b3 −a1 ⎨ 2(b3 −b2 +a2 −a1 ) , a2 = a −2a 3 2 −b2 +2b1 ⎪ ⎪ 2(a3 −a2 −b2 +b1 ) , b2 ⎪ ⎪ ⎩ 1, b2
≤ a1 ; > b2 , b2 < a3 ; > a2 , b2 < a3 ; ≥ a3 .
(5)
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
I ≥ B I ) = Crν (A
≤ a1 ;
I ≤ B I ≤ B I ) ≥ α, Nesν (A I ) ≤ β Nesµ (A
> b2 , b3 > a1 ;
(6)
≥ a3 .
b2 ≤ a1 ;
2 −a1 −b3 +b2 ) 2b2 −2a2 −b1 +a3 2(b2 −b1 +a3 −a2 )
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
, a1 < b2 , b3 < a2 ;
b1 ≥ a3 .
0,
TH
I
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b3 − a1 a2 − b 2 ≥ α, ≤ β. b3 − b 2 + a 2 − a 1 a2 − a 1 + b 3 − b 2
I
I
I
Posµ (A ≤ B ) ≥ α, Posν (A ≥ B ) ≤ β. Now from (1) and (2) I ≤ B I ) ≥ α ⇔ Posµ (A
a3 − b2
b3 − a1 ≥α b3 − b 2 + a 2 − a 1
a2 − b2 ≤β a2 − a 1 + b 3 − b 2
I ≤ x) ≥ α and Posν (A I ≤ x) ≤ β ⇔ Posµ (A a2 −x ≥ α and ≤ β a2 −a1
a3 − a3 − b2 + b1
≤β
I ≤ x) ≥ α and Nesν (A I ≤ x) ≤ β ⇔ Note: Nesµ (A x−a2 a3 −a2
≥ α and
a3 −x
a3 −a2
≤β
I = (a1 , a2 , a3 )(a , a2 , a ) and B I = Lemma 3. If A 1 3 (b1 , b2 , b3 )(b1 , b2 , b3 ) then I ≤ B I ≤ B I ) ≥ α, Crν (A I ) ≤ β Crµ (A ⇔
a3 −2a2 −b2 +2b1 b3 −a1 ≥ α, ≥α 2(b3 −b2 +a2 −a1 ) 2(a3 −a2 −b2 +b1 )
and
2a2 −2b2 +b3 −a1 2(a2 −a1 +b3 −b2 )
≤ β,
a3 −b2
I ≤ B I ≤ B I ) ≥ α, Crν (A I ) ≤ β. Crµ (A
I ≤ B I ) ≥ α Crµ (A ⇔
2(a3 −a2 −b2 +b1 )
Now from (5) and (6)
I ≤ B I ) ≤ β ⇔ Posν (A
x−a1 a2 −a1
I ≤ B I ) ≤ β ⇔ Nesν (A
Proof. Let us consider
and
Note:
b1 − a 2 ≥α a3 − a 2 − b 2 + b 1
I ≤ B I ≤ B I ) ≥ α and Posν (A I ) ≤ β Posµ (A
Proof. Let us consider
I ≤ B I ) ≥ α ⇔ Nesµ (A
, b2 > a2 , a3 > b1 ;
I = (a1 , a2 , a3 )(a , a2 , a ) and B I = Lemma 1. If A 1 3 (b1 , b2 , b3 )(b1 , b2 , b3 ) then
⇔
Now from (3) and (4)
and
b2 −a1
I ≤ B I ≤ B I ) ≥ α, Nesν (A I ) ≤ β. Nesµ (A
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a3 −b1 ⎪ ⎪ 2(a3 −a2 +b2 −b1 ) , a2 < b2 , a3 > b1 ; ⎪ ⎪ ⎩ 0, a3 ≤ b1 .
0,
a3 − b 2 b1 − a2 ≥ α, ≤ β. a3 − a 2 − b 2 + b 1 a3 − a 2 − b 2 + b 1
Proof. Let us consider
⎧ 1, b2 ≤ a1 ; ⎪ ⎪ ⎪ ⎪ ⎨2a2 −2b3 +b2 −a1 , b2 > a1 , a2 > b3 ; 2(a2 −a1 −b3 +b2 )
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2(a
⇔
< a3 , b1 > a2 ;
OR
I ≥ B I ) = Crµ (A
I = (a1 , a2 , a3 )(a , a2 , a ) and B I = Lemma 2. If A 1 3 (b1 , b2 , b3 )(b1 , b2 , b3 ) then
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I ≤ B I ) Crν (A ⎧ 1, b3 ⎪ ⎪ ⎪ ⎪ ⎪ 2a2 −2b2 +b −a ⎪ ⎪ ⎨ 2(a −a +b 3−b 1) , a2 2 2 1 3 = ⎪ a3 −b2 ⎪ ⎪ , b2 ⎪ ⎪ 2(a3 −a2 −b2 +b1 ) ⎪ ⎪ ⎩ 0, b2
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b3 − a1 ≥ α, 2(b3 − b2 + a2 − a1 ) a3 − 2a2 − b2 + 2b1 ≥α 2(a3 − a2 − b2 + b1 )
≤ β.
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D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
and
M
I ≤ B I ) ≤ β Crν (A
⇔
(p)
(p)
j = 1, 2, 3, · · · , N and ∀p
i=1
2a2 − 2b2 + b3 − a1
xij bjI
2(a2 − a1 + b3 − b2 )
≤ β,
p
xij ≥ 0
∀ i, j, p.
(7)
a3 − b 2
2(a3 − a2 − b2 + b1 )
≤ β.
4.2. Model-2: Single objective multi choice single item Atanassov’s intuitionistic fuzzy transportation problem(SMSIFTP)
Note:
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I ≤ x) ≥ α, Crν (A I ≤ x) ≤ β Crµ (A
We propose the mathematical model of single objective (t = 1), k(> 1) choices and single item (p = 1) Atanassov’s intuitionistic fuzzy transportation problem as follows
x − a1 a3 − 2a2 + x ⇔ ≥ α, ≥α 2(a2 − a1 ) 2(a3 − a2 )
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a3 − x 2a2 − x − a1 and ≤ β, ≤ β. 2(a2 − a1 ) 2(a3 − a2 )
min F =
M N
1
2
k
{ cijI , cijI , · · · , cijI }xij
i=1 j=1
4. Atanassov’s intuitionistic fuzzy transportation problem
subject to
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AU
TH
represents items p(= 1, 2, · · · , T ). The variable the unknown quantity to be transported from origin (p) (p) aiI to destination bjI for item p = 1, 2, · · · , T . (p) (p) Here aiI and bjI are Atanassov’s intuitionistic fuzzy numbers.
xij aiI
i = 1, 2, 3, · · · , M
xij bjI
j = 1, 2, 3, · · · , N
j=1
Let p(= 1, 2 · · · , T ) items are to be transported from M origins(or sources) aiI (i = 1, 2, · · · , M), N desti(p) aiI nations(i.e. demands) bjI (j = 1, 2, · · · , N). Let be the product available at ith origin for items p(= (p) 1, 2 · · · , T ), bjI be the demand at jth destination for (p) xij
N
M
(8)
i=1
xij ≥ 0
∀ i, j.
4.3. Model-3: Single objective multi choice multi-item Atanassov’s intuitionistic fuzzy transportation problem(SMMIFTP)
4.1. Model-1: Multi objective multi choice multi item Atanassov’s intuitionistic fuzzy transportation problem(MMMIFTP)
We propose the mathematical model of single objective (t = 1), k(> 1) choices and p(= 1, 2, · · · , T ) items Atanassov’s intuitionistic fuzzy transportation problem as follows
We propose the mathematical model of t(t = 1, 2, · · · , R) objectives, k choices and p(= 1, 2, · · · T ) items Atanassov’s intuitionistic fuzzy transportation problem as follows
min F =
min Ft =
T M N
1(tp)
{ cijI
2(tp)
, cijI
k(tp)
, · · · , cijI
(p)
}xij ,
p=1 i=1 j=1
t = 1, 2, · · · , R subject to
N j=1
(p)
xij aiI
(p)
T M N
1(p)
{ cijI
2(p)
, cijI
k(p)
, · · · , cijI
(p)
}xij
p=1 i=1 j=1
subject to
N
xij aiI
(p)
xij bjI
(p)
(p)
i = 1, 2, 3, · · · , M and ∀p
j=1 M
(p)
j = 1, 2, 3, · · · , N and ∀p
i=1
i = 1, 2, 3, · · · , M and ∀p
(p)
xij ≥ 0 ∀ i, j, p.
(9)
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
4.4. Model-4: Multi objective multi choice single item Atanassov’s intuitionistic fuzzy transportation problem(MMSIFTP)
min F =
min Ft =
(t) I1
(t) I2
subject to
M
{ cij , cij }xij , cij , · · · ,
i = 1, 2, 3, · · · , M
xij bjI
j = 1, 2, 3, · · · , N
(10)
CO
4.7. Model-7: Single objective single choice multi-item Atanassov’s intuitionistic fuzzy transportation problem(SSMIFTP) We propose the mathematical model of single objective, single choice (k = 1) and p(= 1, 2, · · · , T ) items Atanassov’s intuitionistic fuzzy transportation problem as follows
i=1
xij ≥ 0
∀ i, j.
OR
4.5. Model-5: Multi objective single choice multi item Atanassov’s intuitionistic fuzzy transportation problem(MSMIFTP)
min F =
p=1 i=1 j=1
subject to
N
xij aiI
(p)
xij bjI
(p)
(p)
(p)
j = 1, 2, 3, · · · , N and ∀p
i=1 (p)
T M N
(p)
(p)
cijI xij
p=1 i=1 j=1
subject to
N
xij aiI
(p)
i = 1, 2, 3, · · · , M and ∀p
xij bjI
(p)
j = 1, 2, 3, · · · , N and ∀p
(p)
j=1 M
(p)
i=1 (p)
xij ≥ 0 ∀ i, j, p.
(13)
i = 1, 2, 3, · · · , M and ∀p
j=1 M
t = 1, 2, · · · , R
AU
min Ft =
(tp) (p) cijI xij ,
TH
We propose the mathematical model of t(t = 1, 2, · · · , R) objectives, single choice (k = 1) and p(= 1, 2, · · · T ) items Atanassov’s intuitionistic fuzzy transportation problem as follows T M N
(12)
xij ≥ 0 ∀ i, j.
j=1 M
xij bjI j = 1, 2, 3, · · · , N
PY
xij aiI
xij aiI i = 1, 2, 3, · · · , M
i=1
(t) Ik
t = 1, 2, · · · , R N
N j=1
i=1 j=1
subject to
cijI xij
i=1 j=1
We propose the mathematical model of t(t = 1, 2, · · · , R) objectives, k(> 1) choices and and single item (p = 1) Atanassov’s intuitionistic fuzzy transportation problem as follows M N
M N
851
xij ≥ 0 ∀ i, j, p.
(11)
4.6. Model-6: Single objective single choice single item Atanassov’s intuitionistic fuzzy transportation problem(SSSIFTP) We propose the mathematical model of single objective, single choice (k = 1) and single item (p = 1) Atanassov’s intuitionistic fuzzy transportation problem as follows
4.8. Model-8: Multi objective single choice single item Atanassov’s intuitionistic fuzzy transportation problem(MSSIFTP) We propose the mathematical model of t(t = 1, 2, · · · , R) objectives, single choice (k = 1) and single items (p = 1) Atanassov’s intuitionistic fuzzy transportation problem as follows
min Ft =
M N
(t)
cijI xij , t = 1, 2, · · · , R
i=1 j=1
subject to
N j=1
xij aiI
i = 1, 2, 3, · · · , M
852
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
M
xij bjI
Now for the transformation of the objective function involving multi-choice cost parameter under imprecise environments, using [33] and lemma-1, the above model can transform to
j = 1, 2, 3, · · · , N
i=1
xij ≥ 0
∀ i, j.
(14)
(1) f1
Min
(1) (2) + f 2 , f1
(2) + f2 , · · ·
(R) , f1
(R) + f2
5. Solution Methodology Subject to The chance operator is taken as possibility or necessity or credibility measures. We use chance operator to transform the Atanassov’s intuitionistic fuzzy transportation problem into crisp equivalent transportation problem.
(t)
(t)
PY
Zt ≤ f1 + f2 , t = 1, 2, · · · , R N (p) I (p) Chµ ≥ λi xij ai
(16)
j=1
5.1. Model-1 based on chance operator
Chν
N
(p) xij
(p) aiI
≤ ψi
(17)
≥ τj
(18)
to
CO
j=1
Applying Chance operator on (7), model-1 transform (1) (1) (2) (2) (R) (R) Min f1 +f2 ,f1 +f2 ,· · ·,f1 +f2
Chµ
M
(p) xij
(p) bjI
i=1
p=1 i=1 j=1 1 { cijI
(tp)
2 , cijI
p=1 i=1 j=1
Chµ
N
(p) aiI
(p) xij
(p) aiI
(p) xij
(p) bjI
(p) xij
≥ λi
j=1
≤ ψi
j=1
Chµ
M
i=1
Chν
M
xij bjI (p)
(p)
,···
k , cijI
(tp)
(p) }xij
≤
(t) f2
≤β
AU
Chν
N
(tp)
TH
Chν
T M N
OR
Subject to
T M N (tp) (tp) (tp) (p) (t) I1 I2 Ik Chµ { cij , cij , · · · , ≤ f1 ≥α cij }xij
(15)
≥ τj
≤ ηj
i=1 (p) xij
≥ 0 for i = 1, 2, · · · , M and j = 1, 2, · · · , N, t = 1, 2, · · · , R
The abbreviation Chµ and Chν represents chance operator for membership and non-membership function. α, β, λi , ψi , τj and ηj are the predetermined confidence levels such that 0 ≤ λi + ψi ≤ 1, for i = 1, 2, · · · , M, 0 ≤ αt + βt ≤ 1 (t = 1, 2, · · · , R) and 0 ≤ τj + ηj ≤ 1 for j = 1, 2, · · · , N.
Chν
M
(p) xij
(p) bjI
≤ ηj
(19)
i=1 (p)
xij ≥ 0, for i = 1, 2, · · · , M and j = 1, 2, · · · , N (20) which is equivalent to
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
853
The demand amount of items at destination [ bjI ] (p)
1
(1) b2I (2) bI
2
(1) b3I (2) bI
3
⎤ (1) b4I (6, 7, 9)(5, 7, 11) (4, 5, 7)(3, 5, 8) (2, 3, 5)(1.5, 3, 6) (1, 2, 3)(0.5, 2, 4) ⎦= (2) (4, 6, 8)(3, 6, 8) (3, 6, 8)(2, 6, 9) (1, 3, 6)(0.5, 3, 6) (0, 1, 3)(0, 1, 4) b4I
We formulate a MMMIFTP using the above data, where the objective function and constraints are as in Model-1. Now using possibility measure as a chance operator and method explain in solution methodology we get the following crisp transportation model
Z1 , Z2 , · · · , ZR
Min Subject to
(16) − (20) which is a multi objective crisp transportation problem and can be solved using Interactive satisfied method, Goal programming method and Global criteria method. Similarly Model-2, 3, 4, 5, 6, 7 and 8 can be transformed into corresponding equivalent crisp model by above explain method. 6. A real life example
min[Z1 , Z2 ] Subject to
TH
AU
6.1. Input data for model-1,2,3,4:
(1)
(1)
(1)
(1)
(21)
x11 + x21 + x31 ≥ 4 + 2λ4
(2)
(2)
(2)
(2)
(22)
(2) x11
(2) + x21
(2) + x31
≥
(2) 6 − 3ψ4
(23)
(1) x12
(1) + x22
(1) + x32
≥
(1) 4 + λ5
(24)
(1)
(1)
(1)
(1)
(25)
(2)
(2)
(2)
(2)
(26)
(2)
(2)
(2)
(2)
(27)
(1)
(1)
(1)
(1)
(1)
(1)
(2)
(2)
(2)
(2)
(2)
(2)
(1)
(1)
(1)
x11 + x21 + x31 ≥ 7 − 2ψ4
OR
A fruit supplier company of West Bengal, India supply two types of fruits namely mango and orange from four source points namely Darjeeling, Siliguri and Malda of West Bengal, India by lorry or train to the three destinations situated at Kolkata, Ranchi, Durgapur and Kharagpur through twelve routes. The main aims of the proposed models are to minimize the transportation cost(first objective) and packaging cost (second objective), so as to maximize the profit against the market price at different market. Due to fluctuation of fuel price, road tax for different route, political issues, different type of packaging for different route, fluctuation of price packaging material the transportation cost and packaging cost in each route is not fixed. The transportation cost and packaging cost of carrying one unit (500 gm) of mango and orange from source to a destination is treated as a multi objective multi choice multi item transportation problem. Here we have considered the parameters of transportation problem in Atanassov’s intuitionistic fuzzy environment. The input data are given as follows:
PY
(1) b1I ⎣ (2) bI
CO
⎡
x12 + x22 + x32 ≥ 5 − 2ψ5
x12 + x22 + x32 ≥ 3 + 3λ5
x12 + x22 + x32 ≥ 6 − 4ψ5 (1)
x13 + x23 + x33 ≥ 2 + λ6
(1)
x13 + x23 + x33 ≥ 3 − 1.5ψ6 (2)
x13 + x23 + x33 ≥ 1 + 2λ6
2
3
(29) (30)
(2)
x13 + x23 + x33 ≥ 3 − 2.5ψ6 (1)
x14 + x24 + x34 ≥ 1 + λ7
(31) (32)
x14 + x24 + x34 ≥ 2 − 1.5ψ7
(1)
(1)
(1)
(33)
(2) x14
(2) + x24
(2) + x34
≥
(2) λ7
(34)
(2) x14
(2) + x24
(2) + x34
≥
(2) 1 − ψ7
(35)
(tp) xij
(1)
≥ 0∀i, j, t, p.
0 ≤ α(t) + β(t) ≤ 1, for t = 1, 2. 0≤
(p) λi
(p) + βi
(36) (37)
≤ 1, for
(p) p = 1, 2 and i = 1, 2 · · · 7. Amount of items available at origin [ aiI ] (1) (1) (1) (4, 6, 9)(2, 6, 10) (0.5, 1, 3)(0, 1, 5) (8.5, 10, 12)(8, 10, 14) a2I a3I a1I = (1) (1) (1) (5, 8, 9)(4, 8, 10) (0, 2, 4)(0, 2, 6) (6, 8, 10)(5, 8, 11) aI aI aI
1
(28)
(38)
854
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator cijI ] Transportation cost for [ (tp)
cijI
(tp)
i j t p
1 1 1 1
(2,3,4)(1,3,5),(1,2,3)(0,2,4),(5,6,7)(4,6,8)
2 1
cijI
(tp)
t p
(0.5,1,2)(0,1,2),(0,2,3)(0,2,4)
1 2 1 1
(3,4,5)(2,4,6),(1,3,4)(0.5,3,5),(2,4,5)(1,4,6)
2 1
1 3 1 1
(10,11,12)(9,11,13),(5,6,8)(4,6,9)
2 1
(3,4,5)(2,4,7),(2,3,4)(1,3,5),(2,4,6)(1,4,7)
1 4 1 1
(6,7,8)(5,7,9),(2,3,4)(1,3,6)
2 1
(5,6,7)(5,6,8),(4,5,6)(3,5,7)
2 1 1 1
(0,1,2)(0,1,3),(0.5,1,4)(0,1,5),(0,2,3)(0,2,4)
2 1
2 2 1 1
(0,0,0)(0,0,0),(0,0,1)(0,0,2)
2 1
2 3 1 1
(6,7,8)(5,7,8),(1,2,3)(0,2,4)
3 1 1 1
(5,6,8)(4,6,9),(5,5,6)(5,5,7)
2 1
3 2 1 1
(8,9,10)(7,9,11),(9,10,11)(8,10,12),(10,11,12)(9,11,13)
2 1
3 3 1 1
(15,16,17)(14,16,17),(13,14,15)(11,14,16)
2 1
3 4 1 1
(7,8,9)(6,8,10),(6,7,8)(6,7,9)
2 1
(0.5,0.5,2)(0,0.5,3),(0,1,2)(0,1,3)
(1,2,3)(0,2,4),(3,4,5)(2,4,6),(5,6,7)(4,6,8)
1 3 1 2
(7,9,10)(5,9,11),(6,8,9)(5,8,10)
1 4 1 2
(4,5,7)(3,5,8),(3,4,5)(2,4,6),(5,6,7)(4,6,8)
2 1 1 2
(0,1,2)(0,1,3),(1,2,3)(0.5,2,4)
2 3 1 2
(1,2,3)(0,2,4),(0,0.5,2)(0,0.5,3)
(5,5,6)(5,5,7),(4,5,7)(3,5,8)
(15,17,18)(13,17,19),(14,16,17)(12,16,18)
3 1 1 2
(0.5,1,4)(0,1,5),(1,2,3)(0,2,3)
AU
(1,2,4)(0,2,6),(3,4,5)(2,4,6)
(3,4,5)(2,4,6),(2,3,4)(1,3,5)
(6,7,8)(5,7,9),(4,6,8)(3,6,9)
(13,14,16)(12,14,18),(11,12,13)(10,12,13),(10,13,15)(9,13,18)
(6,8,10)(5,8,11),(4,5,6)(3,5,8)
(0,1,2)(0,1,3),(1,2,3)(0.5,2,4)
(1,3,4)(0.5,3,5),(0.5,2,3)(0,2,4)
(2,3,4)(1,3,5),(3,4,5)(2,4,6),(1,2,3)(0.5,2,5)
2 2
(5,5,7)(5,5,8),(4,5,6)(3,5,7),(6,7,8)(5,7,9)
2 2
(0,2,3)(0,2,5),(3,4,5)(2,4,6)
2 2
2 4 1 2
2 2
TH
2 2 1 2
2 2
2 2
1 2 1 2
OR
1 1 1 2
(1,2,3)(0,2,5),(0.5,1,2)(0,1,3)
(7,8,9)(6,8,10),(6,7,8)(5,7,9)
2 1
(13,14,15)(12,14,16),(11,12,13)(10,12,14),(9,10,11)(8,10,12)
(1,2,3)(0,2,4),(1,2,3)(0,2,5),(2,3,4)(1,3,5)
2 1
2 4 1 1
PY
CO
(2,3,4)(1,3,5),(1,2,3)(0,2,5)
(7,8,9)(6,8,10),(5,7,8)(4,7,9),(4,5,6)(3,5,8)
2 2
(12,13,14)(11,13,15),(10,11,12)(9,11,13)
2 2
(9,10,11)(8,10,13),(7,9,10)(6,9,11)
2 2
(2,3,4)(1,3,5),(1,2,3)(0.5,2,6)
3 2 1 2
(2,3,4)(1,3,5),(1,2,3)(0.5,2,4),(0.5,1,2)(0,1,3)
2 2
3 3 1 2
(10,11,12)(9,11,13),(9,10,11)(8,10,14)
2 2
(2,3,4)(1,3,5),(1,2,4)(0.5,2,6),(3,4,5)(2,4,6)
3 4 1 2
(6,7,8)(6,7,9),(5,6,7)(4,6,8)
2 2
(5,6,7)(4,6,8),(4,5,6)(3,5,7)
where Zt for t = 1, 2 has been given in Appendix B. For possibility levels (α(t) = 0.6, β(t) = 0.4) for t = (p) (p) 1, 2 and (λi = 0.5, ψi = 0.4) for i = 1, 2, · · · , 8 and p = 1, 2, Z1 , Z2 are calculated using GRG (LINGO-11.0) and we get Z1U = 150.92, Z1L = 49.20, Z2U = 126.90, Z2L = 49.12, Then we compute the following model to get the interactive satisfied solution,
(1,2,3)(0.5,2,5),(3,4,6)(2,4,8)
⎧ min λ ⎪ ⎪ x ⎪ ⎧ ⎪ ⎨ ⎪ ⎨ Z1 (x) ≤ 150.92 − (µ1 − λ)(150.92 − 49.20) ⎪ Z2 (x) ≥ 126.90 − (µ2 − λ)(126.90 − 49.12) s.t ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ (21) − (38). To get the satisfied solution, we solve the above problem which are listed in the Table-2. The first line of Table-2 lists initial reference of membership function as 1, the value of objective function Z1 (x) and
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
855
Table 2 Interactive satisfied solution of model-1 based on possibility measure Z1
Z2
Value of the decision variables
λ
(1) (1) (1) (1) (1) (1) x11 =6.37, x12 = 1.12, x23 = 2,x31 = 0.12,x32 = 3.37,x33 = 0.5 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =6.5, x12 = 0.99, x13 = 0.006,x23 = 2,x32 = 3.50,x33 = 0.49 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =6.5, x12 = 0.87, x13 = 0.12„x23 = 2,x32 = 3.62,x33 = 0.37 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =6.5, x12 = 0.75, x13 = 0.24,x23 = 2,x32 = 3.74,x33 = 0.25 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =6.5, x12 = 0.67, x13 = 0.32,x23 = 2,x32 = 3.82,x33 = 0.17 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =6.23, x12 = 1.26, x23 = 2,x31 = 0.26,x32 = 3.23,x33 = 0.5 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =6.03, x12 = 1.26, x23 = 2,x31 = 0.46,x32 = 3.23,x33 = 0.5 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =5.83, x12 = 1.66, x23 = 2,x31 = 0.66,x32 = 2.83,x33 = 0.5 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =5.69, x12 = 1.80, x23 = 2,x31 = 0.80,x32 = 2.69,x33 = 0.5 , (1) (2) (2) (2) (2) (2) (2) x34 = 1.5 x11 = 5, x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2,x32 = 0.2
0.253
1
1
74.97
68.82
1
0.98
73.61
69.34
1
0.95
71.69
70.21
1
0.92
69.75
71.07
1
0.90
68.49
71.65
0.98
1
76.32
68.30
0.95
1
78.36
67.53
0.92
1
80.40
66.75
0.90
1
81.76
66.24
PY
µ2
(1)
(1)
(1)
(1)
0.240 0.221 0.202 0.189 0.246 0.236 0.226 0.220
(1)
x13 = 0.41, x23 = 2,x32 = 3.91, x33 = 0.08, x34 = (2) (2) (2) (2) (2) 1.5,x11 = 5,x12 = 1, x13 = 2, x14 = 0.6, x22 = 3.2, (2) x32 = 0.2 . Suppose the tolerance given by decision maker is Z1∗ = 70 and Z2∗ = 65. Then we can use the goal programming method to handle the problem MMMIFTP based on possibility measure and get the following goal programming model ⎧ 2 ⎪ ⎪ ⎪ min wi (di+ + di− ) ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎧ ⎪ ⎨ Z1 (x) + d1+ − d1− = Z1∗ ⎪ ⎪ ⎪ ⎨ Z (x) + d + − d − = Z∗ ⎪ 2 2 2 2 ⎪ ⎪ s.t ⎪ ⎪ ⎪ (21) − (38). ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ + − ⎪ ⎩ di di = 0, di+ , di− ≥ 0, i = 1, 2.
AU
TH
OR
Z2 (x), and its corresponding solution x. If the decision maker hopes that improve Z2 (x) on the basis of sacrifice Z1 (x). We may consider reset the reference value of membership function (µ1 , µ2 ), e.g., we set (µ1 , µ2 ) = (1, 0.98), or (µ1 , µ2 ) = (0.98, 1). The corresponding result are listed in the second and third lines. Suppose that when the reference value of membership function is (µ1 , µ2 ) = (1, 0.9), the decision maker is satisfied, then the interactive process is stopped, so we obtain satisfied solution as x∗ = (1) (1) (1) (1) (1) x11 = 6.5, x12 = 0.67, x13 = 0.32,x23 = 2,x32 = (1) (1) (2) (2) (2) 3.82,x33 = 0.17,x34 = 1.5, x11 = 5, x12 = 1, x13 = (2) (2) (2) 2, x14 = 0.6,x22 = 3.2, x32 = 0.2 and (Z∗ 1, Z2∗ ) = (194.29, 150.18). Moreover the decision maker can modify ZiU , ZiL for i = 1, 2 and built a new reference membership function to obtain his or her satisfactory solution. Applying global criteria method in L2 norm in MMMIFTP based on possibility measure, we get the following result Z1∗ = 67.04 and Z2∗ = 72.30 and (1) (1) optimum solution is x∗ = x11 = 6.50, x12 = 0.58,
CO
µ1
We take w1 = 0.5, w2 = 0.5 and solving the above model, we can obtain the efficient solution as follows (1) (1) (1) (1) x∗ = x11 = 6.50, x12 = 0.76, x13 = 0.23, x23 = 2, (1) (1) (1) (2) (2) x32 = 3.73, x33 = 0.26, x34 = 1.5, x11 = 5, x12 = 1,
Table 3 Optimum results of Model-2, Model-3, Model-4 based on possibility measure Models
Method
Model-2 Model-3
GRG GRG
26.6 49.2
-
Model-4
GLOBAL CRITERIA
55.85
53.61
Z1
Z2
Value of the decision variables x11 = 6.5, x13 = 0.5, x14 = 0.5,x23 = 2,x32 = 4.5,x34 = 1 (1) (1) (1) (1) (1) (1) x11 = 6.5, x13 = 0.5, x14 = 0.5,x23 = 2,x32 = 4.5, x34 = 1, (2) (2) (2) (2) (2) x11 = 5, x12 = 1.78, x14 = 1.19, x23 = 2,x32 = 2.61 x11 = 4.79, x12 = 2.70, x23 = 2,x31 = 1.70,x32 = 0.5, x34 = 1.5
856
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
(2) (2) (2) (2) x13 = 2, x14 = 0.6, x22 = 3.2, x32 = 0.2 . Optimum results of Model-2, Model-3, Model-4 based on possibility measure using three soft computing technique has given in Table-3.
6.2. Input data for Model-5,6,7,8: Let us consider a multi-objective multi-item transportation problem with two types of item (i.e T = 2) 3 origins(i.e M = 3), 4 destinations(i.e N = 4). The parameter are given as follows Transportation cost for 1st objective and 1st item (11) [ cijI ]
⎡
⎤
(11)
CO
(11)
PY
⎡ ⎤ (2, 3, 4)(1, 3, 5) (3, 4, 5)(2, 4, 6) (10, 11, 12)(9, 11, 13) ⎢ (11) (11) (11) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (6, 7, 8)(5, 7, 9) I I I (0, 1, 2)(0, 1, 3) (0, 0, 0)(0, 0, 0) ⎥ c21 c22 ⎢ c14 ⎥ ⎢ ⎥ ⎢ (11) (11) (11) ⎥ = ⎢ I I I (1, 2, 3)(0, 2, 4) (5, 6, 8)(4, 6, 9) ⎥ ⎢ ⎥ ⎣ (6, 7, 8)(5, 7, 8) c c c ⎦ ⎣ 23 24 31 ⎦ (11) I (11) I (11) (8, 9, 10)(7, 9, 11) (15, 16, 17)(14, 16, 18) (7, 8, 9)(6, 8, 10) I c32 c33 c34 (11)
I I I c11 c13 c12
Transportation cost for 1st objective and 2nd item (12) [ cijI ]
⎢ (12) (12) ⎢ I I c21 ⎢ c14 ⎢ (12) (12) I I ⎢ c24 ⎣ c23 (12)
(12)
I I c32 c33
⎡
(21)
(21)
(21)
(21)
I I c32 c33
⎡
⎤
⎡
Transportation cost for 2st objective and 1st item (21) [ cijI ]
⎤ (0.5, 1, 2)(0, 1, 2) (2, 3, 4)(1, 3, 5) (3, 4, 5)(2, 4, 7) ⎥ ⎢ ⎥ I (21) ⎥ (5, 6, 7)(5, 6, 8) (1, 2, 3)(0, 2, 5) (1, 2, 4)(0, 2, 6) ⎥ c22 ⎥ ⎢ ⎢ ⎥ ⎥=⎢ ⎥ I (21) ⎥ (13, 14, 15)(12, 14, 16) (7, 8, 9)(6, 8, 10) (3, 4, 5)(2, 4, 6) c31 ⎦ ⎦ ⎣ (21) (6, 7, 8)(5, 7, 9) (13, 14, 16)(12, 14, 18) (6, 8, 10)(5, 8, 11) I c34 (21)
I I I c11 c12 c13
⎢ (21) (21) ⎢ I I c21 ⎢ c14 ⎢ (21) (21) I I ⎢ c24 ⎣ c23
⎤
⎤ (0.5, 0.5, 2)(0, 0.5, 3) (1, 2, 3)(0, 2, 4) (7, 9, 10)(5, 9, 11) ⎥ ⎢ ⎥ I (12) ⎥ (4, 5, 7)(3, 5, 8) (0, 1, 2)(0, 1, 3) (1, 2, 3)(0, 2, 4) ⎥ c22 ⎥ ⎢ ⎢ ⎥ = ⎥ ⎢ I (12) ⎥ (5, 5, 6)(5, 5, 7) (15, 17, 18)(13, 17, 19) (0.5, 1, 4)(0, 1, 5) ⎥ c31 ⎣ ⎦ ⎦ (12) (2, 3, 4)(1, 3, 5) (10, 11, 12)(9, 11, 13) (6, 7, 8)(6, 7, 9) I c34 (12)
OR
(12)
TH
(12)
I I I c11 c12 c13
AU
⎡
Transportation cost for 2nd objective and 2nd item (22) [cij ] ⎡
⎤
⎤ ⎡ (0, 1, 2)(0, 1, 3) (1, 3, 4)(0.5, 3, 5) (2, 3, 4)(1, 3, 5) ⎢ (22) (22) (22) ⎥ ⎥ ⎢ ⎢ I I I (5, 5, 7)(5, 5, 8) (0, 2, 3)(0, 2, 5) (7, 8, 9)(6, 8, 10) ⎥ c21 c22 ⎥ ⎥ ⎢ ⎢ c14 ⎥ ⎢ (22) (22) (22) ⎥ = ⎢ I I I ⎥ ⎢ ⎣ (12, 13, 14)(10, 13, 15) (9, 10, 11)(8, 10, 13) (2, 3, 4)(1, 3, 5) ⎦ c24 c31 ⎦ ⎣ c23 (22) I (22) I (22) (1, 2, 3)(0.5, 2, 5) (1, 2, 3)(0, 2, 5) (5, 6, 7)(4, 6, 8) cI c c (22)
(22)
(22)
I I I c11 c13 c12
32
33
34
(p)
Amount of items available at origin [ aiI ]
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
(1)
a2I
(1)
a2I
a1I a1I
(1)
a3I
(1)
a3I
(1)
=
(1)
(4, 6, 9)(2, 6, 10) (0.5, 1, 3)(0, 1, 5) (8.5, 10, 12)(8, 10, 14) (5, 8, 9)(4, 8, 10) (0, 2, 4)(0, 2, 6) (6, 8, 10)(5, 8, 11)
857
(p) The demand amount of items at destination [ bjI ]
1
(1) b2I (2) bI
2
(1) b3I (2) bI
3
⎤ (1) b4I (6, 7, 9)(5, 7, 11) (4, 5, 7)(3, 5, 8) (2, 3, 5)(1.5, 3, 6) (1, 2, 3)(0.5, 2, 4) ⎦ = (2) (4, 6, 8)(3, 6, 8) (3, 6, 8)(2, 6, 9) (1, 3, 6)(0.5, 3, 6) (0, 1, 3)(0, 1, 4) b4I
Using the above input data Model-5 can be formulate as
Min
(1)
(1)
(2)
(2)
f1 + f2 , f1 + f2
(1)
(41) − (43)
(1)
(1)
(1)
(1)
(1)
OR
Which is equivalent to (1) (1) (1) (1) (1) min Z1 = (1 − α ) 2x11 + 3x12 + 10x13 + 6x14 (1)
(1)
+ 0.x21 + 0.x22 + 6x23 + x24 + 5x31 (1)
(2)
(2)
+ 8x32 + 15x33 + 7x34 + 0.5x11 + x12 (2)
(2)
(2)
(2)
(2)
(2)
(2)
TH
+ 7x13 + 4x14 + 0.x21 + 1x22 + 5x23 (2)
(2)
+ 15x24 + 0.5x31 + 2x32 + 10x33
(2) (1) (1) (1) (1) + 6x34 +(1−β +α ) 3x11 + 4x12 (1)
(1)
(1)
(1)
(2)
(2)
(2)
(2)
(1)
(1)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(1) (1) (1) + (1 − β(2) + α(2) ) x11 + 3x12 + 4x13 (1)
(1)
(1)
(1)
(2)
(2)
(1)
(1)
(1)
+ 6x14 + 2x21 + 2x22 + 14x23 + 8x24 (1)
(1)
(2)
+ 4x31 + 7x32 + 14x33 + 8x34 + x11 (2)
(2)
(2)
+ 3x12 + 3x13 + 5x14 + 2x21 + 8x22
(1)
(1)
(2)
(2)
(2)
(2)
(2)
(2)
(1)
(1)
(1)
(1)
(2)
(2)
(2)
(1)
(1)
(2)
(1)
(1)
(1)
(1)
(2)
(1)
(2)
(2)
(2)
+ 0.5x12 + x13 + 5x14 + 0.x21 + 6x22
(2)
(1)
(2)
(2)
+ 2x13 + 5x14 + 0.x21 + 7x22 + 12x23
(2) (2) (2) (2) (2) + 9x24 + 2x31 + x32 + x33 + 5x34
+ 2x31 + 5x32 + 12x33 + 5x34 + 0.x11
+ 0.x11 + 0.x12 + 5x13 + 3x14 + 0.x21 (2)
(2)
(2)
(2)
(1)
(2)
(2)
+ 5x14 + 0.x21 + 0.x22 + 12x23 + 6x24
+ 0.x24 + 4x31 + 7x32 + 14x33 + 6x34 (2)
(1)
(1)
(2)
+ 0.x22 + 5x23 + 13x24 + 0.x31 + 1x32
(2) (2) + 9x33 + 6x34
(2)
(2)
(2)
(2)
+ 10x23 + 8x24 + x31 + 0.5x32 + 0.x33
(2) + 4x34
+ 9x13 + 5x14 + 0.x21 + 0.x22 + 5x23 (1)
(1)
(1)
+ 2x22 + 5x23 + 17x24 + x31 + 3x32
(2) (2) (1) (1) (1) x11 + 2x12 + 11x33 + 7x34 + β (1)
(1)
(1)
+ 0.5x11 + 2x12 + 9x13 + 5x14 + x21 (2)
(1)
+ 6x32 + 13x33 + 6x34 + 0.x11 + x12
(2)
+ 2x24 + 6x31 + 9x32 + 16x33 + 8x34 (2)
(1)
(1)
AU
(1)
(1)
+ 13x23 + 10x24 + 3x31 + 2x32 + 2x33
(2) (1) (1) (1) + 6x34 + β(2) 0.x11 + x12 + 2x13
+ 11x13 + 7x14 + x21 + 0.x22 + 7x23 (1)
(1)
+ x21 + x22 + 13x23 + 7x24 + 3x31
Subject to (21) − (38) and
(1) (1) (1) (1) min Z2 = (1 − α(2) ) 0.5x11 + 2x12 + 3x13 + 5x14
PY
(1) bI ⎣ 1(2) bI
CO
⎡
Subject to (21) − (38)
(39)
As the above problem is a multi objective problem we can use the Interactive satisfied method or global criteria method or goal programming method to solve the above transportation problem.
858
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator Table 4 Interactive satisfied solution of model-5 based on possibility measure Value of the decision variables
Z1
Z2
λ
(1) (1) (1) (1) (1) (1) x11 =2.5, x12 = 2.5, x13 = 2.5,x22 = 2,x31 = 4,x34 = 1.5 (2) (2) (2) (2) (2) (2) (2) x11 = 2.75, x12 = 4.2, x13 = 1.04, x14 = 0.6,x21 = 2.24,x23 = 0.95, x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =2.5, x12 = 2.5, x13 = 2.5,x22 = 2,x31 = 4,x34 = 1.5 (2) (2) (2) (2) (2) (2) (2) x11 = 2.81, x12 = 4.2, x13 = 0.98, x14 = 0.6,x21 = 2.18,x23 = 1.01, x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =2.5, x12 = 2.5, x13 = 2.5,x22 = 2,x31 = 4,x34 = 1.5 (2) (2) (2) (2) (2) (2) (2) x11 = 2.70, x12 = 4.2, x13 = 1.09, x14 = 0.6,x21 = 2.29,x23 = 0.9, x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =2.5, x12 = 2.5, x13 = 2.5,x22 = 2,x31 = 4,x34 = 1.5 (2) (2) (2) (2) (2) (2) (2) x11 = 2.84, x12 = 4.2, x13 = 0.95, x14 = 0.6,x21 = 2.15,x23 = 1.04, x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =2.5, x12 = 2.5, x13 = 2.5,x22 = 2,x31 = 4,x34 = 1.5 (2) (2) (2) (2) (2) (2) (2) x11 = 2.67, x12 = 4.2, x13 = 1.12, x14 = 0.6,x21 = 2.32,x23 = 0.87, x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =2.5, x12 = 2.5, x13 = 2.5,x22 = 2,x31 = 4,x34 = 1.5 (2) (2) (2) (2) (2) (2) (2) x11 = 2.89, x12 = 4.2, x13 = 0.9, x14 = 0.6,x21 = 2.10,x23 = 1.09, x32 = 0.2 (1) (1) (1) (1) (1) (1) x11 =2.5, x12 = 2.5, x13 = 2.5,x22 = 2,x31 = 4,x34 = 1.5 (2) (2) (2) (2) (2) (2) (2) x11 = 2.61, x12 = 4.2, x13 = 1.18, x14 = 0.6,x21 = 2.38,x23 = 0.81, x32 = 0.2
195.13
147.60
0.288
194.79
148.63
0.277
195.46
146.57
0.278
194.62
149.15
0.252
195.63
146.06
0.253
194.29
150.18
0.212
195.97
145.03
0.213
1
1
1
0.98
0.98
1
0.98
0.95
0.95
0.98
0.95
0.9
0.9
0.95
PY
µ2
CO
µ1
⎧ min λ ⎪ ⎪ x ⎪ ⎧ ⎪ ⎨ ⎪ ⎨ Z1 (x) ≤ 218.4 − (µ1 − λ)(218.4 − 185.72) ⎪ Z2 (x) ≥ 223.02 − (µ2 −λ)(223.02−117.1) s.t ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ (21) − (38).
For possibility levels (α(t) = 0.6, β(t) = 0.4) for t = (p) (p) 1, 2 and (λi = 0.5, ψi = 0.4) for i = 1, 2, · · · 8 and p = 1, 2, Z1 , Z2 are calculated using GRG and we get
OR
Z1U = 218.4, Z1L = 185.72, Z2U = 223.02, Z2L = 117.1 So we can get the membership function of Z1 and Z2 as follows,
µ2 (Z2 (x)) ⎧ 1 ⎪ ⎨ 223.02−Z2 (x) = 223.02−117.1 ⎪ ⎩ 0
for
185.72 < Z1 (x) < 218.4 for Z1 (x) > 218.4
for for
TH
for
Z1 (x) < 185.72
AU
µ1 (Z1 (x)) ⎧ 1 ⎪ ⎨ 218.4−Z1 (x) = 218.4−185.72 ⎪ ⎩ 0
Z2 (x) < 117.1
117.1 < Z2 (x) < 223.02 for Z1 (x) > 223.02
Then we compute the following model to get the interactive satisfied solution,
To get the satisfied solution, we solve the above problem which are listed in the Table-4. The first line of Table-4 lists initial reference of membership function as 1, the value of objective function Z1 (x) and Z2 (x), and its corresponding solution x. If the decision maker hopes that improve Z2 (x) on the basis of sacrifice Z1 (x). We may consider reset the reference value of membership function (µ1 , µ2 ), e.g., we set (µ1 , µ2 ) = (1, 0.98), or (µ1 , µ2 ) = (0.98, 1). The corresponding result are listed in the second and third lines. Suppose that when the reference value of membership function is (µ1 , µ2 ) = (0.95, 0.9), the decision maker is satisfied, then the interactive process is stopped, so we obtain satisfied solu (1) (1) (1) (1) tion as x∗ = x11 = 2.5, x12 = 2.5, x13 = 2.5,x22 = (1) (1) (2) (2) (2) 2,x31 = 4,x34 = 1.5, x11 = 2.61, x12 = 4.2, x13 = (2) (2) (2) (2) 1.18, x14 = 0.6,x21 = 2.38,x23 = 0.81, x32 = 0.2
Table 5 Optimum results of Model-6, Model-7, Model-8 based on possibility measure Models
Method
Model-6 Model-7
GRG GRG
144 185.72
-
x11 = 2.5, x12 = 4.5, x13 = 0.5,x23 = 2,x31 = 4.5,x34 = 1.5 (1) (1) (1) (1) (1) (1) x11 = 2.5, x12 = 4.5, x13 = 0.5,x23 = 2,x31 = 4, x34 = 1.5, (2) (2) (2) (2) (2) (2) x11 = 4.8, x12 = 3.2, x14 = 0.6, x22 = 1.2,x23 = 2,x31 = 0.2
Model-8
GLOBAL CRITERIA
145.2
108.3
x11 = 2.5, x12 = 3.5, x13 = 1.5,x22 = 1,x23 = 1,x31 = 4 ,x34 = 1.5
Z1
Z2
Value of the decision variables
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
AU
TH
We take w1 = 0.5, w2 = 0.5 and solving the above model, we can obtain the efficient solution as fol(1) (1) (1) (1) lows x∗ = x11 = 2.5, x12 = 2.5, x13 = 2.5,x22 = (1) (1) (2) (2) (2) 2,x31 = 4,x34 = 1.5, x11 = 2.77, x12 = 4.2, x13 = (2) (2) (2) (2) 1.02, x14 = 0.6, x21 = 2.22,x23 = 0.97, x32 = 0.2 . Optimum results of Model-6, Model-7, Model-8 based on possibility measure using three soft computing technique has given in Table-5.
7. Conclusion
The authors sincerely thank the anonymous reviewers and Editor-in-Chief for their careful reading, constructive comments and fruitful suggestions. The first two authors also thank the Ms. Priyanka De, Assistant Professor, Haldia Institute of Technology for advices on grammar, organization of the paper.
PY
References
[1] P. Burillo and H. Bustince, Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems 78 (1996), 305–316. [2] H. Bustince, A. Jurio, A. Pradera, R. Mesiar and G. Beliakov, Generalization of the weighted voting method using penalty functions constructed via faithful restricted dissimilarity functions, European Journal of Operational Research 225 (2013), 472–478. [3] D. Paternain, A. Jurio, E. Barrenechea, H. Bustince, B. Bedregal and E. Szmidt, An alternative to fuzzy methods in decisionmaking problems, Expert Systems with Applications 39 (2012), 7729–7735. [4] N.R. Pal, H. Bustince, M. Pagola, U.K. Mukherjee, D.P. Goswami and G. Beliakov, Uncertainties with Atanassov’s intuitionistic fuzzy sets: Fuzziness and lack of knowledge, Information Sciences 228 (2013), 61–74. [5] M.P. Biswal and S. Acharya, Transformation of a multi-choice linear programming problem, Applied Mathematics and Computation 210 (2009), 182–188. [6] B. Bankian-Tabrizi, K. Shahanaghi and M. Jabalameli, Fuzzy multi-choice goal programming, Applied Mathematical Modelling 36 (2012), 1415–1420. [7] R.J. Hussain and S.P. Kumar, Algorithmic approach for solving intuitionistic fuzzy transportation problem, Applied Mathematical Sciences 6 (2012), 3981–3989. [8] H.C.W. Lau, T.M. Chan, W.T. Tsui, F.T.S. Chan, G.T.S. Ho and K.L. Choy, A fuzzy guided multi-objective evolutionary algorithm model for solving transportation problem, Expert Systems with Applications 36 (2009), 8255–8268. [9] W. El-Waheda and S.M. Lee, Interactive fuzzy goal programming for multi-objective transportation problems, Omega 34 (2006), 158–166. [10] A. Ojha, B. Das, S. Mondala and M. Maiti, A multi-item transportation problem with fuzzy tolerance, Applied Soft Computing 13 (2013), 3703–3712. [11] K. Kapur, Mathematical methods of optimization for multiobjective transportation systems, Socio-Economic Planning Sciences 4 (1970), 451–467. [12] E. Sancak and F. Salman, Multi-item dynamic lot-sizing with delayed transportation policy, International Journal of Production Economics 131 (2011), 595–603. [13] D. Dutta and S. Murthy, Multi-choice goal programming approach for a fuzzy transportation problem, International Journal of Research and Reviews in Applied Sciences 2 (2010), 132–139. [14] A. Nagoorgani and R. Ponnalagu, A new approach on solving intuitionistic fuzzy linear programming problem, Applied Mathematical Sciences 6 (2012), 3467–3474.
OR
⎧ 2 ⎪ ⎪ ⎪ min wi (di+ + di− ) ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎧ ⎪ ⎨ Z1 (x) + d1+ − d1− = Z1∗ ⎪ ⎪ ⎪ ⎨ Z (x) + d + − d − = Z∗ ⎪ 2 2 2 2 ⎪ ⎪ s.t ⎪ ⎪ ⎪ (21) − (38). ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ + − ⎪ ⎩ di di = 0, di+ , di− ≥ 0, i = 1, 2.
Acknowledgments
CO
and (Z1∗ , Z2∗ ) = (194.29, 150.18). Moreover the decision maker can modify ZiU , ZiL for i = 1, 2 and built a new reference membership function to obtain his or her satisfactory solution. Applying global criteria method in L2 norm in MSMIFTP based on possibility measure, we get the following result Z1∗ = 194.86 and Z2∗ = 148.42 and (1) (1) optimum solution is x∗ = x11 = 2.5, x12 = 2.5, (1) (1) (1) (1) (2) x22 = 2,x31 = 4, x34 = 1.5, x11 = x13 = 2.5, (2) (2) (2) (2) 2.8,x12 = 4.2, x13 = 0.99, x14 = 0.6, x21 = 2.1, (2) (2) x23 = 1, x32 = 0.2 . Suppose the tolerance given by decision maker is Z1∗ = 195 and Z2∗ = 148. Then we can use the goal programming method to handle the problem (39) and get the following goal programming model
Multi-objective multi-choice multi-item Atanassov’s intuitionistic fuzzy transportation problem has been modeled and its eight cases has been discussed. Solution methodology of MMMIFTP and its eight cases using chance operator has been developed. To validate the proposed method a real life example is presented and the solution of eight proposed models has been discussed. The optimal results are obtained by using three different soft computing techniques (i) Interactive satisfied method, (ii) Global criteria method and (iii) Goal programming method. The present idea can be extended for solid transportation problem for future research work.
859
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
PY
[31] P.P. Angelov, Optimization in an intuitionistic fuzzy environment, Fuzzy Sets and Systems 86 (1997), 299–306. [32] K. Maity and M. Maiti, Possibility and necessity constraints and ˘ TA ˇ multi-item production-inventory their defuzzificationˆaA scenario via optimal control theory, European Journal of Operational Research 177 (2007), 882–896. [33] R.D. Mahapatra, K.S. Roy and P.M. Biswal, Multi-choice stochastic transportation problem extreme value distribution, Applied Mathematical Modelling 37 (2013), 2230–2240. [34] D. Panda, S. Kar, K. Maity and M. Maiti, A single period inventory model with imperfect production and stochastic demand under chance and imprecise constraints, European Journal of Operational Research 188 (2008), 121–139. [35] B. Das, K. Maity and M. Maiti, A two warehouse supplychain model under possibility/necessity/credibility measures, Mathematical and Computer Modelling 46 (2007), 398–409. [36] G.S. Mahapatra and T.K. Roy, Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations, World Academy of science, Engineering and Technology 50 (2009), 574–581. [37] R. Parvathi and C. Malathi, Intuitionistic fuzzy linear programming problems, World Applied Sciences Journal 17 (2012), 1802–1807. [38] A. Gupta and A. Kumar, A new method for solving linear multi-objective transportation problems with fuzzy parameters, Applied Mathematical Modelling 36 (2012), 1421–1430. [39] S. Isla and T.K. Roy, A new fuzzy multi-objective programming: Entropy based geometric programming and its application of transportation problems, European Journal of Operational Research 173 (2006), 387–404. [40] M.P. Biswal and S. Acharya, Multi-choice multi-objective mathematical programming model for integrated production planning: a case study, International Journal of Systems Science 44 (2012), 1651–1665. [41] S. Pramanik, D.K. Jana and M. Maiti, A multi objective Solid Transportation Problem in Fuzzy, Bi-fuzzy environment via Genetic Algorithm, International Journal of Advanced Operations Management 6 (2014), 4–26. [42] S. Pramanik, D.K. Jana and M. Maiti, Multi-objective solid transportation problem in imprecise environments, Journal of Transportation Security 6 (2013), 131–150.
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[15] A. Ojha, B. Das, S. Mondala and M. Maiti, A transportation problem with fuzzy-stochastic cost, Applied Mathematical Modelling 38 (2013), 1464–1481. [16] W. El-Wahed, A multi-objective transportation problem under fuzziness, Fuzzy Sets and Systems 117 (2001), 27–33. [17] L. Tang and H. Gong, A hybrid two-stage transportation and batch scheduling problem, Applied Mathematical Modelling 32 (2008), 2467–2479. [18] S. Chanas and D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and Systems 82 (1996), 299–305. [19] G.S. Mahapatra and T.K. Roy, Intuitionistic fuzzy number and its arithmetic operation with application on system failure, Journal of Uncertain Systems 7 (2013), 92–107. [20] S. Chanas, W. Kolodziejczyk and A. Machaj, A fuzzy approach to the transportation problem, Fuzzy Sets and Systems 13 (1984), 211–221. [21] K.T. Atanassov, Intuitionistic fuzzy sets, Sofia, Bulgarian, 1983. [22] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96. [23] D. Dubey and A. Mehra, Linear programming with triangular intuitionistic fuzzy number, Advances in Intelligent Systems Research 1 (2011), 563–569. [24] M. Esmailzadeh and M. Esmailzadeh, New distance between triangular intuitionistic fuzzy numbers, Advances in Computational Mathematics and its Applications 2 (2013), 3–10. [25] J. Xu and X. Zhou, Fuzzy-Like Multiple Objective Decesion Making, Springer, 2010. [26] S. Wan and J. Dong, A possibility degree method for intervalvalued intuitionistic fuzzy for decision making, Journal of Computer and System Sciences 80 (2014), 237–256. [27] F.L. Hitchcock, The distribution of a product from several sources to numerous localities, Journal Mathematical Physics 20 (1941), 224–230. [28] R. Parvathi and C. Malathi, Intuitionistic fuzzy simplex method, International Journal of Computer Applications 48 (2012), 39–48. [29] A.N. Gani and S. Abbas, Solving intuitionstic fuzzy transportation problem using zero suffix algorithm, International Journal of Mathematics Sciences & Enggineering Applications 6 (2012), 73–82. [30] A.I. Ban, Intuitionistic fuzzy-valued possibility and necessity measures, Eighth International Conference on IFSs, Varna 10 (2004), 1–7.
CO
860
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
861
8. Appendix A
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1 − α ) 2x11 + 3x12 + 10x13 + 6x14 + 0.x21 + 0.x22 + 6x23 + x24 + 5x31 + 8x32 + 15x33 + 7x34 (1)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
+0.5x11 + x12 + 7x13 + 4x14 + 0.x21 + 1x22 + 5x23 + 15x24 + 0.5x31 + 2x32 + 10x33 + 6x34
(1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) +α(1) 3x11 + 4x12 + 11x13 + 7x14 + x21 + 0.x22 + 7x23 + 2x24 + 6x31 + 9x32 + 16x33 + 8x34 (2) + x21
(2) + 2x22
(2) + 5x23
(2) (2) + 17x24 +x31
(1)
(1)
(1)
(2)
(2)
(2)
(1)
(1)
(1)
(1)
PY
(2) (2) (2) (2) +0.5x11 +2x12 +9x13 +5x14
(2) (2) (2) + 3x32 +11x33 +7x34
(1)
(1)
(1)
(1)
≤ f1
(40)
(1)
(1)
(1)
(2)
(2)
CO
(1 − β ) 3x11 + 4x12 + 11x13 + 7x14 + x21 + 0.x22 + 7x23 + 2x24 + 6x31 + 9x32 + 16x33 + 8x34 (2)
(2)
(2)
(2)
(2)
(2)
(2)
+0.5x11 + 2x12 + 9x13 + 5x14 + x21 + 2x22 + 5x23 + 17x24 + x31 + 3x32 + 11x33 + 7x34
(1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) +β(1) x11 + 2x12 + 9x13 + 5x14 + 0.x21 + 0.x22 + 5x23 + 0.x24 + 4x31 + 7x32 + 14x33 + 6x34
OR
(2) (2) (2) (2) (2) (2) (2) (2) (2) +0.x11 +0.x12 +5x13 +3x14 +0.x21 +0.x22 +5x23 +13x24 +0.x31
(1)
(1)
(1)
(1)
(1)
(1)
(2) + 1x32
(1)
(2) + 9x33
(1)
(2) + 6x34
(1)
(1)
≤ f2
(41)
(1)
(1)
(1)
(2)
(2)
(2)
(2)
TH
(1 − α ) 0.5x11 + 2x12 + 3x13 + 5x14 + x21 + x22 + 13x23 + 7x24 + 3x31 + 6x32 + 13x33 + 6x34 (2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
+0.x11 + x12 + 2x13 + 5x14 + 0.x21 + 7x22 + 12x23 + 9x24 + 2x31 + x32 + x33 + 5x34 (1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(2) + 13x23
(2) + 10x24
(1)
(1)
(1)
(1)
x11 + 3x12 + 4x13 + 6x14 + 2x21 + 2x22 + 14x23 + 8x24 + 4x31 + 7x32 + 14x33 + 8x34
(2) (2) +x11 +3x12
(2) + 3x13
AU
+α
(2)
(2) + 5x14
(2) + 2x21
(2) + 8x22
(2) (2) (2) (2) + 3x31 +2x32 +2x33 +6x34
(2)
≤ f1
(42)
(1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1 − β(2) ) x11 + 3x12 + 4x13 + 6x14 + 2x21 + 2x22 + 14x23 + 8x24 + 4x31 + 7x32 + 14x33 + 8x34 (2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
+x11 + 3x12 + 3x13 + 5x14 + 2x21 + 8x22 + 13x23 + 10x24 + 3x31 + 2x32 + 2x33 + 6x34 +β
(2)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
0.x11 + x12 + 2x13 + 5x14 + 0.x21 + 0.x22 + 12x23 + 6x24 + 2x31 + 5x32 + 12x33 + 5x34
(2) (2) (2) (2) (2) (2) (2) +0.x11 +0.5x12 +x13 +5x14 +0.x21 +6x22 +10x23
(2) + 8x24
(2) + x31
(2) (2) (2) + 0.5x32 +0.x33 +4x34
(2)
≤ f2
(43)
862
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator (11)
g24 = 1(1 − y81 )(1 − y82 ) + y81 (1−y82 )+2(1−y81 )y82 ,
9. Appendix B
(t1) (1) (t1) (1) (t1) (1) (t) g11 x11 + g12 x12 + g13 x13 Zt = 1 − α
(11)
g31 = 5y91 ) + 5(1 − y91 ), (11)
(t1) (1) + g21 x21
(t1) (1) + g22 x22
(t1) (1) + g23 x23
(t1) (1) +g24 x24
(t1) (1) + g31 x31
(t1) (1) + g32 x32
(t1) (1) + g33 x33
(t1) (1) +g34 x34
(t2) (2) + g11 x11
(t2) (2) + g12 x12
(t2) (2) + g13 x13
(t2) (2) +g14 x14
(t2) (2) + g21 x21
(t2) (2) + g22 x22
(t2) (2) + g23 x23
(t2) (2) +g24 x24
(t2) (2) + g31 x31
(t2) (2) + g32 x32
(t2) (2) + g33 x33
(t2) (2)
+g34 x34
1 2 1 2 g32 = 8(1 − y10 )(1 − y10 ) + 9y10 (1 − y10 ) 1 2 )y10 , +10(1 − y10 (11)
1 1 g33 = 15y11 + 13(1 − y11 ), (11)
1 1 g34 = 7y12 + 6(1 − y12 ), (12)
1 2 g11 = 0.5y13 + 0(1 − y13 ), (12)
1 2 1 2 g12 = (1 − y14 )(1 − y14 ) + 3y14 (1 − y14 )
+ 1 − β(t) + α(t)
(t1) (1)
r11 x11
1 2 +2(1 − y14 )y14 ,
(12)
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
1 1 + 6(1 − y15 ), g13 = 7y15
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
1 2 1 2 1 2 g14 = 4(1−y16 )(1−y16 )+3y16 (1−y16 )+5(1−y16 )y16 ,
(t1) (1)
(t1) (1)
(t1) (1)
(t2) (2)
1 1 ) + 1(1 − y17 ), g21 = 0y17
(t2) (2)
(t2) (2)
(t2) (2)
(t2) (2)
1 1 g22 = y18 + 0.(1 − y18 ),
(t2) (2)
(t2) (2)
(t2) (2)
(t2) (2)
1 1 g23 = 5y19 + 4(1 − y19 ),
CO
+r12 x12 + r13 x13 + r14 x14 + r21 x21
(12)
+r22 x22 + r23 x23 + r24 x24 + r31 x31
(12)
+r32 x32 + r33 x33 + r34 x34 + r11 x11
(12)
+r12 x12 + r13 x13 + r14 x14 + r21 x21
(12)
OR
+r22 x22 + r23 x23 + r24 x24 + r31 x31
(t2) (2) (t2) (2) (t2) (2) +r32 x32 + r33 x33 + r34 x34 + β(t)
(12)
1 1 g24 = 15y20 + 14(1 − y20 ), (12)
1 1 g31 = 0.5y21 + 1(1 − y21 ),
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
1 2 )y22 , +0.5(1 − y22
(t1) (1)
(t1) (1)
(t1) (1)
(t1) (1)
1 1 + 9(1 − y23 g33 = 10y23 ),
(t2) (2)
(t2) (2)
(t2) (2)
(t2) (2)
1 1 g34 = 6y24 + 5(1 − y24 ).
(t2) (2)
(t2) (2)
(t2) (2)
(t2) (2) +h31 x31
(t2) (2) + h32 x32
(t2) (2) + h33 x33
h11 x11 + h12 x12 + h13 x13 + h14 x14
(11)
TH
1 2 1 2 g32 = 2(1 − y22 )(1 − y22 ) + y22 (1 − y22 )
+h21 x21 + h22 x22 + h23 x23 + h24 x24 +h31 x31 + h32 x32 + h33 x33 + h34 x34
AU
+h11 x11 + h12 x12 + h13 x13 + h14 x14
(t2) (2)
+h21 x21 + h22 x22 + h23 x23 + h24 x24
(t2) (2) + h34 x34
for t = 1, 2.
(12) (12)
(11)
r11 = 3(1 − y11 )(1−y12 )+2y11 (1−y12 ) + 6(1−y11 )y12 , (11)
r12 = 4(1−y21 )(1−y22 )+3y21 (1−y22 ) + 4(1 − y21 )y22 , (11)
r13 = 11y31 + 6(1 − y31 ), (11)
r14 = 7y41 + 3(1 − y41 ),
where
(11)
(11) g11 = 2(1−y11 )(1−y12 )+y11 (1−y12 ) + 5(1 − y11 )y12 , (11) g12
=
3(1 − y21 )(1 − y22 )+y21 (1−y22 )+2(1−y21 )y22 ,
(11) g13
=
10y31
(11) g21
=
0(1−y51 )(1−y52 )+0.5y51 (1−y52 )+0(1−y51 )y52 ,
(11) g22
=
0y61
(11)
PY
(t1) (1) +g14 x14
(11) + 5(1 − y31 ), g14
+ 0(1 − y61 ),
g23 = 6y71 + (1 − y71 ),
=
6y41
+ 2(1 − y41 ),
r21 = 1(1 − y51 )(1 − y52 )+1y51 (1−y52 )+2(1−y51 )y52 , (11)
r22 = 0y61 + 0(1 − y61 ), (11)
r23 = 7y71 + 2(1 − y71 ), (11)
r24 = 2(1 − y81 )(1 − y82 )+2y81 (1−y82 )+3(1 − y81 )y82 , (11)
r31 = 6y91 ) + 5(1 − y91 ), (11)
1 2 1 2 r32 = 9(1 − y10 )(1 − y10 ) + 10y10 (1 − y10 )
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator 1 2 +11(1 − y10 )y10 ,
(11)
1 1 + 6(1 − y12 ), h34 = 6y12
(11)
1 2 h11 = 0.y13 + 0.(1 − y13 ),
(11)
1 2 1 2 h12 = 0.(1 − y14 )(1 − y14 ) + y14 (1 − y14 )
1 1 r33 = 16y11 + 14(1 − y11 ),
(12)
1 1 r34 = 8y12 + 7(1 − y12 ),
(12)
(12)
1 2 +1(1 − y14 )y14 ,
(12)
1 1 + 5(1 − y15 ), h13 = 5y15
1 2 r11 = 0.5y13 + 1(1 − y13 ), (12)
1 2 1 2 r12 = 2(1 − y14 )(1 − y14 ) + 4y14 (1 − y14 )
(12)
1 2 +3(1 − y14 )y14 ,
1 2 1 h14 = 3(1 − y16 )(1 − y16 ) + 2y16 (1
(12)
1 1 + 8(1 − y15 ), r13 = 9y15
PY
2 1 2 −y16 ) + 4(1 − y16 )y16 , (12)
(12)
1 1 h21 = 0.y17 ) + 0.5(1 − y17 ),
1 2 1 2 r14 = 5(1 − y16 )(1 − y16 ) + 4y16 (1 − y16 )
(12)
1 2 +6(1 − y16 )y16 ,
1 1 h22 = 0.y18 + 0.(1 − y18 ),
(12)
1 1 r21 = y17 ) + 2(1 − y17 ),
(12)
CO
1 1 h23 = 5y19 + 3(1 − y19 ),
(12)
1 1 r22 = 2y18 + 0.5(1 − y18 ),
(12)
1 1 h24 = 13y20 + 12(1 − y20 ),
(12)
1 1 + 5(1 − y19 r23 = 5y19 ),
(12)
1 1 + 0.(1 − y21 h31 = 0.y21 ),
(12)
1 1 r24 = 17y20 + 16(1 − y20 ),
(12)
1 2 1 2 h32 = (1 − y22 )(1 − y22 ) + 0.5y22 (1 − y22 )
(12)
1 2 1 2 r32 = 3(1 − y22 )(1 − y22 ) + 2y22 (1 − y22 )
OR
(12)
1 1 + 2(1 − y21 ), r31 = y21
(12)
(21)
TH
g11 = 0.5z11 + 0.(1 − z11 ),
(12)
1 1 r34 = 7y24 + 6(1 − y24 ). (11)
h11 = (1 − y11 )(1−y12 )+0.y11 (1−y12 )+4(1 − y11 )y12 , (11)
(11) (11)
AU
h12 = 2(1−y21 )(1−y22 )+0.5y21 (1−y22 )+1(1−y21 )y22 ,
h14 = 5y41 + 1(1 − y41 ),
(12)
1 1 + 8(1 − y23 ), h33 = 9y23
1 1 h34 = 6y24 + 4(1 − y24 ).
1 1 + 10(1 − y23 ), r33 = 11y23
(11)
1 2 )y22 , +0.(1 − y22
(12)
1 2 )y22 , +1(1 − y22
h13 = 9y31 + 4(1 − y31 ),
h21 = 0.(1−y51 )(1−y52 )+0.y51 (1−y52 )+0.(1−y51 )y52 ,
(21)
g12 = 2z12 + 1(1 − z12 ), (21)
g13 = 3(1−z13 )(1−z23 ) + 2z13 (1 − z23 ) + 2(1 − z13 )z23 , (21)
g14 = 5z14 + 4(1 − z14 ), (21)
g21 = 1z15 + 0.5(1 − z15 ), (21)
g22 = 1z16 + 3(1 − z16 ), (21)
g23 = 13(1−z17 )(1−z27 )+11z17 (1−z27 )+9(1−z17 )z27 ,
(11)
g24 = 7z18 + 6(1 − z18 ),
(11)
g31 = 3z19 + 2(1 − z19 ),
(11)
g32 = 6z110 + 4(1 − z110 ),
(11)
g33 = 13(1 − z111 )(1 − z211 ) + 11z111 (1 − z211 )
h22 = 0.y61 + 0.(1 − y61 ), h23 = 5y71 + 0.(1 − y71 ), h24 = 0.(1−y81 )(1−y82 )+0.y81 (1−y82 )+1(1−y81 )y82 , h31 = 4y91 ) + 5(1 − y91 ),
(21) (21) (21) (21)
(11)
1 2 1 h32 = 7(1 − y10 )(1 − y10 ) + 8y10 (1 2 1 2 ) + 9(1 − y10 )y10 , −y10
(11)
1 1 h33 = 14y11 + 12(1 − y11 ),
863
+10(1 − z111 )z211 , (21)
g34 = 6z112 + 4(1 − z112 ) (22)
g11 = 0.z113 + 1(1 − z113 ),
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator
(22)
(22)
g12 = z114 + 0.5(1 − z114 ),
r13 = 3(1 − z115 )(1 − z215 ) + 4z115 (1 − z215 )
(22)
+2(1 − z115 )z215 ,
g13 = 2(1 − z115 )(1 − z215 ) + 3z115 (1 − z215 ) (22)
r14 = 5(1 − z116 )(1 − z216 ) + 5z116 (1 − z216 )
+(1 − z115 )z215 ,
+7(1 − z116 )z216
(22)
g14 = 5(1 − z116 )(1 − z216 ) + 4z116 (1 − z216 ) (22)
r21 = 2z117 + 4(1 − z117 ),
+6(1 − z116 )z216 (22)
(22)
g21 = 0.z117 + 3(1 − z117 ),
r22 = 8(1 − z118 )(1 − z218 ) + 7z118 (1 −z218 ) + 5(1 − z118 )z218 ,
(22)
g22 = 7(1 − z118 )(1 − z218 ) + 5z118 (1 − z218 ) (22)
PY
864
r23 = 13z119 ) + 11(1 − z119 ),
+4(1 − z118 )z218 , (22)
r24 = 10z120 + 9(1 − z120 ),
(12)
r31 = 3z121 + 2(1 − z121 ),
(22)
r32 = 2z122 + 4(1 − z122 ),
(22)
r33 = 3(1 − z123 )(1 − z223 ) + 2z123 (1 − z223 )
(12)
g23 = 12z119 ) + 10(1 − z119 ),
(22)
CO
g24 = 9z120 + 7(1 − z120 ),
(22)
g31 = 2z121 + (1 − z121 ),
(22)
g32 = z122 + 3(1 − z122 ), (22)
+4(1 − z123 )z223 ,
g33 = 2(1 − z123 )(1 − z223 ) + z123 (1 − z223 )
(22)
r34 = 6z124 + 5(1 − z124 ).
OR
+3(1 − z123 )z223 , (22)
(21)
g34 = 5z124 + 4(1 − z124 ).
h11 = 0.z11 + 0.(1 − z11 ), (21)
(21)
r11 = z11 + 2(1 − z11 ),
h12 = z12 + 0.(1 − z12 ), (21)
(21)
h13 = 2(1 − z13 )(1 − z23 ) + z13 (1 − z23 ) + (1 − z13 )z23 ,
(21)
TH
r12 = 3z12 + 2(1 − z12 ),
(21)
r14 = 6z14 + 5(1 − z14 ), (21)
r21 = 2z15 + (1 − z15 ), (21)
r22 = 2z16 + 4(1 − z16 ), (21)
AU
r13 = 4(1−z13 )(1−z23 ) + 3z13 (1 − z23 ) + 4(1 − z13 )z23 ,
(21)
h14 = 5z14 + 3(1 − z14 ), (21)
h21 = 0.z15 + 0.(1 − z15 ), (21)
h22 = 0.z16 + 2(1 − z16 ), (21)
h23 = 12(1 − z17 )(1 − z27 ) + 10z17 (1 − z27 ) +8(1 − z17 )z27 ,
r23 = 14(1−z17 )(1−z27 )+12z17 (1−z27 )+10(1−z17 )z27 , (21)
r24 = 8z18 + 7(1 − z18 ),
(21)
h24 = 6z18 + 5(1 − z18 ), (21)
(21)
h31 = 2z19 + (1 − z19 ),
(21)
h32 = 5z110 + 3(1 − z110 ),
(21)
h33 = 12(1 − z111 )(1 − z211 ) + 10z111 (1 − z211 )
r31 = 4z19 + 3(1 − z19 ), r32 = 7z110 + 6(1 − z110 ), r33 = 14(1 − z111 )(1 − z211 ) + 12z111 (1 − z211 )
(21) (21)
+9(1 − z111 )z211 ,
+13(1 − z111 )z211 , (21)
(21)
h34 = 5z112 + 3(1 − z112 )
(22)
h11 = 0.z113 + 0.5(1 − z113 ),
(22)
h12 = 0.5z114 + 0.(1 − z114 ),
r34 = 8z112 + 5(1 − z112 ) r11 = z113 + 2(1 − z113 ), r12 = 3z114 + 2(1 − z114 ),
(22)
(22)
D. Chakraborty et al. / A new approach to solve MMMIFTP using chance operator (22)
h13 = (1 − z115 )(1 − z215 ) + 2z115 (1 − z215 )
865
(22)
h31 = z121 + 0.5(1 − z121 ),
+0.5(1 − z115 )z215 ,
(22)
h32 = 0.5z122 + 2(1 − z122 ),
(22)
h14 = 5(1 − z116 )(1 − z216 ) + 3z116 (1 − z216 )
(22)
h33 = (1 − z123 )(1 − z223 ) + 0.5z123 (1 − z223 )
+5(1 − z116 )z216
+2(1 − z123 )z223 ,
(22)
h21 = 0.z117 + 2(1 − z117 ),
(22)
h34 = 4z124 + 3(1 − z124 ).
(22)
h22 = 6(1 − z118 )(1 − z218 ) + 4z118 (1 − z218 )
yia and zai = 0 or 1 for i = 1, 2, · · · , 24 and a = 1, 2, 1 ≤ yi1 + yi2 ≤ 2 for i = 1, 2, 5, 8, 10, 14, 16, 22, and 1 ≤ z1i + z2i ≤ 2 for i = 3, 7, 11, 15, 16, 18, 23.
PY
+3(1 − z118 )z218 , (22)
h23 = 11z119 ) + 9(1 − z119 ), (12)
AU
TH
OR
CO
h24 = 8z120 + 6(1 − z120 ),
