Uncertain multi-objective multi-product solid transportation problems

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1 Sādhanā Vol. 41, No. 5, May 2016, pp DOI /s x Ó Indian Academy of Sciences Uncertain multi-objective multi-product solid transportation problems DEEIKA RANI* and T R GULATI Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee , India deepikaiitroorkee@gmail.com MS received 17 July 2014; revised 28 April 2015; accepted 18 December 2015 Abstract. The solid transportation problem is an important generalization of the classical transportation problem as it also considers the conveyance constraints along with the source and destination constraints. The problem can be made more effective by incorporating some other factors, which make it useful in real life situations. In this paper, we consider a fully fuzzy multi-objective multi-item solid transportation problem and present a method to find its fuzzy optimal-compromise solution using the fuzzy programming technique. To take into account the imprecision in finding the exact values of parameters, all the parameters are taken as trapezoidal fuzzy numbers. A numerical example is solved to illustrate the methodology. Keywords. Solid transportation problem; fuzzy optimal-compromise solution; fuzzy programming technique; trapezoidal fuzzy number; multi-product (item). 1. Introduction The need of generalization of traditional transportation problem to solid transportation problem arises when different kinds of conveyances are available for the transportation of goods to save time as well as money. It was first stated by Schell [1]and later on Haley [2] described its solution procedure. The multi-objective multi-item solid transportation problem (MOMIST) is further generalization of the solid transportation problem. It deals with optimizing multiple objectives and using different types of conveyances to transport heterogeneous products from the warehouses to the consumer points. This type of transportation problem is very beneficial in many industries, where more than one kind of products are shipped. In multiple objective problems, the objectives are generally conflicting in nature, so the concept of optimal solution is replaced by optimal compromise solution also called efficient solution or pareto optimal solution or non-dominated solution. In the conventional solid transportation problem, it is assumed that all the parameters are known exactly and many algorithms have been developed to solve these problems. But in real world situations, it is not always so. Due to uncontrollable factors like lack of information and uncertainty in judgement, the values of the transportation parameters, i.e., unit cost of transportation, availability and demand are not exact always. This impreciseness in the values of the parameters can be represented by using the fuzzy set theory given by Zadeh [3]. A systematic study of fuzzy mathematical programming has also been given by Bector and Chandra [4]. Many authors have used the fuzzy *For correspondence numbers to represent the uncertainty in transportation parameters and proposed methods to solve them. The MOMISTs in which all the parameters are represented by fuzzy numbers are called fully fuzzy multi-objective multiitem solid transportation problems (FFMOMISTs). Fuzzy programming technique for the multi-objective transportation problems was given by Zimmermann [5]. Bit et al [6] applied the fuzzy programming technique to solve MOST. Li et al [7] solved the multi-objective solid transportation problem (MOST) using the genetic algorithm in which only objective function coefficients are taken as fuzzy numbers. Liu and Liu [8] presented the expected value model in fuzzy programming. Islam and Roy [9] studied the geometric programming approach for the multiobjective transportation problems. Ojha et al [10] proposed methods to solve fuzzy MOST, where all the parameters except decision variables are taken as fuzzy numbers. Gupta et al [11] proposed a method, called Mehar s method, to find the exact fuzzy optimal solution of unbalanced fully fuzzy multi-objective transportation problems. Uncertainty theory based expected constrained programming for the solid transportation problems in uncertain environment is studied by Cui and Sheng [12]. Recently, Kundu et al [13] have proposed a method to find the crisp optimal compromise solution of the fuzzy MOMIST, using the fuzzy programming technique and global criterion method. Solid transportation problems are also studied by Baidya et al [14] and Kundu et al [15]. Ebrahimnejad [16] studied the transportation problem with generalized trapezoidal fuzzy numbers. To the best of our knowledge, no method has been proposed in the literature to find the fuzzy optimal compromise solution of the FFMOMIST. 531

2 532 D Rani and T R Gulati In this paper, a method is proposed to find the fuzzy optimal compromise solution of FFMOMIST. The application of the proposed method is shown by obtaining the fuzzy optimal compromise solution of the numerical example for which Kundu et al [13] found the crisp optimal compromise solution. Since the proposed problem is the generalization of the traditional solid transportation problem, it is also applicable to solve both single and multiobjective solid transportation problems, single or multiobjective transportation problems (both for single and multi-item). (i) ea 1 ea 2 ¼ða 1 þ a 2 ; b 1 þ b 2 ; c 1 þ c 2 ; d 1 þ d 2 Þ (ii) ea 1 ea 2 ¼ða 1 d 2 ; b 1 c 2 ; c 1 b 2 ; d 1 a 2 Þ (iii) kea 1 ¼ ðka 1; kb 1 ; kc 1 ; kd 1 Þ; k 0 ðkd 1 ; kc 1 ; kb 1 ; ka 1 Þ; k 0 (iv) ea 1 ea 2 ¼ða; b; c; dþ, where a ¼ minða 1 a 2 ; a 1 d 2 ; d 1 a 2 ; d 1 d 2 Þ, b ¼ minðb 1 b 2 ; b 1 c 2 ; c 1 b 2 ; c 1 c 2 Þ, c ¼ maxðb 1 b 2 ; b 1 c 2 ; c 1 b 2 ; c 1 c 2 Þ, d ¼ maxða 1 a 2 ; a 1 d 2 ; d 1 a 2 ; d 1 d 2 Þ: 2. reliminaries In this section, some basic definitions and ranking approach for trapezoidal fuzzy numbers are presented. 2.1 Basic definitions [17, 18] Definition 1 A fuzzy number ea defined on the universal set of real numbers R, denoted as ea ¼ða; b; c; dþ, is said to be a trapezoidal fuzzy number if its membership function l ea ðxþ is given by 8 ðx aþ a x\b ðb aþ >< 1 b x c l ea ðxþ ¼ ðx dþ c\x d ðc dþ >: 0 otherwise. Definition 2 A trapezoidal fuzzy number ea ¼ða; b; c; dþ is said to be zero trapezoidal fuzzy number if and only if a ¼ 0; b ¼ 0; c ¼ 0 and d ¼ 0. Definition 3 A trapezoidal fuzzy number ea ¼ða; b; c; dþ is said to be non-negative trapezoidal fuzzy number if and only if a 0. Definition 4 Two trapezoidal fuzzy numbers ea 1 ¼ ða 1 ; b 1 ; c 1 ; d 1 Þ and ea 2 ¼ða 2 ; b 2 ; c 2 ; d 2 Þ are said to be equal if and only if a 1 ¼ a 2 ; b 1 ¼ b 2 ; c 1 ¼ c 2 and d 1 ¼ d 2 ; and is denoted by ea 1 ¼ ea 2. Remark 1 If for a trapezoidal fuzzy number ea ¼ða; b; c; dþ; b ¼ c, then it is called a triangular fuzzy number and is denoted by (a, b, b, d) or (a, b, d) or (a, c, d). 2.2 Arithmetic operations [17] Let ea 1 ¼ða 1 ; b 1 ; c 1 ; d 1 Þ and ea 2 ¼ða 2 ; b 2 ; c 2 ; d 2 Þ be two trapezoidal fuzzy numbers. Then 3. Mathematical model A FFMOMIST with parameters as trapezoidal fuzzy numbers can be stated as a transportation problem with R objectives in which l different items are to be transported from m sources (S i ; 1 i m) to n destinations (D j ; 1 j n) via K different conveyances. Let ea p i denote the fuzzy availability of item p at source S i ; eb p j denote the fuzzy demand of item p at destination D j ; e k be the total fuzzy capacity of kth conveyance, ec rp ijk be the fuzzy penalty for transporting one unit of item p from S i to D j via kth conveyance for rth objective Z r and ex p ijk be the fuzzy quantity of item p to be transported from S i to D j using kth conveyance in order to minimize R objective functions. The problem is mathematically modeled as follows: ðþ Minimize ðez 1 ; ez 2 ;...; ez R Þ X m ex p ijk eap i ; 1 i m; 1 p l; ex p ijk e b p j ; 1 j n; 1 p l; ex p ijk e k ; 1 k K; where ex p ijk for all i; j; k; p and ez r ¼ l m n is a non-negative trapezoidal fuzzy number, K ðecrp ijk ex p ijkþ; 1 r R: For the above problem to be balanced it should satisfy: (i) (ii) m eap i ¼ n e j¼1 b p j ; 1 p l, i.e., for an item, its total availability at all sources should be equal to its demand at all the destinations. l p¼1 m eap i ¼ l p¼1 n e j¼1 b p j ¼ K e k, i.e., overall availability/demand of all the items at all the sources/destinations and total conveyance capacity should be equal.

3 Uncertain multi-objective multi-product solid transportation 533 Definition 5 A fuzzy feasible solution ex ¼fex p ijkg of () is said to be a fuzzy optimal compromise (efficient) solution if there exists no other feasible solution ey ¼fey p ijkg such that, and X l X m p¼1 Xl X l X m p¼1 X m p¼1 \ Xl X m p¼1 4. roposed method Rðec rp ijk eyp ijk Þ Rðec rp ijk exp ijk Þ Rðec rp ijk eyp ijk Þ for all r Rðec rp ijk exp ijkþ for atleast one r: In this section a method has been proposed to find the fuzzy optimal compromise solution of roblem (). The proposed method consists of the following steps: Step 1: Verify whether the problem under consideration is balanced. For all p; 1 p l, find m eap i and n e j¼1 b p j. Let m eap i ¼ðx p ; y p ; z p ; w p Þ and n e j¼1 b p j ¼ðx 0 p ; y0 p ; z0 p ; w0 p Þ. Case 1: m eap i ¼ n e j¼1 b p j for all p; 1 p l.gotostep2. Case 2: m eap i 6¼ n e j¼1 b p j for all or any p, 1 p l. To make m eap i ¼ n e j¼1 b p j, proceed according to the following subcases. One or all may apply. Subcase 2a: Ifx p x 0 p ; y p x p y 0 p x0 p ; z p y p z 0 p y0 p and w p z p w 0 p z0 p (for one or more values of p), then introduce a dummy source with fuzzy availability of the pth item(s) equal to ðx 0 p x p; y 0 p y p; z 0 p z p; w 0 p w pþ. Subcase 2b:Ifx p x 0 p ; y p x p y 0 p x0 p ; z p y p z 0 p y0 p and w p z p w 0 p z0 p (for one or more values of p), then introduce a dummy destination having fuzzy demand of the pth item(s) equal to ðx p x 0 p ; y p y 0 p ; z p z 0 p ; w p w 0 p Þ. Subcase 2c: None of the Subcases 2a or 2b is satisfied. In such a situation add a dummy source with availability of the pth item(s) equal to ðmaxf0; ðx 0 p x pþg; maxf0; ðx 0 p x p Þg þ maxf0; ðy 0 p x0 p Þ ðy p x p Þg; maxf0; ðx 0 p x pþg þ maxf0; ðy 0 p x0 p Þ ðy p x p Þg þ maxf0; ðz 0 p y0 p Þ ðz p y p Þg; maxf0; ðx 0 p x pþg þ maxf0; ðy 0 p x0 p Þ ðy p x p Þg þ maxf0; ðz 0 p y0 p Þ ðz p y p Þg þ maxf0; ðw 0 p z 0 p Þ ðw p z p ÞgÞ and also a dummy destination having demand of these item(s) as ðmaxf0; ðx p x 0 pþg; maxf0; ðx p x 0 p Þg þ maxf0; ðy p x p Þ ðy 0 p x0 p Þg; maxf0; ðx p x 0 p Þg þ maxf0; ðy p x p Þ ðy 0 p x0 p Þg þ maxf0; ðz p y p Þ ðz 0 p y0 p Þg; maxf0; ðx p x 0 p Þg þ maxf0; ðy p x p Þ ðy 0 p x0 p Þg þ maxf0; ðz p y p Þ ðz 0 p y0 pþg þ maxf0; ðw p z p Þ ðw 0 p z0 pþgþ. Now go to Step 2. Step 2: After Step 1, we get s eap i ¼ t e j¼1 b p j for all p; 1 p l, where s ¼ m or m þ 1 and t ¼ n or n þ 1, i.e., l s p¼1 eap i ¼ l t e p¼1 j¼1 b p j ¼ ðu; v; w; aþ (say). Now, check whether l s p¼1 eap i ¼ l t e p¼1 j¼1 b p j ¼ K e k. Let K e k ¼ðu 0 ; v 0 ; w 0 ; a 0 Þ. Case 1: If l s p¼1 eap i ¼ l t e p¼1 j¼1 b p j ¼ K e k, i.e., the total availability of all the items, their demand and the total conveyance capacity are equal, then go to Step 3. Case 2: If l s p¼1 eap i ¼ l t e p¼1 j¼1 b p j 6¼ K e k, then proceed according to the following subcases: Subcase 2a:Ifuu 0 ; v u v 0 u 0 ; w v w 0 v 0 and a w a 0 w 0, then check whether in Step 1 a dummy source and/or a dummy destination have been added and proceed as below: Case (i): If both a dummy source and a dummy destination have been introduced, then increase their total availability and demand by the fuzzy quantity ðu 0 u; v 0 v; w 0 w; a 0 aþ (The demand of only those items is to be increased whose availability has been increased at the dummy source). Case (ii): If only a dummy source (destination) has been introduced, then increase its total availability (demand) by the fuzzy number ðu 0 u; v 0 v; w 0 w; a 0 aþ and also introduce a dummy destination (source) having the demand (availability) of these added items as ðu 0 u; v 0 v; w 0 w; a 0 aþ. Case (iii): If neither a dummy source nor a dummy destination has been added, then introduce a dummy source with the total availability equal to the fuzzy number ðu 0 u; v 0 v; w 0 w; a 0 aþ and a dummy destination with demand of these added items equal to ðu 0 u; v 0 v; w 0 w; a 0 aþ. Subcase 2b: If u u 0 ; v u v 0 u 0 ; w v w 0 v 0 and a w a 0 w 0, then introduce a dummy conveyance having capacity ðu u 0 ; v v 0 ; w w 0 ; a a 0 Þ. Subcase 2c: If neither Subcase 2a nor Subcase 2b applies, then check whether in Step 1 a dummy source or a dummy destination or both have been added and proceed according to the following cases. Let ðu; v; w; aþ ¼ðmaxf0; u 0 ug; maxf0; u 0 ugþmaxf0; ðv 0 u 0 Þ ðv uþg; maxf0; u 0 ug þmaxf0; ðv 0 u 0 Þ ðv uþg þ maxf0; ðw 0 v 0 Þ ðw vþg; maxf0; u 0 ugþ maxf0; ðv 0 u 0 Þ ðv uþg þ maxf0; ðw 0 v 0 Þ ðw vþg þ maxf0; ða 0 w 0 Þ ða wþgþ and ðu 0 ; v 0 ; w 0 ; a 0 Þ¼ ðmaxf0; u u 0 g; maxf0; u u 0 g þmaxf0; ðv uþ ðv 0 u 0 Þg; maxf0; u u 0 gþmaxf0; ðv uþ ðv 0 u 0 Þg þ maxf0; ðw vþ ðw 0 v 0 Þg; maxf0; u u 0 gþ maxf0;

4 534 D Rani and T R Gulati ðv uþ ðv 0 u 0 Þg þ maxf0; ðw vþ ðw 0 v 0 Þgþ maxf0; ða wþ ða 0 w 0 ÞgÞ. Case (i): If both a dummy source and a dummy destination have been introduced, then increase their total availability and demand by the fuzzy quantity ðu; v; w; aþ (The demand of only those items is to be increased whose availability has been increased at the dummy source). Also, introduce a dummy conveyance with capacity ðu 0 ; v 0 ; w 0 ; a 0 Þ. Case (ii): If only a dummy source (destination) has been introduced, then increase its total availability (demand) by the fuzzy number ðu; v; w; aþ and introduce a dummy destination (source) having the demand (availability) of these added items as ðu; v; w; aþ. Also, introduce a dummy conveyance with capacity ðu 0 ; v 0 ; w 0 ; a 0 Þ. Case (iii): If neither a dummy source nor a dummy destination has been added, then introduce both a dummy source as well as a dummy destination with the availability and demand of added items as ðu; v; w; aþ. Also, introduce a dummy conveyance with capacity ðu 0 ; v 0 ; w 0 ; a 0 Þ. Assume the unit transportation costs required due to the dummy source/destination/conveyance to be zero trapezoidal fuzzy number. Now, the problem is balanced and takes the form: ð 0 Þ Minimize ðfz 1 ; fz 2 ;...; fz R Þ 1 p l X l ðx p ijk ; yp ijk ; zp ijk ; wp ijk Þ¼ðap i ; bp i ; cp i ; dp i Þ; 1 i s; ðx p ijk ; yp ijk ; zp ijk ; wp ijk Þ¼ða0p j ; b0p j ; c 0p j ; d 0p j Þ; 1 j t; 1 p l 1 k K; ðx p ijk ; yp ijk ; zp ijk ; wp ijk Þ¼ða00 k ; b00 k ; c00 k ; d00 k Þ; where ez r ¼ X l p¼1 ððarp ijk ; brp ijk ; crp ijk ; drp ijk Þ ðx p ijk ; yp ijk ; zp ijk ; wp ijk ÞÞ; 1 r R; ðxp ijk ; yp ijk ; zp ijk ; wp ijkþ is a nonnegative trapezoidal fuzzy number and ec rp ijk ¼ðarp ijk ; brp ijk ; c rp ijk ; drp ijk Þ; eap i ¼ða p i ; bp i ; cp i ; dp i Þ; eb p j ¼ða 0p j ; b 0p j ; c 0p j ; d0p j Þ; e k ¼ ða 00 k ; b00 k ; c00 k ; d00 k Þ. Step 3: Corresponding to each objective, convert the problem ð 0 Þ into following four crisp problems ð 0 1 Þ ð0 4 Þ: ð 0 1 Þ Minimize X l ð 0 3 Þ Minimize X l p¼1 a rp ijk xp ijk x p ijk ¼ ap i ; 1 i s; 1 p l x p ijk ¼ a0p j ; 1 j t; 1 p l x p ijk ¼ a00 k ; 1 k K x p ijk 0 p¼1 c rp ijk zp ijk z p ijk ¼ cp i ; 1 i s; 1 p l z p ijk ¼ c0p j ; 1 j t; 1 p l z p ijk ¼ c00 k ; 1 k K z p ijk yp ijk ð 0 2 Þ Minimize X l p¼1 b rp ijk yp ijk y p ijk ¼ bp i ; 1 i s; 1 p l y p ijk ¼ b0p j ; 1 j t; 1 p l y p ijk xp ijk ð 0 4 Þ Minimize X l y p ijk ¼ b00 k ; 1 k K p¼1 d rp ijk wp ijk w p ijk ¼ dp i ; 1 i s; 1 p l w p ijk ¼ d0p j ; 1 j t; 1 p l y p ijk ¼ d00 k ; 1 k K w p ijk zp ijk

5 Uncertain multi-objective multi-product solid transportation 535 Solve the four problems sequentially using the optimal solution x p ijk of ð 1Þ in ð 2 Þ; optimal solution y p ijk of ð 2Þ in ð 3 Þ and optimal solution z p ijk of ð 3Þ in ð 4 Þ. Let w p ijk be the optimal solution of ð 4 Þ. This leads to the fuzzy optimal solution ex p ijk ¼ðxp ijk ; yp ijk ; zp ijk ; wp ijk Þ of ð0 Þ. Step 4: Apply the fuzzy programming technique to obtain the optimal-compromise solution and calculate the value of each objective function. 5. Numerical example Consider the multi-objective multi-item solid transportation problem solved by Kundu et al [13]. In this problem, the number of destinations is three, while that of sources, items, conveyances and objectives is two each. The authors have proposed a method to find the crisp optimal compromise solution. Since the fuzzy solution has more information than the crisp one, we solve the same problem using the method proposed by us to find the fuzzy optimal compromise solution. The data of the problem is as follows (tables 1 4): From table 5, we find that for the first item, total availability 2 ea1 i ¼ð21; 24; 26; 28Þð28; 32; 35; 37Þ ¼ð49; 56; 61; 65Þ and total demand 3 e j¼1 b 1 j ¼ð14; 16; 19; 22Þ ð17; 20; 22; 25Þð12; 15; 18; 21Þ ¼ð43; 51; 59; 68Þ: Similarly for the second item, total availability 2 ea2 i ¼ ð57; 62; 67; 72Þ and total demand 3 e j¼1 b 2 j = (51,58,63,71). Table 1. Unit transportation penalties for item 1 in the first objective. Destinations! Sources # D 1 D 2 D 3 Conveyance k ¼ 1 S 1 (5,8,9,11) (4,6,9,11) (10,12,14,16) S 2 (8,10,13,15) (6,7,8,9) (11,13,15,17) Conveyance k ¼ 2 S 1 (9,11,13,15) (6,8,10,12) (7,9,12,14) S 2 (10,11,13,15) (6,8,10,12) (14,16,18,20) Table 2. Unit transportation penalties for item 2 in the first objective. Destinations! Sources # D 1 D 2 D 3 Conveyance k ¼ 1 S 1 (9,10,12,13) (5,8,10,12) (10,11,12,13) S 2 (11,13,14,16) (7,9,12,14) (12,14,16,18) Conveyance k ¼ 2 S 1 (11,13,14,15) (6,7,9,11) (8,10,11,13) S 2 (14,16,18,20) (9,11,13,14) (13,14,15,16) Table 3. Unit transportation penalties for item 1 in the second objective. Destinations! Sources # D 1 D 2 D 3 Conveyance k ¼ 1 S 1 (4,5,7,8) (3,5,6,8) (7,9,10,12) S 2 (6,8,9,11) (5,6,7,8) (6,7,9,10) Conveyance k ¼ 2 S 1 (6,7,8,9) (4,6,7,9) (5,7,9,11) S 2 (4,6,8,10) (7,9,11,13) (9,10,11,12) Table 4. Unit transportation penalties for item 2 in the second objective. Destinations! Sources # D 1 D 2 D 3 Conveyance k ¼ 1 S 1 (5,7,9,10) (4,6,7,9) (9,11,12,13) S 2 (10,11,13,14) (6,7,8,9) (7,9,11,12) Conveyance k ¼ 2 S 1 (7,8,9,10) (4,5,7,8) (8,10,11,12) S 2 (6,8,10,12) (5,7,9,11) (9,10,12,14) Table 5. Availability and demand data. Fuzzy availability Fuzzy demand Conveyance capacity ea 1 1 ¼ð21; 24; 26; 28Þ e b 1 1 ¼ð14; 16; 19; 22Þ e 1 ¼ð46; 49; 51; 53Þ ea 1 2 ¼ð28; 32; 35; 37Þ e b 1 2 ¼ð17; 20; 22; 25Þ e 2 ¼ð51; 53; 56; 59Þ ea 2 1 ¼ð32; 34; 37; 39Þ e b 1 3 ¼ð12; 15; 18; 21Þ ea 2 2 ¼ð25; 28; 30; 33Þ e b 2 1 ¼ð20; 23; 25; 28Þ eb 2 2 ¼ð16; 18; 19; 22Þ ¼ð15; 17; 19; 21Þ eb 2 3 Since 2 ea1 i 6¼ 3 e j¼1 b 1 j and 2 ea2 i 6¼ 3 e j¼1 b 2 j, the problem is unbalanced. Now, first step is to balance the problem. We find that neither Subcase 2a nor Subcase 2b of Step 1 holds for any of the items. So, according to Subcase 2c, we introduce a dummy source (S 3 ) having availabilities of the first and second items as ea 1 3 ¼ð0; 1; 4; 9Þ and ea 2 3 ¼ð0; 2; 2; 5Þ, respectively. Also, we introduce a dummy destination (D 4 ) with demand of the first and second items as eb 1 4 ¼ eb 2 4 ¼ð6; 6; 6; 6Þ so that the total availability and total demand of both the items become equal, i.e., 2 p¼1 3 eap i ¼ 2 4 e p¼1 j¼1 b p j ¼ð106; 121; 134; 151Þ: Since, the total conveyance capacity 2 e k ¼ ð97; 102; 107; 112Þ. Clearly, 2 3 p¼1 eap i ¼ 2 4 p¼1 j¼1

6 536 D Rani and T R Gulati eb p j 6¼ 2 e k. For (106, 121, 134, 151) = ðu; v; w; aþ and (97, 102, 107, 112) = ðu 0 ; v 0 ; w 0 ; a 0 Þ, the condition u u 0 ; v u v 0 u 0 ; w v w 0 v 0 and a w a 0 w 0 is met and so according to Subcase 2b of Step 2, we introduce a dummy conveyance having capacity e 3 = (9,19,27,39). Thus the problem becomes balanced. Since, a dummy source (S 3 ), a dummy destination (D 4 ) and a dummy conveyance are introduced so we assume ec rp 3jk ¼ ecrp i4k ¼ ecrp ij3 ¼ð0; 0; 0; 0Þ for all r ¼ 1; 2; p ¼ 1; 2; i ¼ 1; 2; 3; j ¼ 1; 2; 3; 4 and k ¼ 1; 2; 3. The obtained balanced problem can be written as Minimize ez 1 ¼ð5; 8; 9; 11Þ ex ð9; 11; 13; 15Þ ex ð4; 6; 9; 11Þex1 121 ð6; 8; 10; 12Þex1 122 ð10; 12; 14; 16Þex ð7; 9; 12; 14Þex1 132 ð8; 10; 13; 15Þ ex ð10; 11; 13; 15Þex1 212 ð6; 7; 8; 9Þ ex1 221 ð6; 8; 10; 12Þex ð11; 13; 15; 17Þex1 231ð14; 16; 18; 20Þ ex ð9; 10; 12; 13Þex2 111 ð11; 13; 14; 15Þex2 112 ð5; 8; 10; 12Þex ð6; 7; 9; 11Þ ex2 122 ð10; 11; 12; 13Þ ex ð8; 10; 11; 13Þex2 132 ð11; 13; 14; 16Þex2 211 ð14; 16; 18; 20Þex ð7; 9; 12; 14Þ ex2 221ð9; 11; 3; 14Þ ex ð12; 14; 16; 18Þ ex2 231 ð13; 14; 15; 16Þ ex2 232 ð0; 0; 0; 0Þðex ex1 123 ex1 133 ex1 141 ex1 142 ex1 143 ex1 213 ex ex1 233 ex1 241 ex1 242 ex1 243 ex1 311 ex1 312 ex1 313 ex ex1 322 ex1 323 ex1 331 ex1 332 ex1 333 ex1 341 ex1 342 ex ex2 113 ex2 123 ex2 133 ex2 141 ex2 142 ex2 143 ex2 213 ex2 223 ex ex2 241 ex2 242 ex2 243 ex ex2 312 ex2 313 ex2 321 ex ex2 323 ex2 331 ex2 332 ex2 333 ex2 341 ex2 342 ex2 343 Þ: Minimize ez 2 ¼ð4; 5; 7; 8Þex ð6; 7; 8; 9Þex1 112 ð3; 5; 6; 8Þex ð4; 6; 7; 9Þex1 122 ð7; 9; 10; 12Þex1 131 ð5; 7; 9; 11Þ ex ð6; 8; 9; 11Þ ex1 211 (4,6,8,10) ex ð5; 6; 7; 8Þex1 221 ð7; 9; 11; 13Þex1 222 ð6; 7; 9; 10Þ ex ð9; 10; 11; 12Þex1 232 ð5; 7; 9; 10Þex2 111 ð7; 8; 9; 10Þex ð4; 6; 7; 9Þ ex2 121 ð4; 5; 7; 8Þ ex2 122 ð9; 11; 12; 13Þex ð8; 10; 11; 12Þex2 132 (10,11,13,14) ex ð6; 8; 10; 12Þex2 212 ð6; 7; 8; 9Þex ð5; 7; 9; 11Þ ex ð7; 9; 11; 12Þex2 231 ð9; 10; 12; 14Þex ð0; 0; 0; 0Þðex ex1 123 ex1 133 ex1 141 ex1 142 ex1 143 ex ex1 223 ex1 233 ex1 241 ex ex1 243 ex1 311 ex1 312 ex ex1 321 ex1 322 ex1 323 ex1 331 ex1 332 ex1 333 ex1 341 ex ex1 343 ex2 113 ex2 123 ex2 133 ex2 141 ex2 142 ex2 143 ex ex2 223 ex2 233 ex2 241 ex2 242 ex2 243 ex2 311 ex2 312 ex ex2 321 ex2 322 ex2 323 ex2 331 ex2 332 ex2 333 ex2 341 ex ex2 343 Þ X 2 X 2 X 2 ðx 1 1jk ; y1 1jk ; z1 1jk ; w1 1jkÞ¼ð21; 24; 26; 28Þ; ðx 2 1jk ; y2 1jk ; z2 1jk ; w2 1jkÞ¼ð32; 34; 37; 39Þ ðx 1 2jk ; y1 2jk ; z1 2jk ; w1 2jkÞ¼ð28; 32; 35; 37Þ; ðx 2 2jk ; y2 2jk ; z2 2jk ; w2 2jkÞ¼ð25; 28; 30; 33Þ ðx 1 3jk ; y1 3jk ; z1 3jk ; w1 3jkÞ¼ð0; 1; 4; 9Þ; ðx 2 3jk ; y2 3jk ; z2 3jk ; w2 3jkÞ¼ð0; 2; 2; 5Þ ðx 1 i1k ; y1 i1k ; z1 i1k ; w1 i1kþ¼ð14; 16; 19; 22Þ; ðx 2 i1k ; y2 i1k ; z2 i1k ; w2 i1kþ¼ð20; 23; 25; 28Þ ðx 1 i2k ; y1 i2k ; z1 i2k ; w1 i2kþ¼ð17; 20; 22; 25Þ; ðx 2 i2k ; y2 i2k ; z2 i2k ; w2 i2kþ¼ð16; 18; 19; 22Þ ðx 1 i3k ; y1 i3k ; z1 i3k ; w1 i3kþ¼ð12; 15; 18; 21Þ; ðx 2 i3k ; y2 i3k ; z2 i3k ; w2 i3kþ¼ð15; 17; 19; 21Þ ðx 1 i4k ; y1 i4k ; z1 i4k ; w1 i4kþ¼ð6; 6; 6; 6Þ; ðx 2 i4k ; y2 i4k ; z2 i4k ; w2 i4kþ¼ð6; 6; 6; 6Þ ðx p ij1 ; yp ij1 ; zp ij1 ; wp ij1þ¼ð46; 49; 51; 53Þ; ðx p ij2 ; yp ij2 ; zp ij2 ; wp ij2þ¼ð51; 53; 56; 59Þ ðx p ij3 ; yp ij3 ; zp ij3 ; wp ij3þ¼ð9; 19; 27; 39Þ ðx p ijk ; yp ijk ; zp ijk ; wp ijkþ for all i, j, k, p is a non-negative trapezoidal fuzzy number. We minimize ez 1 by solving the following four problems: Minimize Z1 1 ¼ 5x1 111 þ 9x1 112 þ 4x1 121 þ 6x1 122 þ 10x1 131 þ 7x þ 8x1 211 þ 10x1 212 þ 6x1 221 þ 6x1 222 þ 11x1 231 þ 14x1 232 þ 9x þ 11x2 112 þ 5x2 121 þ 6x2 122 þ 10x2 131 þ 8x2 132 þ 11x2 211 þ 14x þ 7x2 221 þ 9x2 222 þ 12x2 231 þ 13x2 232

7 Uncertain multi-objective multi-product solid transportation 537 X 2 x 1 1jk ¼ 21; x 2 2jk ¼ 25; x 2 3jk ¼ 0; x 1 i2k ¼ 17; x 1 i3k ¼ 12; x 2 i4k ¼ 6;X2 p¼1 x p ij3 ¼ 9; x 2 1jk ¼ 32; x 1 3jk ¼ 0; x 1 i1k ¼ 14; j¼1 x 2 i2k ¼ 16; x 2 i3k ¼ 15; x p ij1 ¼ 46; X 2 xp ijk 0 8i;j;k;p: x 1 2jk ¼ 28; x 2 i1k ¼ 20; x 1 i4k ¼ 6; x p ij2 ¼ 51; X 2 X 2 z 1 1jk ¼ 26; j¼1 z 2 2jk ¼ 30; z 2 3jk ¼ 2; j¼1 z 1 i2k ¼ 22; z 2 i3k ¼ 19; z p ij1 ¼ 51; X 2 z p ij3 ¼ 27; z 2 1jk ¼ 37; z 1 3jk ¼ 4; z 1 i1k ¼ 19; z 2 i2k ¼ 19; z 1 i4k ¼ 6; z p ij2 ¼ 56; zp ijk yp ijk 8 i; j; k; p: z 1 2jk ¼ 35; z 2 i1k ¼ 25; z 1 i3k ¼ 18; z 2 i4k ¼ 6; Minimize Z1 2 ¼ 8y1 111 þ 11y1 112 þ 6y1 121 þ 8y1 122 þ 12y1 131 þ 9y þ 10y1 211 þ 11y1 212 þ 7y1 221 þ 8y1 222 þ 13y1 231 þ 16y1 232 þ 10y þ 13y2 112 þ 8y2 121 þ 7y2 122 þ 11y2 131 þ 10y2 132 þ 13y2 211 þ 16y þ 9y2 221 þ 11y2 222 þ 14y2 231 þ 14y2 232 X 2 X 2 y 1 1jk ¼ 24; y 2 2jk ¼ 28; y 1 i1k ¼ 16; y 2 1jk ¼ 34; y 1 2jk ¼ 32; y 1 3jk ¼ 1;X4 y 2 3jk ¼ 2; y 2 i1k ¼ 23; y 2 i2k ¼ 18;X3 y 1 i3k ¼ 15; y 1 i4k ¼ 6; y p ij1 ¼ 49; X 2 y p ij3 ¼ 19; y 2 i4k ¼ 6; yp ijk xp ijk y p ij2 ¼ 53; 8i;j;k;p: y 1 i2k ¼ 20; y 2 i3k ¼ 17; Minimize Z1 3 ¼ 9z1 111 þ 13z1 112 þ 9z1 121 þ 10z1 122 þ 14z1 131 þ 12z þ 13z1 211 þ 13z1 212 þ 8z1 221 þ 10z1 222 þ 15z1 231 þ 18z1 232 þ 12z þ 14z2 112 þ 10z2 121 þ 9z2 122 þ 12z2 131 þ 11z2 132 þ 14z2 211 þ 18z þ 12z2 221 þ 13z2 222 þ 16z2 231 þ 15z2 232 Minimize Z1 4 ¼ 11w1 111 þ 15w1 112 þ 11w1 121 þ 12w1 122 þ 16w þ 14w1 132 þ 15w1 211 þ 15w1 212 þ 9w1 221 þ 12w1 222 þ 17w þ 20w1 232 þ 13w2 111 þ 15w2 112 þ 12w2 121 þ 11w2 122 þ 13w þ 13w2 132 þ 16w2 211 þ 20w2 212 þ 14w2 221 þ 14w2 222 þ 18w þ 16w2 232 X 2 w p ijk zp ijk w 1 1jk ¼ 28; w 2 2jk ¼ 33; w 1 i1k ¼ 22; w 2 1jk ¼ 39; w 1 2jk ¼ 37; w 1 3jk ¼ 9;X4 w 2 3jk ¼ 5; w 2 i1k ¼ 28; w 2 i2k ¼ 22;X3 w 1 i3k ¼ 21; w 1 i4k ¼ 6; w p ij2 ¼ 59; X 2 8i;j;k;p: w 2 i4k ¼ 6;X2 p¼1 w p ij3 ¼ 39; w 1 i2k ¼ 25; w 2 i3k ¼ 21; j¼1 w p ij1 ¼ 53; x p ijk ; yp ijk and zp ijk are the optimal solutions of the previous problems. On solving these problems sequentially, the obtained values of x p ijk ; yp ijk ; zp ijk, and wp ijk ; for p ¼ 1; 2; i ¼ 1; 2; 3; j ¼ 1; 2; 3; 4 and k ¼ 1; 2; 3 are ex ¼ð9; 9; 9; 9Þ;

8 538 D Rani and T R Gulati ex ¼ð0; 0; 2; 2Þ; ex1 121 ¼ð0; 2; 2; 2Þ; ex1 132 ¼ð12; 12; 12; 12Þ; ex ¼ð0; 1; 1; 3Þ; ex1 211 ¼ð5; 5; 5; 5Þ; ex1 213 ¼ð0; 1; 2; 2Þ; ex ¼ð5; 5; 7; 7Þ; ex1 222 ¼ð12; 12; 12; 12Þ; ex1 223 ¼ð0; 1; 1; 3Þ; ex ¼ð0; 2; 2; 2Þ; ex1 242 ¼ð6; 6; 6; 6Þ; ex1 312 ¼ð0; 1; 1; 4Þ; ex ¼ ð0; 0; 0; 1Þ; ex1 332 ¼ ð0; 0; 3; 3Þ; ex1 333 ¼ ð0; 0; 0; 1Þ; ex ¼ð1; 1; 1; 1Þ; ex2 113 ¼ð0; 2; 2; 4Þ; ex2 121 ¼ð16; 16; 16; 16Þ; ex ¼ð0; 0; 1; 1Þ; ex2 132 ¼ð15; 15; 15; 15Þ; ex2 133 ¼ ð0; 0; 2; 2Þ; ex ¼ ð10; 10; 10; 10Þ; ex2 213 ¼ ð9; 9; 11; 11Þ; ex ¼ ð0; 1; 1; 4Þ; ex ¼ ð0; 2; 2; 2Þ; ex ¼ ð6; 6; 6; 6Þ; ex ¼ð0; 1; 1; 2Þ; ex2 322 ¼ð0; 1; 1; 1Þ; ex2 333 ¼ð0; 0; 0; 2Þ: The remaining variables are zero trapezoidal fuzzy numbers and ez 1 ¼ð590; 791; 961; 1131Þ: Minimizing ez 2 in a similar way and then applying the fuzzy programming technique, the obtained optimal-compromise solution is ex ¼ð9; 9; 9; 9Þ; ex1 121 ¼ ð0; 2; 2; 2Þ; ex ¼ð0; 1; 1; 3Þ; ex1 132 ¼ ð12; 12; 12; 12Þ; ex1 133 ¼ð0; 0; 2; 2Þ; ex ¼ð5; 5; 5; 5Þ; ex1 213 ¼ð0; 1; 1; 1Þ; ex1 221 ¼ð17; 17; 19; 19Þ; ex ¼ð0; 3; 4; 6Þ; ex1 242 ¼ð6; 6; 6; 6Þ; ex1 311 ¼ ð0; 1; 1; 1Þ; ex ¼ð0; 0; 3; 6Þ; ex ¼ð0; 0; 0; 1Þ; ex1 333 ¼ð0; 0; 0; 1Þ; ex ¼ð15; 15; 15; 16Þ; ex2 113 ¼ð0; 0; 2; 2Þ; ex2 122 ¼ð11; 11; 11; 11Þ; ex ¼ð0; 2; 3; 4Þ; ex ¼ð6; 6; 6; 6Þ; ex2 212 ¼ ð5; 5; 5; 5Þ; ex ¼ð0; 1; 1; 2Þ; ex ¼ð5; 5; 5; 5Þ; ex2 233 ¼ ð9; 11; 13; 15Þ; ex ¼ð6; 6; 6; 6Þ; ex2 311 ¼ð0; 0; 0; 1Þ; ex2 312 ¼ ð0; 2; 2; 2Þ; ex ¼ð0; 0; 0; 2Þ: All other variables are zero trapezoidal fuzzy numbers. The values of ez 1 and ez 2 are found to be (635,778,955,1100) and (428,566,724,847), respectively. 6. Interpretation of results In this section, the results of the numerical example obtained by using the proposed method are interpreted graphically. The graph of membership functions of the obtained optimal values of ez 1 and ez 2 are in figures 1 and 2. From figure 1, the following information about the minimum value of the objective function Z 1 can be interpreted: (i) 635 MinZ Figure 1. Optimal value of ez 1. Figure 2. Optimal value of ez 2. (ii) The chances that the minimum value of Z 1 will lie in the range units are maximum. (iii) The overall level of satisfaction for other values of Z 1 (say y) isl ez 1 ðyþ%, where 8 ðy 635Þ 635 y\ >< y 955 l ez ðyþ ¼ 1 ð1100 yþ 955\y 1100 >: otherwise The obtained results of the objective function ez 2 can be interpreted in a similar manner. 7. Advantages of the proposed method (i) In the method proposed by Kundu et al [13], the multi-objective multi-item solid transportation problem with transportation parameters as trapezoidal fuzzy numbers is first converted to the equivalent crisp problem. The obtained results are thus real numbers, while the method proposed in this paper provides the fuzzy optimal compromise solution. (ii) Kumar and Kaur [19] pointed out the limitations of existing methods [10, 20 23, 25] and the shortcomings of the method proposed by Liu [24] to solve the single and multi-objective solid transportation problems. To overcome these limitations and resolve the shortcomings, they have proposed a method to obtain the fuzzy optimal solution of the fuzzy solid transportation problem. They have also solved two existing fuzzy solid transportation problems by their method and showed that the problems which could be solved by the existing methods can also be solved by their method. However, none of the method proposed in the above cited papers can be applied to a FFMOMIST, for which a method has been proposed in the present paper. This method is also applicable to the problems considered in Gen et al [25], [10, 20 23]. We have also solved the examples in Kumar and Kaur [19] by the method proposed by us. The results obtained are same as shown in table 6.

9 Uncertain multi-objective multi-product solid transportation 539 Table 6. Results obtained by using the existing as well as proposed method. Example Existing method roposed method Example 3.1 ([19]) (1800,1900,1900,2800) (1800,1900,1900,2800) Example 5.1 ([19]) (226,540,750,879) (226,540,750,879) Numerical example in section 5 Not applicable ez 1 ¼ ð635; 778; 955; 1100ÞeZ 2 ¼ð428; 566; 724; 847Þ 8. Conclusions In this work, the fuzzy optimal compromise solution is obtained for the multi-objective multi-item solid transportation problem, where all the parameters are represented by trapezoidal fuzzy numbers. As in the real world applications, the transportation parameters are not always precise and the fuzzy numbers handle more information than the crisp ones, the obtained results are more beneficial for the decision maker. Since the proposed method is for the FFMOMIST, same is also applicable to the single and multi-objective solid transportation problems as well as to the single and multi-objective solid transportation problems with fuzzy parameters. Acknowledgments The authors are thankful to the reviewers for their valuable comments and suggestions, which improved the presentation of the paper. The first author is also thankful to CSIR, Government of India, for providing financial support. References [1] Schell E D 1955 Distribution of a product by several properties. In: roceedings of 2nd Symposium in Linear rograming, DCS/comptroller, HQ US Air Force, Washington DC, [2] Haley K B 1962 The solid transportation problem. Oper. Res. 10: [3] Zadeh L A 1965 Fuzzy sets. Inf. Control 8: [4] Bector C R and Chandra S 2010 Fuzzy mathematical programming and fuzzy matrix games, studies in fuzziness and soft computing, 169 Springer, Verlag, Heidelberg [5] Zimmermann H J 1978 Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1: [6] Bit A K, Biswal M and Alam S S 1993 Fuzzy programming approach to multi-objective solid transportation problem. Fuzzy Sets Syst. 57: [7] Li Y, Ida K and Gen M 1997 Improved genetic algorithm for solving multi-objective solid transportation problem with fuzzy numbers. Comput. Ind. Eng. 33: [8] Liu B and Liu Y K 2002 Expected Value of Fuzzy Variable and Fuzzy Expected Value Models. IEEE Trans. Fuzzy Syst. 10: [9] Islam S and Roy T K 2006 A new fuzzy multi-objective programming: Entropy based geometric programming and its application of transportation problems. Eur. J. Oper. Res. 173: [10] Ojha A, Das B, Mondala S and Maiti M 2009 An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality. Math. Comput. Model. 50: [11] Gupta A, Kumar A and Kaur A 2012 Mehar s method to find exact fuzzy optimal solution of unbalanced fully fuzzy multiobjective transportation problems. Optim. Lett. 6: [12] Cui Q and Sheng Y 2012 Uncertain programming model for solid transportation problem. Information 15: [13] Kundu, Kar S and Maiti M 2013 Multi-objective multiitem solid transportation problem in fuzzy environment. Appl. Math. Model. 37: [14] Baidya A, Bera U K and Maiti M 2014 Solution of multiitem interval valued solid transportation problem with safety measure using different methods. OSearch 51: 1 22 [15] Kundu, Kar S and Maiti M 2014 Multi-objective solid transportation problems with budget constraint in uncertain environment, Int. J. Syst. Sci. 45: [16] Ebrahimnejad A 2014 A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 19 (2014) [17] Kaufmann A and Gupta M M 1985 Introduction to fuzzy arithmetics: Theory and applications. Van Nostrand Reinhold, New York [18] Liou T S and Wang M J 1992 Ranking fuzzy number with integral values. Fuzzy Sets Syst. 50: [19] Kumar A and Kaur A 2014 Optimal way of selecting cities and conveyances for supplying coal in uncertain environment. Sādhanā 39: [20] Jimenez F and Verdegay J L 1997 Obtaining fuzzy solutions to the fuzzy solid transportation problem with genetic algorithms. roceedings of sixth IEEE internatonal conference on fuzzy systems, Barcelona, Spain, [21] Jimenez F and Verdegay J L 1998 Uncertain solid transportation problems. Fuzzy Sets Syst. 100: [22] Jimenez F and Verdegay J L 1999 Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach. Eur. J. Oper. Res. 117: [23] Yang L and Liu L 2007 Fuzzy fixed charge solid transportation problem and algorithm. Appl. Soft Comput. 7: [24] Liu S T 2006 Fuzzy total transportation cost measures for fuzzy solid transportation problem. Appl. Math. Comput. 174: [25] Gen M, Ida K, Li Y and Kubota E 1995 Solving bicriteria solid transportation problem with fuzzy numbers by a genetic algorithm. Comput. Ind. Eng. 29: