A new fuzzy multi-objective programming: Entropy based geometric programming and its application of...

European Journal of Operational Research 173 (2006) 387–404

www.elsevier.com/locate/ejor

Continuous Optimization

A new fuzzy multi-objective programming: Entropybased geometric programming and its application

of transportation problems

Sahidul Islam, Tapan Kumar Roy *

Department of Mathematics, Bengal Engineering College (D.U.), Howrah 711103, West Bengal, India

Received 13 April 2004; accepted 19 January 2005Available online 9 April 2005

Abstract

In this paper, we introduce a fuzzy mathematical programming with generalized fuzzy number as objective coeffi-cients. We also examine a transportation problem with additional restriction. There is an additional entropy objectivefunction in the transportation problem besides transportation cost objective function. Using new fuzzy mathematicalprogramming, this multi-objective entropy transportation problem with generalized trapezoidal fuzzy number costshas been reduced to a primal geometric programming problem. Pareto optimal solution of the transportation modelis found. Numerical examples have been provided to illustrate the problem.� 2005 Elsevier B.V. All rights reserved.

Keywords: Multi-objective programming; Transportation; Entropy; Generalized fuzzy number; Geometric programming

1. Introduction

The classical transportation problem (Hitchcock transportation problem) is one of the subclasses of lin-ear programming problem in which all the constraints are equality type. Initially, Hitchcock [10] and Kant-orovich [13] developed a transportation model. Arsham and Khan [1] developed a simplex type algorithmto solve the general transportation problem.

0377-2217/$ - see front matter � 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2005.01.050

* Corresponding author. Fax: +91 033 2668 4564.E-mail address: [email protected] (T.K. Roy).

mailto:[email protected]

388 S. Islam, T.K. Roy / European Journal of Operational Research 173 (2006) 387–404

Very few authors discussed transportation planning with entropy objective function. Here entropy isacted as a measure of dispersal of trips between origin and destinations. The entropy maximization modelhas attracted a good deal of attention in urban and regional analysis as well as in other areas. The deve-lopment and analysis of such models is based to a large extent on the pioneering work of Wilson [21].This development has involved the construction of a dual problem based on the duality theory for convexproblems as characterized by the very general theory of Evans [8] and the geometric programmingapproach of Charnes et al. [5], Jefferson and Scott [12] and Nijkamp and Paelinck [18]. Usefulness ofentropy optimization models in transportation planning problem is illustrated in three well-known books([9,14,15]).

Zadeh [22] first introduced the concept of fuzzy set theory. Then Zimmermann [23] applied the fuzzy settheory concept with some suitable membership functions to solve linear programming problem with severalobjective functions. Bit et al. [3] applied a fuzzy programming technique with linear membership function tosolve the multi-objective transportation problem. Liu and Kao [17] described solving fuzzy transportationproblems based on extension principle. Verma et al. [20] applied a a fuzzy programming technique to solvemultiple objective transportation problems with some nonlinear membership functions.

In general transportation problems are solved with the assumptions that the coefficients or cost param-eters are specified in a precise way i.e. in crisp environment. In real life, there are many diverse situationsdue to uncertainty in judgments, lack of evidence etc. Sometimes it is not possible to get relevant precisedata for the cost parameter. This type of imprecise data is not always well represented by random var-iable selected from a probability distribution. Fuzzy number may represent this data. So fuzzy decision-making method is needed here. Before making a decision, decision-makers have to assess the alternativeswith fuzzy numbers and rank these fuzzy numbers correspondingly. It can be seen that ranking fuzzynumbers is a very important procedure in solving a fuzzy programming problem. In reality, decision-makers having different viewpoints will give different ranking outcomes under the same situation. There-fore, a number of methods have been proposed for ranking fuzzy numbers [4]. A relatively simplecomputation and easily understood ranking method proposed by Liou and Wang [16] is considered inthis study.

In this paper we have introduced fuzzy multi-objective mathematical programming problem with gener-alized fuzzy number costs in objective function. Pareto optimal solution of this problem is established. Weconsider a multi-objective entropy transportation problem with an additional restriction. The transporta-tion model consists of two objectives. They are minimum transportation costs and maximum entropyamount. There is an additional restriction on total amount of penalties. It is other than total transportationcost amount. This multi-objective entropy transportation problem is considered with generalized fuzzynumber costs. It is reduced to primal geometric programming problem. Then primal geometric program-ming problem is solved. Numerical examples have been provided to illustrate the problem.

2. Mathematical model

2.1. Single objective transportation problem

In a typical transportation problem, a homogeneous commodity is to be transported from several origins(or sources) to various destinations in such a way that the total transportation cost is minimum.

Consider m origins Oi (i = 1, 2, . . ., m) and n destinations Dj (j = 1, 2, . . ., n). At each origin Oi let ai be theamount of a homogeneous product, which we want to transport to n destinations Dj to satisfy the demandfor bj units of the product there. A variable Tij represents the unknown quantity to be transported fromorigin Oi to destination Dj. C is the total penalty amount (e.g. total delivery time) incurred by all tripsacross the region.

S. Islam, T.K. Roy / European Journal of Operational Research 173 (2006) 387–404 389

A typical transportation problem can be stated as follows:

Minimize ZðT Þ ¼Xm

i¼1

Xn

j¼1

c1ijT ij; ð2:1:1Þ

subject toXn

j¼1

T ij ¼ ai ði ¼ 1; 2; . . . ;mÞ;

Xm

i¼1

T ij ¼ bj ðj ¼ 1; 2; . . . ; nÞ;

Xm

i¼1

Xn

j¼1

c2ijT ij ¼ C;

T ij P 0 for all i and j

(c1ij; c

2ij represent transportation cost, delivery time respectively to transport a unit product from ith source

to jth destination).We assume that ai > 0, bj > 0, c1

ij, c2ij P 0, "i 2 {1, 2, . . ., m} and/or j 2 {1, 2, . . ., n} and

Pmi¼1ai ¼

Pnj¼1bj

(balanced condition).The balanced condition is necessary and sufficient condition for the existence of a feasible solution to the

transportation problem.

2.2. Entropy

In physics, the word entropy has important physical implications as the amount of ‘‘disorder’’ of a sys-tem. In mathematics, a more abstract definition is used. The (Shannon) entropy of a variable X is defined as

EnðX Þ ¼ �X

x

pðxÞ ln pðxÞ;

where p(x) is the probability that X is in the state x, and p(x) log p(x) is defined as 0 if p(x) = 0.In transportation problem, normalizing the trip numbers Tij by dividing them by the total number of

trips T ¼Pn

j¼1

Pmi¼1T ij, a bivariate probability distribution pij ¼

T ij

T is found. So
Xn
j¼1

Xm

i¼1

pij ln pij ¼ � ln T þ 1

T

Xn

j¼1

Xm

i¼1

T ij ln T ij: ð2:2:1Þ

We have seen an interesting alternative interpretation [21] of entropy objective function.We want to find the matrix M = [Tij] which has the greatest number c(M) associated with it subject to

constraints given above where c(M) denotes the number of assignments leading to trip matrix M. We canobtain the number of states which give rise to a matrix M as follows. Let T is the total number of trans-ported commodities. Firstly we can select from T11 from T, T12 from T � T11, T13 from T � T11 � T12,etc., and so the number of possible assignments or states is the number of ways of selecting T11 from T,T CT 11

, multiplied by the number of ways of selecting T12 from T � T11, T�T 11 CT 12, etc. Thus

cðMÞ ¼ T CT 11

T�T 11 CT 12� � � T�T 11�T 13��T mn�1 CT mn ;

cðMÞ ¼ T !

T 11!ðT � T 11Þ!� ðT � T 11Þ!T 12!ðT � T 11 � T 12Þ!

� � � ðT � T 11 � T 13 � � � � T mn�1Þ!T mn!ðT � T 11 � T 12 � T 13 � � � � � T mnÞ!

¼ T !Qnj¼1

Qmi¼1T ij!


or

ln cðMÞ ¼ ln T !�Xn

j¼1

Xm

i¼1

ln T ij!; ð2:2:2Þ

ln cðMÞ ¼ ln e�T T T� �

�Xn

j¼1

Xm

i¼1

ln e�T ij T T ijij

h i½by using Stirlings approximation formula�

¼ �T þ T ln T þXn

j¼1

Xm

i¼1

T ij �Xn

j¼1

Xm

i¼1

T ij ln T ij;

ln cðMÞ ¼ T ln T �Xn

j¼1

Xm

i¼1

T ij ln T ij: ð2:2:3Þ

Since T is given, maximizing ln c(M)(=En(T)) is equivalent to maximizing entropy, as defined in above.This is one of the reasons why the entropy optimization model is particularly suitable for the trip distribu-tion problem. (Here Tij�s and T are assumed to be sufficiently large. Eq. (2.2.3) is not suitable for small Tij�sand T.)

In transportation model, entropy is acted as a measure of dispersal of trips between origin and destina-tions. So it will be more powerful, if we would like to have minimum transportation penalties as well asmaximum entropy amount.

2.3. Multi-objective entropy transportation problem (MOETP)

Taking additional objective function

Maximize EnðT Þ ¼ �Xn

j¼1

Xm

i¼1

T ij

Tln

T ij

T

� �; ð2:3:1Þ

then (2.1.1) becomes

Maximize ENðT Þ ¼ �Xn

j¼1

Xm

i¼1

T ij

Tlog

T ij

T

� �;


i¼1

Xn

j¼1

c1ijT ij; ð2:3:2Þ

subject to the same constraints and restriction as in (2.1.1).

This problem can be written as

Maximize ENðT Þ ¼ �Xm

i¼1

Xn

j¼1

T ij log T ij;


i¼1

Xn

j¼1

c1ijT ij; ð2:3:3Þ

subject to the same constraints and restriction as in (2.1.1).The multi-objective optimization problem is convex if all the objective functions and the feasible region

are convex. So the problem (2.3.3) is a convex programming problem.


2.4. Prerequisite mathematics

Fuzzy sets first introduced by Zadeh [22] in 1965 as a mathematical way of representing impreciseness orvagueness in everyday life.

Fuzzy set: A fuzzy set eA in a universe of discourse X is defined as the following set of pairseA ¼ fðx; leAðxÞÞ : x 2 Xg. Here leA : X ! ½0; 1� is a mapping called the membership function of the fuzzy

set eA and leAðxÞ is called the membership value or degree of membership of x 2 X in the fuzzy set eA.The larger leAðxÞ is the stronger the grade of membership form in eA.

Convex fuzzy set: A fuzzy set eA of the universe of discourse X is convex if and only if for all x1, x2 in X,

leAðkx1 þ ð1� kx2ÞP minðleAðx1Þ;leAðx2ÞÞ when 0 6 k 6 1:

Normal fuzzy set: A fuzzy set eA of the universe of discourse X is called a normal fuzzy set implying thatthere exist at least one x 2 X such that leAðxÞ ¼ 1.

Generalized fuzzy number (GFN): In [6,7], Chen represented a generalized trapezoidal fuzzy number(GTrFN) eA as eA ¼ ða; b; c; d; wÞ, where 0 < w 6 1, and a, b, c and d are real numbers. The generalized fuzzy

number eA is a fuzzy subset of real line R, whose membership function leA satisfies the following conditions:

(1) leA is a continuous mapping from R to the closed interval [0, 1];(2) leAðxÞ ¼ 0, where �1 < x 6 a;(3) leAðxÞ is strictly increasing with constant rate on [a, b];(4) leAðxÞ ¼ w, where b 6 x 6 c;(5) leAðxÞ is strictly decreasing with constant rate on [c, d];(6) leAðxÞ ¼ 0, where d 6 x < 1.

Note: eA is a convex fuzzy set. It will be normalized for w = 1.If w = 1, the generalized fuzzy number eA is called a trapezoidal fuzzy number (TrFN) denotedeA ¼ ða; b; c; dÞ.(i) If a = b and c = d, then eA is called crisp interval [a, d].

(ii) If b = c, then eA is called a generalized triangular fuzzy number (GTFN) as eA ¼ ða; b; c; wÞ.(iii) If b = c, w = 1 then it is called a triangular fuzzy number (TFN) as eA ¼ ða; b; cÞ.(iv) If a = b = c = d and w = 1, then eA is called a real number a.

Definition. A GTrFN eA � ða; b; c; d; wÞ is a fuzzy set of the real line R whose membership functionleAðxÞ : R! ½0;w� is defined as

lweAðxÞ ¼lw

LeAðxÞ ¼ w x�ab�a

� �for a 6 x 6 b;

w for b 6 x 6 c;

lwReAðxÞ ¼ w d�x

d�c

� �for c 6 x 6 d;

0 otherwise;

8>>>><>>>>: ð2:4:1Þ

where a < b < c < d and w 2 (0, 1].

Fig. 1 shows two different generalized trapezoidal fuzzy numbers eA ¼ ða; b; c; d; w1Þ and eB ¼ða; b; c; d; w2Þ which denote two different decision maker�s opinions. The values w1 and w2 represent the de-grees of confidence of the opinions of the decision makers eA and eB, respectively, where w1 = 0.8 andw2 = 1.

Ã(x)

1.0

0.8

~B

~A

O

μ

a b c d

Fig. 1. Two generalized trapezoidal fuzzy numbers eA and eB.


Because of traditional fuzzy arithmetic operations we can any deal with normalized fuzzy numbers, theynot only change the type of membership function of fuzzy number after arithmetical operations, but alsohave a drawback of requiring troublesome and tedious arithmetical operations. Thus in [6] Chen proposedthe function principle, which could be used as the fuzzy numbers arithmetic operations between generalizedfuzzy numbers, where these fuzzy arithmetic operations can deal with the generalized fuzzy numbers. In[11], Hsieh et al. pointed out that arithmetic operators on fuzzy numbers presented in [6] are not onlydo not change the type of membership function of fuzzy numbers after arithmetic operations, but also theycan reduce the troublesomeness and tediousness of arithmetical operations. Thus in this paper, we useChen�s fuzzy number�s arithmetical operators based on Chen [6] (shown in (1)–(5)) to deal with the fuzzynumber arithmetical operations between generalized fuzzy numbers.

The difference between the arithmetic operations on generalized fuzzy numbers and the traditional fuzzynumbers is that the former can deal with both non-normalized and normalized fuzzy numbers, but the lat-ter can only deal with normalized fuzzy numbers.

Let k 2 [0, 1] be a pre-assigned parameter, called the degree of optimism. The graded mean value (or

total k-integral value) of eA is defined as IkðeAÞ ¼ kIRðeAÞ þ ð1� kÞILðeAÞ where ILðeAÞ and IRðeAÞ are the leftand right interval values of eA defined as

IwLðeAÞ ¼ Z 1

0

ðlwLeAÞ�1ada;

IwRðeAÞ ¼ Z 1

0

ðlwReAÞ�1ada:

ð2:4:2Þ

Now,

ðlwLeAÞ�1a ¼ aþ a

wðb� aÞ;

and ðlwReAÞ�1a ¼ d � a

wðd � cÞ:

ð2:4:3Þ

Therefore the left and right integral values are

IwLðeAÞ ¼ w

aþ b2

� �and Iw

RðeAÞ ¼ wcþ d

2

� �: ð2:4:4Þ

Hence the total k-integral value of eA is

Iwk ðeAÞ ¼ kw

cþ d2

� �þ ð1� kÞw aþ b

2

� �� : ð2:4:5Þ


The left integral value is used to reflect the pessimistic viewpoint and the right integral value is used toreflect the optimistic viewpoint of the decision-maker. The total k-integral value is a convex combination ofright and left integral values through the degree of optimism. A large value of k specifies the higher degree

of optimism. For instance, when k = 1, the total integral value Iw1 ðeAÞ ¼ wðcþd

2Þ ¼ Iw

RðeAÞ represents an opti-

mistic viewpoint. On the other hand, when k = 0, the total k-integral value is Iw0 ðeAÞ ¼ w aþb

2

� �¼ Iw

LðeAÞ rep-

resents a pessimistic viewpoint. When k = 0.5, the total k-integral value is Iw0:5ðeAÞ ¼ 1

2½wðaþb

2Þ þ wðcþd

2Þ� ¼

12½Iw

LðeAÞ þ IwRðeAÞ� reflects a moderately optimistic decision-maker�s viewpoint and is the same as the defuzz-

ification of the fuzzy number eA.

Property 2.4.1

(a) If eU ¼ ðu1; u2; u3; u4; wÞ and y = ku, k > 0 then ~y ¼ keu is a fuzzy number (ku1, ku2, ku3, ku4; w).(b) If y = ku, k < 0 then ~y ¼ k~u is a fuzzy number (ku4, ku3, ku2, ku1; w).

Proof. See Appendix A. h

Property 2.4.2. If eA1 ¼ ða1; b1; c1; d1; w1Þ and eA2 ¼ ða2; b2; c2; d2; w2Þ. then eA1 � eA2 is a fuzzy number

(a1 + a2, b1 + b2,c1 + c2, d1 + d2; min(w1,w2)).

Proof. See Appendix B. h

2.5. Fuzzy model with imprecise cost

In our multi-objective entropy transportation problem (MOETP), we have considered that the cost coef-ficient c1

ij as fuzzy numbers, the above crisp model (2.3.3) reduces to


i¼1

Xn

j¼1

T ij log T ij;


i¼1

Xn

j¼1

~c1ijT ij; ð2:5:1Þ

subject to the same constraints and restriction as in (2.1.1).The generalized trapezoidal fuzzy number here represent the cost coefficient c1

ij. These are~c1

ij ¼ ðcij1; cij2; cij3; cij4; wÞ for i = 1, 2, . . ., n; j = 1, 2, . . ., m.For a fixed value of k the k-integral value Iw

k ð~c1ijÞ ¼ ½kwðcij3þcij4

2Þ þ ð1� kÞwðcij1þcij2

2Þ�. Using the k-integral

values of the fuzzy cost coefficients ~c1ij in the objective function of (2.5.1) we get


i¼1

Xn

j¼1

T ij log T ij;


i¼1

Xn

j¼1

Iwk ð~c1

ijÞT ij; ð2:5:2Þ

subject to the same constraints and restriction as in (2.1.1).


3. Mathematical analysis

3.1. Fuzzy programming technique to solve MONLP problem

A multi-objective non-linear programming (MONLP) or a vector minimization problem (VMP) may betaken in the following form:

Minimize f ðxÞ ¼ ½f1ðxÞ; f2ðxÞ; . . . ; fkðxÞ�T ð3:1:1Þsubject to x 2 X ¼ fx 2 Rn : gjðxÞ 6 or ¼ or P bj for j ¼ 1; . . . ;m; x P 0g;

and li 6 xi 6 ui ði ¼ 1; 2; . . . ; nÞ:
Zimmermann [23] showed that fuzzy programming technique could be used nicely to solve the multi-objec-tive programming problem.
To solve the MONLP (3.1.1) problem, following steps are used:

Step 1: Solve the MONLP (3.1.1) as a single objective non-linear programming problem using only oneobjective at a time and ignoring the others. These solutions are known as ideal solutions.

Step 2: From the results of step 1, determine the corresponding values for every objective at each solutionderived. With the values of all objectives at each ideal solution, pay-off matrix can be formulatedas follows:

f1 (x) f2 (x) fk(x)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

)(...)()(

............

)(...)()(

)(...

...

)()(

...*

21

22*2

21

112

1*1

2

1

kk

kk

k

k

k xfxfxf

xfxfxf

xfxfxf

x

x

x

:

Here x1, x2, . . ., xk are the ideal solutions of the objectives f1(x), f2(x), . . ., fk(x) respectively. So

Ur ¼ maxffrðx1Þ; frðx2Þ; . . . ; frðxkÞg; ð3:1:2Þ
and
Lr ¼ f �r ðxrÞ ð3:1:3Þ
(Lr and Ur are lower and upper bounds of the rth objective function fr(x) for r = 1, . . ., k).
Step 3: Using aspiration levels of each objective of the MONLP (3.1.1) may be written as follows:Find x so as to satisfy

frðxÞ6�

Lr ðr ¼ 1; 2; . . . ; kÞ; x 2 X : ð3:1:4Þ

Here objective functions of (3.1.1) are considered as fuzzy constraints. This type of fuzzy con-straints can be quantified by eliciting a corresponding membership function

lwrr ðfrðxÞÞ ¼ 0 if f rðxÞP Ur

¼ wrl1r ðxÞ if L1

r 6 frðxÞ 6 U r

¼ 1 if f rðxÞ 6 L1r

where L1r ¼ Lr þ 2r and 0 6 2r 6 Ur � Lr

9>>>=>>>; ðr ¼ 1; 2; . . . ; kÞ: ð3:1:5Þ

Here l1r ðxÞ is a strictly monotonic decreasing function with respect to fr(x).

Having elicited the membership functions (as in (3.1.5)) lwrr ðfrðxÞÞ for r = 1, 2, . . ., k, a general

aggregation function


lw~DðxÞ ¼ F ðlw1

1 ðf1ðxÞÞ; lw22 ðf2ðxÞÞ; . . . ; lwk

k ðfkðxÞÞÞ

is introduced.So a fuzzy multi-objective decision making problem can be defined as

Maximizex2X

lw~DðxÞ: ð3:1:6Þ

Fuzzy decision [2] based on convex operator (like Tewari et al. [19]), the problem (3.1.6) isreduced to

Maximize lw~Dðx; xÞ ¼

Xk

r¼1

xrlwrr ðfrðxÞÞ ð3:1:7Þ

subject to x 2 X ;

0 6 lwrr ðfrðxÞÞ 6 wr for r ¼ 1; 2; . . . ; k;

where xr P 0 for all r ¼ 1; 2; . . . ; k;Pk

r¼1xr ¼ 1.Step 4: Solve (3.1.7) to get Pareto optimal solution. Some basic definitions and three theorems on Pareto

optimal solutions are introduced below.

Definition 3.1.1 (Complete optimal solution). x� is said to be a complete optimal solution to the MONLP(3.1.1) if and only if there exists x� 2 X such that fr (x�) 6 fr(x), for r = 1, 2, . . ., k and for all x 2 X.

However, when the objective functions of the MONLP conflict with each other, a complete optimalsolution does not always exist and hence the Pareto Optimality Concept arises and it is defined as follows.

Definition 3.1.2 (Pareto optimal solution). x� is said to be a Pareto optimal solution to the MONLP (3.1.1)if and only if there does not exist another x 2 X such that fr(x

�) 6 fr(x) for all r = 1, 2, . . ., k andfj(x) 5 fj(x

�) for at least one j, j 2 {1, 2, . . ., k}.

Definition 3.1.3 (Local (global) optimal solution). x� 2 X (the feasible set of constrained decisions) is saidto be a local (global) Pareto optimal solution to the MONLP (3.1.1) iff x� is Pareto optimal in X \ N(x�,d)where N(x�,d) denotes the d neighborhood of x� defined by {x 2 Rn :kx � x�k < d,d 2 R+}.

Theorem 3.1.1. The solution of fuzzy additive goal programming (FAGP) problem (3.1.7) is weakly Pareto

optimal.

Proof. Let x� 2 X be a solution of the FAGP problem. Let us suppose that it is not weakly Pareto optimal.In this case, there exists a solution x 2 X such that lwr

r ðxÞ < lwrr ðx�Þ for all r = 1, 2, . . ., k. According to

the assumption set to the weighting coefficients, xj > 0 for at least one j. Thus we havePkr¼1xrlwr

r ðxÞ <Pk

r¼1xrlwrr ðx�Þ. This is a contradiction to the assumption that x� is a solution of the FAGP

problem. Thus x� is weakly Pareto optimal. h

Theorem 3.1.2. The solution of the FAGP problem (3.1.7) is Pareto optimal if the weighting coefficients are

positive, that is xr > 0 for all r = 1, 2, . . ., k.

Proof. Let x� 2 X be a solution of the FAGP problem with positive weighting coefficients. Let us supposethat it is not Pareto optimal. This means that there exists a solution x 2 X such that lwr

r ðxÞ 6 lwrr ðx�Þ for all

r = 1,2, . . ., k and lwjj ðxÞ < lwj

j ðx�Þ for at least one j. Since xr > 0 for all r = 1, 2, . . ., k, we havePkr¼1xrlwr

r ðxÞ <Pk

r¼1xrlwrr ðx�Þ. This contradicts the assumption that x� is a solution of the FAGP problem

and thus, x� must be Pareto optimal solution. h


Theorem 3.1.3. The unique solution of FAGP problem (3.1.7) is Pareto optimal.

Proof. Let x� 2 X be a unique solution of the FAGP problem. Let us suppose that it is not Pareto optimal.In this case, there exists a solution x 2 X such that lwr

r ðxÞlwrr ðx�Þ for all r = 1, 2, . . ., k and lwj

j ðxÞ < lwjj ðx�Þ

for at least one j. Because all the weighting coefficients wi are nonnegative, we havePkr¼1xrlwr

r ðxÞ 6Pk

r¼1xrlwrr ðx�Þ. On the other hand, the uniqueness of x� means that

Pkr¼1xrlwr

r ðx�Þ <Pkr¼1xrlwr

r ðx̂Þ for all x̂ 2 X . The two inequalities above are contradictory and thus, x� must be Paretooptimal. h

3.2. Fuzzy programming technique to solve multi-objective entropy transportation problem (MOETP) based

on primal geometric programming (PGP) method

To solve MOETP (2.3.3), step-1of (3.1) is used. After that according to step-2 pay-off matrix is formu-lated as follows:

.

Now U1, L1, U2, L2 (where L1 6 EN(T) 6 U1 and L2 6 Z(T) 6 U2) are identified.Here, for simplicity linear membership functions l ~EN ðENðT ÞÞ and l~ZðZðT ÞÞ for the objective functions

EN(T) and Z(T) respectively are defined as follows:

lw1~ENðENðT ÞÞ ¼

0 for ENðT Þ 6 L01;

w1ENðT Þ�L0

1

U1�L01

�for L01 6 ENðT Þ 6 U 1;

w1 for ENðT ÞP U 1;

8>><>>: ð3:2:1Þ

and

lw2~ZðZðT ÞÞ ¼

w2 for ZðT Þ 6 L02;

w2U2�ZðT Þ

U2�L02

�for L02 6 ZðT Þ 6 U 2;

0 for ZðT ÞP U 2;

8>><>>: ð3:2:2Þ

where L0i ¼ Li þ ei ði ¼ 1; 2Þ. ei (2(0, Ui � Li)) is a real number.Rough sketches of l ~EN ðENðT ÞÞ and l~ZðZðT ÞÞ are shown in Figs. 2 and 3.

))((~ TENEN

μ

1

0 L1 U1 EN(T)

Fig. 2. Membership function for entropy goal.

))((~ TZZ

μ

1

L2 U2 Z(T)0

Fig. 3. Membership function for transportation cost goal.


According to step-3, having elicited the above membership functions crisp non-linear programmingproblem is formulated as follows:

Max F ¼ x1lw1~ENðENðT ÞÞ þ x2l

w2~ZðZðT ÞÞ ð3:2:3Þ

subject to lw1~ENðENðT ÞÞ ¼ w1

ENðT Þ � L01U 1 � L01

� �;

lw2Z ðzðT ÞÞ ¼ w2

U 2 � ZðT ÞU 2 � L02

� �;

Xn

j¼1

T ij ¼ ai ði ¼ 1; 2; . . . ;mÞ;

Xm

i¼1

T ij ¼ bj ðj ¼ 1; 2; . . . ; nÞ;

Xm

i¼1

Xn

j¼1

c2ijT ij ¼ C;

0 6 lw1~ENðENðT ÞÞ 6 w1;

0 6 lw2Z ðzðT ÞÞ 6 w2;

T ij P 0 ði ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; nÞ:

The problem (3.2.3) can be written as

Max F ¼ x1wENðT Þ � L01

U 1 � L01

� �þ x2w

U 2 � ZðT ÞU 2 � L02

� �ð3:2:4Þ

subject toXn

j¼1

T ij ¼ ai ði ¼ 1; 2; . . . ;mÞ;

Xm

i¼1

T ij ¼ bj ðj ¼ 1; 2; . . . ; nÞ;

Xm

i¼1

Xn

j¼1

c2ijT ij ¼ C;

T ij P 0 ði ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; nÞ;

where w = min(w1, w2) and w 2 (0, 1].


Which is equivalent to

Max F 0 ¼ �aXm

i¼1

Xn

j¼1

T ij ln T ij � bXm

i¼1

Xn

j¼1

c1ijT ij;

subject to the same constraints as in (3.2.4)

ð3:2:5Þ

where

a ¼ x1wU 1 � L01

; b ¼ x2wU 2 � L02

and F ¼ F 0 þ x2wU 2

U 2 � L02� x1wL01

U 1 � L01:

Let us apply the transformation Tij = Txij, the above problem (3.2.5) is reduced to

Max F 0 ¼ �aXm

i¼1

Xn

j¼1

Txij ln Txij � bXm

i¼1

c1ijTxij ð3:2:6Þ

subject toXn

j¼1

Txij ¼ ai; ði ¼ 1; 2; . . . ;mÞ;

Xm

i¼1

Txij ¼ bj; ðj ¼ 1; 2; . . . ; nÞ;

Xm

i¼1

Xn

j¼1

c2ijTxij ¼ C;

xij P 0 ði ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; nÞ:

UsingPm

i¼1

Pnj¼1xij ¼ 1

T

Pmi¼1

Pnj¼1T ijð¼ 1Þ, the problem (3.2.6) is equivalent to

Max F 0 ¼ aT lnYmi¼1

Yn

j¼1

kij

xij

� �xij !

ð3:2:7Þ

subject toXm

i¼1

Xn

j¼1

xij ¼ 1 ðnormality conditionÞ;

TPnj¼1

xij � ciPmi¼1

Pnj¼1

xij ¼ 0 ði ¼ 1; 2; . . . ;mÞ

TPmi¼1

xij � bjPmi¼1

Pnj¼1

xij ¼ 0 ðj ¼ 1; 2; . . . ; nÞ

Pmi¼1

Pnj¼1

c2ijT � C

�xij ¼ 0

9>>>>>>>>=>>>>>>>>;ðorthogonality conditionsÞ;

xij > 0 ði ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; nÞ ðpositivity conditionsÞ;

where kij ¼ e�bac1

ij .As a result of these transformation the MOETP (2.3.3) is equivalent to the dual geometric program

(DGP) with objective function and constraints as (3.2.7). The additional constraint gives rise to the orthog-onality condition. The normality condition is equivalent to normalizing the Tij. In the information-theoreticcharacterization of Charnes et al. [5], the normality condition allows the xij to be interpreted as a frequencydistribution.


For the MOETP (2.3.3), if it is assumed that all (i, j) pairs are defined for i = 1, 2, . . ., m; j = 1, 2, . . ., n,the DGP with objective function and constraints involves n2 variables. Under the assumptionPm

i¼1ai ¼Pn

j¼1bj the system involves (m + n � 1) linearly independent constraints.The primal geometric programming (PGP (3.2.7)) is as follows

TablePareto

Metho

GPFAGP

Minimize g0ðtÞ ¼Xm

i¼1

Xn

j¼1

kij

Ymr¼1

tdijr ;

t ¼ t1; t2; . . . ; tmð ÞT P 0;

which is a posinomial geometric programming problem.

4. A numerical example

Crisp model:

Minimize ZðT Þ ¼X2

i¼1

X2

i¼1

c1ijT ij;

Maximize ENðT Þ ¼ �X2

i¼1

X2

j¼1

T ij

Tln

T ij

T

� �; ð4:0:1Þ

subject to

X2

j¼1

T 1j ¼ 100;X2

j¼1

T 2j ¼ 50;X2

i¼1

T i1 ¼ 70;X2

i¼1

T i2 ¼ 80;

X2

i¼1

X2

i¼1

c2ijT ij ¼ C; T ij P 0; i ¼ 1; 2; j ¼ 1; 2;

where

c111 c1

12

c121 c1

22

" #¼

25 35

20 5

� ;

c211 c2

12

c221 c2

22

" #¼

23 25

26 7

� and C ¼ 890:

The Pareto optimal solutions of the MOETP (4.0.1) by primal geometric programming (PGP) methodand fuzzy additive goal programming (FAGP) method are given in Table 1.

The Pareto optimal solutions of MOETP are presented by PGP and FAGP method with equal weightsin Table 1. The PGP method gives better results than the FAGP method.

The Pareto optimal solutions of the MOETP (4.0.1) by primal geometric programming (PGP) methodwith different weights are given in Table 2.

1optimal solutions of MOETP

d T �11 T �12 T �21 T �22 EN*(T) Z*(T)

17.001 5.96 4.65 41.91 1.43 936.1815.91 8.23 4.76 40.97 1.32 945.87

Table 2Pareto optimal solutions of MOETP with different weights by PGP method

Weights T �11 T �12 T �21 T �22 EN*(T) Z*(T) Type

x1 = 0.7, x2 = 0.3 17.001 5.96 4.65 41.91 1.43 946.17 Ix1 = 0.3, x2 = 0.7 15.91 8.23 4.76 40.97 1.32 935.77 II


The results in PGP method follow a particular pattern as expected. In type-I, entropy is higher than thetype-II as in this case more importance has been fixed to the goal of entropy maximization. In type-II,transportation cost is less than that in type-I as in this case more importance has been fixed to the goalof transportation cost minimization.

Fuzzy model: Considering the GTrFN for cost coefficient ~c1ij of the MOETP (2.5.1) we take following

fuzzy input data instead of crisp coefficient matrix and other input data are same as in crisp model.Input data:

TablePareto

Weigh

x1 =x1 =

TablePareto

Test

OptimAboutModerAboutPessim

~c111 ~c1

12

~c121 ~c1

22

" #¼

~25 ~35~20 ~5

" #;

where

~25 ¼ 22; 24; 26; 27; 0:9ð Þ; ~35 ¼ 31; 34; 35; 37; 0:7ð Þ; ~20 ¼ 17; 19; 21; 23; 0:8ð Þ; ~5 ¼ 2; 3; 5; 7; 0:9ð Þ:

The Pareto optimal solutions of the fuzzy multi-objective entropy transportation problem (FMOETP)with different weights by PGP method are given when k = 0.5 in Table 3.

The Pareto optimal solutions of FMOETP with different weights are presented by PGP method whenk = 0.5 in Table 3. The results in PGP method follow a particular pattern as expected. In type-I, entropyis higher than the other types as in this case more importance has been fixed to the goal of entropy max-imization. In type-II, transportation cost is less than the other types as in this case more importance hasbeen fixed to the goal of transportation cost minimization.

The Pareto optimal solutions of FMOETP for different values of k are presented in Table 4.It can be seen that as k decreases, the total transportation cost slightly increases and entropy decreases.

3optimal solutions of FMOETP with weights by PGP method when k = 0.5

ts T �11 T �12 T �21 T �22 EN*(T) Z*(T) Type

0.8, x2 = 0.2 17.31 4.96 5.65 44.91 1.47 938.67 I0.2, x2 = 0.8 16.93 7.31 6.76 42.73 1.42 928.83 II

4optimal solutions of FMOETP with equal weights by PGP method for different values of k

T �11 T �12 T �21 T �22 EN*(T) Z*(T)

istic, i.e., k = 1 18.01 4.86 5.57 43.91 1.49 923.67optimistic, i.e., k = 0.7 17.93 7.39 6.69 41.93 1.46 927.33ate, i.e., k = 0.5 18.04 5.06 6.59 39.98 1.45 930.65pessimistic, i.e., k = 0.2 17.95 8.08 5.97 43.76 1.42 933.98istic, i.e., k = 0 16.75 6.97 4.73 42.63 1.39 939.72


5. Conclusion

In this paper a multi-objective entropy transportation problem with an additional delivery time con-straint is considered. The problem is then solved by FAGP and PGP methods. The PGP method providesan alternative approach to this problem. Here transportation costs are taken as generalized trapezoidalfuzzy numbers. It is formulated as proposed a new fuzzy multi-objective mathematical programming prob-lem. Its advantages lie in its computational efficiency. Solving the problem using the reduced primal systemshould be computationally easier because it is unconstrained. Here decision-maker may obtain the optimalresults according to his expectation of optimistic/pessimistic/moderate cost components. The method pre-sented is quite general and can be applied to the typical problems in other areas of Operation Research andEngineering Sciences (like assignment problems, Inventory problems and structural optimization, etc.).

Acknowledgements

The authors wish to acknowledge the helpful comments and suggestions of the referee�s. This researchwas supported by C.S.I.R. junior research fellowship in the Department of Mathematics, Bengal Engineer-ing College (A Deemed University). This support is great fully acknowledged.

Appendix A. Proof of Property 2.4.1

(a) When k > 0, with the transformation y = ku, we can find the membership function of fuzzy set~y ¼ k eU by a-cut method.

Let the left-hand a-cut of eU , 0 6 a 6 1 is X ULðaÞ ¼ u1 þ aw ðu2 � u1Þ, the right-hand is

X URðaÞ ¼ u4 �awðu4 � u3Þ i:e: u 2 u1 þ

aw

u2 � u1ð Þ; u4 �aw

u4 � u3ð Þh i

:

So,

yð¼ kuÞ 2 ku1 þaw

ku2 � ku1ð Þ; ku4 �aw

ku4 � ku3ð Þh i

: ð1Þ

Therefore, we have

a ¼ wy � ku1

ku2 � ku1

� �; ku1 6 y 6 ku2; ð2Þ

and

a ¼ wku4 � y

ku4 � ku3

� �; ku3 6 y 6 ku4: ð3Þ

From (2) and (3) we have the membership function of ~y ¼ keu is

lw~y ðyÞ ¼

wy � ku1

ku2 � ku1

� �; ku1 6 y 6 ku2;

w; ku2 6 y 6 ku3;

wku4 � y

ku4 � ku3

� �; ku3 6 y 6 ku4;

0; otherwise:

8>>>>>>><>>>>>>>:ð4Þ


(b) Similarly we can proof that, if y = ku, k < 0 then

l~yðyÞ ¼

wku4 � y

ku4 � ku3

� �; ku4 6 y 6 ku3;

w; ku3 6 y 6 ku2;

wy � ku1

ku2 � ku1

� �; ku2 6 y 6 ku1;

0; otherwise: �

8>>>>>>>>><>>>>>>>>>:ð5Þ

Appendix B. Proof of Property 2.4.2

With the transformation y = x1 + x2, we can find the membership function of fuzzy set ~y ¼ eA1 � eA2 bya-cut method.

Let the left-hand a-cut of eA1, 0 6 a 6 1 is X A1LðaÞ ¼ a1 þ aw1

b1 � a1ð Þ, the right-hand is

X A1RðaÞ ¼ d1 �a

w1

d1 � c1ð Þ; i:e: x1 2 a1 þa

w1

b1 � a1ð Þ; d1 �a

w2

d1 � c1ð Þ�

:

The left-hand a-cut of eA2, 0 6 a 6 1 is X A2LðaÞ ¼ a2 þ aw2

b2 � a2ð Þ, the right-hand is

X A2RðaÞ ¼ d2 �a

w2

d2 � c2ð Þ; i:e: x2 2 a2 þa

w2

b2 � a2ð Þ; d2 �a

w2

d2 � c2ð Þ�

:

So,

yð¼ x1 þ x2Þ 2 a1 þ a2 þaw

b1 � a1ð Þ þ b2 � a2ð Þð Þ; d1 þ d2 �aw

d1 � c1ð Þ þ d2 � c2ð Þð Þh i

; ð6Þ

where w = min(w1, w2). Therefore, we have

a ¼ wy � a1 � a2

b1 þ b2 � a1 � a2

� �; a1 þ a2 6 y 6 b1 þ b2; ð7Þ

and

a ¼ wd1 þ d2 � y

d1 þ d2 � c1 � c1

� �; c1 þ c2 6 y 6 d1 þ d2: ð8Þ

From (7) and (8) we have the membership function of ~y ¼ eA1 � eA2 is

lw~y ðyÞ ¼

w y�a1�a2

b1þb2�a1�a2

�for a1 þ a2 6 y 6 b1 þ b2;

w for b1 þ b2 6 y 6 c1 þ c2;

w d1þd2�yd1þd2�c1�c2

�for c1 þ c2 6 y 6 d1 þ d2;

0 otherwise:

8>>>>>>>><>>>>>>>>:ð9Þ

Thus we have eA1 � eA2 ¼ ða1 þ a2; b1 þ b2; c1 þ c2; d1 þ d2; minðw1;w2ÞÞ.


Note: If we have the transformation y = k1A1 + k2A2, k1, k2m not all zero then the fuzzy set~y ¼ k1

eA1 þ k2eA2 is the following fuzzy number:

(i) k1 > 0; k2 P 0 or k1 P 0; k2 > 0;

~y ¼ ðk1a1 þ k2a2; k1b1 þ k2b2; k1c1 þ k2c2; k1d1 þ k2d2 : wÞ;

k1 > 0; k2 6 0 or k1 P 0; k2 < 0;
(ii)
~y ¼ ðk1a1 þ k2d2; k1b1 þ k2c2; k1c1 þ k2b2; k1d1 þ k2a2 : wÞ;

k1 < 0; k2 P 0 or k1 6 0; k2 > 0;
(iii)
~y ¼ ðk1d1 þ k2a2; k1c1 þ k2b2; k1b1 þ k2c2; k1a1 þ k2d2 : wÞ;

k1 < 0; k2 6 0 or k1 6 0; k2 < 0;
(iv)
~y ¼ ðk1d1 þ k2d2; k1c1 þ k2c2; k1b1 þ k2b2; k1a1 þ k2a2 : wÞ:

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