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ORIGINAL PAPER

Fuzzy chance-constrained geometric programming:the possibility, necessity and credibility approaches

Rashed Khanjani Shiraz1 • Madjid Tavana2,3 •

Hirofumi Fukuyama4 • Debora Di Caprio5,6

Received: 6 August 2015 / Revised: 28 October 2015 /Accepted: 11 November 2015 /

Published online: 10 December 2015

� Springer-Verlag Berlin Heidelberg 2015

Abstract Geometric programming (GP) is a powerful tool for solving a variety of

optimization problems. Most GP problems involve precise parameters. However,

the observed values of the parameters in real-life GP problems are often imprecise

or vague and, consequently, the optimization process and the related decisions take

place in the face of uncertainty. The uncertainty associated with the coefficients of

GP problems can be formalized using fuzzy variables. In this paper, we propose

chance-constrained GP to deal with the impreciseness and the ambiguity inherent to

real-life GP problems. Given a fuzzy GP model, we formulate three variants of

chance-constrained GP based on the possibility, necessity and credibility approa-

ches and show how they can be transformed into equivalent deterministic GP

& Madjid Tavana

[email protected];

http://tavana.us/

Rashed Khanjani Shiraz

[email protected]

Hirofumi Fukuyama

[email protected]

Debora Di Caprio

[email protected]

1 School of Mathematics Science, University of Tabriz, Tabriz, Iran

2 Distinguished Chair of Business Analytics, Business Systems and Analytics Department, La

Salle University, Philadelphia, PA 19141, USA

3 Business Information Systems Department, Faculty of Business Administration and Economics,

University of Paderborn, 33098 Paderborn, Germany

4 Faculty of Commerce, Fukuoka University, Fukuoka, Japan

5 Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

6 Polo Tecnologico IISS G. Galilei, Via Cadorna 14, 39100 Bolzano, Italy

123

Oper Res Int J (2017) 17:67–97

DOI 10.1007/s12351-015-0216-7

http://crossmark.crossref.org/dialog/?doi=10.1007/s12351-015-0216-7&domain=pdf

http://crossmark.crossref.org/dialog/?doi=10.1007/s12351-015-0216-7&domain=pdf

problems to be solved via the duality algorithm. We demonstrate the applicability of

the proposed models and the efficacy of the introduced procedures with two

numerical examples.

Keywords Geometric programming � Chance-constrained programming � Fuzzylogic � Possibility � Necessity � Credibility

1 Introduction

Geometric programming (GP) is a method for solving non-linear optimization

problems with many useful applications in business and engineering (Beightler and

Phillips 1976; Liu 2007b). More precisely, GP provides a structured framework for

solving non-linear optimization problems by converting a non-linear problem with

inequality constraints (primal problem) to an equivalent linear problem with

equality constraints (dual problem) which is much easier to solve than the primal

problem.

A variety of methods have been proposed for solving GP problems, ranging from

the original one proposed by Duffin et al. (1967) to the ellipsoid methods. Some of

these methods are based on primal GP and others are based on dual GP. Duffin et al.

(1967) studied problems involving only a positive coefficient for the cost terms.

Passy and Wilde (1967) extended GP by considering both positive and negative

coefficients for the cost terms and generalized it so as to be applied with

posynomials. Successively, Duffin and Peterson (1973) studied extensions of GP to

asignomials (i.e., the differences of two posynomials). In addition, several

approximation methods have been proposed for solving GP problems (Kortanek

et al. 1996; Boyd and Vandenberghe 2004; Boyd et al. 2005, 2007).

A standard GP problem is a minimization problem with objective function being

a posynomial whose variables can take only positive values and a finite number of

inequality constraints, that is:

GP Primal problem

minXh

p¼1

cpYs

j¼1

xap;jj

s:t:

XkðiÞ

tðiÞ¼1

ci;tðiÞYs

j¼1

xci;tðiÞ;jj � bi; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð1Þ

where xj (j = 1,…,s) are the variables of the problem, cp (p = 1, …, h) and ci,t(i)(i = 1, …, n and t(i) = 1, …, k(i)) are positive constant values, ap,j(p = 1, …, h and j = 1, …, s) and ci,t(i),j (i = 1, …, n, t(i) = 1, …, k(i) and

j = 1, …, s) are arbitrary real numbers, and bi (i = 1, …, n) are positive constant

values.

68 R. Khanjani Shiraz et al.

123

It is well-known by the duality theorem of GP that the primal model and the dual

model have the same objective values. Thus, the most common solution approach

for a GP problem is to solve an equivalent linearly constrained dual program

(Peterson 2001). Duffin et al. (1967) and Beightler and Phillips (1976) have shown

that the dual of a GP program is to be formulated as a maximization problem. The

standard dual form of the GP problem (1) is:

Dual problem

maxYh

p¼1

cp

xp

� �xp Yn

i¼1

YkðiÞ

tðiÞ¼1

ci;tðiÞxi;tðiÞbi

� �xi;tðiÞ Yn

i¼1

kkii

0

@

1

A s:t:

Xh

p¼1

xp ¼ 1;

Xh

p¼1

ap;jxp þXn

i¼1

XkðiÞ

tðiÞ¼1

ci;tðiÞ;jxi;tðiÞ ¼ 0; j ¼ 1; . . .; s

ki ¼XkðiÞ

tðiÞ¼1

xi;tðiÞ; i ¼ 1; . . .; n

xp [ 0; p ¼ 1; . . .; h

xi;tðiÞ � 0; i ¼ 1; . . .; n; tðiÞ ¼ 1; . . .; kðiÞ

ð2Þ

1.1 GP drawbacks and the need for fuzzy coefficients

The conventional GP requires precise values for the coefficients. However, the

values of the parameters observed in real-life problems are often imprecise or

vague. This fact causes the optimization process, as well as the consequent

decisions, to be made in the face of uncertainty. These kind of decision making

problems are usually dealt with by means of stochastic programming (Charnes and

Cooper 1959; Danzig 1955; Kall and Mayer 2005), fuzzy programming (Lai and

Hwang 1992; Rommelfanger 1996; Sakawa 1993) or a combination of fuzziness and

randomness (Luhandjula and Gupta 1996; Sakawa et al. 2012; Wang and Qiao

1993; Yano and Sakawa 2014). Regarding, in particular, the issue of representing

the uncertainty associated with the coefficients of GP problems, the standard fuzzy

approach (Zadeh 1965) was proved to be quite useful.

Fuzzy GP has evolved over the past four decades by means of several studies on

GP with interval and fuzzy coefficients (Cao 1993), GP with T fuzzy coefficients

(Cao 2002), and GP with L - R fuzzy coefficients (Mandal and Roy 2006), among

others. In particular, further assessing the wide applicability of GP, Cao (2002)

proposed fuzzy relational GP. Yang and Cao (2005, 2007) made a significant

contribution to fuzzy relational GP using a wide range of fuzzy operators. Wu

(2008) studied a geometric objective function with max–min fuzzy relational

equations as constraints and proposed an efficient procedure for solving the

problem. Zhou and Ahat (2011) proposed an efficient procedure to find the optimal

Fuzzy chance-constrained geometric programming: the… 69

123

solution by considering a GP problem with a system of max-product fuzzy relational

equations as constraints.

Liu (2006a, 2007a) employed GP techniques for deriving the objective value and

providing useful information for discovering the relationship between profit

maximization and returns to scale. Liu (2006b) developed a procedure to derive

the lower- and upper-bounds of the objective of the posynomial GP problem when

the cost and constraint parameters are uncertain. He transformed the imprecise GP

problem with imprecise parameters represented by intervals into a family of

conventional geometric programs and calculated the corresponding objective value.

Liu (2007b) developed a procedure for deriving the fuzzy objective value of the

fuzzy posynomial GP problem when all the decision variables (in the objective

function and the constraints) and the right-hand sides are fuzzy numbers. He

transformed a pair of two-level mathematical programs into a pair of conventional

GP problems based on the duality algorithm.

Tsai et al. (2007) proposed a method for handling non-positive variables with

integer powers in generalized GP problems. They solved the generalized GP

problem with non-positive variables turning them into positive variables through

variable transformation. Liu (2008) developed a solution procedure for solving GP

problems with interval data for the exponents in the objective function, the cost and

the constraint coefficients, and the right-hand sides. He formulated a pair of two-

level mathematical programs to obtain the upper- and lower-bound of the objective

values. He then used the duality theorem and applied a variable separation technique

to transform the pair of two-level mathematical programs into a pair of ordinary

one-level geometric programs. Both geometric programs were finally solved to

produce the interval corresponding to the objective value. Liu (2011) utilized an

extension principle and developed a pair of two-level mathematical programs to

calculate the upper- and lower-bounds of the profit value. He then transformed the

two-level mathematical programs into a class of one-level signomial geometric

programs to solve the problem following the duality theorem and a variable

separation technique.

1.2 Contribution

In summary, there already exists an ample literature on posynomial GP, most of

which is oriented towards a uncertainty-based approach to GP and its applications.

Thanks to its broad applicability GP modeling has been employed and extended so

as to solve a myriad of problems, chance-constrained and not, whose coefficients are

fuzzy numbers, fuzzy variables or random variables.

However, to the best of our knowledge, there is no previous study dealing with

the formulation of crisp GP models equivalent to specific chance-constrained fuzzy

GP problems.

In this paper, we use fuzzy numbers to account for the unavoidable vagueness of

the parameters characterizing real-world GP problems. We define three chance-

constrained GP models that can be implemented when the coefficients can be

interpreted as L - R fuzzy numbers. Finally, we show that the proposed chance-

constrained GP problems can be transformed into conventional GP problems and


123

the optimal objective values calculated using their dual forms. Thus, the novelty of

our analysis is twofold: (1) in the chance-constraint formulation of a conventional

fuzzy GP model that allows us to consider different risk attitudes on the side of the

decision makers; and (2) in the technique applied to obtain the deterministic

equivalent models.

The advantage of the current approach to fuzzy GP over other formalizations

based on fuzzy programming and fuzzy chance-constrained formulations is the

following. In fuzzy programming and, in particular, in standard fuzzy GP, the

coefficients usually are positive interval coefficients, often corresponding to alpha

cuts of fuzzy numbers. Thus, in order to solve the optimization problem at hand, it is

necessary to define and solve a pair of geometric programs. This allow to identify an

upper and lower bound for the optimal objective value at specific alpha level, but

not necessarily the optimal objective value. On the other hand, we solve the

uncertainty of the problem at hand by constructing equivalent deterministic models.

Thus, our method grants the optimal objective value, not just an approximation of it.

From a more technical viewpoint, to transform the constraints of the chance-

constrained GP models into deterministic constraints we build on a technique

similar to the one employed in a fuzzy DEA environment by Khanjani et al. (2014)

and Tavana et al. (2012).

We show that requiring the possibility, necessity and credibility of an event such

as ~k1 � ~k2, where ~k1 and ~k2 are two positive L - R fuzzy numbers, to satisfy a

certain fixed confidence level is equivalent to impose deterministic inequalities that

dependent on the spread of the fuzzy numbers. This defuzzification method allows

for deterministic solutions against the approximate solutions provided by GP

models with interval coefficients. The reader interested in knowing more about

Fuzzy DEA can refer to Emrouznejad et al. (2014).

We demonstrate the applicability of the proposed models and exhibit the efficacy

of the introduced procedures with two numerical examples.

The remainder of the paper is organized as follows. In Sect. 2, we present some

essential definitions concerning fuzzy sets and L - R fuzzy numbers. In Sect. 3, we

present the possibility-constrained approach to fuzzy GP. In Sect. 4, we propose the

necessity-constrained approach to fuzzy GP. In Sect. 5, we present the credibility-

constrained approach to fuzzy GP. Section 6 provides the duals of the deterministic

GP problems obtained in the previous sections, while Sect. 7 presents two

numerical examples. Finally, in Sect. 8, we present our conclusions and some future

research directions.

2 Definitions

In this section, we present a few definitions that will be used throughout the paper.

Definition 1 Let X be a universal set and A � X. The fuzzy subset ~A of X is

defined by means of its membership function l ~A : X ! 0; 1½ � which assigns to each

element x 2 X a real number l ~AðxÞ belonging to the interval [0, 1], where the value

of l ~AðxÞ at x shows the degree of membership of x in A.


123

Definition 2 (Dubois and Prade 1980): A fuzzy subset ~A of the real line < is a real

fuzzy number if it satisfies:

• Normality: 9x 2 < such that l ~AðxÞ = 1;

• Convexity: 8x; y 2 < and 8k 2 ½0; 1�, l ~Aðkxþ ð1� kÞyÞ�minfl ~AðxÞ; l ~AðyÞg.

Definition 3 (Dubois and Prade 1980): A real fuzzy number ~A is said to be

positive (negative), in symbols ~A[ 0 (~A\ 0), if its membership function l ~A

satisfies l ~AðxÞ = 0 whenever x\ 0 (whenever x[ 0).

Definition 4 (Dubois and Prade 1980): A real fuzzy number ~A is called an

L - R fuzzy number if it has the following membership function:

l ~AðxÞ ¼L

m� x

a

� �; if m� a\x\m

1 if x ¼ m

Rx� m

b

� �; if m\x\mþ b

8>><

>>:ð3Þ

where

• a and b are two non-negative real values known as the left and right spreads of~A, respectively,

• m is a real value called the mean value of ~A,• L and R are two non-increasing continuous functions of [0, 1] into [0, 1] such

that L(0) = R(0) = 1 and L(1) = R(1) = 0, called the left and right functions,

respectively.

An L - R fuzzy number is usually denoted by ~A ¼ ða; m; bÞLR.

Definition 5 (Dubois and Prade 1980): An L - R fuzzy number, ~A ¼ða;m; bÞLR ¼ a;m; bð Þ, is called a triangular fuzzy number (in short: TFN) if

LðxÞ ¼ RðxÞ ¼ 1� x; 0� x� 1

0; otherwise:

�

Definition 6 (Fuzzy Arithmetic) (Dubois and Prade 1980): Let ~A ¼ ða; m; bÞLRand ~B ¼ ð�a; �m; �bÞLR be two positive TFNs, ~A[ 0 and ~B[ 0. Then:

Addition: ~Aþ ~B ¼ ða; m; bÞLR þ ð�a; �m; �bÞLR ¼ ðaþ �a; mþ �m; bþ �bÞLR

Subtraction: ~A� ~B ¼ ða;m; bÞLR � ð�a; �m; �bÞLR ¼ ðaþ �b;m� �m; bþ �aÞLR

Multiplication approximationð Þ:~A� ~B ¼ ða;m; bÞLR � ð�a; �m; �bÞLR ¼ ðm�aþ �ma� a�a;m �m;m�bþ �mbþ b�bÞLR

and, if h is the non-zero real number, then:


123

h ~A ¼ hða; m; bÞLR ¼ ðha; hm; hbÞLR; if h[ 0

ð�hb; hm; �haÞLR; if h\0

�

Definition 7 (Dubois and Prade 1980): The c-cut of a fuzzy set ~A, denoted by ~Ac,

is the crisp set of elements belonging to A with a degree of at least c, that is,~Ac ¼ x 2 Ujl ~AðxÞ� c

� �.

Definition 8 (Klir and Yuan 1995): Given c 2 0; 1½ �, the c-cut of an L - R fuzzy

number ~A ¼ ða;m; bÞLR ¼ a;m; bð Þ is the closed interval defined as follows:

~Ac ¼ xjl ~AðxÞ� c� �

¼ ALc ;A

Rc

h i¼ m� aL�1ðcÞ; mþ bR�1ðcÞ

;

where AcL and Ac

R are the left and right extreme points, respectively.

Definition 9 (Zadeh 1978; Zimmermann 1996): A possibility space is a structure

H;P Hð Þ;Posð Þ, where H is a non-empty set, P(H) is the power set of H, and Pos is

a possibility measure, that is, Pos : H ! ½0; 1� is a set function satisfying the

following axioms:

1. P(Ø) = 0, PðHÞ ¼ 1;2. 8A; B 2 PðHÞ, A � B implies Pos(A) B Pos(B);

3. 8fAw : w 2 Wg � PðHÞ; Posð[wAwÞ ¼ SupwPosðAwÞ:

The elements of P(H) are called fuzzy events.

Definition 10 (Zimmermann 1996; Dubois and Prade 1978, 1988): Let

H;P Hð Þ;Posð Þ be a possibility space. The necessity measure of a fuzzy event

A 2 PðHÞ, denoted by Nec(A), is defined as NecðAÞ ¼ 1� PosðAcÞ where Ac is the

complementary set of A.

The necessity measure satisfies the following properties:

a. Nec (Ø) = 0; Nec Hð Þ ¼ 1

b. 8A 2 PðHÞ; PosðAÞ�NecðAÞ;c. 8A; B 2 PðHÞ; A � B implies Nec(A) B Nec(B);

d. 8A 2 PðHÞ; PosðAÞ\1 ) NecðAÞ ¼ 0;e. 8A 2 PðHÞ; NecðAÞ[ 0 ) PosðAÞ ¼ 1:

Thus, the necessity measure is the dual of the possibility measure, that is,

Pos(A) ? Nec(Ac) = 1.

Definition 11 (Liu and Liu 2002): Let H;P Hð Þ;Posð Þ be a possibility space. The

credibility measure of a fuzzy event A 2 PðHÞ, Cr(A), is defined as

Cr(A) = 0.5(Pos(A) ? Nec(A)).

The credibility measure satisfies the following properties:


123

a. Cr(Ø) = 0, Cr(H) = 1;

b. Monotonicity: 8A; B 2 PðHÞ, A � B implies Cr Að Þ�Cr Bð Þ;c. Self-duality: 8A 2 PðHÞ, Cr Að Þ þ Cr Acð Þ ¼ 1;

d. 8fAw : w 2 Wg � PðHÞ, such that SupwCrðAwÞ\0:5Crð[wAwÞ ¼SupwCrðAwÞ:

e. Subadditivity: 8A; B 2 PðHÞ, Cr A [ Bð Þ�Cr Að Þ þ Cr Bð Þ;f. 8A 2 PðHÞ, Pos Að Þ�Cr Að Þ�Nec Að Þ.

Definition 12 (Liu and Liu 2002): Let ~n be a fuzzy variable on a possibility space

H;P Hð Þ;Posð Þ. The possibility, necessity and credibility of the fuzzy event f~n� rg,where r is any real number, are defined as follows:

Posð~n� rÞ ¼ Supt� r

l~nðtÞ

Necð~n� rÞ ¼ 1� Supt\r

l~nðtÞ

Crð~n� rÞ ¼ 0:5 Posð~n� rÞ þ Necð~n� rÞh i

where l~n : < ! ½0; 1� is the membership function of ~n. Note here that

Crð~n� rÞ ¼ 1� Crð~n\rÞ.

Remark 1 Since in the standard GP model in Eq. (1), the coefficients are assumed

to be positive, in the following subsections we will restrict the analysis to the case

where the coefficients are positive L�R fuzzy number. Nonetheless, the GP model

in Eq. (1) admits a more general formulation that allows to account for situations

where the coefficients can also take non-positive values. For the sake of

completeness, we include the corresponding primal and dual problems.

GP primal problem Dual problem

minPh

p¼1

ep cpQs

j¼1

xap;jj max eoo

Qh

p¼1

cp

xp

� �epxp Qn

i¼1

QkðiÞ

tðiÞ¼1

ci;tðiÞxi;tðiÞbi

� �ei;tðiÞxi;tðiÞ Qn

i¼1

kkii

!eoo

s:t: s:t:PkðiÞ

tðiÞ¼1

ei;tðiÞ ci;tðiÞQs

j¼1

xci;tðiÞ;jj � ei bi; eoo

Ph

p¼1

epxp ¼ 1;

i ¼ 1; . . .; nPh

p¼1

epap;jxp þPn

i¼1

PkðiÞ

tðiÞ¼1

ei;tðiÞci;tðiÞ;jxi;tðiÞ ¼ 0; j ¼ 1; . . .; s

xj [ 0; j ¼ 1; . . .; s ki ¼ eiPkðiÞ

tðiÞ¼1

ei;tðiÞxi;tðiÞ; i ¼ 1; . . .; n

ep; ei;tðiÞ; ei ¼ 1 or � 1 xp [ 0; p ¼ 1; . . .; hxi;tðiÞ � 0; i ¼ 1; . . .; n; tðiÞ ¼ 1; . . .; kðiÞep; ei;tðiÞ; ei ¼ 1 or � 1

eoo is the sign of the primal objective at the optimum


123

3 Possibility-constrained GP model

Zadeh (1978) introduced the idea of fuzzy variable. A fuzzy variable is associated

with a possibility distribution in the same way as a random variable is associated

with a probability distribution. Possibility theory is the statistical counterpart of

probability theory and deals with uncertainty by offering qualitative means for

incomplete knowledge using fuzzy sets. An excellent reference on possibility theory

is provided by Dubois and Prade (1988).

Chance-constrained programming methods deal with problems involving prob-

abilistic uncertainty while fuzzy programming sees each fuzzy coefficient as a fuzzy

variable and each constraint as a fuzzy event.

We propose a method to solve standard GP models [see Eq. (1)] whose

coefficients are real fuzzy numbers and, in particular, TFNs. In this section, we

focus on solving the fuzzy GP problem by using the possibility approach.

Interpreting the constraints as fuzzy events, we can start by formulating the

following chance-constrained GP model.

min u

s:t:

Pos u�Xh

p¼1

~cpYs

j¼1

xap;jj

!� d

PosXkðiÞ

tðiÞ¼1

~ci;tðiÞYs

j¼1

xci;tðiÞ;jj � ~bi

0

@

1

A� d; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð4Þ

where

• d is a predetermined value between 0 and 1 signifying the threshold level;

• for p = 1, …, h, ~cp is a positive L - R fuzzy number cap; cmp ; c

bp

� �characterized

by the following membership function:

l~cpðxÞ ¼L

cmp � x

cap

!; if cmp � cap\x� cmp

Rx� cmp

cbp

!; if cmp � x\cmp þ cbp

8>>>><

>>>>:

ð5Þ

• for i = 1, …, n and t(i) = 1, …, k(i), ~ci;tðiÞ is a positive L - R fuzzy number

cai;tðiÞ; cmi;tðiÞ; c

bi;tðiÞ

� �characterized by the following membership function:


123

l~ci;tðiÞðxÞ ¼

Lcmi;tðiÞ � x

cai;tðiÞ

!; if cmi;tðiÞ � cai;tðiÞ\x� cmi;tðiÞ

Rx� cmi;tðiÞ

cbi;tðiÞ

0

@

1

A; if cmi;tðiÞ � x\cmi;tðiÞ þ cbi;tðiÞ

8>>>>><

>>>>>:

ð6Þ

• for i = 1, …, n, ~bi is a positive L - R fuzzy number ðbai ; bmi ; bbi Þ characterized

by the following membership function:

l~biðxÞ ¼

Lbmi � x

bai

� �; if bmi � bai\x� bmi

Rx� bmi

bbi

!; if bmi � x\bmi þ b

bi

8>>><

>>>:ð7Þ

We will refer to Model (4) as the possibility-constrained GP model.

The objective function of Model (4) is minimized by imposing the constraint

satisfying a threshold level of at least d. Thus, the interpretation of Model (4) at the

optimal solution is that the objective valuePh

p¼1 ~cpQs

j¼1 xap;jj is at least equal to the

value taken by u and all constraints are simultaneously satisfied at the pre-specified

possibility level d.In order to solve the possibility-constrained GP problem (4), we must convert its

constraints into their respective crisp equivalents. In this sense the following lemma

presented by Khanjani et al. (2014) plays a crucial role in solving the proposed

Model (4).

Lemma 1 Let ~k1 ¼ ða1; m1; b1ÞLR and ~k2 ¼ ða2; m2; b2ÞLR be two L - R fuzzy

number with continuous membership functions. For a given confidence level

d 2 0; 1½ �, we have:

Posð~k1 � ~k2Þ� d ) m1 þ b1R�1ðdÞ�m2 � a2R

�1ðdÞ:

Since, the fuzzy coefficients ~cp and ~ci;tðiÞ are fuzzy numbers whose membership

functions are defined in Eqs. (5) and (6), by Zadeh’s extension principle, the fuzzy

numbersPh

p¼1 ~cpQs

j¼1 xap;jj and

PkðiÞtðiÞ¼1

~ci;tðiÞQs

j¼1 xci;tðiÞ;jj are characterized by the

following membership functions:

lPh

p¼1~cpQs

j¼1xap;jj

ðxÞ

¼L

Php¼1 c

mp

Qsj¼1 x

ap;jj � x

Php¼1 c

ap

Qsj¼1 x

ap;jj

!; if

Ph

p¼1

cmpQs

j¼1

xap;jj �

Ph

p¼1

capQs

j¼1

xap;jj \x�

Ph

p¼1

cmpQs

j¼1

xap;jj

Rx�

Php¼1 c

mp

Qsj¼1 x

ap;jjPh

p¼1 cbp

Qsj¼1 x

ap;jj

!; if

Ph

p¼1

cmpQs

j¼1

xap;jj � x\

Ph

p¼1

cmpQs

j¼1

xap;jj þ

Ph

p¼1

cbpQs

j¼1

xap;jj

8>>>>><

>>>>>:

ð8Þ


123

lPkðiÞtðiÞ¼1

~ci;tðiÞQs

j¼1xci;tðiÞ;jj

ðxÞ

¼

L

PkðiÞtðiÞ¼1

cmi;tðiÞQs

j¼1 xci;tðiÞ;jj � x

PkðiÞtðiÞ¼1

cai;tðiÞQs

j¼1 xci;tðiÞ;jj

0

@

1

A; ifPkðiÞ

tðiÞ¼1

cmi;tðiÞQs

j¼1

xci;tðiÞ;jj �

PkðiÞ

tðiÞ¼1

cai;tðiÞQs

j¼1

xci;tðiÞ;jj \x�

PkðiÞ

tðiÞ¼1

cmi;tðiÞQs

j¼1

xci;tðiÞ;jj

Rx�

PkðiÞtðiÞ¼1

cmi;tðiÞQs

j¼1 xci;tðiÞ;jj

PkðiÞtðiÞ¼1

cbi;tðiÞQs

j¼1 xci;tðiÞ;jj

0

@

1

A; ifPkðiÞ

tðiÞ¼1

cmi;tðiÞQs

j¼1

xci;tðiÞ;jj � x\

PkðiÞ

tðiÞ¼1

cmi;tðiÞQs

j¼1

xci;tðiÞ;jj þ

PkðiÞ

tðiÞ¼1

cbi;tðiÞ

Qs

j¼1

xci;tðiÞ;jj

8>>>>>><

>>>>>>:

ð9Þ

Lemma 1 indicates that the deterministic equivalent form of the first constraint in

Model (4), given by the possibility relationship Pos u�Ph

p¼1 ~cpQs

j¼1 xap;jj

� �� d,

yields u�Ph

p¼1 cmp � R�1ðdÞcap� �Qs

j¼1 xap;jj :

Similarly, the chance constraint PosPkðiÞ

tðiÞ¼1~ci;tðiÞ

Qsj¼1 x

ci;tðiÞ;jj � ~bi

� �� d can be

transformed into the following constraint:

XkðiÞ

tðiÞ¼1

cmi;tðiÞ � R�1ðdÞcai;tðiÞ� �Ys

j¼1

xci;tðiÞ;jj � bmi þ R�1ðdÞbbi ; i ¼ 1; . . .; n:

Consequently, Model (4) becomes the following crisp programming model:

min /

s:t:

u�Xh

p¼1

cmp � R�1ðdÞcap� �Ys

j¼1

xap;jj ;

XkðiÞ

tðiÞ¼1


j¼1

xci;tðiÞ;jj � bmi þ R�1ðdÞbbi ; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð10Þ

Since the decision variable u appears only in the first constraint, and minimizing

u is equivalent to minimizingPh

p¼1 cmp � R�1ðdÞcap� �Qs

j¼1 xap;jj , Model (10) can be

transformed into the following problem:

minXh

p¼1

cmp � R�1ðdÞcap� �Ys

j¼1

xap;jj

s:t:

XkðiÞ

tðiÞ¼1


j¼1

xci;tðiÞ;jj � bmi þ R�1ðdÞbbi ; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð11Þ


123

Remark 2 Model (11) is an instance of standard deterministic GP problem [see

Eq. (1)]. Note, in particular, that all the coefficients of this problem are positive.

Indeed, since the coefficients in Model (4) are all positive L - R fuzzy numbers, we

must have that cmi;tðiÞ � cai;tðiÞ [ 0 (i ¼ 1; . . .; n) and cmp � cap [ 0 (p ¼ 1; . . .; h).

Hence, cmi;tðiÞ [ cai;tðiÞ (i ¼ 1; . . .; n) and cmp [ cap (p ¼ 1; . . .; h). At the same time, R is

a non-increasing continuous functions of 0; 1½ � into 0; 1½ �. Thus, it follows that

cmp � R�1ðdÞcap [ 0 (p ¼ 1; . . .; h) and cmi;tðiÞ � R�1ðdÞcai;tðiÞ [ 0 (i ¼ 1; . . .; n).

Finally, it is trivial that bmi þ R�1ðdÞbbi [ 0 8i ¼ 1; . . .; n. h

4 Necessity-constrained GP model

The possibility approach formalizes an optimistic viewpoint and is suitable for

risk-taking decision makers. In contrast, the necessity approach formalizes a

pessimistic viewpoint and is suitable for decision makers who take risk-averse

behavior. Indeed, as already observed, the necessity measure is the dual of the

possibility measure, that is, Nec Acð Þ þ Pos Að Þ ¼ 1 where A 2 P Hð Þ and Ac is the

complement of A in H: In other words, the relation Nec Acð Þ ¼ 1� Pos Að Þ says

that the necessity measure of a set is defined as the impossibility of its

complementary set, that is, an event is sure (necessarily true) when its opposite

event is impossible.

It can also be verified that Pos(A) C Nec(A). This means that an event becomes

possible before becoming necessary (Dubois and Prade 1988). Dubois and Prade

(1983) defined the following indices:

Pos ~a� ~b� �

¼ supu� v

min l~a uð Þ; l~b vð Þ� �

Nec ~a� ~b� �

¼ infusup

u

infv:u� vf g

max 1� l~a uð Þ; l~b vð Þ� �

Pos ~a[ ~b� �

¼ supu

infv:u� vf g

min l~a uð Þ; 1� l~b vð Þ� �

We can now present a method to solve a standard GP model (see Eq. (1)) with

fuzzy coefficients, that is, a fuzzy GP model, using the necessity approach. As in the

possibility approach, we first interpret the fuzzy GP problem as a chance-

constrained GP model obtaining the following model that will be called the

necessity-constrained GP model.


123

min u

s:t:

Nec u�Xh

p¼1

~cpYs

j¼1

xap;jj

!� d

NecXkðiÞ

tðiÞ¼1

~ci;tðiÞYs

j¼1


0@

1A� d; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð12Þ

Afterwards, we convert it into a crisp model. We show the following lemma, which

plays a crucial role in transforming the necessity-constrained GP model (12) into a

deterministic GP problem.

Lemma 2 Let ~k1 ¼ ða1;m1; b1ÞLR and ~k2 ¼ ða2;m2; b2ÞLR be two positive

L - R fuzzy numbers with continuous membership functions. For a given confidence

level d 2 0; 1½ �, we have:

Necð~k1 � ~k2Þ� d ) m1 � a1L�1ð1� dÞ�m2 þ b2L

�1ð1� dÞ:

Proof Using the fuzzy arithmetic (see Definition 6), the L - R fuzzy number~k ¼ ~k1 � ~k2 is equal to (a1 ? b2, m1 - m2, a2 ? b1)LR. By Definition 12, the

necessity of the fuzzy event f~k� 0g is expressed as follows:

Necð~k� 0Þ ¼

1; if 0\ �m� �a

1� L�m

�a

� �; if �m� �a\0� �mþ �b

0; if 0[ �mþ �b

8>><

>>:

where �a ¼ a1 þ b2, �b ¼ a2 þ b1 and �m ¼ m1 � m2: Suppose now that

Necð~k1 � ~k2Þ� d. Then, we have:

d� 1� L�m

�a

� �, ð1� dÞ� L

�m

�a

� �, L�1ð1� dÞ� �m

�a

, ða1 þ b2ÞL�1ð1� dÞ� ðm1 � m2Þ, m1 � a1L

�1ð1� dÞ�m2 þ b2L�1ð1� dÞ: Q:E:D:

Consider the first constraint in Model (12) for the deterministic equivalent.By

Lemma 2, the following equivalence holds:

Nec u�Xh

p¼1

~cpYs

j¼1

xap;jj

!� d , u�

Xh

p¼1

cmp þ L�1ð1� dÞcbp� �Ys

j¼1

xap;jj :

Similarly, Lemma 2 can be applied to the remaining constraints and ultimately

Model (12) is transformed into Model (13).


123

min u

s:t:

u�Xh

p¼1


j¼1

xap;jj ;

XkðiÞ

tðiÞ¼1

cmi;tðiÞ þ L�1ð1� dÞcbi;tðiÞ

� �Ys

j¼1

xci;tðiÞ;jj � bmi � L�1ð1� dÞbai ; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð13Þ

Finally, Model (13) can be transformed into the following problem:

minXh

p¼1


j¼1

xap;jj

s:t:

XkðiÞ

tðiÞ¼1


� �Ys

j¼1

xci;tðiÞ;jj � bmi � L�1ð1� dÞbai ; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð14Þ

Remark 3 Model (14) is an instance of standard deterministic GP problem (see

Eq. (1)). As for Model (11), we note that all the coefficients of this problem are

positive. Since the coefficients in Model (4) are all positive L� R fuzzy numbers,

we also have that bmi � bai [ 0 8i ¼ 1; . . .; n, from which it follows that bmi [ bai8i ¼ 1; . . .; n. Given that L is a non-increasing continuous function of 0; 1½ � into0; 1½ �, we have bmi � L�1ð1� dÞbai [ 0 8i ¼ 1; . . .; n. On the other hand, it is

trivial to check that cmp þ L�1ð1� dÞcbp [ 0 (p ¼ 1; . . .; h) and cmi;tðiÞ þ L�1ð1�dÞcb

i;tðiÞ [ 0 (i ¼ 1; . . .; n). h

5 Credibility-constrained GP model

As we discussed in Sects. 3 and 4, the possibility and necessity approaches proposed

by Dubois and Prade (1980, 1988) reflect extreme optimistic and pessimistic

attitudes, respectively. Therefore, we need a more general measure for fuzzy

problems. In this sense, the optimistic-pessimistic character of a fuzzy event should

be considered to avoid extreme attitudes. Accordingly, Liu (2002) introduced the

credibility measure. This measure adjusts to the varying attitudes of the decision

makers.


123

Using the credibility of the fuzzy events, we proposed to solve a fuzzy GP

model through the formulation of following credibility-constrained GP model.

min /

s:t:

Cr /�Xh

p¼1

~cpYs

j¼1

xap;jj

!� d

CrXkðiÞ

tðiÞ¼1

~ci;tðiÞYs

j¼1


0

@

1

A� d; i ¼ 1; . . .; n

xj [ 0; j ¼ 1; . . .; s

ð15Þ

Lemma 3 (Tavana et al. 2012): Let ~k1 ¼ ða1; m1; b1ÞLR and ~k2 ¼ ða2; m2; b2ÞLRbe two independent L - R fuzzy numbers with continuous membership functions.

For a given confidence level d 2 0; 1½ �, we have:

(a) If d B 0.5, then Crð~k1 � ~k2Þ� d , m1 þ b1R�1ð2dÞ�m2 � a2R�1ð2dÞ.

(b) If d[ 0.5, then Crð~k1 � ~k2Þ� d , m1 � a1L�1ð2ð1� dÞÞ�m2þb2L

�1ð2ð1� dÞÞ.

Using Lemma 3, the first constraint of Model (15) can be rewritten as follows:

(a) If d B 0.5, then

Cr u�Xh

p¼1

~cpYs

j¼1

xap;jj

!� d , u�

Xh

p¼1

cmp � R�1ð2dÞcap� �Ys

j¼1

xap;jj

(b) If d[ 0.5, then

Cr u�Xh

p¼1

~cpYs

j¼1

xap;jj

!� d , u�

Xh

p¼1

cmp þ L�1ð2ð1� dÞÞcbp� �Ys

j¼1

xap;jj

We can apply a similar process to the other constraints of Model (15), i.e.,

CrPkðiÞ

tðiÞ¼1~ci;tðiÞ

Qsj¼1 x

ci;tðiÞ;jj � ~bi

� �� d; i ¼ 1; . . .; n. Thus, Model (15) can be

transformed into the following two models, corresponding to the case where

d B 0.5 and d[ 0.5, respectively.


123

If d� 0:5 If d� 0:5minu

s:t:

minu

s:t:

u�Ph

p¼1

cmp � R�1ð2dÞcap� � Qs

j¼1

xap;jj ; u�

Ph

p¼1

cmp þ L�1ð2ð1� dÞÞcbp� � Qs

j¼1

xap;jj ;

PkðiÞ

tðiÞ¼1

cmi;tðiÞ � R�1ð2dÞcai;tðiÞ� � Qs

j¼1

xci;tðiÞ;jj

PkðiÞ

tðiÞ¼1

cmi;tðiÞ þ L�1ð2ð1� dÞÞcbi;tðiÞ

� � Qs

j¼1

xci;tðiÞ;jj

� bmi þ R�1ð2dÞbbi ; i ¼ 1; . . .; n � bmi � L�1ð2ð1� dÞÞbai ; i ¼ 1; . . .; nxj [ 0; j ¼ 1; . . .; s xj [ 0; j ¼ 1; . . .; s

ð16Þ

These two models can be then transformed into the following problems for

d B 0.5and d[ 0.5, respectively.

If d� 0:5 If d� 0:5

minPh

p¼1

cmp � R�1ð2dÞcap� � Qs

j¼1

xap;jj min

Ph

p¼1

cmp þ L�1ð2ð1� dÞÞcbp� � Qs

j¼1

xap;jj

s:t: s:t:PkðiÞ

tðiÞ¼1

cmi;tðiÞ � R�1ð2dÞcai;tðiÞ� � Qs

j¼1

xci;tðiÞ;jj

PkðiÞ

tðiÞ¼1


� � Qs

j¼1

xci;tðiÞ;jj

� bmi þ R�1ð2dÞbbi ; i ¼ 1; . . .; n � bmi � L�1ð2ð1� dÞÞbai ; i ¼ 1; . . .; nxj [ 0; j ¼ 1; . . .; s xj [ 0; j ¼ 1; . . .; s

ð17Þ

Remark 4 Both models in Eq. (17) are instances of standard deterministic GP

problems (see Eq. (1)). To check that all the coefficients are positive we can reason

as in Remarks 2 and 3. h

6 Solution approach: dual models

In order to solve the models introduced in the previous sections, we follow the

duality theory and transform them into their dual form using the model in Eq. (2). In

all the dual problems below, the variables are xp and xi,t(i).

The dual of the possibility-constrained GP model (Model (11)) is given by:


123

maxYh

p¼1

cmp � R�1ðdÞcbpxp

!xp Yn

i¼1

YkðiÞ

tðiÞ¼1

cmi;tðiÞ � R�1ðdÞcai;tðiÞðbmi þ R�1ðdÞbbi Þxi;tðiÞ

!xi;tðiÞ Yn

i¼1

kkii

s:t:

Xh

p¼1

xp ¼ 1;

Xh

p¼1

ap;jxp þXn

i¼1

XkðiÞ

tðiÞ¼1

ci;tðiÞ;jxi;tðiÞ ¼ 0; j ¼ 1; . . .; s

ki ¼XkðiÞ

tðiÞ¼1

xi;tðiÞ; i ¼ 1; . . .; n

xp [ 0; p ¼ 1; . . .; h

xi;tðiÞ � 0; i ¼ 1; . . .; n; tðiÞ ¼ 1; . . .; kðiÞ

ð18Þ

The dual of the necessity-constrained GP model (Model (14)) is:

maxYh

p¼1

cmp þ L�1ð1� dÞcbpxp

!xp Yn

i¼1

YkðiÞ

tðiÞ¼1


ðbmi � L�1ð1� dÞbai Þxi;tðiÞ

!xi;tðiÞ Yn

i¼1

kkii

s:t:

Constraints of Model ð18Þð19Þ

The dual problems of the credibility-constrained GP models in Eq. (17) are the

following:

If d� 0:5

maxYh

p¼1

cmp � R�1ð2dÞcapxp

!xp Yn

i¼1

YkðiÞ

tðiÞ¼1

cmi;tðiÞ � R�1ð2dÞcai;tðiÞðbmi þ R�1ð2dÞbbi Þxi;tðiÞ

!xi;tðiÞ Yn

i¼1

kkii

s:t:


If d[ 0:5

maxYh

p¼1

cmp þ L�1ð2ð1� dÞÞcbpxp

!xp Yn

i¼1

YkðiÞ

tðiÞ¼1


ðbmi � L�1ð2ð1� dÞÞbai Þxi;tðiÞ

!xi;tðiÞ Yn

i¼1

kkii

s:t:



123

7 Numerical examples

7.1 Example 1

Let us consider the following GP problem:

min ~c1x�11 x�1

2 x�13 þ ~c2x2x3;

s:t:

~a1x1x3 þ ~a2x1x2 � ~b;

x1 [ 0; x2 [ 0; x3 [ 0:

ð22Þ

where ~c1; ~c2; ~a1; ~a2 and ~b are positive L - R fuzzy numbers. Model (22) is an

instance of standard fuzzy posynomial GP problem (Model (1)). The objective

function is composed by h = 2 terms with coefficients ~cp, p = 1, 2. The variables

are s = 3 and there is only one constraint (that is, n = 1) consisting of two terms

(hence, k(1) = 2 and t(1) = 1, 2) whose coefficients ~c1;1 and ~c1;2 correspond to ~a1and ~a2, respectively. Finally, ~b stands for the only bounding fuzzy coefficient ~b1 ofthe problem.

As shown in the previous sections, Model (22) can be solved considering its

chance-constrained formulation under the possibility, necessity or credibility

approach [resp. Models (4), (12) or (15)]: after transforming it in a deterministic

posynomial GP problem [resp. Models (11), (14) or (17)], the solution is obtained

passing to the dual [resp. Models (18), (19) or (20)–(21)].

Assume that the fuzzy coefficients of Model (22) to be positive and symmetrical

TFNs. A TFN is symmetrical when its left and right spreads coincide so that it can

be represented by a pair (m, a)LR, where m is the mean and a the value of both the

left and the right spreads.

By Definition 5, the left and right functions are defined as follows:

LðxÞ ¼ RðxÞ ¼ 1� x; 0� x� 1

0; otherwise:

�

The membership functions are described by Eq. (3), where a = b. The fuzzy

numbers used in this example are reported in Table 1.

Using our solution approach, we can move the problem to its dual form.

In the possibility approach we apply Model (18) and obtain the dual problem

described by Model (23) below. The variables are xp, with p = 1, 2, and x1,t(1),

t(1) = 1, 2.

Table 1 The triangular fuzzy variables

~c1 (50, 20) ~a1 (3, 2)

~c2 (50, 20) ~a2 (2, 2)

~b (3, 2)


123

By the definition of the function R, given a confidence level d 2 [0, 1], we have:

RðdÞ ¼ 1� d , R�1ðdÞ ¼ 1� d:

Hence, we have:

max50� 20ð1� dÞ

x1

� �x1 50� 20ð1� dÞx2

� �x2 3� 2ð1� dÞx1;1 3þ 2ð1� dÞ½ �

� �x1;1

2� 2ð1� dÞx1;2 3þ 2ð1� dÞ½ �

� �x1;2

kk

s:t:

x1 þ x2 ¼ 1

k ¼ x1;1 þ x1;2

� x1 þ x1;1 þ x1;2 ¼ 0

� x1 þ x2 þ x1;2 ¼ 0

� x1 þ x2 þ x1;1 ¼ 0

x1 [ 0; x2 [ 0; x1;1 � 0; x1;2 � 0:

ð23Þ

The duals in the necessity and credibility approaches are obtained from Model

(19) and Models (20) and (21).

In the necessity approach the dual GP problem is given by:

max50þ 20d

x1

� �x1 50þ 20dx2

� �x2 3þ 2dx1;1 3� 2d½ �

� �x1;1 2þ 2dx1;2 3� 2d½ �

� �x1;2

kk

s:t:

Constraints of Model ð23Þ

ð24Þ

In the credibility approach the dual GP problem is given by:

If d� 0:5

max50� 20ð1� 2dÞ

x1

� �x1 50� 20ð1� 2dÞx2

� �x2

3� 2ð1� 2dÞx1;1 3þ 2ð1� 2dÞ½ �

� �x1;1 2� 2ð1� 2dÞx1;2 3þ 2ð1� 2dÞ½ �

� �x1;2

kk

s:t:


ð25Þ


123

If d[ 0:5

max50þ 20ð2d� 1Þ

x1

� �x1 50þ 20ð2d� 1Þx2

� �x2

3þ 2ð2d� 1Þx1;1 3� 2ð2d� 1Þ½ �

� �x1;1 2þ 2ð2d� 1Þx1;2 3� 2ð2d� 1Þ½ �

� �x1;2

kk

s:t:


ð26Þ

Tables 2, 3 and 4 show the computational results in the possibility approach

(Model (11)), the necessity approach (Model (14)) and the credibility approach

(Eq. (17)), respectively, relative to four threshold levels: d = 0.25, d = 0.5,

d = 0.75 and d = 1.

As shown in Table 2, following the possibility approach we obtain the results

corresponding to an optimistic viewpoint. Indeed, the objective values are 35, 60,

90.98 and 171.10 for d = 0.25, d = 0.5, d = 0.75 and d = 1, respectively. On the

other hand, as shown in Table 3, following the necessity approach we obtain results

corresponding to a pessimistic viewpoint: the objective values are 184.58, 259.60,

373.02 and 570.03 for d = 0.25, d = 0.5, d = 0.75 and d = 1, respectively.

Table 2 Objective values and

their corresponding primal

solutions (possibility approach)

d Objective value Corresponding primal solutions

0.25 35.00 x1 ¼ 4:5; x2 ¼ 14; x3 ¼ 1

3

0.50 60.00 x1 ¼ 2; x2 ¼ 1; x3 ¼ 1

0.75 90.98 x1 ¼ 1:10; x2 ¼ 3:18; x3 ¼ 1:06

1.00 171.10 x1 ¼ 0:39; x2 ¼ 0:88; x3 ¼ 0:59



solutions (necessity approach)


0.25 184.58 x1 ¼ 0:0375; x2 ¼ 13:33; x3 ¼ 9:52

0.50 259.60 x1 ¼ 0:24; x2 ¼ 1:39; x3 ¼ 0:625

0.75 373.02 x1 ¼ 0:14; x2 ¼ 1:5; x3 ¼ 1:21

1.00 570.03 x1 ¼ 0:07; x2 ¼ 1:76; x3 ¼ 1:43



solutions (credibility approach)


0.25 90 x1 ¼ 2; x2 ¼ 1; x3 ¼ 0:50

0.50 171.71 x1 ¼ 0:39; x2 ¼ 0:88; x3 ¼ 0:59

0.75 173.07 x1 ¼ 0:24; x2 ¼ 1:39; x3 ¼ 1:04

1.00 244.30 x1 ¼ 0:066; x2 ¼ 1:89; x3 ¼ 0:521


123

In order to avoid extreme attitudes, we need to consider the optimistic-

pessimistic viewpoint which is reflected by the credibility approach. The

computational results for the credibility approach, formalized by the models of

Eq. (17), provide the objective values 90, 171.71, 173.07 and 244.30 for d = 0.25,

d = 0.5, d = 0.75 and d = 1, respectively.

Therefore, the computational results show that the credibility approach provides

quite a flexible way to evaluate fuzzy events as this measure can be adjusted

according to the varying attitudes of the decision makers. This fact is also

represented in Fig. 1 where we have plotted the objective values obtained for

d = 0.25, d = 0.5, d = 0.75 and d = 1 following the possibility, necessity and

credibility approaches. As shown in Fig. 1, the credibility approach avoids extreme

attitudes and is more flexible than the possibility and the necessity approaches.

7.2 Example 2

To further show the efficacy of the procedures developed in this paper, we generate

a data set by fuzzifying the input prices and output of the well-known Banker and

Maindiratta (1988)’s data set. The data set comprises one output (y) and three inputs

corresponding to labor (x1), material (x2) and capital (x3). Table 5 shows the original

data set and the fuzzy data.

To analyze the generated data set in our framework, we assume that the

production function f(x1, x2, x3) is a Cobb–Douglas function, that is:

Fig. 1 Objective values in Example 1 for the possibility, necessity and credibility approaches


123

Table

5Original

dataofBanker

andMaindiratta

(1988)andfuzzydata

DMU

x 1x 2

x 3~ y

~ w1

~ w2

~ w3

130,722.0

38,054.0

8184.00

(94,593,4729.65)

(1,0.5)

(1,0.5)

(1,0.5)

228,365.0

35,795.0

8119.00

(95,921,3836.84)

(1,0.5)

(1,0.5)

(1,0.5)

325,445.0

31,814.0

8079.00

(76,852,2305.56)

(1,0.5)

(1,0.5)

(1,0.5)

430,648.9

41,743.4

8667.62

(94,141,3765.64)

(1.04686,0.5234)

(1.01700,0.5085)

(1.04700,0.5235)

533,279.0

41,364.8

8881.57

(102,132,4085.28)

(1.04688,0.5758)

(1.01700,0.5594)

(1.04700,0.5759)

630,828.1

40,124.9

8744.03

(100,341,4013.64)

(1.04684,0.5234)

(1.01700,0.5085)

(1.04700,0.5235)

727,360.1

32,910.5

8109.84

(81,755,2452.65)

(1.04689,0.5234)

(1.01700,0.5085)

(1.04700,0.5235)

831,544.9

39,720.8

9659.96

(95,154,3806.16)

(0.09571,0.0431)

(1.04600,0.4707)

(1.09400,0.4923)

933,485.8

40,893.9

8889.40

(91,393,3655.72)

(1.09593,0.548)

(1.04600,0.523)

(1.09400,0.547)

10

30,725.8

39,137.7

8808.96

(90,752,3630.08)

(1.09553,0.493)

(1.04600,0.4707)

(1.09400,0.4923)

11

27,881.6

32,143.4

8442.41

(75,033,2250.99)

(1.09628,0.4933)

(1.04600,0.4707)

(1.09400,0.4923)

12

30,042.7

28,737.6

8113.79

(85,681,2570.43)

(1.15742,0.5787)

(1.10900,0.5545)

(1.16000,0.58)

13

24,799.7

32,198.4

6962.07

(87,399,4369.95)

(1.15682,0.5784)

(1.10900,0.5545)

(1.16000,0.58)

14

27,676.7

38,023.4

6887.93

(80,469,3218.76)

(1.15556,0.52)

(1.10900,0.4991)

(1.16000,0.522)

15

25,173.9

30,527.5

7051.72

(65,009,2600.36)

(1.15397,0.577)

(1.10900,0.5545)

(1.16000,0.58)

16

28,634.7

43,111.2

8520.29

(86,443,3457.72)

(1.22135,0.6107)

(1.12400,0.562)

(1.23200,0.616)

17

28,289.2

46,075.6

7384.74

(94,454,4722.7)

(1.22015,0.6711)

(1.12400,0.6182)

(1.23200,0.6776)

18

26,157.7

39,393.2

7344.97

(84,361,3374.44)

(1.21861,0.6093)

(1.12400,0.562)

(1.23200,0.616)

19

23,490.0

33,694.0

7351.46

(76,176,3047.04)

(1.21992,0.61)

(1.12400,0.562)

(1.23200,0.616)

20

23,078.5

31,686.8

7311.08

(75,775,3031)

(1.27937,0.6397)

(1.19400,0.597)

(1.31800,0.659)


123

f ðx1; x2; x3Þ ¼ Axa11 xa22 x

a33 ; A[ 0; a1 [ 0; a2 [ 0; a3 [ 0: ð27Þ

where the parameters a1, a2 and a3 are the cost shares of inputs 1, 2, and 3,

respectively. In (27), we assume a1 ? a2 ? a3 = 1 so that f(x1, x2, x3) is homo-

geneous of degree 1 in x1, x2 and x3. In economics, Cobb–Douglas production

functions have been employed in theoretical and empirical analyses of productivity

and growth. The production technology represented by (27) can be expressed by the

production possibility set T = {(x1, x2, x3, y)|f(x1, x2, x3) C y.}, or the input

requirement set L(y) = {(x1, x2, x3)|f(x1, x2, x3) C y}. Relative to L(y), the cost

function is defined by:

Cðy;w1;w2;w3Þ ¼ minx1;x2;x3

w1x1 þ w2x2 þ w3x3 ðx1; x2; x3Þ 2 LðyÞjf g ð28Þ

where w1; w2 and w3 are the input prices. In the case of deterministic data, we can

estimate the cost function as follows:

Cðy;w1;w2;w3Þ ¼ minx1;x2;x3

w1x1 þ w2x2 þ w3x3

s:t: yx�a11 x�a2

2 x�ð1�a1�a2Þ3 �A;

x1; x2; x3 [ 0:

ð29Þ

Note that the problems (28) and (29) are equivalent. Indeed, since

a1 ? a2 ? a3 = 1, we have:

ðx1; x2; x3Þ 2 LðyÞ , f ðx1; x2; x3Þ� y , Axa11 xa22 x

a33 � y , yx�a1

1 x�a22 x

�ð1�a1�a2Þ3 �A

Incorporating fuzzy variables, problem (29) becomes:

Cð~y; ~w1; ~w2; ~w3Þ ¼ minx1;x2;x3

~w1x1 þ ~w2x2 þ ~w3x3

s:t: ~yx�a11 x�a2

2 x�ð1�a1�a2Þ3 �A;

x1; x2; x3 [ 0:

ð30Þ

where the input prices and the output are fuzzy coefficients. In order to solve (30),

we need first to estimate the parameters A, a1; a2 and a3 appearing in (27). We do

so by ordinary least squares logarithmic regression using the data displayed in

Table 5. That is:

log y=x3ð Þ ¼ logAþ a1 log x1=x3ð Þ þ a2 log x2=x3ð Þestimate 0:510205 0:518 0:361

ðp-valueÞ ð0:0014Þ ð0:0824Þ ð0:0238ÞR2 ¼ 0:5274; F ¼ 9:4857; significance F ¼ 0:0017

The estimate of a3 is obtained subtracting from 1 the sum of the estimates of a1and a2, i.e., the estimate of a3 equals 0.121 = 1 - 0.518 - 0.361. Since the

intercept is 0.510205, the estimate of A is 1.666 & exp (0.510205). Based on these


123

estimates, we solve (30) under the assumption that both the output and the input

prices are positive and symmetrical TFNs:

~y ¼ ym; yað Þ ~w1 ¼ wm1 ;w

a1

� �~w2 ¼ wm

2 ;wa2

� �~w3 ¼ wm

3 ;wa3

� �:

As in the previous sections, the super-indices m and a indicate the mean and both

sides spread of the TFNs, respectively.

Our fuzzy program can be then written as follows:

Cð~y; ~w1; ~w2; ~w3Þ ¼ minx1;x2;x3

~w1x1 þ ~w2x2 þ ~w3x3

s:t: ~yx�0:5181 x�0:361

2 x�0:1213 � 1:666;

x1; x2; x3 [ 0

ð31Þ

The dual of (31) in the possibility approach is obtained applying Model (18). The

variables of the dual program are xp, with p = 1, 2, 3, and x1,t(1), t(1) = 1.

maxwm1 � ð1� dÞwa

1

x1

� �x1 wm2 � ð1� dÞwa

2

x2

� �x2 wm3 � ð1� dÞwa

3

x3

� �x3

ym � ð1� dÞya1:666x1;1

� �x1;1

kk

s:t:

x1 þ x2 þ x3 ¼ 1

k ¼ x1;1

x1 ¼ 0:518x1;1

x2 ¼ 0:361x1;1

x3 ¼ 0:121x1;1

x1 [ 0; x2 [ 0; x3 [ 0; x1;1 � 0:

ð32Þ

In particular, in Model (32), by using the constraint equations, we have:

x1 þ x2 þ x3 ¼ 1 , ð0:518þ 0:361þ 0:121Þx1;1 ¼ 1 , x1;1 ¼ 1:

Thus, incorporating the constraints, the objective of the above problem becomes

wm1 � ð1� dÞwa

1

0:518

� �0:518wm2 � ð1� dÞwa

2

0:361

� �0:361wm3 � ð1� dÞwa

3

0:121

� �0:121

ym � ð1� dÞya1:666

� �:

ð33Þ

For the sake of completeness, we include the duals of (31) in the necessity and

credibility approaches. In the necessity approach, we implement Model (19)

obtaining the following:


123

maxwm1 þ dwa

1

x1

� �x1 wm2 þ dwa

2

x2

� �x2 wm3 þ dwa

3

x3

� �x3 ym þ dya

1:666x1;1

� �x1;1

kk

s:t:


ð34Þ

In the credibility approach, we use Models (20) and (21) and obtain:

If d� 0:5

maxwm1 � ð1� 2dÞwa

1

x1

� �x1 wm2 � ð1� 2dÞwa

2

x2

� �x2

wm3 � ð1� 2dÞwa

3

x3

� �x3 ym � ð1� 2dÞya1:666x1;1

� �x1;1

kk

s:t:


ð35Þ

If d[ 0:5

maxwm1 þ ð2d� 1Þwa

1

x1

� �x1 wm2 þ ð2d� 1Þwa

2

x2

� �x2

wm3 þ ð2d� 1Þwa

3

x3

� �x3 ym þ ð2d� 1Þya1:666x1;1

� �x1;1

kk

s:t:


ð36Þ

The computational results obtained in the possibility, necessity, and credibility

approaches for the four threshold levels of d = 0.25, d = 0.5, d = 0.75 and d = 1,

are reported in Table 6.

It can be observed that for every decision making unit (DMU), the optimal value

of the cost function increases as d increases. For instance, for DMU1, the optimal

cost values are 89,570.5, 108,880.5, 128,655.8 and 148,896.4 for d = 0.25, d = 0.5,

d = 0.75 and d = 1, respectively. A similar pattern can be observed in the necessity

and credibility approaches.

It can also be noted that the minimum cost (optimistic) values obtained in the

possibility approach are much less than the corresponding minimum cost

(pessimistic) values in the necessity approach at the same threshold level. Also

the minimum cost values obtained in the credibility approach are in between the

corresponding ones found in the necessity approach and those produced by the

possibility approach. Figures 2, 3, 4 and 5 provide a graphical representation of the

optimal costs behaviors for the various threshold levels considered, namely,

d = 0.25, d = 0.5, d = 0.75 and d = 1, respectively, allowing for a visual

comparison of such behaviors. In particular, note that for each DMU, at d = 0.25,

the distances between the optimal values in possibility approach and those in

credibility approach are smaller than the distances between the necessity approach

values and the credibility approach ones. As d increases, the minimum cost values


123

Table

6Minim

um

costbased

onthepossibility,necessity

andcredibilityapproaches

DMUs

Possibility

Necessity

Credibility

d=

0.25

d=

0.5

d=

0.75

d=

1d=

0.25

d=

0.5

d=

0.75

d=

1d=

0.25

d=

0.5

d=

0.75

d=

1

189,570.5

108,880.5

128,655.8

148,896.4

169,602.3

190,773.5

212,410

234,511.8

108,880.5

148,896.4

190,773.5

234,511.8

291,535.7

110,975.2

130,792.2

150,986.7

171,558.7

192,508.1

213,834.9

235,539.3

110,975.2

150,986.7

192,508.1

235,539.3

373,905.5

89,367.1

105,055.5

120,970.7

137,112.7

153,481.6

170,077.3

186,899.7

89,367.1

120,970.7

153,481.6

186,899.7

493,072.6

112,837.7

132,986.5

153,519.2

174,435.6

195,735.8

217,419.7

239,487.5

112,837.7

153,519.2

195,735.8

239,487.5

594,910

118,332.6

142,213.3

166,552

191,348.8

216,603.6

242,316.4

268,487.3

118,332.6

166,552

216,603.6

268,487.3

699,200.7

120,267.4

141,743.2

163,628.1

185,922

208,625

231,737

255,258.2

120,267.4

163,628.1

208,625

255,258.2

781,454.1

98,493.7

115,783.3

133,322.8

151,112.4

169,151.8

187,441.3

205,980.7

98,493.7

133,322.8

169,151.8

205,980.7

829,324.9

34,661

40,099.9

45,641.5

51,285.8

57,032.9

62,882.7

68,835.2

34,661

45,641.5

57,032.9

68,835.2

993,963.8

113,920

134,263.7

154,994.9

176,113.6

197,619.9

219,513.6

241,794.8

113,920

154,994.9

197,619.9

241,794.8

10

98,885.7

116,870.5

135,201.5

153,878.8

172,902.2

192,271.9

211,987.9

232,050

116,870.5

153,878.8

192,271.9

232,050

11

82,420.9

97,156.1

112,106.1

127,270.8

142,650.3

158,244.5

174,053.5

190,077.3

97,156.1

127,270.8

158,244.5

190,077.3

12

93,932.7

113,583.8

133,523.2

153,750.9

174,266.8

195,071.1

216,163.6

237,544.4

113,583.8

153,750.9

195,071.1

237,544.4

13

94,320.5

114,654.2

135,478

156,791.6

178,595.3

200,888.9

223,672.5

246,946.1

114,654.2

156,791.6

200,888.9

246,946.1

14

92,715

109,578

126,765.6

144,277.9

162,114.8

180,276.4

198,762.6

217,573.4

109,578

144,277.9

180,276.4

217,573.4

15

70,612.8

85,609.2

100,896.8

116,475.6

132,345.6

148,506.8

164,959.2

181,702.8

85,609.2

116,475.6

148,506.8

181,702.8

16

97,875.7

118,662.3

139,852.6

161,446.5

183,444

205,845.2

228,649.9

251,858.3

118,662.3

161,446.5

205,845.2

251,858.3

17

99,701.7

124,634.5

150,173.5

176,318.5

203,069.7

230,427

258,390.3

286,959.8

124,634.5

176,318.5

230,427

286,959.8

18

95,408.7

115,670.7

136,326

157,374.8

178,817.1

200,652.7

222,881.8

245,504.4

115,670.7

157,374.8

200,652.7

245,504.4

19

86,197.8

104,504.7

123,167

142,184.9

161,558.2

181,286.9

201,371.2

221,810.9

104,504.7

142,184.9

181,286.9

221,810.9

20

90,559.1

109,791.6

129,397.5

149,376.9

169,729.7

190,456

211,555.7

233,028.9

109,791.6

149,376.9

190,456

233,028.9


123

obtained in the credibility approach become closer and closer to those obtained in

the necessity approach until they actually coincide at d = 1.

Finally, it deserves to point out that the reduction in terms of variables that can be

applied to the objective function of this example is strictly related to the fact that the

production function is homogeneous of degree 1. Alternatively, the objective

Fig. 2 Optimal cost values in Example 2 for the possibility, necessity and credibility approaches(d = 0.25)



123

function of Model (32) can be proved equivalent to that in Eq. (33) by imposing the

cost function to be homogeneous of degree 1 in the variables ~w1, ~w2 and ~w3, that is:

Cð~y; h ~w1; h ~w2; h ~w3Þ ¼ hCð~y; ~w1; ~w2; ~w3Þ; 8h[ 0 ð~y fixed coefficientÞ:

The cost function being homogeneous of degree 1 is a standard assumption in the

economics literature. Thus, our solution method can be applied and give rise to an

interesting analysis of the DMUs’ risk attitudes in different economic related




123

contexts. It suffices for the cost function of the dual programming problem to be

homogeneous of degree 1.

8 Conclusion and future research directions

GP is a well-known methodology for solving algebraic non-linear optimization

problems. Due to insufficient information, the parameters in the real-life problems

solvable by GP are often imprecise and ambiguous which is the reason why more

and more studies on fuzzy GP have been developing in the last decades.

Thanks to its broad applicability GP modeling has been employed to solve a

myriad of problems, chance-constrained and not, whose coefficients are fuzzy

numbers, fuzzy variables or random variables. However, to the best of our

knowledge, there is no previous study dealing with the formulation of crisp GP

models equivalent to specific chance-constrained fuzzy GP problems.

In this paper, we introduced three chance-constrained GP models with respect to

possibility, necessity and credibility constraints for solving fuzzy GP problems. We

showed that the proposed chance-constrained GP problems can be transformed into

conventional GP problems and, consequently, the optimal objective values

calculated using their dual forms.

The possibility, necessity and credibility measures used in the paper allow to

consider different risk attitudes on the side of the decision makers.

The capacity of our chance-constrained models to provide an optimal objective

value (not just an approximation of it) constitutes a clear advantage of the model

over other fuzzy and fuzzy chance-constrained models in the literature.

Finally, the technique used to defuzzify the constraints and obtain the crisp

models is inspired to a similar one employed in fuzzy DEA environments (Khanjani

et al. 2014; Tavana et al. 2012).

Two numerical examples were presented to demonstrate the efficacy of the

procedures and the algorithms.

Future extensions of the current research include: (a) developing, from a

theoretical perspective, a GP methodology allowing to deal with situations where

randomness and fuzziness coexist, and (b) addressing, from a practical perspective,

the possible applications of such a methodology to a variety of problems.

Acknowledgments The authors would like to thank the anonymous reviewers and the editor for their

insightful comments and suggestions.

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