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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
http://tavana.us/
Rashed Khanjani Shiraz
Hirofumi Fukuyama
Debora Di Caprio
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
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
70 R. Khanjani Shiraz et al.
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.
Fuzzy chance-constrained geometric programming: the… 71
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:
72 R. Khanjani Shiraz et al.
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:
Fuzzy chance-constrained geometric programming: the… 73
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
74 R. Khanjani Shiraz et al.
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:
Fuzzy chance-constrained geometric programming: the… 75
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
76 R. Khanjani Shiraz et al.
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
cmi;tðiÞ � R�1ðdÞcai;tðiÞ� �Ys
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
cmi;tðiÞ � R�1ðdÞcai;tðiÞ� �Ys
j¼1
xci;tðiÞ;jj � bmi þ R�1ðdÞbbi ; i ¼ 1; . . .; n
xj [ 0; j ¼ 1; . . .; s
ð11Þ
Fuzzy chance-constrained geometric programming: the… 77
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.
78 R. Khanjani Shiraz et al.
123
min u
s:t:
Nec u�Xh
p¼1
~cpYs
j¼1
xap;jj
!� d
NecXkðiÞ
tðiÞ¼1
~ci;tðiÞYs
j¼1
xci;tðiÞ;jj � ~bi
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).
Fuzzy chance-constrained geometric programming: the… 79
123
min u
s:t:
u�Xh
p¼1
cmp þ L�1ð1� dÞcbp� �Ys
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
cmp þ L�1ð1� dÞcbp� �Ys
j¼1
xap;jj
s:t:
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
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.
80 R. Khanjani Shiraz et al.
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
xci;tðiÞ;jj � ~bi
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.
Fuzzy chance-constrained geometric programming: the… 81
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
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
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:
82 R. Khanjani Shiraz et al.
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
cmi;tðiÞ þ L�1ð1� dÞcbi;tðiÞ
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:
Constraints of Model ð18Þð20Þ
If d[ 0:5
maxYh
p¼1
cmp þ L�1ð2ð1� dÞÞcbpxp
!xp Yn
i¼1
YkðiÞ
tðiÞ¼1
cmi;tðiÞ þ L�1ð2ð1� dÞÞcbi;tðiÞ
ðbmi � L�1ð2ð1� dÞÞbai Þxi;tðiÞ
!xi;tðiÞ Yn
i¼1
kkii
s:t:
Constraints of Model ð18Þð21Þ
Fuzzy chance-constrained geometric programming: the… 83
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)
84 R. Khanjani Shiraz et al.
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:
Constraints of Model ð23Þ
ð25Þ
Fuzzy chance-constrained geometric programming: the… 85
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:
Constraints of Model ð23Þ
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
Table 3 Objective values and
their corresponding primal
solutions (necessity approach)
d Objective value Corresponding primal solutions
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
Table 4 Objective values and
their corresponding primal
solutions (credibility approach)
d Objective value Corresponding primal solutions
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
86 R. Khanjani Shiraz et al.
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
Fuzzy chance-constrained geometric programming: the… 87
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)
88 R. Khanjani Shiraz et al.
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
Fuzzy chance-constrained geometric programming: the… 89
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:
90 R. Khanjani Shiraz et al.
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:
Constraints of Model ð32Þ
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:
Constraints of Model ð32Þ
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:
Constraints of Model ð32Þ
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
Fuzzy chance-constrained geometric programming: the… 91
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
92 R. Khanjani Shiraz et al.
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)
Fig. 3 Optimal cost values in Example 2 for the possibility, necessity and credibility approaches(d = 0.5)
Fuzzy chance-constrained geometric programming: the… 93
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
Fig. 4 Optimal cost values in Example 2 for the possibility, necessity and credibility approaches(d = 0.75)
Fig. 5 Optimal cost values in Example 2 for the possibility, necessity and credibility approaches(d = 1.0)
94 R. Khanjani Shiraz et al.
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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