Fuzzy Sets and Systems 158 (2007) 1961 – 1978
www.elsevier.com/locate/fss
Duality results and a dual simplex method for linear programming
problems with trapezoidal fuzzy variables
N. Mahdavi-Amiria, b,∗ , S.H. Nasseria
a Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
b Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
Received 8 October 2005; received in revised form 2 May 2007; accepted 7 May 2007
Available online 25 May 2007
Abstract
Linear programming problems with trapezoidal fuzzy variables (FVLP) have recently attracted some interest. Some methods
have been developed for solving these problems by introducing and solving certain auxiliary problems. Here, we apply a linear
ranking function to order trapezoidal fuzzy numbers. Then, we establish the dual problem of the linear programming problem with
trapezoidal fuzzy variables and hence deduce some duality results. In particular, we prove that the auxiliary problem is indeed the
dual of the FVLP problem. Having established the dual problem, the results will then follow as natural extensions of duality results
for linear programming problems with crisp data. Finally, using the results, we develop a new dual algorithm for solving the FVLP
problem directly, making use of the primal simplex tableau. This algorithm will be useful for sensitivity (or post optimality) analysis
when using primal simplex tableaus.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Fuzzy variable linear programming; Duality; Ranking function; Dual simplex; Trapezoidal fuzzy number
1. Introduction
Fuzzy set theory has been applied to many disciplines such as control theory and management sciences, mathematical modeling and industrial applications. The concept of fuzzy mathematical programming on general level was first
proposed by Tanaka et al. [25] in the framework of the fuzzy decision of Bellman and Zadeh [4]. The first formulation
of fuzzy linear programming (FLP) is proposed by Zimmermann [31]. Afterwards, many authors considered various
types of the FLP problems and proposed several approaches for solving these problems [5–8,16–20,26]. Chanas [7]
showed an application of parametric programming techniques in FLD and obtained the set of solutions maximizing
the objective function, being analytically dependent on a parameter. Delgado et al. [8] studied a general model for FLP
problems which includes fuzziness both in the coefficients and in the accomplishment of the constraints. Buckley and
Feuring [5] considered the extreme case of fully fuzzified linear program with all the parameters and variables as fuzzy
numbers. They turned the problem into a multi-objective FLP one. They showed that the fuzzy flexible programming
could be used to explore the undominated set to the multi-objective problem and proposed an evolutionary algorithm
∗ Corresponding author. Tel.: +98 21 66165607.
E-mail addresses: [email protected], [email protected] (N. Mahdavi-Amiri).
0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2007.05.005
1962
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
to solve the fuzzy flexible program. Some authors used the concept of comparison of fuzzy numbers for solving FLP
problems [6,16]. In effect, most convenient methods are based on the concept of comparison of fuzzy numbers by
use of ranking functions [9,17,18,20]. Of course, ranking functions have been proposed by researchers to suit their
requirements of the problem under consideration [6,18,20], and conceivably there are no generally accepted criteria for
application of ranking functions. Nevertheless, usually in such methods authors define a crisp model which is equivalent
to the FLP problem and then use optimal solution of the model as the optimal solution of the FLP problem. A review of
some common methods for ranking fuzzy numbers can be seen in [27]. Moreover, a review of the literature concerning
fuzzy mathematical programming as well as comparison of fuzzy numbers can be seen in Klir and Yuan [12] and also
Lai and Hwang [13]. Some authors considered types of single and multi-objective linear programming problems in
which the variables and the right-hand sides of the constraints are fuzzy parameters [10,17–19].
The study of duality theory for fuzzy parameter linear programming problems has attracted researchers in fuzzy
decision theory. The duality of fuzzy parameter linear programming was first studied by Rodder and Zimmermann [23].
Verdegay [26] defined the fuzzy dual problem with the help of parametric linear programming and showed that the fuzzy
primal and dual problems both have the same fuzzy solution under some suitable conditions. The fuzzy primal and
dual linear programming problems with fuzzy coefficients were formulated by using the fuzzy scalar product proposed
in [28]. Liu et al. [15] proposed the fuzzy primal and dual problems by considering the fuzzy-max and fuzzy-min
in the objective functions as the crisp pattern of linear programming problems. Bector and Chandra [1] discussed
duality in FLP based on a modification of the dual formulation stated by Rodder and Zimmermann [23]. Afterwards
Bector et al. [2] considered a fuzzy matrix game and proved its equivalence to two crisp linear programming problems
which constitute a primal–dual pair in the sense of duality for linear programming with fuzzy parameters (see also
[3]). Inuiguchi et al. [11] studied FLP duality in the setting of fuzzy relations. Ramik [22,21] discussed a class of FLP
problems based on fuzzy relations and a new concept of duality and deduced the weak and strong duality theorems.
Wu [29] offered a concept of fuzzy scalar product and proved the weak and strong duality theorems using a dual fuzzy
mathematical programming problem.
The fuzzy variable linear programming (FVLP) problems have been explored by Zimmermann’s discussion [32] of
the so-called nonsymmetric flexible linear programming (NFLP) problems, where the problem data are considered to
be crisp but certain constraints are considered to be “fuzzy inequality" constraints (see also [13,14]). It was shown that
under certain conditions the NFLP problem is equivalent to an FVLP problem. Moreover, as we will show later, an
FVLP problem is the dual of a fuzzy number linear programming (FNLP) problem, in which the coefficients of the cost
function are fuzzy. Therefore, methods for solving FVLP problems can be used for solving both the NFLP and FNLP
problems. Maleki et al. [17,18] gave an auxiliary problem, having only fuzzy cost coefficients, for an FVLP problem.
In [18], they obtained some results leading to an algorithm for solving the auxiliary problem as a method for solving
the FVLP problem (we will see later that their algorithm is a dual algorithm for solving a primal FVLP problem). In
[17], they used the algorithm to solve the NFLP problems. Here, we first show that the auxiliary problem given in [18]
is indeed the dual of the FVLP problem. This leads us to duality results for the two problems. Our main contributions
here are the establishment of duality and complementary slackness, upon the use of certain linear ranking functions to
order trapezoidal fuzzy numbers. Using the results, we develop and present a dual simplex algorithm directly using the
primal simplex tableau (as opposed to solving the dual problem considered by Maleki et al. [17]). The new algorithm
tenders the capability for sensitivity (or post optimality) analysis using primal simplex tableaus. In Section 2, we first
give some necessary notations and definitions of fuzzy set theory. Then we provide a discussion of fuzzy numbers and
linear ranking functions for ordering them. In particular, a certain linear ranking function for ordering trapezoidal fuzzy
numbers is emphasized. The definition of the FVLP problem is given in Section 3. Section 4 explains the notion of
fuzzy basic feasible solution. We establish duality for the FVLP problem in Section 5 and deduce the duality results. In
Section 6, we develop and present the dual simplex algorithm, using the primal tableau, for solving the FVLP problems.
We conclude in Section 7.
2. Preliminaries
2.1. Definitions and notations
We review the fundamental notions of fuzzy set theory, initiated by Bellman and Zadeh [4].
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1963
Definition 2.1. A convex fuzzy set à on R is a fuzzy number if the following conditions hold:
(a) Its membership function is piecewise continuous.
(b) There exist three intervals [a, b], [b, c] and [c, d] such that à is increasing on [a, b], equal to 1 on [b, c],
decreasing on [c, d] and equal to 0 elsewhere.
Definition 2.2. Let à = (a L , a U , , ) denote the trapezoidal fuzzy number, where (a L − , a U + ) is the support
of à and [a L , a U ] its core.
Remark 2.1. We denote the set of all trapezoidal fuzzy numbers by F(R).
We next define arithmetic on trapezoidal fuzzy numbers. Let ã = (a L , a U , , ) and b̃ = (bL , bU , , ) be two
trapezoidal fuzzy numbers. Define,
x > 0, x ∈ R;
x ã = (xa L , xa U , x, x),
x < 0, x ∈ R;
x ã = (xa U , xa L , −x, −x),
ã + b̃ = (a L + bL , a U + bU , + , + ).
We point out that the arithmetic on trapezoidal fuzzy numbers follows the Extension Principle (for a discussion
see [13]).
2.2. Ranking functions
One convenient approach for solving the FLP problems is based on the concept of comparison of fuzzy numbers
by use of ranking functions (see [27]). An effective approach for ordering the elements of F(R) is to define a ranking
function R : F(R) −→ R which maps each fuzzy number into the real line, where a natural order exists.
We define orders on F(R) by
ã b̃
if and only if
R(ã) R(b̃),
(2.1)
ã > b̃
if and only if
R(ã) > R(b̃),
(2.2)
ã = b̃
if and only if
R(ã) = R(b̃),
(2.3)
R
R
R
where ã and b̃ are in F(R). Also we write ã b̃ if and only if b̃ ã.
R
R
We restrict our attention to linear ranking functions, that is, a ranking function R such that
R(k ã + b̃) = kR(ã) + R(b̃)
(2.4)
for any ã and b̃ belonging to F(R) and any k ∈ R.
Remark 2.2. For any trapezoidal fuzzy number ã, the relation ã 0̃ holds, if there exist ε 0 and 0 such that
R
ã (−ε, ε, , ). We realize that R(−ε, ε, , ) = 0 (we also consider ã = 0̃ if and only if R(ã) = 0). Thus, without
R
R
loss of generality, throughout the paper we let 0̃ = (0, 0, 0, 0) as the zero trapezoidal fuzzy number.
The following lemma is now immediate.
Lemma 2.1. Let R be any linear ranking function. Then,
(i) ã b̃ if and only if ã − b̃ 0̃ if and only if −b̃ −ã.
R
R
(ii) If ã b̃ and c̃ d̃, then ã + c̃ b̃ + d̃.
R
R
R
R
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N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
We consider the linear ranking functions on F(R) as
R(ã) = cL a L + cU a U + c + c ,
(2.5)
where ã = (a L , a U , , ), and cL , cU , c , c are constants, at least one of which is nonzero. A special version of the
above linear ranking function was first proposed by Yager [30] (see also [9,24]) as follows:
1 1
R(ã) =
(2.6)
(inf ã + sup ã ) d,
2 0
which reduces to
R(ã) =
aL + aU
1
+ ( − ).
4
2
(2.7)
Then, for trapezoidal fuzzy numbers ã = (a L , a U , , ) and b̃ = (bL , bU , , ), we have
ã b̃
R
if and only if
a L + a U + 21 ( − )bL + bU + 21 ( − ).
(2.8)
3. Linear programming problem with trapezoidal fuzzy variables
A linear programming problem with trapezoidal fuzzy variables (FVLP) is defined as [17,18]
min
z̃ = cx̃,
s.t.
Ax̃ b̃,
R
R
x̃ 0̃ = (0, 0, 0, 0),
(3.1)
R
where b̃ ∈ (F(R))m , A ∈ Rm×n , cT ∈ Rn are given and x̃ ∈ (F(R))n is to be determined, and R is a linear ranking
function as defined by (2.5).
Definition 3.1. We say that fuzzy vector x̃ ∈ (F(R))n is a fuzzy feasible solution to (3.1) if x̃ satisfies the constraints
of the problem.
Remark 3.1. It should be emphasized that in solving problem (3.1) , we are concerned with finding a minimizer x̃ so
that Ax̃ b̃, x̃ 0̃ (implying that R(Ax̃)R(b̃), R(x̃) 0). In fact, the operator R plays an implicit role here and
R
R
the concern is finding x̃ to satisfy an equivalent fuzzy equality relation Ax̃ + ỹ = b̃, as described by (5.11) in proof of
R
Theorem 5.3. We will see that in solving problem (3.1), fuzzy basic solutions (x̃B = B −1 b̃, x̃N = 0̃), as explained in the
next section, of a corresponding equality constrained problem play the decisive roles. Every basic solution is, of course,
unique (see Theorem 5.3 in Section 5), in the absence of degeneracy. Thus, while the ranking function has no role in
computing the basic solutions but the constraint relations are satisfied by the ranking function at every basic solution.
˜ is not of interest here because it is not a solution of the constraints
Any other fuzzy vector satisfying R(Ax̃) R(b)
in the first place. However, the ranking function will play its crucial role in ordering the trapezoidal numbers being
used for testing the optimality (inequality) conditions (see Theorem 4.1) and making the decision for pivoting (see
Algorithm 6.1). These have been used in computing the dual simplex tableaus in Example 6.1.
Definition 3.2. A fuzzy feasible solution x̃∗ is a fuzzy optimal solution for (3.1), if for all fuzzy feasible solution x̃ for
(3.1), we have cx̃∗ cx̃.
R
Remark 3.2. In practice, there are several linear programming models that are related to linear programming problems
with fuzzy variables. One such problem is the FNLP problem. In Section 5, we will see that the dual of any FVLP
problem is a linear programming problem with fuzzy cost coefficients. Knowing this, one may use the solutions of the
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1965
FVLP problem to obtain the solutions of the linear programming problem with fuzzy cost coefficients (see Theorem
5.3). Another problem is the NFLP considered by Zimmermann [32] (see also [13,14]).
A method for solving the FVLP problem based on solving an auxiliary problem has been given by Maleki et al.
[17,18]. We will show that the auxiliary problem is indeed the dual of the FVLP problem. This leads us to the concept
of duality in linear programming with trapezoidal fuzzy variables. We state and prove duality results obtained by a
natural extension of the results in crisp linear programming. We finally introduce a dual simplex algorithm for solving
the FVLP problem directly, using the simplex tableau corresponding to the original primal problem (3.1). The new
algorithm will then be useful for post optimality analysis based on the use of primal simplex tableaus.
4. Fuzzy basic feasible solution
Here, we explore the concept of fuzzy basic feasible solution for FVLP problems. Consider the FVLP problem,
min
z̃ = cx̃,
s.t.
Ax̃ = b̃,
R
R
x̃ 0̃,
(4.1)
R
where the parameters of the problem are as defined in (3.1). Let A = [aij ]m×n . Assume rank(A) = m. Partition A as
[B N] where B, m × m, is nonsingular. It is obvious that rank(B) = m. Let yj be the solution to By = aj , where
aj is the jth column of the coefficient matrix A. It is apparent that the basic solution
x̃B = (x̃B1 , . . . , x̃Bm )T = B −1 b̃, x̃N = 0̃
is a solution of Ax̃ = b̃. We call x̃, accordingly partitioned as (x̃BT
R
(4.2)
T )T , a fuzzy basic solution corresponding to basis
x̃N
B. If x̃B 0̃, then the fuzzy basic solution is feasible and the corresponding fuzzy objective value is z̃ = cB x̃B , where
R
R
cB = (cB1 , . . . , cBm ). Now, corresponding to every index j, 1 j n, define
zj = cB yj = cB B −1 aj .
(4.3)
Observe that for any basic index j = Bi , 1 i m, we have B −1 aj = ei where ei = (0, . . . , 0, 1, 0, . . . , 0)T is the ith
unit vector, since Bei = [aB1 , . . . , aBi , . . . , aBm ]ei = aBi = aj , and so we have:
zj − cj = cB B −1 aj − cj = cB ei − cj = cBi − cj = cj − cj = 0.
(4.4)
The following theorem characterizes optimal solutions. The converse part of the result needs the nondegeneracy
assumption of the problem, where all fuzzy basic variables corresponding to every basis B are nonzero (and hence
positive).
Theorem 4.1 (Optimality conditions). Assume the linear programming problem with trapezoidal fuzzy variables (4.1)
is nondegenerate and B is a feasible basis. A fuzzy basic feasible solution x̃B = B −1 b̃ 0̃, x̃N = 0̃ is optimal to (4.1)
if and only if zj = cB B −1 aj cj for all j, 1 j n.
R
T )T is a fuzzy basic feasible solution to (4.1), where x̃ = B −1 b̃, x̃
Proof. Suppose x̃∗ =(x̃BT x̃N
B
N = 0̃. Then the
R
corresponding fuzzy objective value is:
z̃∗ = cx̃∗ = cB x̃B = cB B −1 b̃.
R
R
R
(4.5)
On the other hand, for any fuzzy basic feasible solution x̃ to (4.1), we have
b̃ = Ax̃ = B x̃B + N x̃N .
R
R
(4.6)
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N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
Hence, we can rewrite (4.6) as follows:
x̃B = B −1 b̃ − B −1 N x̃N .
(4.7)
R
Then, for any fuzzy basic feasible solution to (4.1), we have
z̃ = cx̃ = cB x̃B + cN x̃N = cB B −1 b̃ − (cB B −1 N − cN )x̃N
R
R
= cB B
−1
R
b̃ −
n
R
(cB B
−1
aj − cj )x̃j = cB B
−1
R
j =1
b̃ −
n
(zj − cj )x̃j .
j =1
Hence, using (4.4) and (4.5), we have
(zj − cj )x̃j .
z̃ = z̃∗ −
R
(4.8)
j =Bi
Now, if for all j, 1j n we have zj cj , then from feasibility of x̃ we have (zj − cj )x̃j 0̃, and then we obtain
R
0̃.
Therefore,
it
follows
from
(4.8)
that
z̃
(z
−
c
)
x̃
z̃
,
and
so
x̃
is
optimal.
For
“only if”part, let x̃∗ be
j j
∗
∗
j =Bi j
R
R
a fuzzy optimal basic feasible solution to (4.1). For j = Bi , 1 i m, from (4.4) we know that zj − cj = 0. From
(4.8) it is obvious that if for any nonbasic variable x̃j we have zj > cj , then we can enter x̃j into the basis and obtain
z̃∗ > z̃ (because the problem is nondegenerate and x̃j > 0̃ in the new basis). This is a contradiction to z̃∗ being optimal.
R
Hence we must have zj cj , 1 j n.
R
Remark 4.1. A basic solution of the problem, x̃B = ỹ0 = B −1 b̃ = (x̃B1 , . . . , x̃Bm )T , where x̃Bi = (xBLi , xBUi , Bi , Bi ),
x̃N = 0̃ is a unique fuzzy vector due to b̃ being fuzzy. If one is interested in a crisp solution, a natural defuzzification
∗ = 0.
of the trapezoidal fuzzy solution is obtained by setting x ∗ = R(x̃∗ ), that is, xB∗ = R(x̃B ), xN
In the next section, we develop the duality results.
5. Duality
5.1. Formulation of the dual problem
Consider the FVLP problem below:
n
min z̃ = cx̃ =
cj x̃j ,
R
s.t.
R
Ax̃ b̃
R
x̃ 0̃
R
j =1
or
n
aij x̃j b̃i ,
or
x̃j 0̃,
R
i = 1, . . . , m,
R
j =1
j = 1, . . . , n,
(5.1)
where R is a linear ranking function, x̃ = (x̃1 , . . . , x̃n )T ∈ (F(R))n , b̃ = (b̃1 , . . . , b̃m )T ∈ (F(R))m , A = [aij ]m×n ,
c = (c1 , . . . , cn )T ∈ Rn , b̃i = (biL , biU , bi , bi ) ∈ F(R), x̃j = (xjL , xjU , xj , xj ) ∈ F(R).
It is obvious that problem (5.1) is equivalent to problem (5.2) below:
min
s.t.
z = R(cx̃),
R(Ax̃) R(b̃),
R(x̃) 0,
(5.2)
where, R(x̃) = (R(x̃1 ), . . . , R(x̃n
∈ R , R(b̃) = (R(b̃1 ), . . . , R(b̃m
∈R .
U
L
In the above notations, R(x̃j ) = cL xj + c U xj + c xj + c xj , R(b̃i ) = cL biL + c U biU + c bi + c bi , where
cL , cU , c , c are constants, at least one of which is nonzero.
))T
n
))T
m
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1967
Theorem 5.1. Problems (5.1) and (5.2) are equivalent to the following problem:
z = cR(x̃),
min
s.t.
AR(x̃)R(b̃),
R(x̃)0.
(5.3)
Proof. Since, R is a linear ranking function, we have
⎛
⎞
n
n
n
R(cx̃) = R ⎝
cj x̃j ⎠ =
R(cj x̃j ) =
cj R(x̃j ) = cR(x̃),
j =1
j =1
j =1
where, R(x̃) = (R(x̃1 ), . . . , R(x̃n ))T ∈ Rn .
On the other hand,
⎛
⎞
n
n
n
R⎝
aij x̃j ⎠ =
R(aij x̃j ) =
aij R(x̃j ).
j =1
j =1
j =1
So, if we denote the ith row of matrix A by āi , we have
⎞
⎛
⎞⎞T
⎛ ⎛
n
n
amj x̃j ⎠⎠
R(Ax̃) = ((R(ā1 x̃), . . . , R(ām x̃))T = ⎝R ⎝
a1j x̃j ⎠ , . . . , R ⎝
⎛
=⎝
j =1
n
a1j R(x̃j ), . . . ,
j =1
j =1
n
⎞T
amj R(x̃j )⎠
j =1
= (ā1 R(x̃), . . . , ām R(x̃))T = AR(x̃).
We note that for any feasible x̃ ∈ (F(R))n for (5.1) we have
n
aij R(x̃j )R(b̃i ),
i = 1, . . . , m,
j =1
R(x̃j )0,
j = 1, . . . , n,
or, using a linear ranking function,
n
j =1
cL xjL + c U xjU + c xj + c xj 0,
j = 1, . . . , n.
Now, for any optimal x̃ ∗ for (5.1) we have
cx̃ ∗ cx̃
R
for any feasible solution x̃ for (5.1), or
R(cx̃ ∗ ) = cR(x̃ ∗ ) R(cx̃) = cR(x̃),
or
n
j =1
aij (cL xjL + c U xjU + c xj + c xj )cL biL + c U biU + c bi + c bi ,
cj R(x̃j∗ )
n
j =1
cj R(x̃j ),
i = 1, . . . , m,
(5.4)
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N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
or, using a linear ranking function,
n
j =1
cj (cL xj∗ L + c U xj∗ U + c xj∗ + c xj∗ ) n
j =1
cj (cL xjL + cU xjU + c xj + c xj ).
(5.5)
Therefore, x̃ ∗ = (x̃1∗ , . . . , x̃n∗ )T , where x̃j∗ = (xj∗ L , xj∗ U , xj∗ , xj∗ )T ∈ F(R), is optimal for (5.1), if and only if
x ∗ = (x1∗ L , x1∗ U , x1∗ , x1∗ , . . . , xn∗ L , xn∗ U , xn∗ , xn∗ )T ∈ R4n is an optimal solution for (5.2).
We are now in a position to naturally extend the usual definition of a dual problem in linear programming to the
problem with trapezoidal fuzzy variables.
Definition 5.1. The dual of (5.1) is defined to be
max
ũ = y b̃,
s.t.
yAc,
y 0.
R
(5.6)
We name the above problem as the DFVLP problem.
Now, we shall show that the dual of (5.1) is indeed (5.6), using the usual definition of the dual in linear programming.
Using Theorem (5.1) and applying a linear ranking function, we can write (5.1) as follows:
min
n
z=
j =1
s.t.
n
j =1
cj (cL xjL + c U xjU + c xj + c xj ),
aij (cL xjL + c U xjU + c xj + c xj )R(b̃i ),
cL xjL + c U xjU + c xj + c xj 0,
i = 1, . . . , m,
j = 1, . . . , n.
(5.7)
The dual of (5.7) is
max
u=
m
yi (cL biL + c U biU + c bi + c bi ) =
i=1
s.t.
m
m
yi R(b̃i ),
i=1
aij cL yi + cL ym+1 = cL cj ,
i=1
m
aij cU yi + cU ym+1 = cU cj ,
i=1
m
aij c yi + c ym+1 = c cj ,
i=1
m
aij c yi + c ym+1 = c cj ,
j = 1, . . . , n,
i=1
y1 , . . . , ym 0,
ym+1 0.
(5.8)
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1969
Note that if any of cL , cU , c , c is equal to zero then the corresponding constraint is not present, and since at least
one of the values cL , c U , c , c is nonzero, the corresponding constraints reduce to m
i=1 aij yi + ym+1 = cj , for
j = 1, . . . , n. Then we have
m
yi R(b̃i ),
max u =
s.t.
m
i=1
yi aij cj ,
i=1
yi 0,
j = 1, . . . , n,
i = 1, . . . , m,
(5.9)
or, using matrix notations,
max
s.t.
u=y R(b̃),
yA c,
y 0.
(5.10)
Since problems (5.6) and (5.10) are equivalent, then problem (5.6) is the dual of (5.1) in the usual definition of the dual
in linear programming.
Lemma 5.1. The dual of the problem (5.6) is the problem (5.1).
Proof. Problems (5.6) and (5.1) are, respectively, equivalent to problems (5.9) and (5.7). Knowing from linear programming that the dual of (5.9) is (5.7), it follows that the dual of (5.6) is (5.1). 5.2. Relations between FVLP and DFVLP problems
Theorem 5.2 (The weak duality property). If x̃0 and w0 are feasible solutions to FVLP and DFVLP problems, respectively, then cx̃0 w0 b̃.
R
Proof. Multiplying Ax̃0 b̃ on the left by w0 0 and w0 A c on the right by x̃0 0̃ and using Lemma 2.1, we get
cx̃0 w0 Ax̃0 w0 b̃.
R
R
R
R
Corollary 5.1. If x̃0 and w0 are feasible solutions to FVLP and DFVLP problems, respectively, and cx̃0 = w0 b̃, then
x̃0 and w0 are optimal solutions to their respective problems.
Proof. It is straightforward, using Theorem 5.2.
R
The following corollary relates unboundedness of one problem to infeasibility of the other. We use the definition
below.
Definition 5.2. We say FVLP problem (or DFVLP problem) is unbounded if feasible solutions exist with arbitrary
small (or large) ranking function for the fuzzy objective value.
Corollary 5.2. If any one of the FVLP or DFVLP problem is unbounded, then the other problem has no feasible
solution.
Proof. It is straightforward, using Theorem 5.2.
We are now ready to present the strong duality result. The converse part of the result was given in [18] before. Here,
we state and prove a stronger result.
1970
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
Theorem 5.3 (Strong duality). If any one of the FVLP or DFVLP problem has an optimal solution, then both problems
have optimal solutions and the two optimal value of ranking functions for the fuzzy objective values are equal. (In fact,
∗ = 0̃ is optimal solution of the primal problem then the crisp vector w ∗ = c B −1 ,
if x̃ ∗ with x̃B∗ = B −1 b̃, and x̃N
B
where B is the optimal basis, is optimal solution of the dual problem.)
Proof. First, assume that the FVLP problem has a fuzzy optimal solution, and rank(A) = m. Let ỹ 0̃ be the fuzzy
slack variables for the constraints Ax̃ b̃. The new equivalent problem to the FVLP problem is
min
z̃ = cx̃ + 0 ỹ,
s.t.
Ax̃ + ỹ = b̃,
R
R
R
R
x̃ 0̃,
R
ỹ 0̃.
(5.11)
R
Assume B is the optimal basis matrix and x̃∗ =(x̃BT
R
0̃T )T =(b̃T B −T
R
0̃T )T is the fuzzy basic optimal solution corre-
sponding to the FVLP problem. From Theorem 4.1 we have
cB B −1 aj − cj 0,
j = 1, . . . , n, n + 1, . . . , n + m,
or equivalently,
cB B −1 aj cj ,
cB B −1 ei 0,
j = 1, . . . , n,
i = 1, . . . , m.
Hence, we must have
cB B −1 Ac
cB B −1 0.
Now, let w∗ = cB B −1 . Using the above inequalities, we can write,
w∗ Ac,
w∗ 0.
Thus, w∗ is a feasible solution to the DFVLP problem and
w∗ b̃ = cB B −1 b̃ = cB x̃B = cx̃∗
R
R
R
and hence
w∗ b̃ = cx̃∗ .
R
Therefore, the result follows immediately from Theorem 5.2. The proof for the converse is given in [18].
Remark 5.1. We emphasize that the basic solution to the FVLP problem is a fuzzy vector that is uniquely determined
by the optimal basis B. The solution to the dual problem is a unique crisp solution, however, depending solely on the
basis matrix B, with the objective value of the dual having the same rank as the primal (see also Remark 4.1). We realize
that the duality results obtained here are independent of the choice of the linear ranking function. It is clear that we
can use any other linear ranking function, and although the solution obtained may be different but the duality results
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1971
are still valid for the new solution. In particular, the dual simplex algorithm that we present later will find the fuzzy
solution of the primal as well as the crisp solution of the dual problem along with the induced basis corresponding to
the ranking function being used (we point out that the optimal basis corresponding to a linear ranking function does
not change if the ranking function is multiplied by a positive constant). As for the types of the fuzzy data in the model
and the assumption of fuzziness in the variables, the choice and compatibility of the ranking function for the linear
programming model should be the decision maker’s main concerns. For trapezoidal fuzzy numbers and variables, the
linear ranking function (2.5) is deemed to be appropriate (see [9,24]).
For an illustration of the above theorem consider the following FVLP problem and its dual.
Example 5.1.
min
s.t.
z̃ = 6x̃1 + 10x̃2 ,
R
⎧
2x̃1 + 5x̃2 (5, 8, 2, 5),
⎪
⎪
⎪
R
⎪
⎪
⎨
3x̃1 + 4x̃2 (6, 10, 2, 6),
R
⎪
⎪
⎪
⎪
⎪
⎩ x̃1 , x̃2 0̃.
R
max
s.t.
ũ =(5, 8, 2, 5)w1 + (6, 10, 2, 6)w2 ,
R
⎧
2w + 3w2 6,
⎪
⎨ 1
5w1 + 4w2 10,
⎪
⎩
w1 , w2 0.
12 18 19
−2 30 30 38
We see that the fuzzy basic optimal solution for the FVLP problem is x̃1 =( −5
7 , 7 , 7 , 7 ), x̃2 =( 7 , 7 , 7 , 7 )
R
R
16
−62 300 360 418
with R(x̃1 ) = 15
28 , R(x̃2 ) = 7 , and also the optimal objective value is z̃ = ( 7 , 7 , 7 , 7 ) with R(z̃) = 19.0714,
with the optimal basis matrix
5 2
B=
.
4 3
Hence, if we let w = cB B −1 , we obtain w = ( 67 , 10
7 ), which is equal to the optimal solution of the DFVLP problem.
On the other hand, we have R(ũ) = 19.0714. Therefore, both problems have optimal solutions and the two optimal
value of ranking functions for the fuzzy objective values are equal.
Remark 5.2. Note that in Example 5.1, we considered trapezoidal fuzzy number and used linear ranking function as
given in (2.5).
We can now state the fundamental duality result.
Theorem 5.4 (Fundamental theorem of duality). For any FVLP problem and its corresponding DFVLP problem, exactly one of the following statements is true.
(1) Both have optimal solutions x̃∗ and w∗ with cx̃∗ = w∗ b̃.
R
(2) One problem is unbounded and the other is infeasible.
(3) Both problems are infeasible.
Proof. Using the above results, the first and second statements are obviously correct. We give Example 5.2 to show
the third. 1972
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
Example 5.2. Consider the FVLP problem and its corresponding DFVLP problem as follows:
z̃ = x̃1 − 2x̃2 ,
min
R
x̃1 − x̃2 (−1, 1, 4, 2),
s.t.
R
−x̃1 + x̃2 (−2, −1, 3, 7),
R
x̃1 , x̃2 0̃.
R
ũ =(−1, 1, 4, 2)w1 + (−2, −1, 3, 7)w2 ,
max
R
w1 − w2 1,
s.t.
−w1 + w2 − 2,
w1 , w2 0.
We see that both problems are infeasible.
We now state and prove an important result of duality theory, commonly named as complementary slackness.
Theorem 5.5 (Complementary slackness). Let x̃∗ and w∗ be any feasible solutions to FVLP problem and its corresponding DFVLP problem. Then x̃∗ and w∗ are respectively optimal if and only if
(w∗ A − c)x̃∗ = 0̃,
R
w∗ (b̃ − Ax̃∗ ) = 0̃.
R
(5.12)
Proof. Suppose that x̃∗ and w∗ are feasible solutions to FVLP and DFVLP problems, respectively. Therefore,
Ax̃∗ b̃
(5.13)
w∗ Ac.
(5.14)
R
and
Multiplying Ax̃∗ b̃ on the left by w∗ 0 yields
R
w∗ Ax̃∗ w∗ b̃.
(5.15)
R
Multiplying w∗ Ac on the right by x̃∗ 0̃ yields
R
w∗ Ax̃∗ cx̃∗ .
(5.16)
R
Therefore, we will have
w∗ b̃ w∗ Ax̃∗ cx̃∗ .
R
(5.17)
R
On the other hand, since x̃∗ and w∗ are optimal solutions to the primal and dual problems, respectively, then by Theorem
5.2, we have w∗ b̃ = cx̃∗ , and (5.17) is written as
R
w∗ b̃ = w∗ Ax̃∗ = cx̃∗ .
R
(5.18)
R
From, (5.18) we immediately have
(w∗ A − c)x̃∗ = 0̃,
R
w∗ (b̃ − Ax̃∗ ) = 0̃.
R
(5.19)
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1973
The converse of the theorem follows from the fact that (w∗ A − c)x̃∗ = 0̃ and w∗ (b̃ − Ax̃∗ ) = 0̃ imply that cx̃∗ = w∗ b̃.
Therefore, optimality of x̃∗ and w∗ follows from Corollary 5.1.
R
R
R
Remark 5.3. The complimentary slackness conditions (5.12) is equivalent to (below, āi refers to the ith row and aj
refers to the jth column of A)
w∗ aj < cj ⇒ x̃j ∗ = 0̃,
or
x̃j ∗ > 0̃ ⇒ w∗ aj = cj , j = 1, . . . , n,
āi x̃∗ < b̃i ⇒ wi ∗ = 0,
or
wi ∗ < 0 ⇒ āi x̃∗ = b̃i , i = 1, . . . , m.
R
R
R
R
Maleki et al. [17,18] consider the FVLP problem and introduce a new method for solving it. The method proposed,
solves a related problem called the auxiliary problem. In the next section, we introduce a new dual method for solving
the FVLP problem directly, working on the primal problem and its associated simplex tableaus.
6. Dual simplex method
6.1. Primal optimality and dual feasibility
Consider the following FVLP problem,
min
z̃ = cx̃,
s.t.
Ax̃ b̃,
R
R
x̃ 0̃,
(6.1)
R
where x̃ = (x̃1 , . . . , x̃n )T , b̃ = (b̃1 , . . . , b̃m )T , c = (c1 , . . . , cn ), A = [aij ]m×n , b˜i , x˜j ∈ F(R), cj , aij ∈ R, for all
i, j , and R is a linear ranking function.
Now, we may rewrite (6.1) as follows:
min
z̃ = cx̃ + 0 ỹ,
s.t.
Ax̃ + ỹ = b̃,
R
R
x̃ 0̃,
R
ỹ 0̃,
(6.2)
R
where ỹ = (ỹ1 , . . . , ỹm )T .
Define x̄ ∈ (F(R))n+m and c̄ ∈ Rn+m as
x̃j , j = 1, . . . , n,
x̄j =
ỹj −n , j = n + 1, . . . , n + m,
c̄j =
cj , j = 1, . . . , n,
0, j = n + 1, . . . , n + m.
(6.3)
Suppose that a basic solution for (6.2) is given by x̄B = B −1 b̃, with the basis matrix B. Now, let zj = c̄B B −1 āj , ỹ0 =
R
R
B −1 b̃, where c̄B = (c̄B1 , . . . , c̄Bm ), and āj is the j th column of the coefficient matrix [A I ]. Consider Table 1, where
(x̄B )r is the rth fuzzy basic variable, yj = B −1 āj , and R(ỹr0 ) is the real number corresponding to the fuzzy number
ỹr0 obtained by a linear ranking function. Suppose that for j = 1, . . . , n + m, we have
zj − c̄j 0.
(6.4)
Define w = c̄B B −1 , where w = (w1 , . . . , wm ). For j = 1, . . . , n, we have
y0j = zj − c̄j = c̄B B −1 aj − cj = w aj − cj .
Therefore, for j = 1, . . . , n, from zj − cj 0, it follows that waj − cj 0. Then,
wAc.
(6.5)
1974
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
Table 1
A dual feasible simplex tableau
Basis
x̄1
···
x̄l
···
x̄n+m
R.H.S.
R(R.H.S.)
z̃
z1 − c̄1
···
zl − c̄l
···
zn+m − c̄n+m
cB ỹ0
R(cB ỹ0 )
(x̄B )1
.
.
.
(x̄B )r
.
.
.
(x̄B )m
y11
.
.
.
yr1
.
.
.
ym1
···
y1l
.
.
.
yrl
.
.
.
yml
···
y1,n+m
.
.
.
yr,n+m
.
.
.
ym,n+m
ỹ1 0
.
.
.
ỹr 0
.
.
.
ỹm 0
R(ỹ1 0 )
.
.
.
R(ỹr 0 )
.
.
.
R(ỹm 0 )
···
···
···
···
On the other hand, using (6.4), we have,
0 zn+i − c̄n+i = cB B −1 ei − 0 = wei = wi ,
i = 1, . . . , m,
and hence,
w 0,
(6.6)
yielding the dual feasibility. If R(ỹr0 ) 0, for all r = 1, . . . , m, then a fuzzy feasible solution for the FVLP problem
is at hand. Moreover, we will have,
c̄x̄ = c̄B ỹ0 = c̄B B −1 b̃ = w b̃,
R
R
R
and thus, by Corollary 5.1, establishing the optimality of x̄ and w for the FVLP and DFVLP problems, respectively.
Therefore, we have the following result.
Corollary 6.1. The optimality criteria zj − c̄j 0 for all j, for the FVLP problem is equivalent to the feasibility
condition for the DFVLP problem. If, in addition, x̄ corresponding to a basis B is primal feasible then x̄ is optimal for
the FVLP problem and w = c̄B B −1 is optimal to the DFVLP problem.
Now, let’s assume that the DFVLP problem is feasible and x̄, corresponding to a basis B, is dual feasible but primal
infeasible (and hence not optimal). That is, we have,
y0j = zj − c̄j 0,
j = 1, . . . , n + m
and there exists at least one r such that ỹr 0 < 0̃. We can immediately deduce that the primal problem must be finite.
R
Thus, the FVLP problem can be either infeasible (in which case, the DFVLP problem is unbounded), or it has an
optimal solution. In what follows we will show how to work on row r of the simplex tableau corresponding to B, as the
pivoting row, and either (1) detect infeasibility of the FVLP problem (or unboundedness of the DFVLP problem), or
(2) find a column l, as the pivoting column, to pivot on yrl and obtain a new dual feasible tableau with a nondecreasing
primal objective value. We explain these cases as Lemmas 6.1 and 6.2 below.
Lemma 6.1. If in a dual feasible simplex tableau an r exists such that ỹr0 < 0̃ and yrj 0, for all j, then the FVLP
R
problem is infeasible.
Proof. Suppose that a dual simplex tableau is feasible, and an r exists such that ỹr0 < 0̃ and yrj 0, for all j. CorreR
sponding to the rth row of the tableau, we have j yrj x̄j = ỹr0 . Since yrj 0 for all j and x̄j is required to be nonnegR
ative, then j yrj x̄j 0̃ for any fuzzy basic feasible solution. However, ỹr0 < 0̃. This shows that FVLP problem is
infeasible.
R
R
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1975
Lemma 6.2. If in a dual feasible simplex tableau, an r exists such that ỹr0 < 0̃ and there exists a nonbasic index j such
R
that yrj < 0, then a pivoting column l can be found so that pivoting on yrl will yield a dual feasible tableau with a
corresponding nondecreasing objective value.
Proof. We need a criterion for choosing a nonbasic fuzzy variable to enter the basis so that the new simplex tableau
will remain dual feasible and the new objective value is nondecreasing. Assume column l is the pivot column. Pivoting
on the pivot yrl will result in the new 0th row as follows:
y0l
ŷ0j = y0j −
yrj , j = 1, . . . , n + m, j = Bi .
(6.7)
yrl
For the new tableau to be dual feasible we need to have
ŷ0j 0,
j = Bi ,
(6.8)
which, using (6.7), results in
y0l y0j
, ∀j = Bi .
yrl yrj
(6.9)
To satisfy (6.9), it is sufficient to let
y0j
y0l
= min
| yrj < 0 .
j =Bi yrj
yrl
(6.10)
We note that the new objective value is nondecreasing, since
y0l
y0l
ŷ00 = R ỹ00 −
ỹr0 R(ỹ00 ),
ỹr0 = R(ỹ00 ) − R
yrl
yrl
using the fact that
y0l
R
ỹr0 0.
yrl
Now, using the above results, we introduce a new dual algorithm to solve the FVLP problem directly, making use of
the dual feasible simplex tableau. Thus, we refer to the new algorithm as a dual simplex method.
Algorithm 6.1. A dual simplex method
(1) {Dual feasibility} Given a basis B for the FVLP problem such that
y0j = zj − c̄j 0 for all j. Compute the simplex tableau.
(2) If ỹ0 0̃ then Stop (the current solution is optimal)
R
else select the pivot row r with ỹr0 < 0̃ (that is, r so that R(ỹr0 ) < 0).
R
(3) If yrj 0 for all j then Stop (the FVLP problem is infeasible)
else select the pivot column l by the following minimum ratio test:
y0j
y0l
= min
| yrj < 0 .
j =Bi yrj
yrl
(4) Pivot on yrl and go to (2).
Remark 6.1. One suggestion for choice of r in step (2) may be r so that
R(ỹr0 ) = min1 i m {R(ỹi0 )}.
Remark 6.2. Pivoting on yrl in step (4) is the usual Gaussian elimination process that, using yrl , converts column l to
the unit vector er yielding the new simplex tableau corresponding to the new basis (the one that is obtained by replacing
column r of B with al ).
1976
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
6.2. A numerical example
For an illustration of the above method we consider an example used by Maleki [17], which was solved by use of an
auxiliary problem. Note that here we use from linear ranking function (2.5) as a pattern of linear ranking function on
F(R). Moreover, we suppose the decision making variables be trapezoidal fuzzy number. In practice we consider the
known coefficients and the decision making variables are the same type of fuzzy number.
Example 6.1.
min
s.t.
z̃ = 6x̃1 + 10x̃2 ,
⎧R
2x̃1 + 5x̃2 (5, 8, 2, 5),
⎪
⎪
⎪
R
⎨
3x̃1 + 4x̃2 (6, 10, 2, 6),
R
⎪
⎪
⎪
⎩ x̃1 , x̃2 0̃.
R
We may write the first dual feasible simplex tableau as follows:
⎧
−2x̃1 − 5x̃2 + x̃3 =(−8, −5, 5, 2),
⎪
⎪
R
⎨
−3x̃1 − 4x̃2 + x̃4 =(−10, −6, 6, 2),
R
⎪
⎪
⎩ x̃ , . . . , x̃ 0̃,
1
4
R
or equivalently,
Basis
x̃1
x̃2
x̃3
x̃4
R.H.S.
R(R.H.S.)
z̃
x̃3
x̃4
−6
−10
0
0
(0,0,0,0)
0
−2
−3
−5
−4
1
0
0
1
(−8, −5, 5, 2)
(−10, −6, 6, 2)
−7.25
−9
x̃4 is a leaving variable and x̃1 is an entering variable. The new tableau is
Basis
x̃1
x̃2
x̃3
x̃4
R.H.S.
R(R.H.S.)
z̃
0
−2
0
−2
(12,20,4,12)
18
x̃3
0
− 73
1
− 23
(−4, 53 , 19
3 , 6)
15
− 12
x̃1
1
4
3
0
− 13
2
(2, 10
3 , 3 , 2)
3
x̃3 is a leaving variable and x̃2 is an entering variable. The next tableau is
Basis
x̃1
x̃2
x̃3
x̃4
R.H.S.
R(R.H.S.)
z̃
0
0
− 67
− 10
7
300 360 418
( −62
7 , 7 , 7 , 7 )
x̃2
0
1
− 37
2
7
18 19
(− 57 , 12
7 , 7 , 7 )
x̃1
1
0
4
7
− 57
30 38
(− 27 , 30
7 , 7 , 7 )
267
14
15
28
16
7
Therefore, the optimal solution of the FVLP problem obtained by the dual method is x̃1 =
−62 300 360 418 −5 12 18 19
,
,
,
7
7 7 7 and z̃ =
7 , 7 , 7 , 7 .
−2
7
30 38
, 30
7 , 7 , 7 , x̃2 =
N. Mahdavi-Amiri, S.H. Nasseri / Fuzzy Sets and Systems 158 (2007) 1961 – 1978
1977
Now if we consider the optimal solution, showing four decimal places for the fractional parts, we have
x̃1 = (−0.2857, 4.2857, 4.2857, 5.4286),
x̃2 = (−0.7143, 1.7143, 2.5714, 2.7143),
z̃ = (−8.8571, 42.8571, 51.4286, 59.7143)
and
R(z̃) = 19.0714.
The solution is matched with the solution obtained by Maleki [17].
7. Conclusions
We established the dual of linear programming problem with trapezoidal fuzzy variables and hence developed some
duality results, based on certain linear ranking functions, for the fuzzy primal and fuzzy dual problems. The duality
results have been established using certain general linear ranking functions on trapezoidal fuzzy numbers and appeared
to be natural extensions of the results for linear programming problems with crisp data. Using these results and the dual
feasible primal simplex tableau, we introduced a dual simplex algorithm for solving the primal and the dual problems
directly. The capabilities offered here will be useful for sensitivity (or post optimality) analysis using the primal simplex
tableau.
Acknowledgments
This research was supported in part by the Research Council of Sharif University of Technology, and in part by
a grant from IPM (Grant no. 85900040). We are grateful to the three anonymous referees and the editors for their
constructive comments, raising questions, and providing references. Consideration of all improved the presentation.
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