Fuzzy Sets and Systems 119 (2001) 31–40
www.elsevier.com/locate/fss
Interactive fuzzy programming for two-level linear fractional
programming problems
Masatoshi Sakawa ∗ , Ichiro Nishizaki
Department of Industrial and Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama,
Higashi-Hiroshima, Hiroshima, 739-8527 Japan
Received August 1997; received in revised form October 1998
Abstract
This paper presents interactive fuzzy programming for two-level linear fractional programming problems with the essentially
cooperative decision makers. In our interactive method, after determining the fuzzy goals of the decision makers at both
levels, a satisfactory solution is eciently derived by updating the satisfactory level of the decision maker at the upper
level with considerations of overall satisfactory balance between both levels. In an interactive process, optimal solutions to
the formulated programming problems are obtained by combined use of the bisection method and the phase one of linear
programming and the variable transformation by Charnes and Cooper. An illustrative numerical example is provided to
demonstrate the feasibility and eciency of the proposed method. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Two-level linear fractional programming problem; Fuzzy programming; Fuzzy goals; Interactive methods
1. Introduction
A two-level programming problem is formulated
for a problem in which two decision makers (DMs)
make decisions successively. For example, in a decentralized rm, top management, an executive board,
or headquarters makes a decision such as a budget
of the rm, and then each division determines a
production plan in the full knowledge of the budget. We can also cite the Stackelberg duopoly: two
rms supply homogeneous goods to a market. Suppose one rm dominates the other in the market, and
Corresponding author.
E-mail address: sakawa@msl.sys.hiroshima-u.ac.jp (M. Sakawa).
∗
consequently the predominant rm rst decides its
level of supply, and then the other rm determines
that of itself after it realizes that of the predominant
rm.
The Stackelberg solution has been employed as a
solution concept to two-level programming problems,
and a considerable number of algorithms for obtaining
the solution have been developed (e.g. [1,3,6,23]).
We can nd many instances of decision problems [2,4,9,14], which are formulated as two-level
programming problems, and concerning the abovementioned hierarchical decision problem in the
decentralized rm, it is natural that decision makers behave cooperatively rather than noncooperatively.
0165-0114/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 0 6 6 - 4
32
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
Recently, Lai [12] and Shih, Lai and Lee [20] have
proposed a solution concept, which is di erent from
the concept of the Stackelberg solution, for multi-level
linear programming problems such that decisions of
DMs in both levels are sequential and all of the DMs
essentially cooperate with each other.
Their method is based on the idea that the DM at the
lower level optimizes the lower level objective function, taking a goal or preference of the upper level
into consideration. DMs elicit membership functions
of fuzzy goals for their objective functions, and especially, the DM at the upper level also speci es those
of fuzzy goals for decision variables. The DM at the
lower level solves a fuzzy programming problem with
constraints on fuzzy goals of the DM at the upper level.
Unfortunately, there is a possibility that their method
leads to an undesirable nal solution because of inconsistency between the fuzzy goals of the objective
function and the decision variables.
To overcome the problem in the methods of Lai
et al., eliminating the fuzzy goals for decision
variables, we have developed interactive fuzzy
programming for two-level linear programming problems [16].
However, considering hierarchical decision problems such as in rms, it is frequently appropriate
not only to take the cooperative relationship between
decision makers into consideration but also to examine linear fractional objectives rather than linear ones.
Examples of objectives or criteria represented as fractional functions [10,22] are often encountered in the
following situations: for nance or corporate planning,
debt-to-equity ratio, return on investment, current
ratio, risk-assets to capital, actual capital to required
capital, foreign loans to total loans, residential mortgages to total mortgages, etc., for production planning, inventory to sales, actual cost to standard cost,
output per employee, and so forth.
For instance, by adopting a criterion with respect to
nance or corporate planning as an objective function
at the upper level and employing a criterion regarding
production planning as an objective function at the
lower level, a two-level linear fractional programming
problem can be formulated for hierarchical decision
problems in rms.
A usual linear fractional programming problem is
a special case of a nonlinear programming problem,
but it can be transformed into a linear programming
problem by using the variable transformation method
by Charnes and Cooper [8], or it can be solved by
adopting the updated objective function method by
Bitran and Novaes [7], i.e., we can obtain an optimal solution by repeating two steps: computing
the local gradient of the fractional objective function, and solving the resulting linear programming
problem. Concerning multiobjective linear fractional
programming, Kornbluth and Steuer [10,11] presented two di erent approaches to multiobjective
linear fractional programming based on the weighted
Tchebysche norm. Luhandjula [13] proposed a linguistic approach to multiobjective linear fractional
programming by introducing linguistic variables to
represent linguistic aspirations of the decision maker.
In the framework of the fuzzy decision of Bellman
and Zadeh [5], Sakawa and Yumine [19] presented
a fuzzy programming approach for solving multiobjective linear fractional programming problems
by combined use of the bisection method and the
phase one of the simplex method of linear programming. As a generalization of the result in Sakawa
and Yumine [19], Sakawa and Yano [18] proposed
a linear programming-based interactive fuzzy satis cing method for multiobjective linear fractional
programming to derive the satis cing solution for
the DM eciently from a Pareto optimal solution
set by updating the reference membership values of
the DM.
In this paper, we brie y refer to a method for
obtaining the Stackelberg solution to a two-level
linear fractional programming problem. Next we
deal with the two-level linear fractional programming problem with the essentially cooperative DMs
and propose an interactive fuzzy programming for
the problem. In our interactive method, having determined the fuzzy goals of the decision makers
at both levels, a satisfactory solution is eciently
derived by updating the satisfactory degree of the
decision maker at the upper level with considerations of overall satisfactory balance between both
levels. Optimal solutions to the formulated programming problems are obtained by combined use of the
bisection method and phase one of linear programming and the variable transformation by Charnes and
Cooper. An illustrative numerical example demonstrates the feasibility and e ectiveness of the proposed
method.
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
2. Two-level linear fractional programming
problems
A two-level linear fractional programming problem
for obtaining the Stackelberg solution is formulated as
minimize
z1 (x1 ; x2 ) where x2 solves
minimize
z2 (x1 ; x2 )
subject to
A1 x1 + A2 x2 6b;
x1
x2
(1)
where objective function zi (x1 ; x2 ); i = 1; 2 is represented by a linear fractional function
pi (x1 ; x2 ) ci1 x1 + ci2 x2 + ci3
=
;
qi (x1 ; x2 ) di1 x1 + di2 x2 + di3
an optimal solution to the problem occurs at a vertex
of its feasible region.
The DM1’s problem in which the objective function
of DM2 is eliminated from problem (1) is represented
as
c11 x1 + c12 x2 + c13
minimize
d11 x1 + d12 x2 + d13
(3)
subject to
A1 x1 + A2 x2 6b;
x1 ¿0; x2 ¿0:
x1 ¿0; x2 ¿0;
zi (x1 ; x2 ) =
33
(2)
xi ; i = 1; 2 is an ni -dimensional decision variable; ci1
and di1 ; i = 1; 2 are n1 -dimensional constant row vectors; ci2 and di2 ; i = 1; 2 are n2 -dimensional constant
row vectors; ci3 and di3 ; i = 1; 2 are constants; b is an
m-dimensional constant column vector; Ai ; i = 1; 2 is
an m × ni constant matrix; and it is assumed that the
denominators are positive, i.e., qi (x1 ; x2 )¿0; i = 1; 2.
For the sake of simplicity, we use the following notation: x = (x1 ; x2 ) ∈ Rn1 +n2 ; ci = [ci1 ci2 ci3 ]; i = 1; 2;
di = [di1 di2 di3 ]; i = 1; 2, and let DM1 denote the DM
at the upper level and DM2 denote the DM at the
lower level. In the two-level linear fractional programming problem (1), z1 (x1 ; x2 ) and z2 (x1 ; x2 ),
respectively, represent objective functions of DM1
and DM2, and x1 and x2 , represent decision variables
of DM1 and DM2, respectively.
The Kth best method proposed by Bialas and
Karwan [6] is one of vertex enumeration approaches.
The solution search procedure of the method starts
from a point which is an optimal solution to the
problem of DM1 and checks whether it is also an
optimal solution to the problem of DM2 or not. If
the rst point is not the Stackelberg solution, the
procedure continues to examine the second best solution to the problem of DM1 and so forth. We can
apply the idea of the Kth best method to obtaining
the Stackelberg solution to the two-level linear fractional programming problem (1) because, in general,
objective function level curves of a linear fractional
programming problem are not parallel but linear, and
Let x̂[1] ; x̂[2] ; : : : ; x̂[N ] denote the N ordered basic
feasible solutions to the linear fractional programming problem (3) such that c1 x̂[i] =d1 x̂[i] ¿c2 x̂[i+1] =
d2 x̂[i+1] ; i = 1; : : : ; N − 1. For a given x̂1[i] , DM2’s
problem is formulated as
minimize
subject to
c21 x1[i] + c22 x2 + c23
d21 x1[i] + d22 x2 + d23
A1 x1[i] + A2 x2 6b;
(4)
x1 ¿0; x2 ¿0:
Let x̃2 (x̂1[i] ) denote an optimal solution to problem
(4). Then nding the global optimum is equivalent to
seeking the minimal index i such that x̃2 (x̂1[i] ) = x̂2[i] .
Since problems (3) and (4) are linear fractional programming problems, we can solve them by adopting
the variable transformation method by Charnes and
Cooper [8] or by employing the updated objective
function method by Bitran and Novaes [7].
Another popular method for nding the Stackelberg
solution to the two-level linear programming problem
is referred to as the “Kuhn–Tucker” approach. It is
based on the fact that the two-level linear programming problem is equivalent to the nonlinear programming problem created by replacing DM2’s problem
with its Kuhn–Tucker conditions and appending the
resultant system to DM1’s problem. It should be noted
here that the Kuhn–Tucker approach cannot be directly
applied to the two-level linear fractional programming
problem because the objective function of DM2 is a
linear fractional function and it is not convex.
When the Stackelberg solution is employed, it is
assumed that there is no communication between the
two DMs, or they do not make any binding agreement even if there exists such communication. However, the above assumption is not always reasonable
34
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
when we model decision-making problems in a decentralized rm as a two-level programming problem
in which top management is DM1 and an operation
division of the rm is DM2 because it is supposed
that there exists a cooperative relationship between
them.
Consider a computational aspect of the Stackelberg
solution. Even if objective functions of both DMs and
common constraint functions are linear, it is known
that the problem for obtaining the Stackelberg solution is a non-convex programming problem with special structure. Although a large number of algorithms
for obtaining the Stackelberg solution have been developed, it is known that the problem of nding the
solution is strongly NP-Hard [21]. From such diculties, a new solution concept which is easy to compute
and re ects the structure of multi-level programming
problems is expected.
To realize two demands, we develop an algorithm
for obtaining a satisfactory solution to a two-level linear fractional programming problem based on a basic idea that DM1 claims a solution with at least a
minimal satisfactory level speci ed by DM1, taking
overall satisfactory balance between both levels into
consideration, and DM2’s degree of satisfaction is
maximized on condition that the minimal satisfactory
level of DM1 is satis ed.
In contrast to formulation (1), in this paper, we
employ the following representation in order to deal
with a two-level linear fractional programming problem with cooperative DMs:
minimize
z1 (x) =
c11 x1 + c12 x2 + c13
d11 x1 + d12 x2 + d13
minimize
z2 (x) =
c21 x1 + c22 x2 + c23
d21 x1 + d22 x2 + d23
subject to
A1 x1 + A2 x2 6b;
upper level
upper level
(5)
x1 ¿0; x2 ¿0:
3. Interactive fuzzy programming
In this section we deal with two-level linear fractional programming problems in which the DMs are
essentially cooperative and their decisions are sequential. We propose an interactive fuzzy programming
method for obtaining a satisfactory solution to such
problems.
It is natural that the DMs have fuzzy goals for
their objective functions when they take fuzziness of
human judgments into consideration [15]. For each of
the objective functions zi (x), i = 1; 2 of problem (5),
assume that the DMs have fuzzy goals such as “the
objective function zi (x) should be substantially less
than or equal to some value pi ”.
The individual minimum
zimin = min zi (x);
x∈X
i = 1; 2
(6)
and the individual maximum
zimax = max zi (x);
x∈X
i = 1; 2
(7)
of the objective functions are referred to when the
DMs elicit membership functions prescribing the
fuzzy goals for the objective functions zi (x), i = 1; 2,
where X is a feasible region of problem (5). The DMs
determine the membership functions i (zi (x)), which
are strictly monotone decreasing for zi (x), consulting
the variation ratio of satisfactory degree in the interval
between the individual minimum (6) and the individual maximum (7). The domain of the membership
function is the interval [zimin ; zimax ], i = 1; 2, and each
of the DMs speci es the value zi0 of the objective
function for which the satisfactory degree is 0 and
the value zi1 of the objective function for which the
degree of satisfaction is 1. For the value undesired
(larger than zi0 ), it is de ned that i (zi (x)) = 0, and
for the value desired (smaller than zi1 ), it is de ned
that i (zi (x)) = 1.
The following linear function is an example of a
membership function i (zi ) which characterizes the
fuzzy goal of the DM:
if zi (x)¿zi0 ;
0
z (x) − z 0
i
i
(8)
i (zi (x)) =
if zi1 ¡ zi (x)6zi0 ;
zi1 − zi0
1
if zi (x)6zi1 ;
where zi0 and zi1 denote the values of the objective
function zi (x) such that the degrees of the membership
function are 0 and 1, respectively, and it is assumed
that the DMs subjectively assess zi0 and zi1 .
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
Zimmermann proposed the following method for
eliciting the membership function of the fuzzy goal
[24]. Using the individual minimum
function i (·); i = 1; 2, and then problem (13) can be
transformed into the following equivalent problem:
maximize
subject to
A1 x1 + A2 x2 6b;
x;
zimin = zi (xio )
= min{zi (x) | A1 x1 + A2 x2 6b; x1 ¿0; x2 ¿0}
p1 (x)61−1 ()q1 (x);
(9)
0661;
zim = zi (xjo );
i = 1; 2; j =
(
1
if i = 2;
2
if i = 1;
(10)
the DMs determine the linear membership functions
as in (8) by choosing zi1 = zimin , zi0 = zim , i = 1; 2.
The membership function does not need to be a
linear function throughout this paper, but we assume
that it is a continuous and strictly monotone decreasing function. Sakawa and Yumine proposed several
types of nonlinear membership functions: exponential,
hyperbolic, hyperbolic inverse and piecewise linear
membership functions [19].
To derive a satisfactory solution to the problem (5),
the proposed algorithm starts to nd the maximum decision to the fuzzy decision by Bellman and Zadeh
[5], i.e., to solve the following problem for obtaining
a solution which maximizes the smaller degree of satisfaction between the two DMs:
maximize min{1 (z1 (x)); 2 (z2 (x))}:
x∈X
(11)
For a coming interactive process, we de ne a satisfactory degree of both DMs as
= min{1 (z1 (x)); 2 (z2 (x))}:
(12)
The maximin problem (11) can be written as an equivalent maximization problem:
x;
subject to
(14)
p2 (x)62−1 ()q2 (x)
together with
maximize
35
x1 ¿0; x2 ¿0:
By solving problem (14), we can obtain a solution maximizing a smaller satisfactory degree between
those of both DMs. In problem (14), however, even
if the membership function i (·) is linear, problem
(14) is not a linear programming problem because
i−1 ()qi (x), i = 1; 2 is nonlinear. Thus, we cannot
directly apply the linear programming technique, but
from the following fact, we can solve problem (14) by
combined use of the bisection method and the phase
one of linear programming [15,19]. In problem (14),
if the value of , which satis es 0661, is xed,
constraint functions of problem (14) can be reduced
to a set of linear inequalities. Obtaining the optimal
solution ∗ to the above problem is equivalent to determining the maximum value of so that there exists
an admissible set satisfying the constraints of problem
(14).
It should be noted that the above-mentioned
procedure can be applicable not only to linear membership functions but also to nonlinear membership
functions if they are continuous and strictly monotone
decreasing.
After ∗ has been determined, we minimize the
objective function z2 (x) of DM2 subject to the constraints of problem (14) for = ∗ , i.e., we solve the
following problem:
A1 x1 + A2 x2 6b;
1 (z1 (x))¿;
2 (z2 (x))¿;
0661;
x1 ¿0; x2 ¿0:
z2 (x) =
subject to
A1 x1 + A2 x2 6b;
x
(13)
Let i−1 (·), i = 1; 2 be an inverse function of the continuous and strictly monotone decreasing membership
c21 x1 + c22 x2 + c23
d21 x1 + d22 x2 + d23
minimize
(15)
p1 (x)61−1 (∗ )q1 (x);
x1 ¿0; x2 ¿0;
where the constraint with respect to the fuzzy goal of
DM2 is eliminated because ∗ is a solution to problem
(14).
36
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
Let x∗ be an optimal solution to problem (15). It
can be shown in the following that the optimal solution
x∗ is a weak Pareto optimal solution to the problem:
minimize
minimize
subject to
c11 x1 + c12 x2 + c13
d11 x1 + d12 x2 + d13
c21 x1 + c22 x2 + c23
z2 (x) =
d21 x1 + d22 x2 + d23
z1 (x) =
(16)
x ∈ X:
Suppose that there exists x′ ∈ X such that z1 (x′ )
¡ z1 (x∗ ) and z2 (x′ )¡ z2 (x∗ ), that is, x∗ is not one
of the weak Pareto optimal solutions. Because the
membership functions are continuous and strictly
monotone decreasing, we have ∗ 61 (x∗ )¡1 (x′ )
and then x′ becomes a feasible solution of problem
(15). The assumption z2 (x′ )¡ z2 (x∗ ) contradicts that
x∗ is an optimal solution to problem (15). Thus, it
follows that x∗ is a weak Pareto optimal solution.
In order to solve the linear fractional programming
problem (15), if we use the variable transformation
by Charnes and Cooper [8]
t = 1=q2 (x);
(y1 ; y2 ; t)T = (x1 ; x2 ; 1)T t;
(17)
problem (15) can be equivalently transformed as
minimize
c2 y
subject to
[A1 A2 − b] y60;
y
c1 y61−1 (∗ )d1 y;
(18)
y¿0;
where, for the sake of simplicity, we use the notation
y = (y1 ; y2 ; t)T ∈ Rn1 +n2 +1 .
In order to test the Pareto optimality, we solve the
following linear programming problem:
1 + 2
subject to
p1 (x) − 1 = z1 (x∗ )q1 (x);
p2 (x) − 2 = z2 (x∗ )q2 (x);
A1 x1 + A2 x2 6b;
x¿0; ”¿0;
where ” = (1 ; 2 ).
maximize
zj (x)
subject to
i ∈ {i | i = 0};
zi (x) = zi (x);
(19)
(20)
i ∈ {i | i ¿0};
zi (x)¿zi (x);
x¿0:
If DM1 is satis ed with the optimal solution x∗
to problem (13), which is obtained by solving problems (14), (18)–(20), it follows that the optimal solution x∗ becomes a satisfactory solution; however,
DM1 is not always satis ed with the solution x∗ . It is
natural to suppose that DM1 subjectively speci es a
minimal satisfactory level ˆ ∈ [0; 1] for his/her membership function 1 (z1 (x)).
If DM1 is not satis ed with the solution x∗ to
problem (13), in the proposed algorithm, the following problem in which DM2’s membership function
is maximized on condition that DM1’s membership
function 1 (z1 (x)) is larger than or equal to ˆ is
formulated:
maximize
2 (z2 (x))
subject to
A1 x1 + A2 x2 6b;
x
d2 y = 1;
minimize
Let (x;
”)
be an optimal solution to this problem Then, if all i = 0, then x∗ is a Pareto optimal solution, and if at least one i ¿0, then for
j ∈ {j | j ¿0}, solve the following linear fractional
programming problem by using the variable transformation method, and an optimal solution to the
problem is a Pareto optimal solution [19]:
(21)
ˆ
1 (z1 (x))¿;
x1 ¿0; x2 ¿0:
Because the membership function 2 (·) is strictly
monotone decreasing, problem (21) is equivalent to
the linear fractional problem:
minimize
z2 (x)
subject to
A1 x1 + A2 x2 6b;
x
ˆ 1 (x);
p1 (x)61−1 ()q
x1 ¿0; x2 ¿0:
(22)
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
Using the variable transformation again, we can formulate the linear programming problem:
minimize
c2 y
subject to
[A1 A2 − b] y60;
x
ˆ 1 y;
c1 y61−1 ()d
(23)
d2 y = 1;
y¿0:
It should be noted here that, in our formulation (21),
the constraint on the fuzzy goal for decision variables
of DM1 is eliminated while it is involved in the formulations by Lai [12] and Shih et al. [20].
If an optimal solution to problem (21) exists, it follows that DM1 obtains a satisfactory solution having
a satisfactory degree larger than or equal to the minimal satisfactory level speci ed by DM1’s own self.
However, the larger the minimal satisfactory level is
assessed, the smaller DM2’s satisfactory degree becomes. Consequently, a relative di erence between
the satisfactory degrees of DM1 and DM2 becomes
larger and it is feared that overall satisfactory balance
between both levels cannot be maintained.
To take into account the overall satisfactory balance
between both levels, DM1 needs to compromise with
DM2 on DM1’s own minimal satisfactory level. To
do so, a ratio of satisfactory degree between both DMs
is de ned as
=
2 (z2 (x∗ ))
;
1 (z1 (x∗ ))
(24)
which is de ned by Lai [12], is useful.
Let L and U denote the lower bound and the
upper bound of speci ed by DM1. If ¿U ,
i.e., 2 (z2 (x∗ ))¿U 1 (z1 (x∗ )), then DM1 updates
ˆ
the minimal satisfactory level ˆ by increasing .
Then DM1 obtains a larger satisfactory degree and
DM2 accepts a smaller satisfactory degree. Conversely, if ¡L , i.e., 2 (z2 (x∗ ))¡L 1 (z1 (x∗ )),
then DM1 updates the minimal satisfactory level ˆ
ˆ and DM1 accepts a smaller satisby decreasing ,
factory degree and DM2 obtains a larger satisfactory
degree.
At an iteration ‘, let 1 (z1‘ ), 2 (z2‘ ), ‘ and
‘
= 2 (z2‘ )=1 (z1‘ ) denote DM1’s and DM2’s satisfactory degrees, a satisfactory degree of both levels
37
and the ratio of satisfactory degrees between both
DMs, respectively, and let a corresponding solution
be x‘ . The iterated interactive process terminates if
the following two conditions are satis ed and DM1
concludes the solution as a satisfactory solution.
Termination conditions of the interactive process for two-level linear fractional programming
problems.
(1) DM1’s satisfactory degree is larger than or equal
to the minimal satisfactory level ˆ speci ed by
ˆ
DM1, i.e., 1 (z1‘ )¿.
‘
(2) The ratio of satisfactory degrees is in the
closed interval, the lower and the upper bounds
of which are speci ed by DM1.
Condition (1) is DM1’s required condition for solutions, and condition (2) is provided in order to keep
overall satisfactory balance between both levels.
Unless the conditions are satis ed simultaneously,
ˆ
DM1 needs to update the minimal satisfactory level .
Procedure for updating the minimal satisfactory
ˆ
level .
(1) If condition (1) is not satis ed, then DM1 deˆ
creases the minimal satisfactory level .
(2) If the ratio ‘ exceeds its upper bound, then DM1
ˆ Conincreases the minimal satisfactory level .
‘
versely, if the ratio is below its lower bound,
then DM1 decreases the minimal satisfactory
ˆ
level .
The above-mentioned algorithm is summarized as
follows:
Algorithm of the interactive fuzzy programming
for solving two-level linear fractional programming
problems.
Step 1: Set ‘ = 1. DM1 elicits the membership function 1 (z1 ) of the fuzzy goal of DM1, and speci es
the minimal satisfactory level ˆ and the lower and the
upper bounds of the ratio of satisfactory degrees 1 .
Step 2: DM2 elicits the membership function 2 (z2 )
of the fuzzy goal of DM2.
Step 3: The maximin problem (13) is solved
through problems (14), (18)–(20), and then a solution x‘ to problem (13) is proposed to DM1 together
with ‘ , 1 (z1‘ ), 2 (z2‘ ) and ‘ .
38
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
Step 4: If the solution proposed to DM1 satis es
the termination conditions and DM1 concludes the solution as a satisfactory solution, the algorithm stops.
Step 5: DM1 updates the minimal satisfactory level
ˆ in accordance with the procedure of updating minimal satisfactory level.
Step 6: Problem (21) is solved through problem
(23) and an obtained solution is proposed to DM1.
Return to Step 4.
4. A numerical example for two-level linear
fractional programming problems
Consider the following two-level linear fractional
programming problem:
minimize
x
minimize
x
subject to
c1 x1 + c2 x2 + 1
z1 (x) =
d1 x1 + d2 x2 + 1
c3 x1 + c4 x2 + 1
z2 (x) =
d3 x1 + d4 x2 + 1
maximize
subject to
x ∈ X;
x;
z1 (x) + 0:558660
¿;
−1:436379 + 0:558660
(25)
A1 x1 + A2 x2 6b;
xj ¿0;
is a sum of entries of the corresponding row vector of
A1 and A2 multiplied by 0:6. Coecients are shown
in Table 1.
Suppose that DM1 determines the initial minimal
satisfactory level as = 1:0, and the lower and the
upper bounds of as [0:6; 1:0]. The membership
functions (8) of the fuzzy goals are assessed by using values (9) and (10). The individual minima and
the corresponding optimal solutions are shown in
Table 2 and z1m = − 0:558660 and z2m = 0:525274 are
computed.
Then problem (13) for this numerical example can
be formulated as
j = 1; 2; : : : ; 10;
where x1 = (x1 ; : : : ; x5 )T , x2 = (x6 ; : : : ; x10 )T ; each
entry of ve-dimensional coecient vectors ci ; di ,
i = 1; 2; 3; 4, and of 11 × 5 coecient matrices A1 and
A2 is a random value in the interval [−50; 50]; each
entry of the right-hand side constant column vector b
(26)
z2 (x) − 0:525274
¿;
−0:497256 − 0:525274
where X denotes the feasible region of problem (25).
Data of the rst iteration including an optimal solution
to problem (26) are shown in Table 3.
The rst termination condition of the interactive
process is not satis ed because the satisfactory degree
11 = 0:651722 of DM1 does not exceed the minimal
satisfactory level ˆ = 1:0. Consequently, DM1 must
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
39
Table 2
The individual minima and the corresponding solutions
z1min
x1
x2
−1:436379
1.883205
2.361988
0.667962
0.861438
0.017950
0.242107
0.000000
0.000000
0.000000
2.299405
z2min
x1
x2
−0:497256
0.000000
0.223446
0.920781
0.673893
0.859239
0.660960
1.018842
0.000000
0.981706
0.000000
5. Conclusions
change the minimal satisfactory level, and suppose
that DM1 changes the minimal satisfactory level to
ˆ = 0:70 because, neither for ˆ = 0:90 nor for ˆ = 0:80,
any solution satisfying the condition on the upper and
lower bounds of is not obtained. Then a problem
corresponding to problem (21) is formulated as
maximize
2 (z2 (x))
subject to
x ∈ X;
x
In this paper, we have proposed interactive fuzzy
programming for two-level linear fractional programming problems with the essentially cooperative DMs.
In our interactive method, after determining the fuzzy
goals of the decision makers at both levels, a satisfactory solution is eciently derived by updating the
minimal satisfactory levels of decision makers at upper levels with considerations of overall satisfactory
balance between both levels. An illustrative numerical
example for two-level linear fractional programming
problems demonstrated the feasibility and eciency
of the proposed method. It is signi cant to note here
that although the Stackelberg solution does not always
satisfy Pareto optimality, the proposed solution is in
the set of Pareto optimal solutions because the solution
concept is proposed for two-level decision problems
with cooperative DMs.
References
(27)
z1 (x) + 0:558660
¿0:7:
−1:436379 + 0:558660
Data of the second iteration including an optimal solution to problem (27) are shown in Table 4.
At the second iteration, the satisfactory degree
12 = 0:70 of DM1 becomes equal to the minimal satisfactory level ˆ′ = 0:70 and the ratio 2 = 0:816574
of satisfactory degrees is in the valid interval [0:6; 1:0]
of the ratio. Then, if DM1 concludes the solution
as a satisfactory solution, the interactive process
terminates.
[1] J.F. Bard, An ecient point algorithm for a linear two-stage
optimization problem, Oper. Res. 38 (1983) 556 – 560.
[2] J.F. Bard, Coordination of a multidivisional rm through two
levels of management, Omega 11 (1983) 457– 465.
[3] J.F. Bard, J.E. Falk, An explicit solution to the multi-level
programming problem, Comput. Oper. Res. 9 (1982) 77–100.
[4] J.F. Bard, J. Plummer, J.C. Sourie, Determining tax credits
for converting nonfood crops to biofuels: an application
of bilevel programming, in: A. Migdalas, P. Pardalos,
P. Varbrand (Eds.), Multilevel Optimization: Algorithms and
Applications, Kluwer Academic Publishers, Dordrecht, 1997,
pp. 23– 50.
[5] R.E. Bellman, L.A. Zadeh, Decision making in a fuzzy
environment, Management Sci. 17 (1970) 141–164.
[6] W.F. Bialas, M.H. Karwan, Two-level linear programming,
Management Sci. 30 (1984) 1004 –1020.
40
M. Sakawa, I. Nishizaki / Fuzzy Sets and Systems 119 (2001) 31–40
[7] G.R. Bitran, A.G. Novaes, Linear programming with a fractional objective function, Oper. Res. 21 (1973) 22–29.
[8] A. Charnes, W.W. Cooper, Programming with linear fractional functions, Naval Res. Logist. Quart. 9 (1962) 181–186.
[9] J. Fortuny-Amat, B. McCarl, A representation and economic
interpretation of a two-level programming problem, J. Oper.
Res. Soc. 32 (1981) 783–792.
[10] J.S.H. Kornbluth, R.E. Steuer, Goal programming with linear
fractional criteria, European J. Oper. Res. 8 (1981) 58–65.
[11] J.S.H. Kornbluth, R.E. Steuer, Multiple objective linear fractional programming, Management Sci. 27 (1981) 1024 –1039.
[12] Y.J. Lai, Hierarchical optimization: a satisfactory solution,
Fuzzy Sets and Systems 77 (1996) 321–335.
[13] M.K. Luhandjula, Fuzzy approaches for multiple objective
linear fractional optimization, Fuzzy Sets and Systems 13
(1984) 11–23.
[14] P. Marccote, Network design problem with congestion e ects:
a case of bilevel programming, Math. Programming 34 (1986)
142–162.
[15] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993.
[16] M. Sakawa, I. Nishizaki, Y. Uemura, Interactive fuzzy programming for multi-level linear programming problems,
Comput. Math. Appl. 36 (1998) 71–86.
[17] M. Sakawa, I. Nishizaki, Y. Uemura, Interactive fuzzy programming for multi-level linear programming problems with
fuzzy parameters, Fuzzy Sets and Systems 109 (2000) 3–19.
[18] M. Sakawa, H. Yano, An interactive fuzzy satis cing method
for multiobjective linear fractional programming problems,
Fuzzy Sets and Systems 28 (1988) 129 –144.
[19] M. Sakawa, T. Yumine, Interactive fuzzy decision-making
for multiobjective linear fractional programming problems,
Large Scale Systems 5 (1983) 105 –113.
[20] H.S. Shih, Y.J. Lai, E.S. Lee, Fuzzy approach for multi-level
programming problems, Comput. Oper. Res. 23 (1996) 73– 91.
[21] K. Shimizu, Y. Ishizuka, J.F. Bard, Nondi erentiable and
Two-Level Mathematical Programming, Kluwer Academic
Publishers, Boston, 1997.
[22] R.E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986.
[23] D.J. White, G. Anandalingam, A penalty function approach
for solving bi-level linear programs, J. Global Optim. 3 (1993)
397– 419.
[24] H.-J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and
Systems 1 (1978) 45–55.
Qaiser Farooq