Interactive Fuzzy Programming for Two-Level 0-1

Programming Problems with Fuzzy Parameters through

Genetic Algorithms

Masatoshi Sakawa, Ichiro Nishizaki, and Masatoshi Hitaka

Faculty of Engineering, Hiroshima University, Higashi-Hiroshima, Japan 739-8527

SUMMARY

In this paper, an interactive fuzzy programming

method using genetic algorithms has been proposed for

two-level 0-1 programming problems with fuzzy parame-

ters. According to the proposed technique, the decision

maker in each level establishes his fuzzy goals related to the

objective functions, using linear membership functions.

After that, the upper level decision maker establishes, sub-

jectively, the minimal acceptable degree of the degree of

satisfaction for the membership functions and, simultane-

ously, considers the ratio of satisfaction degrees between

the levels; if necessary, the decision maker updates his

minimal acceptability degree interactively. In so doing, a

satisfactory solution is produced by taking into considera-

tion also the achievement balance of the overall satisfaction

degree, while respecting the upper-level decision maker�s

decision. The feasibility and validity of the proposed

method was demonstrated through a numerical example for

a two-level 0-1 programming problem with fuzzy parame-

ters. The algorithm proposed in this paper can be extended

to multilevel problems. © 2000 Scripta Technica, Electron

Comm Jpn Pt 3, 83(6): 40�49, 2000

Key words: Two-level 0-1 programming problem;

fuzzy programming; fuzzy objective; genetic algorithm;

interactive method.

1. Introduction

The static Stackelberg game model has become

widely accepted for two-level programming problems [1�

9]. A Stackelberg solution is a strategy selected by two

decision makers in the following way. There is a decision

maker for the upper level and another one for the lower

level. The upper-level decision maker first chooses a strat-

egy, after which the lower-level decision maker establishes

another strategy. Both decision makers then meet and be-

come aware of each other�s objective functions and con-

straints. Based on a good knowledge of the upper-level

decision maker�s strategy, the lower-level decision maker

selects an optimal strategy for his own objective functions;

under similar assumptions, the upper-level decision maker

does the same for his own objective functions.

Stackelberg game models for two-level programming

problems assume that the decision makers do not mutually

notify their intentions or, if there is the possibility of such

notifications, there does not exist any binding agreement

between them. Yet, we can think of two-level programming

problems where in the modeling of intention settings there

is a relationship between the upper and the lower levels, for

example in an enterprise between the upper-level general

division and each of the business sections. It is safe then to

assume in these situations that the aforementioned premise

is not realistic.

Consider on the other hand the computational aspects

of Stackelberg game models. Though the upper- and lower-

level objective functions are linear, it is known that the

© 2000 Scripta Technica

Electronics and Communications in Japan, Part 3, Vol. 83, No. 6, 2000Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J81-A, No. 6, June 1998, pp. 971�979

40

problem structure is of the nonconvex linear programming

problem even when the constraints given for both levels are

also linear. Several algorithms have been developed for

computing Stackelberg solutions, but it has been shown that

this intrinsically difficult problem is NP-hard [10]. Because

of this computational difficulty, researchers look for analy-

sis concepts that are computationally simpler but reflect the

two-level programming problem structure without neces-

sarily being of the Stackelberg game type.

With such a goal, we have pursued and improved

Shih, Lai, and Lee�s approach [12] to two-level program-

ming problems by attempting an extension to 0-1 program-

ming problems with fuzzy parameters. Namely, in this

paper we formalize the problem according to the following

basic principles. The upper-level decision maker considers

the balance of the achievements for objectives in both levels

and the lower-level decision maker optimizes his own ob-

jective while considering the preference of the upper-level

decision maker.

In general, the upper-level decision maker�s solution

for minimization of his own objective functions under

common constraints is not identical to the lower-level de-

cision maker�s solution for minimization of his own objec-

tive functions under the same common constraints.

Therefore, the upper- and lower-level decision makers must

yield a solution by mutual compromise considering the

hierarchical relations of the problem and the respective

objectives.

Furthermore, we employ the approach of using fuzzy

numbers to represent the parameters of the optimization

problem [13, 14] in order to consider the ambiguity in the

judgment and evaluation of human experts when choosing

the parameters for the formulation of the optimization

problem.

In this paper, we carry out the formulation with fuzzy

parameters for a two-level linear programming problem

with 0-1 decision variables, and we consider a solution

procedure for situations where decision makers choose

strategies cooperatively. Based on Lai and colleagues� tech-

niques, we propose an interactive fuzzy programming

method introducing genetic algorithms to two-level 0-1

programming problems with 0-1 fuzzy parameters. In order

to deal with the difficult issues of Lai�s method, we carry

out a problem formulation where the rules for the member-

ship functions for the decision variables of the upper level

have been eliminated.

2. Interactive Fuzzy Programming Method

for Two-Level 0-1 Programming Problems

We consider in this paper the two-level 0-1 program-

ming problem under cooperative relations, in which the

upper-level decision maker considers the minimization of

his own objective functions together with an achievement

balance of the objectives between levels, and the lower-

level decision maker minimizes his own objective function

while respecting the upper-level decision maker�s inten-

tions. This kind of 0-1 programming problem can be for-

malized as follows:

where xj, j 1, 2 are nj-dimensional vectors of 0-1 vari-

ables, ci,j, i, j 1, 2 are nj-dimensional constant row vec-

tors, b is an m-dimensional constant column vector, and the

Aj are constant m u nj matrices. Hereafter, for the sake of

simplicity, we set x �x1, x2�T � {0, 1}n

1�n

2 and denote as X

the feasible region of problem (1). The superscript T de-

notes transposition. For the two-level programming prob-

lem (1), let z1�x1, x2� and z2�x1, x2� be the objective functions

for minimization in the upper and lower levels, respectively,

and x1 and x2 the 0-1 decision variables of the upper and

lower levels, respectively.

For several of the parameters in the two-level pro-

gramming problem formulated in this manner, we prefer

fuzzy numbers of the type describable as �numbers more

or less, or approximately equal to m� in order to represent

properly the human judgment of experts participating in the

problem formulation, rather than just setting up directly

some traditional heuristic or some subjective method. We

expect in this way to be able to represent more conveniently

the actual decision situations. From this point of view,

assuming that any vagueness of the parameters involved in

the objective functions and the constraints of the two-level

programming problem is an inherent property [13, 14], the

two-level programming problem with fuzzy parameters

that we work with in this paper can be formulated as

follows:

where c~ �c~ij�, b~ �b

~k�, and A

~ �A

~j� i = 1,2, j = 1, 2, k = 1,

. . . , m represent fuzzy parameters. The ambiguity of the

human judgment of the experts involved in the problem

formulation is thus assumed to be a characteristic inherent

to all of the fuzzy numbers of c~, b~, and A

~.

(1)

(2)

41

Next, let us introduce the D-level set �c~, b~

, A~�D as the

set fuzzy number triplets from the fuzzy parameters c~, b~,

and A~

whose membership values are equal or larger than D,

that is,

where Pc~ij,r���, Pb

~k���, and Pd

~j,ki��� are membership functions

of the respective fuzzy numbers.

We may now assume that the decision maker in the

upper level considers a solution correct if the degree of

membership for all membership functions prescribing the

vector of fuzzy numbers involved in the object functions

and constraints of the two-level programming problem are

equal to or greater than some value D. In this case, the

two-level 0-1 programming problem can be formulated as

a conventional two-level 0-1 programming problem de-

pendent on the coefficients �c, b, A� � �c~, b~, A~�D as follows

[13, 14]:

There are an infinite number of such two-level program-

ming problems that may depend on the coefficients

�c, b, A� � �c~, b~, A

~�D. The above problem statement means

that the coefficients (c, b, A) in the D-level set �c~, b~, A

~�D are

arbitrary as long as all of the membership degrees for the

membership functions of the fuzzy number vectors are

equal to or greater than D. Let us look, however, at the

problem from the decision makers� positions: if the selec-

tion of these coefficients (c, b, A) � �c~, b~

, A~�D could be done

arbitrarily, it would be in particular very desirable to set

them so as to minimize the objective functions satisfying

the constraints of the two-level programming problem.

From this point of view, among the coefficients (c, b, A) �

�c~, b~, A

~�D for which all of the degrees of membership of the

membership functions of the fuzzy number vectors are

equal to or greater than D� that is, the coefficients (c, b,

A) with a degree of problem realizability equal to or greater

than D� we should select in particular the best set for both

decision makers. With that purpose in mind we introduce

the following nonfuzzy D-two-level programming prob-

lem:

where D is given by the upper-level decision maker. The

coefficients (c, b, A) in the D-two-level programming prob-

lem are not treated as known coefficients but rather as

variables.

Furthermore, considering that the judgment of hu-

man decision makers is somewhat ambiguous, it is natural

to think that the decision makers of both levels have vague

goals for their respective objective functions. As for the

minimization problem, we might consider that the decision

makers have fuzzy goals such as expecting the objective

function to be approximately equal to or less than a certain

value [13, 15].

For the sake of simplicity in this paper, the member-

ship functions fall within the range of the individual maxi-

mum and minimum values of each objective function, and

are employed in the following linear function connecting

two points such that the objective function values are zi0 and

zi1 and membership function values are 0 and 1, respec-

tively:

where the parameters zi0 and zi

1 are specified by the decisions

makers. These membership functions express the charac-

teristics of the fuzzy goals of the upper- and lower-level

decision makers.

The decision makers choose the parameters zi0 and

zi1 subjectively, but we can employ Zimmermann�s method

[16], that is, zi1 is set as the minimal value of its own

objective function, and then zi0 is selected so as to minimize

the objective function of the other decision maker. For D =

0, if �xio, ci1o , ci2

o � is the optimal solution for the minimization

problem of individual objective functions under the given

constraints, we compute the corresponding minimal values

zimin zi�x

io, ci1o , ci2

o � and zim zi�x

jo, cj1o , cj2

o � where j = 2

when i = 1, and j = 1 when i = 2. For the linear membership

functions of formula (6) we set zi1 zi

min and

zi0 zi

m, i 1, 2.

(6)

(3)

(4)

(5)

42

After determining the linear membership functions

for the fuzzy goals of each level decision maker, the upper-

level decision maker indicates the satisfaction level for his

own fuzzy goals, and then the lower-level decision maker

presents to the upper-level decision maker the solution that

optimizes his own fuzzy goal satisfaction degree while

keeping the satisfaction degree of the upper-level fuzzy

goals as stated by the upper-level decision maker. Taking

into consideration a balance between the satisfaction de-

grees, the upper-level decision maker then decides if the

proposed solution is adequate. If the balance between the

satisfaction degrees is not appropriate, the upper-level de-

cision maker updates his own levels of satisfaction degree.

The lower-level decision maker computes again a solution

that optimizes the degree of satisfaction of his fuzzy goals

taking into consideration the updated level of the degree of

satisfaction from the upper level and shows his solution to

the upper-level decision maker. By repeating this process,

the upper-level decision maker�s intentions are respected

and a solution is obtained which considers a balance in the

achievement of the satisfaction degrees between the levels.

In the final solution �x1, x2�, the upper-level decision

maker�s selection becomes x1, and the lower-level decision

maker�s choice is x2.

Let us now detail the solution process outlined above.

Let the level of satisfaction degree for the membership

functions of the upper-level decision maker be called mini-

mal acceptable degree, denoted by G^ � [0, 1]. This parame-

ter is subjectively chosen by the upper-level decision maker.

In connection with this selection, on the premise that the

upper-level decision maker�s membership function

P1�z1�x, c11, c12�� satisfies the condition of being equal or

larger than G^, the lower-level decision maker tries to maxi-

mize his own membership function under the given con-

straint. Thus, the problem that the lower-level decision

maker must solve becomes

In Lai�s formulation [11, 12], there are also constraints on

the fuzzy goals for the decision variables. In problem (7),

these constraints are eliminated.

Of course, if a solution exists for this problem, the

upper-level decision maker can obtain a satisfactory solu-

tion that complies with the minimal acceptable degree G~ set

up by himself. However, as the value chosen for the minimal

acceptable degree G~ becomes higher, the lower-level deci-

sion maker�s satisfaction degree appears to decrease, and

the difference in the relative satisfaction degrees of the

upper- and lower-level decision makers becomes larger,

thereby worsening the balance of the relative satisfaction

degrees for the decision makers.

In order to consider also a relative balance of the

satisfaction degrees for both decision makers, it is advisable

to search a well-balanced satisfactory solution that properly

maximizes satisfaction degrees for the fuzzy goals of the

upper- and lower-level decision makers. From this stand-

point, we intend in this paper to determine a decision

maker�s satisfactory solution in which, while satisfying the

minimal acceptable degree G^ set up by the upper-level

decision maker, this decision maker also considers to com-

promise this acceptability value in order to maintain the

balance between P1�z1�x, c11, c12�� and P2�z2�x, c21, c22��.

To achieve this purpose, we define the overall degree

of satisfaction as

and consider the following problem instead of the lower-

level problem (7):

We start by solving the auxiliary problem relative to

problem (9):

By solving problem (10) we obtain a solution that

maximizes the overall satisfaction degree within the D-level

sets �c~, b~, A

~�D relative to an D value set by the upper-level

decision maker. Yet, even if all of the membership functions

Pi�zi�x, ci1, ci2��, i, 1, 2 in problem (10) were linear, the

problem is still unfortunately not a linear 0-1 programming

problem since the coefficient vectors (c, b, A) are the

decision variables. Fortunately enough, from the charac-

(7)

(8)

(9)

(10)

43

teristics of the D-level sets for the fuzzy numbers

c~11, c~

12, c~

21, c~

22, b~, A~1, and A

~2, we know that the feasible

regions of c11, c12, c21, c22, bi, A1, and A2 are limited by the

respective left and right extreme points of the correspond-

ing D-level set. Using these extreme points, the closed

intervals may be expressed as [c11L , c11

R ], [c12L , c12

R ],

[c21L , c21

R ], [c22L , c22

R ], [bjL, bj

R], [A1L, A1

R], and [A2L, A2

R], and the

optimal solution of problem (10) can be obtained by solving

the following problem:

If the optimal solution x* of the auxiliary problem

(10) obtained through problem (11) satisfies the condition

P1�z1�x , c11

L , c12L �� t G

^, then the upper-level decision maker

has arrived at a satisfactory solution. However, the solution

of problem (10) does not necessarily satisfy this condition,

and in such a case it is useful to have some information

relative to the ratio of satisfaction degrees between levels.

Such a ratio is defined similar to Lai�s ratio [11] as

Using this ratio ', if ' > 1, that is,

P2�z2�x , c21

L , c22L �� ! P1�z1�x

, c11L , c12

L ��, then the upper-

level decision maker updates his own minimal acceptable

degree G by increasing its value. As for the lower-level

decision maker, he solves problem (7) with respect to the

updated G value and his satisfaction degree is decreased;

hence, in this case the upper-level decision maker�s satis-

faction degree is increased and the lower-level decision

maker�s satisfaction degree is decreased. If, on the contrary,

' < 1, that is, P2�z2�x , c21

L , c22L �� � P1�z1�x

, c11L , c12

L ��, then

the upper-level decision maker�s satisfaction degree is re-

duced by diminishing the upper-level decision maker�s

minimal acceptable degree value, and the lower-level deci-

sion maker�s satisfaction degree is increased by solving

problem (7) for the updated G value. Of course, if ' = 1,

both decision maker�s satisfaction degrees are equal.

Now, let us define the following notations for the

l-iteration: lower-level decision maker�s satisfaction degree

P2�z2l �, upper-level decision maker�s satisfaction degree

P1�z1l �, ratio of satisfaction degree levels 'l P2�z2

l � /P1�z1l �,

overall degree of satisfaction Ol. With these notations, when

a solution is proposed to the upper-level decision maker, a

satisfactory solution is obtained if the solution fulfills the

following termination condition:

Termination condition for the two-level 0-1

programming problem:

The algorithm stops when the following two condi-

tions are satisfied simultaneously:

(1) For the minimal acceptable degree G set by the

upper-level decision maker, P1�z1l � t G

^.

(2) 'l falls in a range between upper and lower

bounds selected by the upper-level decision maker.

Termination condition (1) is an upper-level decision

maker�s self-imposed condition for his solution; termina-

tion condition (2) is the condition set by the upper-level

decision maker to achieve a balance between the lower-

level decision maker�s satisfaction degree and the upper-

level decision maker�s satisfaction degree.

When these termination conditions are not fulfilled,

the upper-level decision maker updates his minimal accept-

able degree G. Let us denote the updated value of G by Gc.

The updating of G is done as follows.

Upgrading of G^

(1) If termination condition (1) is not satisfied, the

upper-level decision maker reduces the value of G to a value

G^c.

(2) When 'l surpasses the upper bound in termina-

tion condition (2), Gc is the updated value of G that has been

increased by the upper-level decision maker. On the con-

trary, if 'l falls below the lower bound, G is reduced and

becomes Gc, thereby loosening the limitations for the lower-

level decision maker.

For such updated Gc, the lower-level decision maker

solves the following maximization problem:

Similar to the relationship between problems (10) and (11),

the optimal solution for problem (13) can be obtained by

solving the following problem:

(11)

(12)

(13)

(14)

44

The fuzzy programming for the two-level 0-1 pro-

gramming problem with fuzzy parameters which has been

explained is summarized as follows.

Step 1. The upper-level decision maker determines

the value of D-level, the membership functions of the fuzzy

goals, the minimal acceptable degree G, and the upper and

lower bounds of '.

Step 2. The lower level decision maker determines

the membership functions of the fuzzy goals.

Step 3. The lower-level decision maker solves the

auxiliary problem (10) by solving problem (11), and shows

the optimal solution obtained and the related values

x , �z1

l , z2l �, Ol, P1�z1

l �, P2�z2l �, and 'l to the upper-level deci-

sion maker.

Step 4. If the solution given to the upper-level de-

cision maker satisfies the termination conditions, and the

upper-level decision maker is satisfied by the proposed

solution, then a satisfactory solution is attained and the

process stops.

Step 5. The upper-level decision maker sets an up-

dated value G^c of G

^ according to the updating rules of G

^c.

Step 6. The lower-level decision maker solves

problem (13) by solving problem (14), and proposes the

solution to the upper-level decision maker. Go to step 4.

Here, if the constraints of the formulated two-level

0-1 programming problem are linear, and all of the coeffi-

cients are positive, it is possible to apply to this 0-1 pro-

gramming problem the genetic algorithms with double

strings proposed by Sakawa and colleagues [17�19].

3. Genetic Algorithms with Double Strings

3.1. Double string coding

The double string coding of Fig. 1, where the ele-

ments in the upper row are the variable indices s�i� and those

in the lower row the values of the variables xs�i�, has been

proposed for 0-1 programming problems with linear con-

straints Ax d b, with all of the components of A and b being

positive [17, 18].

If we decode the double string according to the fol-

lowing algorithm, we can generate feasible solutions only.

Here, let the length of the string be n, the position of a string

i, the index of the variable s�i�, the value of the variable

xs�i�, the corresponding column vector of coefficients of the

constraints as�i�, and the constraints 6i 1nas�i�xs�i� d b. Let us

denote by ps�i� the value decoded from xs�i�.

Step 1: Set i = 1, sum = 0

Step 2: If xs�i� 0, let ps�i� 0, i i � 1, and go to

step 4; if xs�i� 1, go to step 3.

Step 3: If sum + as�i� d b, take ps�i� 1, sum = sum

+ as�i�, i i � 1, and go to step 4; otherwise, set

ps�i� 0, i i � 1, and go to step 4.

Step 4: If i ! n, stop. Otherwise, go back to step 2.

In this algorithm, starting from the left-hand side of the

strings the elements for which the variable xs�i� is 1 are set

with 1 as long as the constraints are satisfied; the subsequent

elements are set with 0.

The problems to be actually solved in the proposed

fuzzy programming method are problems (11) and (14). For

problem (11), the constraints on the fuzzy goals can be

treated as the objective function min�P1, P2�. Furthermore,

if Zimmermann�s method to set up the functions is em-

ployed, we have 0 d Pi d 1, i 1, 2, and the condition

0 d O d 1 is satisfied. Therefore, if each element of A1L, A2

L,

and bR is positive, it is possible to generate a feasible

solution. The objective function value min�P1, P2� can be

used directly as the fitness.

Since problem (14) has a constraint on P1, feasible

solutions could not always be generated only by decoding

of the double strings. Hence, for the individuals not satis-

fying the constraints, we assign a penalty �1, and use the

following fitness function:

3.2. Genetic algorithms

In the proposed fuzzy programming method we util-

ize the following genetic operators: reproduction, cross-

over, and mutations and inversions.

Reproduction

In this work we use the elicit roulette wheel selection.

If the fitness of an individual in the past populations is larger

than that of every individual in the current population,

preserve this individual into the current generation. The

roulette wheel selection allocates offsprings using a roulette

wheel with slots given by the probabilities according to the

fitness values.

(15)

Fig. 1. Double string.

45

Crossover

Since in double string coding there are letters other

than 0 and 1 in the upper row of indices, when doing the

traditional one point and multiple point crossovers it is

possible to get the same index numbers of the variables

coming out in the upper row of indices. To avoid this

inconvenience, we use in this work the partially matched

crossover (PMX) [19]. The PMX for double strings is given

by the following sequence.

Step 1: X and Y denote two double strings. Let the

elements of index i in the strings X and Y be sX�i�, sY�i� and

let xsX�i�, xsY�i� be the corresponding values of the variables.

Select two crossover points at random and determine the

exchanging substrings.

Step 2: Use the operations (1), (2), and (3) below

and let Xc be the result of changing the values of the indices

and variables of X.

(1) Let h and k (> h) denote the first and last ends

of the changing substring and let i h.

(2) Determine j such that sY�i� sX�j� and exchange

the i-th column �sX�i�, xsX�i��T of X with the j-th column

�sX�j�, xsX�j��T of X and set i i � 1.

(3) If i ! k, stop. Otherwise go to (2).

Proceed similar ly to obtain Yc.

Step 3: Let X* be the str ing that results when sub-

stituting the transformed substring of Y into the respective

Xc. Determine similarly Y*. Having obtained X* and Y* by

crossover, stop.

Mutations and inversions

In individuals coded by single strings, mutations are

carried out by exchanging the elements of two arbitrary

positions in the string. However, in the representation of

individuals by double strings, decoding is realized giving

preference in a sequence from the left-hand elements, so it

becomes difficult to yield a better individual by just ex-

changing elements of the indices in the upper row of the

double strings. In order to deal with such a problem, it is

convenient to append the so-called inversion genetic opera-

tor that reverses the order in a substring of some length. The

inversion for double strings is expressed by the following

steps:

Step 1: Select two positions in the double string, k

and h (k > h), and in the upper row which is index section

in the double strings select the substring going from posi-

tion h to position k.

Step 2: Rearrange the substring between positions

h and k in reverse order.

Step 3: Replace the original substring in the upper

row of the double string with the reversed substring and

stop.

4. Numerical Example

Let us consider the following two-level programming

problem for the numerical example:

where x1 �x1, . . . , x10�T and x2 �x11, . . . , x20�

T. The co-

efficients cij, i, j 1, 2, A1 and A2 were generated using

random numbers in the interval (0, 100), while b is set up

by multiplying the sum of all elements in the rows of A1 and

A2 by 0.6. In addition, out of cij, i, j 1, 2, A1, A2, and b,

90% were made fuzzy parameters. The values obtained are

presented in Table 1. For these values, we set G^ 1.0 as the

initial minimal acceptable degree of the satisfaction degree

of the upper-level decision maker; the upper bound for '1

is 1.0 and the lower bound 0.6. Since all of the constraint

coefficients in this example are positive, we apply the

genetic algorithm with double strings shown in the previous

section. The parameters for this genetic algorithm are a

crossover probability of 0.7, mutation probability of 0.05,

100 individuals, 1000 generations. We carry out 10 trials.

The determination of the membership functions is

done with Zimmermann�s method [16]. The optimal solu-

tions of the individual minimization problems for the ob-

jective functions in each level and the minimal values of the

objective functions zimin zi�x

io, ci1o , ci2

o �, i 1, 2 are shown

in Table 2. We have z1m �449 and z2

m 81. Let us now take

Table 1. Coefficients of the two-level 0-1 programming

problem

46

the D-level degree of the coefficient vectors for this example

as D = 0.9. Problem (10) then becomes

The solution for this problem is shown in Table 3.

Here, the upper-level decision maker�s membership func-

tion obtained P1l = 0.634146 corresponds to a minimal

acceptable degree G^ = 1.0, so the termination condition is

not satisfied. Even if we update the upper-level minimal

acceptable degree to G^ 0.9, the condition on the upper

bound of 'l is not satisfied. Thus, there is another updating

to G^ = 0.8. In this case, the corresponding problem (13) is

as follows:

The solution obtained for this problem is shown in Table 4.

The upper-level decision maker�s membership value is P1l

= 0.809756, which is larger than the minimal acceptable

degree G^ = 0.8. Also, 'l = 0.610030, which is in the chosen

range of [0.6, 1.0]. Therefore, the solution obtained com-

plies with the termination conditions and is a satisfactory

solution. Thus, the algorithm stops.

5. Conclusions

In this paper, an interactive fuzzy programming

method using genetic algorithms has been proposed for

two-level 0-1 programming problems with fuzzy parame-

ters. According to the proposed technique, the decision

maker in each level establishes his fuzzy goals related to

the objective functions, using linear membership functions.

After that, the upper-level decision maker establishes, sub-

jectively, the minimal acceptable degree of the degree of

satisfaction for the membership functions and, simultane-

ously, considers the ratio of satisfaction degrees between

the levels; if necessary, the decision maker updates his

minimal acceptability degree interactively. In so doing, a

satisfactory solution is produced by taking into considera-

tion also the achievement balance of the overall satisfaction

degree, while respecting the upper-level decision maker�s

decision. The feasibility and validity of the proposed

method was demonstrated through a numerical example for

a two-level 0-1 programming problem with fuzzy parame-

ters. The algorithm proposed in this paper can be extended

to multilevel problems.

Future problems include further considerations about

the updating method for the minimal degree of accept-

ability, the determination of the ratio of satisfaction degrees

between levels, a method applying genetic algorithms with-

out using the penalty of Eq. (15) for problem (14), and

others. Current research is being done extending the pro-

posed method to the case where the objective functions of

each level are multiobjective, and this is the topic of another

report in these transactions.

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47

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AUTHORS

Masatoshi Sakawa received his B.E. and D.Eng. degrees in applied mathematics and physics from Kyoto University in

1970 and 1975. He then joined Kobe University as a research associate in the Department of Systems Engineering, and became

an associate professor in 1981. He joined Iwate University as a professor in 1987, and moved to the Department of Industrial

and Systems Engineering of Hiroshima University in 1990. His main research interests are decision making methods in

multiobjective and fuzzy systems. He is author of the Japanese books Optimization of Linear Systems, Optimization of Nonlinear

Systems, Fundamental Concepts of Fuzzy Theory and Applications, Fundamentals of Management Mathematical Systems, and

Soft Optimization and Genetic Algorithms, as well as Fuzzy Sets and Interactive Multiobjective Optimization (Plenum Press).

48

Ichiro Nishizaki received his B.E. and his M.Sc. degrees in systems engineering from Kobe University in 1982 and 1984.

He holds a D.Eng. degree. He worked for Nippon Steel Corporation as a systems engineer from 1984 to 1990, became a research

associate at the Institute of Economic Research, Kyoto University, from 1990 to 1993, an associate professor in the Faculty of

Business Administration and Informatics, Setsunan University, from 1993 to 1997. He is an associate professor in the

Department of Industrial and Systems Engineering, Faculty of Engineering, Hiroshima University. His research interests include

game theory in fuzzy and multiobjective environments, as well as decision making.

Masatoshi Hitaka received his B.E. degree in industrial and systems engineering from Hiroshima University in 1997,

and entered the graduate school there. His research interests include multilevel 0-1 programming methods using genetic

algorithms.

AUTHORS (continued) (from left to right)

49