interactive fuzzy stochastic two-level integer programming through fractile criterion optimization

$: Interactive fuzzy stochastic two-level integer programming through fractile criterion optimization$
ORI GIN AL PA PER

Interactive fuzzy stochastic two-level integerprogramming through fractile criterion optimization

Masatoshi Sakawa • Hideki Katagiri • Takeshi Matsui

Received: 10 September 2009 / Revised: 25 June 2010 / Accepted: 31 July 2010 /

Published online: 14 August 2010

� Springer-Verlag 2010

Abstract In this paper, we focus on stochastic two-level integer programming

problems with cooperative decision makers. Using the fractile criterion optimization

model in chance constrained programming, the formulated stochastic two-level

integer programming problems are transformed into deterministic ones. Taking into

account vagueness of judgments of the decision makers, we present an interactive

fuzzy programming method to derive a satisfactory solution through interactions

with the upper-level decision maker in consideration of the cooperative relation to

the lower-level decision maker. For solving transformed deterministic problems

efficiently, we also introduce genetic algorithms with double strings for nonlinear

integer programming problems. An illustrative numerical example is provided to

demonstrate the feasibility and efficiency of the proposed method.

Keywords Two-level integer programming � Stochastic programming � Interactive

fuzzy programming � Chance constraints � Fractile criterion optimization � Genetic

algorithms � Double strings

1 Introduction

There exist many approaches for two-level programming problems depending on

situations which the decision makers (DMs) are placed in (Sakawa and Nishizaki

2009). Under the assumption that these DMs do not have motivation to cooperate

mutually, the Stackelberg solution (Shimizu et al. 1997) is adopted as a reasonable

solution for the situation. On the other hand, in the case of a project selection

problem in the administrative office of a company and its autonomous divisions, the

M. Sakawa (&) � H. Katagiri � T. Matsui

Faculty of Engineering, Hiroshima University,

1-4-1, Kagamiyama, Higashi-Hiroshima 739-8527, Japan

e-mail: [email protected]

123

Oper Res Int J (2012) 12:209–227

DOI 10.1007/s12351-010-0085-z

situation that these DMs can cooperate with each other seems natural rather than the

noncooperative situation.

Lai (1996) and Shih et al. (1996) proposed solution concepts for two-level linear

programming problems or multi-level ones such that decisions of DMs in all levels

are sequential and all of the DMs essentially cooperate with each other. In their

methods, the DMs identify membership functions of fuzzy goals for their objective

functions. In addition, the DM at the upper level also specifies 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. Although

their methods are interesting from the viewpoint of the introduction of fuzzy

concepts for the easy derivation of satisfactory solutions for DMs, there

unfortunately exists a possibility that their methods may lead undesirable final

solution because of the inconsistency between fuzzy goals of objective functions

and those of decision variables. In order to overcome the problem in their methods,

by eliminating the fuzzy goals for the decision variables, Sakawa et al. have

proposed interactive fuzzy programming for two-level or multi-level linear

programming problems to obtain a satisfactory solution for DMs (Sakawa et al.

1998, 2000). The subsequent works on two-level or multi-level programming have

been developing (Lee 2001; Sakawa and Nishizaki 2002a, b, 2009; Sakawa et al.

2001, 2002; Sinha 2003; Pramanik and Roy 2007; Abo-Sinna and Baky 2007;

Roghanian et al. 2007).

In actual decision making situations, however, we must often make a decision on

the basis of vague information or uncertain data. For such decision making

problems involving uncertainty, there exist two typical approaches: probability-

theoretic approach and fuzzy-theoretic one. Stochastic programming, as an

optimization method based on the probability theory, have been developing in

various ways (Stancu-Minasian 1990), including two stage problems considered by

Dantzig (1955) and chance constrained programming proposed by Charnes et al.

(1959). Fuzzy mathematical programming representing the vagueness in decision

making situations by fuzzy concepts have been studied by many researchers

(Rommelfanger 1996; Sakawa 1993). Fuzzy multiobjective linear programming,

first proposed by Zimmermann (1978), have been also developed by numerous

researchers, and an increasing number of successful applications has been appea-

ring (Sakawa et al. 1987; Luhandula 1987; Zimmermann 1987; Slowinski and

Teghem 1990; Lai and Hwang 1992; Sakawa 1993, 2000, 2001; Slowinski 1998).

In particular, after reformulating stochastic multiobjective linear programming

problems using several models for chance constrained programming, Sakawa et al.

(2002, 2003, 2004) presented an interactive fuzzy satisficing method to derive a

satisficing solution for the DM as a generalization of their previous results (Sakawa

and Yano 1985, 1989, 1990; Sakawa et al. 1987; akawa 1993; Sakawa and Kato

2008).

Genetic algorithms (GA) (Holland 1975), initiated by Holland, his colleagues and

his students at the University of Michigan in the 1970s, as stochastic search

techniques based on the mechanism of natural selection and natural genetics, have

received a great deal of attention regarding their potential as optimization

techniques for solving discrete optimization problems or other hard optimization

210 M. Sakawa et al.

123

problems. Although genetic algorithms were not much known at the beginning, after

the publication of Goldberg’s book (Goldberg 1989), genetic algorithms have

recently attracted considerable attention in a number of fields as a methodology for

optimization, adaptation and learning. As we look at recent applications of genetic

algorithms to optimization problems, especially to various kinds of single-objective

discrete optimization problems and/or to other hard optimization problems, we can

see continuing advances (Michalewicz 1992; Gen and Cheng 1996; Back 1996;

Back et al. 1997; Deb 2001; Coello et al. 2002; Eiben and Smith 2003). Focusing on

multiobjective mathematical problems, Sakawa et al., have been advancing genetic

algorithms to derive satisficing solutions to multiobjective problems (Sakawa 2000,

2001).

Under these circumstances, in this paper, we deal with two-level integer

programming problems with random variable coefficients in both objective

functions and constraints. The main contribution of this paper is to provide a novel

decision making methodolgy including a new model, solution concept and solution

algorithm to deal with more realistic problems in the real world, by simultaneously

considering various concepts such as hierarchy structure, fuzziness, randomness,

integer decision variables and interactive fuzzy programming, while most of

previous papers dealt with either of the concepts or a part of them.

Following the concept of the chance constrained programming, the stochastic

two-level linear programming problems are transformed into deterministic nonlin-

ear integer programming ones through the fractile criterion optimization model or

Kataoka’s model (Kataoka 1963). By considering the fuzziness of human

judgments, we present interactive fuzzy programming for deriving a satisfactory

solution for the decision makers. In the proposed interactive method, after

determining the fuzzy goals of the decision makers at both levels, a satisfactory

solution is derived efficiently by updating the satisfactory degree of the decision

maker at the upper level with considerations of overall satisfactory balance among

all the levels. For solving transformed deterministic problems efficiently, we also

propose genetic algorithms with double strings for nonlinear integer programming

problems.

2 Interactive fuzzy stochastic two-level integer programming

Consider two-level integer programming problems with random variable coeffi-

cients formulated as:

minimizefor DM1

z1ðx1; x2;xÞ ¼ c11ðxÞx1 þ c12ðxÞx2

minimizefor DM2


subject to A1x1 þ A2x2� bðxÞx1j1 2 f0; 1; . . .; m1j1g; j1 ¼ 1; 2; . . .; n1

x2j2 2 f0; 1; . . .; m2j2g; j2 ¼ 1; 2; . . .; n2

9>>>>>=

>>>>>;

ð1Þ

where x1 is an n1 dimensional integer decision variable column vector for the

decision maker at the upper level (DM1), x2 is an n2 dimensional integer decision

Interactive fuzzy stochastic two-level integer programming 211

123

variable column vector for the decision maker at the lower level (DM2), Aj, j = 1, 2

are m 9 nj coefficient matrices, vlji , l = 1, 2, jl = 1, 2,…, nl are positive integer

values, cljðxÞ, l = 1, 2, j = 1, 2 are nj dimensional Gaussian random variable

row vectors with mean vectors �clj and covariance matrices Vlpq, p = 1, 2, q = 1, 2,

and they are independent of each other, and bðxÞ is a random variable vector

whose joint distribution function is F(�). It should be noted here that the two

objective functions z1 and z2 are those of DM1 and DM2, respectively, and

‘‘minimizefor DM1

’’ and ‘‘minimizefor DM2

’’ mean that DM1 and DM2 are minimizers for their

objective functions.

Since (1) contains random variable coefficients, solution methods for ordinary

mathematical programming problems cannot be applied directly. Consequently, we

first deal with the constraints in (1) as chance constraints (Charnes and Cooper

1959) which mean that the constraints need to be satisfied with a certain probability

(satisficing level) and over. Namely, replacing constraints in (1) by chance

constraints with a satisficing level b, the problem can be transformed as:

minimizefor DM1


minimizefor DM2


subject to PrfA1x1 þ A2x2� bðxÞg� bx1j1 2 f0; 1; . . .; m1j1g; j1 ¼ 1; 2; . . .; n1

x2j2 2 f0; 1; . . .; m2j2g; j2 ¼ 1; 2; . . .; n2

9>>>>>=

>>>>>;

: ð2Þ

The first constraint in (2) is rewritten as:

PrfA1x1 þ A2x2� bðxÞg� b

, Fðða11x1 þ a12x2Þ; . . .; ðam1x1 þ am2x2ÞÞ� bð3Þ

where F(�) is the joint distribution function for bðxÞ and aij is the i th row vector of

Aj, j = 1, 2.

Therefore, (2) can be rewritten as:

minimizefor DM1


minimizefor DM2


subject to Fðða11x1 þ a12x2Þ; . . .; ðam1x1 þ am2x2ÞÞ� bx1j1 2 f0; 1; . . .; m1j1g; j1 ¼ 1; 2; . . .; n1

x2j2 2 f0; 1; . . .; m2j2g; j2 ¼ 1; 2; . . .; n2

9>>>>>=

>>>>>;

: ð4Þ

In the following, for notational convenience, the feasible region of (4) is denoted

by X.

In addition to the chance constraints, it is now appropriate to consider the fractile

criterion optimization model (Kataoka 1963) for the objective functions of (4).

By adopting the model, permissible levels hl, l = 1, 2 such that the probability

which each objective function is better than hl is greater than or equal to some given

threshold hl under the chance constraints are substituted for the original objective

functions zlðx1; x2;xÞ ¼ cl1ðxÞx1 þ cl2ðxÞx2; l = 1, 2 in (4). As a result, the

problem (4) can be transformed as:


123

minimizefor DM1

h1

minimizefor DM2

h2

subject to Pr c11ðxÞx1 þ c12ðxÞx2� h1½ � � h1

Pr c21ðxÞx1 þ c22ðxÞx2� h2½ � � h2

ðxT1 ; x

T2 Þ

T 2 X

9>>>>>=

>>>>>;

: ð5Þ

In (5), the first two constraints can be converted as:

Pr cl1ðxÞx1 þ cl2ðxÞx2� hlf g� hl

, Prcl1ðxÞx1 þ cl2ðxÞx2 � ð�cl1x1 þ �cl2x2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq � hl � ð�cl1x1 þ �cl2x2Þ


ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq

8><

>:

9>=

>;� hl

,Uhl � ð�cl1x1 þ �cl2x2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq

0

B@

1

CA� hl

where Vl is defined as:

Vl ¼Vl11 Vl12

Vl21 Vl22

� �

:

In the above inequality, U(�) is the probability distribution function of the

standard Gaussian distribution. From the monotonicity of the distribution function,

we can define the inverse function U-1(�) of U(�). Then, the above inequalities are

expressed as:

Uhl � ð�cl1x1 þ �cl2x2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq

0

B@

1

CA� hl

, hl � ð�cl1x1 þ �cl2x2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq �U�1ðhlÞ

, hl�ð�cl1x1 þ �cl2x2Þ þ U�1ðhlÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq

:

Letting Kthetal= U-1(hl) and noting that the equality

hl ¼ ð�cl1x1 þ �cl2x2Þ þ Khl


ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq

holds at the minimum of hl, problem (5) is equivalent to the following problem:

minimizefor DM1

f1ðx1; x2Þ ¼ ð�c11x1 þ �c12x2Þ þ Kh1


ðxT1 ; x

T2 ÞV1ðxT

1 ; xT2 Þ

Tq

minimizefor DM2

f2ðx1; x2Þ ¼ ð�c21x1 þ �c22x2Þ þ Kh2


ðxT1 ; x

T2 ÞV2ðxT

1 ; xT2 Þ

Tq

subject to ðxT1 ; x

T2 Þ

T 2 X

9>>>=

>>>;

: ð6Þ


123

In general, it seems natural that the decision makers have fuzzy goals for their

objective functions when they take fuzziness of human judgments into consider-

ation. For each of the objective functions flðx1; x2Þ, l = 1, 2 in (6), assume that the

decision makers have fuzzy goals such as ‘‘flðx1; x2Þ should be substantially less

than or equal to some specific value.’’ Then, (6) can be rewritten as:

maximizeforDM1

l1ðf1ðx1; x2ÞÞmaximize

forDM2l2ðf2ðx1; x2ÞÞ


T2 Þ

T 2 X

9>=

>;ð7Þ

where ll(�) is a membership function to quantify a fuzzy goal for the lth objective

function in (6) and it is assumed to be monotonically decreasing.

Although the membership function does not always need to be linear, for the sake

of simplicity, we adopt a linear membership function. To be more specific, if the

DM feels that flðx1; x2Þ should be less than or equal to at least fl,0 and

flðx1; x2Þ� fl;1ð\fl;0Þ is satisfactory, the linear membership function llðflðx1; x2ÞÞis defined as:

llðflðx1; x2ÞÞ ¼1; flðx1; x2Þ\fl;1flðx1;x2Þ�fl;0

fl;1�fl;0; fl;1� flðx1; x2Þ� fl;0

0; flðx1; x2Þ[ fl;0

8><

>:ð8Þ

and it is depicted in Fig. 1.

Zimmermann (1978) suggests a method for assessing the parameters of the

membership function. In his method, parameters fl,1, l = 1, 2 are determined as

f1;1 ¼ f1;min ¼ f1ðx11;min; x

12;minÞ ¼ min

ðxT1;xT

2ÞT2X

f1ðx1; x2Þ

f2;1 ¼ f2;min ¼ f2ðx21;min; x

22;minÞ ¼ min

ðxT1;xT

2ÞT2X

f2ðx1; x2Þ

and parameters fl,0, l = 1, 2 are specified as:

f1;0 ¼ f1ðx21;min; x

22;minÞ; f2;0 ¼ f2ðx1

1;min; x12;minÞ

where ðxl1;min; x

l2;minÞ is an optimal solution to the following problem:

Fig. 1 Linear membershipfunction


123

minimize flðx1; x2Þ ¼ ð�cl1x1 þ �cl2x2Þ þ Khl


ðxT1 ; x

T2 ÞVlðxT

1 ; xT2 Þ

Tq


T2 Þ

T 2 X

)

: ð9Þ

Then, by setting the parameters as described above, the linear membership

functions (8) is identified.

To derive an overall satisfactory solution to the membership function maximi-

zation problem (7), we first find the maximizing decision of the fuzzy decision

proposed by Bellman and Zadeh (1970). Namely, the following problem is solved

for obtaining a solution which maximizes the smaller degree of satisfaction between

those of the two decision makers:

maximize minfl1ðf1ðx1; x2ÞÞ; l2ðf2ðx1; x2ÞÞgsubject to ðxT

1 ; xT2 Þ

T 2 X

�

: ð10Þ

Solving problem (10), we can obtain a solution which maximizes the smaller

satisfactory degree between those of both decision makers.

If DM1 is satisfied with the optimal solution ðx�1; x�2Þ to (10), it follows that the

optimal solution ðx�1; x�2Þ becomes a satisfactory solution; however, DM1 is not

always satisfied with the solution ðx�1; x�2Þ. It is quite natural to assume that DM1

specifies the minimal satisfactory level d̂ 2 ð0; 1Þ for the membership function

l1ðf1ðx1; x2ÞÞ subjectively.

Consequently, if DM1 is not satisfied with the solution ðx�1; x�2Þ to problem (10),

the following problem is formulated:

maximize l2ðf2ðx1; x2ÞÞsubject to l1ðf1ðx1; x2ÞÞ� d̂

ðxT1 ; x

T2 Þ

T 2 X

9=

;ð11Þ

where DM2’s membership function l2ðf2ðx1; x2ÞÞ is maximized under the condition

that DM1’s membership function l1ðf1ðx1; x2ÞÞ is larger than or equal to the

minimal satisfactory level d̂ specified by DM1.

If there exists an optimal solution ðx�1; x�2Þ to problem (11), it follows that DM1

obtains a satisfactory solution having a satisfactory degree larger than or equal to

the minimal satisfactory level specified by DM1’s self. However, the larger the

minimal satisfactory level d̂ is assessed, the smaller the DM2’s satisfactory degree

becomes when the membership functions of DM1 and DM2 conflict with each

other. Consequently, a relative difference between the satisfactory degrees of DM1

and DM2 becomes larger, and it follows that the overall satisfactory balance

between both decision makers is not appropriate.

In order to take account of the overall satisfactory balance between both decision

makers, DM1 needs to compromise with DM2 on DM1’s own minimal satisfactory

level. To do so, the following ratio of the satisfactory degree of DM2 to that of DM1

is helpful:

D ¼ l2ðf2ðx1; x2ÞÞl1ðf1ðx1; x2ÞÞ

which is originally introduced by Lai (1996).


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DM1 is guaranteed to have a satisfactory degree larger than or equal to the

minimal satisfactory level for the fuzzy goal because the corresponding constraint is

involved in problem (11). To take into account the overall satisfactory balance

between both decision makers, DM1 specifies the lower bound Dmin and the upper

bound Dmax of the ratio D, and D is evaluated by verifying whether or not it is in the

interval [Dmin, Dmax]. The condition that the overall satisfactory balance is

appropriate is represented by

D 2 ½Dmin;Dmax�:At the iteration k, let ðxk

1; xk2Þ, fk

l ¼ flðxk1; x

k2Þ, llðfk

l Þ and Dk ¼ l2ðfk2Þ=l1ðfk

1Þdenote the current solution, DM1’s objective function value, DM1’s satisfactory

degree and the ratio of satisfactory degrees of the two decision makers, respectively.

The interactive process terminates if the following two conditions are satisfied and

DM1 concludes the solution as an overall satisfactory solution.

2.1 Termination conditions of the interactive process

Condition 1 DM1’s satisfactory degree is larger than or equal to the minimal

satisfactory level d̂ specified by DM1’s self, i.e., l1ðfk1Þ� d̂.

Condition 2 The ratio Dk of satisfactory degrees lies in the closed interval

between the lower and the upper bounds specified by DM1, i.e., Dk 2 ½Dmin;Dmax�.Condition 1 ensures the minimal satisfaction to DM1 in the sense of the

attainment of the fuzzy goal, and condition 2 is provided in order to keep overall

satisfactory balance between both decision makers. If these two conditions are not

satisfied simultaneously, DM1 needs to update the minimal satisfactory level d̂.

The updating procedures are summarized as follows.

2.2 Procedure for updating the minimal satisfactory level d̂

Case 1 If condition 1 is not satisfied, then DM1 decreases the minimal

satisfactory level d̂.

Case 2 If the ratio Dk exceeds its upper bound, then DM1 increases the minimal

satisfactory level d̂. Conversely, if the ratio Dk is below its lower bound, then

DM1 decreases the minimal satisfactory level d̂.

Case 3 Although conditions 1 and 2 are satisfied, if DM1 is not satisfied with the

obtained solution and judges that it is desirable to increase the satisfactory degree

of DM1 at the expense of the satisfactory degree of DM2, then DM1 increases the

minimal satisfactory level d̂. Conversely, if DM1 judges that it is desirable to

increase the satisfactory degree of DM2 at the expense of the satisfactory degree

of DM1, then DM1 decreases the minimal satisfactory level d̂.

In particular, if condition 1 is not satisfied, there does not exist any feasible solution

for problem (11), and therefore DM1 has to moderate the minimal satisfactory level.

Now we are ready to propose interactive fuzzy programming for deriving a

satisfactory solution by updating the satisfactory degree of the decision maker at the

upper level with considerations of overall satisfactory balance among all the levels.


123

2.3 Computational procedure of interactive fuzzy programming

Step 1: Ask the decision maker at the upper level, DM1, to subjectively determine

a satisficing level b [ (0, 1) for constraints in (2) and probability levels hl, l = 1, 2.

Go to Step 2.

Step 2: Solve (9) and calculate fl,min via optimal solutions ðxl1;min, xl

2;minÞ, l = 1, 2

to (9). Then, identify the linear membership function llðflðx1; x2ÞÞ of the fuzzy

goal for the corresponding objective function using the Zimmermann method

(Zimmermann 1978). Go to Step 3.

Step 3: Set k := 1. Solve the maximin problem (10) for obtaining a solution which

maximizes the smaller degree of satisfaction between those of the two decision

makers and calculate fkl ¼ flðxk

1; xk2Þ, llðfk

l Þ, l = 1, 2 and Dk ¼ l2ðfk2Þ=l1ðfk

1Þ via

the optimal solution ðxk1; x

k2Þ to (10). If DM1 is satisfied with the optimal solution

to (10), the optimal solution becomes a satisfactory solution and this interaction

procedure is terminated. Otherwise, ask DM1 to subjectively set the minimal

satisfactory level d̂ 2 ð0; 1Þ for the membership function l1ðf1ðx1; x2ÞÞ. Further-

more, ask DM1 to set the upper bound Dmax and the lower bound Dmin for D. Go

to Step 4.

Step 4: Set k := k ? 1. Solve problem (11) for finding a solution to maximize

DM2’s membership function l2ðf2ðx1; x2ÞÞ under the condition that DM1’s

membership function l1ðf1ðx1; x2ÞÞ is larger than or equal to the minimal

satisfactory level d̂ and calculate fkl ¼ flðxk

1; xk2Þ, llðfk

l Þ, l = 1, 2 and Dk ¼l2ðfk

2Þ=l1ðfk1Þ via the optimal solution ðxk

1; xk2Þ to (11). Go to Step 5.

Step 5: If the current solution ðxk1; x

k2Þ satisfies the termination conditions and

DM1 accepts it, then the procedure stops and the current solution is determined to

be a satisfactory solution. Otherwise, ask DM1 to update the minimal satisfactory

level d̂, and return to Step 4.

In the proposed interactive fuzzy programming method, it is required to solve the

nonlinear integer programming problems (9), (10) and (11), which is apparently

difficult to solve compared to linear integer programming problems and 0–1

nonlinear programming problems. In order to solve such difficult problems, we

propose genetic algorithms designed for nonlinear integer programming problems,

called Genetic Algorithms with Double Strings for Nonlinear Integer Programming

(GADSNIP). We do not employ the state-of-the-art multiobjective evolutionary

algorithms such as NSGA-II and SPEA2 because they are mainly weighted toward

finding the Pareto optimal solution set so as to cover the whole set, whereas the

goal of this paper is to derive a compromise or satisficing solution for the decision

maker.

As one of the most promising methods for solving combinatorial optimization

problems, tabu search have been developed. From the point of view of our

experience, however, in most situations, the Genetic Algorithm with Double String

(GADS) is better than Tabu Search (TS) methods in the case of nonlinear integer

programming problems, whereas TS methods are probably superior to GA-based

methods in the case of 0–1 decision variables. The reason is related to the two


123

computational aspects in metaheuristics; one is the diversity of the solutions

generated from these methods, and the other is the computational cost for finding

the promising direction to obtain better solutions.

As for the first issue, it should be noted here that in general, the diversity of the

solutions explored in integer programming problems should be greater than that in

0–1 programming problems since the feasible region of a 0–1 programming

problem is definitely smaller than that of an integer programming problem if

the objective functions and constraints in both the problems are the same except for

the condition for the taken values of decision variables. Since the diversity of the

solutions generated by GA-based methods is generally greater than that by TS-based

methods, if the range of the integer values taken by decision variables is wider, the

performance of GA-based methods has a tendency to be better than that of TS-based

methods.

Moreover, when using TS-based methods for solving nonlinear problems, the

computational cost for finding the best searching direction clearly becomes far more

expensive compared to the linear cases because the differences of the objective

function values between the adjacent solutions (or moves) are very easily calculated

in the linear cases. Therefore, as the nonlinearity of the problems to be solved

becomes stronger, the degree of superiority of TS-based methods over GA-based

methods decreases.

Considering that the goal of this paper is to solve the problems where the

objective functions are nonlinear and the decision variables are integer, we employ a

GA-based method to derive a satisficing solution for the decision maker.

3 Genetic algorithm with double strings for nonlinear integer programming

As discussed above, in this section, for solving the transformed deterministic

nonlinear integer programming problems efficiently, we propose Genetic Algo-

rithms with Double Strings for Nonlinear Integer Programming (GADSNIP). As an

efficient approximate solution method, GADSNIP are designed for nonlinear integer

programming problems formulated as:

minimize f ðxÞsubject to giðxÞ� 0; i ¼ 1; 2; . . .;m

xj 2 f0; 1; . . .; mjg; j ¼ 1; 2; . . .; n

9=

;ð12Þ

where x is an n dimensional integer decision variable column vector. Furthermore,

f(�) and gi(�), i = 1, 2,…, m may be nonlinear. If mj = 1 for all j [ {1, 2,…, n},

(12) is called a nonlinear 0–1 programming problem.

Quite similar to GADS (Sakawa 2001), an individual is represented by a double

string shown in Table 1. In Table 1, for a certain j, s(j) [ {1, 2,…, n} represents an

Table 1 Double strings(1) s(2) … s(n)

ys(1) ys(2) … ys(n)


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index of a decision variable xs(j) in the solution space, while ys(j), j = 1, 2,…, n does

the integer value among {0, 1,…, mj} of the s(j) th decision variable xs(j).

In GADSNIP, as with GADS (Sakawa 2001), a feasible solution used in

decoding, called a reference solution, must be found before the execution of the

genetic algorithm. One possible way to obtain a feasible solution to (12), is to

maximize the exponential function for the violation of constraints defined by:

GðxÞ ¼ exp �hXm

i¼1

MgiðxÞ

�gi

� �" #

ð13Þ

where

MðnÞ ¼ n; n� 0

0; n\0

�

;

�gi’s are parameters for normalization and h is a positive constant. Namely, for

obtaining a feasible solution, solve an unconstrained maximization problem

maximize GðxÞsubject to xj 2 f0; 1; . . .; mjg; j ¼ 1; 2; . . .; n

�

ð14Þ

through GADSNIP (without using the decoding algorithm) by regarding GðxÞ as the

fitness function of an individual S. A solution x0 such as Gðx0Þ ¼ 1 to this problem

is regarded as a reference solution x� :¼ x0.

In the following, we construct the decoding algorithm for GADSNIP using a

reference solution x�, where N is the number of individuals.

3.1 Decoding algorithm using a reference solution

Step 1: Let j := 1, x :¼ 0, l := 0. Go to Step 2.

Step 2: Let xs(j) := ys(j). Go to Step 3.

Step 3: If giðxÞ� 0, i = 1, 2,…, m, let l := j, j := j ? 1 and go to Step 4.

Otherwise, let j := j ? 1 and go to Step 5.

Step 4: If j B n, go to step 2. Otherwise, go to Step 5.

Step 5: If l [ 0, go to Step 6. Otherwise, go to Step 7.

Step 6: For j such that 1 B j B l, let xs(j) := ys(j). Then, for j such that

(l ? 1) B j B n, let xs(j) := 0. Since the resulting x is a feasible solution, quit the

process.

Step 7: Let j := 1, x :¼ x�. Go to Step 8.

Step 8: Let xs(j) := ys(j). If ysðjÞ ¼ x�sðjÞ, let j := j ? 1 and go to Step 10. Otherwise,

if ysðjÞ 6¼ x�sðjÞ, go to Step 9.

Step 9: If giðxÞ� 0, i = 1, 2,…, m, let j := j ? 1 and go to Step 10. Otherwise,

let xsðjÞ :¼ x�sðjÞ, j := j ? 1 and go to Step 10.

Step 10: If j B n, return to Step 8. Otherwise, since the resulting x is a feasible

solution, quit the process.

It is significant to realize that the diversity of solutions x greatly depends on the

reference solution used in the above decoding algorithm. To overcome such


123

situations, we propose the following reference solution updating procedure such

that the current reference solution is updating by another feasible solution if the

diversity of solutions seems to be lost. To do so, for every generation, check

the dependence on the reference solution through the calculation of the mean of

the Hamming distance between all solutions decoded from individuals and the

reference solution are checked. If the dependence on the reference solution is

strong, the reference solution is replaced by the solution corresponding to an

individual having maximum Hamming distance. Let N, x�, g(\ 1.0) and xr

respectively denote the number of individuals, the reference solution, a parameter

for reference solution updating and a feasible solution decoded by the r th

individual, then the reference solution updating procedure can be described as

follows.

3.2 Reference solution updating procedure

Step 1: Set r := 1, rmax := 1, dmax := 0 and dsum := 0. Go to Step 2.

Step 2: Calculate dr ¼Pn

j¼1 jxrj � x�j j and let dsum :¼ dsum þ dr. If dr [ dmax and

f ðxrÞ\f ðx�Þ, let dmax := dr, rmax := r and r := r ? 1, and go to Step 3.

Otherwise, let r := r ? 1 and go to Step 3.

Step 3: If r [ n, go to Step 4. Otherwise, return to Step 2.

Step 4: If dsum=ðN �Pn

j¼1 mjÞ\g, then update the reference solution as x� :¼ xrmax ,

and stop. Otherwise, stop without updating the reference solution.

It should be observed here that when the constraints of the problem are strict, there

exist a possibility that all of the individuals are decoded in the neighborhood of the

reference solution. To avoid such a possibility, in addition to the reference solution

updating procedure, after every P generations, the reference solution is replaced by

another feasible solution.

In the proposed method, we determine values of individuals of the initial

population by using the property that the optimal solution to an integer

programming problem is close to that to the corresponding continuous relaxation

problem. To be more specific, when we determine initial values of the lower string

of each individual, ys(j), j = 1, 2,…, n, we use Gaussian random numbers with mean

x̂j and variance r2. Here x̂j is the j th element of the optimal solution to the following

continuous relaxation problem of (12).

minimize f ðxÞsubject to giðxÞ� 0; i ¼ 1; 2; . . .;m

0� xj� mj; j ¼ 1; 2; . . .; n

9=

;ð15Þ

If (15) is a convex programming problem, we can obtain an optimal solution by an

existing convex programming technique. Otherwise, we find an approximate optimal

solution by using some nonconvex programming technique, e.g., GENOCOP V

(Kozieł and Michalewicz 1999), a genetic algorithm for general nonlinear program-

ming problems.


123

3.3 Generation of initial population by using continuous relaxation

Step 1: Let r := 1. Go to Step 2.

Step 2: Generate a uniform random number rand() [ [0,1). If it is less than or

equal to the prefixed parameter R, go to Step 3. Otherwise, go to Step 7.

Step 3: Let j := 1. Go to Step 4.

Step 4: Select a number from among {1, 2,…, n} randomly, and let it s(j). Note

that s(j) must not be equal to s(j0), j0 = 1, 2,…, j - 1. Go to Step 5.

Step 5: For the current s(j), using the Gaussian integer random number with mean

x̂sðjÞ and variance r2, determine the value of ys(j) as ysðjÞ :¼ Gaussðx̂sðjÞ; r2Þ. Let

j := j ? 1 and go to Step 6.

Step 6: If j [ n, let r := r ? 1 and go to Step 11. Otherwise, return to Step 4.

Step 7: Let j := 1. Go to Step 8.

Step 8: Select a number from among {1, 2,…, n} randomly, and let it s(j). Note

that s(j) must not be equal to s(j0), j0 = 1, 2,…, j - 1. Go to Step 9.

Step 9: For the current s(j), determine the value of ys(j) by using the uniform

integer random number on the set {0, 1,…, ms(j)}, and let j := j ? 1. Go to

Step 10.

Step 10: If j [ n, let r := r ? 1 and go to Step 11. Otherwise, return to Step 8.

Step 11: If r [ N, quit this process. Otherwise, return to Step 2.

As a reproduction operator, elitist expected value selection, which is the

combination of expected value selection and elitist preserving selection, is adopted.

In Sakawa et al. 2005, elitist expected value selection is defined as a combination of

elitism and expected value selection as mentioned below.

Elitism: If the fitness of the best individual Sbest (elite) in all past populations is

less than that of the best individual S� in the current population, preserve it as the

elite, i.e., Sbest :¼ S�. Otherwise, incorporate Sbest into the current population.

Expected value selection: For a population consisting of N individuals, the expected

value of the number of each individual Sr in the next population

Nr ¼f ðSrÞ

PNr¼1 f ðSrÞ

� N

is calculated. Then, the integral part of Nr means the deterministic number of Sr

preserved in the next population. While, the decimal part of Nr is regarded as the

probability that one Sr can survive, i.e., N �PN

r¼1 Nr individuals are determined on

the basis of this probability.

If either the single-point crossover or the multi-point crossover is directly applied

to individuals of double string type, the k th element of an offspring may take the

same number that the k0 th element takes. Similar violation occurs in solving

traveling salesman problems or scheduling problems through genetic algorithms as

well. In order to avoid this violation, a crossover method called partially matched

crossover (PMX) was proposed (Goldberg and Lingle 1985) PMX suitable for

double strings can be constructed as follows.


123

3.4 PMX for double string

Step 1: Select two individuals X, Y from the population as parent individuals and

prepare copies X0 and Y0 of X and Y, respectively. Go to Step 2.

Step 2: Choose two crossover points at random on these strings, say, h and

k(h \ k). Go to Step 3.

Step 3:

(a) Set j = h.

(b) Find j0 such that sX’(j0) = sY(j).

Then, interchange ðsX0 ðjÞ; ysX0 ðjÞÞT

with ðsX0 ðj0Þ; ysX0 ðj0ÞÞT

and

set j = j ? 1.

(c) If j [ k, stop. Otherwise, go to (b).

Go to Step 4.

Step 4: Replace the part from h to k of X0 with that of Y and let X0 be the offspring

of X. Go to step 5.

Step 5:

(a) Set j = h.

(b) Find j0 such that sY’(j0) = sX(j).

Then, interchange ðsY 0 ðjÞ; ysY 0 ðjÞÞT

with ðsY 0 ðj0Þ; ysY 0 ðj0ÞÞT

and

set j = j ? 1.

(c) If j [ k, stop. Otherwise, return to (b). Go to Step 6.

Step 6: Replace the part from h to k of Y0 with that of X and let Y0 be the offspring

of Y.

It is considered that mutation plays the role of local random search in genetic

algorithms. In this paper, two mutation operators (bit-reverse type and inversion) are

used. A direct extension of mutation for 0–1 programming problems is to change the

value of ys(j) at random in [0,ms(j)] uniformly, when mutation occurs at ys(j). The

mutation operator is further refined by using the information about the solution of

the continuous relaxation problem x̂. To be more explicit, the following algorithm is

carried out.

3.5 Mutation of bit reverse type for double strings

Step 1: Set r := 1.

Step 2: Set j := 1.

Step 3: Generate a uniform random number rand() [ [0,1). If it is less than or

equal to the given mutation rate pm, then go to step 4. Otherwise, go to step 7.

Step 4: Generate a uniform random number rand() [ [0,1) again. If it is less than

or equal to the prefixed parameter R, go to step 5. Otherwise, go to step 6.

Step 5: Using the Gaussian integer random number with mean x̂sðjÞ and variance

s2, determine the value of ys(j) as ysðjÞ :¼ Gaussðx̂sðjÞ; s2Þ and go to step 7.


123

Step 6: Determine the value of ys(j) by using the uniform integer random number

on the set {0, 1,…, ms(j)} and go to step 7.

Step 7: If j \ n, set j := j ? 1 and return to step 3 Otherwise, go to step 8.

Step 8: If r \ N, set r := r ? 1 and return to step 2. Otherwise, stop.

3.6 Inversion operator

Step 1: After determining the two inversion points h and k (h \ k), pick out a part

of the upper row of a double string from h to k.

Step 2: Arrange the substring in the reverse order.

Step 3: Put the arranged substring back in the double string.

From the results discussed thus far, it is now appropriate to present the following

computational procedures of GADSNIP.

3.7 Computational procedures of GADSNIP

Step 1: Determine values of the parameters used in the genetic algorithm: the

population size N, the generation gap G, the crossover rate pc, the mutation rate pm,

the inversion rate pi, the minimal search generation number Imin, the maximal

search generation number Imax([ Imin), the scaling constant cmult, the convergence

criterion e, the degree of use of information about solutions to linear programming

relaxation problems R, the parameter for reference solution updating g and the

penalty constant h. Set the generation counter t at 0. Go to Step 2.

Step 2: Generate the initial population consisting of N individuals based on the

information of the optimal solution to the continuous relaxation problem. Go to

Step 3.

Step 3: Decode each individual in the current population and calculate its fitness

based on the corresponding solution. Go to Step 4.

Step 4: If the termination condition is fulfilled, stop. Otherwise, let t := t ? 1 and

go to Step 5.

Step 5: Apply reproduction operator using elitist expected value selection after

linear scaling. Go to Step 6.

Step 6: Apply crossover operator, called PMX (Partially Matched Crossover) for

double string. Go to Step 7.

Step 7: Apply mutation based on the information of a solution to the continuous

relaxation problem. Go to Step 8.

Step 8: Apply inversion operator. return to Step 3.

4 Numerical example

To demonstrate the feasibility and efficiency of the proposed method, consider the

following two-level integer programming problem involving random variable

coefficients.


123

minimizefor DM1


minimizefor DM2


subject to a11x1 þ a12x2� b1ðxÞa21x1 þ a22x2� b2ðxÞ

..

.

a101x1 þ a102x2� b10ðxÞx1j1 2 f0; 1; . . .; 30g; j1 ¼ 1; . . .; 15

x2j2 2 f0; 1; . . .; 30g; j2 ¼ 1; . . .; 15

9>>>>>>>>>>>>=

>>>>>>>>>>>>;

ð16Þ

where x1 ¼ ðx11; . . .; x115ÞT , x2 ¼ ðx21; . . .; x215ÞT , bðxÞ is also a Gaussian random

vector whose mean is given as:

�b ¼ ð4776; 2790; 5082;�1116; 4944; 2160; 3390; 1608; 3408; 3540Þ

and clðxÞ ¼ ðcl1ðxÞ; cl2ðxÞÞ, l = 1, 2 are Gaussian random vectors whose mean are

given as Table 2. Upper bounds of all decision variables are equal to 30.

Parameters of GADSNIP are set as: population size N = 100, generation gap

G = 0.9, crossover rate pc = 0.9, mutation rate pm = 0.05, inversion rate

pi = 0.05, minimal search generation number Imin = 500, maximal search gener-

ation number Imax = 1000, scaling constant cmult = 1.6, convergence criterion

e = 0.01, degree of use of information about solutions to linear programming

relaxation problems R = 0.9, parameter for reference solution updating g = 0.2,

penalty constant h = 5.

Suppose that DM1 specifies a satisficing level b and probability levels hl, l = 1, 2

as b = 0.70, h1 = 0.75, h2 = 0.70. Using GADSNIP minimal values fl,min of

objective functions flðx1; x2Þ under the chance constraints corresponding to the

satisficing level b are calculated as f1;min ¼ f1ðx11;min; x

12;minÞ ¼ �1080:02,

f2;min ¼ f2ðx21;min; x

22;minÞ ¼ �508:68.

Suppose that the DMs employ the linear membership function (8) whose

parameters are determined by the Zimmermann method (Zimmermann 1978). Then,

parameter values fl,1 and fl,0, l = 1, 2 characterizing membership functions ll(�) are

becomes:

f1;1 ¼ �1080:02; f1;0 ¼ 628:54; f2;1 ¼ �508:68; f2;0 ¼ 1251:00:

The maximin problem is solved by GADSNIP and the obtained result is shown at

the column labeled ‘‘1st’’ in Table 3. Since DM1 is not satisfied with the current

solution, DM1 sets the minimal satisfactory level d̂ 2 ð0; 1Þ for l1ðf1ðx1; x2ÞÞ to

0.70 so that l1ðf1ðx1; x2ÞÞ will be improved from its current value. Furthermore, the

Table 2 Means of coefficients of objective functions

�c11 9 -1 10 -6 4 -6 3 -9 10 -5 7 -4 8 -9 3

�c12 -9 5 -8 2 -10 3 -2 4 -3 5 -2 9 -9 6 -6

�c21 10 2 -5 7 2 -3 -2 7 7 9 8 -5 6 7 9

�c22 4 2 -6 6 3 4 -2 9 -3 -10 2 8 -7 8 -9


123

upper bound and the lower bound of the ratio of satisfactory degrees D are set as

Dmax = 0.80 and Dmin = 0.70.

Problem (11) for d̂ ¼ 0:70 is solved using GADSNIP, and the obtained result of

the second iteration is shown at the column labeled ‘‘2nd’’ in Table 3.

Since the ratio of satisfactory degrees D2 exceeds Dmax = 0.80, the second

condition of termination of the interactive process is not fulfilled. Suppose that DM1

feels that l1 should be considerably better than l2, and DM1 updates the minimal

satisfactory level d̂ from 0.70 to 0.80 in order to improve l1. Consequently, problem

(11) for d̂ ¼ 0:80 is solved by using GADSNIP, and the obtained result is shown at

the column labeled ‘‘3rd’’ in Table 3.

Since the ratio of satisfactory degrees D3 is less than Dmin = 0.70, the second

condition of termination of the interactive process is not fulfilled. Hence, DM1

updates the minimal satisfactory level d̂ from 0.80 to 0.75 for improving l2 at the

sacrifice of l1. As a result, problem (11) for d̂ ¼ 0:75 is solved by GADSNIP, and

the obtained result is shown at the column labeled ‘‘4th’’ in Table 3.

In this example, at the fourth iteration, since the current solution satisfies all

termination conditions of the interactive process and DM1 is satisfied with the

current solution, the satisfactory solution is obtained.

5 Conclusions

In this paper, we focused on two-level integer programming problems involving

random variable coefficients. Using the fractile criterion optimization model in

chance constrained programming, the formulated stochastic two-level integer

programming problems were transformed into deterministic ones. Taking into

account vagueness of judgments of the decision makers, and the assumption that

the upper-level decision maker and the lower-level one in the problem are

mutually cooperative, interactive fuzzy programming has been presented. In the

proposed interactive method, after determining the fuzzy goals of the decision

makers at both levels, a satisfactory solution can be derived efficiently by

updating the satisfactory degree of the decision maker at the upper level with

considerations of overall satisfactory balance among all the levels. Genetic

algorithms designed for nonlinear integer programming problems, called Genetic

Table 3 Interaction process

Interaction 1st 2nd 3rd 4th

d̂ – 0.70 0.80 0.75

f1ðxk1; x

k2Þ -499.65 -571.51 -750.63 -653.35

f2ðxk1; x

k2Þ 85.22 149.97 318.24 255.69

l1ðf1ðxk1; x

k2ÞÞ 0.660313 0.702376 0.807210 0.750273

l2ðf2ðxk1; x

k2ÞÞ 0.662494 0.625696 0.530071 0.565616

Dk 1.003306 0.890828 0.656671 0.753880


123

Algorithm with Double Strings for Nonlinear Integer Programming (GADSNIP),

were also introduced for solving transformed deterministic ones efficiently.

An illustrative numerical example was provided to demonstrate the feasibility and

efficiency of the proposed method. Extensions to two-level integer programming

problems involving fuzzy random variable coefficients will be required in the near

future.

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