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Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 14, 657 – 669

On the Solution of a Special Type of Large Scale

Integer Linear Vector Optimization Problems

with Uncertain Data through TOPSIS Approach

Tarek Hanafi M. Abou-El-Enien

Department of Operations Research & Decision Support

Faculty of Computers & Information, Cairo University

5 Dr. Ahmed Zoweil St. - Orman –Postal Code 12613 - Giza - Egypt

[email protected]

Abstract

This paper focuses on the solution of a Large Scale Integer Linear Vector

Optimization Problemswith chance constraints (CHLSILVOP) of a special type

through the Technique for Order Preference by Similarity Ideal Solution (TOPSIS)

approach, where such problems has block angular structure of the constraints.

Chance constraints involve random parameters in the right hand sides. TOPSIS is

extended to solve CHLSILVOPwith constraints of block angular structure.

Compromise (TOPSIS) control minimizes the measure of distance, provided that

the closest solution should have the shortest distance from the positive ideal

solution (PIS) as well as the longest distance from the negative ideal solution

(NIS). As the measure of “closeness” dP-metric is used. Thus, I reduce a k-

dimensional objective space to a two-dimensional space by a first-order

compromise procedure. The concept of a membership function of fuzzy set theory

is used to represent the satisfaction level for both criteria. Moreover, I derive a

single objective largescale Integer programming problem using the max–min

operator for the second-order compromise operation. Also, An interactive

decision making algorithm for generating integer Pareto optimal (compromise)

solution for CHLSILVOP through TOPSIS approach is provided where the

decision maker (DM) is asked to specify the relative importance of

objectives.Finally, a numerical illustrative example is given to clarify the main

results developed in this paper.

Mathematics Subject Classification: 03E72, 90C06, 90C10, 90C15, 90C29

658 T. H. M. Abou-El-Enien

Keywords: TOPSIS;Interactive decision making; Large Scale Systems; Vector

Optimization Problems;Integer programming;Fuzzy set theory; Stochastic

programming; Block angular structure; Compromise (satisfactory solution);

Positive ideal solution; Negative ideal solution

(1) Introduction

Stochastic programming deals with a class of optimization models and algorithms

in which some or all of the data may be subject to significant uncertainty. In real

world decision situations, when formulating a Large Scale Vector Optimization

problem (LSVOP), some or all of the parameters of the optimization problem are

described by stochastic (or random or probabilistic) variables rather than by

deterministic quantities [20,22,23,24]. Also, various variables of the real system

may be constrained to take only integer values [13,25].

Most CHLSILVOParising in applications have special structures that can be

exploited. One familiar structure is the block angular structure to the constraints,

and several kinds of decomposition methods for linear and nonlinear

programming problems with block angular structure have been proposed[13,16,

22].An algorithm for parametric large scale integer linear multiple objective

decision- making problems with block angular structure has beenpresented in

[13].Recently, a significant number of studies have indeed been reported on single

and multiple objective linear and nonlinear programming problems [2, 3, 4, 5, 7, 8,

9, 13, 14, 15, 16, 17, 18, 20, 21, 24,26,28].

In the last decade, a significant number of papers have been published on the

TOPSIS approach. Abo-Sinna[1] extends TOPSIS approach to solve multi-

objective dynamics programming (MODP) problems. He shows that using the

fuzzy max-min operator with nonlinear membership functions, the obtained

solutions are always nondominated solutions of the original MODP

problems.Deng etal. [12] formulate the inter-company comparison process as a

multi-criteria analysis model, and presents an effective approach by modifying

TOPSIS for solving such problem.Chen [7] extends the concept of TOPSIS to

develop a methodology for solving multi-person multi-criteria decision-making

problems in fuzzy environment. Chen and Tzeng[8] combine the grey relation

model with the ideas of TOPSIS for selecting an expatriate host country. Yang

and Chou [27] use the TOPSIS approach to solve the multiresponce simulation-

optimization problem. Chenget al [9] proposea hierarchy multiple criteria

decision-making (MCDM) model based on fuzzy-sets theory to deal with the

supplier selection problems in the supply chain system. According to the concept

of the TOPSIS, a closeness coefficient is defined to determine the ranking order of

all suppliers by calculating the distances to the both fuzzy positive-ideal solution

(FPIS) and fuzzy negative-ideal solution (FNIS) simultaneously.

Abo-Sinna and Amer[2]extend the TOPSIS approach to solve multi-objective

large-scale nonlinear programming problems.Abo-Sinna and Abou-El-Enien [3]

extend the TOPSIS approach to solve large scale multiple objective decision

Vector optimization problems 659

making problems with block angular structure. Also, they [4] extend the TOPSIS

approach to solve large scale multiple objective decision making problems under

fuzzy environment. Abo-Sinna et al [5] extend TOPSIS approach to solve large

scale multi-objective non-linear programming problems with block angular

structure.

In this paper, TOPSISis extended for solving CHLSILVOP. TOPSIS was first

developed by Hwang and Yoon[16] for solving a multiple attribute decision

making problem. It is based upon the principle that the chosen alternative should

have the shortest distance from the positive ideal solution (PIS) and the farthest

from the negative ideal solution (NIS). The single criterion of the shortest distance

from the given goal or the PIS may be not enough to decision makers. In practice ,

we might like to have a decision which not only makes as much profit as

possible, but also avoids as much risk as possible. A similar concept has also been

pointed out by Zeleny[30].

The formulation of CHLSILVOPwith block angular structure is introduced in

section 2. The family of dp-distance and its normalizationis discussed in section 3.

The TOPSIS approach for CHLSILVOPis presented in section 4. In section 5, an

interactive decision making algorithm for generating a Pareto optimal

(compromise) integer solution for CHLSILVOPthrough TOPSISapproach is

provided where the decision maker (DM) is asked to specify the relative

importance of objectives.For the sake of illustration, a numerical example is given

in section 6. Finally, conclusions will be given in section 7.

2) Formulation of the Problem)

Consider the following CHLSILVOPwith a block angular structure of the

constraints:

( ( ) ( ) ( )) ( ) subject to

* *∑∑ +

*∑ +

+ ( ) where

k : the number of objective functions,

q : the number of subproblems,

m : the number of constraints,

n : the numer of variables,

nj: the number of variables of the jth

subproblem, j=1,2,…,q,

mo: the number of the common constraints represented

by *∑ ∑ +


mj: the number of independent constraints of the jth

subproblem

represented by

*∑ +

: the set of all real numbers,

X : an n-dimensional column vector of variables,

* +, * +,

* ( ) + .

In addition, P means probability, and are a specified probability levels. For

the sake of simplicity, consider that the random parameters, and are

distributed normally and independently of each other with known means E{ } and

E{ } and variances Var{ } and Var{ }.

If the objective functions are linear, then the objective function can be written as

follows:

( ) ∑

∑

( )

Using the chance constrained programming technique [22, 23], the deterministic

version of the CHLSILVOP problem (1) can be written as follows :

( ( ) ( ) ( )) ( ) subject to

* ∑ ∑

{ } √ { }

∑ { } √ * +

+ ( )

where is the standard normal value such that Φ(

)=

and Φ represents the cumulative distribution function of the

standard normal distribution.

(3) Some Basic Concepts of distance Measures

The compromise programming approach [29] has been developed to perform

multiple objective decision making problems, reducing the set of nondominated

solutions. The compromise solutions are those which are the closest by some

distance measure to the ideal one. The point (

) ∑ ( )

in the criteria space is called the ideal point

(reference point). As the measure of “closeness”, dp-metric is used. The dp-metric


defines the distance between two points, ( ) ∑ ( ) and (

)

∑ ( )

(the reference point) in k-dimensional space [3]as:

(∑( )

+

(∑(∑

∑

)

)

( )

where p≥1.

Unfortunately, because of the incommensurability among objectives, it is

impossible to directly use the above distance family. To remove the effects of the

incommensurability, we need to normalize the distance family of equation (4) by

using the reference point [3,29,30] as :

(∑(∑

∑

∑

+

+

⁄

( )

where

To obtain a compromise solution for problem (1), the global criteria

method[3,15,21] for large scale problems uses the distance family of equation(3)

by the ideal solution being the reference point. The problem becomes how to

solve the following auxiliary problem :

(∑(∑ (

) ∑ ( )

∑ ( )

+

)

( )

where is the PIS and

Usually, the solutions based on PIS are different from the solutions based on NIS.

Thus, both ( ) and ( ) can be used to normalize the distance family

and obtain [3,16,19] :

(∑(∑

∑

∑

∑

+

+

( )

where .

In this study, the concept of TOPSIS is extended to obtain a Pareto Optimal

(compromise) integersolution for CHLSILVOP.

(4) TOPSIS for CHLSILVOP

Consider the following CHLSILVOP:

( ( ) ( ) ( )) ( ) subject to

( ) where

∑ ( ) : Objective Function for Maximization, ,


∑ ( ) : Objective Function for Minimization, .

In order to use the distance family of equation (7) to resolve problem (8), we must

first find ( ) and ( ) which are[3]:

( )

∑

( )( ∑

( ), ( ) ( )

( )

∑

( )( ∑

( ), ( ) ( )

where . (

) (

)are the

individual positive(negative) ideal solutions.

Using the PIS and the NIS, we obtain the following distance functions from them,

respectively:

(∑

(∑

∑ ( )

∑

∑

+

∑

(∑ ( ) ∑

∑

∑

+

)

⁄

( )

and

(∑

(∑ ( ) ∑

∑

∑

+

∑

(∑

∑ ( )

∑

∑

+

)

⁄

( )

where are the relative importance (weights) of objectives,and

In order to obtain a compromise solution, transfer problem (8) into the following

bi-objective problem with two commensurable (but often conflicting)

objectives[3,19]:

( )

( )

subject to (11)

where

Since these two objectives are usually conflicting to each other, we can

simultaneously obtain their individual optima. Thus, we can use membership


functions to represent these individual optima. Assume that the membership

functions ( ( )and ( )) of two objective functions are linear. Then, based on

the preference concept, we assign a larger degree to the one with shorter distance

from the PIS for ( ) and assign a larger degree to the one with farther distance

from NIS for ( ). Therefore, as shown in figure 1, ( ) and ( ) can be

obtained as the following[3, 22, 31]:

( )

{

( ) ( )

( ) (

)

( )

(

) (

)

( ) (

)

(

) ( )

and

( )

{

( ) ( )

( ) (

)

( )

(

) (

)

( ) (

)

(

) ( )

where

( )

( )

( )

( )

( )

( ) ( )

( )

▲ ( ), ( )

1

max-min

◄-----solution

0 ----------------------------------------------------------------------------► dp(X)

( )

(

) (

) (

)

Figure 1. The membership functions of ( ) and ( )


Now, by applying the max-min decision model which is proposed by Bellman and

Zadeh[6] and extended by Zimmermann [31], we can resolve problem (11). The

satisfying decision, X*, may be obtained by solving the following model :

( ) { ( ( ) ( ))}

( )

Finally, if ( ( ), ( )), the model (13) is equivalent to the form

ofTchebycheff model [11], which is equivalent to the following model:

( ) subject to

( ) ( ) ( ) ( )

, - ( ) where is the satisfactory level for both criteria of the shortest distance from the

PIS and the farthest distance from the NIS.

(5) The algorithm of TOPSIS for solving CHLSILVOP

Thus, I can introduce the following algorithm togernerate a set of Pareto Optimal

(compromise) integer solutions for the CHLSILVOP:

The algorithm (Alg-I):

Step 1. Formulate CHLSILVOP (1) which have linear objective functions as

in Eq. (2)

Step2. Use the standard normal distribution table to find ,

Step 3. Transform problem (1) to the form of problem (3)

Step 4. Transform problem (3) to the form of problem (8)

Step 5 Ignore the integer requirement.

Step 6 Constructthe PIS payoff table of problem (8) by using the decomposition

algorithm [10, 20, 24] and problem(9-a). Thus, (

) , the

individual positive ideal solutions are obtained.

Step 7.Constructthe NIS payoff table of problem (8) by using the

decompositionalgorithm [10, 20, 24]and problem(9-b).Thus, (

), the individualnegative ideal solutions are obtained.

Step 8. Use equation (10) and the above step to construct and

.

).1) to problem (18Transform problem (. 9Step

Step 10. Ask the DM to select * +, Step 11. Ask the DM to select

where∑

): 1oblem (1Construct the payoff table of pr .3Step 1

At p=1, use the decomposition algorithm [10, 20, 24].

At p≥2, use the generalized reduced gradient method [20,21] and

obtain: ((

) (

) ) ((

) (

) )


Step 14. Construct problem (14) by the use of the membership functions (12).

).4Solve problem (1 .5Step 1

Step 16. If the solution of problem (14) yields optimal integer solution, then go to

step 18.Otherwise go to step 17.

Step 17.Use the Branch-and-Bound method [25]. Go to step 15.

Step 18. If the DM is satisfied with the current solution, go to step 19. Otherwise, go

to step10.

Step 19. Stop.

(6) An illustrative numerical example

Consider the following CHLSILVOP which has the angular structure:

( ) ( ) ( ) ( ) subject to

* + ( ) * + ( )

* + ( ) ( ) where

( ) ( ) ( ) ( ) ( ) ( ) Use (Alg-I) to solve the above problem.

By using problem (3), problem (15) can be written as follows:

( ) ( ) ( ) ( )

subject to

( ) ( ) ( ) ( )

Obtain PIS and NIS for problem (16), (Tables 1&2).

Table 1 : PIS payoff table of problem (17)

______________________________________________________________

______________________________________________________________

( ) 18* -7 1 4

( ) 16 -8* 0 4

______________________________________________________________

PIS: =(18 , -8)

666 T. H. M. Abou-El-Enien Table 2 : NIS payoff table of problem (17)

______________________________________________________________

______________________________________________________________

( ) 0 0 0

( ) 4 2 0

______________________________________________________________

NIS: =(0 , 2)

Next, compute equation (10) and obtain the following equations:

*

( ( )

*

( ( ) ( )

( ))

+

⁄

*

( ( )

*

( ( )

( ))

+

⁄

Thus, problem (11) is obtained.In order to get numerical solutions, assume that

=

=0.5 and p=2 ( Table 3),

Table 3 : PIS payoff table of equation (11), when p=2.

______________________________________________________________

d2PIS

d2NIS

z1 z2 x1 x2

______________________________________________________________

11.3888 -2.3056 0.5339 4

10 -7 1 4

______________________________________________________________

=(0.0365 , 0.6592) ,

=(0.05 , 0.6577).

Now, it is easy to compute (14) :

subject to

( ( )

) (

( )

) , -

The maximum “satisfactory level” (δ=1) is achieved for the solution x1=1,

x2=1.

(7) Conclusions

In this paper, a TOPSIS approach has been extended to solve CHLSILVOP. The

CHLSILVOP using TOPSIS approach provides an effective way to find the

compromise ( satisfactory) solution of such problems. Generally TOPSIS,


provides a broader principle of compromise for solving Vector Optimization

Problems. It transfers k-objectives (criteria), which are conflicting and non-

commensurable, into two objectives ( the shortest distance from the PIS and the

longest distance from the NIS), which are commensurable and most of time

conflicting . Then, the bi-objective problem can be solved by using membership

functions of fuzzy set theory to represent the satisfaction level for both criteria

and obtain TOPSIS, compromise solution by a second–order compromise. The

max-min operator is then considered as a suitable one to resolve the conflict

between the new criteria (the shortest distance from the PIS and the longest

distance from the NIS).

Also, in this paper, an algorithm of generating Pareto Optimal (compromise)

integer solutions of CHLSILVOP problem has been presented. It is based on the

decomposition algorithm of CHLSILVOP with block angular structure via

TOPSIS approach for p=1 and Generalized reduced gradient method for p≥2 .

This algorithm combines CHLSILVOP and TOPSIS approach to obtain

TOPSIS's compromise solution of the problem, Finally, a numerical illustrative

example clarified the various aspects of both the solution concept and the

proposed algorithm.

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Received: September, 2010

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