Solving Integer Goal Programming Problems Based on a Reference Direction Algorithm

(1)

(2)

(3)

(4)

(5)

(6)

Solving integer goal programming problems

based on a reference direction algorithm

Sawaluddin

1

Department of Mathematics The University of Sumatera Utara

Abstract. Integer goal programming problems arise quite naturally in many real-world applications. In this paper, we propose a reference direction approach and interactive algorithm to solve integer goal programming problem. We use analytic hierarchy process to get the reference direction. At each iteration, only one integer linear programming problem is solved to get an efficient solution. Through analytic hierarchy process the decision maker has to provide the preference point such that the original problem has been transformed into linear integer programming model.

Key words: AHP, goal programming

1 Introduction

Multiple objective programming, developed by Lee [9] and Ignizio [6] is an

extension of the linear programming model employed in solving optimal-mix

problems subject to some specified goal constraints. The objective function is

designed to minimize the sum of goal deviations with a view to minimizing the

cost associated with goal under achievements and goal over achievements.

Integer goal programming (IGP) assumes greater importance as a model

capable of handling multiple decision criteria, in which some of the decision

variables are assigned to integer values. Integer goal programming problems

arise quite naturally in many real-world applications. For example, in media

selection, capital budgeting and several location problems, the decision

variables can take on only integer values. [8] Illustrate an IGP approach to the

remanufacturing supply chain model, in the context of environmentally

conscious manufacturing. They present a quantitative methodology to

determine the allowable tolerance limits of planned/unplanned inventory in a

remanufacturing supply chain environment based on the decision maker’s

unique preferences, by applying an integer GP model that provides an unique

solution for the allowable inventory levels.

Gupta and Evans [4] used the model lo demonstrate ihe importance of goal

ranking and weighting of multiple goal priority factors. In formulating the integer

goal programming problem, priority coefficients are used to rank the goals.

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)

(7)

At present, a number of methods are available to solve the general

problem. The algorithms of [1] , [7], [14], and [15] have been designed to find

the set of all efficient solutions. Interactive methods for solving the problem

have been developed by [3], [5]. For survey of the problem, the interested

reader may refer to [12,16].

The investigation in the field of multiple objective linear programming e.g.

[2], [12] and [13] have shown that interactive algorithms are the most

promising for solving multiple objective programming problems. However, the

available interactive algorithms for solving problem require an excessive

amount of computational resources, both in terms of time and storage space

requirements. A few of the algorithm require specialized software for their

implementation; some put too many demands on the decision maker (DM),

whereas others may generate dominated solutions. Therefore, at present there is

a need to develop an effective and efficient algorithm to solve the problem.

Since integer programming problems are NP-hard, it is imperative to

minimize the number of single objective (mixed) integer programming

problems that have to be solved to find an acceptable compromise solution.

The extent to which an algorithm achieves this objective may, to a large part,

determine its applicability/acceptability. Furthermore, it is desirable that the

demands placed on the DM be kept to a minimum.

We use Analytical Hierarchy Process (AHP) to be used by DM in deciding

the priority of the multi objectives in such a way transform the original

problem to become mixed integer programming problem. Our objective in this

paper is to develop an algorithm that solves only one mixed integer

programming problem at each iteration and does not place too many demands

on the DM.The rest of the paper is organized as follows: next we give the

problem statement and some results, then we state the proposed algorithm and

illustrate it with a numerical example. We conclude the paper with a few

remarks.

2 Problem Statement

The integer goal programming (IGP) problem can be stated mathematically as:

1 2

c

1

and

c

2

are vectors of weights placed on the violation of constraints.

p

i

and

m

i

are variables showing by how much a given goal is violated.

(8)

Note that in a given constraint either

p

i

or

m

i

is certain to be zero in an optimal

solution.

As we knew that goal programming is a problem structure of multiobjective

programming. Therefore in deriving the method for solving the integer goal

programming we start from solving the multiobjective integer programming

(MOIP).

AHP was proposed by Saaty [10,11] twenty years ago and is a widely used technique for multi-attribute decision making. It is based upon pairwise subjective judgment of elements which are used to complete a matrix. The eigenvalue for each element is then used to assess the contribution of that element to the overall component. As the name suggests, a hierarchy of matrices can be used where components are themselves elements of a higher order component. A typical example might be choosing a supplier on the basis of several criteria such as cost and quality. We would need to determine the relative contributions of cost and quality to the overall decision and also the relative degree to which each supplier possesses each criterion. It is normal to proceed from the more general to the more concrete.

Assume that there are n elements, then we require (n(n-1))/2 pairwise judgements to complete the matrix, where each judgement reflects the perception of the ratio of the relative contributions of elements i and j to the overall component be assessed so

(

/

)

ij i j

a



w w

, subject to the following constraints;

a

_ij



0,

a

_ij



1, and

a

_ij



(1/

a

_ji

)

. Saaty argues that the technique can only be effectively used where the elements are homogeneous, that is within the same order of magnitude, hence the ratios must range from 1

9

to 9

.

In order to make the comparison process easier, some researchers attach semantic

labels such as “equal” where the ratio is 1, “slightly more important” where it is 2 and so forth. For instance, if we considered quality to be “slightly more important” than

(9)

cost, one would assign the value two to the appropriate cell in the matrix. In this case, the matrix would be completed as follows:

Quality Cost

Quality 1 2

Cost 0.5 1

Each component has a priority scale, that is a derive ratio scale, to measure the contribution of each element to that component. This is based upon the approximate eigenvalue (i.e. divide the sum of the row by n) of each element.

One problem that can occur, especially since the judgements are subjective, is that values assigned are inconsistent. For example, one would expect to observe transitivity. Consistency can be measured as the deviation of the principal eigenvalue of the matrix from the order of the matrix.

The consistency index, CI, is calculated as follows:

max

(

) /(

1)

CI







n



where



_max is the maximum principal eigenvalue of the judgments matrix. The nearer CI is to zero the more consistent the judgements. The CI can be compared with the consistency index of a random matrix (RI). The ratio (CI/RI) is known as the consistency ratio (CR). Saaty suggests CR should be less than 0.1, although one should be cautions about attaching undue significance to this value.

Definition 1. The reference direction is defined by the difference between the reference point given by the DM and the last solution of the problem.

Let fk denote an arbitrary value of the objective function of (1) and fk denote the

(10)

( ) ,

functions kH. That is, it tries to take us as far as possible from the current solution. When we solve (2), the values off the objective functions that belong to set H increase whereas those that belong to set L may decrease. This way, function (1) exists only in

k k which define the single objective problem (2). When the sets H and L are non-empty, then the optimal solutions of (2) obtained for various values of  are weak efficient solutions for (1), see Theorem 1 in the Appendix. It is useful to note that the last solution of (1) is a feasible solution for (2);this is important when solving (2) by an exact algorithm. Further, the feasible solutions of (2) lie close to the efficient surface of (1) which allows us to use an approximate algorithm to solve (2).

Since the objective function of (2) is not linear, no standard algorithm for solving linear or linear integer programming problems can be used to solve it. However, the problem can be started as the following equivalent mixed integer linear programming problem.

When (2) has no solution, then problem (5) also has no solution. This is due to the fact that both problems have the same constraints. When (2) has a solutions, then (5) has a solution and the optimal values of their objective functions are equal, see the lemma in the Appendix. Since problems (2) and (5) are equivalent, the optimal solution of (5) is a weak efficient solution of (1) .

(11)

The solution of (2) (or equivalently (B)) is a weak efficient solution for (1).

where  is an arbitrary small positive number.

In Theorem 4 (see the Appendix) we prove that the optimal solution of (11) is an

The proposed algorithm consists of the following three steps.

Steps 1. Determine an initial (weak) efficient solution.

Steps 2. Show the solution to the DM. if DM is satisfied with the solution,

Stop: otherwise, ask the DM to specify a new reference point f_k, using AHP and go to step 3.

Steps 3. Based on the values of fk and

f

k (the last solution), solve (5) (or

(11)) and find a new intermediate weak efficient (or efficient) solution

f x

k

( )

; go to Step 2.

The initial weak efficient (or efficient) solution in Step 1 of the proposed algorithm is obtained by solving (5) and (14) for =0, f_k 0,kK and f_k 1,kK. If the

(12)

values of some

f k

_k

,



K

can be negative, then problem (5) or (14) may be solved by replacing y by y1-y2 where y1,y2 0.

Since (2) and (14) are mixed integer linear programming problems, the may be solved by any standard exact algorithm. The branch-and-bound algorithms are the most appropriate for this purpose. Since we start with an initial feasible solution for the problem, the solution time in subsequent iterations (Step 3) can be considerable decreased. It may be noted that we start by solving the problem for =0 and may continue the solution procedure parametrically for several new values of .

Problems (5) and (14) are NP-hard. The exact algorithms may take considerable time to solve problems of large dimensions. Therefore, it may be desirable to use an approximate algorithm to solve the problems in Step 3. It may be noted that, based on the theorems, the feasible solutions obtained by an approximate algorithm lie close to or on the weak-efficient (or efficient) surface; they are also used to formulate problem (5) or (14) for the next iteration. The preceding statements are also true when an exact algorithm is used to solve (5) and (14).

It may be pointed out that if at any iteration the aspiration levels desired by the DM exceed the objective function values (for all objectives) obtained at a previous iteration. i.e. f_k  f_k, for every kK, the solution of problem (5) (or (14)) will be the same as the one obtained at the previous iteration. There are two ways to avoid this problem. One way is to require the DM to state the aspiration levels such that f_k  f_k

for at least one kK. However, this puts an extra constraint on the DM. The second way to avoid the problem is to solve (22) (or(26)) instead of (5) (or (14)) .

(13)

1 problems or whether they will lie near the efficient surface for (1).

5 Numerical Example

For the sake of clarity and ease of understanding, we illustrate the proposed algorithm with a simple example where the objective function space is also the variable space. solution which is non-dominated. After using AHP, DM found out that f2 would be his

first priority, therefore DM would like to increase the value of f2 and is willing to

For the sake of simplicity, we solve the problems for  =0 only. The solution of this mixed integer programming problem is ( ,f f₁ ₂)(3,9)( ,x x₁ ₂) and

y



2 / 3

. This is a non dominated solution for the original problem. Suppose the DM now wants

(14)

to increase the value of

f

₁, reduce the value of

f

₂ and provide ( ,f f₁ ₂)(4.5, 6.5)

as the aspiration vector. Now we solve the following problem corresponding to (11):

2 stops; otherwise the DM provides a new aspiration vector and the process continues.

6 Appendix

We give proofs of the theorems and a lemma mentioned in the text.

Theorem 1. The optimal solution x* for (2) is a weak-efficient solution for (1).

proof. If

E





, the proof is obvious. Let

E





. Since x* is an optimal solution for

Since (32) contradicts (31);

x

* is a weak efficient solution for (1).

Lemma 1. The optimal values of the objective function of (2) and (5) are equal, i.e.

(15)

( )

because otherwise y can be increased further. Furthermore, the right-hand side of the preceding equality is equal to

max min

_k

( )

_k

/

_k _k

k H

x X 

f x



f



f

which proves the lemma.

Next, we consider cases when (B) has a solution for an arbitrary value of parameter , since these cases have particular importance for practical applications of the proposed algorithm.

Theorem 2. If

f x

_k

( )



f x

_k

( ),



k



K

, is an accessible weak non-dominated point (x

is a weak-efficient solution), then for any value of parameter



0, (5) has an optimal solution and the optimal value of the objective function in non-negative.

proof. Suppose that the solution x for which

f x

_k

( )

 

f

_k is a weak efficient solution. For



0, the constraints (7) and (8) are satisfied. From (33), it follows that the maximum value of y is equal to zero.

Since the optimal solution of the problem (5) is a weak efficient solution, then in the worst case if there is no other solution, the optimal solution

x

* will coincide with

x. If the optimal solution of the problem is not x, then from the lemma it follows

(16)

The optimal solution of the problem

x

* is a weak efficient solution and

[1] G.L.Chalmet, L.Lemonidis and D.Elzinga (1986) “An algorithm for the bicriterion integer programming problem”. Europ. J. Opl Res. 25, 192-300.

[2] G.W.Evans (1984) “An overview of techniques for solving multi-objective mathematical programs”. Mgmt Sci. 30, 1263-1282.

[3] D.Gabbani and M.J.Magazie (1986) An iterative heuristic approach for multi-objective programming problems”.Europ. J. Opl Res.Soc. 37,285-291.

[4] Gupta, A. and W.G. Evans, 2009. “A goal programming model for the operation of closed-loop supply chains”. Eng. Optimizat.,41: 713-735.

[5] J.J.Gonzales, G.R.Reeves and L.S.Franz (1993) “An interactive procedure for multiple objective integer linear programming problems”. In Decision Making with Multiple Objective (Y.Y.Haimes and V.Changkong, Eds). Pp 250-260. Springer Verlag Berlin. [6] J.P. Ignizo, 1976.” Goal Programming and Extensions”.Heath, Lexington Books,

Lexington, Mass, Lexington,

[7] D.Klein and E.Hannan (1982) “An algorithm for multi-objective zero-one linear programming problem”. Europ. J. Opl Res. 9, 378-385.

[8] E.Kongar and S.M.Gupta (2003), “A goal programming approach to the remanufacturing supply chain Model”. Northeastern University, Boston.

[9] S.M. Lee, 1972. “Goal Programming for Decision Analysis Philadelphia Pennsylvania”. Auerbach Publishing, Boston.

[10] T.L.Saaty., “The Analytic Hierarchy Process”. McGraw-Hill,New York, 1980.

[11] T.L.Saaty., “Hightlights and critical points in the theory and application of the analytic Hierarchy Process”. EJOR 74,3 (1994), 426-447.

[12] R.Stever (1986) “Multiple Criteria Optimization Theory, Computation and Application”. Wiley, New York.

[13] J.Teghem and P.L.Kunsch (1985) “Interactive methods for multi objective integer linear programming. In Large Scale Modeling and Interactive Decision Analysis”. (G.Fandel, M.Grauer, A.Kurzhansk and A.P.Wierrzbieki, Eds) pp 75-87 Springer Verlag, Berlin. [14] J.Teghem and P.L.Kunsch (1986) “A survey of techniques for finding efficient solutions

to multi-objective integer linear programming”. Asia Pacific J. Opns. Res. 3, 95-108. [15] B.Villareal and M.H.Karwan (1981) “Multi-criteria integer programming: A (hybrid)

dynamic programming recursive approach”. Math Prog. 21, 204-223.

(17)

[16] W.Wongthatsanekorn (2009) “A goal programming approach for plastic recycling system

in Thailand”. World Academy of Science and Technology 40 513-517.

[17] S.ZIONTS (1977) “Integer linear programming with multiple objectives”. Ann. Discrete Math.i1, 551-562.