Solving Integer Goal Programming Problems Based on a Reference Direction Algorithm

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Contents
Message from UTAR President

I

Message fromFES Dean

2

Message from the Chair

3

.

Keynote Lecturrcs and Invited Tatks

4

Biodata of Invited Speakers

4

Keynote

[rcture : Wavelets, Multiwavelets

and Wavelet Frames for Periodic

Functions
Say Song

Goh

.

lrcture : Data Depttr for New Nonparamenic Inference Schemes
and Beyond (Abstract

Keynote

ffiy)

ReginaY.liu

22

Keynote l-ecture : Stochastic Mixed Integer Nonlinem Programming (Abstract Only)

Herman Mawenglwng

23

Keynote Lecture: Contemporary Statistical Data Visualization

Junji Nakano

24

Keynote l,ecture: Insurance Risk Models: V/ith and Without Dividends
(Abstract Only)

Hailiang Yang

47

Invited Talk Some One-sided Multivariate Tests
Sarnruarn

Chongcharuen

Invited Thlk: On Dynamical Systems and Phase Transitions for
padic Potts Model on the Cayley Tree

FamtkhMukhamedov

48

e

* l-state
.....68

Talk Quartic-Nonnal Distributions (Abstract Only)
AhHinPooi .
....
Invited Talk Rssearch Collaboration Network Analysis of the Joumal of
Invited

.

Bz

Finance (Abstract Only)

Kurunathan

Ratnavelu

83

r_

CONTENTS

lnvited ralk rhe ultimate solution Approach to Intractable problems
Abdellah Salhi .

84

contributed rhlks : Pure Mathematics / combinatorics I Algebra
Analysis/Graph Theory

/

Minimal Realization of B.L-General Fuzzy Automata
Klmdijeh Abolpour and Mohammad Mehdi Znhedi

94

Comparative Study of Geomeric product and Mixed product
Md. Shah Alam and Sabar Bauk

110

on chromatic uniqueness and Equivalence of Ka-Homeomorphic Graphs
Sabina Catada-Ghimire and Roslan Hasni

115

constructions of Non-comrnutative Generalized Latin squares of order 5
H. V Chen, A. Y M. Chin, and. Shereen
.

Shnrmini

.

IZO

Spectral Corrections for a Class of Eigenvalue problems
Mohamed K El Daou

131

On Special Solvents of Some Nonlinear Matrix Equations
frn-huan Han and Hyun-min Kim

144

n-fold Commutative Hyper K-ideals
Mona Pirasghari, Parvaneh Babari and Mohammad Mahdi znhedi

150

On {"-quadratic Stochastic Operators in 2-dimensional Simplex
Famtkh Mukhamedov and Afifah Haruum Mohd Jamal . .

159

.

single Folygon counting for m Fixed Nodes over cayley Tree of order
Chin Hee Pah and Mansoor Saburov .

Ttrvo

173

n-fold Positive Implicative Hyper K-ideals
Paruaneh Babari, Mona Pir*sghari and Mohammad Mahdi znhedi 1g6
On *p-closed Fuzzy Sets, Fuzzy *p-closed Maps, Fazzy *p-irresolute
Maps and *p-homeomorphism Mappings in Fuzzy
Topologicar spaces

SadanandN.Patil

..ZAI

AnFazzy.gp,-closed Maps, Fazzy g;^r-continuous Maps andFuzzy gtrr_
irresolute Mappings in Fuzzy Topological Spaces
Sa.danandN. Patil, A. S. Madabhavi S. R. Sadugol and G.n. S. B.

Madagi

....2I4

on Graph-(super)magic Labelings of a path-amalgamation of Isomorphic
Graphs

A.N.M. Salman andT.K.

Maryati

A Survey on Equations in Group Ring
Tai Wei Hang and Denis Wong Chee

- . ZZg

Keong

Groups with Small Conjugacy Classes
Yean Nee Tbn, Guan Aun How and Miin

Huq

Ang

.....

234

. . . . . ZM

CONTENTS

vlr

A Goal Programming Model for the Recycling Supply Chain Problem
Putri K.NasutioraRimnAprilia"Amalia,HermanMawengkang .

. . 903

An Active Set Method with Central Measure on Removing Impulse Noise

ManickNeri

.....917

Inner Solution for Oscillatory Free Convection about a Sphere Embedded
in a Porous Medium

Lai 7he Phooi, Rozaini Roslary Ishak Hashim, and hinodin Haji

Jubok

.....926

The use of Adomian Decompositiou Method for Solving Generalised
Ricc ati Differential Equations

T.R.RameshRao

...935

Subclasses Discriminant Analysis by Fuzzy Clusoer Algorithm
Ghasem Rekabdar Naser Haddadzadeh and Davood Seifipoor

...

942

A Multi-stage Stochastic Optimization Model for Water Resources Management

EJly&osmaioi,...!:.!

.!i.

,.951

Stochastic hogramming Model for Land Management Problems

SitiRusdiana

Predator-prey Model in a Bioreactor with Death Coefficient

ZubaidahSadikinandNormahSalim

.....9G2
....973

Recommending a Hybrid Method for Solving the Ordered Crossover
Problem

BahadorSakctandFarnazBehrang

.....984

Solving Integer Goal Programming Problems Based on a Reference Direction Algorithm

Sawaluddin

. . 1000

Integer Programming Model for Supply Chain with Market Selection
Selamat Siregati Agusma4 Sindak Siturnorang, Lisbet Marbun, Ab-

dul Jalil, Herman

Mawengkang

. . . l0lz

Autom+ttq Gddding for DNA Microarr.ay hu4gp using Im4gg,prpj..qftrpq
Profile

JolCISiswantoru

Efficient Reduction of Fuzzy Finite Tree Automata
Somaye Moghari, Mohammad Mehdi Tnhedi and Reza

. ..1028
Ameri . . .

1034

Minimization of Fuzzy Finite Tree Automata
Sornaye Moghari, Mohawnad Mehdi 7flhedi and Rezs Ameri . . . 1044
An Improved Strategy for Solving Quadratic Assignment problems
Susiana, Nunik Ardiana, Wahab Y. S.Hasibuan, Herman Mawenglcang
1053

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1000

Solving integer goal programming problems
based on a reference direction algorithm
Sawaluddin1
1

Department of Mathematics
The University of Sumatera Utara

Abstract. Integer goal programming problems arise quite naturally in many realworld 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)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1001

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:
Minimize (c1 p  c2 m)
subject to Ax  I p  I m  b
x integer
where
c1 and c2 are vectors of weights placed on the violation of constraints.
pi and mi are variables showing by how much a given goal is violated.

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1002

Note that in a given constraint either pi or mi 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).
Consider the MOIP problem:

max[ f k ( x), k  K ]

(1)

subject to

where K  {1, 2,

x  X  x | Ax  b, x [0, r ], x integer
, k}, A is an m  m matrix coefficients of the constraints; b is

an m-vector of the right hand side, b  Rm ; f k ( x), k  K , is a linear function of
the decision variables, x is an n-vector of decision variables, x  R n and r is an
upper bound on x. By max we mean that all objective functions have to be
maximized simultaneously.

3 Analytic Hierarchy Process
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
aij  (wi / w j ) , subject to the following constraints; aij  0, aij  1,and aij  (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

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1003

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
1
0.5

Quality
Cost

Cost
2
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:

CI  (max  n) /(n  1)

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 f k denote the
aspiration level. Further let,

H   k  K | f k  f k 
L   k  K | f k  f k 
E   k  K | f k  f k 
where K  H  L  E . To find the next solution of (1), we solve the following
single objective surrogate problem:

max s1 ( x)  max min( f k ( x)  f k ) /( f k  f k )
xX

xX

kK

(2)

subject to

f k ( x)  f k   ( f k  f k ), k  L,

(3)

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

f k ( x)  f k , k  E

1004

(4)

where  is a non-negative parameter and f k , k  K is the objective function value
found in the last solution.
The objective function (2), max xX s1 ( x) maximizes the smallest standardized
difference between the current solution fx(x) and the last solution fk for all objective
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
f k  f k for at least one k , k  K . This also implies that the reference point

f k , k  K , does not have to be dominated by the previous solution of (1).
Depending upon the values of f k and f k , k  K , sets H, L and E are created
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.

max y

(5)

subject to

f k ( x)  ( f k  f k ) y  f k , k  H ,

(6)

f k ( x)  f k   ( f k  f k ), k  H ,

(7)

f k ( x)  f k , k  E ,
x X
y0

(8)
(9)
(10)

where y is a scalar.
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) .

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1005

The solution of (2) (or equivalently (B)) is a weak efficient solution for (1).
However, if it is desired to obtain an efficient solution, then we may solve the
following single objective surrogate problem:


max T ( x)  max min
xX
xX
 kH

fk ( x ) fk
fk  fk


   ( f k ( x)  f k ) 
kK


(11)

subject to

f k ( x)  f k   ( f k  f k ), k  L
f k ( x)  f k , k  E

(12)
(13)

where  is an arbitrary small positive number.
In Theorem 4 (see the Appendix) we prove that the optimal solution of (11) is an
efficient solution of (1).
Problem (11) can be reduced to the following equivalent mixed integer linear
programming problem.



max  y    yk 
kK



(14)

f k ( x)  f k  yk , k  H

(15)

f k ( x)  f k   yk , k  L

(16)

f k ( x)  ( f k  f k ) y  f k , k  H

(17)

f k ( x)  ( f k  f k )  f k , k  L

(18)

f k ( x)  f k , k  E
x X
y, yk  0, k  K

(19)

subject to

(20)
(21)

4 Proposed Algorithm
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 f k and f k (the last solution), solve (5) (or
(11)) and find a new intermediate weak efficient (or efficient)
solution f k ( x) ; 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

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1006

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)) .
That is,
max( z1  z2 )
(22)
subject to
(23)
f k ( x)  ( f k  f k)( z1  z2 )  f k   ( f k  f k ), k  K

x X
z1 , z2  0

(24)
(25)

or



max  z1  z2    yk 
kK


f k ( x)  f k  yk , k  K

f k ( x)  ( f k  f k)( z1  z2 )  f k   ( f k  f k ), k  K
x X,
yk  0, k  K
z1 , z2  0,
where

k1  arg max( f k  f k )
and

(26)
(27)
(28)
(29)
(30)
(31)

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1007

 f  ( f k  f k ) / 2, k  k1
f k   k
k  k1
 fk ,
Unfortunately, the last weak-efficient (or efficient) solution may not be feasible for
(22) (or (26)); and nothing definite can be said about the feasible solutions of these
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.
Consider the following:

max f1 ( x)  x1
max f 2 ( x)  x2
subject to

x1  4 x2  45
9 x1  2 x2  18
2.8 x1  3.1x2  39
5 x1  4 x2  60
4 x1  2 x2  35
x1  7.5
x1 , x2 integer
Let X denote the feasible set. We obtain f  ( f1 , f 2 )  (6,5)  ( x1 , x2 ) as the initial
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
lower the value of f1. That is, the DM provides ( f1 , f 2 )  (3,11) as the aspiration
vector. To find a weak efficient solution, we solve the following problem
corresponding to (5):

max y
x2  6 y  5,
x1  3  3 ,

( x1 , x2 )  X ,
y0
For the sake of simplicity, we solve the problems for  =0 only. The solution of this
mixed integer programming problem is ( f1 , f 2 )  (3,9)  ( x1 , x2 ) and y  2 / 3 .
This is a non dominated solution for the original problem. Suppose the DM now wants

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1008

to increase the value of f1 , reduce the value of f 2 and provide ( f1 , f 2 )  (4.5,6.5)
as the aspiration vector. Now we solve the following problem corresponding to (11):

max y
x2  1.5 y  3,

x2  6.5  2.5 ,
( x1 , x2 )  X ,
y0
As before, we solve the problem for  = 0 only and obtain ( f1 , f 2 )  (5,7)  ( x1 , x2 )
and y  4 / 3 as the solution. If the DM is satisfied with this solution, the process
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
(1).
(31)
s1 ( x* )  s1 ( x) for each x  X
*
let us assume that x is not a weak-efficient solution for (1). Then there exist a point
for
which
for
and
f k ( x* )  f k ( x)
x  X
k H

f k ( x)  f k   ( f k  f k ) for k  K . Now

s1 ( x)  min
kH

f k ( x)  f k
fk  fk

[ f k ( x)  f k ( x* )]  [ f k ( x* )  f k ]
 min
kH
fk  fk
 min
kH

(32)

f k ( x* )  f k
fk  fk

i.e. s1 ( x)  s1 ( x* ) .
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.

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

max y  max min
xX

xX

kH

1009

f k ( x)  f k
fk  fk

proof. From (6), it follows that

y  ( f k ( x)  f k ) / f k  f k , k  H
since this inequality is valid for each k  H , it follows that
y  min( f k ( x)  f k ) / f k  f k , k  H
kH

*

if x is an optimal solution for (5), then

max y  min f k ( x* )  f k / f k  f k
xX

kH

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

max min f k ( x)  f k / f k  f k
xX

kH

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 k ( x)  f k ( x), 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 k ( x)  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
that y  0 .

Theorem 3. If f k , k  K is an accessible and dominated solution, then problem (5)
has an optimal solution for   0 and the optimal value of the objective function is
positive.

proof. If f k , k  K dominates f k , k  K , then L is an empty set and the problem (5)
always has feasible solutions.

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)
Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

1010

The optimal solution of the problem x* is a weak efficient solution and
f k ( x* ), k  K is a weak non-dominated solution. Since f k ( x* ), k  K dominates

f k ( x), k  K .
min
kH

[ f k ( x* )  f k ]
fk  fk

it follows from the lemma that y > 0.
If f k , k  K , does not dominate f k , k  K then the sets H and L are non-empty sets.
Since

f k , k  K , is a dominated point, f k , k  K is also a dominated point.

Therefore f k , k  L are accessible for y > 0.

References
[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.

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Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

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[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.