On some stability notions of multiobjective integer linear fractional
programming problem under randomness
Omar M. Saad and Eman F.Abdellah
Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, P.o. Box 11795, Cairo, Egypt.
E-mail: omarsd@gmail.com
Abstract
In this paper a solution algorithm to chance-constrained multiobjective integer linear
fractional programming problem is suggested. We assume that there is randomness in the
right-hand side of the constraints only and that the random variables are normally distributed.
Some stability notions for the problem of concern are defined and characterized. An
illustrative numerical example is provided to clarify the developed theory and the suggested
solution algorithm.
Keywords: Multiobjective optimization; Integer programming; Stochastic programming; Fractional
programming; Stability.
MSC NO (2001): 90C29, 90C10, 90C15, 90C32, 90C31.
1. Introduction
Fractional programming (FP), which has been used as an important planning tool for
the last four decades, is applied to different disciplines such as engineering, business,
economics. Fractional programming is generally used for modeling real life problems
with fractional objective such as profit / cost, inventory / sales, actual cost / standard
cost and output / employee etc [1, 2 and 3].
Integer linear fractional programming problem with multiple objectives (MOILFP) is
an important field of research and has not received as much attention as did multiple
objective linear fractional programming since some computational difficulties are
posed in solving such problems.
Stability of efficient solutions becomes more and more attractive in the area of
multiobjective mathematical programming. Publications on this topic usually
investigate the impact of parameter changes (in the right-hand side or/and in the
objective functions or/and in the left-hand side or/and the domination structure) on the
solution in various models of vector optimization problems.
Stability study allows the decision maker to take on decisions under various changes
keeping the solution of problem under consideration the same in a specified solution
domain, which has great importance in management decision making as well as
outside of it.
1
We list the some frequently occurring fractional objectives in which the stability of
solutions can be investigated under randomness: a multi-facility location –queuing
problem; financial planning with multiple fractional goals; an application in
computational geometry leading to a convex/convex quadratic problem; fractional
semi-infinite programming problems; applications in engineering give rise to such a
problem when a lower bound for the smallest eigenvalue of an elliptic differential
operator is to be investigated.
According to our experiences, it is believed that the stability in stochastic
multiobjective integer linear fractional programming problems has not been treated
and discussed in the literature before.
Previous papers in the literature do not discuss directly proper efficiency and the
stability of solutions in stochastic multiobjective integer linear fractional
programming problems, but they could be interesting reference works and have to be
reported here [4-8].
The structure of this paper is as follows:
Section 2 contains the mathematical formulation of multiobjective integer linear
fractional programming problem with chance constraints. In addition, a deterministic
version corresponding to the formulated model is stated along with the concept of
efficiency. Section 3 outlines the solution algorithm of such problem described in
finite steps. Some basic stability notions for the problem of concern are defined and
characterized in Section 4. An illustrative numerical example is provided in Section 5.
Finally, the presented paper is concluded in Section 6.
2. Problem Formulation and the Solution Concept
The purpose of this paper is to develop an algorithm for solving the following chanceconstrained multiobjective integer linear fractional programming problem
(CHMOILFP ) :
(CHMOILFP ) :
cr x   r
max z r ( x) 
, r  1,2,..., k ,
drx r
Subject to
x X,
where, in the above problem, k  2, c r , d r are 1 n - row vectors;  r ,  r are scalars
for each r  1,2,..., k.In addition the feasible region X is defined as:
2
n


n
x

R
/
P
{
g
(
x
)

aij x j  bi }   i , (i  1,2,..., m);

i


X 

j 1


x j  0 and integer , ( j  1,2,..., n)

In the above problem, x  R n is the vector of the integer decision variables and
zr ( x), (r  1,2,..., k ) are linear real-valued fractional objective functions to be
maximized over X. Furthermore, P means probability and  i is a specified
probability value. This means that the linear constraints may be violated some of the
time and at most 100(1-  i ) % of the time.
For the sake of simplicity, we assume that the random parameters bi, (i =1, 2… m) are
distributed normally with known means E{bi} and variances Var{bi} and
independently of each other.
The concept of the efficient solution of problem (CHMOILFP ) is given in the
following definition:
Definition 1.
A point x *  X is said to be an efficient solution to problem (CHMOILFP) with
m
*
probability   i if there does not exist another x X such that z r ( x)  z r ( x ) and
i 1
z r ( x)  z r ( x* ) holding for at least one r.
The basic idea in treating problem (CHMOILFP) is to convert the probabilistic nature
of this problem into a deterministic version. Here, the idea of employing deterministic
version will be illustrated by using the interesting technique of chance-constrained
programming [9, 10]. In this case, the set of constraints X can be rewritten in the
equivalent deterministic form as:
n


n
 x  R / g i ( x)   aij x j  E{bi }  K Var{bi }, (i  1,2,..., m);

i
X 

j 1


x j  0 and integer , ( j  1,2,..., n)

~
where Ki is the standard normal value such that ( K ) 1  i ; and  (a ) represents
i
the “cumulative distribution function” of the standard normal distribution evaluated
at a.
Thus, problem (CHMOILFP) can be understood as the following deterministic multiobjective integer linear fractional problem:
3
( MOILFP ) :
cr x   r
max z r ( x) 
, r  1,2,..., k ,
drx r
Subject to
~
x X .
Now, a suitable scalaization technique for treating problem (MOILFP) is to use the
  constraint method [11]. For this purpose, we consider the following integer linear
fractional programming problem with single-objective function as:
Ps ( ) :
max z s ( x) 
cs x  s
d sx   s
,
Subject to

n
 x  R / z r ( x) 
~
X ( )  


cr x   r
drx r
~
x X

  r ; r  K  {s},

,


where s  K  {1,2,..., k} which can be taken arbitrary.
We should point out that the efficiency concept of a solution x * for the formulated
problem (CHMOILFP) has been given in Definition 1 earlier and this efficient
solution can be found by solving the scalar problem Ps ( ). This can be done when the
minimum allowable levels (1,  2 ,...,  s 1,  s 1,...,  k ) for the (k  1)  objectives
( z1 ( x), z 2 ( x),..., z s 1 ( x), z s 1 ( x),..., z k ( x)) are determined in the feasible region of
~
solutions X ( ) defined above.
It is clear from [11, 12] that a systematic variation of  r ; r  K  {s}will yield a set of
efficient solutions to problem (MOILFP).
On the other hand, the resulting scalar problem Ps ( ) can be solved at a certain vector
of parameters    *  ( *1 ,  *2 ,...,  * s 1 ,  * s 1 ,...,  k* ) using the Charnes and Cooper
Transformation [13] together with the branch and bound method [14, 15], where the
LINGO software packages are used in the computational process.
~
It should be noted that if x*  X ( * ) is a unique optimal integer solution to problem
Ps ( * ) , then this solution becomes an efficient to problem (MOILFP) [11] or an
efficient solution to the formulated problem (CHMOILFP) with probability levels
 i* , (i  1,2,..., m).
4
3. Solution Algorithm
In this section, we propose an algorithm described in finite number of steps to solve
problem (CHMOILFP). The suggested algorithm can be summarized in the following
manner:
Step1.
Determine the means E {bi} and variances Var{bi}; (i=1, 2,…,m ).
Step2.
Convert the original set of constraints X of problem (CHMOILFP) into the equivalent
~
set of constraints X .
Step3.
Formulate the deterministic multiobjective integer linear fractional programming
problem (MOILFP) equivalent to problem (CHMOILFP).
Step4.
Formulate the integer linear fractional programming problem with single-objective
function Ps ( ) .
Step5.
Solve k-individual integer linear fractional programming problems:
cr x   r
max z r ( x) 
, r  1,2,..., k ,
drx r
Subject to
~
x X .
to find k- optimal integer solutions of the k-objectives. This can be done using the
Charnes and Cooper Transformation together with the branch-and-bound method and
the LINGO software packages are used in the computational process.
Step6.
Construct the payoff table and determine nr , M r (the smallest and the largest numbers
in the r th column in the payoff table).
Step7.
Determine the  r' s from the formula:
 r  nr 
t
( M r  nr ); t  0,1, 2,..., N  1,
N 1
5
where t is the number of all partitions of the interval [ nr , M r ] and N is the number of
different values of  r' s that will be used in the generation of efficient solutions.
Step8.
Find the set D  {  R k 1 / nr   r  M r ; r  K  {s}}.
Step9.
Choose  r*  D and solve the integer linear fractional problem Ps ( * ) as mentioned
in Step 5 above to find its optimal integer solution x * .
4. Parametric Analysis on Problem Ps ( )
Now and before go any further, we can write problem Ps ( ) in the following singleobjective relaxed parametric problem as:
cs x  s
max z s ( x) 
,
d sx   s
Subject to
PsR ( ) :


cr x   r
n
  r ; r  K  {s}, 
 x  R / z r ( x)  r
d xr




~
n


g i ( x)   aij x j  Ci ; i  1,2,..., m, ,
X R ( )  


j 1


l j  x j  u j ; j  J  {1,2,..., n}.





where the constraints l j  x j  u j ; j  J  {1,2,..., n} are additional constraints on the
decision variables x j and have been added to the original set of constraints of
problem Ps ( ) for obtaining the optimal integer solution x * using the branching and
bounding process.
In addition, it is assumed that:
Ci 
n
 aij x j  E{bi }  K i
j 1
Var{bi }, (i  1,2,..., m).
In what follows, definitions of some basic stability notions are given for the relaxed
problem PsR ( ) defined above. We shall be essentially concerned with three basic
notions: (i) the set of feasible parameters; (ii) the solvability set and (iii) the stability
set of the first kind (SSK1).
6
The feasibility condition of problem PsR ( ) is given below in the following definition.
Definition 2.
(i) The set of feasible parameters
The set of feasible parameters of problem PsR ( ) , which is denoted by A, is defined
~
A  {  R k 1 / X R   }.
by:
Definition 3.
(ii) The solvability set
The solvability set of problem PsR ( ) , which is denoted by B, is defined by:
B  {  A / Problem PsR(ε ) has an optimal integer solution x* }.
Definition 4.
(iii) The stability set of the first kind
Suppose that  *  B with a corresponding optimal integer solution x * , then the
stability set of the first kind of problem PsR ( ) , which is denoted by S ( x * ) , is defined
by:
S ( x* )  {  B / x* remains optimal integer solution of problem PsR ( ) for all   R k 1 }.
(iv) Utilization of the Kuhn-Tucker Necessary Optimality Conditions for Problem
PsR ( )
Now, given an optimal integer solution x * to problem PsR ( ) at certain  *  B ,
which may be found as described earlier. The question is: For what values of the
 r ; r  K  {s}, the Kuhn-Tucker necessary optimality conditions for the relaxed
parametric problem PsR ( ) are utilized?
In what follows, the Kuhn-Tucker necessary optimality conditions corresponding to
the relaxed parametric problem PsR ( ) will be written in the form:
7
z s ( x) k
z r ( x) m g i ( x)
  r
  i
  j   j  0, ( j  1,2, ,..., n,
x j
x j
x j
r 1
i 1
rs
z r ( x) 
cr x   r
  r ; r  K  {s},
drx r
g i ( x)  Ci , (i  1,2,..., m,
x j  u j , ( j  1,2,..., n),
x j  l j , ( j  1,2,..., n),
 r [  z r ( x)   r ]  0, r  K  {s},
 i [ g i ( x)  Ci ]  0, (i  1,2,..., m,
 j [ x j  u j ]  0, ( j  1,2,..., n),
 j [ x j  l j ]  0, , ( j  1,2,..., n),
 r ,  i ,  j , j  0.










(*)










where all the relations of the above system (*) are evaluated at the optimal integer
solution x * and  r ,  i ,  j , j are the Lagrange multipliers.
The first equality and the last four inequalities of the above system (*) represent a
Polytope in      space for which its vertices can be determined using any
algorithm based on the Simplex method, e.g., Balinski [18].
According to whether any of the variables  r ,  i ,  j , j are zero or positive, the set
of parameters  r ; r  K  {s} for which the Kuhn-Tucker necessary optimality
conditions are utilized, will be determined and is denoted by T ( x* ). Choosing another


 r   r  D  T ( x* ) will yield another T ( x * ) and so on. The above procedure
terminates when the range of the set D  {  R k 1 / nr   r  M r ; r  K  {s}} is
fully exhausted and the stability set of the first kind S ( x * ) is given as:
S ( x* ) 
k i
 Ti ( x
*
).
i 1
5. An Illustrative Numerical Example
In this section, an illustrative example is provided to clarify the proposed solution
algorithm. This example is adapted from one appearing in Saad etc.[8] and the
LINGO software package is used in the computational process.
8
The problem to be solved here is the following chance –constrained multiobjective
integer linear fractional programming problem:
2 x1  3x2
max z1 ( x1 , x2 )  (
),
x

4
x

6
1
2
(CHMOILFP):
max z 2 ( x1 , x2 )  (
3x1  4 x2
),
6 x1  4 x2  3
max z3 ( x1 , x2 )  ( x1  x2 ),
Subject to
P{ 2 x1  x2  b1 }  0.95,
P{ x1 3x2  b2 }  0.90,
x1 , x2  0 and integers.
where bi, (i =1, 2) are independent, normally distributed random parameters with the
following means and variances:
E {b1} = 1, E {b2} = 9, Var {b1} = 25, Var {b2} = 4.
and from the standard normal tables, we have: K  1  1.645 , K  2  1.285 .
Problem (CHMOILFP) can be understood as the following deterministic multiobjective integer linear fractional programming problem (MOILFP):
max z1 ( x1 , x2 )  (
2 x1  3x2
),
x1  4 x2  6
max z 2 ( x1 , x2 )  (
3x1  4 x2
),
6 x1  4 x2  3
(MOILFP):
max z3 ( x1 , x2 )  ( x1  x2 ),
Subject to
2 x1  x2  9.225,
 x1 3x2  11.57,
x1 , x2  0 and integers.
With the help of the branch-and-bound method [17], an equivalent multiobjective
linear fractional programming problem (MOLFP) corresponding to the deterministic
problem (MOILFP) can be formulated as follows:
max z1 ( x1 , x2 )  (
(MOLFP):
9
2 x1  3x2
),
x1  4 x2  6
max z 2 ( x1 , x2 )  (
3x1  4 x2
),
6 x1  4 x2  3
max z3 ( x1 , x2 )  ( x1  x2 ),
Subject to
2 x1  x 2  9.225,
 x1 3 x 2  11.57,
x1  4,
x2  3
x1 , x 2  0 .
If the problem (MOLFP) is solved for each objective function one by one using the
Charnes and Cooper Transformation [13] together with the branch and bound method
[14, 15], where the LINGO software packages are used in the computational process ,
therefore we get: z1 4,0  0.8 , z 2 0,3  0.8, z 3 4,0  4 .
Now, using the   constraint method [11], the following integer linear fractional
programming problem with single-objective function will be considered as:
P1 ( ) :
max z1 ( x1 , x2 )  (
2 x1  3x2
),
x1  4 x2  6
Subject to
z 2 ( x1 , x2 )  (
3 x1  4 x2
)  2,
6 x1  4 x2  3
z3 ( x1, x2 )  ( x1  x2 )   3 ,
2 x1  x 2  9.225,
 x1 3 x 2  11.57,
x1  4,
x2  3
x1 , x 2  0 .
From the payoff table, we get:
0.444   2  0.8,3   3  4,
Choosing  2   2*  0.444,  3   3*  4, the above problem can be solved again using
the Charnes and Cooper Transformation [13] to obtain the unique optimal integer
solution ( x1* , x2* )  (4,0) with the optimum objective value functions: z1* 4,0  0.8 ,
z 2* (4,0)  0.444, z3* 4,0   4 . This obtained solution is an efficient for the illustrative
problem (CHMOILFP) and the Kuhn-Tucker necessary optimality conditions
10
corresponding to the relaxed parametric problem P1 ( ) evaluated at the optimal
integer solution ( x1* , x2* )  (4,0) will be written in the following form:
1

1   2  21   2   1  1  0, 
81

20
 0.02 
1   2  1  3 2   2   2  0,

243

0.444   2 , 
4  3, 

1 (0.444   2 )  0, 
 2 (4   3 )  0, 

 1.225 1  0, (*)
 15.57  2  0, 

(0) 1  0, 

(3) 2  0, 
(4)1  0, 

(0) 2  0, 
1 ,  2 , 1 ,  2 ,  1 ,  2 ,1 , 2  0.



0.12 
Solving the above system (*), we obtain:
1  0, 2  0,1  0, 2  0, ,  1 ,  2  0,1  0,2  0 ,
and
 2  0.444,  3   3  4,
Therefore, the stability set of the first kind corresponding to the integer optimal
solution ( x1* , x2* )  (4,0) is given by:

(  ,  ,  ,  ,  ,  , , )  R 8 / 1 ,  2  0, 1  0,  2  0,
S (4, 0)  T (4, 0)   1 2 1 2 1 2 1 2

 1 ,  2  0, 1  0, 2  0,  2  0.444,  3   3  4






ACKNOWLEDGMENT
This paper based on a part of Eman F.Abdellah Ph.D. thesis at Helwan University,
Cairo, Egypt. The helpful comments and suggestions of anonymous referees are
gratefully acknowledged.
11
6. Conclusions
In this paper, a powerful approach based the   constraint method to solve chanceconstrained
multiobjective
integer
linear
fractional
programming
problem
(CHMOILFP) has been suggested. It has been assumed that there was randomness in
the right-hand side of the constraints only and that the random variables were
normally distributed. Some stability notions for the problem under consideration have
been defined and characterized.
Certainly, there are many other points for future research in the area of stability of
stochastic multiobjective integer linear fractional programming problems and should
be studied and explored. One may have to tackle the following open points for future
research:
(i)Stability of chance-constrained bi-level multiobjective integer linear fractional
programming problems.
(ii) Stability of chance constrained multi-level multiobjective integer linear fractional
programming problems.
(iii) Stability of chance constrained bi-level multiobjective mixed - integer and zeroone nonlinear fractional programming problems.
(iv) Stability of chance constrained bi-level large-scale multiobjective mixed - integer
and zero-one nonlinear fractional programming problems.
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12
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