Solving the Interval-Valued Linear Fractional Programming Problem ()
1. Introduction
While modeling practical problems in real world, it is observed that some parameters of the problem may not be known certainly. Specially for an optimization problem it is possible that the parameters of the model be inexact. For example in a linear programming problem we may have inexact right hand side values or the coefficients in objective function may be fuzzy (e.g. [1]).
There are several approaches to model uncertainty in optimization problems such as stochastic optimization and fuzzy optimization. Here we consider an optimization problem with interval valued objective function. Stancu, Minasian and Tigan ([2,3]), investigated this kind of optimization problem. Hsien-Chung Wu ([4,5]) proved and derived the Karush-Kuhn-Tucker (KKT) optimality conditions for an optimization problem with interval valued objective function.
A fractional programming problem is the optimizing one or several ratios of functions (e.g. [6]). Such these models arise naturally in decision making when several rates need to be optimized simultaneously such as production planning, financial and corporate planning, health care and hospital planning. Several methods were suggested for solving this problem such as the variable transformation method [7] and the updated objective function method [8]. Several new methods are proposed ( e.g. [9-11]). The first monograph [12] in fractional programming published by the first author in 1978 extensively covers applications, theoretical results and algorithms for single-ratio fractional programs (see [13,14]).
Here first we introduce a linear fractional programming problem with interval valued parameters. Then we try to convert it to an optimization problem with interval valued objective function.
In Section 2 we state some required preliminaries from interval arithmetic. In Section 3 the interval valued linear fractional programming problem is introduced. In Section 4 we solved numerical examples. Finally Section 5 contains some conclusions.
2. Preliminaries
We denote by
the set of all closed and bounded intervals in
. Suppose
, then we write
and also
. We have the following operations on
(note that throughout this paper our intervals considered to be bounded and closed):
(i) 
(ii)
;
(iii) 
where
is a real number and so we have

Definition 2.1. If
and
are bounded, real intervals, we define the multiplication of
and
as follows:
where
. For example if
and
are positive intervals (i.e.
and
) then we have:
(1)
and if
and
then we have:
(2)
There are several approaches to define interval division. Following Ratz (see [15]) we define the quotient of two intervals as follows:
Definition 2.2. Let
and
be two real intervals, then we define:

We observe that the quotient of two intervals is a set which may not itself be an interval. For example,
. Given definition 2.2, The Ratz formula [15] is given by the following Theorem:
Theorem 2.1. ([15]) Let
and
be two nonempty bounded real intervals.
Then if
we have:
(3)
Theorem 2.2. (see [16]) If
and
are nonempty, bounded, real intervals, then so are
, and
. In addition, if
does not contain zero, then
is also a nonempty, bounded, real interval as well.
Definition 2.3. A function
is called an interval valued function (because
for each
is a closed interval in
). Similar to interval notation, we denote the interval valued function
with
where for every 
are real valued functions and

Proposition 2.1. Let
be an interval valued function defined on
. Then
is continuous at
if and only if
and
are continuous at c.
Now, here we introduce weakly differentiability.
Definition 2.4. Let
be an open set in
. An interval valued function
with
is called weak differentiable at
if the real valued functions
and
are differentiable (usual differentiability) at
.
Definition 2.5. We define a linear fractional function
as follows:
(4)
where
and
are real scalars.
Remark 2.1. Note that every real number
can be considered as an interval
.
Definition 2.6. To interpret the meaning of optimization of interval valued functions, we introduce a partial ordering
over I. Let
,
be two closed, bounded, real intervals
, then we say that
, if and only if
and
. Also we write
, if and only if
and
. In the other words, we say
if and only if:
or
or 
3. Interval-Valued Linear Fractional Programming (IVLFP)
Consider the following linear fractional programming problem:
(5)
First consider the linear fractional programming problem (5). Suppose that

where
, we denote
and
the lower bounds of the intervals
and
respectively (i.e.
and also
where
and
are real scalars for
) and
, similarly we can define
and
. Also
,
. So we can rewrite (5) as follows:
(6)
where
and
are interval-valued linear functions as 
and
. So for example we have:
and
. Finally from (6) we have:
(7)
To introduce an interval-valued linear fractional programming problem, we can consider another kind of possible linear fractional programming problems as follows:
(8)
where
and
are linear fractional functions (as in definition 2.5). Also we may have interval-valued linear fractional programming in the form (7):
(9)
Theorem 3.1. Any IVLFP in the form IVLFP(2) (see Equation (9)) under some assumptions can be converted to an IVLFP in the form IVLFP(1) (see Equation (8)).
Proof. The objective function in (9) is a quotient of two interval valued functions (
and
). To convert (9) to the form (8), we suppose that
for each feasible point
, so we should have:
(10)
or
(11)
for each feasible point
. Using theorem 2.1, because the denominator doesn’t contain zero we can rewrite the objective function in (9) as:
(12)
Now we can consider two possible states:
Case (1). When
, we have two possibilities:
(i) When
, using Definition 2.1, we have:
(13)
(ii) When
, by Definition 2.1, we have:
(14)
Case (2). When
, we have two possibilities:
(i) When
, by Definition 2.1, we have:
(15)
(ii) When
, by Definition 2.1, we have:
(16)
(Note that the subcase
easily can be derived from above cases, because in this state,
implies that
). Now according to theorem 2.2, and considering above cases, the objective function in (7) can be rewritten as follows:
(17)
where the objective function is an interval valued function and
and
are linear fractional functions (according to the corresponding case (13) - (16)), and this completes the proof.
Now following Wu [5], we interpret the meaning of minimization in (17):
Definition 3.1. (see [5]) Let
be a feasible solution of problem (17). We say that
is a nondominated solution of problem (17), if there exist no feasible solution x such that
. In this case we say that
is the nondominated objective value of
.
Now consider the following optimization problem (corresponding to problem (17)):
(18)
To solve problem (17), we use the following theorem from [5].
Theorem 3.2. If
is an optimal solution of problem (18), then
is a nondominated solution of problem (17).
Proof. See [5].
4. Numerical Examples
This section contains three numerical examples which are solved by the new proposed approach. Example 4.3 introduces an application of IVLFP.
Example 4.1. Consider the following optimization problem:
(19)
We see that here

and
.
So because
we have
and also
so we should apply case (1)(i). Finally we will have the following optimization problem:
(20)
Now to obtain a nondominated solution for (20), we use theorem 3.2. and solve the following optimization problem:
(21)
The optimal solution is
with optimal value
.
Example 4.2. Now consider the following optimization problem:
(22)
By Theorem 3.1, we can convert (22) to the following problem:
(23)
Now we can apply Theorem 3.2, and solve the optimization problem:
(24)
Finally a nondominated solution for (22) is
with
, which is the optimal solution of (24).
Example 4.3. Consider the following applied problem from [17]:
A company manufactures two kinds of products
,
with a uncertain profit of
,
dollar per unit respectively. .However the uncertain cost for each one unit of the above products is given by
,
dollar. It is assumed that a fixed cost of
dollars is added to the cost function due to expected duration through the process of production and also a fixed amount of
dollar is added to the profit function. If the objectives of this company is to maximize the profit in return to the total cost, provided that the company has a raw materials for manufacturing and suppose the material needed per pounds are 1, 3 and the supply for this raw material is restricted to 30 pounds, it is also assumed that twice of production of
is more than the production of
at most by 5 units. In this case if we consider
and
to be the amount of units of
,
to be produced then the above problem can be formulated as
(25)
The optimal solution is
,
with the objective value 
5. Conclusion
In this paper, first we introduced two possible types (Equations (8), (9)) of linear fractional programming problems with interval valued objective functions. Then we proved that we can convert the problem of the form (9) to the form (8). By solving (8), we obtained a nondominated solution for original linear fractional programming problem with interval valued objective function. Work is in progress to apply and check the approach for solving nonlinear fractional programming problems as well as for quadratic fractional programming problems.