Inventory Model with Price-Dependent Demand Rate and No Shortages: An Interval-Valued Linear Fractional Programming Approach

Operations Research and Applications : An International Journal (ORAJ), Vol.2, No.4, November 2015
DOI : 10.5121/oraj.2015.2402 17
Inventory Model with Price-Dependent Demand
Rate and No Shortages: An Interval-Valued
Linear Fractional Programming Approach
Dr. Pavan Kumar
Assistant Professor, Department of Mathematics,
Faculty of Science and Technology,
IIS University, Jaipur-302020, India.
Abstract
In this paper, an interval-valued inventory optimization model is proposed. The model involves the price-
dependent demand and no shortages. The input data for this model are not fixed, but vary in some real
bounded intervals. The aim is to determine the optimal order quantity, maximizing the total profit and
minimizing the holding cost subjecting to three constraints: budget constraint, space constraint, and
budgetary constraint on ordering cost of each item. We apply the linear fractional programming approach
based on interval numbers. To apply this approach, a linear fractional programming problem is modeled
with interval type uncertainty. This problem is further converted to an optimization problem with interval-
valued objective function having its bounds as linear fractional functions. Two numerical examples in crisp
case and interval-valued case are solved to illustrate the proposed approach.
Keywords
Inventory model, Price-dependent demand, Interval-valued function, Fractional programming.
1. INTRODUCTION
For many practical inventory problems, some parameters are uncertain. Therefore to obtain more
realistic results, the uncertainty in parameters must be considered, and the corresponding
uncertain optimization methods must be constructed. Some techniques have been developed to
solve uncertain optimization problems. The fuzzy and stochastic approaches are normally applied
to describe imprecise characteristics. In these two types of techniques, the membership function
and probability distribution play important roles.
This is, however, practically difficult to specify a precise probability distribution or membership
function for some uncertain parameter. Over the past few decades, the interval number
programming approach has been developed to deal with such type of problems in which the
bounds of the uncertain parameters are only required. An optimization problem with interval
coefficients is referred as the interval-valued optimization problem. In this case, the coefficients
are assumed as closed intervals.
Charnes et al (1977) proposed a concept to solve the linear programming problems in which the
constraints are considered as closed intervals. Steuer (1981) developed some algorithms to solve
the linear programming problems with interval objective function coefficients. Alefeld and
Herzberger (1983) introduced the concept of computations on interval-valued numbers.
Ishibuchi and Tanaka (1990), and then Chinneck and Ramadan (2000) proposed the linear

18
programming problem (LPP) with interval coefficients in objective function. Tong (1994) studied
the interval number and fuzzy number linear programming, and considered the case in which the
coefficients of the objective function and constraints are all interval numbers. The possible
interval-solution was obtained by taking the maximum value range and minimum value range
inequalities as constraint conditions.
Moore and Lodwick (2003) further extended the theory of interval numbers and fuzzy numbers.
Mráz (1998) proposed an approach to determine the exact bounds of optimal values in LP with
interval coefficients. In the interval number programming, the objective function and constraints
may not always be linear, but are often nonlinear. Levin (1999) introduced the nonlinear
optimization under interval uncertainty. Sengupta et al (2001) studied the Interpretation of
inequality constraints involving interval coefficients and a solution to interval linear
programming.
The optimality conditions of Karush–Kuhn–Tucker (KKT) possess a significant role in
optimization theory . In literature, various approaches to interval-valued optimization problems
have been proposed in details, while few papers studied the KKT conditions for interval-valued
optimization problems. Wu (2007) used the KKT conditions to solve an optimization problem
with interval-valued objective function. Wu (2008a) studied the interval-valued nonlinear
programming problems, and then Wu (2008b) & (2010) established the Wolfe duality for
optimization problems with interval-valued objective functions. Bhurjee and Panda (2012)
suggested a technique to obtain an efficient solution of interval optimization problem.
In single objective optimization, the aim is to find the best solution which corresponds to the
minimum or maximum value of a single objective function. As far as the applications of
nonlinear programming problems are considered, a ratio of two functions is to be maximized or
minimized under certain number of constraints. In some other applications, the objective function
involves more than one ratio of functions. The problems of optimizing one or more ratios of
functions are referred as fractional programming problems. Normally, most of the multi objective
fractional programming problems are first converted into single objective fractional programming
problems and then solved. The decision maker (DM) must altogether optimize these conflicting
goals in a framework of fuzzy aspiration levels. Charnes and Cooper (1962) proposed that a
linear fractional program with one ratio can be reduced to a linear program using a nonlinear
variable transformation. Chanas and Kuchta (1996) generalized the solution concepts of the
linear programming problem with interval coefficients in the objective function based on
preference relations between intervals.
In the recent years, a kind of new applications of fractional programming was found in inventory
optimization problems. Sadjadi et al (2005) proposed a fuzzy approach to solve a multi objective
linear fractional inventory model. They considered two objective functions as fractional with two
constraints: space capacity constraint, and budget constraint, and aimed to simultaneously
maximize the profit ratio to holding cost and to minimize the back orders ratio to total ordered
quantities. Carla Oliveira et al (2007) presented an overview of multiple objective linear
programming models with interval coefficients. Some applications of multi objective linear
fractional programming in inventory were proposed by Toksarı (2008). Effati et al (2012)
presented an approach to solve the interval-valued linear fractional programming problem.
Liu (2006) derived a computational method for the profit bounds of inventory model with
interval demand and unit cost. Liu and Wang (2007) developed a method for numerical solution
to interval quadratic programming. Liu (2008) discussed the posynomial geometric programming
with interval exponents and coefficients.

19
Mishra (2007) studied some problems on approximations of functions in Banach Spaces, and
then Deepmala (2014) studied the fixed point theorems for nonlinear contractions and its
applications. Recently, Vandana and Sharma (2015) proposed an EPQ inventory model for
non-instantaneous deteriorating item under trade credit policy. The concept of fuzzy was also
used in case of fractional inventory model. Kumar and Dutta (2015) proposed a multi-objective
linear fractional inventory model of multi-products with price-dependant demand rate in fuzzy
environment.
This paper aims at extending the application of linear fractional programming to inventory
optimization problems in interval uncertainty. We consider an interval-valued fractional objective
function and interval-valued constraints for the proposed inventory problem. The paper is
organized as follows. Some preliminaries are given in Section 2. In Section 3, the notations and
assumptions are given. In Section 4, a linear fractional programming inventory problem is
proposed in crisp sense. In Section 5, we formulate above problem considering interval
uncertainty. Two numerical examples are solved in Section 6. Finally, we conclude in Section 7.
2. PRELIMINARIES
Interval-Valued Number (Moore & Lodwick (2003)):
Let ∗ ∈ {+, −, ., /} be a binary operation on real line ℜ. Let A and B be two closed intervals.
Then, a binary operation is defined as
A ∗ B = {a ∗ b : a ∈ A, b ∈ B}
In the case of division, it is assumed that 0∉B. The operations on intervals used in this paper are
defined as:
a) kA = ൜
ሾ݇ܽ௅, ݇ܽோሿ ; ݂݅ ݇ ≥ 0
ሺ݇ܽோ, ݇ܽ௅ሻ ; ݂݅ ݇ < 0
(1)
b) A + B = [ܽ௅ , ܽோ] + [ܾ௅ , ܾோ]
= [ܽ௅ + ܾ௅, ܽோ + ܾோ] (2)
c) A − B = [ܽ௅ , ܽோ] − [ܾ௅ , ܾோ]
= [ܽ௅ , ܽோ]+[−ܾோ ,−ܾ௅]
= [ܽ௅−ܾோ, ܽோ−ܾ௅] (3)
d) A × B = [min{ܽ௅ܾ௅, ܽ௅ܾோ, ܽோܾ௅, ܽோܾோ}, max{ܽ௅ܾ௅, ܽ௅ܾோ, ܽோܾ௅, ܽோܾோ}] (4)
e)
୅
୆
= A ×
ଵ
஻
= [ܽ௅ , ܽோ] × [
ଵ
௕ೃ
,
ଵ
௕ಽ
] , provided 0 ∉ [ܾ௅ , ܾோ],
Alternatively, division of interval numbers can be defined as
୅
୆
=
ሾ௔ಽ , ௔ೃሿ
ሾ௕ಽ , ௕ೃሿ
= [min{
௔ಽ
௕ಽ
,
௔ಽ
௕ೃ
,
௔ೃ
௕ಽ
,
௔ೃ
௕ೃ
}, max{
௔ಽ
௕ಽ
,
௔ಽ
௕ೃ
,
௔ೃ
௕ಽ
,
௔ೃ
௕ೃ
}], (5)
Alternatively, division of interval numbers can be defined as
୅
୆
=
ሾ௔ಽ , ௔ೃሿ
ሾ௕ಽ , ௕ೃሿ
= [
௔ಽ
௕ೃ
,
௔ೃ
௕ಽ
], where 0∉B, 0 ≤ ܽ௅ ≤ ܽோ, 0 < ܾ௅ ≤ ܾோ. (6)

20
f) ‫ܣ‬௞
= ሾܽ௅, ܽோሿ௞
=
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬
ሾ1, 1ሿ ; ݂݅ ݇ = 0
ൣܽ௅
௞
, ܽோ
௞
൧; ݂݅ ܽ௅ ≥ 0 or k odd
ൣܽோ
௞
, ܽ௅
௞
൧; ݂݅ ܽோ ≤ 0 or k even
ൣ0, max൫ܽ୐
୩
, ܽୖ
୩
൯൧; if ܽ௅ ≤ 0 ≤ ܽோ, and ݇ሺ> 0ሻis even
(7)
Interval-Valued Function: Let ℜ௡
be an n-dimensional Euclidean space. Then a function
݂: ℜ௡
→ ‫ܫ‬
is called an interval valued function (because ݂ሺ‫ݔ‬ሻ for each x∊ ℜ௡
is a closed interval in ℜ).
Similar to interval notation, we denote the interval-valued function ݂ with
݂ሺ‫ݔ‬ሻ = ሾ݂௅ሺ‫ݔ‬ሻ, ݂௎ሺ‫ݔ‬ሻሿ,
where for each x∊ ℜ௡
, ݂௅ሺ‫ݔ‬ሻ, ݂௎ሺ‫ݔ‬ሻ are real valued functions defined on ℜ௡
, called the lower
and upper bounds of ݂ሺ‫ݔ‬ሻ, and satisfy the condition:
݂௅ሺ‫ݔ‬ሻ ≤ ݂௎ሺ‫ݔ‬ሻ.
Proposition 1. Let ݂ be an interval valued function defined on ℜ௡
. Then ݂ is continuous at c
∊ ℜ௡
if and only if ݂௅
ܽ݊݀ ݂௎
are continuous at c.
Definition 1. We define a linear fractional function F(x) as follows:
F(x) =
௖௫ାఈభ
ௗ௫ାఈమ
(8)
Where x = ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, … , ‫ݔ‬௡ሻ௧
∊ ℜ௡
, c = ሺܿଵ, ܿଶ, … , ܿ௡ሻ ∊ ℜ௡
, d = ሺ݀ଵ, ݀ଶ, … , ݀௡ሻ ∊ ℜ௡
, and ߙଵ,
ߙଶ are real scalars.
Definition 2. Let F : V →I be an interval-valued function, deﬁned on a real vector space V. Now,
we consider the following interval-valued optimization problem
(IVP) Min≼ ‫ܨ‬ሺ‫ݔ‬ሻ = [‫ܨ‬௅ሺ‫ݔ‬ሻ, ‫ܨ‬௎ሺ‫ݔ‬ሻ]
Subject to x ∊ Y ⊆ V, (9)
where Y is the feasible set of problem (IVP) and ≼ is the partial ordering on the set of integers.
Definition 3. Let ≼ be a partial ordering on the set of integers. Then, for ܽ = ሾܽ௅
, ܽ௎ሿ, and
ܾ = ሾܾ௅
, ܾ௎ሿ, we write
ܽ ≼ ܾ
if and only if ܽ௅
≤ ܾ௅
and ܽ௎
≤ ܾ௎
.
This means that ܽ is inferior to ܾ or ܾ is superior to ܽ.
Definition 4. Let F :V →I be an interval-valued function, deﬁned on a real vector space V, and
let Y be a subset of V. Suppose that we are going to maximize F. We say that F (‫̅ݔ‬) is a non-
dominated objective value if and only if there exists no x (≠ ‫̅ݔ‬) ∊ ܻ such that F(‫̅ݔ‬) ≺ ‫ܨ‬ሺ‫ݔ‬ሻ.

21
Interval-Valued Linear Fractional Programming (IVLFP)
Minimize f(x) =
௖௫ାఈభ
ௗ௫ାఈమ
Subject to Ax = b (10)
x ≥ 0
Suppose c = ሺܿଵ, ܿଶ, … , ܿ௡ሻ , d = ሺ݀ଵ, ݀ଶ, … , ݀௡ሻ, where ܿ௝, ݀௝ ∊ ‫,ܫ‬ j = 1,2, ..., n.
Define ܿ௅
= ሺܿଵ
௅
, ܿଶ
௅
, … , ܿ௡
௅
ሻ , ܿ௎
= ሺܿଵ
௎
, ܿଶ
௎
, … , ܿ௡
௎ሻ, ݀௅
= ሺ݀ଵ
௅
, ݀ଶ
௅
, … , ݀௡
௅ሻ, ݀௎
=
ሺ݀ଵ
௎
, ݀ଶ
௎
, … , ݀௡
௎ሻ, where ܿ௝
௅
, ܿ௝
௎
, ݀௝
௅
, ݀௝
௅
are real scalars for j = 1,2, ..., n. Also, ߙଵ = ሾߙଵ
௅
, ߙଵ
௎
ሿ,
ߙଶ = ሾߙଶ
௅
, ߙଶ
௎
ሿ.
Then IVLFP (10) can be re-written as:
Minimize f(x) =
௣ሺ௫ሻ
௤ሺ௫ሻ
x ≥ 0,
where p(x) and q(x) are interval-valued linear functions and are given by
p(x) = [‫݌‬௅ሺ‫ݔ‬ሻ, ‫݌‬௎ሺ‫ݔ‬ሻ] = [ܿ௅
‫ݔ‬ + ߙଵ
௅
, ܿ௎
‫ݔ‬ + ߙଵ
௎
],
q(x) = [‫ݍ‬௅ሺ‫ݔ‬ሻ, ‫ݍ‬௎ሺ‫ݔ‬ሻ] = [݀௅
‫ݔ‬ + ߙଶ
௅
, ݀௎
‫ݔ‬ + ߙଶ
௎
].
Therefore, IVLFP (11) can re-written as
Minimize f(x) =
ሾ௖ಽ௫ାఈభ
ಽ
, ௖ೆ௫ାఈభ
ೆ
ሿ
ሾௗಽ௫ାఈమ
ಽ, ௗೆ௫ାఈమ
ೆሿ
Subject to Ax = b
x ≥ 0 (12)
Further, using the concept of division of two interval-valued numbers, IVLFP (12) can re-written
as
Minimize f(x) =[݂௅ሺ‫ݔ‬ሻ, ݂௎ሺ‫ݔ‬ሻ]
x ≥ 0
where ݂௅ሺ‫ݔ‬ሻ ܽ݊݀ ݂௎ሺ‫ݔ‬ሻ are linear fractional functions.
Definition 5. (Wu (2008)) Let ‫ݔ‬∗
be a feasible solution of IVLFP (13). Then, ‫ݔ‬∗
is a non-
dominated solution of IVLFP (13) if there exist no feasible solution x such that ݂(x) ≺ ݂ሺ‫ݔ‬∗
ሻ. In
this case, we say that ݂ሺ‫ݔ‬∗
ሻ is the non-dominated objective value of ݂.
Corresponding to IVLFP (13)), consider the following optimization problem:
Minimize g(x) = ݂௅ሺ‫ݔ‬ሻ + ݂௎ሺ‫ݔ‬ሻ
x ≥ 0
To solve IVLFP (14), we use the following theorem (Wu (2008)).
Theorem. Wu (2008) If ‫ݔ‬∗
is an optimal solution of IVLFP(14), then ‫ݔ‬∗
is a non-dominated
solution of IVLFP(13).
Proof. Let us refer to Wu(2008).

22
3. NOTATIONS AND ASSUMPTIONS
To develop the proposed model, we considered the following notations and assumptions:
Notations
λ : Fixed cost per order
B : Maximum available budget for all products
W : Maximum available space for all products
For ith
product: (i = 1, 2, …, n)
ܳ௜ : Ordering quantity of product i,
ℎ௜ : Holding cost per product per unit time for ith
product
ܲ௜ : Purchasing price of ith
product
ܵ௜ : Selling price of ith
product
‫ܦ‬௜ : Demand quantity per unit time of ith
product
݂௜ : Space required per unit for ith
product
ܱ‫ܥ‬௜ : Ordering cost of ith
product.
Assumptions
1. Multiple products are considered in this model.
2. Infinite time horizon is considered. Further, there is only one period in the cycle time.
3. Lead time is zero, and so rate of replenishment is infinite.
4. Holding cost is known and constant for each product.
5. Demand is taken here as inversely related to the selling price of the product, that is,
‫ܦ‬௜ = ‫ܦ‬௜(ܵ௜) = ݉ଵܵ௜
ି௠మ
,
where ݉ଵ > 0 is a scaling constant, and ݉ଶ > 1 is price-elasticity coefficient. For
notational simplicity,
we will use ‫ܦ‬௜ and ‫ܦ‬௜(ܵ௜) interchangeably.
6. Shortages are not allowed.
7. Purchase price is constant for each product, i.e. no discount is available.
4. CRISP LINEAR FRACTIONAL PROGRAMMING INVENTORY
PROBLEM
A multi-product inventory model under resources constraints is introduced as a linear fractional
programming problem. This model refers to a multi-product inventory problem, with limited
capacity of warehouse and constraints on investment in inventories. For each product, we impose
the constraint on ordering cost.
We practically observe the inventory models with more than one objective functions. These
objectives may be in conflict with each other, or may not be. In such type of inventory models,
the decision maker is interested to maximize or minimize two or more objectives simultaneously
over a given set of decision variables. We call this model as linear fractional inventory model.
Without loss of generality, we assume there is only one period in the cycle time. Then,
Total Profit = ∑ ሺܵ௜ − ܲ௜ሻܳ௜
௡
௜ୀଵ (15a)
Holding cost = ∑
௛೔ொ೔
ଶ
௡
௜ୀଵ (15b)

23
Ordering cost = ∑
λ஽೔
ொ೔
௡
௜ୀଵ = ∑
λ௠భௌ೔
ష೘మ
ொ೔
௡
௜ୀଵ (15c)
Back ordered quantity = ∑ ሺ‫ܦ‬௜ − ܳ௜
௡
௜ୀଵ ሻ = ∑ ሺ݉ଵܵ௜
ି௠మ
− ܳ௜
௡
௜ୀଵ ሻ (15d)
Total ordered quantity = ∑ ܳ௜
௡
௜ୀଵ (15e)
In this paper, the crisp linear fractional programming inventory model is formulated as
Maximize: Z =
୘୭୲ୟ୪ ୔୰୭ϐ୧୲
ୌ୭୪ୢ୧୬୥ ୡ୭ୱ୲
,
Subject to, (i) Total Budget Limit,
(ii) Storage Space Limit,
(iii) Limit on Ordering Cost of Each Product.
(iv) Non-Negative Restriction.
Formulation of Constraints
(i) Total Budget Limit: A budget constraint is all about the combinations of goods and
services that a consumer may purchase given current prices within his or her given
income. For the proposed model, the budget constraint is
∑ ܲ௜ܳ௜
௡
௜ୀଵ ≤ ‫ܤ‬ (16a)
(ii) Storage Space Limit: ∑ ݂௜ܳ௜
௡
௜ୀଵ ≤ ܹ (16b)
(iii) Limit on Ordering Cost of Each Product: We impose the upper limit of ordering
cost as a constraint. Since ܱ‫ܥ‬ଵ, ܱ‫ܥ‬ଶ, …, ܱ‫ܥ‬௡ are the ordering cost of 1st
product, 2nd
product,…, nth
product, we can express the concerned constrained as follows:
For first product,
λ஽భ
ொభ
≤ ܱ‫ܥ‬ଵ
⇒ λ‫ܦ‬ଵ − ܱ‫ܥ‬ଵ. ܳଵ ≤ 0
For second product,
λ஽మ
ொమ
≤ ܱ‫ܥ‬ଶ
⇒ λ‫ܦ‬ଶ − ܱ‫ܥ‬ଶ. ܳଶ ≤ 0
Similarly, for nth
product, λ‫ܦ‬௡ − ܱ‫ܥ‬௡. ܳ௡ ≤ 0
⇒ λ݉ଵܵ௡
ି௠మ
− ܱ‫ܥ‬௡. ܳ௡ ≤ 0 (16c)
(iv) Non-Negative Restriction:
ܳଵ, ܳଶ , …, ܳ௜ ≥ 0 (16d)
5. INTERVAL-VALUED LINEAR FRACTIONAL PROGRAMMING
INVENTORY PROBLEM
In real life cases, it is observed that the selling price is uncertain due to dynamic behavior of the
market. So it may be considered to vary in an interval. Let the selling price be represented by the
interval [ܵ௜௅, ܵ௜௎], where ܵ௜௅ & ܵ௜௎ are the left limit & right limit of interval for ܵ௜.
Mathematically, we can express as
ܵ௜ = [ܵ௜௅, ܵ௜௎], with ܵ௜௅ ≤ ܵ௜௎ (17)

24
Similarly, the purchasing, holding cost, and demand would be expressed as interval-valued
numbers as
ܲ௜ = [ܲ௜௅, ܲ௜௎], with ܲ௜௅ ≤ ܲ௜௎ (18)
ℎ௜ = [ℎ௜௅, ℎ௜௎], with ℎ௜௅ ≤ ℎ௜௎ (19)
‫ܦ‬௜ = [݉ଵܵ௜௅
ି௠మ
, ݉ଵܵ௜௎
ି௠మ
], with ݉ଵܵ௜௅
ି௠మ
≤ ݉ଵܵ௜௎
ି௠మ
(20)
Accordingly, the constraints transform to the interval-valued form as follows
Total Budget Limit: ∑ ሾܲ௜௅, ܲ௜௎ሿܳ௜
௡
௜ୀଵ ≤ ‫ܤ‬ (21)
Storage Space Limit: ∑ ݂௜ܳ௜
௡
௜ୀଵ ≤ ܹ (22)
Limit on Ordering Cost of each Product:
For nth
product, we have
λሾ݉ଵܵ௡௅
ି௠మ
, ݉ଵܵ௡௎
ି௠మ
ሿ ≤ ܳ௡[ܱ‫ܥ‬௡௅, ܱ‫ܥ‬௡௎ሿ (23)
Non-Negative Restriction:
ܳଵ, ܳଶ , …, ܳ௜ ≥ 0 (24)
In this case, we define the following interval-valued fuzzy linear fractional programming problem
for the proposed inventory model,
(IVP1) Maximize: Z =
∑ ሺሾௌ೔ಽ, ௌ೔ೆሿିሾ௉೔ಽ, ௉೔ೆሿሻொ೔
೙
೔సభ
∑
ሾ೓೔ಽ, ೓೔ೆሿೂ೔
మ
೙
೔సభ
, (25)
Subject to ∑ ሾܲ௜௅, ܲ௜௎ሿܳ௜
௡
௜ୀଵ ≤ B ,
∑ ݂௜ܳ௜
௡
௜ୀଵ ≤ W,
ܳ௜[ܱ‫ܥ‬௜௅, ܱ‫ܥ‬௜௎ሿ ≥ λሾ݉ଵܵ௡௅
ି௠మ
, ݉ଵܵ௡௎
ି௠మ
ሿ, (for i = 1, 2,.., n)
and ܳ௜ ≥ 0, (for i = 1, 2,.., n)
Using the arithmetic operations of intervals, the interval-valued fractional objective function can
be written as
Z =
∑ ሾௌ೔ಽି௉೔ೆ, ௌ೔ೆି௉೔ಽሿொ೔
೙
೔సభ
∑ ሾ
೓೔ಽೂ೔
మ
,
೓೔ೊ೔
మ
ሿ೙
೔సభ
=
ሾ∑ ሺௌ೔ಽି௉೔ೆሻொ೔
೙
೔సభ , ∑ ሺௌ೔ೆି௉೔ಽሻொ೔
೙
೔సభ ሿ
ሾ∑
೓೔ಽೂ೔
మ
೙
೔సభ , ∑
೓೔ೊ೔
మ
೙
೔సభ ሿ
= [
∑ ሺௌ೔ಽି௉೔ೆሻொ೔
೙
೔సభ
∑
೓೔ೊ೔
మ
೙
೔సభ
,
∑ ሺௌ೔ೆି௉೔ಽሻொ೔
೙
೔సభ
∑
೓೔ಽೂ೔
మ
೙
೔సభ
ሿ
= [ܼ௅ሺܳଵ, ܳଶ, … , ܳ௡ሻ, ܼ௎ሺܳଵ, ܳଶ, … , ܳ௡ሻ] (26)
where ܼ௅ሺܳଵ, ܳଶ, … , ܳ௡ሻ =
೙
೔సభ
∑
೓೔ೊ೔
మ
೙
೔సభ
,
and ܼ௎ሺܳଵ, ܳଶ, … , ܳ௡ሻ =
೙
೔సభ
∑
೓೔ಽೂ೔
మ
೙
೔సభ
(27)
Thus the single-objective function consisting of interval-valued number coefficients is
transformed into multi-objective functions given by (27) where, in each objective function, the
coefficients are crisp numbers. Hence, the interval-valued optimization problem (IVP1),
transforms to following optimization problem as follows,

25
(IVP2) Maximizeሾܼ௅ሺܳଵ, ܳଶ, … , ܳ௡ሻ , ܼ௎ሺܳଵ, ܳଶ, … , ܳ௡ሻ (28)
Subject to ∑ ሾܲ௜௅, ܲ௜௎ሿܳ௜
௡
௜ୀଵ ≤ B ,
∑ ݂௜ܳ௜
௡
௜ୀଵ ≤ W,
ܳ௜[ܱ‫ܥ‬௜௅, ܱ‫ܥ‬௜௎ሿ ≥ λሾ݉ଵܵ௜௅
ି௠మ
, ݉ଵܵ௜௎
ି௠మ
ሿ, (for i = 1, 2,.., n),
And ܳ௜ ≥ 0, (for i = 1, 2,.., n).
To solution this problem, the techniques of classical linear fractional programming cannot be
applied if and unless the above interval-valued structure of the problem be reduced into a
standard linear fractional programming problem.
To deal with interval inequality constraints, we use the Tong’s Approach [Tong (1994) &
Sengupta et al (2001)]. Tong deals with interval inequality constraints in a separate way.
According to Tong’s Approach, each interval inequality constraint is transformed into 2௡ାଵ crisp
inequalities. Let us take a simple inequality constraint with a single variable:
[10, 20]‫ݔ‬ ≤ [5, 35].
According to Tong (1994), the interval inequality generates 2ଵାଵ
crisp inequalities:
10‫ݔ‬ ≤ 5 ⇒ ‫ݔ‬ ≤ 0.5
10‫ݔ‬ ≤ 35 ⇒ ‫ݔ‬ ≤ 3.5
20‫ݔ‬ ≤ 5 ⇒ ‫ݔ‬ ≤ 0.25
20‫ݔ‬ ≤ 35 ⇒ ‫ݔ‬ ≤ 1.75
Hence, the interval-valued linear fractional inventory problem (IVP2) can be transformed to a
non-interval optimization problem as follows:
(IVP3) Maximize ሾܼ௅ሺܳଵ, … , ܳ௡ሻ, ܼ௎ሺܳଵ, … , ܳ௡ሻ] (29)
where ܼ௅ሺܳଵ, ܳଶ, … , ܳ௡ሻ =
೙
೔సభ
∑
೓೔ೊ೔
మ
೙
೔సభ
ܼ௎ሺܳଵ, ܳଶ, … , ܳ௡ሻ =
೙
೔సభ
∑
೓೔ಽೂ೔
మ
೙
೔సభ
Subject to: ∑ ܲ௜௅ܳ௜
௡
௜ୀଵ ≤ ‫ܤ‬
∑ ܲ௜௎ܳ௜
௡
௜ୀଵ ≤ ‫ܤ‬
∑ ݂௜ܳ௜
௡
௜ୀଵ ≤ ܹ
ܱ‫ܥ‬௜௅ܳ௜ ≥ λ݉ଵܵ௜௅
ି௠మ
, (for i = 1, 2,.., n)
ܱ‫ܥ‬௜௎ܳ௜ ≥ λ݉ଵܵ௜௅
ି௠మ
, (for i = 1, 2,.., n)
ܱ‫ܥ‬௜௅ܳ௜ ≥ λ݉ଵܵ௜௎
ି௠మ
, (for i = 1, 2,.., n)
ܱ‫ܥ‬௜௎ܳ௜ ≥ λ݉ଵܵ௜௎
ି௠మ
, (for i = 1, 2,.., n)
ܳ௜ ≥ 0, (for i = 1, 2,.., n)
Now, let us write the corresponding optimization problem of (IVP3) as follows:
(IVP4) Maximize g(x) = ܼ௅ሺܳଵ, ܳଶ, … , ܳ௡ሻ + ܼ௎ሺܳଵ, ܳଶ, … , ܳ௡ሻ (30)

26
To solve the (IVP3), theorem (Wu (2008)) is conveniently used. According to this theorem, the
non-dominated solution of (IVP3), would be determined by solving its corresponding
optimization problem (IVP4).
6. NUMERICAL EXAMPLES
To illustrate the proposed approach, we solve the following two numerical examples in crisp and
interval-value cases:
Example1. Crisp Case Consider an inventory model with following input data (in proper
units):
Table 1: Input data (Crisp Case)
Interval-Values Case Consider an inventory model with following input data:
Table 2: Input data (Interval-Valued Case)

27
The corresponding optimization problem would be:
Hence, the bounds of profit are 8084.79 (lower bound), and 12355.989 (upper bound).
Example2. Crisp Case Consider an inventory model with following input data (in proper
units):
Table 3: Input data (Crisp Case)

28
Interval-Values Case Consider an inventory model with following input data:
Table 4: Input data (Interval-Valued Case)
The corresponding optimization problem would be:
7. CONCLUSIONS
In this paper, the author presented an approach to solve interval-valued inventory optimization
problem based on linear fractional programming. The uncertainty in inventory parameters is
represented by interval-valued numbers. Using interval arithmetic, the interval-valued
optimization problem is changed into a crisp multi-objective linear fractional programming
problem. LINGO package is used to solve the subsequent optimization problem. The optimal
order quantity and profit bounds are determined.

29
For future directions, the paper can be extended to inventory model with shortages case and
inventory model with price-discount. Moreover, it can also be extended to interval-valued multi-
objective optimization case.
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