STUDIES ON IMPRECISE ECONOMIC ORDER QUAN

(1)

Journal of Global Research in Mathematical Archives

UGC Approved Journal

RESEARCH PAPER

Available online at http://www.jgrma.info

STUDIES ON IMPRECISE ECONOMIC ORDER QUANTITY MODEL USING

INTERVAL PARAMETER

Asim Kumar Das

1*

,Tapan Kumar Roy

2

.

* Email id: asd.math@gmail.com

Department of Applied Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah,

West-Bengal, India, 711103.

Abstract: In this paper, we introduce an imprecise economic order quantity (EOQ) model with demand, holding cost and set up cost are assumed as an interval number. We consider the parameters of the proposed model with imprecise data as form of interval number. The proposed EOQ model is presented with impreciseness of parameters by introducing parametric functional form of interval number and then solves the problem by geometric programming technique. Numerical example is presented to support of the proposed approach.

Keywords: EOQ model, holding cost, set up cost, Interval number, Geometric programming

INTRODUCTION

An inventory deals with decision that minimize the cost function or maximize the profit function. For this purpose the task is to construct a suitable mathematical model of the real life Inventory system, such a mathematical model is based on various assumption and approximation. This type of imprecise data is not always well represented by random variables selected from probability distribution. So decision making methods under uncertainty are needed. To deal with this uncertainty and imprecise data, the concept of fuzziness can be applied.

(2)

© JGRMA 2016, All Rights Reserved 17 with or without shortage[5]. S. Islam, T.K. Roy (2006) presented a fuzzy EPQ model with flexibility and reliability consideration and demand depended unit Production cost under a space constraint[9-10]. Geometric programming plays an important role for optimization for all above models developed by the authors.

The rest of the paper is organized as follows: In section II, we introduce some basic concepts and definition of interval number. Imprecise EOQ model using interval number is discussed in section III. In section IV, numerical example has been presented for different values of the parameter p



(0, 1) to illustrate the proposed model. Finally, conclusion and future research are drawn in Section V.

II. Some basic concept and definition

Pre-requisite mathematics

In this section we discuss some preliminary mathematics which we have used to study the imprecise EOQ model.

Definition 1 (Interval number): An interval number A is represented by closed interval [ and defined by A=[ = {x:



R}, where R is the set of real numbers and are the left and right limit of the interval number respectively.

Now we define interval-valued function which will be used to present an interval number.

Definition 2 (Interval-valued function): Let c, d >0 and consider the interval is of the form [c, d], the interval-valued function of the interval is represented as h (p) = for p



[0, 1].

Now we present some arithmetic operations on interval valued functions. Let A=[ and B= [ be two interval number so that > 0.

Addition: A+B = [ +[ = [ . The interval-valued function for the interval number A +B is given by h (p) = where and .

Subtraction: A



B =[



[ = [ . Provided > 0. The interval-valued function for the interval number A



B is given by h (p) = , where and .

Scalar multiplication: βA = β[ ={[

[ provided > 0. The interval-valued function for the interval

number βA is given by h (p) = _{if β ≥ 0 and}_{h (p) =} _{if β < 0}_where _and _,



a

l and



a

u.

III. IMPRECISE ECONOMIC ORDER QUANTITY (EOQ) OR ECONOMIC LOT SIZE (ELS) MODEL

Economic Order Quantity (EOQ) or Economic Lot Size (ELS) model with uniform rate of demand infinite production rate and has no shortages.

We derive an imprecise EOQ formula and the minimum total average cost under the following assumption and notation:

1) The inventory system involves only one item.

2) The demand rate is known constant and occurs uniformly but imprecise, is presented in terms of interval number. 3) Holding cost per unit quantity per unit time is known constant but imprecise, is presented in terms of interval number. 4) Fixed ordering cost or set up cost per order is known constant but imprecise is presented in terms of interval number. 5) Production or the re-supply of the item is instantaneous (i.e. production or re-supply rate is infinite.)

6) Lead time is zero.

7) Shortages are not allowed. Notations:



D

ˆ

[ : Demand rate, units per unit time.



1

ˆ

(3)



3

ˆ

C

[ : Fixed ordering cost or set up cost per order ( or production run)

Q : Order quantity (or lot size),i.e., number of units ordered per order (units)

U : Purchasing cost per unit quantity.

q(t) : Inventory level at any time, t≥0.

T: Cycle of length of the given inventory.

TAC(T): Total average cost per unit time.

The parametric functional form of the interval numbers

D

ˆ

,

C

ˆ

₁ ,

C

ˆ

₃are presented as follows

D(p) = , = , = for

p



[0,1]

For p= 0 and 1 we have the left end and right end value of the interval.

Now q(t) is the inventory level at time [ , the differential equation for the instantaneous inventory level q(t) at any time t over [ is

D

ˆ

for (3.1)

With initial condition q(0) = Q (3.2)

And boundary condition q(T) = 0 (3.3)

From (4.1) , using initial condition (3.2) , we get

D

ˆ

(3.4)

Using boundary condition (3.3) in (3.4), we get Q =

_D

ˆ

_฀T (3.5)

Holding Cost (HC)

C

ˆ

₁_∫

C

ˆ

1∫ (

D

ˆ

)

C

ˆ

₁

D

ˆ

[ using (3.5) ] (3.6)

Total cost = Set up cost + Holding cost + Purchasing cost

C

ˆ

₃

C

ˆ

₁

D

ˆ

(3.7)

Total Average cost i.e. TAC(T) [

C

ˆ

₃

C

ˆ

₁

D

ˆ

]

3

ˆ

C

1

ˆ

C

D

ˆ

D

ˆ

[using(3.5)] (3.8)

So, Problem is

D

ˆ

3

ˆ

C

1

ˆ

C

D

ˆ

(3.9)

Such that .

(4)

(3.10) can be taken as a primal geometric programming problem with degree of difficulty (DD)

Degree of Difficulty (DD)

Its dual geometric programming problem is

(

C

ˆ

3)

C

ˆ

1

D

ˆ

; are dual variables

Again from primal dual relations, we get

3

The representation of T*, Q* and TAC*(T) in terms of interval-valued function is as follows

(5)

A manufacturing company produce an item whose demand is almost 50-60 units per year. The production cost of one item is $ 200 and the holding cost per item is near about $ (10-15) per year. The replacement is instantaneous and no shortages are allowed.

We shall now calculate for different values of p



฀(0,1)

1) The economic lot size,( Q)

2) Optimal total average cost (TAC)

3) Optimal time period (T)

Where the set up cost is assumed almost $(100-110).

For p= 0 and 1 we have the left end and right end value of the interval, which is as our classical problem, here we study only for intermediate values of in between 0 and 1.

This is given in terms of tables representation and graphical representation.

TABLE: Optimum value of T*, Q* and

TAC

*

(

T

)

for different values of ‘p’

rough sketch of p versus T* graph

values of p

rough sketch of p versus Q* graph

values of p

rough sketch of p versus Q* graph

(6)

© JGRMA 2016, All Rights Reserved 21 imprecise constant in realistic sense. This impreciseness are represented here with an interval number which are introduced in terms of parametric functional form of an interval. This new approach of handling the impreciseness has an advantage that when p varies from 0 to 1, we get lower bound, intermediate value and upper bound of the interval valued number to express the nature of the model. The objective goals are not always precise. The authority allows some flexibility to attain his target. Using this procedure an authority can achieve their target by varying the level of optimistic value of p from 0 and 1. The model is illustrated with a practical example. Geometric Programming (GP) method is used here to solve the problem. The model can be easily extended to any other inventory problems with other constraints. The method presented here is quite general and can be applied to the real life inventory problems faced by the practitioners in industry or in other areas.

References

2) Cheng.T.E.C “An economic order quantity model with demand-dependent unit cost”, European Journal of Operation Research, 40(1989), 252-256.

3) Donaldson, W.A., “Inventory replenishment policy for a linear trend in demand - an analytical solution”, Operational Research Quarterly, 28 (1977) 663-670.

4) Duffin, R. J., Peterson, E. L. and Zener, C. Geometric Programming-Theory and Application. New York: John Wiley. 1967.

5) D. Dutta, Pravin Kumar, Fuzzy inventory without shortages using trapezoidal fuzzy number with sensitivity analysis, IOSR Journal of mathematics, 4 (3) (2012) 32-37.

6) Geunes.J, Shen.J.Z, Romeijn.H.E, Economic ordering Decision with Market Choice Flexibility, DOI 10.1002/nav.10109, June 2003.

7) Kicks.P,and Donaldson, W.A., “Irregular demand: assessing a rough and ready lot size formula”, Journal of Operational Research Society, 31 (1980) 725-732.

8) Kochenberger, G. A. Inventory models: Optimization by geometric programming. Decision Sciences. 1971. 2: 193–205.

9) Islam.S, Roy.T.K, A fuzzy EPQ model with flexibility and reliability consideration and demand depended unit Production cost under a space constraint: A fuzzy geometric programming approach, Applied Mathematics and Computation, 176 (2) (2006) 531-544.

10) Islam.S , Roy.T.K, Modified Geometric programming problem and its applications, J. Appt. Math and computing, 17 (1) (2005) 121-144.

11) Liu, S. T. Using geometric programming to profit maximization with interval coefficients and quantity discount. Applied Mathematics and Computation. 2009. 209: 259–265.

12) Mahapatra,G.S. Mandal,T.K, “Posynomial parametric Geometric programming with Interval Valued Coefficient”, J Optim Theory Appl.(2012) 154: 120-132

13) Ritchie. E., “Practical inventory replenishment policies for a linear trend in demand followed by a period of steady

demand”, Journal of Operational Research Society, 31 (1980) 605-613.

14) Ritchie. E., “The EOQ for linear increasing demand: a simple optimal solution” Journal of Operational Research Society, 35 (1984) 949-952.

15) Roy.T.K, Maity.M, “A fuzzy EOQ model with demand-dependent unit under limited storage capacity”, European Journal of Operation Research, 99(1997) 425-432.

16) Silver.E.A“A simple inventory replenishment decision rule for a linear trend in demand”, Journal of Operational Research Society, 30 (1979) 71-75.

17) Silver. E.A., and Meal. H.C., “A simple modification of the EOQ for the case of a varying demand rate”, Production and Inventory Management, 10(4) (1969) 52-65.

18) Taha. A.H, Operations Research: An Introduction,chapter11/8th edition, ISBN 0-13-188923·0.