Deteriorated Economic Production Quantity (EPQ) · PDF fileDeteriorated economic production quantity (EPQ) model 3609 it is used and this is highly correlated to the return of investment

Applied Mathematical Sciences, Vol. 8, 2014, no. 73, 3607 - 3618

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.43178

Deteriorated Economic Production Quantity (EPQ)

Model for Declined Quadratic Demand with Time-

value of Money and Shortages

Bhanupriya Dash

Kamala Nehru Womens’ College

Bhubaneswar, India

Monalisha Pattnaik

Dept. of Business Administration

Utkal University, Bhubaneswar

India-751004

Hadibandhu Pattnaik

Dept. of Mathematics

KIIT University, Bhubaneswar, India

Copyright © 2014 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik. This is an

open access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

Abstract

A deteriorating inventory model using time-value of money with price

dependent declined quadratic demand is developed for a deterministic inventory

system. This study applied the discounted cash flows (DCF) approach for problem

analysis. The objective of this model is to maximize the net present value profit so

as to determine the optimal time period and order quantity. The numerical

analysis shows that an appropriate policy can benefit the retailer and that policy is

important, especially for deteriorating items. Finally, sensitivity analysis of opti-

3608 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik

mal solution with respect to the major parameters are also studied to draw some

decisions with managerial implications for competitive advantage.

Keywords: Economic Production quantity (EPQ), Declined quadratic demand,

Deteriorating items, time-value of money, shortages

1 Introduction

Inventory modelling is an important part of operations research, which

may be used in variety of problems. To make it applicable in real life situation

researchers are engaged in modifying existing models on different parameters

under various circumstances. Inventories play a major role in the united stages

economy and have been in excess of 22% of the nation’s gross national product

over the past few decades. As millions of dollars are tied up in inventories, proper

management of these inventories can prove to be very profitable. A major concern

of inventory management is to know when and how much to order or manufacture

so that the total cost per unit time is minimized. The total cost consists of

carrying, shortage, replenishment or setup cost and the purchase or production

cost. Usually the time value of money is not considered explicitly in analysing

inventory systems, although the cost of capital tried up in inventories is included

in the carrying cost. Most of the literature on inventory control and production

planning has dealt with the assumption that the demand for product will continue

infinitely in the future either in a deterministic or in a stochastic fashion. This

assumption does not always hold true. Inventory management plays a significant

role for production system in business since it can help companies reach the goal

of ensuring prompt delivery, avoiding shortages, helping sales at competitive

prices and so forth for achieving competitive advantage in the globe. However,

excessive simplification of assumptions results in mathematical models that do

not represent the inventory situation to be analyzed.

The classical analysis builds a model of an inventory system and

calculates the EOQ which minimize the costs satisfying minimization criterion.

One of the unrealistic assumption is that items stocked preserve their physical

characteristics during their stay in inventory for long run. Items in stock are

subject to many possible risks, e.g. damage, spoilage, dryness, vaporization etc.,

those results decrease of usefulness of the original one and a cost is incurred to

account for such risks of the product. The problem of deteriorating

inventory has received considerable attention in recent years. This is a realistic

trend since most products such as medicine, diary products and chemicals starts to

deteriorate once they are produced.

Most researches in inventory do not consider the time-value of money.

This is unrealistic, since the resource of an enterprise depends vary much on when

Deteriorated economic production quantity (EPQ) model 3609

it is used and this is highly correlated to the return of investment. Therefore,

taking into account the time-value of money should be critical especially when

investment and forecasting are considered. Wee (2001) developed a replenishment

policy for items with a price-dependent demand and a varying rate of

deterioration. Sarkar et al. (1994) assumed a finite replenishment model and

analysed the effects of inflation and time-value of money on order quantity when

shortages and allowed. Hariga (1995) extended the study to analyze the effect in

inflation and time-value of money of an inventory model with time-dependant

demand rate and shortages. Bose et al. (1993) developed an EOQ model for

deteriorating items with linear time-dependent demand rate and shortages under

inflation and time discounting. Abad (1996), Chang et al. (1999), Goyal et al.

(2001) Raafat (1991), Singh et al. (2013) and Chung et al. (1993) investigated

deterministic deteriorated economic order quantity models. Vandeltt et al. (1995)

proposed a simple economic order quantity for inventory with a short and

stochastic life time. Their approach was performed in the framework of the total

discounted cost criterion. Pattnaik (2011), (2012) and (2013) derived different

types of deterministic inventory models for deteriorating items in finite horizon.

The objective of this paper is to maximize the net present-value profit so

as to determine the optimal period and order quantity by using time-value of

money and price dependent declined quadratic demand allowing shortages.

Replenishment decision during planning horizon due to deterioration for

maximizing the net present value profit in response to change the market demand

and is also considered. Controlling the demand of customer through manipulation

of selling price for maximizing total net value profit. The major assumptions used

in the above research article summarized in Table-1. The remainder of paper

organized in section 2 is assumptions and notations for development of the model.

The mathematical model is developed in section 3. Optimization is given in

section 4. Numerical example is presented to illustrate the development of model

in section 5. The sensitivity analysis carried out in section 6 to observe the

changes in optimal solution. Finally section 7 deal with conclusion.

Table 1 Summary of Related Researches Authors

Publishe

d Year

Model

Structur

es

Demand Demand

Pattern

Deteriorati

on

Allowin

g

shortag

e

Time

value

of

Mone

y

Planni

ng

Model New form

Wee et

al. 2001

EOQ Price Declined

Linear

Yes

Weibull

Yes Yes Finite Profit DCF

approach

Pattnaik

2013

EOQ Constant

Deteriorati

on

Constant Yes

Constant

Wasting

No. No. Finite Profit Unit lost due

to

deterioration

and various

ordering cost

Hariga

1995

EOQ Time Decreas

e

No Yes Yes Finite Profit Inflation

Present

Paper

2014

EPQ Price Declined

Quardrat

ic

Yes

Weibull

Yes Yes Finite Profit DCF

approach


2 Assumption and Notation

The distribution of time until deterioration the item follows a tow-parameter

Weibull distribution.

Deterioration occurs as soon as the items received into inventory.

There is no replacement or repair of deteriorating items during the period

under consideration.

The demand rate is a decreasing quadratic function of selling price.

The replenishment rate is instantaneous; the order quantity and the

replenishment cycle is same for each period.

The system operates for a prescribed period of a planning horizon.

Shortages are completely back ordered.

The order quantity, inventory level, replenishment epoch and demand are

treated as continuous variables while the number of replenishments is

restricted to an integer variable.

Continuous cost compounding is implemented throughout the analysis.

Product transactions are followed by instantaneous cash flow.

S Per unit selling price of the items (s/unit) where m

lsc

d(s) Demand rate, 2)( smslsd ,

m,l Constant values where l>0 and m>0

C Per unit cost of items (s/unit)

)( 11 tI Inventory level at any time 1t : 110 Tt (Positive inventory)

)t(I 22 Inventory level at any time, 120 TTt (Negative inventory)

H Planning horizon

T Replenishment cycle

N Number of replenishment during planning horizon; T

HN

1T Time with positive inventory

1TT Time when shortage occurs

R Interest rate

Q The 2nd

, 3rd

, .... Nth replenishment size (units)

Im maximum inventory level

Cost per replenishment when t= 0 ($)

Per unit holding cost per unit time (s unit / unit time)

Per unit shortage cost per unit time (s / unit / unit time)

One distribution that has been used extensively in literature to pattern a varying

rate of deterioration is the Weibull distribution. The two parameter Weibull

density function is e .)( t-1 ttf (1)

here t is the time to deterioration, t>0, f(t) probability density function the scale parameter, t>0 and the shape parameter, 0 . This probability


density function represent the hand inventory deterioration that may have an

increasing, decreasing or constant rate depending on the value of . When

1 , deteriorating rate increases with time e.g. Fish and vegetables. When

1 deteriorating rate decreases with time e.g.: light bulb where the initial

deterioration breakdown rate may be higher due to irregular voltages and

handling. When 1 deterioration rate is constant; e.g. electronic products.

Here, the two-parameter Weibull distribution is reduced to an exponential

distribution. The instantaneous rate of deterioration of the hand inventory is

given by 1 t .

3 Mathematical Model

The changes in inventory level against time are depicted in Fig. 1.

Fig. 1 The Inventory system when shortage is allowed.

The first replenishment lot size of Im is replenished at t=0. During the

period 1T , the inventory level decreases due to demand an deterioration until it is

zero at 1Tt . During the time interval, )TTT(T 122 , shortages occurred are

accumulated until Tt before they are backordered. The inventory system at any

time t can therefore be represented by the following equations.

1111

1

1

1

11 0),()()(

TtsdtItdt

tdI (2)

12

2

22 0 TTt),s(ddt

)t(dI

(3)

The first-order differential equation can be solved by using the boundary

conditions 021 )O(IIm,)O(I ,

110

11 0;)(Im

)(1

1

Tte

duesdtI

t

tu

(4)

T1

Q

0 t2

t1

T

T=H


12222 0 TTt;t)s(d)t(I (5)

Since 011 )T(I , one can derive from (4) the maximum inventory level as

1 1

0 0 0

1

1

0 )1(!)(

!)()(

T T

n

nn

n

nnu

mnn

Tsddu

n

usdduesdI

(6)

Assuming a very small value ).( 050 the approximate solution is

found by neglecting the second and higher – order terms of , one has

1)(Im

1

11

TTsd (7)

The total cost in this model includes the replenishment cost, material cost,

holding cost and shortage cost. The objective is to maximize the total profit when

the time-value of money with compounding interest rate is considered. The

detailed analysis of each cost function is given below.

Present – Value Sales Profit

During the period T1, the replenished inventory is being consumed due to

demand and deterioration. At t=T1, all the shortages during the period T-T1, are

backordered with an instantaneous cash transactions during sales, the present-

value sale profit is

1

0 0

21

11 1

11

)()( TTer

esdsdtsdsedtesdsR rt

rTT TT

rTrt

(8)

Assuming a very small r value 080.r approximate solutions can be

found by neglecting the second and higher-order terms of r.

TrTrT

rTT)s(dsR 1

22

1

2 (9)

Present-Value Ordering Cost

Since replenishment is each cycle is done at the start of each cycle, the

present-value replenishment cost is 10 cC 10)

Present Value Inventory Cost

Inventory occurs during period 1T , therefore, the present value inventory

cost during the period is

1

0

1

00

1

0

1

1

1

12

01

00

20

1112

!!1!)(

)()(

1

11

1

11

11

dtn

rt

n

t

nn

tTsdc

dtee

dueduesdcdtetIcC

n

nT

n

n

n

nnn

Trt

t

tu

Tu

Trt

H


Assuming very small and r value (see above) the approximate solution

is found by neglecting second and higher-order terms of and r and terms

containing r . Consequently,

2162

21

31

21

2

TrTT)s(dCCH

(11)

Present-value shortage cost

The maximum shortage level )TT)(s(dIb 1 . All shortage during

)TT( 1 be completely backordered at T. The present value shortage cost for the

period is

]1[)(

))(()(

1

121

121

12

3

02

)(

230

2

)(

223

rTrT

TTtTr

TTtTr

S

erTrTer

sdc

dtetsdcdtetIcC

Assuming a very small r value (see above) approximate solutions are

founded by neglecting second and higher order terms of r, on has

31

211

231

23 332636

rTTTrTrTTTT)s(dC

CS 12)

Present-value item cost

Replenishment is done at t=0 and T; the replenishment items are

consumed by demand as deterioration during 1T . The present-value cost, Cp,

therefore includes item cost and deterioration cost, one has

1

02

TTrTmP dt)s(dCeCIC (13)

Assuming very small and r values (see above) the approximate solution

is found by neglecting the second and higher order terms of and r and the

terms containing r. Consequently,

1

1

11

2

TTrTrTT)s(CdCP

(14)

The first cycle present-value net profit is 1 PSHO CCCCR (15)

There are N cycles during the planning horizon. Some inventory is

assumed to start and end at zero, an extra replenishment at T=H is required to

satisfy the back orders of the last cycle in the planning horizon. Therefore, the

total number of replenishment = N+1 times; the first replenishment lot are = Im

and the 2nd

, 3rd

, Nth

replenishment lot size

1

02

TTdt)s(dIml

(16)

and the last or (N+1)th

replenishment lot size 1

02

TTdt)s(d (17)

The time-value of money affects all the replenishment periods and

therefore must be considered separately, the total net present-value profit for the

planning horizon is


rH

rT

rnTN

n

rHrnT

rHrTNrTrTrT

ece

eece

eceeeeNTt

11

1

0

1

1

132

11

1

1

.....1),,(

(18)

where, T=H/N and 1 is derived by substituting (8) to (14) into (15). The

optimization problem of this study can be formulated by maximizing (18) subject

to m

lSc and TT 10 .

4 Model Analysis

The following heuristic technique is derived the optimal S, T1 and N

values;

Step 1:Start by choosing a discrete variable N, where N is any integer number

equal or greater than 1;

Step 2:Take the partial derivatives of ),,( NTS with respect to 1T and S, and

equate the results to zero, the necessary conditions for optimality are

0),,(1

NTS

T and

0),,( 1

NTS

S

Step 3:For different integer N values, derive *T1 and *S from above two equations

substitute N,T,S **1 into (18) to derive NTS ,, *

1

* Step 4:Repeat step 2 and 3 for all possible N values within the lower and upper

bound until the maximum NTS ,, *

1

* is found. The *** N,T,S 1 and

**

1

* ,, NTS values constitute the optimal solution and they satisfy the

following conditions:

1,,0,, **

1

***

1

* NTSNTS (19)

Where **

1

***

1

***

1

* ,,1,,,, NTSNTSNTS substitute *** N,T,S 1

into (16) to derive the 2nd

, 3rd

, ..., Nth replenishment lot size. If the

objective function is concave, the following sufficient condition must

be satisfied;

02

2

2

1

22

1

STTS

(20)

and any one of the following 02

1

2

T

, 0

2

2

S

(21)


Since the total net present-value profit for the planning horizon ~ is a very

complicated function due to high-power expression of the exponential function, it

is not possible to show analytically the validity of the above sufficient conditions,

a search procedure is used instead. The computation al results are show in the

following illustrative example.

5 Numerical Examples

Optimal replenishment and pricing policies for the maximum present-

value profit may be derived by using the methodology given in the preceding

sections; this will help managers to improve their replenishment and pricing

decisions. The replenishment cost, c1 is $80/order, the annual inventory cost c2 is

$0.6/unit/year, the annual shortage cost c3 is $1.4/unit/year. The unit item cost c is

$5/unit, the scale and the shape parameters of the deterioration rate are =0.05

and =1.5 respectively. The annual interest rate, r is 0.08, the yearly demand rate

d(s) is 1000-4S-S2 unit/year and the planning horizon, H is 10 years.

Table – 2 Optimal Values of the Proposed Model Model Iteratio

n

N S T1 T d(s) Q Profit =

Quadrati

c d(s)

58 25.9315

6

18.5235

5

0.264951

2

0.385630

4

570.379

8

220.083

3

74703.1

3

Linear

d(s)

83 57.5486

8

127.539

6

0.158247

8

0.173765

9

489.841

7

85.2273

3

591039.

6

% change - 121.93 588.53 -40.27 -54.94 -14.12 -61.27 691.18

For the given data the total net value profit for the planning horizon is

Rs.74703.13, the number of replenishment during planning horizon N, is 26, per

unit selling price of the items is Rs. 18.52, time with positive inventory T1, is

0.2649512, replenishment cycle T is 0.3856304, selling price dependent declining

quadratic demand d(s) is 570.3798 and the replenishment size Q is 220.0833. The

total number of order is therefore N+1=27. All the decision parameters are

compared with the other model related to the declined demand d(S) which is also

related linearly to the selling price. It is observed that demand rate d(s), order

quantity Q, time with positive inventory 1T and replenishment cycle T are more

than the compared model. But only the number of replenishment during planning

horizon, N, selling price S, unit selling prices and the total present value profit are less from the compared model. Fig.2 represents the relationship between the

unit selling price S and declined quadratic demand rate d(S). Similarly Fig. 3

depicts the mesh plot of T, T1 and total present value profit .


Fig. 2 Unit Selling Price S and Demand Rate d(S). Fig. 3 Mesh Plot of T, T1 and .

Table – 3 Sensitivity Study

Parameter Value Iteration N S T1 T d(s) Q Profit =

Change in

Profit

1.3 56 26.04150 18.82446 0.2610136 0.3840024 570.3421 219.2651 74680.85 0.029824720

1.7 57 25.83612 18.82296 0.2681136 0.3870550 570.4044 220.8428 74718.97 0.021203930

1.9 53 25.75498 18.82257 0.2707048 0.3882745 570.4204 221.5129 74730.31 0.036384017

R 0.06 42 23.43569 18.82469 0.2903966 0.4266995 570.3324 243.4881 75090.75 0.518805834

0.07 47 24.71444 18.82401 0.2767682 0.4046217 570.3605 230.9078 74891.83 0.252599857

0.09 66 27.09511 18.82325 0.2545707 0.3690703 570.3924 210.6424 74523.27 0.240766350

H 8 57 20.74525 18.82355 0.2649512 0.3856304 570.3798 220.0833 59746.50 20.021423470

11 89 28.52772 18.82355 0.2649512 0.3856304 570.3798 220.0833 82181.44 10.01070504

12 92 31.11787 18.82355 0.2649512 0.3856304 570.3798 220.0833 89659.76 20.021423470

c1 75 67 26.77670 18.82198 0.2567588 0.3734590 570.4451 213.1654 74835.87 0.177690011

85 93 25.16201 18.82507 0.2728813 0.3974246 570.3164 226.7853 74574.42 0.172295324

90 62 24.45739 18.82655 0.2805722 0.4088743 570.2548 233.2901 74449.40 0.339651096

c2 0.5 53 25.61640 18.82151 0.2756066 0.3903750 570.4649 222.8228 74756.55 0.071509721

0.8 91 26.50602 18.82707 0.2459799 0.3772728 570.2333 215.2610 74606.14 0.129833917

0.9 81 26.76853 18.82858 0.2374960 0.3735730 570.1701 213.1276 74561.97 0.188961292

c3 1.2 76 25.78575 18.82142 0.2595902 0.3878111 570.4687 221.3617 74725.32 0.029704243

1.5 83 25.99868 18.82451 0.2674014 0.3846350 570.3399 219.5002 74692.94 0.013640660

1.7 61 26.12278 18.82624 0.2719040 0.3828076 570.2677 218.4303 74674.15 0.038793555

C 4 79 26.29748 18.44723 0.2706690 0.3802646 585.9108 222.9321 80442.99 7.683560247

6 59 25.51993 19.20582 0.2598668 0.3913507 554.3131 217.3319 69125.82 7.465965616

8 59 24.56800 19.98819 0.2514508 0.4070322 520.5196 211.9847 58479.41 21.717590680

0.04 52 25.83192 18.82325 0.2678795 0.3871179 570.3921 220.8820 74714.09 0.014671406

0.06 54 26.02814 18.82384 0.2621229 0.3841996 570.3679 219.3363 74692.42 0.014336748

0.08 54 26.21276 18.82438 0.2567432 0.3814937 570.3452 217.9960 74671.72 0.042046430

6 Sensitive Analysis

It is interesting to investigate the influence of major parameters

.,,,,,,, 321 ccccHr

N , S , and d(s) are insensitive to the parameter . T1 , T , Q and are

moderately sensitive to the parameter .

N, T1 , T and Q are moderately sensitive to parameter r. S and d(s) are

insensitive to r but is sensitive to the parameter r.

0 10 20 30 40 50 60 70 80-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4

s: Unit Selling Price

d(s): D

eclined Q

uadratic D

em

and R

ate, d(s)=

1000-4s-s

2


N , T1 and T are moderately sensitive to parameter H but S , d(s) and Q are

insensitive to the parameter H and is sensitive to the parameter H.

N, S, 1T and d(s) are insensitive to the parameter C1 but T is moderately

sensitive and Q and are sensitive to the parameter C1.

N, S and d(s) are insensitive to the parameter C2 but T1 and T are moderately

sensitive to C2 and Q and are sensitive to the parameter C2.

N, S, and d(s) are insensitive to the parameter C3 but T1 and T, Q and are

moderately sensitive to the parameter C3.

N is sensitive, S, T1 and T and Q are moderately sensitive to parameter C. d(s)

and are sensitive to the parameter c.

N, S and d(s) are insensitive to the parameter . T1 , T , Q and are

moderately sensitive to the parameter .

7 Conclusion

In this paper, an EPQ model is introduced which investigates the optimal

replenishment quantity. Unit selling price, replenishment cycle, time with positive

invention the total value net profit with finite planning horizon for deteriorating

items. The model considers the impact of price dependant quadratic demand, and

shortages and varying rate of deterioration. The model can be used for electronics

and other luxury products which are more likely to have the above characteristics.

This paper provides a useful property for finding the optimal net present value

profit with finite planning horizon for deteriorating items. A new mathematical

model with decline quadratic demand is developed and compared to the other

EPQ model with decline linear demand and numerically. The economic

replenishment quantity Q* and net present value profit

* for the present model

were found to be more than that of the compared model respectively. Hence the

utilization of selling price dependent declined quadratic demand makes the scope

of application broader. Lingo 13.0 version software is used to derive the optimal

number of replenishment and unit price to maximize the total present value net

profit. Further, a numerical example is presented to illustrate the theoretical

results, and some observations are obtained from sensitivity analysis with respect

to the major parameters controlling the market demand through the manipulation

of selling price is an important strategy for increasing profit. This can be achieved

by using the joint optimal replenishment and pricing strategy developed in this

study. In the future study, it is hoped further incorporate the proposed model into

several situations such as stochastic market demand, fuzzy decision parameters,

partial back logging, and selling price.

References [1] B.R. Sarkar and H. Pan, Effects of inflation and time value of money on order

quantity and allowable shortage. International Journal of Production

Economics, 35, (1994), 65-72.


[2] C. H. Vandeltt and J.P. Vital, Discounted costs obsolescent and planned stock

outs with the EOQ formula. International Journal of Production Economics, 44

(1995), 255-265.

[3] F. Raafat, Survey of Literature on continuously deteriorating inventory models.

Journal of Operation Research Society, 40, (1991), 27-37.

[4] H.J. Chang and L.Y. Dye, An EOQ model for deteriorating items with time

varying demand and partial backlogging. Journal of Operational Research

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[5] H.M. Wee and S. Law, Replenishment and Pricing policy for deteriorating

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Production Economics, 71 (2001), 213-220.

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[9] M. Pattnaik, Entropic Order quantity (EnOQ) Model under Cash Discounts.

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[13] M. Pattnaik, Wasting of Percentage on hand inventory of an Instantaneous

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[14] P.L. Abad, Optimal Pricing and lot sizing under conditions of perish ability and

partial back ordering . Management Science, 42 (1996), 1093-1104.

[15] S. Bose, S.B. Goswami and K.S. Choudhury, An EOQ model for deteriorating

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[16] S. K. Goyal and B. C. Giri, Recent trends in modelling of deteriorating

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Received: March 15, 2014

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Deteriorated Economic Production Quantity (EPQ) · PDF fileDeteriorated economic production quantity (EPQ) model 3609 it is used and this is highly correlated to the return of investment