Effects of inspection on retailer's ordering policy for deteriorating items with time-dependent demand under inflationary conditions

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Effects of inspection on retailer's ordering policy fordeteriorating items with time-dependent demandunder inflationary conditionsChandra K. Jaggi a , Mandeep Mittal b & Aditi Khanna aa Department of Operational Research, Faculty of Mathematical Sciences , New AcademicBlock, University of Delhi , Delhi – 110007 , Indiab Amity School of Engineering & Technology , 580, Delhi Palam Vihar Road, Bijwasan , NewDelhi – 110061 , IndiaPublished online: 11 Jul 2012.

To cite this article: Chandra K. Jaggi , Mandeep Mittal & Aditi Khanna (2013): Effects of inspection on retailer's orderingpolicy for deteriorating items with time-dependent demand under inflationary conditions, International Journal of SystemsScience, 44:9, 1774-1782

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International Journal of Systems ScienceVol. 44, No. 9, September 2013, 1774–1782

Effects of inspection on retailer’s ordering policy for deteriorating items with time-dependent

demand under inflationary conditions

Chandra K. Jaggia*, Mandeep Mittalb and Aditi Khannaa

aDepartment of Operational Research, Faculty of Mathematical Sciences, New Academic Block, University of Delhi,Delhi – 110007, India; bAmity School of Engineering & Technology, 580, Delhi Palam Vihar Road, Bijwasan,

New Delhi – 110061, India

(Received 7 September 2011; final version received 21 May 2012)

In this article, an Economic Order Quantity (EOQ) model has been developed with unreliable supply, where eachreceived lot may have random fraction of defective items with known distribution. Thus, the inspection of lotbecomes essential in almost all the situations. Moreover, its role becomes more significant when the items aredeteriorating in nature. It is assumed that defective items are salvaged as a single batch after the screeningprocess. Further, it has been observed that the demand as well as price for certain consumer items increaseslinearly with time, especially under inflationary conditions. Owing to this fact, this article investigates the impactof defective items on retailer’s ordering policy for deteriorating items under inflation when both demand andprice vary with the passage of time. The proposed model optimises the order quantity by maximising the retailer’sexpected profit. Results are demonstrated with the help of a numerical example and the sensitivity analysis is alsopresented to provide managerial insights into practice.

Keywords: inventory; deterioration; inspection; time-dependent demand; inflation

1. Introduction

Generally, every manufacturing system wants to pro-

duce 100% perfect quality items. However, in reality

certain fraction of products may be of imperfect quality

due to labour problem or machine breakdown. Thus,

the inspection of lot becomes essential in all public and

private sector organisation. To address this very fact,

researchers devoted a great amount of effort to develop

Economic Production Quantity (EPQ)/Economic

Order Quantity (EOQ) models for defective items

(Porteus 1986; Rosenblatt and Lee 1986; Lee and

Rosenblatt 1987; Schwaller 1988; Zhang and Gerchak

1990). Salameh and Jaber (2000) developed an EOQ

model where each order contains a random fraction of

imperfect quality items with a known probability

distribution. They also considered that the imperfect

quality items are sold at a discounted price as a single

batch by the end of the screening process. Cardenas-

Barron (2000) corrected the optimum order size for-

mula obtained by Salameh and Jaber (2000) by adding a

constant parameter. Further, Goyal and Cardenas-

Barron (2002) presented a simple approach for deter-

mining economic production quantity for imperfect

quality items and compared the results based on the

simple approach with an optimal method suggested by

Salameh and Jaber (2000), which results in almost zero

penalty. Papachristos and Konstantaras (2006) exam-

ined Salameh and Jaber’s (2000) paper closely and

rectified the proposed conditions to ensure that short-

ages do not occur. They further extended their model to

the case in which withdrawing takes place at the end of

the planning horizon.Further, when the items are deteriorating in nature,

the role of inspection becomes more prominent.In fact, in all the aforementioned papers the authorshave not considered deterioration factor in theirmodelling, whereas deterioration is a well-establishedfact in the literature, due to which utility of an itemdoes not remain same over the period of time. Ghareand Schrader (1963) were first to present an economicorder quantity model for exponentially decayinginventory. Thereafter, several interesting papers forcontrolling the deteriorating items have appeared indifferent journals (Raafat, Wolfe, and Eldin 1991;Cardenas-Barron 2009a, 2009b, 2011; Cardenas-Barron, Smith, and Goyal 2010; Widyadana,Cardenas-Barron, and Wee 2011).

Recently, Jaggi and Mittal (2011) formulated aninventory model for deteriorating items with imperfectquality. They assumed that the screening rate is morethan the demand rate. This assumption helps one tofulfil the demand, out of the products which are found

*Corresponding author. Email: [email protected]; [email protected]

ISSN 0020–7721 print/ISSN 1464–5319 online

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to be of perfect quality, along with the screeningprocess.

All the above-mentioned articles are based on theassumption that the demand is known and uniform,but this assumption is not always pertinent to manyinventory items like basic consumer goods, namely,food items like paddy, wheat, potato, onion, etc.,because there may be variation in demand. Manyproducts experience a period of rising demand duringthe growth phase of their product life cycle. On theother hand, the demand of some products may declinedue to the introduction of more attractive productsinfluencing customers’ preference. Moreover, the ageof inventory has a negative impact on demand due tothe loss of consumer confidence on the quality of suchproducts and physical loss of the materials. This factprompted many researchers to develop deterioratinginventory models with time-dependent demand. Indeveloping inventory models, two types of time-dependent demands have been considered so far:(i) continuous-time and (ii) discrete-time. Most of thecontinuous-time inventory models have been devel-oped considering either linearly increasing/decreasingdemand or exponential increasing/decreasing demandpatterns. Owing to this fact, during the past few years,a lot of research work has been done on inventorymodels with time-dependent demand, which has beensummarised by Goyal and Giri (2001).

In all the aforementioned models, inflation andtime value of money has not been taken into account,whereas in the last three decades, the economicsituation of most of the countries have changed verymuch. Most of countries have suffered from large-scaleinflation and sharp decline in the purchasing power ofmoney over last several years. Besides this, inflationalso influences demand of certain products. As infla-tion rate increases, the value of money goes down,which erodes the future worth of savings and forcesone for more current spending. This expenditure maybe on clothes, accessories, peripherals or daily house-hold items. As a result, while determining the optimalinventory policy, the effect of inflation should not beignored. Many authors have developed differentinventory models under inflationary conditions withdifferent assumptions. The fundamental result in thedevelopment of the EOQ model with inflation is that ofBuzacott (1975), who discussed the EOQ model withinflation subject to different types of pricing policies.In the same year, Misra (1975) also developed an EOQmodel incorporating inflationary effects. Several otherinteresting and relevant papers in this direction areDatta and Pal (1991), Sarker and Pan (1994), Hariga(1995), Hariga and Ben-Daya (1996).

Further, Ray and Chaudhuri (1997) developed theEOQ model incorporating the effects of inflation, time

value of money and a linear time-dependent demandrate. Sana and Chaudhuri (2003) extended the Ray andChaudhuri (1997) model for deteriorating productsfollowing the Weibull distribution with dependent ratebeing constant up to a fixed time, after that it varieslinearly with time. Recently, Jaggi and Mittal (2007)developed an EOQ model for spoilable items thatfollows constant deterioration and time-dependentdemand rate under inflation and time value of money.Their model suggests that the total average profit anddemand rate increases after certain time period due tothe effect of inflation and time value of money.

However, none of the researcher has explored theeffects of in-process inspection on the retailer’s order-ing policy so far, when items are deteriorating in natureand demand and price varies with time under infla-tionary conditions.

Motivated with this aspect, this article deals withretailer’s inventory model for deteriorating imperfectquality items under inflationary conditions, wheredemand and price are functions of time. This modelis highly beneficial for products like crops (paddy,wheat, potato, onion, ginger, etc.), where demand aswell as price varies linearly with time. Moreover, in thepresent model the screening rate is assumed to be morethan the demand rate in order to meet the demandparallel to the screening process by the good qualityitems. The proposed model optimises retailer’s orderquantity by maximising his total expected profit.A comprehensive sensitivity analysis also has beenperformed to study the effects of deterioration, infla-tion and expected number of imperfect quality items onthe order quantity and expected total profit.

2. Assumptions and notations

The following assumptions are used in developing themodel:

(1) Demand rate varies with time.(2) Deterioration rate is constant.(3) Selling price per unit also varies with time.(4) Lead time is zero.(5) Shortages are not allowed.(6) Inflation and time value of money is considered.

The following notations are also used:

I(t) On-hand inventory at time t(>0)D(t) Demand rate at time t(>0)Sp(t) Selling price per unit at time t

r Discount rate representing the timevalue of money

i Inflation rate per unit timeR Net inflation rate¼ (r� i)c Purchase cost per unit

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Ch Holding cost per unit per unit timeA0 Ordering cost per orderT Duration of a cycleD0 Initial demand rateP0 Initial selling price per unit� Unit screening costa Rate of change of selling price per

unit with respect to tb Rate of change of demand rate

with respect to t� Deterioration rateps Salvage value per defective item,

ps 5 c� Screening rate per unit timet1 Screening time� Percentage of defective items in Q

f(�) Probability density function of �E(�) Expected value operator

E ð� Þ Expected value of �, which is equaltoR d2d1 �f ð�Þd�, d1��� d2

I1ðtÞ Inventory level during timeinterval, 0 � t � t1

I2ðtÞ Inventory level during time inter-val, t1 � t � �

I3ðtÞ Inventory level during time inter-val, � � t � T

Ieffðt1Þ Effective inventory level at time t1excluding the defective items

�ðTÞ Present worth of total profitE½�ðTÞ� Present worth of expected total

profit

3. Mathematical model

In this model, we have considered time-dependentdemand rate, D(t), which is constant up to a certaintime t¼�, after which it increases linearly with time.This type of demand pattern is commonly observed incrops (like paddy, wheat, potato, onion, ginger, etc.).In the beginning, the businessmen stock the items incold stores or warehouses; as a result, the demand ofthe items remains constant up to a certain period (0,�)in the market. After that certain period (0,�) thedemand automatically increases in the market becauseall customers are bound to purchase their essentialcommodities due to decrease of the harvested item.A similar behaviour is also followed by the price ofthese commodities. Since the demand rate increases theselling price, Sp(t) also increases with time. Hence, wedefine D(t) and Sp(t) as follows:

DðtÞ ¼ D0 þ bðt� �ÞH ðt� �Þ ð1Þ

SpðtÞ ¼ P0 þ aðt� �ÞH ðt� �Þ ð2Þ

where

Hðt� �Þ ¼1, if t � �0, if t5�

�ð3Þ

At the beginning Q items are procured. It is

presumed that each lot may have some defective

items. Let � be the percentage of defective items with

a known probability density function, f(�), which can

be calculated from the past data. The whole lot has to

pass through screening process at the rate of � units perunit time, which is assumed to be greater than the

demand rate (D) during the period 0 to t1. During

the screening process, the demand occurs parallel

to the screening which is fulfilled from the good

quality items. After the inspection is complete at time

t1, all the defective items are sold as a single lot at a

discounted price. After time t1 the effective inventory

level reduces to Ieffðt1Þ. Now, up to time �, the demand

is constant, thereafter it will increase linearly with time.

Further, inventory level gradually decreases mainly

due to demand and partially due to deterioration and

reaches zero at time T, which is depicted in Figure 1.Now, in order to avoid shortages during the

screening time (t1), screened lot has to be greater

than the demand during t1, i.e.

ð1� E½��ÞQ � D0t1 ) E½�� � 1�D0

�ð4Þ

where � is a random variable and uniformly distributed

in the range [d1, d2], d1��� d2 and the expected

value of � is E[�]. Screening time is

t1 ¼Q

�ð5Þ

Let I1(t) be the inventory level at any time t,

(0� t� t1). The differential equation that describes the

t1

µ T

Q αQ

Time

Figure 1. Inventory scenario with screening process.

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instantaneous states of I1(t) over the period (0, t1) is

given by

dI1ðtÞ

dtþ �I1ðtÞ ¼ �DðtÞ, 0 � t � t1 ð6Þ

Substituting Equation (1) and (3) in Equation (6),

we have

dI1ðtÞ

dtþ �I1ðtÞ ¼ �DðtÞ ¼ �D0, 0 � t � t1 ð7Þ

The solution of the above differential equation (7)

along with the boundary condition

t ¼ 0, I1ð0Þ ¼ Q is

I1ðtÞ ¼ Qe��t þD0

�½e��t � 1�

ð8Þ

Inventory level at time t1, including the defective

items is

I1ðt1Þ ¼ Qe��t1 þD0

�½e��t1 � 1� ð9Þ

Since after the screening process, the number of

defective items at time t1 is �Q. Hence the effective

inventory level at t¼ t1, after removing the defective

items the effective inventory is given by

Ieffðt1Þ ¼ Qe��t1 þD0

�½e��t1 � 1� � �Q ð10Þ

Now again the inventory level at any time (t1, m) isgiven by

dI2ðtÞ

dtþ �I2ðtÞ ¼ �D0, t1 � t � � ð11Þ

Solution of the above differential equation (11)

with the boundary condition, at t ¼ t1,

I2ðt1Þ ¼ Ieffðt1Þ is

I2ðtÞ ¼ Qe��t þD0

�½e��t � 1� � �Qe��ðt�t1Þ

ð12Þ

Further, during (�,T) the inventory level is again

given by

dI3ðtÞ

dtþ �I3ðtÞ ¼ �D0 � bðt� �Þ, � � t � T ð13Þ

Solutions of Equation (13) with the boundary

condition, t ¼ �, I3ð�Þ ¼ I2ð�Þ is

I3ðtÞ ¼ Qe��t þD0

�½e��t � 1� þ

b

��� t½ �

�b

�2e��ðt��Þ � 1� �

� �Qe��ðt�t1Þ ð14Þ

Now at t ¼ T, I3ðTÞ ¼ 0; hence the order quantity

Q is given by

Q ¼1

ð1� �e�t1 Þ

D0

�e�T � 1� �

þbe�T

�T� �ð Þ

�

�b

�2e�T � e��� ��

ð15Þ

Also from Equation (5), we have Q ¼ �t1.Thus, from Equations (5) and (15), we get

�t1 ¼1

ð1� �e�t1Þ

D0

�e�T � 1� �

þbe�T

�T� �ð Þ

�

�b

�2e�T � e��� ��

) ð1� �e�t1 Þt1

¼1

�

D0

�e�T � 1� �

þbe�T

�T� �ð Þ �

b

�2e�T � e��� �� �

ð16Þ

Now in order to get a closed form expression of t1,

we use the Taylor series expansion and ignoring the

second and higher order powers of � t1 (� t1� 1),

we get

t1ð1� �� ��t1Þ

¼1

�

D0

�e�T � 1� �

þbe�T

�T� �ð Þ �

b

�2e�T � e��� �� �

ð17Þ

Moreover, the defective percentage lies between 0

and 0.1, hence �� t1 again is a very small quantity

which can be ignored and Equation (17) reduces to

t1 ¼1

ð1� �Þ�

D0

�e�T � 1� �

þbe�T

�T� �ð Þ

�

�b

�2e�T � e��� ��

ð18Þ

Now, the retailer’s total profit during a cycle is

given by

�ðTÞ ¼ Sales revenue�Ordering cost

� Purchasing cost� Screening cost

�Holding cost ð19Þ

Furthermore, the present model has been devel-

oped under inflationary conditions. Thus, by using

continuous compounding of inflation and discount

rate, the present worth of the various costs are

evaluated as follows:

(1) Present worth of the total sales revenue is the

sum of revenue generated by the demand met

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during the time period (0,T) and sale of

imperfect quality items is

¼

Z �

0

P0D0e�Rt dt

þ

Z T

�

D0 þ bðt� �Þ

P0 þ aðt� �Þe�Rt dt

þ ps�Qe�Rt1

¼ P0D01� e�RT

R

� �

þ ðD0aþ P0bÞe�R�

R2�e�RT

R2�ðT� �Þe�RT

R

� �

þ ab �ðT� �Þ2e�RT

R

�

�2ðT� �Þe�RT

R2þ2e�RT

R3�2e�R�

R3

� ��þ ps�Qe�Rt1

ð20Þ

2: Present worth of Ordering Cost ¼ A ð21Þ

(Since ordering is made at time t¼ 0, the inflation does

not affect the ordering cost)

3: Present worth of Purchase Cost ¼ cQ ð22Þ

4: Present worth of Screening Cost ¼ �Qe�Rt1

ð23Þ

(1) Present worth of holding cost for the replen-

ishment cycle is

¼Ch

"Z t1

0

I1ðtÞe�Rtdtþ

Z �

t1

I2ðtÞe�Rtdtþ

Z T

�

I3ðtÞe�Rtdt

#

¼Ch

Q1� e�ð�þRÞT

ð�þRÞ

� þD0

�

1� e�ð�þRÞT

ð�þRÞ�1� e�RT

R

�

þ�Qe�t1e�ð�þRÞT� e�Rt1

ð�þRÞ

�

þb�

�þ

b

�2

� e�R�� e�RT

R

�

�b

�

R�e�R��RTe�RTþ e�R�� e�RT

R2

�

�b

�2e��

e�ð�þRÞ�� e�ð�þRÞT

ð�þRÞ

�

2666666666666666666664

3777777777777777777775ð24Þ

Substituting the values from Equations (20) to (24)

in Equation (19), the present worth of total profit,

�ðTÞ, becomes

�ðTÞ ¼P0D01�e�RT

R

� �

þðD0aþP0bÞe�R�

R2�e�RT

R2�ðT��Þe�RT

R

� �

þab �ðT��Þ2e�RT

R

�

�2ðT��Þe�RT

R2þ2e�RT

R3�2e�R�

R3

� ��þps�Qe�Rt1

�A0� cQ��Qe�Rt1

�Ch

Q1� e�ð�þRÞT

ð�þRÞ

�

þD0

�

1� e�ð�þRÞT

ð�þRÞ�1� e�RT

R

�

þ�Qe�t1e�ð�þRÞT� e�Rt1

ð�þRÞ

�

þb�

�þ

b

�2

� e�R�� e�RT

R

�

�b

�

R�e�R��RTe�RTþ e�R�� e�RT

R2

�

�b

�2e��

e�ð�þRÞ��e�ð�þRÞT

ð�þRÞ

�

2666666666666666666664

3777777777777777777775

ð25Þ

Since � is a random variable with a known

probability density function, f ð�Þ, the present worth

of expected total profit, E ½�ðTÞ� , is

E½�ðTÞ� ¼P0D01� e�RT

R

� �

þðD0aþP0bÞe�R�

R2�e�RT

R2�ðT��Þe�RT

R

� �

þab �ðT��Þ2e�RT

R

�

�2ðT��Þe�RT

R2þ2e�RT

R3�2e�R�

R3

� ��þpsE½��Qe�Rt1 �A0� cQ��Qe�Rt1

�Ch

Q1� e�ð�þRÞT

ð�þRÞ

�

þD0

�

1� e�ð�þRÞT

ð�þRÞ�1� e�RT

R

�

þE½��Qe� t1e�ð�þRÞT� e�Rt1

ð�þRÞ

�

þb�

�þ

b

�2

� e�R�� e�RT

R

�

�b

�

R�e�R��RTe�RTþ e�R�� e�RT

R2

�

�b

�2e��

e�ð�þRÞ�� e�ð�þRÞT

ð�þRÞ

�

26666666666666666666664

37777777777777777777775

ð26Þ

where R¼ r� i.

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4. Solution procedure

Our objective is to find the optimal values of T¼T*

(say) which maximise the expected total profit func-

tion, E½�ðTÞ�. Therefore, the necessary and sufficient

conditions for E½�ðTÞ� to be optimal are dE½�ðTÞ�dT ¼ 0 and

d2E½�ðTÞ�dT2 5 0Now,

dE½�ðTÞ�

dT¼P0D0e

�RTþ P0bþD0að Þ ðT��Þe�RT� �

þab ðT��Þ2e�RT� �

þpsE½��e�Rt1 ðY�QRXÞ�cY

��Ye�Rt1 þ�QRe�Rt1X

�ChY

ð�þRÞð1� e�ð�þRÞTÞþQe�ð�þRÞT

� �

�ChD0

�e�ð�þRÞT� e�RT� �

�ChE½��Ye�t1e�ð�þRÞT� e�Rt1

ð�þRÞ

� �

�ChE½��QRe�Rt1Xþð�X�ð�þRÞÞe�t1e�ð�þRÞT

ð�þRÞ

� �

�Chb�

�þ

b

�2

� e�RTþ

ChbTe�RT

�

þChb

�2e��e�ð�þRÞT¼ 0 ð27Þ

where

t1 ¼1

�ð1� �Þ

D0

�ðe�T � 1Þ þ

b

�e�TðT� �Þ

�

�b

�2ðe�ðT��Þ � 1Þ

�and Q ¼ �t1

X ¼dt1dT¼

1

�ð1� �ÞD0e

�T þb

�e�T

�

þbðT� �Þe�T �b

�e�ðT��Þ

�

Y ¼dQ

dT¼ �

dT1

dT¼ �X

The optimal value of T is obtained by solving

Equation (27).Further, for the expected total profit function

E½�ðTÞ� to be concave, the following sufficient condi-

tion must be satisfied:

d2E½�ðTÞ�

dT25 0 ð28Þ

Since the second derivative of the retailer’s expected

total profit function (E½�ðTÞ�) is complicated, it is very

difficult to prove concavity mathematically. Thus, the

concavity of the expected total profit function has been

established graphically, which is shown in Figure 2.

5. Numerical example

The model has been validated with the followingdata: A0¼ 250/cycle, D0¼ 3000 units/year, P0¼ $50,c¼ $30, a¼ 0.25, b¼ 3.5, r¼ 0.12, i¼ 0.06, R¼ 0.06,Ch¼ $3/unit/year, ps¼ $15, �¼ 1.2, �¼ 0.1, d1¼ 0,d2¼ 0.8, E[�]¼ 0.04, �¼ 0.5/unit and�¼ 175,000 units/year.

First of all, we check the condition on � given byEquation (4) for the given data which ensurethe shortages are not there during the screeningprocess,

i:e:E ½�� � 0:983

Now, the optimal value of T can be obtained bysolving Equation (27), i.e. T*¼ 2.02 years. Substitutingthe optimal value of T* in Equations (18), (15) and(26), we get, t�1¼ 0.040 years, Q*¼ 6995 units andE [�(T*)]¼ $57,205.

6. Sensitivity analysis

Sensitivity analysis has been performed on the keyparameters to look into the robustness of themodel, i.e. net inflation rate (R), deterioration rate(�) and expected number of imperfect quality items(E[�]) in order to study their effect on the Q, t1,T and E ½�ðTÞ� . The results are summarised inTables 1–3.

Based on the computational results, as shown in theTables 1–3, we obtain the following managerialphenomena:

. It is observed from Table 1 that when theinflation rate i increases, the optimal orderquantity and cycle length increases. Becauseunder the inflationary situation the value ofgoods goes up, this fact encourages the retailerto order more, which eventually results inhigher profit.

. Table 2 indicates, as the deterioration rate �increases, the profit and the cycle lengthdecreases. This suggests that it is beneficialfor the retailer to order more frequently,which helps him to manage the loss due todeterioration effectively.

. Further, Table 3 illustrates that as theexpected number of imperfect quality items(E½��) increases, the optimal order quantityand the cycle length decreases marginally,whereas the retailer’s expected profitdecreases significantly. This clearly signalsthe retailer that he should not ignore theinspection process and critically examine

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the sources of supply in order to sustain themarket pressure.

7. Conclusion

In this article, inventory models have been developedunder the push-and-pull effect of inflation and deteri-oration, for imperfect quality items when thedemand and selling price are the functions of time.The screening rate is assumed to be more than thedemand rate and inspection is performed on the wholelot. Its role becomes more prominent when items aredeteriorating in nature. Moreover, it has been observedthat the demand as well as price for certain consumeritems increases linearly with time (like wheat, paddy,etc.), especially under inflationary conditions. Hence, itis important to consider the effects of inflation andtime value of money while formulating the inventoryreplenishment policy for such products. The proposedmodel can be very useful in retail business of consum-able goods and other products that are likely to havethese characteristics.

Lastly, a numerical example has been solved tovalidate the results. Sensitivity analysis on differentparameters has been carried out in order to examinetheir impact on the optimum ordering policy. Findingsclearly suggests important managerial insights to theretailers for deciding the appropriate ordering policyunder varying situations, namely (a) when the defectiveitems increases in the ordered lot, the retailer should be

Figure 2. Concavity of the expected total profit function.

Table 1. Impact of i on Q, t1, T and E ½�ðTÞ� .

i Q t1 T E ½�ðTÞ�

0.02 5870 0.034 1.72 48,1120.04 6383 0.036 1.86 52,2640.06 6995 0.040 2.02 57,2050.08 7737 0.044 2.21 63,1810.10 8655 0.049 2.44 70,560

Table 3. Impact of E½�� on Q, t1, T and E ½�ðTÞ� .

E[�] Q t1 T E ½�ðTÞ�

0.01 7026 0.040 2.08 60,6110.02 7017 0.040 2.06 59,4860.03 7007 0.040 2.04 58,3500.04 6995 0.040 2.02 57,2050.05 6982 0.040 2.00 56,049

Table 2. Impact of � on Q, t1, T and E ½�ðTÞ� .

� Q t1 T E ½�ðTÞ�

0.05 8611 0.049 2.58 71,7880.10 6995 0.040 2.02 57,2050.15 5889 0.034 1.66 47,5690.20 5084 0.029 1.41 40,726

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more vigilant while ordering; (b) for highly deterioratingitems, the retailer should order more frequently and (c)in the highly inflationary market, the retailer shouldorder a large quantity so as to increase his profits.For future study, it is desirable to extend the proposedmodel for stock-dependent demand with trade creditand shortages.

Acknowledgements

The authors are grateful to the anonymous referees for theirvaluable suggestions and comments which helped immenselyin improving the article. The first author would like toacknowledge the support of the Research Grant No.:Dean(R/R&D/2011/423), provided by the University ofDelhi, Delhi, India for conducting this research.

Notes on contributors

Chandra K. Jaggi is an AssociateProfessor in the Department ofOperational Research, Faculty ofMathematical Sciences, University ofDelhi, India. His research interest liesin the field of analysis of InventoryManagement. He has published morethan 70 papers in various interna-tional/national journals including

IJPE, EJOR, JORS, IJSS, TOP etc. He has guided 7 PhDand 15 MPhil candidates in Operations Research. He isEditor-in-Chief of International Journal of Inventory Controland Management and Associate Editor of InternationalJournal of Systems Assurance Engineering and Management,Springer, Co-Editor/Reviewer-In-Charge of The GSTFJournal of Mathematics, Statistics and Operations Researchand on the Editorial Board of the International Journal ofServices Operations and Informatics, American Journal ofOperational Research, International Journal of EnterpriseComputing And Business Systems, Journal of Applied SciencesResearch, Australian Journal of Basic and Applied Sciences.He was awarded Shiksha Rattan Puraskar (for meritoriousservices, outstanding performance and remarkable role) in2007 by India International Friendship Society. He hastravelled extensively in India and abroad and deliveredinvited talks.

Mandeep Mittal is an AssistantProfessor in the Department ofComputer Science and Dean(Student’s Activities) at AmitySchool of Engineering andTechnology, New Delhi, India. Hehas completed his PhD (InventoryManagement) in 2012 and MSc(Applied Mathematics) in 2000 from

IIT Roorkee, India. He has six research papers published inInternational Journal of Industrial Engineering Computations,International Journal of Strategic Decision Sciences, RevistaInvetigacion Operacional, International Journal of AppliedIndustrial Engineering, International Journal of ServicesOperations and Informatics and Indian Journal ofMathematics and Mathematical Sciences

Aditi Khanna is working as anAssistant Professor in theDepartment of Mathematics, KeshavMahavidayalya, University of Delhi,India. She has completed her PhD(Inventory Management) in 2010,MPhil (Inventory Management) in2004, and MSc (OperationalResearch) in 2002 from the

University of Delhi. She has seven Research papers publishedin Canadian Journal of Pure & Applied Sciences, InternationalJournal of Applied Decision Sciences, International Journal ofServices Operations and Informatics, International Journal ofProcurement Management, International Journal ofOperational Research, International Journal of Mathematicsin Operational Research, OPSEARCH and MathematicsToday; and one research paper is to be appear inInternational Journal of System Sciences.

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