Pricing and Replenishment Policies for Imperfect Quality Deteriorating Items under Inflation and...

Madjid TavanaLa Salle University, USA

Management Theories and Strategic Practices for Decision Making

Management theories and strategic practices for decision making / Madjid Tavana, editor. p. cm. Includes bibliographical references and index. Summary: “This book brings together a collection of research that provides communication for the management of new methodologies to solve strategic problems and apply decision making approaches”--Provided by publisher. ISBN 978-1-4666-2473-3 (hardcover) -- ISBN 978-1-4666-2474-0 (ebook) -- ISBN 978-1-4666-2475-7 (print & perpetual access) 1. Decision making. 2. Strategic planning. 3. Business logistics. I. Tavana, Madjid, 1957- HD30.23.M362 2013 658.4’032--dc23 2012023348

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Chapter 10

Chandra K. JaggiUniversity of Delhi, India

Satish K. GoelUniversity of Delhi, India

Mandeep MittalAmity School of Engineering and Technology, India

Pricing and Replenishment Policies for Imperfect Quality

Deteriorating Items under Inflation and Permissible

Delay in Payments

ABSTRACT

Usually it is assumed that all items in a lot are of good quality, but in reality this assumption may not always be pertinent. Thus, the inspection of lots becomes essential in almost all organizations. Moreover, its role becomes more vital when the items are deteriorating in nature. Owing to this fact, this paper investigates the impact of initial inspection on retailer’s pricing and ordering policy for deteriorating items under inflation and permissible delay in payments using discounted cash flow approach over a finite planning horizon. Demand rate is assumed to be a function of selling price. The proposed model jointly optimizes the number of replenishments and price by maximizing the retailer’s total profit. Results have been demonstrated with the help of a numerical example, and sensitivity analyses are also presented to provide managerial insights into practice.

DOI: 10.4018/978-1-4666-2473-3.ch010

171

Pricing and Replenishment Policies for Imperfect Quality Deteriorating Items

1. INTRODUCTION

It is generally experienced that imperfect quality of goods has got direct implication on inventory management. This aspect has been recognized well but has not received the due attention of the researchers. The quantity received should be subject to 100% inspection in almost all types of concern and its role becomes more prominent when items are deteriorating in nature. Deterioration is a well established fact in literature, due to which utility of an item does not remain same over the period of time. Ghare and Schrader (1963) were first who presented an economic order quantity model for exponentially decaying inventory. Co-vert and Phillip (1973) extended the model with the assumption of weibull distribution deteriora-tion. Thereafter, several interesting papers for controlling the deteriorating items have appeared in different journals (Dave & Patel, 1981; Sachan, 1984; Chung & Ting, 1994; Hargia & Benkherouf, 1994; Hargia, 1995; Chakrabarti & Chaudhuri, 1997, Wu et al., 2006).

Further, price is an obvious strategy to influ-ence demand, therefore, inventory models for deteriorating items with price-dependent demand has received a significant attention in the literature. Specifically Lau and Lau (1988) designed a model in which demand can be increased by decreasing selling price. Polatoglu (1991) proposed a model in which the pricing and procurement decisions are determined simultaneously. Moreover, Khouja and Mehrez (1996) investigated a multi-product multi-discount constrained newsboy problem and provided an algorithm for solving the problem. Their model contains discount prices but not deci-sion parameters. Chen and Chu (2001) considered a related production planning problem in which demand can be controlled via pricing strategies during the selling period. Wee (1997) studied an in-ventory model in which demand is price-dependent and inventory deteriorates at a varying rate, and proposed an algorithm for determining the maxi-mum net profit. You and Chen (2007) developed

an EOQ model of seasonal goods with spot and forward purchase demands. Additionally, Chen and Chen (2007) investigated periodic pricing and replenishment policy for continuously decaying inventory with multivariate demand. Since then many articles had appeared under different realistic situations (Arcelus et al., 2002; Jaggi et al., 2007; Mondal et al., 2003; You, 2006). Later, Jiangtao et al. (2009) explore the inventory replenishment policy for the kind of items in which the demand is sensitive to stock and selling price.

Most of the above inventory models work under the assumption that the payment will be made to the supplier for the goods immediately after receiving the consignment. But this assump-tion may not be true in the day-to-day dealing, as most of the supplier offer certain fixed period for settling the account in order to stimulate their demand. During this period, the supplier charges no interest, but beyond this period interest is be-ing charged by the supplier under the terms and conditions agreed upon. Owing to this fact, during the past few years, a lot of research work has been conducted on inventory models with permissible delay in payment. Goyal (1985) presented the model by introducing permissible delay in pay-ments for fixed time period. Aggarwal and Jaggi (1995) extended his work for deteriorating items. Since then many articles had appeared under different realistic situations. The review article by Chang et al. (2008) gives a complete and up-to-date survey of published inventory literature under trade credits.

All the above articles are based on the assump-tion that the cost involved in inventory system remains constant over the planning horizon. This assumption may not be true in the real life, as many countries experience high annual inflation rate. As a result, while developing the inventory model, the effect of inflation cannot be ignored. Buzacott (1975) has developed an EOQ model with inflation subject to different types of pricing policies. Several other interesting and relevant papers exist in this direction (Datta & Pal, 1991;

https://www.researchgate.net/publication/32022656_A_Comment_on_Economic_Order_Quantity_under_Conditions_of_Permissible_Delay_in_Payments_Response?el=1_x_8&enrichId=rgreq-402c8bdd-e867-4187-8d51-1f312cf8be5e&enrichSource=Y292ZXJQYWdlOzI2MTk4ODM0NjtBUzoxMDE1Njc3NTE1MjQzNTZAMTQwMTIyNzA0NzYyNQ==

172


Sarkar & Pan, 1994; Hargia & Ben-Daya, 1996; Sarkar et al., 2000; Chung & Lin, 2001; Wee & Law, 2001; Jaggi & Goel, 2005; Jaggi et al., 2006; Moon et al., 2005). They have investigated the effects of inflation, time value of money and deterioration on inventory models.

In this paper, an attempt has been made to develop an inventory model with initial inspection under inflation and permissible delay in payment, in which units are deteriorating at a constant rate and the demand is a function of price, over a finite planning horizon using discounted cash flow (DCF) approach. The proposed model jointly optimizes retailer’s unit price and the number of replenishments by maximizing the total profit. Further, a comprehensive sensitivity analysis has also been performed to study the effects of inspec-tion, deterioration, permissible delay in payments and demand parameters on the optimal solution.

2. ASSUMPTIONS AND NOTATIONS

The following assumptions are used in develop-ing the model:

1. Demand rate is assumed to be a function of selling price, p, and is represented by

D (p) =k p e− , where k and e are the positive constants.

2. A constant fraction θ θ( )0 1≤ ≤ of the on-hand inventory deteriorates per unit time.

3. There is no repair or replenishment of the deteriorated items during the inventory cycle.

4. Rate of replenishment is infinite and lead-time is negligible.

5. Shortages are not allowed.6. The supplier allowed a fixed credit period,

M, to the retailer to settle the account.7. By assuming continuous compounding of

inflation, the ordering cost, purchasing cost,

out-of-pocket inventory carrying cost, and inspection cost at any time t is A t A e t( ) ,= 0

α C t C e t( ) ,= 0

α h t h e t( )= 0α a n d

In t In e t( )= 0α respectively, where A C h0 0 0, ,

and In0 are the ordering cost, the unit pur-chasing cost, the out-of-pocket inventory carrying cost and inspection cost respectively at time zero.

8. The effect of inflation on retailer’s selling price, p, is not considered.

9. A Discount Cash Flow (DCF) approach is used to consider the various costs at various times.

10. Planning horizon is finite.

The notations adopted in this paper are as below:

• H: Length of finite planning horizon• T: Inventory cycle length• n: No. of replenishment over the planning

horizon, n HT=

• N: Number of items received before inspection

• β: Fraction of non-defective items (0 ≤ β ≤ 1)

• Nβ: Number of non defective item at t = 0• Ie Interest earned per unit per unit time• Ip Interest paid per unit per unit time,

I Ip e≥• r: Discount rate (time value of money) per

unit per unit time• α: Inflation rate per unit per unit time, α<r• R r-α: The net discount rate per unit per

unit time• π1

H n p( , )Total profit for planning horizon H for case 1, when M ≤ T

• π2H n p( , )Total profit for planning horizon

H for case 2, when T ≤ M

173


3. MATHEMATICAL FORMULATION

The planning horizon (H) has been divided into n equal cycles of length T (i.e. T=H/n), and let us consider any ith cycle, i.e.t t ti i− ≤ ≤1 , where t0 0= , t Hn = t t Ti i− =−1 and t iT i ni = =( , , ........, ).1 2 At the beginning of the ith cycle, a batch of N units enters the inventory system. After inspection at time ti− =1 0, the effective inventory (Nβ) be the number of non-defective items retained to fulfill the demand and rest of the defective items ((1-β)N) returned to the supplier. The fraction of non-defective items β (0 ≤ β ≤ 1) is known, as it can easily be estimated from the past data. Thereafter, as time passes, the inventory level gradually decreasing mainly due to demand and partly due to deterioration and reaches zero at time ti (Figure 1).

Let I ti ( ) be the inventory level of the ith cycle at any time t t t t i ni i, ( , , , ......., ).− ≤ ≤ =1 1 2 The differential equation describing the instantaneous states of I ti ( ) over ( , )t ti i−1 is

dI t

dtI t D p k p

t t t i n

ii

e

i i

( )( ) ( ) ,

, , , ....., .

+ = − =−

≤ ≤ =

−

−

θ

1 1 2 (1)

The solution of the above differential equation along with the boundary condition

I t I Ni i i( )− = =1 0 β is

I t I eD p

e

t t t i

i it t t t

i i

i i( )( )

,

, , , ..

( ) ( )= + −{ }≤ ≤ =

− −− −

−

0

1

1 1 1

1 2

θ θ

θ...., .n

(2)

Since, I ti i( ) ,= 0 Equation (2) gives

N ID p

e i nit ti iβ

θθ= = −

=− −

01 1 1 2

( ), , .....,( )

(3)

or N D pe i nH n= −

=

( ), , .....,( / )

βθθ 1 1 2

(4)

Figure 1. Inventory scenario for Case 1: M ≤ T and Case 2: M≥ T

174


Substituting I i0 from Equation (3) in Equation (2) give

I tD p

e t t t i nit t

i ii( )

( ), , , , .....,( )= −

≤ ≤ =−

−θθ 1 1 2 31

(5)

The present worth of the retailer’s total profit ( ( , ), , )πjH n p j = 1 2 over the planning horizon (H)

is

πjH n p( , )= Sales revenue (Re )vH + Interest

earned ( )IeH − Order cost ( )AH − Purchase cost ( )PH − Holding cost ( )hH − Inspection cost ( )InH − Interest paid ( ),IpH where j = 1,2. (6)

1. Present worth of sales revenue (Re )vi for the ith cycle, is

Re ( )v pe D p e dtir t r t

t

ti

i

i

= − −−

−∫1

1

and the present worth of sales revenue (Re )vH over the planning horizon H is given by

Re Re ( ) //

v v pD p eee

Hi

i

nrH n

rH

rH n= = −

−−

=

−

−∑1

111

(7)

2. Present worth of ordering cost (Ai) for ith cycle, is

A A e e A e i nit r t Rti i i= = =− − −− −

0 01 1 1 1 2α , , , ......,

and the present worth of ordering cost ( )AH over the planning horizon H,is given by

A A Aee

Hi

i

n RH

RH n= =

−−=

−

−∑1

0

11 /

(8)

3. Present worth of effective inventory (Nβ) purchase cost (Pi) for ith cycle, is

P I C e e I C e i ni it r t

iRti i i= = =− − −− −

0 0 0 01 1 1 1 2α , , , ......,

and the present worth of purchase cost ( )PH over the planning horizon H,is given by

P P I C e

D p Ce

e

Hi

i

n

iRt

i

n

H nRH

i= =

= −

−

=

−

=

−

∑ ∑ −

10 0

1

0

1

111

( ) ( / )

θθ

−− −e RH n/

(9)

4. Present worth of inventory holding cost (hi) for ith cycle, is

h h t e I t e dt

h e I t e

i ir t

i

t

t

Rt

Rti

t

t

i

i

i

i

i

i

=

=

−− −

−

−

−

−

−

∫

∫

( ) ( )

( )

1

0

1

1

1

1

−−Rt dt

and the present worth of holding cost (hH ) over the planning horizon H is given in Equation 10 in Box 1.

5. Present worth of inspection cost (Ini) for ith cycle, is

Box 1.

h hh D p e e

ReR

Hi

i

n H n RH n RH n

= =−+

+−

−

=

− −

∑1

0 1 1( ) ( ) ( ) (/ / /

θ θ

θ eee

RH

RH n

−

−−)

( )/1 (10)

175


In In e e I In e I i nit r t

iRt

ii i i= = =− − −− −

0 0 0 01 1 1 1 2α , , , ......,

and the present worth of inspection cost ( InH ) over the planning horizon His given by

In InIn D p

eee

Hi

i

nH n

RH

RH n= = −

−−=

−

−∑1

0 111

( ) ( )( )

( / )/θ

θ

(11)

Since this model assumes permissible delay in payment, therefore, there will be two cases i.e. Case 1: M ≤ T and Case 2: M≥ T:

Case 1: M ≤ T

In this case, the retailer can earn interest on revenue generated from the sales up to M. He has to settle the account at M, for that, he has to arrange money at some specified rate of interest in order to get his remaining stocks financed for the period M to T.

6. Present worth of interest paid (Ipi) for ith cycle, is

Ip C t Ipe I t e dt C Ipe I t e dti ir t Rt

M

tRt Rt

M

Ti

ii= =−

− − − −− −∫( ) ( ) ( )1 01 1 ∫∫

and the present worth of interest paid ( IpH ) over the planning horizon H is given in Equation 12 in Box 2.

7. Present worth of interest earned (Iei) for ith cycle, is

Ie p Ie e D p t e dtir t r t

t

Mi

i

= − −−

−∫1

1

( )

and the present worth of interest earned ( IeH ) over the planning horizon H is given by

Ie Ie p IeD pr r

e M eee

Hi

i

nrM rM

rH

= = −{ }−

−−=

− −−

∑1

1 11

11

( )( )( −−rH n/ )

(13)

Substitute the values from Equations (7), (8), (9), (10), (11), (12), and (13) in Equation (6), the present worth of total total profit ( ( , ))π1

H n p becomes Equation 14 in Box 3.

Case 2: M≥ T

Here, the retailer can earn interest on revenue generated from the sales up to permissible time period M and pays no interest for the items kept in stock.

8. Present worth of interest earned (Iei) for ith cycle, is

Ie p Ie e D p t e dt

p Ie e D p Te e

ir t r t

t

t

r t rT r t

i

i

i

i

=

+

− −

− − −

−

−

−

∫11

1

( )

( ( ) ) ddtt

M

i∫

and the present worth of interest earned ( )IeH over the planning horizon H is given in Equation 15 in Box 4.

Substitute the values from Equations (7), (8), (9), (10), (11), and (15) in Equation (6), the pres-ent worth of total profit (π2

H n p( , ) ) becomes Equation 16 in Box 5.

Therefore, the retailer’s total profit over the planning horizon H can be expressed as

ππ

πH

H

Hn p

n p M T

n p M T( , )

( , ),

( , ),=

≤

≥

1

2

(17)

which is a function of two variables n and p, where n is discrete and p is continuous variable.

176


Box 2.

Ip IpC IpD p e e

Re eH

ii

n H n M RM RH n RH n RM

= =−

++

−

=

− − − − −

∑1

0 ( )

( )

( / ) / /

θ θ

θ

RRee

RH

RH n

−−

−

−

( )( )/11

(12)

Box 3.

π1 11 1

1H rH n rM rMn p pD p e p IeD pr r

e M e( , ) ( ) ( )/= −

+ −{ }−

− −

−−

−

+ −

−

−

11

100

ee

AD p C

e

rH

rH n

H n

/

( / )( )

θθ

+−+

+−

+

− −h D p e eR

eR

In D pe

H n RH n RH n0

0

1( ) ( ) ( )

( )

/ / /

θ θ

θ

θ

θθ

θ

θ θ

( / )

( / ) / /( )

( )

H n

H n M RM RH n RH nC IpD p e eR

e e

−

+−

++

−− − − −

1

0−−

−

−

−−RM

RH

RH

R

ee

( )(11 // )n

(14)

Box 4.

Ie Iep IeD pr r

eHne TH

ii

nrH n rH n= = −{ }−

+

=

− −∑1

11

( ) / / ee e eee

rH n rH n rMrH

rH n− − −

−

−−{ }

−−

/ //

( )( )11

(15)

Box 5.

π2

1

11

H

rH n

rH n rH nn p

pD p e

p IeD pr r

eHne

( , )

( )

( )

/

/ /=

−

+ −{ }− − −

+ −{ }

− − −Te e erH n rH n rM/ /

1−−−

−

+ −

+

−

−

ee

AD p C

e

h D p e

rH

rH n

H n

1

100

0

/

( / )( )

( ) (

θ

θ

θ

θθ

θ

θ

θ

H n RH n RH n

H n

eR

eR

In D pe

/ / /

( / )

) ( )

( )

−+

+−

+ −

− − 1

10

−−

−

−

( )( )/11ee

RH

RH n

(16)

177


4. SOLUTION PROCEDURE

Our objective is to find the optimal values of n and p which maximize total profit functions, πjH n p j( , ), , .= 1 2 Since the total profit functions, πjH n p j( , ), ,= 1 2 are the functions of two vari-

ables n and p, where n is discrete and p is con-tinuous variable, therefore, for any given value of n=n0 (say), the necessary condition for πjH n p j( , ), ,= 1 2 to be maximum are

∂

∂= =

πjH n p

pj

( , ), ,0 0 1 2 (18)

Further, πjH n p j( , ), ,= 1 2 are concave func-

tions in p for a fixed value of n (Appendix). Therefore, the Equation (18) will provide the optimal values of p for n = n0. In order to find the optimal value of n and p simultaneously, the fol-lowing algorithm is proposed:

Algorithm:

Step 1: Initialize n = n0, where n0 >1.Step 2: Calculate T = H/ n0. If M T≤ then go

to step 3 else go to step 5.Step 3: Obtain the optimal value of p (say p*) by

solving∂∂

=π1 0 0H n p

p

( , ).

Step 4: If π π1 0 1 0 1H Hn p n p( , ) ( , ),* *> − increment the value of n0 by 1 and go to step 2, else the present value of n0 is optimal (say n*) and the corresponding optimal values of price and total profit are p* and π1

H n p( , ).* *

Step 5: Obtain the optimal value of p (say p*) by

solving∂∂

=π2 0 0H n p

p

( , ).

Step 6: If π π2 0 2 0 1H Hn p n p( , ) ( , ),* *> − increment the value of n0 by 1 and go to step 2, else the present value of n0 is optimal (say n*) and the corresponding optimal values of price and total profit are p* and π2

H n p( , ).* *

Step 7: Set πH n p( , )* * = max [( ( , ), ( , )].* * * *π π1 2

H Hn p n p Determine the optimal value of T using, T* = H/n* and lot size (Nβ) using Equation (3).

5. NUMERICAL EXAMPLE

Let A0 = $80 /order, k = 400000, e = 2.0, θ = 0.1, Ip = 0.18 / year, Ie = 0.16 /year, C0 = $3 /unit, h0 = 0.36 /year, In0 = $0.15 /unit, and M = 0.17 year, β = 0.85, H = 5 year, r = 0.07 /year, and α = 0.03 /year, R = 0.04 /year.

Results are obtained using the proposed algo-rithm as: n* = 35, T* = 0.143 year, p* = $6.78, Nβ = 1253 units and πH n p( , )* * = $1, 25,084.

6. SENSITIVITY ANALYSIS

In this section, sensitivity analysis has been per-formed to study the impact of demand parameters (K and e), deterioration (θ), permissible payment period (M), net discount rate (R), inflation rate (α), inspection rate (β) on the optimal number of cycles (n*), optimal cycle length (T*), effective lot size (Nβ) and present worth of retailer’s total profit, πH n p( , )* * and results are displayed in Tables 1, 2, 3, 4, 5, and 6 respectively.

Based on the computational results shown in Tables 1, 2, 3, 4, 5, and 6, the following manage-rial insights are obtained.

1. Table 1 clearly shows that as the value of k increases, the selling price decreases margin-ally but the effective lot size and the present worth of total profit increases significantly. Further, it is evident from Table 1 that as the value of e decreases, there is considerable increase in selling price, effective lot size and present worth of total profit. Findings suggest that the retailer should be more vigilant while ordering.

178


Table 1. Impact of demand parameters (k and e) on the optimal replenishment policy

k ↓ e n p TNβ πH n p( , )

300000 3.0 16 5.201 0.313 677 14885

2.5 21 5.721 0.238 923 36435

2.0 30 6.799 0.167 1091 93222

1.7 35 8.230 0.143 1199 171993

1.5 37 10.156 0.135 1261 268760

350000 3.0 17 5.188 0.294 748 17565

2.5 24 5.698 0.208 951 42780

2.0 33 6.785 0.152 1161 109140

1.7 38 8.218 0.132 1291 201096

1.5 40 10.142 0.125 1363 314022

400000 3.0 19 5.166 0.263 774 20260

2.5 26 5.685 0.192 1008 49149

2.0 35 6.778 0.143 1253 125084

1.7 40 8.210 0.125 1404 230228

1.5 43 10.130 0.116 1451 359315

Table 2. Impact of θ on the optimal replenishment policy

θ 0.1 0.2 0.3

n 35 41 46

p 6.78 6.80 6.82

T 0.143 0.122 0.109

Nβ 1253 1068 950

πH n p( , ) 125084 124246 123514

Table 3. Impact of M on the optimal replenishment policy

M(Days) 40 50 60 70 80

n 23 24 26 27 27

p 5.76 5.72 5.69 5.66 5.63

T 0.217 0.208 0.192 0.185 0.185

Nβ 1106 1075 1008 983 993

πH n p( , )48182 48646 49149 49690 50238

179


2. Table 2 shows that as deterioration rate increases then there is significant decrease in cycle length (which suggests that retailer should procure more frequently), and the present worth of total profit (this happens because the cost of deteriorated units in-creases). Results suggests that by improving the storing conditions, the retailer can slow down the deterioration process which will help him to reduce the number of deteriorated units and hence increase his total profit.

3. From Table 3, one can easily observe that as permissible delay period increases sell-ing price and cycle length decreases but the present worth of total profit increases. From, economical point of view, the retailer should order more frequently to take benefit of delay in payments, which results in higher profit.

4. The effect of net discount rate can be ob-served from Table 4, as the net discount rate decreases then the order quantity decreases but the present worth of total profit increases,

Table 4. Impact of R on the optimal replenishment policy

R 0.05 0.04 0.03 0.02 0.01

n 33 35 37 39 40

p 6.776 6.778 6.779 6.781 6.786

T 0.152 0.143 0.135 0.128 0.125

Nβ 1330 1253 1184 1122 1093

πH n p( , )122531 125084 127733 130478 133321

Table 5. Impact of α on the optimal replenishment policy

α 0.05 0.04 0.03 0.02 0.01

n 37 41 46 52 60

p 7.10 6.92 6.75 6.58 6.42

T 0.135 0.122 0.109 0.096 0.083

Nβ 1080 1025 960 893 813

πH n p( , )119106 122121 125204 128357 131584

Table 6. Impact of β on the optimal replenishment policy

β 0.85 0.90 0.95

n 35 35 35

P 6.778 6.757 6.738

T 0.143 0.143 0.143

Nβ 1253 1261 1268

πH n p( , ) 125084 125476 125829

180


which clearly shows that in the highly liquid market the retailer should order less but more frequently that eventually increases his present worth of total profit.

5. Table 5 suggests that if the inflation rate increases then the cycle length as well as order quantity increases but the present worth of total profit decreases. Results show that in the highly inflationary market, e.g. developing countries where inflation runs in 2-digits, the retailer should procure more quantity but less frequently.

6. It is observed from Table 6 that if the frac-tion of non-defective items increases, then selling price remains unchanged but effec-tive lot size and present worth of total profit increases marginally. It suggests that if the fraction of non-defective items increases then the retailer should order more quantity.

7. CONCLUSION AND FUTURE RESEARCH DIRECTIONS

This paper presents a profit maximizing imperfect-quality inventory model of deteriorating items for price sensitive demand under inflation and permissible delay in payments over the finite planning horizon using the discounted cash flow (DCF) approach. This approach provides the proper and exact recognition of various costs in the inventory analysis. Concavity of the present worth of total profit function is proved. The model jointly optimizes the number of replenishments and the selling price with the help of proposed algorithm. A comprehensive sensitivity analysis is also conducted to explore the effects of the key parameters (viz. k, e, θ, M, R, α, β) on the optimal results. The findings provide important managerial insights to the retailers for deciding the appropriate ordering policy under varying situations, viz. (a) when the demand is highly

elastic then the retailer should be more vigilant while ordering, (b) in the highly liquid market the retailer should order less but more frequently that eventually increases his present worth of total profit, (c) for highly deteriorating items the retailer should order more frequently, and (d) in the highly inflationary market, the retailer should procure more but less frequently.

Further, the present study can be extended for more realistic situations such as the defective items may be reworked, stock dependent and stochastic demand with and without shortages.

ACKNOWLEDGMENT

The authors are thankful to anonymous referees and the editor for their constructive comments and valuable suggestions that improved the presenta-tion of the paper.

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183


APPENDIX

The Concavity

In this section, concavity of the total profit function, πH p n( , ) on appropriate domain for both the cases is shown.

Case 1: M<T

The total profit function (π1H n p( , ) ) for this case for any positive values of n= n0 (say) is

π1 00 1

1 11H rH n rM rMn p pD p e pD p Ie

r re M e( , ) ( ) ( )/= −

+ −{ }−

− −

−−

−

+ −

−

−

1

1

1

0

00

0

e

e

AD p C

e

rH

rH n

H n

/

( / )( )

θθ

+

−+

+−

+

− −h D p e eR

eR

In

H n RH n RH n0

0 0 0 1( ) ( ) ( )/ / /

θ θ

θ

00 00

0 0

1D p

eC IpD p e e

RH n

H n M RM RH n( ) ( )

( )( / )

( / ) /

θ θ θθ

θ

−

−+

− − −

++−

−−− −

−

−e e

R

e

eRH n RM

RH

RH n/ /

( )

(0

1

1 00 ),

(1)

(1) can be rewritten as,

π1 0 0

1

1 0

HRH

RH nn p pD p A A

e

eD p B( , ) ( )

( )

( )( )

/= −

−−

−−

− (2)

where

A e Ier r

e M eerH n rM rMr

= −

+ −{ }−

−− −

−/ 0 1

1 11

1 HH

rH ne1 0−

− /

B

Ce

h e eR

eR

H nH n RH n RH n

=

−

+

−+

+− − −

0 00

0 0 0

11

θ θ θθ

θ( / )

/ / /( ) ( )

+ −

+

−− − −Ine

C Ip e eH nH n M RM RH n

0 00

0 0

1θ θ θ

θθ

( / )( / ) /

( +++

−

−−− −

−

−

Re e

R

e

eRH n RM

RH

R

)

( )

(/ 0

1

1 HH n/ )0

Differentiate π1 0H n p( , )w.r.t. p, we get

d n p

dpD p pD p A D p B

Hπ1 0( , ) ( ) ( ) ( ) ,= + ′ − ′

184


and

d n p

dpD p A D p pA B

H21 0

22

π ( , )( ) ( )( ),= ′ + ′′ −

Sincer > α andp > 0, ⇒ −−

>

−−

> −−

−

−

−

1

101

10 1

0 0

0e

e

e

ee

rH

rH n

RH

RH nrH n

/ //,

( )

( ),

< 0 and

Ier r

e M erM rM1 11 0−{ }−

<− − thenA and B< >0 0

⇒ <d n p

dp

H21 0

20

π ( , ), for any integer n0 0>

Hence, π1 0H n p( , ) is a concave function of p for any positive integer.

Case 2: M>T

Here, the total profit function (π2H n p( , ) ) for any positive values of n= n0 (say) is

π2 0

0

0

0

1

11

H

rH n

rH n rn p

pD p e

pD p Ier r

eHne

( , )

( )

( )

/

/=

−

+ −{ }−− − HH n rH n rH n rMTe e e/ / /0 0 0

+ −{ }

− − −

−−

−+ −

−

−

1

1

1

0

00

0

e

e

AD p C

e

rH

rH n

H n

/

( / )( )

θθ

+−+

+−

+

− −h D p e eR

eR

In D

H n RH n RH n0

0

0 0 0 1( ) ( ) ( )

(

/ / /

θ θ

θ

ppe

e

eH n

RH

RH n)

( )

( )( / )

/

θθ 0

0

1

1

1−

−−

−

−

(3)

(3) can be rewritten as,

π2 0 0

1

1 0

HRH

RH nn p pD p C A

e

eD p D( , ) ( )

( )

( )( )

/= −

−−

−−

− (4)

where,

C eIer r

eHnerH n rH n rH n= −

+ −{ }−

− −/ / /0 0 0111

0

++ −{ }

−−

− − −−

−Te e ee

erH n rH n rM

rH

rH n/ /

/0 0

1

1 00

DCe

h e eR

eR

H nH n RH n RH n

= −

+

−+

+− − −

0 00

0 0 0

11

θ θ θθ

θ( / )

/ / /( ) ( )

+ −

−−

−

−

Ine

e

eH n

RH

RH n0 0

01

1

1θθ( / )

/

( )

( )

185


Differentiate π2 0

H n p( , )w.r.t. p, we get

d n p

dpD p pD p C D p D

Hπ2 0( , ) ( ) ( ) ( ) ,= + ′ − ′

and

d n p

dpD p C D p pC D

H22 0

22

π ( , )( ) ( )( ),= ′ + ′′ −

Sincer > α andp > 0,

⇒−−

>

−−

> −−

−

−

−

1

101

10 1

0 0

0e

e

e

ee

rH

rH n

RH

RH nrH n

/ //,

( )

( ),

< 0

and 1 1 00 0

0re

HnerH n rH n−{ }− <− −/ / then, C < 0 and D > 0

⇒ <d n p

dp

H22 0

20

π ( , ), for any integer n0 0>

Hence, π2 0H n p( , ) is a concave function of p for any positive integer, n0 0> .

This work was previously published in the International Journal of Strategic Decision Sciences, Volume 2, Issue 2, edited by Madjid Tavana, pp. 20-35, copyright 2011 by IGI Publishing (an imprint of IGI Global).

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