An EOQ Model for A Deteriorating Item with Time Dependent Exponential Demand Under Permissible Delay in Payment

(1)

An EOQ Model for A Deteriorating Item with Time Dependent Exponential Demand Under Permissible

Delay in Payment

TRAILOKYANATH SINGH

¹

, SUDHIR KUMAR SAHU

²

and AMEEYA KUMAR NAYAK

³

1

Department of Mathematics,

C. V. Raman College of Engineering, Bhubaneswar, Odisha, INDIA

2

Department of Statistics, Sambalpur University, Odisha, INDIA

3

Department of Mathematics,

Ravenshaw University, Cuttack, Odisha, INDIA

(Received on: June 3, 2012)

ABSTRACT

This paper deals with an EOQ (Economic Order Quantity) model for deteriorating items with time-dependent demand when the delay in payment is permissible. The deterioration rate is assumed to be constant and the time varying demand rate is taken to be an exponential function of time. It is based on items like compute chips, mobile phones, fashion and seasonal goods etc. having a short market life. Mathematical models are also derived under two different circumstances, i.e., Case-I: the credit period is less than or equal to the cycle time for setting the account and Case-II: the credit period is greater than the cycle time for setting the account.

We then develop an algorithm for retailers to determine the minimum average cost in the economic order quantity. Numerical examples justify the validity of the theoretical results.

Keywords: Inventory, Deterioration, Exponential demand, Permissible delay in payment.

INTRODUCTION

Deteriorating items refer to items that become decayed, damaged, expired, invalid, evaporative, devaluation and so on

through time. Factors such as demand,

deteriorating rate, price discount, allow

shortage or not, inflation and so on should

be taken into consideration in the

deteriorating inventory study. Among them,

(2)

demand is one of the key factors that should be taken into consideration in an inventory study.

In traditional EOQ (Economic Order Quantity) model, it assumes that buyers must pay for the items purchased as soon as the items are received. However, a supplier permits the buyer a period of time called credit period, to settle the total amount owed to him. Usually, interest is not charged for the outstanding amount if it is paid within the permissible delay period. Wagner and Whitin

¹

developed an inventory model for fashionable products deteriorating at the end of a prescribed storage period. In 1915, the classical EOQ was developed in which the demand rate was taken as constant. In the classical inventory models, the demand rate is assumed to be either constant or time- dependent. In a buyer-seller situation, the depletion of the inventory is caused by constant demand rate. But in real life situation, the demand rate of any product is always in a dynamic state. In this respect, many inventory researchers have paid their attention for time-dependent demand. Silver and Meal

²

first suggested a simple modification of the case of a varying demand. Many researchers like Donaldson

³

, Silver

⁴

, Ritchie

^5,6,7

, Mitra et al.

⁸

etc., made valuable contributions in this direction.

Researchers like Dave and Patel

⁹

, Bahari- Kasani

¹⁰

, Gowsami and Chaudhuri

¹¹

, Chung and Ting

¹²

, Hariga

¹³

, Jalan, Giri and Chaudhuri

¹⁴

, Giri, Goswami and Chaudhuri

¹⁵

, Jalan and Chaudhuri

¹⁶

, Lin, Tan and Lee

¹⁷

, etc., then developed inventory models for deteriorating items with trended demands.

Researchers like Wee

¹⁸

and Jalan and Chaudhuri

¹⁹

developed their model taking the demand pattern as an exponentially time

varying demand. Between the two types of time-dependent demands, namely linear and exponential, a linearly time-varying demand indicates a uniform change in demand rate of item per unit time which seldom occurs in the real market situation and the other indicates a rapid change in demand rate. It is often seen in the supermarket when the new computer chips, mobile phones, fashion and seasonal goods are come to the market. In real life situation, we see that suppliers offer their customers a certain credit period with interest during the permissible delay period.

Goyal

²⁰

first studied the inventory models with permissible delay in payment.

Researchers like Shinn et al.

²¹

, Chu et al.

²²

and Chung, Chang and Yang

²³

also extended Goyal’s (1985) models for the case of deteriorating item. Researchers like Davis and Gaither

²⁴

, Mandal and Phaujder

²⁵

, Aggarwal and Jaggi

²⁶

, Chang, Hung and Dye

²⁷

and Chung and Lio

²⁸

developed the inventory models considering the permissible delay in payment. Sana and Chaudhury

²⁹

developed a more general EOQ model with delay in payments, price-discount effects and different types of demand rates.

Recently, Khanra, Ghosh and Chaudhuri

³⁰

developed an EOQ model for a deteriorating item with time dependent quadratic demand under permissible delay in payment.

The purpose of this paper is to analyze an EOQ model for a deteriorating item with time dependent exponential demand under permissible delay in payment.

This model is based on the products like

computer chips, mobile phones, fashion and

seasonal goods found in the super market

with a short market life. The demand rate in

this case is of the exponential form which is

realistic to discuss such model.

(3)

Mathematical models have been derived to illustrate the two different circumstances:

Case-I: The credit period is less than or equal to the cycle time for settling the account and Case-II: The credit period is greater than the cycle time for settling the account. Numerical examples are provided to illustrate the model. Also the sensitivity analysis is carried out to study the optimal solution to changes in the values of the different parameters involved in the system.

ASSUMPTIONS AND NOTATIONS

The following assumptions are made in developing the model.

(i) The demand rate for the item is represented by an exponential and continuous function of time.

(ii) Replenishment rate is infinite, i. e. , replenishment rate is instantaneous.

(iii) Shortage is not allowed.

(iv) The deterioration rate is constant on the on-hand inventory per unit time and there is no repair or replenishment of the deteriorated items within the cycle.

(v) Time horizon is infinite.

The following notations have been used in developing the model.

(i) The time dependent demand rate is

^ఒ௧

, 0, 0.

(ii) is the unit purchase cost of item.

(iii)

௣

is the inventory holding cost (excluding interest charges) per rupee of unit purchase cost per unit time.

(iv) 0 1 is the constant rate of deterioration of an item.

(v) is the replenishment cost.

(vi)

௣

is the interest charges per rupee investment in stock per year.

(vii)

௘

is the interest earned per rupee in a year.

(viii)

ଵ

is the permissible period (in year) of delay in settling the accounts with the supplier.

(ix) is the time interval (in year between two successive orders).

MATHEMATICAL FORMULATION AND SOLUTION OF THE PROBLEM The instantaneous inventory level

at any time during the cycle time is governed by the following differential equation

ௗூሺ௧ሻ

ௗ௧

, 0 , (1) with 0

଴

, 0 and

^ఒ௧

. The solution of Eq. (1) is

_ఒାఏ^௄

^{ሺఒାఏሻ்ିఏ௧}

^ఒ்

, 0 (2)

If 0 , then Eq. (2) represents the instantaneous inventory level at any time for the constant demand rate.

The initial order quantity is

଴

0

_ఒାఏ^௄

^{ሺఒାఏሻ்}

1 . (3) The total demand during the cycle period

0, is

_଴^்

^௄_ఒ

^ఒ்

1 . (4)

(4)

Number of deteriorating units is

଴ ଴^் _ఒାఏ^ଵ ^{ሺఒାఏሻ்} 1 ^ଵ_ఒ ^ఒ் 1

.

(5)

Deterioration cost for the cycle 0, is (number of deteriorating units)

_ఒାఏ^ଵ ^{ሺఒାఏሻ்} 1 ^ଵ_ఒ^ఒ் 1

. (6) Total holding cost for the cycle 0, is

_଴^் _ఒାఏ^௄௛ ^ଵ_ఏ^{ሺఒାఏሻ்} ^ఒ்

ଵ

ఒ^ఒ் 1

where

௣

. (7) Case-1: Let

ଵ

.

Since the interest is payable during the time

ଵ

, the interest payable in any cycle 0, is

ଵ ௣ ௧^்భ ^௣ூ_ఒାఏ^೛^௄^ଵ_ఏ^ఒ்^{ఏሺ்ି௧}^భ^ሻ 1

ଵ

ఒ^ఒ் ^ఒ௧^భ

. (8) Interest earned in the cycle period 0, is

ଵ ௘ _଴^் ^௣ூ_ఒ^೐^௄^ఒ்

ଵ

ఒ^ఒ் 1

. (9)

Total variable cost per cycle

=replenishment cost +inventory holding cost +deterioration cost +interest payable during the permissible delay period –interest earned during the cycle.

The total variable cost per cycle pert unit time is

ଵ ^஺_்_{்ሺఒାఏሻ}^௄௛ ^ଵ_ఏ ^{ሺఒାఏሻ்} ^ఒ் ^ଵ_ఒ ^ఒ் 1

^௣௄_் _ఒାఏ^ଵ ^{ሺఒାఏሻ்} 1 ^ଵ_ఒ^ఒ் 1

_{்ሺఒାఏሻ}^௣ூ^೛^௄ _ఏ^ଵ^ఒ்^{ఏሺ்ି௧}^భ^ሻ 1

ଵ

ఒ^ఒ் ^ఒ௧^భ ^௣ூ_்ఒ^೐^௄^ఒ் ^ଵ_ఒ^ఒ் 1 .

(10) Our aim is to find minimum variable cost per unit time.

The necessary and sufficient condition to minimize

_ଵ

for a given value

ଵ

are respectively

^ௗ௓^భ^ሺ்ሻ

ௗ்

0 and

^ௗ^మ_ௗ்^௓^భ^ሺ்ሻ_మ

0 . Now

^ௗ௓^భ^ሺ்ሻ

ௗ்

0 gives the following non- linear equation in :

^ఒ்^{ሺ௛ାఏ௣ሻ}_ఏ ^ఏ் 1 ^௣ூ_ఏ^೛ ^{ఏሺ்ି௧}^భ^ሻ 1 ௘

ሺఒାఏሻ^௄௛ _ఏ^ଵ^{ሺఒାఏሻ்} ^ఒ் ^ଵ_ఒ^ఒ் 1

_ఒାఏ^ଵ ^{ሺఒାఏሻ்} 1 ^ଵ_ఒ^ఒ் 1

_{ሺఒାఏሻ}^௣ூ^೛^௄^ଵ_ఏ^ఒ்^{ఏሺ்ି௧}^భ^ሻ 1 ^ଵ_ఒ^ఒ் ^ఒ௧^భ

^௣ூ_ఒ^೐^௄^ఒ் ^ଵ_ఒ^ఒ் 1

.

(11)

To get the optimal cycle length

ଵ

, we have to solve Eq. (11) provided it satisfies the following condition

ௗ^మ௓_మሺ்ሻ

ௗ்^మ

0 .

The EOQ in this case is as follows:

଴

ଵ

_ఒାఏ^௄

!

^{ሺఒାఏሻ்}^భ

1" . (12) The minimum annual variable cost

_ଵ

ଵכ

is obtained from Eq. (7) for

ଵ

. Case-2: Let

ଵ

.

In this case the customer earns

interest on the sales revenue up to the

(5)

permissible delay period and no interest payable during the period for the item kept in the stock.

Interest earned for the time period 0, is

௘ _଴^் ^௣ூ_ఒ^೐^௄^ఒ்^ଵ_ఒ^ఒ் 1

.

(13)

Interest earned for the permissible delay period ,

ଵ

is

௘ଵ _଴^் ^௣ூ^೐^ሺ௧^భ_ఒ^ି்ሻ௄^ఒ் 1

.

(14)

Hence the total interest earned during the cycle= Interest earned for the time period 0, + Interest earned for the permissible delay period ,

ଵ

, i. e.,

ଶ^௣ூ_ఒ^೐^௄^ఒ்^ଵ_ఒ^ఒ் 1 ^௣ூ^೐^ሺ௧^భ_ఒ^ି்ሻ௄^ఒ் 1 ^௣ூ_ఒ^೐మ^௄ଵ 1^ఒ் 1

. (15) In this case, the total variable cost per cycle

=replenishment cost +inventory holding cost +deterioration cost –interest earned during the cycle.

Hence the total variable cost per cycle per unit time is

ଶ ^஺_்_{்ሺఒାఏሻ}^௄௛ ^ଵ_ఏ ^{ሺఒାఏሻ்} ^ఒ் ^ଵ_ఒ ^ఒ் 1

^௣௄

் _ఒାఏ^ଵ ^{ሺఒାఏሻ்} 1

ଵ

ఒ^ఒ் 1 ^௣ூ_்ఒ^೐మ^௄ଵ 1^ఒ் 1

.

(16)

As before we have to minimize

_ଶ

for a given value of

ଵ

.

The necessary and sufficient condition to minimize

_ଶ

for a given value

ଵ

are respectively

^ௗ௓^మ^ሺ்ሻ

ௗ்

0 and

^ௗ^మ_ௗ்^௓^మ^ሺ்ሻ_మ

0 .

Now

^ௗ௓^మ^ሺ்ሻ

ௗ்

0 gives the following non- linear equation in :

^ఒ்^{ሺ௛ାఏ௣ሻ}_ఏ ^ఏ் 1 ௘ଵ^ଵ_ఒ^ିఒ் 1

ሺఒାఏሻ^௄௛ ^ଵ_ఏ ^{ሺఒାఏሻ்} ^ఒ் ^ଵ_ఒ ^ఒ் 1

_ఒାఏ^ଵ ^{ሺఒାఏሻ்} 1 ^ଵ_ఒ ^ఒ் 1

^௣ூ_ఒ^೐^మ^௄ଵ 1 ^ఒ் 1

.

(17)

To get the optimal cycle length

ଶ

, we have to solve Eq. (17) provided it satisfies the following condition

ௗ^మ௓_మሺ்ሻ

ௗ்^మ

0 .

The EOQ in this case is as follows:

଴

ଶ

_ఒାఏ^௄

!

^{ሺఒାఏሻ்}^మ

1" . (18) The minimum annual variable cost

_ଶ

ଶכ

is obtained from Eq. (7) for

ଶ

. Case-3: Let

ଵ

.

ଵ

ఒ^ఒ௧^భ 1

^௣௄_௧

భ _ఒାఏ^ଵ ^{ሺఒାఏሻ௧}^భ 1

ଵ

ఒ^ఒ௧^భ 1 ^௣ூ_௧^೐^௄

భఒ ଵ^ఒ௧^భ ^ଵ_ఒ^ఒ௧^భ 1

. (19) The EOQ in this case is as follows:

଴

ଵ

_ఒାఏ^௄

!

Step 2: Determine

ଶכ

from (17). If

ଵכ

ଵ

, evaluate

_ଵ

ଵכ

from Eq. (16).

Step 3: If the condition

ଵכ

ଵ

ଶכ

is satisfied, go to Step-4. Otherwise go to Step-5.

Step 4: Compare

_ଵ

ଵכ

and

ଶ

_ଶ

ଶכ

is the minimum cost.

Step 6: Compute

଴כ

ଵ

or

଴כ

ଶ

for the minimum cost.

NUMERICAL EXAMPLES

The numerical examples given below cover all the three cases that arise in the model.

Example 1: (case I and Case II) Minimum average cost is

ଵ

ଵכ

.

Let us consider the parameter values of the inventory system as 1000 units per year, 1 unit per year, #$. 200 per order,

௣

0.15 per year,

௘

0.13 per year, 0.12 per year, #$. 20 per unit,

0.20 and

ଵ

0.25 year.

Solving Eq. (11), we have

ଵכ

0.314 year and minimum average cost is

ଵ

ଵכ

975.726 .

Again, solving Eq. (17), we have

ଶכ

0.224 year and minimum average cost is

ଶ

ଶכ

1023.27 .

Here the condition

ଵכ

ଵ

ଶכ

is satisfied.

Comparing

_ଵ

ଵכ

and

ଶ

ଶכ

, the minimum average cost in this case is

ଵ

ଵכ

975.726 where the optimum cycle length is

0.15 per year,

௘

0.13 per year, 0.12 per year, #$. 20 per unit,

0.20 and

ଵ

0.15 year.

Solving Eq. (11), we have

ଵכ

0.276 year and minimum average cost is

ଵ

ଵכ

1100.38 .

Again, solving Eq. (17), we have

ଶכ

0.221 year and minimum average cost is

ଶ

ଶכ

1314.46 .

Here the condition

ଵכ

ଵ

holds, but not

ଶכ

ଵ

, so only Case-I holds.

Hence the minimum average cost in this case is

_ଵ

ଵכ

1100.38 where the optimum cycle length is

ଵכ

0.274 year.

The economic order quantity is given by

଴כ

ଵכ

164.348 .

Example 3: (case I and Case II) Minimum

average cost is

ଶ

ଶכ

.

(7)

Let us consider the parameter values of the inventory system as 1000 units per year, 1 unit per year, #$. 200 per order,

௣

0.15 per year,

௘

0.13 per year, 0.12 per year, #$. 20 per unit,

0.20 and

ଵ

0.35 year.

Solving Eq. (11), we have

ଵכ

0.363 year and minimum average cost is

ଵ

ଵכ

928.909 .

Again, solving Eq. (17), we have

ଶכ

0.228 year and minimum average cost is

_ଶ

ଶכ

728.511 .

Here the condition

ଵכ

ଵ

ଶכ

is satisfied.

Comparing

_ଵ

ଵכ

and

ଶ

ଶכ

, the minimum average cost in this case is

ଶ

ଶכ

728.511 where the optimum cycle length is

0.15 per year,

௘

0.13 per year, 0.12 per year, #$. 20 per unit,

0.20 and

ଵ

0.45 year.

Solving Eq. (11), we have

ଵכ

0.419 year and minimum average cost is

ଵ

ଵכ

939.516 .

Again, solving Eq. (17), we have

ଶכ

0.232 year and minimum average cost is

ଶ

ଶכ

439.153 .

0.15 per year,

௘

0.13 per year, 0.12 per year, #$. 20 per unit,

0.20 and

ଵ

year.

Solving Eq. (19), we have

ଵכ

ଶכ

ଵכ

0.229 year which satisfies Case-III.

Here the minimum average cost in this case is

_ଶ

ଶכ

649.563 where the optimum cycle length is

ଶכ

0.229 year.

The economic order quantity is given by

଴כ

ଶכ

263.935 .

SENSITIVITY ANALYSIS

We now study the study the effects of changes in the values of the system parameters , ,

௣

,

௘

, , ,

ଵ

, and on the optimal total cost and number of reorder.

The sensitivity analysis is performed by

changing each of the parameters by 50%,

10%, 10% and 50% , taking one

parameter at a time and keeping the

remaining parameters unchanged. The

analysis is based on the example-3 and the

results are shown in the Table-1. The

following points are observed.

(8)

(i)

ଵכ

&

ଶכ

decrease while

_ଵ

ଵכ

&

ଶ

ଶכ

increase with the increase in the value of the parameter . Both

ଵ

ଵכ

&

ଶ

ଶכ

have low sensitivity to changes in and

ଵ

ଵכ

_ଵ

ଵכ

&

ଶ

ଶכ

decrease with the increase in the value of the parameter

௘

. Here

ଵכ

,

_ଵ

ଵכ

&

ଶ

ଶכ

are moderately sensitive while

(vi)

ଵכ

&

ଶכ

decrease while

_ଵ

ଵכ

&

ଶ

ଶכ

increase with the increase in the value of the parameter . Here

ଵכ

,

ଶכ

,

ଵ

ଵכ

&

ଶ

(ix)

ଵכ

,

ଶכ

,

_ଵ

ଵכ

&

ଶ

ଶכ

increase with the increase in the value of the parameter . Here

ଵכ

&

ଶכ

have low sensitivity to changes in and

ଵ

ଵכ

&

ଶ

ଶכ

have moderately sensitive to changes in .

However, these outcomes may be due

to the choice of the particular parameter

values in this numerical example. From the

Table-1, it is found that the optimum cost

decreases rapidly with the increase of the

parameter

௘

and . These justify the real

market situation.

(9)

Table-1

Parameter %

change ଵכ ଵଵכ ଶכ ଶଶכ Remark Solution % change

50 10 -10 -50

0.283 0.306 0.323 0.384

1129.28 1008.84 940.907 781.608

0.187 0.215 0.235 0.303

1049.43 1034.56 1008.03 889.862

ଵכ ଵ ଶכ

ଵכ ଵ

ଵଵכ

+15.73 +3.39 -3.56 -19.89

50 10 -10 -50

0.301 0.311 0.317 0.331

1012.99 983.24 968.174 937.482

0.218 0.223 0.226 0.232

1034.11 1025.53 1020.96 1011.17

ଵכ ଵ ଶכ

ଵଵכ

+3.81 +0.76 -0.77 -3.91

௣ 50 10 -10 -50

0.301 0.311 0.317 0.335

986.521 978.257 972.959 958.613

0.224 0.224 0.224 0.224

1023.27 1023.27 1023.27 1023.27

ଵכ ଵ ଶכ

ଵଵכ

+1.11 +0.25 -0.28 -1.75

௘ 50 10 -10 -50

0.365 0.323 0.306 0.280

699.875 924.425 1025.45 1210.95

0.211 0.222 0.227 0.240

812.524 981.718 1064.51 1025.98

ଵכ ଵ ଶכ

ଵଵכ

-28.27 -5.25 +5.10 +24.11 50

10 -10 -50

0.312 0.314 0.315 0.300

987.587 978.106 973.346 1068.22

0.223 0.224 0.225 0.217

1031.2 1024.86 1021.68 1085.54

ଵכ ଵ ଶכ

ଵଵכ

+1.21 +0.24 -0.24 +9.48

50 10 -10 -50

0.285 0.307 0.323 0.380

1118.74 1006.53 943.372 788.841

0.188 0.215 0.235 0.300

1042.97 1033.05 1009.71 901.174

ଵכ ଵ ଶכ

ଵכ ଵ

ଵଵכ

+14.66 +3.15 -3.31 -19.15

ଵ 50 10 -10 -50

0.376 0.325 0.303 0.269

926.818 957.645 998.65 1146.64

0.229 0.225 0.223 0.220

658.479 950.386 1096.12 1387.17

ଵכ ଵ ଶכ

ଶଶכ

ଵଵכ

-32.51 -2.59 +2.34 +17.51

50 10 -10 -50

0.263 0.301 0.329 0.419

1340.68 1054.96 892.438 500.241

0.196 0.218 0.232 0.272

1272.66 1076.2 968.503 725.844

ଵכ ଵ ଶכ

ଵכ ଵ

ଶଶכ

ଵଵכ

+30.43 +8.12 -8.54 -48.73

50 10 -10 -50

0.352 0.322 0.306 0.266

1275.68 1038.57 911.183 631.709

0.268 0.234 0.214 0.164

1429.32 1110.56 932.04 510.133

ଵכ ଵ ଶכ

ଵଵכ

ଶଶכ

+30.74 +6.44 -6.62 -47.71

(10)

CONCLUSION

In this paper, an EOQ model for a deteriorating item with time dependent exponential demand under permissible delay in payment is considered. This type of model is useful for the items like computer chips, mobile phones, fashion and seasonal goods etc. which deteriorate with time having a short market life. Thus in the real market situation, we see that the suppliers offer their customers a certain credit period without interest during the permissible delay time period. As a result, customers order more quantities due to delay in payment. The model proposed in this paper can be extended in constant deterioration rate to variable deterioration rate and time- dependent deterioration.

REFERENCES

1. H. M. Wagner, T. M. Whitin, “Dynamic version of the economic lot size Model”, Management Science, 5, 89-96 (1958).

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

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

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

5. E. Ritchie, “Practical inventory replenishment policies for a linear trend

in demand followed by a period of steady demand”, Journal of Operational Research Society 31, 605–613 (1980).

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

7. E. Ritchie, “Stock replenishment quantities for unbounded linear increasing demand: an interesting consequence of the optimal policy”, Journal of Operational Research Society 36, 737–739 (1985).

8. A. Mitra, J.F. Fox, R.R. Jessejr, “A note on determining order quantities with a linear trend in demand”, Journal of Operational Research Society 35, 141–

144 (1984).

9. U. Dave, L.K. Patel, (T, Si) policy inventory model for deteriorating items with time proportional demand, Journal of Operational Research Society 32, 137–142 (1981).

10. H. Bahari-Kashani, Replenishment schedule for deteriorating items with time-proportional demand, Journal of Operational Research Society 40 75–81, (1989).

11. A. Goswami, K.S. Chaudhuri, An EOQ model for deteriorating items with a linear trend in demand, Journal of Operational Research Society 42, (12), 1105–1110 (1991).

12. K.J. Chung, P.S. Ting, A heuristic for replenishment of deteriorating items with a linear trend in demand, Journal of Operational Research Society 44 (12), 1235–1241 (1993).

13. M. Hariga, An EOQ model for

deteriorating items with shortages and

time-varying demand, Journal of

(11)

Operational Research Society 46, 398–

404 (1995).

14. A.K. Jalan, R.R. Giri, K.S. Chaudhuri, EOQ model for items with Weibull distribution deterioration, shortages and trended demand, International Journal of Systems Science 27 (9), 851–855 (1996).

15. B.C. Giri, A. Goswami, K.S. Chaudhuri, An EOQ model for deteriorating items with time varying demand and costs, Journal of Operational Research Society 47, 1398–1405 (1996).

16. A.K. Jalan, K.S. Chaudhuri, Structural properties of an inventory system with deterioration and trended demand, International Journal of Systems Science 30 (6), 627–633 (1999).

17. C. Lin, B. Tan, W.C. Lee, An EOQ model for deteriorating items with time- varying demand and shortages, International Journal of Systems Science 31 (3), 391–400 (2000).

18. H.M. Wee, A deterministic lot-size inventory model for deteriorating items with shortages and a declining market, Computer & Operations Research 22 (3), 345–356 (1995).

19. A.K. Jalan, K.S. Chaudhuri, An EOQ model for deteriorating items in a declining market with SFI policy, Korean Journal of Computational and Applied Mathematics 6 (2), 437–449 (1999).

20. S. K. Goyal, EOQ under conditions of permissible delay in Payments, Journal of Operation Research Society 36, 335–

338 (1985).

21. S.W. Shinn, H.P. Hwang, S. Sung, Joint price and lot size determination under condition of permissible delay in

payments and quantity discounts for freight cost, European Journal of Operational Research 91, 528–542 (1996).

22. P. Chu, K.J. Chung, S.P. Lan, Economic order quantity of deteriorating items under permissible delay in payments, Computer & Operations Research 25, 817–824 (1998).

23. K.J. Chung, S.L. Chang, W.D. Yang, The optimal cycle time for exponentially deteriorating products under trade credit financing, The Engineering Economist 46, 232–242 (2001).

24. R.A. Davis, N. Gaither, Optimal ordering policies under conditions of extended payment privileges, Manage- ment Science 31, 499–509 (1985).

25. B.N. Mandal, S. Phaujdar, Some EOQ models under conditions of permissible delay in payments, International Journal of Management Science 5 (2), 99–108 (1989).

26. S.P. Aggarwal, C.K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, Journal of Operation Research Society 36, 658–

662 (1995).

27. H.J. Chang, C.H. Hung, C.Y. Dye, An inventory model for a deteriorating items with linear trend in demand under conditions of permissible delay in payments, Production Planning and Control 12, 272–282 (2001).

28. K.J. Chung, J.J. Lio, Lot-sizing decisions under trade credit depending on the ordering quantity, Computer &

Operations Research 31, 909–928 (2004).

29. S.S. Sana, K.S. Chaudhuri, A

deterministic EOQ model with delays in

(12)

An EOQ Model for A Deteriorating Item with Time Dependent Exponential Demand Under Permissible Delay in Payment