Cmjv03i03p0358

J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

An EOQ Model for A Deteriorating Item with Time Dependent Exponential Demand Under Permissible Delay in Payment TRAILOKYANATH SINGH1, SUDHIR KUMAR SAHU2 and AMEEYA KUMAR NAYAK3 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, Permissible delay in payment.

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

Exponential

demand,

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,

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

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Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

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 Whitin1 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 timedependent. 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 Meal2 first suggested a simple modification of the case of a varying demand. Many researchers like Donaldson3, Silver4, Ritchie5,6,7, Mitra et al.8 etc., made valuable contributions in this direction.

Researchers like Dave and Patel9, BahariKasani10, Gowsami and Chaudhuri11, Chung and Ting12, Hariga13, Jalan, Giri and Chaudhuri14, Giri, Goswami and Chaudhuri15, Jalan and Chaudhuri16, Lin, Tan and Lee17, etc., then developed inventory models for deteriorating items with trended demands.

Researchers like Wee18 and Jalan and Chaudhuri19 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.

Goyal20 first studied the inventory models with permissible delay in payment.

Researchers like Shinn et al.21, Chu et al.22 and Chung, Chang and Yang23 also extended Goyal’s (1985) models for the case of deteriorating item. Researchers like Davis and Gaither24, Mandal and Phaujder25, Aggarwal and Jaggi26, Chang, Hung and Dye27 and Chung and Lio28 developed the inventory models considering the permissible delay in payment. Sana and Chaudhury29 developed a more general EOQ model with delay in payments, price-discount effects and different types of demand rates.

Recently, Khanra, Ghosh and Chaudhuri30 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.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

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.

360

is the replenishment cost. ௣ is the interest charges per rupee investment in stock per year. (vii) ŕŻ˜ 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). (v) (vi)

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 ŕŻ—ŕŻ‚áˆşŕŻ§áˆť

ௗ௧

,

0 ,

(1)

with 0 ଴ , 0 and ఒ௧ .

The solution of Eq. (1) is ௄

ŕ°’ŕŹžŕ°? áˆşŕ°’ŕŹžŕ°?áˆťŕŻ?ିŕ°?௧ ŕ°’ŕŻ? , 0 (2)

The following notations have been used in developing the model.

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

(i)

The initial order quantity is

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.

௄

଴ 0 ŕ°’ŕŹžŕ°? áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? 1 .

(3)

The total demand during the cycle period 0, is �

௄

଴ ŕ°’ ŕ°’ŕŻ? 1 .

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

(4)

361

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012) ଵ ஺

Number of deteriorating units is �

଴ ଴

ଵ

�

ŕ°’

.

(5)

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

ଵ

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? 1 ŕ°’ŕŻ? 1 . ଵ

ŕ°’ŕŹžŕ°?

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? 1 ŕ°’ŕŻ? 1 ଵ

ŕ°’ŕŹžŕ°?

ŕ°’

(6)

ଵ ŕ°’

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? ŕ°’ŕŻ? ŕ°’ŕŻ? 1

௄௛

ଵ

ଵ

ŕŻ?áˆşŕ°’ŕŹžŕ°?áˆť ŕ°?

௣௄

ŕ°’

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? 1 ŕ°’ŕŻ? 1

ଵ

ଵ

ŕŻ? ŕ°’ŕŹžŕ°? ௣ூ೛ ௄ ଵ

ŕŻ?áˆşŕ°’ŕŹžŕ°?áˆť ŕ°? ௣ூŕł? ௄

ఒ� ఒ௧భ

ŕ°’

ŕ°’ŕŻ? ŕ°?áˆşŕŻ?ŕŹżŕŻ§ŕ°­áˆť 1

�ఒ

ŕ°’ŕŻ? ŕ°’ŕŻ? 1 . ଵ ŕ°’

(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 ௗ௓భ áˆşŕŻ?áˆť ௗŕŻ?

0 and

ௗఎ ௓భ áˆşŕŻ?áˆť ௗŕŻ? ŕ°Ž

0.

Total holding cost for the cycle 0, is

respectively

଴

Now భ 0 gives the following nonௗ� linear equation in :

�

ଵ ŕ°’

௄௛

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? ŕ°’ŕŻ? ଵ

ŕ°’ŕŹžŕ°? ŕ°?

ఒ� 1

where ௣ .

Case-1:

(7)

Since the interest is payable during the time ଵ , the interest payable in any cycle 0, is

ଵ ŕ°’

ŕ°’ŕŻ? ŕ°?áˆşŕŻ?ŕŹżŕŻ§ŕ°­áˆť 1

�

௣ூ೛ ௄ ଵ

ŕ°­

ŕ°’ŕŹžŕ°? ŕ°?

ఒ� ఒ௧భ

.

(8)

Interest earned in the cycle period 0, is ଵ ŕŻ˜ ଴ ŕŻ?

ଵ ŕ°’

ఒ�

1 .

áˆşŕŻ›ŕŹžŕ°?ŕŻŁáˆť

ఒ�

ŕ°?

�� 1

௣ூ� ௄ ఒ

ఒ�

(9)

௣ூ೛ �

ŕ°?áˆşŕŻ?ŕŹżŕŻ§ŕ°­áˆť 1 ŕŻ˜

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? ŕ°’ŕŻ? ŕ°’ŕŻ? 1 áˆşŕ°’ŕŹžŕ°?áˆť ŕ°?

ŕ°’

ଵ ଵ áˆşŕ°’ŕŹžŕ°?áˆťŕŻ?

1 ఒ� 1

ŕ°’ŕŹžŕ°?

ŕ°’ ௣ூ೛௄ ଵ ŕ°’ŕŻ? ŕ°?áˆşŕŻ?ି௧భ áˆť 1 ଵ ŕ°’ŕŻ? ఒ௧భ áˆşŕ°’ŕŹžŕ°?áˆť ŕ°?

ŕ°’

௣ூŕł? ௄ ଵ ŕ°’ŕŻ? ŕ°’ŕŻ?

1

ఒ ఒ ௄௛

Let ଵ .

ଵ ௣ ௧

ௗ௓ áˆşŕŻ?áˆť

ଵ

ଵ

.

(11)

To get the optimal cycle length ଵ , we have to solve Eq. (11) provided it satisfies the following condition ௗఎ ௓ఎ áˆşŕŻ?áˆť ௗŕŻ? ŕ°Ž

0.

The EOQ in this case is as follows: ௄

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 ŕ°’ŕŹžŕ°? ! áˆşŕ°’ŕŹžŕ°?áˆťŕŻ?ŕ°­ 1" .

(12) The minimum annual variable cost ଵ ଵ ‍ כ‏ 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

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

362

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012) ௗ௓ áˆşŕŻ?áˆť

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

Now ఎ 0 gives the following nonௗ� linear equation in :

Interest earned for the time period 0, is

ఒ�

ŕŻ˜ ଴ ŕŻ?

௣ூ� ௄ ఒ

ŕ°’ŕŻ? ŕ°’ŕŻ? 1 . ଵ

ŕ°’

(13) Interest earned for the permissible delay period , ଵ is ŕŻ˜ ଵ ଴ ŕŻ?

௣ூŕł? áˆşŕŻ§ŕ°­ ିŕŻ?áˆťŕŻ„ ŕ°’

ఒ� 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.,

ŕ°?ŕŻ? 1 ŕŻ˜ ଵ ŕŹżŕ°’ŕŻ? 1

áˆşŕŻ›ŕŹžŕ°?ŕŻŁáˆť ŕ°? ௄௛

ଵ ŕ°’

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? ŕ°’ŕŻ? ŕ°’ŕŻ? 1

áˆşŕ°’ŕŹžŕ°?áˆť ŕ°? ŕ°’

ଵ áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? 1 ଵ ŕ°’ŕŻ? 1

. ŕ°’ŕŹžŕ°? ŕ°’

௣ூ ௄ ŕł?ŕ°Ž ଵ 1 ŕ°’ŕŻ? 1

ଵ

ଵ

ŕ°’

(17)

To get the optimal cycle length ଶ , we have to solve Eq. (17) provided it satisfies the following condition ௗఎ ௓ఎ áˆşŕŻ?áˆť ௗŕŻ? ŕ°Ž

0.

The EOQ in this case is as follows: ௄

ଶ

௣ூ� ௄

ఒ ௣ூ� ௄ ఒఎ

ଵ

௣ூŕł? áˆşŕŻ§ŕ°­ ିŕŻ?áˆťŕŻ„

ŕ°’

ŕ°’

ఒ� ఒ� 1

ଵ 1 ŕ°’ŕŻ? 1 .

ఒ� 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 ஺ ŕŻ?

௄௛

ŕ°’

ଵ

ఒ� 1

௣௄

ŕ°’

ଵ

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? 1

ŕŻ? ŕ°’ŕŹžŕ°? ௣ூŕł? ௄ ଵ ŕŻ?ŕ°’ŕ°Ž

1 ఒ� 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

ௗ௓ఎ áˆşŕŻ?áˆť ௗŕŻ?

(18)

The minimum annual variable cost ଶ ଶ ‍ כ‏ is obtained from Eq. (7) for ଶ .

Let ଵ .

Case-3:

For ଵ , both the cost function

and ଵ ଶ are identical and the cost function is obtained by putting ଵ either in Eq. (10) or Eq. (16) and is given by

ଵ

ŕŻ?áˆşŕ°’ŕŹžŕ°?áˆť ŕ°?

ଵ

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ? ŕ°’ŕŻ? ŕ°’ŕŻ?

1

଴ ଶ ŕ°’ŕŹžŕ°? ! áˆşŕ°’ŕŹžŕ°?áˆťŕŻ?ŕ°Ž 1" .

0 and

ௗఎ ௓ఎ áˆşŕŻ?áˆť ௗŕŻ? ŕ°Ž

0.

ଷ ଵ ଵ ŕ°’

ଵ ŕ°’

஺ ௧భ

ఒ௧భ 1

ఒ௧భ 1

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ§ŕ°­ ఒ௧భ

௄௛

ଵ

௧భ áˆşŕ°’ŕŹžŕ°?áˆť ŕ°?

௣௄

ଵ

áˆşŕ°’ŕŹžŕ°?áˆťŕŻ§ŕ°­ 1

௧భ ŕ°’ŕŹžŕ°? ௣ூ ௄ ŕł? ଵ ఒ௧భ ௧ ŕ°’ ŕ°­

ఒ௧భ 1 ଵ ŕ°’

.

(19)

The EOQ in this case is as follows: ଴ ଵ

௄ ! áˆşŕ°’ŕŹžŕ°?áˆťŕŻ§ŕ°­ ŕ°’ŕŹžŕ°?

1" .

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

(20)

363

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

SOLUTION FOR ECONOMIC ORDER POLICY: ALGORITHM The following steps are to be followed to find the optimum cost and economic order quantity unless ଵ .

Step 1: Determine ŕŹľâ€Ť כ‏from (11). If ŕŹľâ€Ť ×›â€ŹŕŹľ , evaluate ଵ ŕŹľâ€Ť כ‏from Eq. (10).

Step 2: Determine ŕŹśâ€Ť כ‏from (17). If ŕŹľâ€Ť ×›â€ŹŕŹľ , evaluate ଵ ŕŹľâ€Ť כ‏from Eq. (16).

Step 3: If the condition ŕŹľâ€Ť ×›â€ŹŕŹľ ŕŹśâ€Ť כ‏is satisfied, go to Step-4. Otherwise go to Step-5.

Step 4: Compare ଵ ŕŹľâ€Ť כ‏and find the minimum cost.

‍כ‏ ଶ ଶ

and

Step 5: If the condition ŕŹľâ€Ť ×›â€ŹŕŹľ is satisfied, but ŕŹśâ€Ť ×›â€ŹŕŹľ , then ଵ ŕŹľâ€Ť כ‏is the minimum cost, else if ŕŹľâ€Ť ×›â€ŹŕŹľ is satisfied, but ŕŹśâ€Ť כ‏ ଵ , then ଶ ŕŹśâ€Ť כ‏is the minimum cost.

Step 6: Compute ŕŹ´â€Ť ×›â€ŹŕŹľ or ŕŹ´â€Ť ×›â€ŹŕŹś 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 ŕŹľâ€Ť כ‏.

Let us consider the parameter values of the inventory system as 1000 units per year, 1 unit per year, #$. 200 per order, ௣ 0.15 per year, ŕŻ˜ 0.13 per year, 0.12 per year, #$. 20 per unit, 0.20 and ଵ 0.25 year.

Solving Eq. (11), we have ŕŹľâ€Ť כ‏0.314 year and minimum average cost is ‍כ‏ ଵ ଵ 975.726 .

Again, solving Eq. (17), we have ŕŹśâ€Ť כ‏0.224 year and minimum average cost is ଶ ŕŹśâ€Ť כ‏1023.27 .

Here the condition ŕŹľâ€Ť ×›â€ŹŕŹľ ŕŹśâ€Ť כ‏is satisfied. ‍כ‏ ‍כ‏ Comparing and the ଵ ଵ ଶ ଶ , minimum average cost in this case is ‍כ‏ ଵ ଵ 975.726 where the optimum cycle length is ŕŹľâ€Ť כ‏0.314 year.

The economic order quantity is given by ŕŹ´â€Ť ×›â€ŹŕŹľ ‍ כ‏291.549 .

Example 2: (case I) Minimum average cost is ଵ ŕŹľâ€Ť כ‏.

Let us consider the parameter values of the inventory system as 1000 units per year, 1 unit per year, #$. 200 per order, ௣ 0.15 per year, ŕŻ˜ 0.13 per year, 0.12 per year, #$. 20 per unit, 0.20 and ଵ 0.15 year.

Solving Eq. (11), we have ŕŹľâ€Ť כ‏0.276 year and minimum average cost is ‍כ‏ ଵ ଵ 1100.38 .

Again, solving Eq. (17), we have ŕŹśâ€Ť כ‏ 0.221 year and minimum average cost is ‍כ‏ ଶ ଶ 1314.46 .

Here the condition ŕŹľâ€Ť ×›â€ŹŕŹľ holds, but not ŕŹśâ€Ť ×›â€ŹŕŹľ , so only Case-I holds.

Hence the minimum average cost in this ‍כ‏ case is ଵ ଵ 1100.38 where the optimum cycle length is ŕŹľâ€Ť כ‏0.274 year.

The economic order quantity is given by ŕŹ´â€Ť ×›â€ŹŕŹľ ‍ כ‏164.348 .

Example 3: (case I and Case II) Minimum average cost is ଶ ŕŹśâ€Ť כ‏.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

Let us consider the parameter values of the inventory system as 1000 units per year, 1 unit per year, #$. 200 per order, ௣ 0.15 per year, ŕŻ˜ 0.13 per year, 0.12 per year, #$. 20 per unit, 0.20 and ଵ 0.35 year.

Solving Eq. (11), we have ŕŹľâ€Ť כ‏0.363 year and minimum average cost is ‍כ‏ ଵ ଵ 928.909 .

Again, solving Eq. (17), we have ŕŹśâ€Ť כ‏0.228 year and minimum average cost is ଶ ŕŹśâ€Ť כ‏728.511 .

Here the condition ŕŹľâ€Ť ×›â€ŹŕŹľ ŕŹśâ€Ť כ‏is satisfied. ‍כ‏ ‍כ‏ Comparing and the ଵ ଵ ଶ ଶ , minimum average cost in this case is ‍כ‏ ଶ ଶ 728.511 where the optimum cycle length is ŕŹśâ€Ť כ‏0.228 year.

The economic order quantity is given by ŕŹ´â€Ť ×›â€ŹŕŹś ‍ כ‏262.466 .

Example 4: (case II) Minimum average cost is ଶ ŕŹśâ€Ť כ‏.

Let us consider the parameter values of the inventory system as 1000 units per year, 1 unit per year, #$. 200 per order, ௣ 0.15 per year, ŕŻ˜ 0.13 per year, 0.12 per year, #$. 20 per unit, 0.20 and ଵ 0.45 year.

Solving Eq. (11), we have ŕŹľâ€Ť כ‏0.419 year and minimum average cost is ‍כ‏ ଵ ଵ 939.516 .

Again, solving Eq. (17), we have ŕŹśâ€Ť כ‏ 0.232 year and minimum average cost is ‍כ‏ ଶ ଶ 439.153 .

Here the condition ŕŹśâ€Ť ×›â€ŹŕŹľ , but not ŕŹľâ€Ť ×›â€ŹŕŹľ is satisfied. Comparing ଵ ŕŹľâ€Ť כ‏and ଶ ŕŹśâ€Ť כ‏, the minimum average cost in this case is

364

‍כ‏ ଶ ଶ

439.153 where the optimum cycle length is ŕŹśâ€Ť כ‏0.232 year.

The economic order quantity is given by ŕŹ´â€Ť ×›â€ŹŕŹś ‍ כ‏267.811 .

Example 5: (case III) Let us consider the parameter values of the inventory system as 1000 units per year, 1 unit per year, #$. 200 per order, ௣ 0.15 per year, ŕŻ˜ 0.13 per year, 0.12 per year, #$. 20 per unit, 0.20 and ଵ year.

Solving Eq. (19), we have ŕŹľâ€Ť ×›â€ŹŕŹśâ€Ť ×›â€ŹŕŹľâ€Ť כ‏ 0.229 year which satisfies Case-III. Here the minimum average cost in this case is ଶ ŕŹśâ€Ť כ‏649.563 where the optimum cycle length is ŕŹśâ€Ť כ‏0.229 year.

The economic order quantity is given by ŕŹ´â€Ť ×›â€ŹŕŹś ‍ כ‏263.935 .

SENSITIVITY ANALYSIS We now study the study the effects of changes in the values of the system parameters , , ௣ , ŕŻ˜ , , , 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.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

365 (i)

(ii)

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏decrease while ଵ ŕŹľâ€Ť& כ‏ ‍כ‏ ଶ ଶ increase with the increase in the value of the parameter . Both ‍כ‏ ‍כ‏ ଵ ଵ & ଶ ଶ have low sensitivity to changes in and ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏are moderately sensitive to changes in . ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏decrease while ଵ ŕŹľâ€Ť& כ‏ ‍כ‏ ଶ ଶ increase with the increase in the value of the parameter . Both ‍כ‏ ‍כ‏ ଵ ଵ & ଶ ଶ have low sensitivity to changes in and ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏are moderately sensitive to changes in .

(iii) ŕŹľâ€Ť כ‏decreases while ଵ ŕŹľâ€Ť כ‏increases with the increase in the value of the parameter ௣ . ଵ ŕŹľâ€Ť כ‏is low sensitive while ŕŹľâ€Ť כ‏is moderately sensitive to changes in ௣ . Here ŕŹśâ€Ť & ×›â€ŹŕŹś ŕŹśâ€Ť כ‏ have no effect with changing value of ௣ . (iv)

ŕŹľâ€Ť כ‏increases while ŕŹśâ€Ť כ‏, ଵ ŕŹľâ€Ť& כ‏ ‍כ‏ ଶ ଶ decrease with the increase in the value of the parameter ŕŻ˜ . Here ŕŹľâ€Ť×›â€Ź , ଵ ŕŹľâ€Ť & ×›â€ŹŕŹś ŕŹśâ€Ť כ‏are moderately sensitive while ŕŹśâ€Ť כ‏is very low sensitive to changes in ŕŻ˜ .

(v)

ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏increase while ଵ ŕŹľâ€Ť& כ‏ ‍כ‏ ଶ ଶ decrease with the increase in the value of the parameter . Here ŕŹśâ€Ť×›â€Ź has low sensitivity to the changes in while ŕŹľâ€Ť כ‏, ଵ ŕŹľâ€Ť & ×›â€ŹŕŹś ŕŹśâ€Ť כ‏are all moderately sensitive to changes in .

(vi)

ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏decrease while ଵ ŕŹľâ€Ť& כ‏ ‍כ‏ ଶ ଶ increase with the increase in the value of the parameter .

Here ŕŹľâ€Ť כ‏, ŕŹśâ€Ť כ‏, ଵ ŕŹľâ€Ť & ×›â€ŹŕŹś ŕŹśâ€Ť כ‏are all moderately sensitive to changes in .

(vii) ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏increase while ଵ ŕŹľâ€Ť& כ‏ ‍כ‏ ଶ ଶ decrease with the increase in the value of the parameter ଵ . Both ŕŹľâ€Ť×›â€Ź ‍כ‏ & ଵ ଵ are moderately sensitive and ଶ ŕŹśâ€Ť כ‏is highly sensitive to changes in ଵ while ŕŹśâ€Ť כ‏is insensitive to changes in ଵ. (viii) ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏decrease while ଵ ŕŹľâ€Ť& כ‏ ‍כ‏ ଶ ଶ increase with the increase in the value of the parameter . Here ŕŹśâ€Ť×›â€Ź has low sensitivity to the changes in , ‍כ‏ ŕŹľâ€Ť& כ‏ have moderately ଶ ଶ sensitivity to changes in and ଵ ŕŹľâ€Ť כ‏ has highly sensitivity to changes in . (ix)

ŕŹľâ€Ť כ‏, ŕŹśâ€Ť כ‏, ଵ ŕŹľâ€Ť & ×›â€ŹŕŹś ŕŹśâ€Ť כ‏increase with the increase in the value of the parameter . Here ŕŹľâ€Ť & ×›â€ŹŕŹśâ€Ť כ‏have low sensitivity to changes in and ଵ ŕŹľâ€Ť כ‏ & ଶ ŕŹśâ€Ť כ‏have 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 ŕŻ˜ and . These justify the real market situation.

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Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

366

Table-1 Parameter

௣

௘

ଵ

% change 50 10 -10 -50 50 10 -10 -50 50 10 -10 -50 50 10 -10 -50 50 10 -10 -50 50 10 -10 -50 50 10 -10 -50 50 10 -10 -50 50 10 -10 -50

ଵ‫כ‬

ଵ ଵ‫ כ‬

ଶ‫כ‬

ଶ ଶ‫ כ‬

Remark

Solution

0.283 0.306 0.323 0.384 0.301 0.311 0.317 0.331 0.301 0.311 0.317 0.335 0.365 0.323 0.306 0.280 0.312 0.314 0.315 0.300 0.285 0.307 0.323 0.380 0.376 0.325 0.303 0.269 0.263 0.301 0.329 0.419 0.352 0.322 0.306 0.266

1129.28 1008.84 940.907 781.608 1012.99 983.24 968.174 937.482 986.521 978.257 972.959 958.613 699.875 924.425 1025.45 1210.95 987.587 978.106 973.346 1068.22 1118.74 1006.53 943.372 788.841 926.818 957.645 998.65 1146.64 1340.68 1054.96 892.438 500.241 1275.68 1038.57 911.183 631.709

0.187 0.215 0.235 0.303 0.218 0.223 0.226 0.232 0.224 0.224 0.224 0.224 0.211 0.222 0.227 0.240 0.223 0.224 0.225 0.217 0.188 0.215 0.235 0.300 0.229 0.225 0.223 0.220 0.196 0.218 0.232 0.272 0.268 0.234 0.214 0.164

1049.43 1034.56 1008.03 889.862 1034.11 1025.53 1020.96 1011.17 1023.27 1023.27 1023.27 1023.27 812.524 981.718 1064.51 1025.98 1031.2 1024.86 1021.68 1085.54 1042.97 1033.05 1009.71 901.174 658.479 950.386 1096.12 1387.17 1272.66 1076.2 968.503 725.844 1429.32 1110.56 932.04 510.133

ଵ‫ כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ‫כâ€à¬µ ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ‫כâ€à¬µ ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ‫כâ€à¬µ ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כâ€à¬µâ€« כ‬ଵ ଶ‫כ‬

ଵ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬¶ ଶ‫ ×›â€à¬¶ ଶ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬¶ ଶ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬µ ଵ‫ ×›â€à¬¶ ଶ‫ כ‬

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

% change +15.73 +3.39 -3.56 -19.89 +3.81 +0.76 -0.77 -3.91 +1.11 +0.25 -0.28 -1.75 -28.27 -5.25 +5.10 +24.11 +1.21 +0.24 -0.24 +9.48 +14.66 +3.15 -3.31 -19.15 -32.51 -2.59 +2.34 +17.51 +30.43 +8.12 -8.54 -48.73 +30.74 +6.44 -6.62 -47.71

367

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

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 timedependent deterioration.

6.

7.

8.

9.

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 replenishment 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

10.

11.

12.

13.

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

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

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).

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).

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).

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

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).

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).

M. Hariga, An EOQ model for deteriorating items with shortages and time-varying demand, Journal of

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

14.

15.

16.

17.

18.

19.

20.

21.

Operational Research Society 46, 398– 404 (1995).

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).

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).

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).

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

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).

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).

S. K. Goyal, EOQ under conditions of permissible delay in Payments, Journal of Operation Research Society 36, 335– 338 (1985).

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

22.

23.

24.

25.

26.

27.

28.

29.

368

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

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).

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).

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

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).

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).

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).

K.J. Chung, J.J. Lio, Lot-sizing decisions under trade credit depending on the ordering quantity, Computer & Operations Research 31, 909–928 (2004).

S.S. Sana, K.S. Chaudhuri, A deterministic EOQ model with delays in

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

369

Trailokyanath Singh, et al., J. Comp. & Math. Sci. Vol.3 (3), 358-369 (2012)

payments and price discount offers, European Journal of Operational Research 184, 509-533 (2008).

30. S. Khanra, S. K. Ghosh and K. S. Chaudhuri,

An

EOQ

model

for

deteriorating item with time dependent quadratic demand under permissible delay in payment. Applied Mathematics and Computation, 1-9 (2011).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)