An EOQ Model for Deteriorating Items with Power-Form ... - CiteSeerX

Information and Management Sciences Volume 16, Number 1, pp. 1-16, 2005

An EOQ Model for Deteriorating Items with Power-Form Stock-Dependent Demand Jinn-Tsair Teng

Liang-Yuh Ouyang

The William Paterson University

Tamkang University

of New Jersey

R.O.C.

U.S.A. Mei-Chuan Cheng Tamkang University R.O.C.

Abstract Datta and Pal [4] established an EOQ model in which the demand rate is a power function of the on-hand inventory until down to a certain stock level, at which the demand rate becomes a constant. In this paper, we extend their EOQ model to allow for not only deteriorating items but also non-zero ending inventory. Due to the complexity of the demand function, the inventory problem here has three possible cases. We then establish the necessary and sufficient conditions for each case. Moreover, we propose an algorithm to determine the optimal replenishment cycle time and ordering quantity such that the total profit per unit time is maximized. Finally, we provide some numerical examples to illustrate the proposed algorithm, and obtain the effects of the parameters on the replenishment time and ordering quantity.

Keywords: Inventory, Deteriorating Items, Stock-Dependent Demand.

1. Introduction In daily life, the deteriorating of goods is a common phenomenon. Pharmaceuticals, foods, vegetables and fruit are a few examples of such items. Therefore, the loss due to deterioration cannot be neglected. Deteriorating inventory models have been widely studied in recent years. Ghare and Schrader [6] were the two earliest researchers to Received April 2003; Revised June 2003; Accepted March 2004. This research was supported by the ART for Research and a Summer Research Funding from the William Paterson University of New Jersey. In addition, the second author’s research was partially supported by the National Science Council of the Republic of China under Grant NSC-92-2213-E-007065, and a two-month research grant in 2003 by the Graduate Institute of Management Sciences in Tamkang University in Taiwan.

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Information and Management Sciences, Vol. 16, No. 1, March, 2005

consider continuously decaying inventory for a constant demand. Later, Shah and Jaiswal [18] presented an order-level inventory model for deteriorating items with a constant rate of deterioration. Aggarwal [1] developed an order-level inventory model by correcting and modifying the error in Shah and Jaiswal’s analysis [18] in calculating the average inventory holding cost. Covert and Philip [3] used a variable deterioration rate of twoparameter Weibull distribution to formulate the model with a constant demand rate and without shortages. Then, Philip [15] extended the model by considering a variable deterioration rate of three-parameter Weibull distribution. Goyal and Giri [9] presented a detailed review of the advances of deteriorating inventory literature since the early 1990s. It is a common belief that a large pile of goods attracts more customers in the supermarket. This phenomenon is termed as stock-dependent demand rate. In the last several years, there has been a considerable body of literature in the operational research area on how inventory-level dependent demand should affect inventory control policies. Levin et al. [10] observed that “large piles of consumer goods displayed in a supermarket will lead the customer to buy more.” Silver and Peterson [19] also noted that sales at the retail level tend to be proportional to the amount of inventory displayed. In order to quantify this, Baker and Urban [2] established an economic order quantity model for a power-form inventory-level-dependent demand pattern (i.e., the demand rate at time t is D(t) = α[I(t)]β , where I(t) is the inventory level, α > 0, and 0 < β < 1). Mandal and Phaujdar [11] then developed an economic production quantity model for deteriorating items with constant production rate and linearly stock-dependent demand (i.e., D(t) = α + βI(t), where both α and β > 0). Later on, Datta and Pal [4] presented an EOQ model in which the demand rate is dependent on the instantaneous stocks displayed until a given level of inventory S0 is reached, after which the demand rate becomes constant (i.e., D(t) = α[I(t)]β , if I(t) > S0 and D(t) = αS0β , if 0 ≤ I(t) ≤ S0 ). In addition, they assumed the replenishment cycle ends with zero stock. Urban [21] then relaxed the assumption of zero ending inventory because it may be desirable to order larger quantities, resulting in stock remaining at the end of the cycle, due to the potential profits resulting from the increased demand. Other papers related to this area are: Giri et al. [8], Padmanabhan and Vrat [13], Pal et al. [14], Ray and Chaudhuri [16], Urban and Baker [22], Ray et al. [17], Giri and Chaudhuri [7], Datta and Paul [5], Ouyang et al. [12], Teng and Chang [20] and others. We summarize the major assumptions used in the previous articles in Table 1.

An EOQ Model for Deteriorating Items with Power-Form Stock-Dependent Demand

3

Table 1. Summary of Stock-Dependent Demand Related Literature Author(s) and Year

EOQ

Pricing

or EPQ

Demand Patterns

Non-zero Deteriorating Ending

Items

Inventory Baker and Urban (1988)

EOQ

No

power form

Yes

No

Datta and Pal (1990)

EOQ

No

power form

No

No

Datta and Paul (2001)

EOQ

Yes

power form

Yes

No

Giri et al. (1996)

EOQ

No

power form

Yes

Yes

Giri and Chaudhuri (1998)

EOQ

No

power form

No

Yes

Mandal and Phaujdar (1989)

EPQ

No

linear form

No

Yes

Ouyang et al. (2003)

EOQ

No

linear form

No

Yes

Padmanabhan and Vrat (1995)

EOQ

No

linear form

No

Yes

Pal et al. (1993)

EOQ

No

power form

Yes

Yes

Ray and Chaudhuri (1997)

EOQ

No

power form

No

No

Ray et al. (1998)

EOQ

No

power form

No

No

Teng and Chang (2003)

EPQ

Yes

linear form

No

No

Urban(1992)

EOQ

No

power form

Yes

No

Urban and Baker (1997)

EOQ

Yes

power form

Yes

No

Present Paper

EOQ

No

power form

Yes

Yes

In this paper, we extend the EOQ inventory model by Datta and Pal [4] to allow for deteriorating items and non-zero ending inventory. Our object is to find the maximum profit by simultaneously optimizing the length and the order quantity for each ordering cycle. Due to the complexity of the demand function (i.e., D(t) = α[I(t)] β , if I(t) > S0 and D(t) = αS0β , if 0 ≤ I(t) ≤ S0 ), the proposed inventory problem here has three possible cases. First of all, if the initial stock-level is less than or equal to S 0 (i.e., S ≤ S0 ), then the demand is constant, and it is the classical EOQ model for deteriorating items with constant demand rate. Secondly, if the initial stock-level is higher than S 0 and S0 is higher than the ending inventory level i T (i.e., S > S0 > iT ), then the demand rate initially is a power form of on-hand inventory, and then becomes a constant after the inventory level reaches S0 . Thirdly, if the ending inventory i T is higher than or equal to S0 (i.e. S > iT ≥ S0 ), then the demand rate is a power function of on-hand inventory throughout the entire replenishment cycle. We then establish the necessary

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and sufficient conditions to obtain the optimum solution for each case. Moreover, we propose an algorithm to obtain the optimal replenishment timing and ordering quantity. Finally, we provide some numerical examples to illustrate those three cases and the solution algorithm.

2. Assumptions and Notation The mathematical model in this paper is developed on the basis of the following assumptions and notation. Assumptions: 1. The inventory system involves only one item and the planning horizon is infinite. 2. The replenishment occurs instantaneously at an infinite rate. 3. The deteriorating rate θ(0 ≤ 0 ≤ θ < 1) is constant and there is no replacement or repair of deteriorated units during the period under consideration. 4. The demand rate D(i) is deterministic and its functional form is given by D(i) =

  α(i(t))β ,  D,

i(t) > S0 , 0 ≤ i(t) ≤ S0 ,

where α > 0, 0 < β < 1, D = αS0β , and both α and β are known as scale and shape parameters, respectively. Note that if S 0 is sufficiently large, then the demand is constant, and the problem is reduced to the classical EOQ model. 5. Shortages are not allowed. Notation : c1 : holding cost of the inventory item, $/per unit/per unit time c2 : purchase cost, $/per unit c3 : ordering cost, $/per order p : selling price, $/per unit S : the maximum inventory level for the ordering cycle t1 : time when the inventory level falls to S 0 T : length of the ordering cycle i(t) : inventory level at time t ∈ [0, T ] iT : inventory level at the end of the cycle , i.e., i T = i(T )


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3. Model Formulation and Solution Depending on the initial stock-level S, the constant S 0 , and the ending inventory level iT , the inventory problem here has three possible cases: (i) S ≤ S 0 , (ii) S > S0 > iT , and (iii) S > iT ≥ S0 . Case 1. S ≤ S0 Obviously, it is the classical EOQ model for deteriorating items with constant demand rate. The differential equation describing the instantaneous state of i(t) is given by: di(t) + θi(t) = −D, dt

0 ≤ t ≤ T.

(1)

with the boundary condition i(T ) = 0, the solution of Equation (1) is given by i(t) =

D θ(T −t) (e − 1), θ

0 ≤ t ≤ T.

(2)

By using the condition i(0) = S and Equation (2), we get the maximum inventory level S=

D θT (e − 1). θ

(3)

The inventory holding cost per cycle is HC =

Z

T 0

c1 i(t)dt =

c1 D θT c1 DT (e − 1) − . 2 θ θ

(4)

Therefore, the total profit per unit time is Z1 (T ) = (revenues − holding cost − purchasing cost − ordering cost) / length of ordering cycle ( "Z

T

= p

0

#

Ddt − HC − c2 S − c3

)

.

T

. c1 D θT c2 D θT = pDT − 2 (e − 1 − θT ) − (e − 1) − c3 T. θ θ

(5)

Remark 1. When θ = 0 (i.e., for non-deteriorating item), Equation (5) is reduced to Z1 (T ) = pD − c1 DT /2 − c2 D − c3 /T , in which it becomes the classical EOQ model. The objective of the model here is to determine the optimal value of T in order to maximize the total profit Z1 (T ) subject to the inequality constraint S ≤ S 0 . Thus, the mathematical model of this problem can be expressed as follows: Maximizing Z1 (T ) subject to

S0 − S ≥ 0.

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If T ∗ is the optimal solution of the problem, then there exist a value λ ∗ such that T ∗ and λ∗ satisfy the following Kuhn-Tucker conditions: S0 − S ≥ 0,

(6)

λ(S0 − S) = 0, ∂(S0 − S) ∂Z1 (T ) +λ = 0. ∂T ∂T

(7) (8)

From Equations (3) and (5), Equation (8) can be reduced to eθT − 1 c3 D c1 ( + c2 )( − eθT ) + 2 − λDeθT = 0. T θ Tθ T

(9)

eθT − 1 c3 D c1 ( + c2 )( − eθT ) + 2 = 0. T θ Tθ T

(10)

If λ = 0, then

By assuming S0 − S > 0, we obtain λ∗ = 0 from Equation (7), and the value of T ∗ from Equation (10). Using (3) and T ∗ , we get the value of S (we denote it by S ∗ ). Furthermore, we need to check whether S ∗ satisfies the condition S ∗ < S0 . If yes, then T ∗ and S ∗ are the optimal solutions to the problem, and the maximum total profit per unit time can be obtained by using Equation (5). We denote this value by Z 1∗ (i.e., Z1∗ = Z1 (T ∗ )). Otherwise, we set S ∗ = S0 , substitute this value into Equation (3) to obtain T ∗ , and then determine the value Z1∗ = Z1 (T ∗ ) from Equation (5). Case 2. S > S0 > iT During the period [0, t1 ], the inventory decreases due to both the stock-dependent demand and deterioration. After t1 , the inventory level decreases owing to constant demand rate as well as deterioration. The differential equations describing the instantaneous states of i(t) in this case are given by di(t) + θi(t) = −α(i(t))β , dt di(t) + θi(t) = −D, dt

0 ≤ t ≤ t1 ,

(11)

t1 ≤ t ≤ t1 + k,

(12)

where k ≥ 0 and t1 ≤ T ≤ t1 +k. The solution to Equation (11) with the initial condition i(0) = S is as follows:

i(t) = e

(β−1)θt

S

1−β

α α + e(β−1)θt − θ θ

1 1−β

,

0 ≤ t ≤ t1 .

(13)


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Using the condition i(t1 ) = S0 and Equation (13), we get α S= + (e(1−β)θt1 − 1) θ Substituting Equation (14) into Equation (13), we obtain

S01−β e(1−β)θt1

1 1−β

.

(14)

1

α 1−β α i(t) = e + )− , 0 ≤ t ≤ t1 . (15) θ θ Next, from the boundary condition i(t 1 + k) = 0, we know that the solution to Equation

(1−β)θ(t1 −t)

(S01−β

(12) is as follows: D θ(t1 +k−t) [e − 1], t1 ≤ t ≤ t1 + k. (16) θ 0 From the condition i(t1 ) = S0 and Equation (16), we obtain k = θ1 ln( θS D +1). Therefore, i(t) =

Equation (16) can be rewritten as D D i(t) = (S0 + )eθ(t1 −t) − , θ θ and the ending inventory level is given by

t1 ≤ t ≤ t 1 +

iT = i(T ) = (S0 +

1 θS0 ln( + 1), θ D

(17)

D θ(t1 −T ) D )e − . θ θ

(18)


Z

T 0

c1 i(t)dt

= c1

"Z

= c1

"Z

t1

i(t)dt + 0

0

t1

Z

T

i(t)dt t1

#

α α e(1−β)θ(t1 −t) (S01−β + )− θ

θ

1 1−β

D D 1 dt− (S0 + )(eθ(t1 −T ) −1)− (T −t1 ) θ θ θ

#

(19) Therefore, the total profit per unit time is ( "Z

Z2 (t1 , T ) = p

( "Z

t1

β

α(i(t)) dt + 0 t1

= p

α e 0

"Z

t1

Z

T t1

(1−β)θ(t1 −t)

#

Ddt − HC − c2 S − c3

(S01−β

α α + )− θ θ

1 1−β

(1−β)θ(t1 −t)

−c2 S01−β e(1−β)θt1

(S01−β

α + (e(1−β)θt1 − 1) θ

.

T

dt + D(T − t1 )

#

1

α α 1−β −c1 e + )− dt θ θ 0 D D 1 − (S0 + )(eθ(t1 −T ) −1)− (T −t1 ) θ θ θ

)

1 1−β

− c3

)

.

T.

(20)

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Remark 2. When i(T ) = 0 (i.e., the ending inventory level is zero) and θ = 0, Case 2 is reduced to Datta and Pal’s [4] model. And it can be shown that Z 2 (t1 , T ) is the same as the Equation (11) in Datta and Pal [4]. The objective of the model is to determine the optimal values of t 1 and T in order to maximize the total profit Z2 (t1 , T ) subject to inequality constraints S > S 0 > iT and 0 T ≤ t1 + θ1 ln( θS D + 1). Hence, the mathematical model in this case is:

Maximizing Z2 (t1 , T ) subject to

S − S0 − A21 = 0, S0 − iT − A22 = 0, 1 θS0 t1 + ln( + 1) − T − A23 = 0, θ D

where A21 > 0, A22 > 0 and A23 ≥ 0 are the surplus variables. If (t∗1 , T ∗ , A∗1 , A∗2 , A∗3 ) is the optimal solution of the problem, then, there exist values λ∗1 , λ∗2 and λ∗3 such that (t∗1 , T ∗ , A∗1 , A∗2 , A∗3 ) and (λ∗1 , λ∗2 , λ∗3 ) satisfy the following KuhnTucker conditions: S − S0 − A21 = 0,

(21)

S0 − iT − A22 = 0, 1 θS0 t1 + ln( + 1) − T − A23 = 0, θ D ∂Z2 (t1 , T ) ∂(S − S0 − A21 ) ∂(S0 − iT − A22 ) + λ1 + λ2 ∂A1 ∂A1 ∂A1 ∂ 1 θS0 2 t1 + ln( +λ3 + 1) − T − A3 = 0, ∂A1 θ D ∂(S − S0 − A21 ) ∂(S0 − iT − A22 ) ∂Z2 (t1 , T ) + λ1 + λ2 ∂A2 ∂A2 ∂A2 ∂ 1 θS0 +λ3 t1 + ln( + 1) − T − A23 = 0, ∂A2 θ D ∂(S − S0 − A21 ) ∂(S0 − iT − A22 ) ∂Z2 (t1 , T ) + λ1 + λ2 ∂A3 ∂A3 ∂A3 ∂ 1 θS0 2 +λ3 t1 + ln( + 1) − T − A3 = 0, ∂A3 θ D ∂(S − S0 − A21 ) ∂(S0 − iT − A22 ) ∂Z2 (t1 , T ) + λ1 + λ2 ∂t1 ∂t1 ∂t 1 ∂ 1 θS0 +λ3 t1 + ln( + 1) − T − A23 = 0, ∂t1 θ D ∂(S − S0 − A21 ) ∂(S0 − iT − A22 ) ∂Z2 (t1 , T ) + λ1 + λ2 ∂T ∂T ∂T

(22) (23)

(24)

(25)

(26)

(27)


∂ 1 θS0 +λ3 t1 + ln( + 1) − T − A23 = 0, ∂T θ D

9

(28)

From Equations (14), (18) and (20), we can reduce Equations (24)-(28) to as follows: −2λ1 A1 = 0,

(29)

−2λ2 A2 = 0,

(30)

−2λ3 A3 = 0,

(31)

  2β−1  1−β pα  β α α α S + θβ(S01−β + )eθ(1−β)(t1 −t) (S01−β + )eθ(1−β)(t1 −t) −  T  0 θ θ β  β pD c  α α α 1−β

−

T

−

1

S0 +θ(S01−β + )eθ(1−β)(t1 −t) (S01−β + )eθ(1−β)(t1 −t) − T  θ θ β

1 D D c2 1−β (1−β)θt1 α (1−β)θt1 − (S0 + )(eθ(t1 −T ) −1)− (T −t1 ) − S0 e + (e −1) θ θ θ T θ

c3 α α − + λ1 e(1−β)θt1 (θS01−β + α) e(1−β)θt1 (S01−β + )− T θ θ θ(t1 −T ) −λ2 (θS0 + D)e + λ3 = 0,

1 T

pD − c1 (S0 + 

D (θ(t1 −T ) D )e − θ θ

β 1−β

(32)

β Z t1 α α 1−β 1  1−β (1−β)θ(t1 −t) α e (S0 + )− dt + pD(T − t1 ) − 2 p T  0 θ β

−c1

"Z

0

t1

e

(1−β)θ(t1 −t)

−c2 S01−β e(1−β)θt1

α α (S01−β + )− θ

1 1−β

θ

α + (e(1−β)θt1 − 1) θ

1 1−β

1 1−β

1 D D dt− (S0 + )(eθ(t1 −T ) −1)− (T −t1 ) θ θ θ

#

)

−c3 +λ2 (θS0 + D)eθ(t1 −T ) −λ3 = 0. (33)

Equations (29) and (30) imply λ∗ = 0 and λ∗2 = 0 because A21 > 0 and A22 > 0. ∗

∗

From Equations (21) - (23) and (29) - (33), we can get the values of (t ∗1 , T ∗ , A∗1 , A∗2 , A∗3 ) and (λ∗1 , λ∗2 , λ∗3 ). Consequently, we can obtain the values of S ∗ , i∗T from Equations (14) and (18), respectively. If S ∗ > S0 > i∗T , then, from Equation (20), we can obtain the maximum total profit per unit time, Z 2∗ = Z2 (t∗1 , T ∗ ). Otherwise, we set Z2∗ = 0. Because Equations (21) - (23), and Equations (29) - (33) are highly nonlinear, we can not find the closed solution of (t ∗1 , T ∗ , S ∗ ) by the analytic method. We solve these equations numerically by using Mathematica 4.2.

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Case 3. S > iT ≥ S0 It is to be noted that Case 3 arises only if the reorder point occurs before the constant demand takes place. The differential equation describing the instantaneous state of i(t) is given by:

di(t) + θi(t) = −α(i(t))β , 0 ≤ t ≤ T, dt with the boundary condition i(0) = S. The solution of Equation (34) is given by

i(t) = e(β−1)θt S 1−β +

α (β−1)θt α e − θ θ

(34)

1 1−β

,

0 ≤ t ≤ T.

(35)

Thus, the ending inventory level is given by

iT = i(T ) = e

(β−1)θT

S

1−β

α α + e(β−1)θT − θ θ

1 1−β

.

(36)


Z

T 0

c1 i(t)dt = c1

hZ

T 0

e(β−1)θT S 1−β +

α (β−1)θT α e − θ θ

1 1−β

dt

i

(37)

Therefore, the total profit per unit time is ( "Z

Z3 (T, S) = p

T

β

#

α(i(t)) dt − HC − c2 S − c3

0

)

.

T

   β  Z T 1−β α α dt = p α e(β−1)θt S 1−β + e(β−1)θt −  θ θ 0 "Z # 1 T

−c1

e

(β−1)θt

S

0

1−β


1−β

dt − c2 S − c3

)

.

T. (38)

The objective of the model in this case is to determine the optimal values of T and S in order to maximize the total profit Z 3 (T, S) subject to the inequality constraint iT ≥ S0 . Thus, the mathematical model in this case can be expressed as follows: Maximizing Z3 (T, S) subject to

iT − S0 ≥ 0.

If (T ∗ , S ∗ ) is the optimal solution of the problem, then there exist a value µ ∗ such that (T ∗ , S ∗ ) and µ∗ satisfy the following Kuhn-Tucker conditions: iT − S0 ≥ 0,

(39)

µ(iT − S0 ) = 0,

(40)


µ ≥ 0, ∂(iT − S0 ) ∂Z3 (T, S) +µ = 0, ∂T ∂T ∂Z3 (T, S) ∂(iT − S0 ) +µ = 0. ∂S ∂S

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(41) (42) (43)

From Equations (36) and (38), Equations (42) and (43) can be reduced to 

αp e

(β−1)θT

S


1−β

β 1−β

−c1 e

(β−1)θT

S

1−β

   β  Z T 1−β α α dt − p α e(β−1)θt S 1−β + e(β−1)θt −  θ θ 0 "Z # 1 T

−c1

e

(β−1)θt

S

1−β

0

+µθ(β − 1)e

(β−1)θT

(S


1−β


1−β

dt − c2 S − c3

)

.

α α α + ) e(β−1)θT S 1−β + e(β−1)θT − θ θ θ



1 1−β



.

T

T2

β 1−β

= 0,

(44)

and   

αβp

Z

T

S

−β (β−1)θt

T

e

0

c1 − 1−β

Z

0

e

(β−1)θt

S

1−β


α α e(β−1)θt S 1−β + e(β−1)θt − θ θ

+µS −β e(β−1)θT

e(β−1)θT S 1−β +

2β−1 1−β

β 1−β

α (β−1)θT α e − θ θ

dt − c2

β 1−β

dt

 . 

= 0,

T

(45)

respectively. From Equations (40), (44) and (45), we can get the values of T ∗ , S ∗ and µ∗ . Next, by using T ∗ and S ∗ , we can obtain the value of i∗T from Equations (36). Further, we check whether i∗T satisfies the condition i∗T ≥ S0 . If yes, then T ∗ and S ∗ are the optimal solutions to the problem, and the maximum total profit per unit time can be obtained by using Equation (38). We denote this value by Z 3∗ (i.e., Z3∗ = Z3 (T ∗ )). Otherwise, we set Z3∗ = 0. 4. Search Algorithm Now, we summarize the above results and establish the following algorithm to find the optimal solution of our problem.

12


Algorithm Step 1: In Case 1, let λ∗ = 0 and determine T ∗ by solving Equations (10), and then obtain S ∗ using Equation (3) and T ∗ . If S ∗ < S0 , obtain Z1∗ from Equation (5). Otherwise, we set S ∗ = S0 , and determine T ∗ from Equation (3), and then evaluate Z1∗ from Equation (5). Step 2: In Case 2, determine (t∗1 , T ∗ , A∗1 , A∗2 , A∗3 ) and (λ∗1 , λ∗2 , λ∗3 ) by solving Equations (21) - (23) and (29) - (33), and then obtain S ∗ , i∗T from Equation (14) and (18), respectively. If S ∗ > S0 > i∗T , evaluate Z2∗ from Equation (20). Otherwise, we set Z2∗ = 0. Step 3: In Case 3, determine T ∗ , S ∗ and µ∗ by solving Equations (40), (44) and (45) and then obtain i∗T from Equation (36). If i∗T ≥ S0 , obtain Z3∗ from Equation (38). Otherwise, we set Z3∗ = 0. Step 4: Comparing Z1∗ , Z2∗ and Z3∗ , we obtain the maximum total profit per unit time Z ∗ = max{Z1∗ , Z2∗ , Z3∗ }, and then get the optimal ordering strategy from the corresponding case.

5. Numerical Examples and Sensitivity Analysis According to the results of Section 3, we will provide examples to explain how the solution procedure works. We take the parameters of the inventory system as follows: α = 25, β = 0.4, θ = 0.02, p = 20, c1 = 0.5, c2 = 10, c3 = 70 and S0 = 90 in appropriate units. Example 1. Under given parameter values, from Case 1, we obtain λ = 0, T = 1.4127 and S = 174.579. We find that S > S0 , which implies (T, S) is infeasible. Therefore, we take S ∗ = S0 = 90 and obtain T ∗ = 0.5916 and Z1∗ = $1362.54 from Equations (3) and (5), respectively. Next, from Case 2, we obtain the optimal A 21 = 8227.14, A22 = 80, ∗

∗

A23 = 0, λ∗1 = 0, λ∗2 = 0, λ∗3 = −29.7844, t∗1 = 12.8011, T ∗ = 13.3927. And hence, ∗

we get S ∗ = 8317.14, i∗T = 0 and Z2∗ = $3423.49. Further, from Case 3, we obtain the optimal µ∗ = 0, S ∗ = 8222.6 and T ∗ = 12.4678. Therefore, we get i∗T = 130.84 and Z3∗ = $3447.48 from Equations (36) and (38), respectively. From above results, we find that Z3∗ > Z2∗ > Z1∗ . Thus, Case 3 is the optimal situation; i.e., the optimal ordering strategy is as follows: the maximum inventory level S ∗ = 8222.6 units, the length of ordering cycle T ∗ = 12.4678 unit time, and the maximum total profit per unit time Z ∗ = $3447.48.


13

Example 2. In this example, we study the effects of the changes in the model parameter α on the optimal solutions. The results of this sensitivity analysis are presented in Table 2. Table 2. The sensitivity analysis on parameter α. α

T∗

S∗

i∗T

Z∗

20

12.4784

5675.72

90.033

2375.02

23

12.4713

7158.65

113.793

2999.46

25

12.4678

8222.60

130.840

3447.48

Example 3. In this example, we study the effects of the changes in the model parameter β on the optimal solutions. The computational results are presented in Table 3. Table 3. The sensitivity analysis on parameter β. β

T∗

S∗

i∗T

Z∗

8222.60

130.840

3447.48

0.45 14.1476 21573.80 397.700

7193.30

0.40 12.4678

0.47 14.8427 33177.90 639.710 10098.10 Example 4. In this example, we study the effects of the changes in the model parameters p and θ on the optimal solutions. The sensitivity analysis is performed by changing p or θ while keeping remaining parameters unchanged. The results are presented in Table 4. Table 4. The sensitivity analysis on different parameters p and θ. p

θ

T∗

S∗

i∗T

Z∗

20 0.010 15.0095 10565.70 172.199 3834.77 0.015 13.6199

9277.03

149.258 3628.29

0.020 12.4678

8222.60

130.840 3447.48

23 0.010 18.6220 16147.60 377.525 5983.52 0.015 16.8171 14187.10 326.340 5659.69 0.020 15.3343 12580.60 285.440 5376.45 25 0.010 20.8731 20340.80 570.414 7628.07 0.015 18.7927 17879.20 492.177 7213.73 0.020 17.0937 15860.10 429.852 6851.63

14


On the basis of the results shown in Tables 2 - 4, the following observations can be made. 1. As the values of α or β increase, S ∗ , i∗T and Z ∗ increase. It implies that when the demand increases, we need to order more, and so is the maximum total profit per unit time. In addition, we find that S ∗ , i∗T and Z ∗ are highly sensitive to changes in β. 2. For fixed θ, as the value of p increases, T ∗ , S ∗ , i∗T and Z ∗ increase. It indicates that if we increase the selling price, then the optimal length of ordering cycle, the optimal maximum inventory level, the optimal ending inventory level, and the maximum total profit per unit time will be increased. We also find that T ∗ , S ∗ , i∗T and Z ∗ are highly sensitive to the changes in p. 3. For fixed p, as the value of θ increases, T ∗ , S ∗ , i∗T and Z ∗ decrease. It tells us that when the deteriorating rate θ increases, the optimal length of ordering cycle, the optimal ordering quantity, the optimal ending inventory level and the maximum total profit per unit time will be decreased. 6. Conclusions and Future Research It is well known that the stock level has a motivational effect on the customers in a supermarket; i.e. the demand rate may go up or down if the on-hand inventory level increases or decreases. Such a situation generally arises for a consumer-goods type of inventory. Therefore, it may be desirable to order large quantities, resulting in stock remaining at the end of the cycle, due to the potential profits resulting from the increased demand. In this paper, we develop EOQ inventory model for deteriorating items with powerform stock-dependent demand rate followed by a constant demand rate, in which the terminal condition of zero inventory at the end of the cycle has been relaxed. Our research reveals that the optimal situation always exists in Case 3. Besides, after studying the effects of the changes in the model parameters, we find that all the optimal solutions are highly sensitive to changes in the shape parameter of demand rate, β, and the selling price, p. Finally, the proposed model can be extended in several ways. For instance, it could be of interest to relax the restriction of constant deterioration rate. Also, we could generalize the demand as a function of price and advertising or incorporate quantity discounts.


15

Acknowledgements The authors would like to thank the anonymous referees for their detailed and constructive comments.

References [1] Aggarwal, S. P., A note on an order-level model for a system with constant rate of deterioration, Opsearch, Vol. 15, pp.184-187, 1978. [2] Baker, R. C. and Urban, T. L., A deterministic inventory system with an inventory level dependent demand rate, Journal of the Operational Research Society, Vol. 39, pp.823-831, 1988. [3] Covert, R. B. and Philip, G . S., An EOQ model with Weibull distribution deterioration, AIIE Transactions, Vol. 5, pp.323-326, 1973. [4] Datta, T. K. and Pal, A. K., A note on an inventory model with inventory level dependent demand rate, Journal of the Operational Research Society, Vol. 41, pp.971-975, 1990. [5] Datta, T. K. and Paul, K., An inventory system with stock-dependent, price-sensitive demand rate, Production planning & Control, Vol. 12, No.1, pp.13-20, 2001. [6] Ghare, P. M. and Schrader, G. H., A model for exponentially decaying inventory system, International Journal of Production Research, Vol. 21, pp.449-460, 1963. [7] Giri, B. C. and Chaudhuri, K. S., Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost, European Journal of Operational Research, Vol. 105, pp.467-474, 1998. [8] Giri, B. C., Pal, S., Goswami, A. and Chaudhuri, K. S., An inventory model for deteriorating items with stock-dependent demand rate, European Journal of Operational Research, Vol. 95, pp.604-610, 1996. [9] Goyal, S. K. and Giri, B. C., Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, Vol. 134, pp.1-16, 2001. [10] Levin, R. I., McLaughlin, C. P., Lamone, R. P. and Kottas, J. F., Productions / Operations Management: Contemporary Policy for Managing Operating Systems, McGraw-Hill, New York, p.373, 1972. [11] Mandal, B. N. and Phaujdar, S., An inventory model for deteriorating items and stock-dependent consumption rate, Journal of the Operational Research Society, Vol. 40, pp.483-488, 1989. [12] Ouyang, L. Y., Hsieh, T. P., Dye, C. Y. and Chang, H. C., An inventory model for deteriorating items with stock-dependent demand under the conditions of inflation and time-value of money, The Engineering Economist, Vol. 48, No.1, pp.52-68, 2003. [13] Padmanabhan, G. and Vrat, P., EOQ models for perishable items under stock dependent selling rate, European Journal of Operational Research, Vol. 86, pp.281-292, 1995. [14] Pal, S., Goswami, A. and Chaudhuri, K. S., A deterministic inventory model for deteriorating items with stock-dependent rate, International Journal of Production Economics, Vol. 32, 291-299, 1993. [15] Philip, G. C., A generalized EOQ model for items with Weibull distribution, AIIE Transactions, Vol. l6, pp.159-162, 1974. [16] Ray, J. and Chaudhuri, K. S., An EOQ model with stock-dependent demand, shortage, inflation and time discounting, International Journal of Production Economics, Vol. 53, pp.171-180, 1997. [17] Ray, J., Goswami, A. and Chaudhuri, K. S., On an inventory model with two levels of storage and stock-dependent demand rate, International Journal of Systems Science, Vol. 29, pp.249-254, 1998. [18] Shah, Y. K. and Jaiswal, M. C., An order-level inventory model for a system with constant rate of deterioration, Opsearch, Vol. 14, pp.174-184, 1977.

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[19] Silver, E. A. and Peterson, R., Decision Systems for Inventory Management and Production Planning, 2nd edition, Wiley, New York, 1985. [20] Teng, J. T. and Chang, C. T., Economic production quantity models for deteriorating items with price- and stock- dependent demand, to appear in Computers & Operations Research. [21] Urban, T. L., An inventory model with an inventory-level-dependent demand rate and relaxed terminal conditions, Journal of the Operational Research Society, Vol. 43, pp.721-724, 1992. [22] Urban, T. L. and Baker, R. C., Optimal ordering and pricing policies in a single-period environment with an multivariate demand and markdowns, European Journal of Operational Research, Vol. 103, pp.573-583, 1997.

Authors’ Information Jinn-Tsair (James) Teng is a professor in the Department of Marketing and Management Sciences at William Peterson University of New Jersey in USA. He received a B.S. degree in Mathematical Statistics from Tamkang University in Taiwan, an M.S. degree in Applied Mathematics from National Tsing Hua University in Taiwan, and a Ph.D. in Industrial Administration from Carnegie Mellon University in USA. His research interests are supply chain management and marketing research. He has published research articles in Management Sciences, Marketing Science, Journal of the Operational Research Society, Operations Research Letters, Naval Research Logistics, European Journal of Operational Research, and others. Department of Marketing and Management Sciences College of Business, The William Paterson University of New Jersey, Wayne, New Jersey 07470, U.S.A. E-mail: [email protected]

TEL: +973-720-2651

FAX: +973-720-2809

Liang-Yuh Ouyang is a Professor in the Department of Management Sciences & Decision Making at Tamkang University in Taiwan. He earned his B.S. in Mathematical Statistics, M.S. in Mathematics and Ph.D. in Management Sciences from Tamkang University. His research interests are in the field of Production/Inventory Control, Probability and Statistics. He has publications in Journal of the Operational Research Society, Computers and Operations Research, European Journal of Operational Research, Computers and Industrial Engineering, International Journal of Production Economics, IEEE Transactions on Reliability, Sankh¯ a, Metrika, Production Planning & Control, Journal of the Operations Research Society of Japan, Opsearch, Journal of Statistics & Management Systems, Journal of Interdisciplinary Mathematics, International Journal of Information and Management Sciences, International Journal of Systems Science, Yugoslav Journal of Operations Research, The Engineering Economist, Mathematical and Computer Modelling and Applied Mathematical Modelling. Department of Management Sciences & Decision Making, Tamkang University, Tamsui, Taipei 251, Taiwan 251, R.O.C. E-mail: [email protected]

TEL: +886-2-2621-5656 ext.2075

Mei-Chuan Cheng is currently a Ph.D. student at Tamkang University in Taiwan. She received her M.B.A. degree in statistics from Tamkang University and a B.S. degree in statistics from Fu Jen Catholic University. Her research interest lies in the field on the analysis of inventory system, Probability and Statistics. Graduate Institute of Management Sciences, Tamkang University, Tamsui, Taipei 251, Taiwan 251, R.O.C. E-mail: [email protected]

TEL: +886-2-2621-5656 ext.2464