The production–inventory problem of a product with time varying demand, production and deterioration rates

European Journal of Operational Research 147 (2003) 549–557 www.elsevier.com/locate/dsw

Production, Manufacturing and Logistics

The production–inventory problem of a product with time varying demand, production and deterioration rates S.K. Goyal a

a,*

, B.C. Giri

b

Faculty of Commerce and Administration, Department of Decision Sciences and MIS, Concordia University, 1455, De Maisonneuve Blvd. West, Montreal, Quebec Canada, H3G 1M8 b Department of Industrial Engineering, Pusan National University, Pusan 609-735, South Korea Received 12 March 2001; accepted 26 February 2002

Abstract In this paper, we consider the production–inventory problem in which the demand, production and deterioration rates of a product are assumed to vary with time. Shortages of a cycle are allowed to be backlogged partially. Two models are developed for the problem by employing different modeling approaches over an infinite planning horizon. Solution procedures are derived for determining the optimal replenishment policies. A procedure to find the nearoptimal operating policy of the problem over a finite time horizon is also suggested.
2002 Elsevier Science B.V. All rights reserved. Keywords: Inventory; Deterioration; Time-varying demand; Partial backlogging

1. Introduction The production–inventory systems of deteriorating items (e.g. medicines, volatile liquids, food stuffs etc.) are most common in reality and a number of researchers have investigated the problem of determining economic replenishment policy of such items. Aggarwal and Bahari-Kashani [1] looked at the flexible production rate for deteriorating items in a declining market. Wee and Wang [10] studied a production–inventory system

*

Corresponding author. Fax: +1-514-848-2824. E-mail address: [email protected] (S.K. Goyal).

for deteriorating items with time varying demand, finite production rate and shortages over a known planning horizon. Balkhi and Benkherouf [2] developed a fixed production schedule for deteriorating items where demand and production are allowed to vary with time in an arbitrary way and the rate of deterioration is constant. They [3] also considered a similar model without allowing shortages over a finite planning horizon and suggested a method for finding the optimal replenishment schedule. Wee and Law [11] developed a deterministic inventory model for deteriorating items with price-dependent demand rate, finite production rate, and time varying deterioration rate taking into account the time value of money over a fixed time horizon. Balkhi [4] analyzed an inventory

0377-2217/03/$ - see front matter
2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-2217(02)00296-5

550

S.K. Goyal, B.C. Giri / European Journal of Operational Research 147 (2003) 549–557

model for integrated production system by considering the production, demand and deterioration rates of the finished product and the deterioration rates of the raw materials as known functions of time. He did not allow shortages for both raw materials and final product. Interested readers can find more information regarding production–inventory models of deteriorating items in the review articles by Raafat [9] and Goyal and Giri [8]. Recently Yan and Cheng [13] considered a perishable single-item production–inventory system in which production rate, product demand rate and item deterioration rate are all assumed to be known functions of time. They developed the model over an infinite planning horizon allowing shortages which are backlogged partially. Balkhi [5] pointed out the errors in the formulation of the total cost in Yan and ChengÕs [13] model and also suggested a solution procedure to the corrected total cost model. However, Balkhi et al. [6] provided the exact mathematical treatment for finding the optimal solution of the model. Fig. 1 illustrates a typical production–inventory cycle according to Yan and Cheng [13]. The principal drawbacks of Yan and ChengÕs [13] modeling approach are as follows: • In infinite planning horizon model, the shortage cost of the backlogged items during the period [S; T2 ] of shortage is carried over to the next cycle.

• In finite planning horizon model, all the demands during the shortage period [S; T2 ] of the last cycle are lost. In order to overcome these disadvantages, we reconsider and study the problem under the modeling approaches proposed by Balkhi and Benkherouf [2] and Goyal et al. [7]. The paper is organized as follows. The assumptions and notations are presented in Section 2. The formulations and the solution procedures of the proposed models are given in Section 3. Section 4 deals with the procedure of finding the near-optimal replenishment policies of the developed models over a finite planning horizon. The models are illustrated with the help of a numerical example in Section 5. Finally, conclusions are given in Section 6. 2. Assumptions and notations We consider the following basic assumptions: (a) A single-item inventory is considered over an infinite planning horizon. (b) The production rate pðtÞ and the demand rate qðtÞ are known functions of time and are such that pðtÞ > qðtÞ. (c) The items deteriorate continuously over time which is a known function of time, denoted by hðtÞ. (d) There is no repair or replacement of the deteriorated items during the production cycle. (e) Lead time is assumed to be zero. (f) Shortages are allowed. (g) Only a fraction bð0 6 b 6 1Þ of the demand during the stock out period is backlogged and the remaining fraction (1  b) is lost. In addition, we use the following notations: A c h b

Fig. 1. A typical production-inventory cycle of Yan and ChengÕs [13] modeling approach.

l

setup cost per setup, unit production cost, unit holding cost per unit time, unit shortage cost per unit time for backlogged items, opportunity cost per unit item due to lost sales.

S.K. Goyal, B.C. Giri / European Journal of Operational Research 147 (2003) 549–557

Additional notations will be mentioned in the text whenever needed.

3. Model formulation

551

Solving the above system of differential equations with the initial and boundary conditions IðT0 Þ ¼ 0, IðSÞ ¼ 0 and IðT Þ ¼ 0, we get Z t dðtÞ IðtÞ ¼ e edðuÞ fpðuÞ  qðuÞgdu; T0 6 t 6 T1 ; T0

3.1. Model I: Balkhi and Benkherouf’s [2] approach A typical behaviour of the inventory in a cycle is depicted in Fig. 2. The inventory starts and ends with zero stock. The production starts at T0 and stops at time T1 so that the maximum inventory level occurs at T1 . Inventory level drops to zero at S. The production process then restarts at T2 to meet the unsatisfied demands during the shortage period ½S; T2  and stops at time T. The variation of the inventory level IðtÞ with respect to time t can be described by the following differential equations: dIðtÞ þ hðtÞIðtÞ ¼ pðtÞ  qðtÞ; dt dIðtÞ þ hðtÞIðtÞ ¼ qðtÞ; dt dIðtÞ ¼ bqðtÞ; dt

T0 6 t 6 T1 ;

T1 6 t 6 S;

T2 6 t 6 T :

ð2Þ ð3Þ

ð5Þ t

¼  edðtÞ edðuÞ qðuÞdu; T1 6 t 6 S; S Z t ¼ b qðuÞdu; S 6 t 6 T2 S Z t ¼ fpðuÞ  qðuÞgdu; T2 6 t 6 T ;

ð6Þ ð7Þ ð8Þ

T

R where dðtÞ ¼ hðtÞdt. The holding cost for carrying inventory over the periods ½T0 ; T1  and ½T1 ; S are given respectively by Z t  Z T1 dðtÞ dðuÞ e e fpðuÞ  qðuÞgdu dt HC1 ¼ h T0

and HC2 ¼ h

S 6 t 6 T2 ;

dIðtÞ ¼ pðtÞ  qðtÞ; dt

ð1Þ

Z

Z

T0

S

e

dðtÞ

Z

S

e

T1

dðuÞ

 qðuÞdu dt

t

The shortage cost over the period ½S; T2  is given by  Z T2  Z t qðuÞdu dt SC1 ¼ bb S

Z

¼ bb

ð4Þ

S T2

ðT2  tÞqðtÞdt

S

and the shortage cost over the period ½T2 ; T  is given by  Z T Z T fpðuÞ  qðuÞgdu dt SC2 ¼ b T2

¼b

Z

t

T

ðt  T2 ÞfpðtÞ  qðtÞgdt:

T2

Also, the production cost during the production periods ½T0 ; T1  and ½T2 ; T  are respectively Z T1 pðtÞdt PC1 ¼ c T0

and Fig. 2. A typical production–inventory cycle of Balkhi and BenkheroufÕs [2] modeling approach.

PC2 ¼ c

Z

T

pðtÞdt: T2

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S.K. Goyal, B.C. Giri / European Journal of Operational Research 147 (2003) 549–557

Moreover, the opportunity cost due to lost sales during the period ½S; T2  is given by Z T2 qðtÞdt: OC ¼ lð1  bÞ S

The total inventory cost during the period ½T0 ; T  is the sum of the production costs (PC1 ; PC2 ), holding costs (HC1 ; HC2 ), shortage costs (SC1 ; SC2 ), opportunity cost (OC) and setup cost (A). Therefore, the total variable cost ðR1 Þ per unit time during the production cycle ½T0 ; T  is given by R1 ¼

 Z c

1 T  T0 Z þh

T1

pðtÞdt þ c T0

T1

edðtÞ

T0 S

þh

Z

edðtÞ

T1

Z

Z

Z

Z

T

pðtÞdt

 ð pðuÞ  qðuÞÞedðuÞ du dt

T0 S

T1

pðtÞedðtÞ dt ¼

T0

T2

T2

The above equation can be rewritten as  Z T1 Z T 1 R1 ¼ c pðtÞdt þ c pðtÞdt T  T0 T0 T2 Z T1 þh fDðT1 Þ  DðtÞgrðtÞedðtÞ dt T0 Z S þh fDðtÞ  DðT1 ÞgqðtÞedðtÞ dt T1 Z T2 þ bb ðT2  tÞqðtÞdt þ lð1  bÞ  Z T2 S Z T

qðtÞdt þ b ðt  T2 ÞrðtÞdt þ A ; S

T2

ð9Þ R where DðtÞ ¼ edðtÞ dt and pðtÞ  qðtÞ ¼ rðtÞ. Our problem is to find T1 , S, T2 and T that minimize R1 subject to the constraints: Z S Z T1 fpðtÞ  qðtÞgedðtÞ dt ¼ qðtÞedðtÞ dt ð10Þ T0

T1

fpðtÞ  qðtÞgdt ¼ b

Z

T2

qðtÞdt; S

ð11Þ

S

qðtÞedðtÞ dt:

Since pðtÞ > qðtÞ, therefore from above, when T0 < T1 , we have S > T1 . Again from Eqs. (7) and (8), we have Z S

qðuÞedðuÞ du dt

Z

T0



t

S

T2

Z

IðT2 Þ ¼ b

þ bb ðT2  tÞqðtÞdt þ lð1  bÞ S  Z T Z T2 qðtÞdt þ b ðt  T2 ÞfpðtÞ  qðtÞgdt þ A :

and Z T

Eq. (10) implies that S is a function of T1 when T0 is known. Then from Eq. (11) we find T as a function of S and T2 , that is, a function of T1 and T2 . Moreover, Eq. (10) can be rewritten as

T2

t

ð12Þ

T0 < T1 < S < T2 < T :

T2

qðtÞdt ¼

Z

T2

fpðtÞ  qðtÞgdt < 0: T

This implies that T2 > S and T > T2 . Hence when T1 > T0 , we have the relation T0 < T1 < S < T2 < T . Thus utilizing Eqs. (10) and (11), the problem of finding the optimal replenishment schedule reduces to the problem of finding T1 and T2 that minimize R1 subject to the constraint T1 > T0 . If k be the Lagrange multiplier associated with the constraint, then the Kuhn–Tucker (KT) necessary conditions for the minimum of R1 give oR1 oR1  k ¼ 0; ¼ 0; oT1 oT2 kðT1  T0 Þ ¼ 0; T1 > T0 :

k P 0;

ð13Þ

Since the constraint T1  T0 > 0 is linear, the KT conditions (13) are also sufficient for the unique global minimum of R1 . Also it is clear from the above that k ¼ 0 as T1 > T0 . This means that the corresponding resource constraint is not scarce and consequently it does not affect the value of R1 . Therefore, in order to find the optimal values of T1 and T2 , we have to solve the equations oR1 =oT1 ¼ 0 and oR1 =oT2 ¼ 0 which give respectively dS hfDðSÞ  DðT1 ÞgedðsÞ dT1

 bbðT2  SÞ  lð1  bÞ þ ½cpðT Þ oT ¼ 0; þ bðT  T2 ÞrðT Þ  R1  oT1

cpðT1 Þ þ qðsÞ

ð14Þ

S.K. Goyal, B.C. Giri / European Journal of Operational Research 147 (2003) 549–557

Z bb

553

T2

qðtÞdt þ lð1  bÞqðT2 Þ  cpðT2 Þ Z T b rðtÞdt þ ½cpðT Þ þ bðT  T2 ÞrðT Þ  R1  S

T2

oT

¼ 0; oT2

ð15Þ

where dS pðT1 ÞedðT1 Þ oT bqðSÞ dS ¼ ; ¼ ; dT1 oT1 rðT Þ dT1 qðsÞedðSÞ oT rðT2 Þ þ bqðT2 Þ : ¼ oT2 rðT Þ

and

It is difficult to solve the nonlinear equations (14) and (15) analytically. However, they can be solved numerically by using any suitable bivariate search technique for given production rate pðtÞ, demand rate qðtÞ and deterioration rate hðtÞ. Note that in Balkhi and BenkheroufÕs [2] modeling approach, all the unfilled demands are made up directly within the same cycle. Also the possibility of lost sale at the end of the time horizon for finite planning horizon model does not arise as every cycle ends with zero inventory. But the disadvantage of their modeling approach is that the production process has to restart at T2 to meet the unsatisfied demands during the stock out period ½S; T2 . So in finite time horizon model, an additional setup cost needs to be accounted in the total cost and as a result, the model may not be economically desirable. However, the modeling approach proposed by Goyal et al. [7] addresses all the disadvantages of Yan and ChengÕs [13] model. 3.2. Model II: Goyal et al.’s [7] approach Fig. 3 shows a typical production–inventory cycle in which the inventory starts with zero stock at time S0 . So the shortage begins to accumulate at the early stage in inventory. The production starts only at time T0 to meet the current and backlog demands. T01 is the time when shortage level reaches to zero; afterwards the positive level of inventory begins to build up. T1 is the time when the production process stops; the inventory level then starts declining. Finally the cycle ends with zero stock at time S.

Fig. 3. A typical production–inventory cycle of Goyal et al.Õs [7] modeling approach.

The instantaneous states of the inventory level IðtÞ at time tðS0 6 t 6 SÞ can be described by the following differential equations: dIðtÞ ¼ bqðtÞ; S0 6 t 6 T0 ; IðS0 Þ ¼ 0; ð16Þ dt dIðtÞ ¼ pðtÞ  qðtÞ; dt

T0 6 t 6 T01 ;

IðT01 Þ ¼ 0; ð17Þ

dIðtÞ þ hðtÞIðtÞ ¼ pðtÞ  qðtÞ; dt IðT01 Þ ¼ 0; dIðtÞ þ hðtÞIðtÞ ¼ qðtÞ; dt

T01 6 t 6 T1 ; ð18Þ

T1 6 t 6 S;

IðSÞ ¼ 0:

ð19Þ The solution of the above system of differential equations (16)–(19) gives Z t IðtÞ ¼  b qðuÞdu; S0 6 t 6 T0 ð20Þ ¼

Z

S0 t

f pðuÞ  qðuÞgdu;

T01

¼e

dðtÞ

Z

T0 6 t 6 T01

ð21Þ

t

edðuÞ f pðuÞ  qðuÞgdu;

T01 6 t 6 T1 ;

T01

Z

ð22Þ t

¼  edðtÞ edðuÞ qðuÞdu; S R where dðtÞ ¼ hðtÞdt.

T1 6 t 6 S;

ð23Þ

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S.K. Goyal, B.C. Giri / European Journal of Operational Research 147 (2003) 549–557

Proceeding in the same way as in Model I, the expression for the total variable cost ðR2 Þ per unit time during the production cycle ½S0 ; S can be derived as  Z T1 Z T01 1 c pðtÞdt þ b ðt  T0 ÞrðtÞdt R2 ¼ S  S0 T0 T0 Z T1 þh fDðT1 Þ  DðtÞgrðtÞedðtÞ dt T01 S

þh

Z

and R2

oS ¼ cpðT1 Þ þ hedðT1 Þ oT1

Z

T1

rðtÞedðtÞ dt

T01

þ hfDðSÞ  DðT1 ÞgqðSÞedðSÞ  he

dðT1 Þ

Z

oS oT1

S

qðtÞedðtÞ dt:

ð28Þ

T1

Using (26), the above two equations can be simplified as

fDðtÞ  DðT1 ÞgqðtÞedðtÞ dt

T1

þ bb

Z

T0

R2

ðT0  tÞqðtÞdt

S0

þ lð1  bÞ

Z

T0

 qðtÞdt þ A ;

ð24Þ

S0

R where DðtÞ ¼ edðtÞ dt and pðtÞ  qðtÞ ¼ rðtÞ: Using the relations Z T0 Z T01 b qðtÞdt ¼ fpðtÞ  qðtÞgdt ð25Þ S0

and Z T1

T0

½ pðtÞ  qðtÞedðtÞ dt ¼

T01

oS dT01 ¼ cpðT0 Þ þ bðT01  T0 ÞrðT01 Þ oT0 dT0 Z T01 dT01 b rðtÞdt þ hDðT01 ÞrðT01 ÞedðT01 Þ dT0 T0 oS þ hDðSÞqðsÞedðSÞ oT0 Z T0 þ bb qðtÞdt þ lð1  bÞqðT0 Þ ð29Þ S0

and Z

R2

S

qðtÞedðtÞ dt;

ð26Þ

T1

it is not difficult to show that R2 is a function of two independent variables T0 and T1 and when T0 > S0 , the production starting time T0 and the production stopping time T1 are connected by the relation T0 < T01 < T1 < S. Therefore, our objective is to minimize R2 ðT0 ; T1 Þ subject to the constraint T0 > S0 . As before, the Kuhn–Tucker (KT) necessary conditions for the minimum of R2 give

oS oS ¼ cpðT1 Þ þ hfDðSÞ  DðT1 ÞgqðSÞedðSÞ ; oT1 oT1 ð30Þ

where dT01 =dT0 can be obtained implicitly from Eq. (25); oS=oT0 and oS=oT1 from Eq. (26). Let pðtÞ and qðtÞ be known linearly increasing functions of time such that pðtÞ ¼ p0 þ p1 t and qðtÞ ¼ q0 þ q1 t where p0 P q0 P 0, p1 > q1 > 0; the rate of deterioration hðtÞ ¼ hðt  T01 Þ, t P T01 . Then we have from Eq. (25) Z T0 Z T01 b ðq0 þ q1 tÞdt ¼ ðr0 þ r1 tÞdt; S0

oS dT01 ¼ cpðT0 Þ þ bðT01  T0 ÞrðT01 Þ R2 oT0 dT0 Z T01 b rðtÞdt  hfDðT1 Þ  DðT01 Þg

where r0 ¼ p0  q0 and r1 ¼ p1  q1 . This gives pffiffiffiffiffiffiffiffiffiffiffiffi r0 T01 ¼ f ðT0 Þ ¼ /ðT0 Þ  ; r1

T0

dT01 þ hfDðSÞ  DðT1 Þg dT0 Z T0 oS þ bb qðtÞdt

qðsÞedðSÞ oT0 S0

rðT01 ÞedðT01 Þ

þ lð1  bÞqðT0 Þ

T0

where /ðT0 Þ ¼ ð27Þ

 2 r0 b T0 þ þ 2q0 ðT0  S0 Þ r1 r1

þ q1 ðT02  S02 Þ > 0 as T0 > S0 :

S.K. Goyal, B.C. Giri / European Journal of Operational Research 147 (2003) 549–557

Therefore,    dT01 r0 b 1=2 þ ðq0 þ q1 T0 Þ > 0; ¼ f/ðT0 Þg T0 þ dT0 r1 r1 implying that T01 is an increasing function of T0 . 4. The finite time horizon model The models developed in the previous section can also be considered over a finite planning horizon H by suitably adjusting the production cycles. The following are the steps to be followed: 1. Continue to find the optimal replenishment polPn icies P of the successive cycles until CL i6 i¼1 nþ1 H 6 i¼1 CLi where CLi denotes the ith cycle-length, i ¼ 1; 2; . . . 2. For n production-cycles, increase each cycle proportionally to finish the nth cycle exactly at H. Then modify the values of the decision variables accordingly and evaluate the total cost TCðnÞ by summing up the costs over n cycles. 3. Similarly for ðn þ 1Þ cycles, decrease each cycle proportionally to finish the ðn þ 1Þth cycle at H and evaluate TCðn þ 1Þ. 4. Finally, determine TCnear opt: ¼ minifTCðnÞ; TCðn þ 1Þg. 5. Numerical example In order to obtain the optimal replenishment policies of the developed models we consider the following example-data: pðtÞ ¼ p0 þ p1 t ¼ 15 þ 5t; qðtÞ ¼ q0 þ q1 t ¼ 10 þ 2t;

555

hðtÞ ¼ hðt  T01 Þ ¼ 0:01ðt  T01 Þ; T01 ¼ 0, for the first cycle; c ¼ 2; h ¼ 0:7; b ¼ 1:0; l ¼ 1:2; b ¼ 0:7; A ¼ 80 in appropriate units. To solve the nonlinear equations derived in Section 3, we take help of the numerical computational software MATHEMATICA version 3.0. We write computer programs taking the built-in object ‘‘FindRoot’’ which uses NewtonÕs method or versions of NewtonÕs method [12]. It is already mentioned in Section 3 that the first cycle in Model I requires two production setups. So we take the setup cost equal to 2A in the first cycle and proceed to find the optimal replenishment policies of the successive cycles in Model I. Table 1 shows the optimal results of consecutive five cycles. The optimal results of the first five cycles in Model II are presented in Table 2. It is to be noted from Tables 1 and 2 that the total variable cost per unit of time in Model II is less than that of Model I and the time periods covering the first five cycles in Models I and II are respectively 20.71532 and 19.17399. So to make a comparison of the total inventory costs of the two models over a fixed time period, say 20.71532, we increase the values of S0 , T0 , T01 , T1 and S in Model II by the factor 1.08039 and determine the total cost. The modified results are shown in Table 3. Tables 1 and 3 show that the total inventory cost in Model I is greater than that of Model II over the time period 20.71532. This is further illustrated for the case of the finite horizon problem by determining the economic policy using the two approaches for the time period H ¼ 14. The results of our analysis are given in Table 4. From Tables 1, 3 and 4, it is clear that Goyal et al.Õs [7] modeling approach outperforms Balkhi

Table 1 Optimal results of Model I Cycles

T0

T1

S

T2

T

R1

Total cost

1st 2nd 3rd 4th 5th

0.0 6.36681 10.34030 14.00958 17.45209

2.48338 7.37753 11.19576 14.77359 18.15309

3.76751 8.37284 12.14524 15.67732 19.01597

5.39986 9.63342 13.35611 16.83932 20.13460

6.36681 10.34030 14.00958 17.45209 20.71532

65.4069 82.1716 100.9149 117.9707 133.8009

416.4333 326.5080 370.2850 406.1153 436.6231

Total cost

1955.9647

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S.K. Goyal, B.C. Giri / European Journal of Operational Research 147 (2003) 549–557

Table 2 Optimal results of Model II Cycles

S0

T0

T01

T1

S

R2

Total cost

1st 2nd 3rd 4th 5th

0.0 4.57702 8.64984 12.38622 15.87607

1.39210 5.93425 9.92625 13.59481 17.02928

2.42684 6.74555 10.63851 14.24579 17.63693

3.64511 7.69634 11.46342 14.99098 18.32500

4.57702 8.64984 12.38622 15.87607 19.17399

51.5167 73.7072 92.9867 110.4084 126.4996

235.7930 300.1962 347.4336 385.3088 417.1856

Total cost

1685.9172

Table 3 Modified results of Model II ðmÞ

ðmÞ

ðmÞ

ðmÞ

ðmÞ

Cycles

S0

T0

T01

T1

S ðmÞ

R2

Total cost

1st 2nd 3rd 4th 5th

0.0 4.94495 9.34517 13.38190 17.15229

1.50401 6.41128 10.72419 14.68765 18.39820

2.62193 7.28780 11.49370 15.39096 19.05470

3.93813 8.31502 12.38492 16.19605 19.79808

4.94495 9.34517 13.38190 17.15229 20.71532

51.61481 75.57540 96.36932 115.15131 132.49479

255.2327 332.5484 389.0169 434.1653 472.0829

Total cost

1883.0462

Table 4 Near-optimal results over the planning horizon H ¼ 14 Model

n

TC

I

2 3 3 4

1106.1101 1112.2654 1048.7017 1070.1784

II a

na

TCa

2

1106.1101

3

1048.7017

denotes the near-optimal result.

and BenkheroufÕs [2] approach in terms of the least expensive replenishment policy.

6. Concluding remarks In this paper, we consider the inventory problem of time varying demand, production and deterioration rates with shortages being allowed and backlogged partially over an infinite planning horizon. We describe two modeling approaches – Balkhi and BenkheroufÕs [2] approach and Goyal et al.Õs [7] approach for determining economic operating policies. We also demonstrate the advantage of using Goyal et al.Õs [7] approach as compared to the approach of Balkhi and Benkherouf [2] for the finite time horizon problem.

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