A forward recursive algorithm for inventory lot-size models with power-form demand and shortages

European Journal of Operational Research 137 (2002) 394±400

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Production, Manufacturing and Logistics

A forward recursive algorithm for inventory lot-size models with power-form demand and shortages Hui-Ling Yang a, Jinn-Tsair Teng b

b,*

, Maw-Sheng Chern

c

a Center of General Education, Hung Kuang Institute of Technology, Shalu 433, Taichung, Taiwan, ROC Department of Marketing and Management Sciences, College of Business, The William Paterson University of New Jersey, Wayne, NJ 07470-2103, USA c Department of Industrial Engineering and Engineering Management, National Tsing-Hua University, Hsinchu 30043, Taiwan, ROC

Received 18 January 2000; accepted 1 March 2001

Abstract Barbosa and Friedman (L.C. Barbosa, M. Friedman, Management Science 24 (8) (1978) 819) establish an optimal replenishment policy for power-form demand rate. In this paper, we extend their inventory lot-size model to allow for shortages. The goal is to ®nd the optimal number and time of replenishments in order to keep the total relevant cost as low as possible during a ®nite planning horizon. We develop a simple forward recursive algorithm to determine the optimal replenishment timing. Furthermore, we propose an intuitively accurate estimate for the optimal number of replenishments, which signi®cantly reduces computational complexity in ®nding the optimal replenishment number. A numerical example is provided to illustrate the solution procedure. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Inventory; Optimization; EOQ; Power-form demand; Shortages

1. Introduction The classical economic order quantity (EOQ) model is widely used principally because it is simple to use and apply. However, a major problem in using the EOQ is that it assumes a constant demand pattern. The deviations from the as-

* Corresponding author. Tel.: +1-973-720-2651; fax: +1-973720-2809. E-mail address: [email protected] (J.-T. Teng).

sumption cause varying magnitudes of inaccuracy. In reality, the demand may vary with time. For the discrete case of time-varying demand pattern, it can be solved by dynamic programming (e.g., Wagner and Whitin, 1958). For the continuous time-varying demand pattern, Resh et al. (1976) proposed an algorithm to ®nd the optimal replenishment number and time scheduling for timeproportional demand (i.e., f …t† ˆ bt, with b > 0). Concurrently, Donaldson (1977) also derived an analytical solution to a similar model in which the demand trend is linear (i.e., f …t† ˆ a ‡ bt, with

0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 1 5 4 - 0

H.-L. Yang et al. / European Journal of Operational Research 137 (2002) 394±400

a; b 6ˆ 0). Barbosa and Friedman (1978) further generalized the solutions for various power-form demand rates (i.e., f …t† ˆ btr , with b > 0, r > 2). Henery (1979) then extended the demand function to be any log-concave form (i.e., f …t† is log-concave). Recently, Triantaphyllou (1992) presented a sensitivity analysis of the linear demand model under various conditions. The computational and conceptual complexities of Donaldson's optimal analytical approach prompted many researchers to search for heuristic methods to solve the problem. Silver (1979) o€ered a heuristic algorithm, which provided the ®rst replenishment point to minimize the total relevant cost per unit time. In comparison with Donaldson's examples, Silver concluded that the cost penalties of using his heuristic were likely to be very low. For computational simplicity, Phelps (1980) proposed an inventory policy with a constant replenishment period, which gives only slightly higher costs than the optimal policy with varying replenishment periods. For conceptual simplicity, Mitra et al. (1984) modi®ed the EOQ model to accommodate the case of a linear demand pattern. Their technique was simple in concept, and easy to apply without computational iterative schemes as Silver's. Lately, Teng (1994) presented a hybrid solution method to the problem by using an approximate total cost to ®nd the number of replenishments, and then by applying Donaldson's analysis to obtain the optimal time for replenishments. All of the above models assumed that shortages were prohibited. Following the approach of Donaldson, Dave (1989a,b) developed an exact replenishment policy for an inventory model with shortages and a linear trend in demand. To reduce the complexity, Dave (1989a,b) also extended Silver's heuristic (1979) to solve the problem. By assuming that successive replenishment cycle lengths were in arithmetic progression, Datta and Pal (1991) established a more accurate approach than Dave's. Hariga (1994) provided some insightful properties for the problem, and developed an iterative procedure for both growing and declining markets. Teng (1996) proposed a simple and computationally ecient optimal method in recursive fashion to solve the problem. Recently,

395

Teng et al. (1997) investigated all four possible shortage policies with linearly increasing demand, and provided a forward recursive algorithm without iterative schemes to solve them. Other recent papers related to this area are written by Benkherouf and Mahmoud (1996), Hariga and Goyal (1995), Goyal et al. (1992), Goyal and Giri (2000), and Yang et al. (2001). In reality, the demand growth model, in general, is S-shaped such as Bass di€usion models for durable products (e.g., Bass, 1969). Consequently, the power-form demand is more applicable than a linear form because S-shaped demand patterns (in the growth stage of the product life cycle) can be better approximated by a power-form demand than a linear form. In addition, the mathematical inventory model without shortages is simply a constrained form of the inventory model that allows for shortages (with shortage cost of in®nity). Thus, in contrast to the others, we assume here that not only the demand is a power-form, but also shortages are permitted. We then propose a simple and computationally ecient method in a forward recursive manner to ®nd the optimal replenishment timing. Furthermore, we develop an intuitively accurate estimate for the optimal number of replenishments, which signi®cantly reduces computational complexity in ®nding the optimal replenishment number. A numerical example is provided to illustrate the proposed algorithm. Finally, we summarize the results and provide ways to extend the model for future studies. 2. Assumptions and notation The mathematical model of the inventory replenishment problem here is based on the following assumptions: 1. Lead time is zero. 2. Shortages are allowed and completely backlogged. 3. The initial inventory level is zero. In general, when shortages occur, some customers would like to wait for backlogging, but others would not. To encourage complete backlogging, ®rms need to provide incentives (which are the shortage costs to the ®rms) to customers.

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Otherwise, the unsatis®ed demand will be lost. Likewise, if the suppliers cannot deliver products to retailers on time, then the suppliers should provide retailers sucient incentives to accept backlogging. Of course, the retailers will decide to accept or reject the suppliers' incentives based on their customers' choices. In practice, the shortage cost has two possible extreme values. The upper limit is the standard no-shortage solution (i.e., the shortage cost is in®nitely large). For example, a blood bank must maintain a certain level of inventory because the shortage cost in this case is extremely high. On the other hand, when mailordering Christmas gifts, Valentine gifts, and others, as long as the gifts are delivered on time, we do not care about shortages at time of the order. Similarly, when we mail order tulips, da€odils, etc., we do not need them to be delivered right away. It is ®ne as long as we can receive them in October for the fall planting season. In this case, the shortage cost is negligible. Consequently, the ®rms can use a ``just-in-time'' inventory policy of satisfying orders only at the next re-order point (i.e., the fall planting season) so that no inventory is actually kept. In addition, the following notation is used throughout this paper. H the time horizon under consideration. f …t† …a ‡ bt†r , the demand function at time t, where a; b, and r are non-negative constants, and 0 6 t 6 H . co the ®xed replenishment cost per order. ch the inventory carrying cost per unit per unit time. cs the shortage cost per unit per unit time. n the number of replenishments over ‰0; H Š (a decision variable). ti the ith replenishment time, i ˆ 1; 2; . . . ; n, with t1 P 0 and tn‡1 ˆ H (a decision variable). Ki the fraction of no-shortage in the ith cycle [ti ;ti‡1 ), where 06 Ki ˆ …si ti †=…ti‡1 ti †61; i ˆ 1;2;...;n. si the time at which the inventory level reaches zero in the ith cycle (ti ; ti‡1 ) (a decision variable), where si ˆ Ki ti‡1 ‡ …1

Ki †ti ;

i ˆ 1; 2; . . . ; n:

…1†

3. Mathematical model and solution The objective of this inventory problem is to determine the number of replenishments n, and the timing of the reorder points fti g and the shortage points fsi g during the whole planning horizon in order to keep the sum of replenishment, inventory and shortage costs as low as possible. The ith replenishment is made at ti , the quantity received at ti is partly used to meet the accumulated shortages in the previous cycle from time si 1 to ti (with si 1 < ti ), and the inventory at ti gradually reduces to zero at si (with si > ti ). Consequently, based on whether the inventory is permitted to start and/or end with shortages, we have four possible models as shown in the recent articles by Teng et al. (1997, 1999). Since the other three models are special cases of the proposed model that allows for shortages not only at the initial point but also in the ®nal cycle, we simply use the proposed model here. This is depicted graphically in Fig. 1. The reader can easily obtain similar results for the other three models. Next, we consider the level of inventory at time t; I…t†, ti 6 t 6 si . The inventory level at time t; I…t†, during the ith replenishment cycle is governed by the following di€erential equation: dI…t† ˆ dt

f …t†;

ti 6 t 6 si ;

…2†

with the boundary condition I…si † ˆ 0. Solving the di€erential equation (2), we have Z si I…t† ˆ f …u† du; ti 6 t 6 si : …3† t

As a result, the cumulative inventory level during the ith cycle is Z si Z si I…t† dt ˆ …t ti †f …t† dt; Ii ˆ …4† t1 ti i ˆ 1; 2; . . . ; n: Similarly, the cumulative shortage during the ith cycle is Z ti‡1 Si ˆ …ti‡1 t†f …t† dt; i ˆ 0; 1; . . . ; n: …5† si

H.-L. Yang et al. / European Journal of Operational Research 137 (2002) 394±400

397

Fig. 1. Graphical representation of inventory level.

Therefore, the total relevant cost of the inventory system during the planning horizon H when n orders are placed is as follows: C…n; fsi g; fti g† ˆ nco ‡ ch

n X

Ii ‡ cs

iˆ1

n X

Si :

…6†

iˆ0

and si ˆ

1 V ti‡1 ‡ ti ; 1‡V 1‡V

i ˆ 1; 2; . . . ; n:

…10†

Next, integrating both sides of (8), we obtain 1‡V r‡1 …a ‡ bti † V i ˆ 1; 2; . . . ; n:

…a ‡ bsi †

r‡1

1 r‡1 …a ‡ bsi 1 † ; V

ˆ

Thus, the problem here is to ®nd an integer n and a vector of 2n components ht1 ; s1 ; t2 ; . . . ; tn ; sn i with 0 ˆ s0 < t1 < s1 <


< tn < sn < tn‡1 ˆ H such that the total relevant cost in (6) is minimized. For a ®xed value of n, the necessary conditions for C…n; fsi g; fti g† to be minimized are as follows:

After applying (10) into (11), we obtain the following relation between ti 's:

oC…n; fsi g; fti g†=osi ˆ 0;

‰…a ‡ bti‡1 † ‡ V …a ‡ bti †Šr‡1

i ˆ 1; 2; . . . ; n;

ˆ f…1 ‡ V †r‡2 …a ‡ bti †r‡1

and

r‡1

oC…n; fsi g; fti g†=oti ˆ 0;

‰…a ‡ bti † ‡ V …a ‡ bti 1 †Š i ˆ 2; 3; . . . ; n;

i ˆ 1; 2; . . . ; n;

si

si †;

ti

f …t† dt;

i ˆ 1; 2; . . . ; n;

i ˆ 1; 2; . . . ; n;

…7†

…8†

1

respectively, where V ˆ ch =cs . Substituting (1) into (7), we have Ki

…12†

r‡1

ti † ˆ cs …ti‡1

and Z si Z V f …t† dt ˆ ti

g=V ;

and

which lead to ch …si

…11†

ˆ cs =…ch ‡ cs † ˆ 1=…1 ‡ V †  K;

…9†

‰…a ‡ bt2 † ‡ V …a ‡ bt1 †Š ˆ ‰…1 ‡ V †

r‡2

…a ‡ bt1 †

r‡1

ar‡1 Š=V :

…13†

4. A forward recursive algorithm For simplicity, let us de®ne gi by gi ˆ …a ‡ bti †=…a ‡ bt2 †; i ˆ 1; 2; . . . ; n ‡ 1:

…14†

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H.-L. Yang et al. / European Journal of Operational Research 137 (2002) 394±400

Applying (14) into (12) and (13), we have

Next, we establish a solution procedure for ®nding the optimal number of replenishments, n . For simplicity, let

…1 ‡ Vg1 †r‡1 ˆ ‰…1 ‡ V † g2 ˆ 1;

r‡2 r‡1 g1

…a=…a ‡ bt2 ††

r‡1

Š=V ; …15†

…gi‡1 ‡ Vgi †r‡1 ˆ ‰…1 ‡ V †r‡2 gir‡1

C…n† ˆ C…n; fsi g; fti g†:

…gi ‡ Vgi 1 †r‡1 Š=V ;

i ˆ 2; 3; . . . ; n: Since tn‡1 ˆ H , we know from (14) that the optimal replenishment time fti g can be easily obtained as follows: ti ˆ ‰…gi =gn‡1 †…a ‡ bH † i ˆ 1; 2; . . . ; n:

…17†

It has been proven from Hariga (1996) and Teng et al. (1999) that the total relevant cost is a strictly convex function of the number of replenishments. Therefore, the search for the optimal number of replenishments is simpli®ed to ®nd a local minimum. Now, we propose an accurate estimate of the optimal number of replenishments as shown in Teng (1996). n ˆ rounded integer of

aŠ=b; …16†

It is clear from (10) and (16) that the optimal solution fsi g and fti g not only exists but is also unique (i.e., the optimal values of fsi g and fti g are uniquely determined by Eqs. (10) and (16)). Consequently, it reduces the 2n-dimensional problem of ®nding fsi g and fti g to a one-dimensional problem. From (15), we only need to ®nd t2 to generate gi ; i ˆ 1; 2; . . . ; n ‡ 1, uniquely by repeatedly using (15). For any chosen t2 , if gn‡1 ˆ …a ‡ bH †=…a ‡ bt2 †, then t2 is chosen correctly. Otherwise, we can easily ®nd the optimal t2 by standard search techniques. We then apply (10) to obtain fsi g. For any given value of n, the solution procedure for ®nding fti g is summarized in the following algorithm. Algorithm 1. For ®nding optimal replenishment timing {ti } Step 0. Choose two trial values of t2 , say L ˆ …1 ‡ 2V †H =‰n…1 ‡ V †Š and U ˆ 2…1 ‡ 2V †H = ‰n…1 ‡ V †Š. Compute the corresponding values of gn‡1 by using (15), say M and W , respectively. Step 1. Let t2 ˆ L ‡ …U L†…H M†=…W M†, and calculate the corresponding gn‡1 . Step 2. If gn‡1 …a ‡ bH †=…a ‡ bt2 † is suciently small, then use (16) to ®nd fti g and stop. Step 3. If gn‡1 …a ‡ bH †=…a ‡ bt2 † is signi®cantly larger than zero, then set U ˆ t2 , W ˆ gn‡1 , and go to Step 1. Otherwise, set L ˆ t2 ; M ˆ gn‡1 , and go to Step 1.

f‰ch cs Q…H †H =‰2co …ch ‡ cs †Šg1=2 ;

…18†

where Z Q…H † ˆ

H 0

f …t† dt

ˆ ‰…a ‡ bH †

r‡1

ar‡1 Š=‰b…r ‡ 1†Š:

…19†

It is obvious that searching for n by starting with n in (18) instead of n ˆ 1 will reduce the computational complexity signi®cantly. Thus, we propose the following procedure for ®nding the optimal replenish number and schedule. Algorithm 2. For ®nding optimal number and schedule Step 0. Choose two initial trial values of n , say n as in (18) and n 1. Use Algorithm 1 to search for fti g, then obtain fsi g by (10), and compute the corresponding C…n† and C…n 1†, respectively. Step 1. If C…n† P C…n 1†, then compute C…n 2†;C…n 3†;...; until we ®nd C…k† < C…k 1†. Set n ˆ k and stop. Step 2. If C…n† < C…n 1†, then compute C…n ‡ 1†;C…n ‡ 2†;..., until we ®nd C…k† < C…k ‡ 1†. Set n ˆ k and stop.

5. A numerical example To illustrate the results, we apply the proposed methods to solve the following numerical example.

H.-L. Yang et al. / European Journal of Operational Research 137 (2002) 394±400

399

Acknowledgements

Table 1 Optimal solution of Example 1 i

gi

ti

si

0 1 2 3 4 5 6 7 8

± 0.6648 1.0000 1.2564 1.4714 1.6601 1.8301 1.9861 2.1310

± 0.0826 0.2923 0.4528 0.5873 0.7053 0.8117 0.9093 1.0000

0.0000 0.2457 0.4171 0.5574 0.6791 0.7881 0.8876 0.9798 ±

Example 1. Let r ˆ 2, a ˆ 10, b ˆ 30, H ˆ 1, co ˆ 4:5, ch ˆ 1, cs ˆ 3:5 in the appropriate units. From (18), we search for n starting with the two trial values 7 and 6 as described in Algorithm 2, and the corresponding minimum total relevant costs: C…6† ˆ 67:34 and C…7† ˆ 66:13. Since C…8† ˆ 66:34, we know that the optimal replenishment number is 7. By using Algorithm 1, we obtain the optimal replenishment schedule as shown in Table 1. 6. Conclusions In general, the demand growth model is Sshaped such as Bass di€usion models for durable products (e.g., Bass, 1969). Consequently, the power-form demand function is more applicable than a linear form in the growth stage of the product life cycle. In this paper, we assume that the demand function is power-form. We then develop not only a simple forward recursive algorithm to determine the optimal replenishment timing, but also an intuitively accurate estimate for the optimal number of replenishments, which signi®cantly reduces computational complexity in ®nding the optimal replenishment number. The model can be extended in several ways. For example, we may consider the deterministic case as well as stochastic case in which demand patterns are ¯uctuating. Moreover, we may extend the model to incorporate with deterioration rate, quantity discounts, in¯ation factor, and others. Finally, in a later paper (Teng et al., 2000), we generalize the model to allow for ¯uctuating demand as well as unit purchase cost.

The authors would like to thank the referees for their valuable comments. This research was partially supported by the National Science Council of the Republic of China under Grant NSC-89-2213E-007-135. The second author's research was supported by the Assigned Released Time for research from the William Paterson University of New Jersey. References Barbosa, L.C., Friedman, M., 1978. Deterministic inventory lot size models ± a general root law. Management Science 24 (8), 819±826. Bass, F.M., 1969. A new product growth model for consumer durables. Management Science 15 (1), 215±227. Benkherouf, L., Mahmoud, M.G., 1996. On an inventory model for deteriorating items with increasing time-varying demand and shortages. Journal of the Operational Research Society 47 (1), 188±200. Datta, T.K., Pal, A.K., 1991. A note on a replenishment policy for an inventory model with linear trend in demand and shortages. Journal of the Operational Research Society 43 (10), 993±1001. Dave, U., 1989a. A deterministic lot-size inventory model with shortages and a linear trend in demand. Naval Research Logistics 36, 507±514. Dave, U., 1989b. On a heuristic inventory-replenishment rule for items with a linearly increasing demand incorporating shortages. Journal of the Operational Research Society 40 (9), 827±830. Donaldson, W.A., 1977. Inventory replenishment policy for a linear trend in demand: An analytical solution. Operational Research Quarterly 28 (3), 663±670. Goyal, S.K., Giri, B.C., 2000. Note on an optimal recursive method for various inventory replenishment models with increasing demand and shortages. In: Teng et al. (Eds.), Naval Research Logistics 47, 602±606. Goyal, S.K., Morin, D., Nebebe, F., 1992. The ®nite horizon trended inventory replenishment problem with shortages. Journal of the Operational Research Society 43 (12), 1173± 1178. Hariga, M.A., 1994. The inventory lot-sizing problem with continuous time varying demand and shortages. Journal of the Operational Research Society 45 (4), 827±837. Hariga, M.A., 1996. Optimal EOQ models for deteriorating items with time-varying demand. Journal of the Operational Research Society 47 (10), 1228±1246. Hariga, M.A., Goyal, S.K., 1995. An alternative procedure for determining the optimal policy for an inventory item having linear trend in demand. Journal of the Operational Research Society 46 (4), 521±527.

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