See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/4869899 A forward recursive algorithm for inventory lot-size models with power-form demand and shortages Article in European Journal of Operational Research · February 2002 DOI: 10.1016/S0377-2217(01)00154-0 · Source: RePEc CITATIONS READS 22 163 authors: Hui-Ling Yang Jinn-Tsair Teng Hungkuang University William Paterson University 27 PUBLICATIONS 1,474 CITATIONS 101 PUBLICATIONS 6,377 CITATIONS SEE PROFILE Maw-Sheng Chern National Tsing Hua University 61 PUBLICATIONS 2,386 CITATIONS SEE PROFILE All content following this page was uploaded by Hui-Ling Yang on 07 January 2019 The user has requested enhancement of the downloaded file SEE PROFILE European Journal of Operational Research 137 (2002) 394±400 www.elsevier.com/locate/dsw 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 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 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: tengj@wpunj.edu (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 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 a bt, with 0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V All rights reserved PII: S 7 - 2 ( ) 0 - H.-L Yang et al / European Journal of Operational Research 137 (2002) 394±400 a; b T 0) Barbosa and Friedman (1978) further generalized the solutions for various power-form demand rates (i.e., f t btr , with b > 0, r > À2) 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) oered 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 ecient 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 diusion 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 ecient 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 Assumptions and notation The mathematical model of the inventory replenishment problem here is based on the following assumptions: Lead time is zero Shortages are allowed and completely backlogged 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 396 H.-L Yang et al / European Journal of Operational Research 137 (2002) 394±400 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 sucient 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 not care about shortages at time of the order Similarly, when we mail order tulips, daodils, etc., we 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 btr , the demand function at time t, where a; b, and r are non-negative constants, and t 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 (a decision variable) ti the ith replenishment time, i 1; 2; ; n, with t1 P and tn1 H (a decision variable) Ki the fraction of no-shortage in the ith cycle [ti ;ti1 ), where 06 Ki si À ti = ti1 À ti 61; i 1;2; ;n si the time at which the inventory level reaches zero in the ith cycle (ti ; ti1 ) (a decision variable), where si Ki ti1 1 À Ki ti ; i 1; 2; ; n: 1 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 siÀ1 to ti (with siÀ1 < 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 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 t si The inventory level at time t; I t, during the ith replenishment cycle is governed by the following dierential equation: dI t Àf t; dt ti t si ; 2 with the boundary condition I si Solving the dierential equation (2), we have Z si I t f u du; ti t si : 3 t As a result, the cumulative inventory level during the ith cycle is Z si Z si I t dt t À ti f t dt; Ii 4 t1 ti i 1; 2; ; n: Similarly, the cumulative shortage during the ith cycle is Z ti1 Si ti1 À tf t dt; i 0; 1; ; n: 5 si H.-L Yang et al / European Journal of Operational Research 137 (2002) 394±400 397 Fig 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 nco ch n X Ii cs i1 n X Si : 6 i0 and sÃi V ti1 ti ; 1V 1V i 1; 2; ; n: 10 Next, integrating both sides of (8), we obtain 1V r1 r1 a bti À a bsiÀ1 ; V V i 1; 2; ; n: a bsi r1 Thus, the problem here is to ®nd an integer n and a vector of 2n components ht1 ; s1 ; t2 ; ; tn ; sn i with s0 < t1 < s1 < Á Á Á < tn < sn < tn1 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 0; a bti1 V a bti r1 i 1; 2; ; n; f 1 V r2 a bti r1 and r1 oC n; fsi g; fti g=oti 0; À a bti V a btiÀ1 i 2; 3; ; n; i 1; 2; ; n; 12 r1 ch si À ti cs ti1 À si ; and Z si Z V f t dt ti siÀ1 f t dt; i 1; 2; ; n; i 1; 2; ; n; 7 8 respectively, where V ch =cs Substituting (1) into (7), we have Kià g=V ; and which lead to ti 11 cs = ch cs 1= 1 V K; 9 a bt2 V a bt1 1 V r2 a bt1 r1 À ar1 =V : 13 A forward recursive algorithm For simplicity, let us de®ne gi by gi a bti = a bt2 ; i 1; 2; ; n 1: 14 398 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, nà For simplicity, let 1 Vg1 r1 1 V g2 1; r2 r1 g1 À a= a bt2 r1 =V ; gi1 Vgi r1 C n C n; fsÃi g; ftià g: 15 1 V r2 gir1 À gi VgiÀ1 r1 =V ; i 2; 3; ; n: Since tn1 H , we know from (14) that the optimal replenishment time ftià g can be easily obtained as follows: 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 rounded integer of tià gi =gn1 a bH À a=b; i 1; 2; ; n: 17 16 It is clear from (10) and (16) that the optimal solution fsÃi g and ftià g not only exists but is also unique (i.e., the optimal values of fsÃi g and ftià g are uniquely determined by Eqs (10) and (16)) Consequently, it reduces the 2n-dimensional problem of ®nding fsÃi g and ftià g to a one-dimensional problem From (15), we only need to ®nd t2à to generate gi ; i 1; 2; ; n 1, uniquely by repeatedly using (15) For any chosen t2à , if gn1 a bH = a bt2 , then t2à is chosen correctly Otherwise, we can easily ®nd the optimal t2à by standard search techniques We then apply (10) to obtain fsÃi g For any given value of n, the solution procedure for ®nding ftià g is summarized in the following algorithm Algorithm For ®nding optimal replenishment timing {tÃi } Step Choose two trial values of t2 , say L 1 2V H =n 1 V and U 2 1 2V H = n 1 V Compute the corresponding values of gn1 by using (15), say M and W , respectively Step Let t2 L U À L H À M= W À M, and calculate the corresponding gn1 Step If gn1 À a bH = a bt2 is suciently small, then use (16) to ®nd ftià g and stop Step If gn1 À a bH = a bt2 is signi®cantly larger than zero, then set U t2 , W gn1 , and go to Step Otherwise, set L t2 ; M gn1 , and go to Step fch cs Q H H =2co ch cs g1=2 ; 18 where Z Q H H f t dt a bH r1 À ar1 =b r 1: 19 It is obvious that searching for nà by starting with n in (18) instead of n will reduce the computational complexity signi®cantly Thus, we propose the following procedure for ®nding the optimal replenish number and schedule Algorithm For ®nding optimal number and schedule Step Choose two initial trial values of nà , say n as in (18) and n À Use Algorithm to search for ftià g, then obtain fsÃi g by (10), and compute the corresponding C n and C n À 1, respectively Step If C n P C n À 1, then compute C n À 2;C n À 3; ; until we ®nd C k < C k À 1 Set nà k and stop Step If C n < C n À 1, then compute C n 1;C n 2; , until we ®nd C k < C k 1 Set nà k and stop 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 Optimal solution of Example i gi tià sÃi ± 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 Let r 2, a 10, b 30, H 1, co 4:5, ch 1, cs 3:5 in the appropriate units From (18), we search for nà starting with the two trial values and as described in Algorithm 2, and the corresponding minimum total relevant costs: C 6 67:34 and C 7 66:13 Since C 8 66:34, we know that the optimal replenishment number is By using Algorithm 1, we obtain the optimal replenishment schedule as shown in Table Conclusions In general, the demand growth model is Sshaped such as Bass diusion 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 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