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ARTICLE IN PRESS Int. J. Production Economics 88 (2004) 105–119 The stochastic dynamic production/inventory lot-sizing problem with service-level constraints S. Armagan Tarima, Brian G. Kingsmanb,* a Department of Computer Science, University of York, York YO10 5DD, UK b Department of Management Science, The Management School, Lancaster University, Lancaster LA1 4YX, UK Received 17 April 2003; accepted 26 May 2003 Abstract This paper addresses the multi-period single-item inventory lot-sizing problem with stochastic demands under the ‘‘static–dynamic uncertainty’’ strategy of Bookbinder and Tan (Manage. Sci. 34 (1988) 1096). In the static-dynamic uncertainty strategy, the replenishment periods are ﬁxed at the beginning of the planning horizon, but the actual orders are determined only at those replenishment periods and will depend upon the demand that is realised. Their solution heuristic was a two-stage process of ﬁrstly ﬁxing the replenishment periods and then secondly determining what adjustments should be made to the planned orders as demand was realised. We present a mixed integer programming formulation that determines both in a single step giving the optimal solution for the ‘‘static–dynamic uncertainty’’ strategy. The total expected inventory holding, ordering and direct item costs during the planning horizon are minimised under the constraint that the probability that the closing inventory in each time period will not be negative is set to at least a certain value. This formulation includes the effect of a unit variable purchase/production cost, which was excluded by the two-stage Bookbinder–Tan heuristic. An evaluation of the accuracy of the heuristic against the optimal solution for the case of a zero unit purchase/production cost is made for a wide variety of demand patterns, coefﬁcients of demand variability and relative holding cost to ordering cost ratios. The practical constraint of non- negative orders and the existence of the unit variable cost mean that the replenishment cycles cannot be treated independently and so the problem cannot be solved as a stochastic form of the Wagner–Whitin problem, applying the shortest route algorithm. r 2003 Elsevier B.V. All rights reserved. Keywords: Inventory; Chance-constrained programming; Integer programming; Modelling; Optimisation 1. Introduction The study of lot-sizing began with Wagner and Whitin (1958), and there is now a sizeable *Corresponding author. Tel.: 44-1524-593-848; fax: 44-1524- literature in this area extending the basic model 844-885. E-mail addresses: at@cs.york.ac.uk (S.A. Tarim), to consider capacity constraints, multiple items, b.kingsman@lancaster.ac.uk, b.kingsman@uk.ac.lancaster multiple stages, etc. However, most previous work (B.G. Kingsman). on lot-sizing has been directed towards the 0925-5273/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0925-5273(03)00182-8 ARTICLE IN PRESS 106 S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 deterministic case. The reader is directed to De optimality under the ‘‘static–dynamic uncertainty’’ Bodt et al. (1984), Potts and Van Wassenhove strategy of Bookbinder and Tan. (1992), Kuik et al. (1994) and Kimms (1997) for a The optimal solution to the problem is the (s; S) review of lot-sizing techniques. policy with different values for each period in the The practical problem is that in general much, if time varying demand situation. Where the demand not all, of the future demands have to be forecast. level changes slowly, it is usually satisfactory to Point forecasts are typically treated as determinis- use a steady-state analysis with constant S and s tic demands. However, the existence of forecast values updated once per year. However, this is errors radically affects the behaviour of the lot- inappropriate if the average demand can change sizing procedures based on assuming the determi- signiﬁcantly from period to period. This presents a nistic demand situation. Forecasting errors lead non-stationary problem where the two control both to stockouts occurring with unsatisﬁed parameters change from period to period. The demands and to larger inventories being carried uncertainty in the timing of future replenishments than planned. The introduction of safety stocks in caused by an (s; S) policy may be unattractive turn generates even larger inventories and also from an operational standpoint. more orders. It is reported by Davis (1993) that a Although the inventory problems with station- study at Hewlett-Packard revealed the fact that ary demand assumption are well known and 60% of the inventory investment in their manu- extensively studied, very little has appeared on facturing and distribution system is due to demand the non-stationary stochastic demand case. Re- uncertainty. cently, Sox (1997) and Martel et al. (1995) have There has been increasing recognition as illu- described static control policies under the non- strated by Wemmerlov (1989) that future lot-sizing stationary stochastic demand assumption in a studies need to be undertaken on stochastic and rolling horizon framework. Sox (1997) presents a dynamic environments that have at least a mixed integer non-linear formulation of the modicum of resemblance to reality. Inevitably, dynamic lot-sizing problem with dynamic costs, the forecast errors have to be taken into account in and develops a solution algorithm that resembles planning the future lot-sizes. Similar concerns have the Wagner–Whitin algorithm but with some been expressed by Silver: ‘‘One should not additional feasibility constraints. Martel et al. necessarily use a deterministic lot-sizing rule when (1995) transform the multiple item procurement signiﬁcant uncertainty exists. A more appropriate problem into a multi-period static decision pro- strategy might be some form of probabilistic blem under risk. Other notable works on non- modelling.’’ stationary stochastic demand adopt (s; S) or Silver (1978) suggested a heuristic procedure for base-stock policies and are due to Iida (1999), the stochastic lot-sizing problem assuming that the Sobel and Zhang (2001), and Gavirneni and Tayur forecast errors are normally distributed. A similar (2001). In Iida (1999), the periodic review dynamic heuristic, having a different objective function, was inventory problem is considered and it is shown presented by Askin (1981). Bookbinder and Tan that near myopic policies are sufﬁciently close to (1988) proposed another heuristic, under the optimal decisions for the inﬁnite horizon inventory ‘‘static–dynamic uncertainty’’ strategy. In this problem. In Sobel and Zhang (2001), it is assumed strategy, the replenishment periods are ﬁxed at that demand arrives simultaneously from a deter- the beginning of the planning horizon and the ministic source and a random source, and proven actual orders at future replenishment periods are that a modiﬁed (s; S) policy is optimal under determined only at those replenishment periods general conditions. Gavirneni and Tayur (2001) depending upon the realised demand. The total use the derivative of the cost function that can expected cost is minimised under the minimal handle a much wider variety of ﬂuctuations in the service-level constraint. In this paper, we propose a problem parameters. The above studies have mixed integer programming formulation to solve adopted either static control policies in a rolling the stochastic dynamic lot-sizing problem to horizon framework or dynamic control policies ARTICLE IN PRESS S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 107 like (s; S), although the static-dynamic uncertainty vestigation of the minimal service-level criterion is model is a more accurate representation of given by Chen and Krass (2001). For a discussion industrial practice as pointed out by Sox (1997). of different service constraints see Van Houtum Most companies use MRP in some form for and Zijm (2000) and Rosling (1999). Remember their production planning and thus ordering on that demand is not deterministic with different suppliers. They typically issue advance schedules values in each period. Thus, the service-level of requirements. In talking to companies in the criterion in this case is deﬁned as specifying a supply chain who are in this situation a common minimum probability, say a; that at the end of complaint is that their customers continually every period the net inventory will not be negative. change their schedules, the timing of orders as In general, ﬁnding the solution to this problem can well as the size of orders. It is the changing of the be formulated as solving a chance-constrained timing that they ﬁnd worse. This issue of system programming model (see for example Birge and nervousness is an active current research area. If Louveaux, 1997). It can be expressed as the there is to be more co-operation and co-ordination minimisation of the total expected cost, EfTCg; in supply chains, then a model that attempts to over the N-period planning horizon subject to the determine a schedule for the timing of orders in service-level constraints, as given below: advance taking account of the stochastic demand, minimise EfTCg which remains ﬁxed, is a contribution of practical Z Z Z X N interest. This need to ﬁx the deliveries in advance, ¼ ? ðadt þ hIt þvXt g whilst allowing reasonable ﬂexibility in the order d1 d2 dN t¼1 size has been at the heart of the problem of buying Â g1 ðd1 Þg2 ðd2 ÞygN ðdN Þdðd1 Þdðd2 ÞydðdN Þ ð1Þ raw materials on ﬂuctuating price markets. You have to determine which future months or half subject to months in which to have your delivery of wheat or ( cocoa, etc. Once this is ﬁxed, suppliers do not 1 if Xt > 0; dt ¼ t ¼ 1; y; N; ð2Þ allow it to be changed, although the quantities 0 otherwise; ordered can be varied to some extent. Waiting until the month or half month of delivery and X t using the standard (s; S) model to decide whether It ¼ I0 ðXi À di Þ; t ¼ 1; y; N; ð3Þ i¼1 or not to place an order and then its size is not a good option, as the price you pay tends to be much PrfIt X0gXa; t ¼ 1; y; N; ð4Þ higher as sellers know you are desperate for the material. Thus, investigating models and solution Xt ; It X0; t ¼ 1; y; N; procedures for the static-dynamic uncertainty strategy is potentially important from the practical where N is the number of periods in the planning application perspective. horizon, Xt the replenishment order placed and received (i.e., no lead-time) in period t; dt a {0,1} variable that takes the value of 1 if a replenishment 2. Problem statement order is placed in period t and 0 otherwise, It the inventory level at the end of period t; I0 the stock The stochastic dynamic lot-sizing problem can on hand at the beginning of period 1, dt the be formulated either by assuming a penalty cost demand in period t; a the ﬁxed procurement cost, n for each stockout and unsatisﬁed demand or by the marginal cost of purchasing an item, h the minimising the ordering and inventory costs linear holding cost incurred on any unit carried in subject to satisfying some customer service-level inventory over from one period to the next. criterion. In this paper we consider meeting a The demand dt in period t is considered as a speciﬁed customer service case, where we use a random variable with known probability density ‘‘minimal service-level’’ criterion. A detailed in- function, gt ðdt Þ: The distribution of demand may ARTICLE IN PRESS 108 S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 vary from period to period. Demands in different nistic lot-sizing model given below: time periods are assumed independent. A ﬁxed XN procurement (ordering or set-up) cost, a; is minimise EfTCg ¼ ðadt þ hEfIt g þ vXt Þ ð5Þ incurred each time a replenishment order is placed, t¼1 whatever the size of the order. In addition to the subject to ﬁxed ordering cost, a variable purchasing cost is ( incurred depending on the size of the order, vXt : A 1 if Xt > 0; dt ¼ t ¼ 1; y; N; ð6Þ replenishment order is assumed to arrive instanta- 0 otherwise; neously at the beginning of each period, before the demand in that period occurs. A linear holding X t X t EfIt g ¼ I0 þ Xi À Efdi g; t ¼ 1; y; N; cost h is incurred on any unit carried in inventory i¼1 i¼1 over from one period to the next. For simplicity in ð7Þ the analysis, inventory holding costs are assumed to be incurred only on the inventory at the end of X t À1 each period. The extension to taking the average Xi XGDðtÞ ðaÞ À I0 ; t ¼ 1; y; N; ð8Þ inventory over a period is straightforward. The i¼1 probability that at the end of each and every time Xt ; EfIt gX0; t ¼ 1; y; N: period the net inventory will not be negative is set to be at least a: Hence, it is implicitly assumed that, The above formulation is solved by the Wagner– since normally the desired service level is quite Whitin algorithm. Bookbinder and Tan give the Pt À1 high, the value a incorporates the perception of the equivalent linear constraint, i¼1 Xi XGDðtÞ ðaÞ À cost of backorders, so that shortage cost can be I0 ; for the service-level constraint PrfIt Z0gXa; ignored in the model. The above assumptions are where GDðtÞ ¼ Gd1 þd2 þ?þdt ð:Þ is the cumulative valid for the rest of the paper. probability distribution function of d1 þ d2 þ ? þ dt : It is assumed that G is strictly increasing, therefore G À1 is uniquely deﬁned. They also examine the analogy between the deterministic 3. The Bookbinder–Tan strategy for the model and the ‘‘static uncertainty’’ model, and probabilistic lot-sizing problem conclude that there is correspondence between the respective terms of the two models. One of the few attempts to produce more À1 À1 À1 Since Gd1 ðaÞpGd1 þd2 ðaÞp?pGDðNÞ ðaÞ; the implementable near optimal solutions to the Pt À1 constraint i¼1 Xi XGDðtÞ ðaÞ À I0 is binding for t ¼ stochastic dynamic lot-size problem is given by PN À1 Bookbinder and Tan (1988). In Bookbinder and N and gives t¼1 Xt ¼ GDðNÞ ðaÞ À I0 ; which im- PN Tan’s paper three strategies were analysed, named plies that the total direct item cost, t¼1 vXt ; is ‘‘static uncertainty’’, ‘‘dynamic uncertainty’’ and constant. Therefore, Bookbinder and Tan ignore ‘‘static–dynamic uncertainty’’. In all cases, their the total direct item cost term, which has no effect analysis was based on a service-level constraint on on determination of the best schedule, as in the the probability of a stockout at the end of each deterministic problem. period. Another approach for determining the timing and The ‘‘static uncertainty’’ decision rule assumes size of the replenishment orders is the ‘‘dynamic that the values of all the decision variables, the uncertainty’’ strategy. In the ‘‘dynamic uncertainty’’ timing and size of replenishment orders, are strategy decisions are made every period on the basis determined at the beginning of the planning of the demands that have become known as you horizon. All the Xt can be considered as determi- move forward through time. This approach ignores nistic decision variables. The values and timing of the ordering cost a; and may require a replenishment the replenishment orders is determined by initially order almost every period. Obviously, for large reducing the problem to the equivalent determi- ratios of a=h this result is clearly undesirable. ARTICLE IN PRESS S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 109 Bookbinder and Tan then combined the features to period Ti in the desired buffer stock level and is of both strategies to give a two-stage heuristic, given as follows: which they called the ‘‘static–dynamic uncer- tainty’’ approach. The ﬁrst stage was to ﬁx the XT1 ¼ YT1 ; timing of the replenishment orders using the XT2 ¼ YT2 þ dT1 þ ? þ dT2 À1 ; ‘‘static uncertainty’’ model. Exploiting the analogy ð10Þ ^ between the deterministic model and the ‘‘static XTm ¼ YTm þ dTmÀ1 þ ? þ dTm À1 : uncertainty’’ model, the replenishment periods can be determined by using dynamic programming Hence, the replenishment size XTi can be deter- (Wagner and Whitin, 1958), network ﬂows (Zang- mined by means of the deterministic decision PTi À1 will, 1968) or integer programming (Karni, 1981). variable YTi after the demand k¼TiÀ1 dk is The second stage was then to calculate the realised. It follows from Eq. (10) that adjustments made to those orders at the times they were scheduled as the demand was realised X i X i X Ti À1 over the planning horizon. These adjustments were X Tj ¼ YTj þ dk ; i ¼ 1; y; m; ð11Þ expressed as margins to add to the total demand j¼1 j¼1 k¼1 received over all the periods since the immediately preceding order was received. and that the inventory level It at the end of period The Bookbinder and Tan analysis for the second t; given by Eq. (9), can be expressed as stage is brieﬂy as follows. Consider a review X i X t schedule, which has m reviews over the N period It ¼ I0 þ Y Tj À dk ; planning horizon with orders arriving at j¼1 k¼Ti fT1 ; T2 ; y; Tm g; whereTj > TjÀ1 ; Tm pN: Remem- Ti ptoTiþ1 ; i ¼ 1; y; m: ð12Þ ber that the Ti have been determined at the ﬁrst stage. For convenience T1 ¼ 1 is deﬁned as the Therefore, the random components of the closing start of the planning horizon and Tmþ1 ¼ N þ 1 inventory level, It ; depends only on the demand the period immediately after the end of the since the most recent stock review in period Ti ; horizon. The review schedule may be generalised i ¼ 1; y; m: Note that YTi is not a random to consider the case where T1 > 1; if the opening variable; rather, it is a deterministic decision stock, I0 ; is sufﬁcient to cover the immediate needs variable whose value is determined using a linear at the start of the planning horizon. The associated programming model. stock reviews will take place at the beginning of As in the static model, Bookbinder and Tan say periods Ti ; i ¼ 1; y; m: In the considered dynamic that the probabilistic constraints, review and replenishment policy, clearly the orders Xt are all equal to zero except at the replenishment X j À1 periods T1 ; T2 ; y; Tm : The inventory level It Yti XGdt þdt þ1 þdt þ2 þ?þdt ðaÞ À I0 ; j j j jþ1 À1 carried from period t to period t þ 1 is the opening i¼1 stock plus any orders that have arrived up to and j ¼ 1; 2; y; m; including period t less the total demand to date. Hence it is given as hold as an equality at optimality. By substituting À1 Pm X i X t Gdtm þdtm þ1 þdtm þ2 þ?þdt À1 ðaÞ À I0 for i¼1 Yti ; we mþ1 It ¼ I0 þ XTj À dk ; observe that their total unit variable cost expres- j¼1 k¼1 nP Ptm À1 o m sion, v i¼1 Yti þ i¼1 Efdi g ; takes a constant Ti ptoTiþ1 ; i ¼ 1; y; m: ð9Þ value. In other words, we can conclude that the Bookbinder and Tan transform the decision unit variable cost has no role in replenishment variable XTi to a new variable YTi AR; which policies determined according to the Bookbinder– may be interpreted as the change from period TiÀ1 Tan heuristic. ARTICLE IN PRESS 110 S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 4. An optimisation model for the probabilistic reviews are still variables to be determined and not lot-sizing problem preset values. P Deﬁning RTi ¼ I0 þ ij¼1 YTj and substituting The Bookbinder and Tan static-dynamic proce- for It ; Eq. (12) becomes dure splits the solution into two independent X t stages, ignoring the interactions between them. It It ¼ RTi À dk ; is therefore a heuristic solution procedure. The k¼Ti question then arises as to its accuracy and the Ti ptoTiþ1 ; i ¼ 1; y; m: ð13Þ likely level of cost penalty that it incurs. To evaluate the accuracy of the heuristic, the Note that RTi may be interpreted as an order-up- problem must be formulated in a way that gives to-level to which stock should be raised after the optimal solution under the ‘‘static–dynamic receiving an order at the ith review period Ti ; and P uncertainty’’ strategy. This will allow the simulta- RTi À t i dk is the end of period inventory. k¼T neous determination of the number and timing of Thus, instead of working in terms of decision the replenishments and the information necessary variables YTi ; as in the model proposed by to determine the size of the replenishment orders Bookbinder and Tan, the problem can be ex- that will minimise the expected costs of meeting pressed in terms of these new decision variables demand over some ﬁnite planning horizon, given a RTi : The problem is to determine the number of set of forecasts of the demands and a service-level reviews, m; the Ti ; and the associated RTi for i ¼ constraint on the probability of a stockout. This is 1; y; m: done in this section. It basically models the If there is no replenishment scheduled for period Bookbinder and Tan transformation process in a t; then Rt equals the opening inventory level in way that allows the simultaneous determination of period t: Now, Eq. (13) can be expressed more the timing and size of the orders as a single step. simply as The initial decisions depend upon what informa- I t ¼ R t À dt ; t ¼ 1; y; N: ð14Þ tion can be expected to become available in the future and how best to react to this information. It follows that the variable Rt must be equal to ItÀ1 Therefore, the replenishment times to use are if no order is received in period t and equal to the determined at the beginning of the planning order-up-to-level if there is a review and the receipt horizon considering the interdependency between of an order. The ﬁrst case applies if no stock the stock levels to have available at the start of review takes place in period t; which is indicated by those periods and their timing. The actual order the integer variable dt ¼ 0: If dt ¼ 0 then Rt must quantities at future replenishment periods are equal ItÀ1 : This is achieved by the two linear determined only at those replenishment periods inequalities given by Eq. (15), since the constraints and will depend upon the demand that is realised become Rt pItÀ1 and Rt XItÀ1 ; respectively, for period by period over the planning horizon. It is dt ¼ 0: assumed that negative orders are not allowed, so Rt À ItÀ1 pMdt ; that if the actual stock exceeds the order-up-to- t ¼ 1; y; N; ð15Þ Rt XItÀ1 ; level for that review, this excess stock will be carried forward and not returned to the supply where M is some very large positive number. source. Whilst if dt ¼ 1 then the constraints require Rt to lie between inﬁnity and ItÀ1 satisfying the other condition on Rt : The values for the order-up-to- 4.1. Reformulating the constraints of the level variables, Rt ; when dt ¼ 1 are then those that chance-constrained model give the minimum expected costs EfTCg: The desired opening stock levels, as required for the We start with Eq. (12) of Section 3, but with the solution to the problem, will then be those values difference that the number and timing of the of Rt for which dt ¼ 1: ARTICLE IN PRESS S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 111 It is, therefore, clear from the above explana- À1 zero elsewhere, then GdT þdT þ1 þ?þdt ðaÞ can be i i tions that constraints (6) and (7) can be replaced expressed as with Eqs. (15) and (14), respectively. X t As mentioned before, a is the desired minimum À1 À1 GdT þdT þ1 þ?þdt ðaÞ ¼ GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞPtj ; i i probability that the net inventory level in any time j¼1 period will actually be non-negative and deter- t ¼ 1; y; N; mined subjectively. With this regard the chance constraint and similarly Eq. (19) can be expressed as! Xt X t PrfIt X0gXa; t ¼ 1; y; N ð16Þ It X À1 GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ À dk Ptj ; j¼1 k¼tÀjþ1 can, using Eq. (13), be written alternatively as ( ) t ¼ 1; y; N: ð20Þ X t Pr RTi X dk Xa; t ¼ 1; y; N; ð17Þ The result Ptt ¼ 1 means that the stock review was k¼Ti in period 1, the start of the planning horizon, which implies whilst Pt1 ¼ 1 means that the stock review was at the start of period t itself. There can at most be only GdTi þdTi þ1 þ?þdt ðRTi ÞXa; one most recent order received prior to period t: Ti ptoTiþ1 ; i ¼ 1; y; m; ð18Þ Thus, the Ptj must satisfy and X t Ptj ¼ 1; t ¼ 1; y; N: ð21Þ X t j¼1 À1 It XGdT þdT þ1 þ:::þdt ðaÞ À dk ; i i k¼Ti Three other conditions as given below are neces- sary to identify uniquely the period in which the Ti ptoTiþ1 ; i ¼ 1; y; m: ð19Þ most recent review prior to any period t took The right-hand side of Eq. (19) can be calculated place. or possibly read from a table, once the form of gt ð:Þ Pt tÀjþ1 ¼ 1 and k¼tÀjþ2 dk ¼ 0; so all sub- * If d is selected. À1 sequent dk for k ¼ t À j þ 2; t À j þ 3; y; t are In Eq. (19), GdT þdT þ1 þ?þdt ð:Þ can only be i i 0, then we must have Ptj ¼ 1; as in these determined after the replenishment periods Ti circumstances period t À j þ 1 had the most have been determined. But, as these are chosen recent stock review prior to period t: Pt to minimise the expected costs, the stock replen- * If d tÀjþ1 ¼ 0 and k¼tÀjþ2 dk ¼ 0; then Ptj ¼ 0 ishment periods cannot be determined until the À1 since the most recent review prior to period t appropriate GdT þdT þ1 þ?þdt ð:Þ values to use in the i i must have been earlier than t À j þ 1: Pt model are known. There is an obvious circularity tÀjþ1 ¼ 1 and k¼tÀjþ2 dk X1; then Ptj ¼ 0; * If d here in trying to solve the problem. Since Book- since other reviews prior to period t occur after binder and Tan separate the determination of period t þ j À 1: the timing of the replenishment orders and the adjustments to those orders, they avoid the All of these three conditions can be satisﬁed by circularity by sacriﬁcing optimality. One way to the equality condition given in Eq. (21) and the overcome this problem, by not sacriﬁcing optim- single constraint given below, which are designed ality, is to formulate it as a mixed integer linear to identify uniquely the periods in which the most programming model. recent order prior to t took place: Since the problem has a ﬁnite planning horizon À1 of N periods, GdT þdT þ1 þ?þdt ð:Þ can be calculated X t i i for all relevant cases. If the 0/1 integer variable Ptj Ptj XdtÀjþ1 À dk ; k¼tÀjþ2 is deﬁned as taking a value of 1 if the most recent order prior to period t was in period t À j þ 1 and t ¼ 1; y; N; j ¼ 1; y; t: ð22Þ ARTICLE IN PRESS 112 S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 4.2. The mixed integer programming model to X t Ptj ¼ 1; t ¼ 1; y; N; ð28Þ determine the optimal policy under the static j¼1 dynamic uncertainty strategy X t The chance-constrained programming model Ptj XdtÀjþ1 À dk ; can be expressed as minimising the objective k¼tÀjþ2 function given earlier as Eq. (1) subject to the t ¼ 1; y; N; j ¼ 1; y; t; ð29Þ constraints given by Eqs. (14), (15), (20)–(22) and the non-negativity conditions and the 0/1 integer EfIt g; EfRt gX0; dt ; Ptj Af0; 1g; values for dt and Ptj : Since the decision rule for the t ¼ 1; y; N; j ¼ 1; y; t: above stochastic optimisation problem is chosen to be the ‘‘static–dynamic uncertainty’’ strategy, the This model thus determines the optimum number timing of the replenishments, obtained from dt and of replenishments as well as the optimum replen- Ptj ; must be decided once and for all, before any of ishment schedule, the timing of the replenish- the demands, dt ; become known. Therefore, the ments, together with the optimum values to use for expectation operator must be applied to the dynamically determining the sizes of the replen- stochastic variables It ; Rt and dt in the constraint ishment orders as demand is realised, that give the equations and objective function. In the chance- minimum expected total costs. The problem is to constrained programming model given below, the determine the values of the 0/1 integer variables, expected value of It and Rt are denoted by EfIt g dt for t ¼ 1; y; N and Ptj for j ¼ 1; y; t; t ¼ and EfRt g; respectively. 1; y; N; and the non-negative continuous vari- Such an analysis is completed at the beginning ables EfIt g and EfRt g for t ¼ 1; y; N; that of the horizon by taking expectations. Hence the minimise the objective function. To comply with deterministic equivalent model for the chance- the non-negativity constraint on the lot-sizes of the constrained programming model under the static- original model, we must have EfRt g À EfItÀ1 gX0; dynamic uncertainty strategy may be obtained by t ¼ 1; y; N which is already given in Eq. (15) and taking expectations (see Bookbinder and Tan). incorporated into the model. The times of the The model then is stock reviews are given by the values of i such that X N di ¼ 1: The associated order-up-to-levels, for each minimise EfTCg ¼ ðadt þ hEfIt g i; are given as EfRi g: t¼1 In Section 3, it was shown that the Bookbinder– þ vEfRt g À vEfItÀ1 gÞ; ð23Þ Tan heuristic ignores the unit variable cost in PN where the total expected direct item cost determination of the best schedule as in the PN t¼1 vEfXt g is written as deterministic problem. However, in contrast to t¼1 ðvEfRt g À vEfItÀ1 gÞ following the deﬁnition of order-up-to-levels, Rt ; the Bookbinder–Tan heuristic, the above formula- subject to tion does not ignore the unit variable cost. A rearrangement of the unit variable cost terms in EfIt g ¼ EfRt g À Efdt g; t ¼ 1; y; N; ð24Þ the objective function gives EfRt gXEfItÀ1 g; t ¼ 1; y; N; ð25Þ X N EfRt g À EfItÀ1 gpMdt ; t ¼ 1; y; N; ð26Þ v ðEfRt g À EfItÀ1 gÞ t¼1 X t X NÀ1 À1 EfIt gX GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ ¼ ÀvI0 þ v ðEfRt g À EfIt gÞ þ vEfRN g j¼1 t¼1 ! X t X N À1 À Efdk g Ptj ; t ¼ 1; y; N; ð27Þ ¼ ÀvI0 þ v ðEfdt gÞ þ vEfRN g k¼tÀjþ1 t¼1 ARTICLE IN PRESS S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 113 using Eq. (24) for t ¼ 1; y; N À 1: Thus, one can Table 2, the calculated values of the coefﬁcient write the total unit variable cost component as f þ À1 GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ are given. For instance, ele- vEfRN g; where f is a constant with a value equal ment (t ¼ 7; j ¼ 3) gives the value of to v multiplied by the given opening stock plus the À1 Gd5 þd6 þd7 ð0:95Þ ¼ 2833 which corresponds to the expected demand over the ﬁrst N À 1 periods. It is opening inventory level in period t À j þ 1 ¼ 5 to clear that the total unit variable cost is a function satisfy the demands of periods 5–7 with prob- of the model variable EfRN g; which is a variable ability of at least a ¼ 95%: These values can be rather than a constant. Thus the solution will be calculated using any spreadsheet program. The affected by the variable production cost. solution to the problem is calculated using both BT heuristic and the optimisation model derived in the previous section, denoted by (TK). 5. A numerical example The ﬁrst step in the two-step BT heuristic is the determination of the replenishment times. The To illustrate the technique we shall use the replenishment times are ﬁxed by means of the demand forecasts presented in Table 1. The initial ‘‘static uncertainty’’ model, applying the Wagner– inventory level is taken as zero. It is assumed that Whitin algorithm to the problem formulated as a the demand in each period is normally distributed deterministic one using the analogy of Bookbinder about the forecast value with a constant coefﬁcient and Tan. This gives periods 1, 5 and 7 as the of variation, C ¼ st =mt ¼ 0:333: The other para- replenishment periods. Following that the ‘‘static– meters of the problem are a ¼ $2500 per order, dynamic uncertainty’’ model is used to determine h ¼ $1 per unit per period, and a ¼ 0:95 the order-up-to-levels for these replenishment (za¼0:95 ¼ 1:645). Since BT heuristic ignores the periods. The results are given in Table 3 and Fig. 1. unit variable cost, the example takes into account The same problem is also solved by means only the holding and ordering costs (i.e., v ¼ 0). In of the deterministic equivalent mixed integer Table 1 Forecasts of period demands Period (k) 1 2 3 4 5 6 7 8 9 10 Efdk g 800 850 700 200 800 700 650 600 500 200 Table 2 P P 1=2 Calculated GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ ¼ t À1 k¼tÀjþ1 E fdk g þ z0:95 C t 2 k¼tÀjþ1 E fdk g values t j 1 2 3 4 5 6 7 8 9 10 1 1239 2 1316 2290 3 1084 2154 3096 4 310 1299 2364 3304 5 1239 1452 2293 3304 4223 6 1084 2083 2293 3106 4096 5003 7 1006 1874 2833 3042 3841 4818 5718 8 929 1735 2568 3508 3716 4507 5475 6370 9 774 1528 2307 3127 4056 4264 5050 6013 6904 10 310 995 1742 2518 3335 4264 4471 5256 6219 7110 ARTICLE IN PRESS 114 S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 Table 3 Replenishment policy of BT Period (t) 1 2 3 4 5 6 7 8 9 10 Order-up-to-level (EfRt gdt ¼1 ) 3304 — — — 2083 — 2518 — — — Exp. opening inv. (EfRt g) 3304 2504 1654 954 2083 1283 2518 1868 1268 768 Exp. closing inv. (EfIt g) 2504 1654 954 754 1283 583 1868 1268 768 568 Total expected cost 19,704 3500 arrive in periods 1, 3, 5 and 8. As stated the 3000 opening stock at the time of a replenishment is Inventory Level 2500 known and set by the policy. However, the sizes of 2000 the replenishment orders are determined dynami- 1500 1000 cally as demand is realised. A given period’s lot- 500 size, Rt À ItÀ1 ; cannot be found until the realised 0 demand is known. In other words one waits for the 1 2 3 4 5 6 7 8 9 10 11 demands to become known, having decided in Period advance how this knowledge will be used. Simi- Fig. 1. Replenishment policy of BT. larly the actual opening stocks realised in the intermediate periods would depend on the realised demands. On the assumption of independence of programming model presented in Section 4.2. The successive period demands, Figs. 1 and 2 show results are presented in Table 4 and Fig. 2. It is what the average opening stock would be if the seen that this optimal solution has four rather than actual 10 period situation were repeated many three replenishments which occur in periods 1, 3, 5 times. and 8. In the above example, the unit variable cost has Figs. 1 and 2 show the differences between the been set to zero. At this point, the effect of having near optimal BT and the optimal TK policies unit variable cost in the problem on the inventory under the ‘‘static–dynamic uncertainty’’ strategy. policy should be investigated. To do so, the above The probability of shortage occurring in each problem is solved again for v ¼ 4 using both BT time period is given for both policies in Table 5. and TK. Since BT heuristic ignores the unit Since the probability of a shortage occurring variable cost in both the ‘‘static uncertainty’’ and never exceeds 5% in any period, it is clear from ‘‘static–dynamic uncertainty’’ models, the replen- Table 5 that both policies satisfy the service-level ishment periods, which were {1,5,7}, and the constraints that at the end of each period the calculated expected lot-sizes remain the same. probability that the net inventory will not be Hence, Fig. 1 still depicts the replenishment policy negative is at least a ¼ 95%: The heuristic BT for BT. On the other hand, TK approach replaces method costs 1.55% more than the expected cost the replenishment periods f1; 3; 5; 8gv¼0 with of the optimal policy. f1; 3; 5; 7; 9gv¼4 : Fig. 3 shows the revised replen- The opening stocks in Fig. 1 (the BT heuristic ishment policy. A comparison of the statistics for policy) for periods 1, 5 and 7 when replenishments both approaches is given in Table 6. arrive are values given by the policy. The opening From Table 6, the difference between the stocks for intermediate periods are expected expected total inventory costs of BT and TK values, assuming that the period demands occur solutions for v ¼ 0 is $300 ($19,704–$19,404). The at their expected value. It is the same for Fig. 2 introduction of unit variable cost, v ¼ 4; increases (the optimal TK solution) where replenishments this difference to $939 ($45,975–$45,036). ARTICLE IN PRESS S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 115 Table 4 Replenishment policy of TK Period (t) 1 2 3 4 5 6 7 8 9 10 Order-up-to-level (EfRt gdt ¼1 ) 2290 — 1299 — 2833 — — 1742 — — Exp. opening inv. (EfRt g) 2290 1490 1299 599 2833 2033 1333 1742 1142 642 Exp. closing inv. (EfIt g) 1490 640 599 399 2033 1333 683 1142 642 442 Total expected cost 19,404 3500 optimal solution for v ¼ 0 (v ¼ 4). It is useful to 3000 gain some idea of how well it would perform over Inventory Level 2500 a wider set of examples. This section presents the 2000 1500 results of a cost comparison of the BT heuristic 1000 with the optimal solution, the TK model, for a 500 wide range of problems. The planning horizon is 0 set to 20 periods with no initial inventory in all 1 2 3 4 5 6 7 8 9 10 11 the experiments. The service level is set at a Period constant a ¼ 0:95 for all periods and the ordering Fig. 2. Replenishment policy of TK. cost a ¼ $1000 per order. The unit variable cost is ignored. The problems selected are deﬁned in terms of: Table 5 Probability of shortage (%) * The holding cost, which will affect the average number of periods covered by an order, the Period 1 2 3 4 5 6 7 8 9 10 values being $1, $2, $3, $4, $5, $7.5 or $15 per BT policy 0.0 0.0 1.8 5.0 0.0 5.0 0.0 0.0 1.7 5.0 unit per period. TK policy 0.0 5.0 0.5 5.0 0.0 0.0 5.0 0.0 0.7 5.0 * The coefﬁcient of variation, showing the effect of the size of random variation in demand about the mean. The values selected were 1/3, 1/4, 1/5 and 1/10 and were the same for each 3500 3000 period’s demand. 2500 * The pattern of the mean demands over time. Inventory Level 2000 There are four different demand patterns taken 1500 from Berry (1972), as given in Figs. 4a–d. These 1000 range from a constant level, through a contin- 500 uous sinusoidal changes to a very erratic 0 1 2 3 4 5 6 7 8 9 10 11 pattern. Demand is normally distributed about Period the forecast value under the non-stationarity Fig. 3. Replenishment policy of TKv¼4 : assumption. The number of test problems, generated for seven different holding costs, four different coefﬁ- 6. Numerical comparisons cient of variations and four different mean data sets, therefore totals to 112. For each test problem In the illustrative example in the previous the percentage cost difference between the BT section, it is seen that the Bookbinder and Tan heuristic and the optimal solution (TK method) is heuristic costs 1.55% (2.09%) more than the calculated and the results are listed in Table 7. ARTICLE IN PRESS 116 S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 Table 6 A comparison of v ¼ 0 and 4 cases BTv=0 BTv=4 TKv=0 TKv=4 Replenishment periods {1,5,7} {1,5,7} {1,3,5,8} {1,3,5,7,9} E(average inventory level) 1520.4 1520.4 1240.3 1035.4 E(average buffer stock level) 64.5 64.5 54.5 48.0 Total E(order quantity) 6568 6568 6442 6295 E(total cost) 19,704 45,975 19,404 45,036 Mean Demand Set 1 Mean Demand Set 2 120 200 100 150 80 60 100 40 50 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Mean Demand Set 3 Mean Demand Set 4 400 500 300 400 300 200 200 100 100 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fig. 4. Demand patterns. The results indicate that as the demand becomes gives the optimal solution in 36% of cases. The more erratic, i.e. more non-stationary, either in average cost penalty is 1.24%, similarly to the ﬁrst terms of the mean demand pattern or the size of two data sets. However, the worst cost penalty is the coefﬁcient of variation, the cost penalty above signiﬁcantly higher at 9.5%. In the highly erratic the optimal solution from using the BT heuristic demand data set 4, the BT heuristic is only optimal tends to become larger. For the stationary case, in 11% of cases. The average penalty cost incurred data set 1, there are 78% of cases where the BT is 2.4%, so much higher than the other three cases. heuristic gives the optimal solution. The average The worst penalty is 9.3%, much the same as for penalty cost above the optimal solution incurred is data set 3. 0.5%, whilst the worst penalty cost is 3.3%. For The percentage cost difference between BT and data set 2, the sinusoidal pattern to demand over the optimal solution tends to become smaller as time, the BT heuristic gives the optimal solution the coefﬁcient of variation decreases. This is easy for 14% of cases. The average cost penalty is again to explain. The suboptimality of the BT heuristic only 1.3%, whilst the worst penalty is 3.8%, very arises from the ﬁrst step of the method. Since the similar to case 1. For the more erratic almost replenishment periods are ﬁxed under the ‘‘static sinusoidal pattern of data set 3 the BT heuristic uncertainty’’ strategy and the dynamic nature of ARTICLE IN PRESS S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 117 Table 7 Percentage cost increases of BT above TK solutions (for v ¼ 0) s=m h¼1 h¼2 h¼3 h¼4 h¼5 h ¼ 7:5 h ¼ 15 Set 1 1/3 0.0 0.0 2.6 0.0 0.0 3.3 0.0 1/4 0.0 2.5 2.6 0.0 0.0 0.2 0.0 1/5 0.0 0.0 1.8 0.0 0.0 0.0 0.0 1/10 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Set 2 1/3 1.9 3.5 3.8 2.2 3.3 1.6 0.1 1/4 1.0 2.5 2.6 1.7 2.3 0.8 0.0 1/5 0.5 2.0 2.0 1.3 1.4 0.3 0.0 1/10 0.0 0.4 0.8 0.4 0.0 0.1 0.0 Set 3 1/3 0.8 3.3 9.5 0.1 1.1 0.7 0.0 1/4 0.1 2.0 6.8 0.0 0.5 0.5 0.0 1/5 0.0 1.1 5.1 0.0 0.1 0.4 0.0 1/10 0.0 0.0 2.3 0.0 0.1 0.1 0.0 Set 4 1/3 5.6 1.0 1.9 5.9 9.3 2.1 0.2 1/4 4.3 0.3 0.7 3.8 7.5 1.6 1.0 1/5 3.5 0.0 1.4 2.4 6.1 1.4 0.9 1/10 1.5 0.0 0.6 0.0 2.6 0.4 0.4 the problem is ignored, in general the replenish- cost performance of the heuristic changes drama- ment schedule is not optimal. Therefore, a tically. decrease in the coefﬁcient of variation improves If the unit variable cost is ignored then the the performance of the ‘‘static uncertainty’’ stochastic lot-sizing problem can be modelled as a approach and yields a better replenishment sche- stochastic form of the Wagner–Whitin problem dule. It should be noted that if the replenishment and solved by a shortest route algorithm, where schedule is optimal, then the second step of the the arc cost (i; j) corresponds to the minimum cost method produces the optimal solution for the of placing an order in period i to cover the next adjustments to be made in the lot-sizes. j À i þ 1 periods and satisfy the service-level In all four demand data sets it can be seen constraint. However, the shortest route approach clearly that the cost penalty increases as the considers the replenishment cycles independent of coefﬁcient of variation increases. There is no such each other, and as a consequence of this the simple pattern as the holding cost increases. For closing stock of one cycle may be above the example, in the deterministic demand data set 1, opening stock for the next cycle. It is clear that this the BT heuristic gives the optimal solution for the gives a negative replenishment level, and therefore, smallest, middle and largest holding costs but not is an infeasible solution. Such infeasible solutions for all of the others. This is mainly due to the are observed particularly in the erratic demand discrete nature of the model used. The replenish- case, demand data set 4. In this case, the shortest ment schedule obtained by the ‘‘static uncertainty’’ route method gives infeasible solution for all model will remain unaltered over some range problems when s=m is in {1/3,1/4,1/5} and hX14: for h: As a result of this, for a speciﬁc replenish- This does not happen with the Bookbinder and ment schedule the cost performance deteriorates as Tan heuristic nor the optimal mixed integer h approaches to the limits of its range. One programming model presented in this paper. observes that at the limits the ‘‘static uncertainty’’ Moreover, if the unit variable cost cannot be ig- model gives new replenishment policies and the nored, then it is not possible to treat replenishment ARTICLE IN PRESS 118 S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 cycles independently and therefore to apply the treating the replenishment cycles independently shortest route algorithm. and applying the shortest route algorithm. All experiments were done on a 1.2 GHz Although, it has been assumed that the replen- Pentium III, 512 MB RAM machine. The gener- ishment lead-time is zero, it is possible to extend al-purpose solver CPLEX 8.0 (2002) is used with the model for the non-zero replenishment lead- the default settings and in all cases the optimal time situation without any loss of generality. A solution to the mixed integer programming model similar model, incorporating the shortage cost, in is found in less than a quarter of a minute. place of service-level constraints is currently being Although the BT heuristic provides an almost developed by the authors. Further work could be immediate solution, since the computation time invested in evaluating the performance of the required by the TK model is not excessive, it may optimal ‘‘static–dynamic uncertainty’’ strategy in be worth having the optimal solution. the rolling horizon environment compared to Bookbinder and Tan’s heuristic. 7. Conclusions In this paper, the stochastic dynamic lot-sizing References problem with service-level constraints has been modelled under the ‘‘static–dynamic uncertainty’’ Askin, R.G., 1981. A procedure for production lot-sizing with probabilistic dynamic demand. AIIE Transactions 13, strategy of Bookbinder and Tan. A mixed integer 132–137. programming model for the approach has been Berry, W.L., 1972. Lot sizing procedures for requirements formulated. This gives the optimal solution allow- planning systems: A framework for analysis. Production ing the simultaneous determination of the number and Inventory Management Journal 13, 19–34. and timing of the replenishments and the informa- Birge, J.R., Louveaux, F., 1997. Introduction to Stochastic tion necessary to determine the size of the Programming. Springer, New York. Bookbinder, J.H., Tan, J.Y., 1988. Strategies for the probabil- replenishment orders, from the replenishment istic lot-sizing problem with service level constraints. levels for the periods when stock reviews will take Management Science 34, 1096–1108. place. Unlike the Bookbinder and Tan model this Chen, F.Y., Krass, D., 2001. Inventory models with minimal new MIP model includes a unit variable purchas- service level constraints. European Journal of Operational ing/production cost. This model allows an estima- Research 134, 120–140. Davis, T., 1993. Effective supply chain management. Sloan tion of the accuracy of the Bookbinder and Tan Management Review 34, 35–46. heuristic for solving the ‘‘static–dynamic uncer- De Bodt, M.A., Gelders, L.F., Van Wassenhove, L.N., 1984. tainty’’ approach to be made, by setting the unit Lot-sizing under dynamic demand conditions: A review. cost equal to zero. If the demand data sets, Engineering Costs and Production Economics 8, 165–187. Gavirneni, S., Tayur, S., 2001. An efﬁcient procedure for non- coefﬁcients of variation and relative holding cost stationary inventory control. IIE Transactions 33, 83–89. to ordering values used in the numerical experi- Iida, T., 1999. The inﬁnite horizon non-stationary stochastic ments could be regarded as typical of what occurs inventory problem: Near myopic policies and weak in practice we could conclude that the BT heuristic ergodicity. European Journal of Operational Research has a cost performance that is close to the optimal 116, 405–422. solution. Overall, it gave the optimal solution in ILOG CPLEX 8.0, 2002. User’s Manual, ILOG. Karni, R., 1981. Integer linear programming formulation of the 36% of cases and had an average penalty cost of material requirements planning problem. Journal of Opti- 1.34%. Compared to the optimal solution, which mization Theory and Applications 35, 217–230. is a mixed integer programming model, the Kimms, A., 1997. Multi-Level Lot-Sizing and Scheduling solution times are fast. However, the penalty cost Methods for Capacitated Dynamic and Deterministic for the BT heuristic will be higher for a non-zero Models. Physica Verlag Series on Production and Logistics. Springer, Berlin. unit purchase/production cost. Moreover, in such Kuik, R., Salomon, M., Van Wassenhove, L.N., 1994. Batching cases the problem cannot be modelled as a decisions: Structure and models. European Journal of stochastic form of the Wagner–Whitin problem, Operational Research 75, 243–263. ARTICLE IN PRESS S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119 119 Martel, A., Diaby, M., Boctor, F., 1995. Multiple items Sox, C.A., 1997. Dynamic lot-sizing with random demand and procurement under stochastic nonstationary demands. non-stationary costs. Operations Research Letters 20, European Journal of Operational Research 87, 74–92. 155–164. Potts, C.N., Van Wassenhove, L.N., 1992. Integrating schedul- Van Houtum, G.J., Zijm, W.H.M., 2000. On the relation ing with batching and lot-sizing: A review of algorithms and between cost and service models for general inventory complexity. Journal of the Operational Research Society 43, systems. Statistica Neerlandica 54, 127–147. 395–406. Wagner, H.M., Whitin, T.M., 1958. Dynamic version Rosling, K., 1999. The square-root algorithm for single-item of the economic lot size model. Management Science 5, inventory optimization. Working Paper, Lund University. 89–96. Silver, E.A., 1978. Inventory control under a probabilistic time- Wemmerlov, U., 1989. The behaviour of lot-sizing procedures varying demand pattern. AIIE Transactions 10, 371–379. in the presence of forecast errors. Journal of Operations Sobel, M.J., Zhang, R.Q., 2001. Inventory policies for systems Management 8, 37–47. with stochastic and deterministic demand. Operations Zangwill, W.I., 1968. Minimum concave cost ﬂows in certain Research 49, 157–162. networks. Management Science 14, 429–450.