Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times

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1 44 Int. J. Inventory Research, Vol. 1, No. 1, 2008 Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times Søren Glud Johansen Department of Operations Research, University of Aarhus, Ny Munkegade, Bldg. 530, DK-8000 Aarhus C, Denmark Anders Thorstenson* CORAL Centre for OR Applications in Logistics, Department of Business Studies, Aarhus School of Business, University of Aarhus, Fuglesangs Allé 4, DK-8210 Aarhus V, Denmark *Corresponding author Abstract: We extend well-known formulae for the optimal base stock of the inventory system with continuous review and constant lead time to the case with periodic review and stochastic, sequential lead times. Our extension uses the notion of the extended lead time. The derived performance measures are exact for Poisson demands. Keywords: inventory control; optimisation; periodic review; sequential lead times; single echelon; single item; stochastic demand. Reference to this paper should be made as follows: Johansen, S.G. and Thorstenson, A. (2008) Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times, Int. J. Inventory Research, Vol. 1, No. 1, pp Biographical notes: Søren Glud Johansen is an Associate Professor at the Department of Operations Research, University of Aarhus, Denmark. His main research interests are in production economics, inventory control, Markov decision theory, and costing and pricing. He has published in Advances in Applied Probability, European Journal of Operational Research, International Journal of Production Economics, Journal of the Operational Research Society and Operations Research. Anders Thorstenson is a Professor and Director of the research centre CORAL at Aarhus School of Business, Denmark. His main research interests are in supply chain management, production and inventory control, operations management and performance management. He has published in European Copyright 2008 Inderscience Enterprises Ltd.

2 Optimal base-stock policy 45 Journal of Operational Research, International Journal of Production Economics, International Journal of Production Research, Journal of the Operational Research Society and Managerial and Decision Economics. 1 Introduction We consider a single-item inventory system with periodic review. We assume that the system is controlled by a base-stock policy and that unsatisfied demands are backordered and satisfied as soon as possible. The inventory position (stock on hand + stock on order amount backordered) is monitored at each review instant, when the policy places a replenishment order to restore the base stock s. This type of system is characteristic of, e.g., frequently used spare parts in the automotive industry and some durable consumer goods in the retail industry. The periodic control feature might also be attractive for coordination purposes when the single-item inventory system is embedded in a larger system consisting of multiple items and/or echelons. We assume that demands occur in continuous time and are generated by a Poisson process with rate λ. The holding and shortage costs are incurred accordingly as in Hadley and Whitin (1963, Section 5-3). It is common in models of periodic review systems to assume that demands occur only periodically. Such models can be analysed by direct analogy to continuous review systems; see, for example, Axsäter (2006, Section 5.12). However, as stated in Rosling (2002), demand arriving in continuous time is more realistic. The base-stock policy with review interval R is commonly referred to as an (R, S) policy, where S = s. This policy is frequently used in practice, and it is often applied as a benchmark in theoretical studies. Furthermore, for the inventory system described and with a standard cost structure, the (R, S) policy is the optimal control policy (Zipkin, 2000, Section ). Note, however, that in our model costs are incurred in continuous time. Furthermore, we assume that the lead times L of replenishment orders are generated by an exogenous, sequential supply system (Zipkin, 2000, Section 7.4). This assumption implies that different replenishment orders do not cross in time. Similar ideas have been applied in, e.g., Hadley and Whitin (1963, Chapters 4 5), Kaplan (1970), and Ehrhardt (1984). The sequential supply system is formalised by Zipkin (1986). Our contribution is the provision of necessary and sufficient optimality conditions under stochastic, sequential lead times by using the notion of the extended lead time and the analogy to a continuous-review system. The rest of this paper is organised as follows. In Section 2 we introduce the notion of the extended lead time and characterise the probability mass function (pmf) of extended lead-time demand. Performance measures and optimality conditions for the periodic review system considered are provided in Section 3, which also includes a numerical example. Section 4 contains concluding remarks.

3 46 S.G. Johansen and A. Thorstenson 2 The extended lead time Like Zipkin (2000, Section 6.7.3) and Rao (2003), we model the times from the demand instants until the next review instant as realisations of a random variable U which, because of the Poisson demand process, is uniformly distributed over the interval from 0 to R. Similar approaches are used in Hadley and Whitin (1963, Section 5-3) and Axsäter (1993). The generic variable L' for the time from a demand instant until the delivery instant of the replenishment order triggered by this demand can be specified as U + L. Hadley Whitin, Zipkin and Rao assume that lead time L is constant, while Axsäter considers lead-time uncertainty due to possible shortages at the upstream level in a two-echelon system with no order crossing allowed. Hence, these approaches all involve special cases of the sequential supply system referred to in the Introduction. We call L' the extended lead time and observe that like the ordinary lead times L, which we allow to be stochastic, the extended lead times for customer demands can also be seen as generated by an exogenous, sequential supply system. This observation provides that the exact results derived for the corresponding inventory system with continuous review can be used to derive similar results for the system with periodic review (see also a similar remark in Yano, 1985). The results for the latter system are then obtained by focusing on customer demands and letting the lead times in the former system be distributed as L' rather than as L. This way the net inventory and hence the cost of the base-stock policy in the periodic review system are replicated exactly by the continuous review system in which the inventory position is constant at the base-stock level s. The generic variable D L for the demand during the extended lead time equals D U + D L, where D U is the generic variable for the demand during an interval of random length U and D L is the generic variable for the ordinary lead-time demand. The pmf of D U is R x y 1 λu λu λr 1 ( ) ( ) λr fu ( x) = e du e, x 0,1, R = = (1) x! y! 0 y= x+ 1 The pmf f U (x) is strongly unimodal. A proof is provided in the Appendix. Assuming that lead time L is gamma distributed, D L has the negative binomial distribution with parameters a = (E[L]) 2 /Var[L] and p = λ/(λ + a/e[l]) (Zipkin, 2000, Section ). Hence, if Var[L] > 0, we have the following useful recursion for the pmf of D L α 1+ y fl( y) = pfl( y 1), y = 1,2,, (2) y where f L (0) = (1 p) α. If Var[L] = 0, D L has the Poisson distribution with mean λe[l]. In either case, the pmf f L (y) is also strongly unimodal (Keilson and Gerber, 1971). Finally, because D L is the convolution of D U and D L, the pmf of D L is z f ( z) = f ( x) f ( z x), z = 0,1, (3) L U L x= 0 The following proposition summarises the conditions for strong unimodality of f L (z).

4 Optimal base-stock policy 47 Proposition: If demand is Poisson and if (ordinary) lead times are gamma distributed, then the pmf specified in equation (3) of the extended lead-time demand is strongly unimodal. Proof: The result follows immediately from Theorem 2 in Keilson and Gerber (1971) which states that the pmf of a convolution of strongly unimodal pmfs is also strongly unimodal. Remark: Obviously, the proposition is true for any lead-time distribution that results in a strongly unimodal pmf of (ordinary) lead-time demand. 3 Performance measures Because of the equivalence to the continuous-review system with sequential extended lead times, important performance measures of the considered periodic inventory system can now be obtained using the standard arguments in Zipkin (2000, Corollary 7.4.3). Hence, the stock-out frequency is A() s = Pr{ D s}, (4) L the average backorders are Bs ( ) = E[max{ D s,0}], (5) L and the average inventory is I() s = s E[ D ] + B() s = E[max{0, s D }]. (6) L L Let h and b denote the unit holding and backorder costs per unit time, respectively, and let π denote the cost incurred for each unit backordered. If there are no further costs, then the long-run average cost incurred per unit time is Cs () = hi() s+ bbs () + πλ As (). (7) The base-stock model with this cost structure corresponds to Model 2, Case (ii) in Rosling (2002). Hence, based on his Proposition 2-2, we may conclude that C(s) is quasi-convex in s, because the pmf of D L is strongly unimodal, as stated in the Proposition above. (Quasi-convexity of C(s) is equivalent to the negative of C(s) being unimodal.) Then, for Cs ( ) = Cs ( + 1) Cs ( ) = ( h+ b) Pr{ D s} ( b+ πλ Pr{ D = s}), (8) the optimal base-stock level is obtained as L L s* = min{ s C( s) > 0}. (9) An example is depicted in Figure 1 which shows that C(s) is quasi-convex in s. Figure 1 also shows the misrepresentation that occurs if lead times were assumed to be independent, implying by Palm s theorem (Palm, 1938) that the approximate cost rate

5 48 S.G. Johansen and A. Thorstenson C a (s) depends on the distribution of L' only through its mean E[L ] = R/2 + E[L]. Hence, the approximate cost rate is given as C () s = hi () s + bb () s + πλ A (), s (10) a a a a where the performance measures Aa(), s Ba() s and Ia() s are specified analogously to A(), s B() s and I() s but with (the stochastic) L replaced everywhere by its (constant) mean E[L ]. Figure 1 The base-stock cost-rate function C(s) and its approximation C a (s) assuming independent lead times. Example with parameter values h = 2, b = 5, π = 15, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 For the example depicted in Figure 1, the optimal base-stock levels and their corresponding cost rates are shown in Table 1. This table also contains base-stock levels and cost rates obtained in two special cases discussed below. The shortage costs have been chosen such that for sequential lead times the same optimal base-stock level is obtained in all three cases. Note that in all cases the approximate approach assuming independent lead times yields a misrepresentation both in the optimal base-stock level and in the cost rate. The relative magnitude of these misrepresentations is not negligible. In particular, the optimal base-stock level and its associated cost rate are lower for independent lead times than for sequential lead times. This is a consequence of the fact that lead-time variability has no impact when lead times are independent, as also remarked in Svoronos and Zipkin (1991). A similar effect is noted in Robinson et al (2001) and in Bradley and Robinson (2005) for the periodic review and demand system with order crossing due to independent lead times. Table 1 Examples with fixed parameter values h = 2, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 b = 5, π = 15 b = 21, π = 0 b = 0, π = 19 Example s* C(s*) C a (s*) s* C(s*) C a (s*) s* C(s*) C a (s*) Sequential lead times Independent lead times (approximation)

6 Optimal base-stock policy 49 For π = 0, we obtain the extended standard solution b s* = min s Pr{ DL s} >. h+ b The exact and approximate cost rates in this special case are presented in Figure 2 for the example in Figure 1 with the adjusted shortage costs. In this case, it can easily be inferred from equation (8), and analogously for the approximate cost rate, that both cost rates are in fact convex discrete functions of the base-stock level. This is also illustrated by the two cost functions shown in Figure 2. The optimal solutions and their associated costs are displayed in Table 1. In this particular example, the cost of using the approximate solution exceeds the optimal cost by approx. 29%. (11) Figure 2 The base-stock cost-rate function C(s) and its approximation C a (s) assuming independent lead times. Example with parameter values h = 2, b = 21, π = 0, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 As another special case, consider the case when b = 0 (and π > 0). Equation (9) may then be written as Pr( DL = s) h s* = min s <. Pr{ DL s} πλ The ratio Pr{ DL = s} Pr{ DL s}, is non-increasing in s, because the pmf of D L is strongly unimodal (Rosling, 2002). Note that equation (12) is an extension to the case of periodic review and stochastic lead times of the expression found in the well-known textbook by Silver et al. (1998, Section 8.3.1) in the case of continuous review and a constant lead time. The exact and approximate cost rates in this special case are presented in Figure 3 for the example in Figure 1 with the adjusted shortage costs. In this case, however, it cannot be concluded that the cost rates are convex discrete functions of the base-stock level. This is also illustrated by the two cost functions shown in Figure 3. It is a consequence of having the unit backorder cost π > 0 in the expressions for the cost rates. The optimal solutions and their associated costs are displayed in Table 1. In this particular example, the cost of using the approximate solution exceeds the optimal cost by approx. 9%. (12)

7 50 S.G. Johansen and A. Thorstenson The optimality conditions in equations (9), (11), and (12) are intuitively appealing, since they are obtained from extending the standard continuous-review formulae by simply replacing D L, the demand during the ordinary lead time, by D L, the demand during the extended lead time. However, the numerical examples have illustrated that considering the extended lead time only through its mean, as when assuming that lead times are independent, may lead to considerable misrepresentation of both the optimal base-stock level and the corresponding cost rate. Figure 3 The base-stock cost-rate function C(s) and its approximation C a (s) assuming independent lead times. Example with parameter values h = 2, b = 0, π = 19, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 4 Concluding remarks In this study the extended lead time has been defined as the time between a customer demand instant and the delivery instant of the replenishment order triggered by that demand. By applying this concept, exact performance measures for the base-stock inventory system with continuous review have been extended to a system with periodic review and stochastic, sequential lead times. Simple conditions for optimisation have also been derived. The crucial assumption for these results is that lead times are sequential so that orders do not cross in time. The results obtained can be generalised further to accommodate compound Poisson demand (Johansen and Thorstenson, 2005). Acknowledgments This work was developed while the second author was visiting at the Department of Logistics, Hong Kong Polytechnic University. The authors wish to thank two anonymous referees whose comments significantly improved the presentation of the paper.

8 References Optimal base-stock policy 51 Axsäter, S. (1993) Optimization of order-up-to S policies in two-echelon inventory systems with periodic review, Naval Research Logistics, Vol. 40, pp Axsäter, S. (2006) Inventory Control, 2nd ed., Springer s International Series, New York. Bradley, J.R. and Robinson, L.W. (2005) Improved base-stock approximations for independent stochastic lead times with crossover, Manufacturing and Service Operations Management, Vol. 7, No. 4, pp Ehrhardt, R. (1984) (s, S) policies for a dynamic inventory model with stochastic lead times, Operations Research, Vol. 32, pp Hadley, G. and Whitin, T.M. (1963) Analysis of Inventory Control Systems, Prentice-Hall, Englewood Cliffs, NJ. Johansen, S.G. and Thorstenson, A. (2005) Base-Stock Policy for the Lost-Sales Inventory System with Periodic Review, Working Paper, available from the authors on request. Kaplan, R.S. (1970) A dynamic inventory model with stochastic lead times, Management Science, Vol. 16, No. 7, pp Keilson, J. and Gerber, H. (1971) Some results for discrete unimodality, Journal of the American Statistical Association, Vol. 66, pp Palm, C. (1938) Analysis of the Erlang traffic formula for busy signal assignment, Ericsson Technics, Vol. 5, pp Rao, U.S. (2003) Properties of the periodic review (R, T) inventory control policy for stationary, stochastic demand, Manufacturing and Service Operations Management, Vol. 5, No. 1, pp Robinson, L.W., Bradley, J.R. and Thomas, L.J. (2001) Consequences of order crossover under order-up-to policies, Manufacturing and Service Operations Management, Vol. 3, No. 3, pp Rosling, K. (2002) Inventory cost rate functions with nonlinear shortage costs, Operations Research, Vol. 50, No. 6, pp Silver, E.A., Pyke, D.F. and Peterson, R. (1998) Inventory Management and Production Planning and Scheduling, 3rd ed., John Wiley & Sons, New York. Svoronos, A. and Zipkin, P.H. (1991) Evaluation of one-for-one replenishment policies for multiechelon inventory systems, Management Science, Vol. 37, No. 1, pp Yano, C.A. (1985) New algorithms for (Q, r) systems with complete backordering using a fill-rate criterion, Naval Research Logistics Quarterly, Vol. 32, pp Zipkin, P.H. (1986) Stochastic leadtimes in continuous-time inventory models, Naval Research Logistics Quarterly, Vol. 33, pp Zipkin, P.H. (2000) Foundations of Inventory Management, McGraw-Hill, Boston, MA. Appendix Lemma: The pmf f U (x) as defined in Section 2 is strongly unimodal. Proof: By Theorem 3 in Keilson and Gerber (1971), a necessary and sufficient condition for the pmf of the discrete variable D U to be strongly unimodal is that it is log-concave, i.e., that 2 [ fu( x)] fu( x 1) fu( x 1), x 1,2, + = (13)

9 52 S.G. Johansen and A. Thorstenson The proof is by contradiction: Assume that condition (13) is not true. From equation (1) we then have for some x that 2 y 1 y 1 y 1 ( λr) λr ( λr) λr ( λr) λr e < e e y= x+ 1 y! y= x+ 2 y! y= x y! ( λr) ( λr) ( λr) ( λr) ( x 1)! y! y! x! y y 1 ( λr) λr ( λr) λr e < e. x+ 1 y= x+ 1 y! y= x+ 1 ( y+ 1)! x y 1 y 1 x 1 λr λr e < e + y= x+ 1 y= x+ 2 As can be inferred from its last member, condition (14) is not satisfied. Hence, the assertion is false and condition (13) must be true. (14)

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