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

Transcription

1 WORKING PAPER L Søren Glud Johansen & Anders Thorstenson Note: Optimal base-stock polic for the inventor sstem with periodic review, backorders and sequential lead times

2 Note: Optimal base-stock polic for the inventor sstem with periodic review, backorders and sequential lead times Søren Glud Johansen a and Anders Thorstenson b, October 3, 2005 a Department of Operations Research, Universit of Aarhus, N Munkegade, Bldg. 530, DK-8000 Aarhus C, Denmark b Logistics/SCM Research Group, Aarhus School of Business, Fuglesangs Allé 4, DK-8210 Aarhus V, Denmark Abstract We show that well-known textbook formulae for determining the optimal base stock of the inventor sstem with continuous review and constant lead time can easil be extended to the case with periodic review and stochastic, sequential lead times. The provided performance measures and conditions for optimalit are exact. Kewords: Inventor control, Optimisation, Stochastic 1. Introduction We consider a single-item inventor sstem with periodic review. We assume that the sstem is controlled b a base-stock polic and that unsatisfied demands are backordered and satisfied as soon as possible. The inventor position (stock on hand + stock on order - amount backordered) is monitored at each review instant, when the polic places a replenishment order to restore the base-stock s. Corresponding author: ath@asb.dk; tel.: ; fax:

3 The demand is Poisson with rate λ and the lead times L of replenishment orders are generated b an exogenous, sequential suppl sstem (Zipkin, 2000, Section 7.4). The latter assumption implies that different replenishment orders do not cross in time. The base-stock polic with review interval R is commonl referred to as an (R, S) polic, where S = s. This polic is frequentl used in practice, and it is often applied as a benchmark in theoretical studies. Furthermore, for the inventor sstem described and with the standard cost structure assumed, the (R, S) polic is the optimal control polic (Zipkin, 2000, Section ). 2. The extended lead time Like Zipkin (Zipkin, 2000, Section 6.7.3) and Rao (Rao, 2003), we model the times from the demand instants until the next review instant as realizations of a random variable U which is uniforml distributed over the interval from 0 to R. Therefore, the generic variable L' for the time from a demand instant until the deliver instant of the replenishment order triggered b this demand can be specified as U + L. Zipkin and Rao both assume that lead time L is constant. We call L' the extended lead time and observe that like the ordinar lead times, the extended lead times for customer demands can be seen as generated b an exogenous, sequential suppl sstem. This observation provides that the exact results derived for the corresponding inventor sstem with continuous review can be used to derive similar results for the sstem with periodic review. The results for the latter sstem are obtained b focusing on customer demands and letting the lead times in the former sstem be distributed as L' rather than as L. 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 ordinar lead time demand. The probabilit mass function (pmf) of D U is R 0 x ( λu) ( λr) 1 λu λr fu ( x) = e du e, x 0,1,... R = = (1) x!! = x

4 The pmf f U (x) is strongl unimodal. A proof is provided in the Appendix. Assuming that lead time L is gamma distributed, the variable D L has the negative binomial distribution 2 with parameters a = ( E [ L] ) Var [ L] and p λ ( λ ael [ ]) = + (Zipkin, 2000, Section ). Hence, if Var[L] > 0, we have the following useful recursion for the pmf of D L α 1+ fl( ) = pfl( 1 ), = 1,2,..., (2) 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 () is also strongl unimodal (Keilson and Gerber, 1971). Finall, 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 which is strongl unimodal b Theorem 2 in Keilson and Gerber (ibid.). 3. Performance measures Important performance measures of the considered inventor sstem can now be obtained from Zipkin (Zipkin, 2000, Corollar 7.4.3). The stock-out frequenc is ( ) { } A s = Pr D s, (4) L' the average backorders are ( ) = { } B s E max D s,0, L' (5) and the average inventor is ( ) = [ ] + ( ) = { } I s s E D B s E max 0, s D. L' L' (6) 3

5 Let h and b denote the unit holding and backorder costs per unit time, respectivel, and let π denote the cost incurred for each unit backordered. If no further costs are incurred, then the long-run average cost incurred per unit time is ( ) ( ) ( ) πλ ( ) C s = hi s + bb s + A s. (7) The base-stock model with this cost structure corresponds to Rosling s Model 2, Case (ii) (Rosling, 2002). Hence, based on his Proposition 2-2, we ma conclude that C(s) is quasiconvex in s, because the pmf of D L' is strongl unimodal, as noted above. (Quasiconvexit of C(s) is equivalent to -C(s) being unimodal.) Then, since ( ) ( ) ( ) ( ) { L' } ( πλ { L' }) C s = C s+ 1 C s = h+ b Pr D s b+ Pr D = s, (8) the optimal base-stock level is obtained as { ( ) } s* = min s C s > 0. (9) An example is depicted in Figure 1 which shows that C(s) is not convex in s. Figure 1 also shows the misrepresentation that occurs if lead times are assumed to be independent, impling b Palm's theorem (Palm, 1938) that the approximate cost rate C a (s) depends on the distribution of L' onl through its mean E[L']=R/2+E[L]. 4

6 Cost rates Ca(s) C(s) Base-stock level s 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 giving s* = 42, C(s*) and s a * = 35, C a (s a *) For π = 0, we obtain the extended standard solution b s* = min s Pr{ DL' s} >. (10) h + b As another special case, consider the case when b = 0 (and π > 0). Eq. (9) ma then be written as The ratio { D s} Pr{ D s} L' = L' { L' = } { } Pr D s h s* = min s <. (11) Pr DL ' s πλ Pr is non-increasing in s, because the pmf of D L' is strongl unimodal (Rosling, 2002). Note that Eq. (11) is an extension to the case of periodic review and stochastic lead times of the expression found in the well-known textbook b 5

7 Silver et al (Silver et al, 1998, Section 8.3.1) in the case of continuous review and a constant lead time. The extensions in Eqs. (9)-(11) are intuitivel appealing, since the are obtained from the standard continuous-review formulae b simpl replacing D L, the demand during the ordinar lead time, b D L', the demand during the extended lead time. 4. Concluding remarks The extended lead time has been defined as the time between a customer demand instant and the deliver instant of the replenishment order triggered b that demand. B appling this concept, exact performance measures for the base-stock inventor sstem with continuous review have been extended to a sstem with periodic review and stochastic, sequential lead times. Simple conditions for optimization 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 generalized further to accommodate compound Poisson demand (Johansen and Thorstenson, 2005). Acknowledgment This work was written while the second author was visiting at the Department of Logistics, Facult of Business, Hong Kong Poltechnic Universit, Hung Hom, Kowloon, HK. References Johansen, SG and Thorstenson, A (2005). Base-stock polic for the lost-sales inventor sstem with periodic review, Unpublished manuscript. Keilson, J and Gerber, H (1971). Some results for discrete unimodalit, J Am Stat Assoc, 66: Palm, C (1938). Analsis of the Erlang traffic formula for bus signal assignment, Ericsson Technics, 5:

8 Rao, US (2003). Properties of the periodic review (R, T) inventor control polic for stationar, stochastic demand, Manuf & Serv Opns Mngt 5(1): Rosling, K (2002). Inventor cost rate functions with nonlinear shortage costs, Opns Res, 50(6): Silver, EA, Pke, DF, and Peterson, R (1998). Inventor Management and Production Planning and Scheduling, 3 rd ed., J. Wile & Sons, New York. Zipkin, PH (2000). Foundations of Inventor Management, McGraw-Hill, Boston. Appendix Proposition: The pmf f U (x) as defined in Section 2 is strongl unimodal. Proof: B Theorem 3 in Keilson and Gerber (Keilson and Gerber, 1971), a necessar and sufficient condition for the pmf of the discrete variable D U to be strongl unimodal is that it is log-concave, i.e., that 2 ( ) ( ) ( ) fu x fu x+ 1 fu x 1, x= 1,2,.... (A1) The proof is b contradiction: Assume that Condition (A1) is not true. From Eq. (1) we then have for some x that ( λr) λr ( λr) λr ( λr) λr e < e e = x+ 1! = x+ 2! = x! ( ) ( ) ( ) ( ) ( ) λ λ λ λ x 1 1 x 1 R R λr R λr R e < e + 1! = x+ 1! = x+ 2!! x x ( λr) λr ( λr) e < x ( ) 1 λr e. x+ 1! + 1! = x+ 1 = + 1 (A2) As can be inferred from its last member, Condition (A2) is not satisfied. Hence, the assertion is false and Condition (A1) must be true. 7

9 Working Papers from Logistics/SCM Research Group L L L L L L L L L Søren Glud Johansen & Anders Thorstenson: Note: Optimal base-stock polic for the inventor sstem with periodic review, backorders and sequential lead times. Christian Larsen & Anders Thorstenson: A comparison between the order and the volume fill rates for a base-stock inventor control sstem under a compound renewal demand process. Michael M. Sørensen: Polhedral computations for the simple graph partitioning problem. Ole Mortensen: Transportkoncepter og IT-støtte: et undersøgelsesoplæg og nogle foreløbige resultater. Lars Relund Nielsen, Daniele Pretolani & Kim Allan Andersen: K shortest paths in stochastic time-dependent networks. Lars Relund Nielsen, Daniele Pretolani & Kim Allan Andersen: Finding the K shortest hperpaths using reoptimization. Søren Glud Johansen & Anders Thorstenson: The (r,q) polic for the lostsales inventor sstem when more than one order ma be outstanding. Erland Hejn Nielsen: Streams of events and performance of queuing sstems: The basic anatom of arrival/departure processes, when focus is set on autocorrelation. Jens Lsgaard: Reachabilit cuts for the vehicle routing problem with time windows.

10 ISBN Department of Accounting, Finance and Logistics Aarhus School of Business Fuglesangs Allé 4 DK-8210 Aarhus V - Denmark Tel Fax