Online Appendix to Accompany Further Improvements on Base-Stock Approximations for Independent Stochastic Lead Times with Order Crossover

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1 10.187/msom Olie Appedix to Accompay Further Improvemets o Base-Stock Approximatios for Idepedet Stochastic Lead Times with Order Crossover Lawrece W. Robiso Johso Graduate School of Maagemet Corell Uiversity Ithaca, New York lwr@corell.edu James R. Bradley Maso School of Busiess College of William ad Mary Williamsburg, Virgiia james.bradley@maso.wm.edu Jauary, 007

3 This olie appedix cotais three techical topics i support of the paper Further Improvemets o Base-Stock Approximatios for Idepedet Stochastic Lead Times with Order Crossover. The first sectio describes how to fit a beta distributio to ay four momets. The secod sectio provides the proof of Theorem 1, while the third sectio summarizes the cost performace of the various heuristics. 1 Fittig the Parameters of a Beta Distributio Give the first four momets (μ L, σ L, α 3,adα 4 ) of a probability distributio, the trasformatio of Elderto ad Johso(1969), as cited i Johso et al. (1995), ca be used to compute the four parameters (α, β,, u) thatdefie the matchig beta distributio as follows: α. = 1 ρ 1 (ρ +)α 3 β where ρ. = ρ α,. = 6(α 4 α 3 1) 6+3α 3 α 4 = α + β,ad. = μ L wα/ρ q (ρ +) α 3 +16(ρ +1) u =. + w, where w =. r ρ +1 σ L ρ αβ = u. Proof of Theorem 1 µ c Theorem 1 For ay discrete distributio of the o-egative lead time, σ N v +1/9 μ c v +1 L. Proof: Because Bradley ad Robiso (005) showed that the quasi-uiform discrete distributio for the lead time maximized the variace of the umber of orders outstadig σ N,itissufficiet to show that Theorem 1 holds for the family of quasi-uiform distributios. Note that we ca rewrite this upper boud as µ µ c v +1/9 8μ L μ c v +1 L = 1 9(σ L + μ L ) μ L ; (1) takig the partial derivative with respect to μ L yields, after some simplificatio, 1

4 µ µ 8μ L μ 1 μ L 9(σ L + μ L ) μ L = L /3 σ L σ L + 0, μ L showig that the boud is icreasig i μ L. Thisimpliesthatweeedtesttheboudoly for those quasi-uiform distributios that have a positive mass o zero. (We ca traslate other distributios towards zero, which would tighte this upper boud while leavig σ N uchaged.) It is possible to defie these quasi-uiform probability distributios by two parameters p ad as follows. Defie p [0, 1) to be a poit-mass o 0, with the remaiig mass of 1 p uiformly distributed over [0,] ad the aggregated oto the earest iteger. (Because of this aggregatio, p ad are slightly differet tha their couterparts p 0 ad 0 i the cotiuous distributio.) For ay real umber, defie hi. = b +0.5c to be its earest iteger. The the discrete distributio {f l } defied by (p, ) will be: f 0 = p + 1 p f l = 1 p l =1,..., hi 1 f hi = 1 p hi 1, with f l =0outsideofthisrage. Iordertoallowf 0 to assume values i the rage (0, 1 ). we geeralize the defiitio of p toallowittotakeoegativevalues,sothatp 1, 1 ; 1 f l ad f hi are both always well-defied whe >1.5. As a fuctio of p ad, hi 1 X µ µ 1 p 1 p μ L = l + hi hi 1 l=1 hi = (1 p) hi, ad σ L + μ L = = = hi 1 X µ µ 1 p 1 p l + hi hi 1 l=1 µ hi 1 p (hi 1) hi ( hi 1) + hi hi µ 1 p hi 1+6hi 4hi, 6

5 with σ N = = = X F l (1 F l ) hi 1 X µ 1 p 4 p + 1 p + l hi 1 X µ 1 p (p +1 p)( 1) 1 p 1 p l µ 1 p µ hi 1 1 p X + [ (p +1 p)+(1 p)( 1)] l 4 µ hi 1 1 p X 4(1 p) l 4 µ 1 p = hi 1 p +1 p +6hi (1 p) 4 hi (1 p) 1 µ 1 p = μ L "1 Ã!# 1 [ hi]+4hi 1. () 6 hi I order to prove Theorem 1, we eed to compare () with (1): µ 1 p μ L "1 Ã!# 1 [ hi]+4hi µ 1 8μ L 1 6 hi 9(σ L + μ L ) μ L µ Ã! 1 p 1 [ hi]+4hi 1 8μ L hi 3(σ L + μ L ) 1 [ hi]+4hi 1 hi 8 hi [ hi] 1+6hi 4 hi. Note that p has dropped out of this compariso. Cross-multiplyig, expadig, ad simplifyig leads i a straightforward if tedious maer to the equivalet iequality: 8 hi [ hi] 3 +1[ hi] +6hi [ hi]+hi 1 0. (3) At this poit we replace hi with ε, represetig the differece betwee ad its closest iteger value. It is clear that ε 1, 1. With this substitutio, (3) becomes 8 hi ε 3 +1ε +6hi ε +hi 1 0. (4) 3

6 We eed to show that (4) holds for all hi 1 ad all ε 1, 1 ; we start with the latter. Takig the partial derivative of (4) with respect to ε yields 4 hi ε +4ε +6hi = 6 (hi 1) 4ε +1 +(ε +1), whichiso-egativesicehi 1. Thus the left-had side of (4) is icreasig i ε; to show that (4) holds for all ε, itissufficiet to show that it holds for the lowest feasible value of ε: ε = 1. Makig this substitutio, (4) becomes [hi 1] 0, which is always true, completig the proof. 3 Heuristic Cost Performace For ease of compariso, the umerical results from Bradley ad Robiso (005) are repeated i the first six lies of Table 1. As a aside, ote that may of the umbers give here are trivially differet tha theirs; this is because they had prematurely trucated the lead-time distributio at l =100. This trucatio was icosequetial for μ L =6ad 10, but ot for the highly skewed lead-time distributio with μ L =,wheref 100 was oly 99.95% i the worst case (σ L =8). These mior correctios do ot alter their coclusios. 4

7 Table 1 Summary of the Percetage Cost Icrease Across the Test Bed of 145,800 Parameter Combiatios (All umbers are i percet.) Distributio Variace 1 Mea Std Dev 95%-ile 99%-ile Worst Case Prob{ =0} Prob{ 1%} Prob{ 5%} Normal LT: σ L Normal Old: ˆσ N Normal True: σ N Neg. Bi. LT: σ L , Neg. Bi. Old: ˆσ N Neg. Bi. True: σ N Neg. Bi. New: σ N Normal New: σ N Logormal New: σ N Gamma New: σ N Beta New: σ N Beta 3 New: σ N This colum idetifiesthevariaceusediplaceofσ N whe calculatig σ SF. For these beta distributios, α 3 = ad α 4 = For these beta distributios, α 3 ad α 4 are approximated from c v through the liear regressios ˆα 3 = c v,ad ˆα 4 = c v. 5

8 Refereces [005] Bradley, J. R., L. W. Robiso. Improved base-stock approximatios for idepedet stochastic lead times with order crossover. Mfg. ad Service Oper. Mgmt., [1969] Elderto, W. P., N. L. Johso Systems of Frequecy Curves. Cambridge Uiversity Press, Cambridge. [1995] Johso, N. L., S. Kotz, N. Balakrisha Cotiuous Uivariate Distributios, volume. Joh Wiley & Sos, New York. 6

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio: