Spare parts inventory management: new evidence from distribution fitting

Transcription

1 Spare parts inventory management: new evidence from distribution fitting Laura Turrini Joern Meissner IIF Workshop on Supply Chain Forecasting for Operations Lancaster 29. Juni 2016

2 FORECASTING SPARE PARTS: WHICH DISTRIBUTION? 700" 600" 500" 400" 300" 200" Intermittent demand 100" 0" 1" 5" 9" 13" 17" 21" 25" 29" 33" 37" 41" 45" 49" 53" 57" 61" 65" 69" Demand'data' Inventory performance metrics: Achieved service level Average inventory on hand Which is the right distribution? Which is the right test to measure goodness-of-fit? 1

3 DISTRIBUTIONS IN THE LITERATURE z 2 p t : inter-demand interval z t : demand size Compound distributions: Discrete time p t ~ Ge(p) z t ~ N(m,s), Continuous time p t ~ R(l) z t ~ Ge(p), Log(p), z 1 p 1 p 2 p 3 z 3 Classic distributions: Normal, Gamma, Poisson, Figure: Syntetos et al. (2012) 2

4 KOLMOGOROV-SMIRNOV TEST F(x) F 0 (x) F n (x) unknown real cumulative distribution of the data supposed distribution empirical distribution H 0 : F = F 0 K-S statistic: D = sup F n (x) - F 0 (x) X Standard critical values are distribution-free for fully specified, continuous distribution Conservative test for discrete data Massey (1951): The Kolmogorov-Smirnov Test for Goodness of Fit Journal of the American Statistical Association 46(253)

5 MODIFIED TESTS WITH FOCUS ON TAILS: 2 A MODIFIED ANDERSON-DARLING TEST Classic A-D statistic: A = n + [F n (x) - F(x)] 2 j(f(x))dx with j(u) = ò - 1 u(1- u) The modified A-D test uses j(u) = 1 1- u weight only on the right tail! f [1,k] discrete with support p j probability of f (x) = j S j count of observations assuming value T expected number of observations j assuming value j Z j = S j -T j H j = T j n j Modified A-D statistic: AU 2 = 1 n k å Z j 2 p j j =1 1- H j Sinclair et al. (1990): Modified Anderson Darling test Communications in Statistics-Theory and Methods 19(10) Choulakian et al. (1994): Cramer-von Mises statistics for discrete distributions Canadian Journal of Statistics 22(1)

6 MODIFIED TESTS WITH FOCUS ON TAILS: 2 A MODIFIED K-S TEST 1,2% 1% 0,8% 0,6% 0,4% F 0 (x) F n (x) D +,D - F 0 (x) F n (x) D +,D - 0,2% Classic K-S test Modified K-S test 0% 5% 6% 7% 8% 0% 9% 1% 10% 2% 3% 4% 5% 6% 7% 8% 9% 10% "x Þ D(x) = F n (x) - F 0 (x) x ÎA i Þ D(x) = a i F n (x) - F 0 (x)!0,2% D = sup X D(x) 5

7 CV 2 CV 2 DATASETS Dataset SKUs 3 years weekly data Mixed intermittency Dataset 2 - RAF 5000 SKUs 7 years monthly data Very intermittent! p Min Median Max p Min Median Max CV CV p p 6

8 A-D modif. K-S modif. K-S classic FITTING DEMAND PER PERIOD DATASET 1 Poisson Normal Gamma NBD Stutt. P. Str. 70% Good 3% No 27% Str. 0.6% Good 0.2% No 99% Str. 42% Good 8% No 50% Str. 79% Good 1% No 20% Str. 96% Good 1% No 3% Str. 74% Good 4% No 22% Str. 0.5% Good 0.2% No 99% Str. 62% Good 8% No 30% Str. 76% Good 2% No 22% Str. 92% Good 1% No 7% Str. 39% Good 5% No 56% Str. 0.4% Good 0.3% No 99% Str. 73% Good 13% No 14% Str. 79% Good 1% No 20% Str. 87% Good 3% No 10% 7

9 K-S modif. CLASSIFICATION SCHEME FIT Figure: Syntetos et al. (2012) Fit: D1: 85% D2: 68% Poisson Normal Gamma NBD Stutt. P. 8

10 A-D modif. K-S modif. K-S classic FITTING LEAD TIME DEMAND DATASET 2 Poisson Normal Gamma NBD Stutt. P. Str. 31% Good 4% No 65% Str. 4% Good 7% No 89% Str. 5% Good 7% No 88% Str. 59% Good 9% No 32% Str. 75% Good 10% No 15% Str. 26% Good 4% No 70% Str. 3% Good 5% No 92% Str. 16% Good 13% No 70% Str. 37% Good 11% No 52% Str. 45% Good 13% No 42% Str. 16% Good 6% No 78% Str. 6% Good 10% No 84% Str. 37% Good 22% No 41% Str. 60% Good 14% No 26% Str. 62% Good 19% No 19% 9

11 INVENTORY SIMULATION Inventory-up-to policy with backordering SBA method Initialization time: 4 years (48 obs.) 9 implemented policies: Poisson BestKSClassic Normal BestKSModified Gamma BestADModified NBD StuttP ClassificationRule Classic version + Teunter and Duncan s correction: ˆm L = ˆm L ˆm L = ẑ + ˆm (L -1) ˆ s L = ˆ s L ˆ s L = ˆ s L 10

12 Achieved Service Level EFFICACY OF INVENTORY POLICIES Classic version Teunter and Duncan s correction Target Service Level 11

13 Average inventory on hand Achieved Service Level 12

14 Average inventory on hand Achieved Service Level 13

15 Average inventory on hand Achieved Service Level 14

16 Average inventory on hand Achieved Service Level 15

17 Average inventory on hand Achieved Service Level 16

18 Average inventory on hand Achieved Service Level 17

19 Average inventory on hand Achieved Service Level 18

20 Average inventory on hand Achieved Service Level 19

21 Average inventory on hand Achieved Service Level 20

22 Average inventory on hand Achieved Service Level 21

23 CONCLUSIONS AND FURTHER RESEARCH Distribution of demand per period mostly fits compound Poisson distributions. Lead time demand also mostly fits compound distributions, but results are worse using the modified tests Right to use for inventory? BestKSModified policy is the most efficient EXCEPT for very high service levels (over 94%), where BestADModified is better. Need for classification rules for inventory applications. 22

24 THANK YOU FOR YOUR ATTENTION! 23