Revisiting the safety stock estimation problem. A supply chain risk point of view.

Transcription

1 Revisiting the safety stock estimation problem. A supply chain risk point of view. Juan R. Trapero 1 Manuel Cardos 2 Nikolaos Kourentzes 3 1 Universidad de Castilla-La Mancha. Department of Business Administration 2 Universidad Politécnica de Valencia. Department of Business Administration 3 Lancaster University. Department of Management Science IIF Workshop on Supply Chain Forecasting for Operations, 28 & 29 June 2016, Lancaster, UK Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 1/31

2 Outline Motivation 1 Motivation Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 2/31

3 Why is it important? Motivation Supply chain risk management (SCRM) is becoming an attractive area Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 3/31

4 Why is it important? Motivation Supply chain risk management (SCRM) is becoming an attractive area Definition Supply chain risk is the potential loss for a supply chain in terms of its target values of efficiency and effectiveness evoked by uncertain developments of supply chain characteristics whose changes were caused by the occurrence of triggering-events (Heckmann et al., 2015) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 3/31

5 Why is it important? Motivation Supply chain risk management (SCRM) is becoming an attractive area Definition Supply chain risk is the potential loss for a supply chain in terms of its target values of efficiency and effectiveness evoked by uncertain developments of supply chain characteristics whose changes were caused by the occurrence of triggering-events (Heckmann et al., 2015) Demand is subject to uncertainty. Variance or standard deviation is used as a risk metric. In order to mitigate such risks, safety stocks may be employed Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 3/31

6 Literature review Motivation If demand forecasting error is Gaussian (Thanos,Christina presentation) i.i.d. with zero mean and constant variance, the safety stock (SS) for a 100p % target service level is: SS = kσ L (1) k = Φ 1 (1 p) is the safety factor. Φ( ) denotes the standard normal cumulative distribution function σ L stands for the standard deviation of the forecast error for a certain lead time L The main problem is how to estimate σ L. Theoretical approach. An estimation of σ 1 is provided. Then, an analytical expression that relates σ L and σ 1 is used. Empirical approach. σ L is estimated on the basis of lead time forecast errors (John presentation). Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 4/31

7 Theoretical approach. Estimation of σ 1 Traditional textbooks suggest to compute σ 1 based on forecasting error metrics. (Silver et al., 1998) uses MSE MSE t = 1 n (y t F t ) 2 n t=1 y t is the actual value at time t and F t is the forecast value n is the sample size σ 1 can be approximated by MSE t+1 MSE can be updated when new observations are available: MSE t+1 = α (y t F t ) 2 + (1 α )MSE t (2) α is a smoothing constant, values between 0.01 and 0.1 are commonly employed. Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 5/31

8 Theoretical approach. Estimation of σ L According to (Silver, 1998), such an exact relationship can be complicated to obtain: σ L = L c σ 1 c is a coefficient estimated empirically. We assume the lead time (L) constant and deterministic (Axsater, 2007) : σ L = Lσ 1 (Hyndman, 2008) a ETS(A,N,N) with parameter α σ L = σ 1 L 1 + α(l 1) α2 (L 1)(2L 1) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 6/31

9 Estimation of σ L. A deeper literature review Last expression was also found by (Graves, 1999) for an inventory system that is subject to an IMA(0,1,1) demand process, a deterministic replenishment lead-time and an adaptive base-stock policy. In an addendum to his paper, acknowledges that the same result was obtained by (Wecker, 1979). Wecker manuscript was not published but was referenced by (Eppen, 1998) Graves, S. C., 1999a. Addendum to a single-item inventory model for a nonstationary demand process. Manufacturing & Service Operations Management 1 (2), 174 Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 7/31

10 Estimation of σ L. An empirical approach Alternatively, if we doubt about the validity of the forecasting model and/or about the normally i.i.d. error assumptions. σ L = ni=1 (e t (L) ē(l)) 2 e t (L) = L h=1 y t+h L h=1 F t+h is the lead time forecast error ē(l) is the average error for L. Promotions, judgmental forecasting, special events, complex customer demand patterns, etc. make reasonable to use empirical approaches to validate i.i.d. initial assumptions (triggering-events) n Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 8/31

11 Research objectives Motivation 1 Investigate empirical approaches to determine the safety stock Explore non-parametric approaches as Kernel density estimators if forecast errors might be not normal. Explore time-varying parametric volatility estimators (ES and GARCH) if forecast errors might be not independent. 2 Compare theoretical and empirical approaches 3 Use prediction interval metrics and relate them with traditional stock control metrics 4 Explore combination methods if any of the assumptions is hold. Idea Optimal combination based on maximizing conditional coverage Christoffersen test p-value (Christoffersen, 1998). Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 9/31

12 First: a prediction interval point of view To use conditional coverage Christoffersen test, we need to pose our problem as a prediction interval Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 10/31

13 First: a prediction interval point of view To use conditional coverage Christoffersen test, we need to pose our problem as a prediction interval Parametric (normal): [L L, U L ] = [0, F L + k σ L ] [L L, U L ]: lower and upper interval F L : lead time point forecast σ L : lead time standard deviation We will use ES and GARCH to estimate σ L (These techniques are commonly used in financial risk management) Non-parametric:[L L, U L ] = [0, F L + Q L (1 p)] Q L (p): 100p % lead-time forecast error quantile We will use Kernel density estimator and empirical percentile to determine Q L (p). Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 10/31

14 Methods Motivation Lead time point Forecast (F L ): Single exponential smoothing F t+1 = αy t + (1 α)f t F L = LF t+1 Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 11/31

15 Methods Motivation Lead time point Forecast (F L ): Single exponential smoothing F t+1 = αy t + (1 α)f t F L = LF t+1 Parametric-Normal: Single Exponential Smoothing: σ 2 L,t+1 = α ɛ 2 L,t + (1 α )σ 2 L,t GARCH(1,1): σ 2 L,t+1 = ω + a 1ɛ 2 L,t + β 1σ 2 L,t Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 11/31

16 Methods Motivation Lead time point Forecast (F L ): Single exponential smoothing F t+1 = αy t + (1 α)f t F L = LF t+1 Parametric-Normal: Single Exponential Smoothing: σ 2 L,t+1 = α ɛ 2 L,t + (1 α )σ 2 L,t GARCH(1,1): σl,t+1 2 = ω + a 1ɛ 2 L,t + β 1σL,t 2 Non-parametric: Kernel Density estimator f (x) = 1 Nh ( ) N j=1 K x Xj h f (x) is probability density function of the lead time forecast errors N is the sample size K( ) is the kernel smoothing function h is the bandwidth. All methods were implemented in MATLAB (financial toolbox and statistics toolbox) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 11/31

17 Performance measurement Prediction intervals will be assessed on these metrics: Forecast coverage (hit rate), proportion of times that a real value falls inside the prediction interval given a certain target coverage level. Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 12/31

18 Performance measurement Prediction intervals will be assessed on these metrics: Forecast coverage (hit rate), proportion of times that a real value falls inside the prediction interval given a certain target coverage level. Average prediction interval width. In our particular case the scaled safety stock: the prediction interval at time t minus the point forecast at the same time, all divided by the in-sample demand average (kσ L /ȳ) Backorders are the sum of values out of the prediction interval across time and subsequently averaged across SKUs. Conditional converage: we need a good coverage and independence (Christoffersen, 1998). Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 12/31

19 Performance measurement Prediction intervals will be assessed on these metrics: Forecast coverage (hit rate), proportion of times that a real value falls inside the prediction interval given a certain target coverage level. Average prediction interval width. In our particular case the scaled safety stock: the prediction interval at time t minus the point forecast at the same time, all divided by the in-sample demand average (kσ L /ȳ) Backorders are the sum of values out of the prediction interval across time and subsequently averaged across SKUs. Conditional converage: we need a good coverage and independence (Christoffersen, 1998). The ideal situation is a method that provides a forecast coverage close to the target coverage with a low interval width and pass the conditional coverage test. Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 12/31

20 Performance measurement Forecast coverage and interval width from prediction intervals theory provide similar information that customer service level and inventory investment, respectively, from stock control performance metrics (trade-off curves). Main difference: stock control metrics require a stock control policy, whereas prediction interval metrics do not Recall that we want to use prediction intervals to use Christoffersen test. To validate P.I. metrics, we will compare them with stock control metrics by simulating an order-up-to stock control policy. Target coverage levels are set to 85 %, 90 %, 95 % and 99 % Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 13/31

21 . Design of experiment Is there any relationship between P.I. metrics and stock control metrics? Simulate a demand y = 50 + x t, where x t is an ARIMA(0,1,1) with θ = 0,75 and σ = 2. Lead time L = 4. We have divided the total sample in three parts. First part (40 %): to estimate the initial value and ES parameter to provide the mean forecasts. Second part (40 %): To estimate the initial value and exponential smoothing parameter to provide volatility forecast (σ 1 ) and then compute σ L based on forecasting parameters and lead time value (theoretical approach). The rest of the data is kept as hold-out sample. Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 14/31

22 . Design of experiment Stock control policy: Order up to level implemented in SIMULINK (Dejonckheere et al., 2003) The ordering decision for an order-up-to policy is O t = S t IP t O t, ordering decision S t, order-up-to level IP t inventory position IP t = NS t + WIP t NS t : Net Stock (inventory on hand minus backorders) WIP t : work in progress S t = F L + kσ L Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 15/31

23 Forecasting system + stock policy in SIMULINK Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 16/31

24 . Design of experiment The customer service level is calculated as the proportion of times that the NS is greater than zero. Backorders are the sum of negative values of NS across time and subsequently averaged across SKUs. Inventory investment is the NS average. Did you realize that... Traditional stock control metrics (trade-off curves) do not evaluate independence of the forecasting errors. Conditional coverage from prediction interval theory does! Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 17/31

25 Forecasting system + stock policy in SIMULINK coverage (pred interval) y=1.01x -1.1 backorders (pred interval) y=1x realised service level (stock control) backorders (stock control) safety stock (pred interval) y=99.9x inventory investment (stock control) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 18/31

26 Experimental setup Motivation The data comes from a major UK fast moving consumer goods manufacturer specialized in the production of household and personal care products (Barrow and Kourentzes, 2016). In total 229 products with 173 weekly observations per product are available. We have removed 6 SKUs because of GARCH convergence problems (GARCH less robust than ES) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 19/31

27 Experimental setup Motivation The data has been split into three parts. The first part (20 % of the data) has been used to compute both the ES parameter and the initial value to determine the point demand forecast. The second part (60 % of the data) was employed to optimize the volatility models (parametric and non-parametric). The last part (20 % of the data) was devoted to test the probabilistic forecasts of the considered methods. It should be noted that only single exponential smoothing is used to produce the point forecast Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 20/31

28 Experimental setup. What a mess! Coverage (%) SES Kernel GARCH Percentile Theo aprox Theo exact scaled safety stock Backorders SES Kernel GARCH Percentile Theo aprox Theo exact scaled safety stock Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 21/31

29 Theoretical approaches Theoretical approaches Coverage (%) Theo aprox Theo exact Scaled safety stock (%) 20 Theo aprox Theo exact Backorders Scaled safety stock (%) Theo aprox: σ L = Lσ 1 Theo exact: σ L = σ 1 L 1 + α(l 1) α2 (L 1)(2L 1) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 22/31

30 Empirical and parametric approaches Coverage (%) Empirical-parametric SES GARCH Scaled safety stock (%) 20 SES GARCH Backorders Scaled safety stock (%) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 23/31

31 Empirical and non-parametric approaches Coverage (%) Empirical and non-parametric Kernel Percentile Scaled safety stock (%) Backorders Kernel Percentile Scaled safety stock (%) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 24/31

32 Champions Motivation Champions among theoretical and empirical (parametric and non-parametric) Coverage (%) Kernel GARCH Theo exact Scaled safety stock (%) Backorders Kernel GARCH Theo exact Scaled safety stock (%) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 25/31

33 Individual results Motivation Theoretical approaches: Use exact formulation if a ES is the forecasting technique. In general terms, theoretical models perform worse than empirical approaches. Although, theoretical models can be robust to sample size. KERNEL is the champion of non-parametric empirical approaches, it captures potential asymmetries. GARCH is the champion of parametric empirical approaches, it captures potential error autocorrelation. Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 26/31

34 Why don t we combine them? Combined prediction interval [ˆL c, Û c ] : ˆL c = 0 Û c = w Û Kernel + (1 w) Û GARCH where 0 < w < 1 is a constant that maximizes the conditional coverage Christoffersen test p-value. The conditional coverage test is the combination of the tests for unconditional coverage and independence We also test against 50 %-50 % combination (w = 0,5) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 27/31

35 Combination results Motivation Coverage (%) Kernel GARCH Christof Scaled safety stock Backorders Kernel GARCH Christof Scaled safety stock Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 28/31

36 Motivation Demand uncertainty play a crucial role in supply chain risk modeling. Empirical approaches enhance the determination of the safety stock with respect to theoretical approaches. Prediction interval metrics are suggested to evaluate forecasting performance. This work proposes a novel approach that combines a non-parametric (Kernel) and a parametric approach (GARCH) The combination weight is obtained by maximizing the Christoffersen conditional coverage test p-value. The results show a good compromise between coverage (service level) and average interval width (safety stock/inventory investment). Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 29/31

37 Thank you for your attention! blog: This work has been supported by the Spanish Ministerio de Economía y Competitividad, under Research Grant no. DPI R (MINECO/FEDER/UE) Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 30/31

38 Some key works Motivation 1 Heckmann, I., Comes, T., Nickel, S., A critical review on supply chain risk - definition, measure and modeling. Omega 52, Christoffersen, P. F., Evaluating interval forecasts. International economic review 39 (4), Dejonckheere, J., Disney, S. M., Lambrecht, M. R., Towill, D. R., Measuring and avoiding the bullwhip effect: A control theoretic approach. European Journal of Operational Research 147, Silver, E., Pyke, D., Peterson, R., Inventory Management and Production Planning and Scheduling. Wiley. Juan R. Trapero, Manuel Cardos, Nikolaos Kourentzes 31/31