Modeling the Evolution of Demand Forecasts with Application to Safety Stock

Modeling the Evolution of Demand Forecasts with Application to Safety Stock Analysis in Production/Distribution Systems Overview • Problem • Methodology • Key Results Multi-product Multi-location Multi-period with Capacity and Trans-shipment N=5 Products Multi- Plants L=8 DCs Demand of each product at each DC Planning Sequence of Events 3. Implement 2. Deterministic the current LP derive month production plan production plan Month 0 Month 1 1. At the end of the month forecast demand M-month out Month 2 4. Roll horizon and re-forecast Solving the Inventory Problem • Find an economical safety stock factor for each product at each DC for each month – A DP approach difficult – A simulation approach ƒ Use MMFE to simulate forecast ƒ Use a similar LP to simulate decisions ƒ Tally costs and service level for different safety stock factor The Martingale Model of Forecast Evolution (Additive) Future Month t Month 0 Current Month s Month 1 Month 0 D0,0 Month 1 D0,1 Month 2 D0,2 Month 3 D D1,1 D1,2 D1,3 ε1,1= D1,1-D0,1 ε1,2= D1,2-D0,2 ε1,3= D1,3-D D2,2 D2,3 ε2,2= D2,2-D1,2 ε2,3= D2,3-D1,3 Month 2 i.i.d. multivariate normal with mean 0 Justifying ε are i.i.d. multivariate normal with mean 0 ε s ,t = Ds ,t − Ds −1,t ε s = (ε s ,t )t+=∞s • Information set Fs grows with time s • εs is uncorrelated with all εu for u ≤ s-1 and E[εs]=0 • εs is a stationary process • εs is normal Why is it called a Martingale Model? If forecast is conditional expectation based on current information set Fs Ds ,t = E [Dt ,t | Fs ] Then E [Ds ,t | F0 ,..., Fs −1 ] = E [E [Dt ,t | F0 ,..., Fs ] | F0 ,..., Fs −1 ] = E [Dt ,t | F0 ,..., Fs −1 ] = Ds −1,t Thus, Ds,t is a martingale and ε s ,t = E [Dt ,t | Fs ] − E [Dt ,t | Fs −1 ] is uncorrelated with Fs-1 E [ε s ,t ] = 0 Conditional Expectation as Best MeanSquare Predictor of Dt,t Using the best Mean-square predictor definition of conditional probability E [(Dt ,t − Du ,t )w ] = E [(Dt ,t − Ds ,t )w ] = 0 ∀w ∈ r .v . observed at u Then E [(Dt ,t − Du ,t )w ] = E [(Ds ,t − Du ,t )w ] = 0 Thus, true also under linear predictor The Multiplicative Model ν s ,t = log(Ds ,t ) − log(Ds −1,t ) Rs ,t = exp(ν s ,t ) = Ds ,t / Ds −1,t ν s = (ν s ,t )t+=∞s i.i.d. multivariate normal with mean of each coordinate being the negative of one half of its variance Characterize the Simulation System • The variance-covariance matrix Σ for εs (MN Х MN) – estimated using past demand and forecast • The initial state of the system (D0,0, D0,1, …, D0,M, D) Why not Simulate the Time Series of Forecast Directly • “Simulate the complicated forecast process based on past demand, competitors’ prices, weather forecast etc. is no more credible than assuming the forecast process is MMFE and estimating the variance-covariance matrix” Simulating the Forecast Evolution Using the Variance-covariance Matrix Σ • Properties of Σ – Σ is symmetric and PSD – Σ = CC’ = (UD1/2)(UD1/2)’ where D is the diagonal matrix with eigenvalues sorted in decreasing order • The standard multivariate normal representation: εs= CZ where Z is a standard normal random vector Forecast Variability Resolving Over Time • The 1st column of C captures the 1st order magnitude of εs • The signs and values of entries in the 1st column of C reveals how forecast variability resolves over time and how they are correlated – e.g. C0,1 = -.5511, C0,41 = -.4343 Æ 61.7% variability resolved in month of sale, 38.3% variability resolved 1 month out – e.g. C0,1 = .1616, C0,41 = .3143, C0,81 = -.0880, C0,121 = -.2636 Æ 12%, 49%, 4%, 34% The Experiment • Traditional Forecast Method (2 month out) – 80 X 80 Σ from four years of past forecast and actual demand data • Statistical Forecast Method (4 month out) – (160 X 160) Σ from two years actual demand and simulated forecast • Initial state – forecast for the year of 19901991 fiscal year • Cost and service metrics averaged over 10 – 20 simulated years The Results • Safety stock factor can be reduced without sacrificing much fill rate, if using the Statistical Forecast Method Æ resulting in significant cost savings • Reducing safety stock factor using the Traditional Forecast Method does not show much benefit • More important to increase forecast accuracy than to increase capacity Recap • MMFE to model forecast evolution • Simulate the system using MMFE • Evaluate the performance of the two systems with two different forecast methods