University:
Massachusetts Institute of Technology
Course:
15.992 | S-Lab: Laboratory for Sustainable Business
Academic year:

2023
Views:

146

Pages:

24
Author:

Javier Weaver

Analysis of a ForecastingProduction-Inventory System with Stationary Demand L. Beril Toktay and Lawrence W. Wein Presented by Guillaume Roels Operations Research Center Forecasting-ProductionInventory Demand forecast update Make-to-Stock environment Ct ~ N ( µ , σ ) 2 C Rt Order release Dt It Qt −1 Pt = min{Qt −1 , Ct } Objective Minimize steady-state { { Inventory holding costs h Shortage penalty costs b Recap: MMFE Model Rolling horizon H Forecast Dt ,t +i i = 0,… , H Forecast Update ε t ,t +i = Dt ,t +i − Dt −1,t +i Assumptions { { { Stationary Demand with rate λ Unbiased Forecasts Uncorrelated Forecast Updates Production-Inventory Model MRP-Type Release Policy: H −1 Rt = ∑ ε t ,t +i + Dt ,t + H = eT ε t + λ i =0 Inventory Policy H Qt + I t − ∑ Dt ,t +i = sH i =1 It Production Policy Forecast-corrected base-stock policy ⎧⎪Ct if sH > I t −1 + Ct ⎫⎪ P ( I t −1 ) = ⎨ ⎬ ⎩⎪ sH − I t −1 if sH ≤ I t −1 + Ct ⎭⎪ * State-dependent Optimal Policy ( Dt −1,t , Dt −1,t +1 ,… , Dt −1,t + H −1 , λ ) Benchmark: Myopic Policy Do not use available forecast information Rt = Dt Constant Inventory Qt + I t = sm Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion In Heavy Traffic, the WIP has an exponential distribution Average Excess Capacity 2( µ − λ ) v= T 2 e Σe + σ C Variance of the forecasts Variance of the production WIP at time n units n X n = ∑ ( Rt − Ct ) t =1 Qn = X n − inf1≥t ≥ n X t R2 − C2 time Heavy traffic analysis 1 Consider a sequence of systems k s.t.: k (λ (k ) λ (k ) →λ µ (k ) →µ −µ )→c<0 (k ) Heavy traffic analysis 2 (k ) n Q k X (k ) ⎢⎣ kt ⎥⎦ = = X X (k ) n k (k ) ⎢⎣ kt ⎥⎦ + −m − inf1≥t ≥ n X (k ) t k (k ) ⎣⎢ kt ⎦⎥ k k (k ) (k ) (k ) where m = λ − µ + m (k ) ⎢⎣ kt ⎥⎦ k Heavy traffic analysis 2 σ B(t ) X (k ) ⎢⎣ kt ⎥⎦ k = BM (c, σ ) 2 X (k ) ⎢⎣ kt ⎥⎦ −m (k ) k ct ⎢⎣ kt ⎥⎦ + m (k ) ⎢⎣ kt ⎥⎦ k Heavy traffic analysis 3 (k ) n Q k = X (k ) n k + − inf1≥t ≥ n X k RBM (c, σ 2 ) Reflected Brownian Motion on the nonnegative halfline Estimated by an exponential random variable (k ) t Steady-state WIP distribution P(Q∞ = 0) = 1 − e P(Q∞ > x) = e − vx − vβ impulse (x + β ) β is a correction term, coming from Random Walks Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion Determination of the BaseStock levels 1 Myopic Policy ⎛ b ⎞ sm = F ⎜ ⎟ ⎝b+h⎠ −1 Q∞ Newsboy quantity… … considering the distr. of the WIP! 1 ⎛ b⎞ sm = ln ⎜1 + ⎟ − β v ⎝ h⎠ Determination of the BaseStock levels 2 MRP-type Policy ⎛ b ⎞ sH = F ⎜ ⎟ ⎝b+h⎠ −1 W W = max{Q∞ + Y0 , max1≤ k ≤ H Yk } Y0 is the difference between: { { Total Forecast Error over the horizon H and Total Capacity Determination of the BaseStock levels 3 MRP-type policy: asymptotic b h ⎛1⎞ 2 s = s + µY 0 + ⎜ ⎟ σ Y0 v ⎝2⎠ Proportional to the variance! Good approximation when { b / h large a H { * m High utilization rate Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion Capacity – Stock Trade-Off No advance information s = s − Hλ a H * m Full advance information Interchangeability Capacity/Safety Stock Demand variability Capacity 2 ⎛ ⎞ σ * a D s H = sm − H λ − H ( µ − λ ) ⎜ 2 2 ⎟ ⎝ σ D +σC ⎠ Discussion * ↑→ v ↓→ s Correlation m ↑ σ Y20 is the system variability over H not resolved at the beginning of the horizon { Preference for accurate early forecasts Optimize over all planning horizons H −1 Rt = Greater costs due to { { ∑ε i=0 t , t +1 + Dt , t + H misspecification of the forecast model than misuse of the information in production Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion Conclusion Integrated view { { { Forecast Production Inventory Lots of improvement for current MRP systems Is heavy traffic of practical value?