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# Analysis Of A Forecasting Production Inventory System With Stationary Demand - Forecast Evolution Models - Guillaume Roels

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Analysis Of A Forecasting Production Inventory System With Stationary Demand - Forecast Evolution Models - Guillaume Roels

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### Analysis Of A Forecasting Production Inventory System With Stationary Demand - Forecast Evolution Models - Guillaume Roels

1. 1. Analysis of a Forecasting- Production-Inventory System with Stationary Demand L. Beril Toktay and Lawrence W. Wein Presented by Guillaume Roels Operations Research Center This summary presentation is based on: Toktay, L. Beryl, and Lawrence M. Wein. quot;Analysis of a Forecasting- Production-Inventory System with Stationary Demand.quot; Management Science 47, no. 9 (2001).
2. 2. Forecasting-Production- Inventory Demand Make-to-Stock environment forecast Ct ~ N ( µ , σ ) 2 update Rt C Order Dt release It Qt −1 Pt = min{Qt −1 , Ct }
3. 3. Objective Minimize steady-state Inventory holding costs h Shortage penalty costs b
4. 4. 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
5. 5. 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
6. 6. 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 )
7. 7. Benchmark: Myopic Policy Do not use available forecast information Rt = Dt Constant Inventory Qt + I t = sm
8. 8. Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion
9. 9. In Heavy Traffic, the WIP has an exponential distribution Average Excess Capacity 2( µ − λ ) v= T e Σe + σ C 2 Variance of the forecasts Variance of the production
10. 10. WIP at time n n X n = ∑ ( Rt − Ct ) units t =1 Qn = X n − inf1≥t ≥ n X t R2 − C2 time
11. 11. Heavy traffic analysis 1 Consider a sequence of systems k s.t.: λ →λ (k ) µ →µ (k ) k (λ −µ )→c<0 (k ) (k )
12. 12. Heavy traffic analysis 2 − inf1≥t ≥ n X (k ) (k ) (k ) Q X = + n n t k k k −m ⎢⎥ (k ) (k ) (k ) ⎢ kt ⎥ ⎣ kt ⎦ (k ) X X ⎣⎦ m ⎢ kt ⎥ ⎢ kt ⎥ = + ⎣⎦ ⎣⎦ k k k where m = λ − µ (k ) (k ) (k )
13. 13. Heavy traffic analysis 2 σ B(t ) ct −m ⎢ kt ⎥ (k ) (k ) (k ) ⎢ kt ⎥ ⎣⎦ (k ) X X ⎣⎦ m ⎢ kt ⎥ ⎢ kt ⎥ ⎣⎦ ⎣⎦ = + k k k BM (c, σ ) 2
14. 14. Heavy traffic analysis 3 − inf1≥t ≥ n X (k ) (k ) (k ) Q X = + n n t k k k RBM (c, σ )2 Reflected Brownian Motion on the nonnegative halfline Estimated by an exponential random variable
15. 15. Steady-state WIP distribution − vβ P(Q∞ = 0) = 1 − e impulse (x + β ) − vx P(Q∞ > x) = e β is a correction term, coming from Random Walks
16. 16. Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion
17. 17. Determination of the Base- Stock levels 1 Myopic Policy ⎛b⎞ −1 sm = F ⎜ ⎟ ⎝b+h⎠ Q∞ Newsboy quantity… … considering the distr. of the WIP! 1 ⎛ b⎞ sm = ln ⎜1 + ⎟ − β v ⎝ h⎠
18. 18. Determination of the Base- Stock levels 2 MRP-type Policy ⎛b⎞ −1 sH = F ⎜ ⎟ ⎝b+h⎠ W W = max{Q∞ + Y0 , max1≤ k ≤ H Yk } Y0 is the difference between: Total Forecast Error over the horizon H and Total Capacity
19. 19. Determination of the Base- Stock levels 3 MRP-type policy: asymptotic b h ⎛1⎞ 2 s = s + µY 0 + ⎜ ⎟ σ Y0 v a * H m ⎝2⎠ Proportional to the variance! Good approximation when b / h large High utilization rate
20. 20. Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion
21. 21. Capacity – Stock Trade-Off No advance information s = s − Hλ a * H m Full advance information ⎛σ ⎞ 2 s = s − H λ − H (µ − λ ) ⎜ 2 a * 2⎟ D ⎝ σ D +σC ⎠ H m Interchangeability Capacity/Safety Stock Demand variability Capacity
22. 22. Discussion Correlation ↑→ v ↓→ sm ↑ * σ Y0 is the system variability over H not 2 resolved at the beginning of the horizon Preference for accurate early forecasts Optimize over all planning horizons H −1 ∑ε Rt = + Dt , t + H t , t +1 i=0 Greater costs due to misspecification of the forecast model than misuse of the information in production
23. 23. Outline Model Steady-State Distribution of WIP Base-Stock Levels Discussion Conclusion
24. 24. Conclusion Integrated view Forecast Production Inventory Lots of improvement for current MRP systems Is heavy traffic of practical value?
25. 25. Thank you Questions?