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Analysis of a Forecasting-
     Production-Inventory System with
     Stationary Demand

              L. Beril Toktay and...
Forecasting-Production-
        Inventory
                                              Demand
          Make-to-Stock env...
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 ...
Production-Inventory Model

 MRP-Type Release Policy:
           H −1
     Rt = ∑ ε t ,t +i + Dt ,t + H = eT ε t + λ
     ...
Production Policy

  Forecast-corrected base-stock policy

                   ⎧Ct           if sH > I t −1 + Ct ⎫
        ...
Benchmark: Myopic Policy

 Do not use available forecast
 information

           Rt = Dt
 Constant Inventory

          Q...
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 − ...
WIP at time n      n
             X n = ∑ ( Rt − Ct )
units
                   t =1


                          Qn = X n −...
Heavy traffic analysis 1

Consider a sequence of systems k s.t.:

                     λ          →λ
                     ...
Heavy traffic analysis 2
                                         − inf1≥t ≥ n X
             (k )            (k )        ...
Heavy traffic analysis 2

                               σ B(t )                        ct

                              ...
Heavy traffic analysis 3
                              − inf1≥t ≥ n X
    (k )           (k )                         (k )...
Steady-state WIP distribution

                           − vβ
    P(Q∞ = 0) = 1 − e                 impulse

            ...
Outline

  Model
  Steady-State Distribution of WIP
  Base-Stock Levels
  Discussion
  Conclusion
Determination of the Base-
Stock levels 1

  Myopic Policy

                ⎛b⎞
                −1
         sm = F ⎜   ⎟
 ...
Determination of the Base-
Stock levels 2
  MRP-type Policy
                ⎛b⎞
                  −1
         sH = F ⎜   ⎟...
Determination of the Base-
Stock levels 3

   MRP-type policy: asymptotic
          b    h
                         ⎛1⎞ 2
...
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...
Discussion
 Correlation ↑→ v ↓→ sm ↑
                                *

 σ Y0 is the system variability over H not
   2

 ...
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 heav...
Thank you


     Questions?
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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?

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