Reducing the Cost of Demand
           Uncertainty through Accurate
             Response to Early Sales
                 ...
Fashion Industry
Long lead time
Unpredictable demand
Complex Supply Chain
Enormous inventory loss
  25% of retail sales
Lo...
Quick Response
Lead time reduction through
  Efforts in IT,
  Logistics improvement
  Reorganization of production process...
This Paper
Provides a model for response-based
production planning
A two-stage stochastic program
Use relaxations to obtai...
Time Line
                                    Full Price      Markdown
                                     season        ...
Notation
n
xi0
xi
Di0                 (For notation descriptions, see page
                    90 of the Fisher and Raman ...
Notation (2)
fi(Di0,Di)
gi(Di0)
hi(Di|Di0)   (For notation descriptions, see page
f(D0,D)      90 of the Fisher and Raman ...
Model
Expected cost of demand mismatch:
        ci ( xi , Di ) = Oi ( xi − Di ) + + U i ( Di − xi ) +
                    ...
Z(x0) is convex
               Z 1 ( x, D0 ) = ED|D0 c( x, D)
               Z 2 ( x0 , D0 ) = min Z 1 ( x, D0 )
   (P2)  ...
Approximation to P
“Replace the capacity constraint in 2nd period with a lower limit on total
1st period production”
     ...
Solve (P) for a Given L
1. Given hi(Di|Di0),
                                                     W ( L) = min Z ( x0 )   ...
Choosing L
x0 (L) – value of x0 that solves P with given L
Use Monte Carlo generation of D0 to evaluate
Z(x0 (L))
Choose L...
Lower bounds on W(L)
Relaxing the constraint x >= x0 , we get:
               W 2 ( L) = ED0 min ED|D0 c( x, D)
          ...
Minimum Production Quantities
Let Sj defines the sets of products that satisfies a
min. initial production level Mj0, j=1,...
Minimum Production Quantities
The modified problem can be solved as before with 2 changes:
   Only allow movement away fro...
Sport Opermeyer
                                                            Full Price     Markdown
                      ...
Sport Obermeyer Application
L = 0.4D
Ui = wholesale price – variable cost
Oi = variable cost – salvage value
     (a conse...
µ




     Demand Density Estimation
    Di0 and Di follow a bivariate normal distrn




    Estimate µi0, µi, σi0, σi and...
Estimate
  Assume that ρi is the same for all products
within 5 major product categories
  Estimate ρI as the correlation ...
Proposition 1
           ⎡          (1 − ρi2 ) ⎤
σ io = σ i ⎢ ρi − δ i         2⎥
                      (1 − δ i ) ⎥
     ...
Closed-form solution
Assume that all products have the same Ui and Oi,
and have normally distributed demand with the same ...
Results



(See Table 1 on page 97 of the
  Fisher and Raman paper.)
Bounding Results


(See Figures 6 and 7 on page 98 of the
      Fisher and Raman paper.)
Summary
Provides a model for response-based
production planning
A two-stage stochastic program
Use relaxations to obtain f...
Critique
Clear and practical motivation
Take advantage of both expert opinion and early
demand information
Simple model
So...
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Reducing The Cost Of Demand Uncertainty Through Accurate Response To Early Sales Short Life Cycle Product Management - Hongmin Li

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Reducing The Cost Of Demand Uncertainty Through Accurate Response To Early Sales Short Life Cycle Product Management - Hongmin Li

  1. 1. Reducing the Cost of Demand Uncertainty through Accurate Response to Early Sales Fisher and Raman 1996 Operations Research, vol. 44 (1), 87-99 Presentation by Hongmin Li This summary presentation is based on: Fisher, Marshall, and A. Raman. quot;Reducing the Cost of Demand Uncertainty Through Accurate Response to Early Sales.quot; Operations Research 44, no. 1 (1996): 87-99.
  2. 2. Fashion Industry Long lead time Unpredictable demand Complex Supply Chain Enormous inventory loss 25% of retail sales Lost Sales
  3. 3. Quick Response Lead time reduction through Efforts in IT, Logistics improvement Reorganization of production process Complications Production planning (how much and when) Need a method to incorporate observed demand information
  4. 4. This Paper Provides a model for response-based production planning A two-stage stochastic program Use relaxations to obtain feasible solutions and bounds Implementation at Sport Obermeyer
  5. 5. Time Line Full Price Markdown season season Mid Feb May Sept.1 Dec. 31 spring Reorders Place Orders LT=1 (filled from avail. Inv) Jan.1 Sept.1 Production period
  6. 6. Notation n xi0 xi Di0 (For notation descriptions, see page 90 of the Fisher and Raman paper) Di Oi Ui K ci(xi,Di) = Oi(xi–Di)+ + Oi (Di -xi)+
  7. 7. Notation (2) fi(Di0,Di) gi(Di0) hi(Di|Di0) (For notation descriptions, see page f(D0,D) 90 of the Fisher and Raman paper) g( D 0 ) h(D|D0)
  8. 8. Model Expected cost of demand mismatch: ci ( xi , Di ) = Oi ( xi − Di ) + + U i ( Di − xi ) + n c( x, D) = ∑ ci ( xi , Di ) i =1 ∞ ED| D0 c( x, D) = ∫ c( x, D)h( D | D0 )dD 0 Choose x0, observe D0, then choose x to minimize total over and under production cost: Z * = min Z ( x0 ) = EDo min ED| D0 c ( x, D) x0 ≥ 0 x0 ≥ 0 n n (P) ∑x ≤ K + ∑ xi 0 i i =1 i =1
  9. 9. Z(x0) is convex Z 1 ( x, D0 ) = ED|D0 c( x, D) Z 2 ( x0 , D0 ) = min Z 1 ( x, D0 ) (P2) x ≥ x0 n n ∑x ≤ K + ∑ xi 0 i i =1 i =1 However, no closed ci(xi,Di) is convex in xi ⇒ c(x, D) convex in x form expression for x ⇒ Z1(x, D0) convex in x in D0, so convex opt. ⇒ Z2 convex in x0 is not tractable
  10. 10. Approximation to P “Replace the capacity constraint in 2nd period with a lower limit on total 1st period production” n ∑ Define L = * x i*0 x0* - optimal value of x0 i =1 W ( L) = min ED0 min ED|D0 c( x, D) W ( L) = min Z ( x0 ) x0 ≥ 0 x 0≥0 x ≥ x0 n n ∑x = L where Z ( x0 ) = ∑ EDi 0 min EDi|Di 0 ci ( xi , Di ) n n n ∑ xi 0 = L∑ xi ≤ K + ∑ xi 0 i0 xi ≥ xi 0 i =1 i =1 i =1 i =1 i =1 (P) (P)
  11. 11. Solve (P) for a Given L 1. Given hi(Di|Di0), W ( L) = min Z ( x0 ) x0 ≥ 0 xi* (Di0) = argmin ci(xi,Di) n n ∑ xi 0 = L where Z ( x0 ) = ∑ EDi 0 min EDi|Di 0 ci ( xi , Di ) xi ≥ xi 0 (opt. newsboy solution) i =1 i =1 xi = max(xi* (Di0), xi0) solves min EDi|Di0 ci(xi,Di) (P) Then Z(x0) can be computed using numerical integration 2. If Di0 and Di are positively correlated, There exists a Di0* s.t. for Di0≤Di0*, xi = xi0 and for Di0>Di0* , xi = xi* (Di0) Partials of Z can be approximated with differences => xi0*
  12. 12. Choosing L x0 (L) – value of x0 that solves P with given L Use Monte Carlo generation of D0 to evaluate Z(x0 (L)) Choose L by line search algorithm to the problem minL>=0 Z(x0 (L))
  13. 13. Lower bounds on W(L) Relaxing the constraint x >= x0 , we get: W 2 ( L) = ED0 min ED|D0 c( x, D) x ≥0 (P2) n ∑x ≤K+L i i =1 W(L) is a lower bound W2(L) is a lower bound (a constrained Newsboy problem that can be solved with lagrangian methods.) W(L) is smallest when L=0, W2(L) is largest when L=0 ⇒ minL>=0 max ( W(L), W2(L) ) is a nontrivial bound on Z*
  14. 14. Minimum Production Quantities Let Sj defines the sets of products that satisfies a min. initial production level Mj0, j=1,…, m (See equations on page 92, left hand column, of Fisher and Raman paper) The RHS is evaluated by the following problem: (See equations on can be solved page 92, left hand similarly as P2 column, of Fisher and Raman paper)
  15. 15. Minimum Production Quantities The modified problem can be solved as before with 2 changes: Only allow movement away from x0j =0 to a point that satisfies the min production constraints Change the estimation of the derivative of W(L) w.r.t. xij at xij =0 to: (See equations on page 92, left hand column, of Fisher and Raman paper) Zj* has the same form as (P), thus can be solved similarly
  16. 16. Sport Opermeyer Full Price Markdown season season Mid Feb May Sept.1 Dec. 31 spring LT=1 Place Orders Reorders (filled from avail. Inv) Nov. Sept. Oct. (1yrs ago) (1yrs ago) (2yrs ago) Commit 40% Forecast Design Sept.1 Jan.1 Production period
  17. 17. Sport Obermeyer Application L = 0.4D Ui = wholesale price – variable cost Oi = variable cost – salvage value (a conservative measure) In general, Ui = 3~4Oi
  18. 18. µ Demand Density Estimation Di0 and Di follow a bivariate normal distrn Estimate µi0, µi, σi0, σi and ρi (correlation) yij sales estimate of product i by member j Actual deviation is well correlated with predicted standard deviation
  19. 19. Estimate Assume that ρi is the same for all products within 5 major product categories Estimate ρI as the correlation between total and initial demand in the previous season Let k be the fraction of season sales in period 1 => µ i 0 = ku i Let δ be the correlation coeff. between Di0 and (Di - Di0 )=> σ = σ ⎡ ρ − δ (1 − ρ ) ⎤ (Proposition 1) 2 ⎢ ⎥ i i0 i ⎢ (1 − δ ) ⎥ i i 2 ⎣ ⎦ i
  20. 20. Proposition 1 ⎡ (1 − ρi2 ) ⎤ σ io = σ i ⎢ ρi − δ i 2⎥ (1 − δ i ) ⎥ ⎢ ⎣ ⎦ (For equations and explanation, see Proposition 1 on page σi2 - std of (Di - Di0 ) 95 of the Fisher and Raman paper) Q-Q plot shows that the student’s t distribution with t=2 may be better approximation.
  21. 21. Closed-form solution Assume that all products have the same Ui and Oi, and have normally distributed demand with the same ρ (See Theorem 2 on page 96 of the Fisher and Raman paper.)
  22. 22. Results (See Table 1 on page 97 of the Fisher and Raman paper.)
  23. 23. Bounding Results (See Figures 6 and 7 on page 98 of the Fisher and Raman paper.)
  24. 24. Summary Provides a model for response-based production planning A two-stage stochastic program Use relaxations to obtain feasible solutions and bounds Results at Sport Obermeyer
  25. 25. Critique Clear and practical motivation Take advantage of both expert opinion and early demand information Simple model Some details of the model is not explained very clearly Accuracy of the lower bounds is not discussed Application did not use the approximate method developed in the paper

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