Reducing the Cost of Demand Uncertainty through Accurate Response to Early Sales

Transcription

1 Reducing the Cost of Demand Uncertainty through Accurate Response to Early Sales Fisher and Raman 1996 Operations Research,, vol. 44 (1), Presentation by Hongmin Li This summary presentation is based on: Fisher, Marshall, and A. Raman. "Reducing the Cost of Demand Uncertainty Through Accurate Response to Early Sales." Operations Research 44, no. 1 (1996):

2 Fashion Industry Long lead time Unpredictable demand Complex Supply Chain Enormous inventory loss 25% of retail sales Lost Sales

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 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 Time Line Full Price season Markdown season Mid Feb May Sept.1 Dec. 31 spring LT=1 Place Orders Reorders (filled from avail. Inv) Jan.1 Sept.1 Production period

6 Notation n x i0 x i D i0 D i (For notation descriptions, see page 90 of the Fisher and Raman paper) O i U i K c i (x i,d i ) = O i (x i D i ) + + O i (D i -x i ) +

7 Notation (2) f i (D i0,d i ) g i (D i0 ) h i (D i D i0 f(d 0,D) g(d 0 ) h(d D 0 ) i0 ) (For notation descriptions, see page 90 of the Fisher and Raman paper)

8 Model Expected cost of demand mismatch: c ( x, D) = O( x D) + U ( D x ) + + i i i i i i i i i n cxd (, ) = c( x, D) i= 1 i i i E c( x, D) = c( x, D) h( D D ) dd DD Choose x 0, observe D 0, then choose x to minimize total over and under production cost: (P) * Z = Z x0 = ED ED D c x D x 0 x 0 0 n min ( ) min (, ) o 0 0 n x K + x i i= 1 i= 1 i0

9 Z(x 0 ) is convex (P 2 ) 1 Z x D0 = EDD c x D (, ) (, ) Z ( x, D ) = min Z ( x, D ) n x x n x K + x i i= 1 i= i0 c i (x i,d i ) is convex in x i c(x, D) convex in x Z 1 (x, D 0 ) convex in x Z 2 convex in x0 However, no closed form expression for x in D 0, so convex opt. is not tractable

10 Approximation to P Replace the capacity constraint in 2 nd period with a lower limit on total 1 st period production x 0* - optimal value of x 0 Define L * = n i = 1 * x i 0 W( L) = mine min E c( x, D) D x 0 x x D D n n n 0 x = L x K + x i0 i i0 i= 1 i= 1 i= 1 W( L) = min Z( x ) x 0 n 0 0 x = L where Z( x ) = E min E c ( x, D ) i0 0 Di Di Di0 i i i xi xi0 i= 1 i= 1 n (P) (P)

11 Solve (P)( ) for a Given L 1. Given h i (D i D i0 ), x i* (D i0 ) = argmin c i (x i,d i ) (D i0 (opt. newsboy solution) x i = max(x i* (D i0 ), x i0 ) solves min E Di Di0 c i (x i,d i ) Then Z(x(x 0 ) can be computed using numerical integration 2. If D i0 and D i are positively correlated, There exists a D i0* s.t. for D i0 and for D i0 i0 >D i0 i0 D i0 (D i0) i0*, x i = x i* (D i0*, x i = x i0 W( L) = min Z( x ) x 0 n 0 0 x = L where Z( x ) = E min E c ( x, D) i0 0 Di0 Di Di0 i i i xi xi0 i= 1 i= 1 Partials of Z can be approximated with differences => x * i0 n (P)

12 Choosing L x 0 (L) value of x 0 that solves P with given L Use Monte Carlo generation of D 0 to evaluate Z(x 0 (L)) Choose L by line search algorithm to the problem min L>=0 Z (x 0 (L))

13 Lower bounds on W(L) Relaxing the constraint x >= x 0, we get: (P 2 ) W ( L) = E min E c( x, D) n i= 1 2 D D D x 0 x K + L i 0 0 W(L) is a lower bound W 2 (L) is a lower bound (a constrained Newsboy problem that can be solved with lagrangian methods.) W(L) is smallest when L=0, W 2 (L) is largest when L=0 min L>=0 max ( W(L), W 2 (L) ) is a nontrivial bound on Z*

14 Minimum Production Quantities Let S j defines the sets of products that satisfies a min. initial production level M j0, 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 page 92, left hand column, of Fisher and Raman paper) can be solved similarly as P 2

15 Minimum Production Quantities The modified problem can be solved as before with 2 changes: Only allow movement away from x 0j =0 to a point that satisfies the min production constraints Change the estimation of the derivative of W(L) w.r.t. x ij at x ij =0 to: (See equations on page 92, left hand column, of Fisher and Raman paper) Z j* has the same form as (P), thus can be solved similarly

16 Sport Opermeyer Full Price season Markdown season Mid Feb May Sept.1 Dec. 31 spring Sept. (2yrs ago) Design Oct. (1yrs ago) Forecast Nov. (1yrs ago) Jan.1 LT=1 Commit 40% Place Orders Sept.1 Reorders (filled from avail. Inv) Production period

17 Sport Obermeyer Application L = 0.4D U i = wholesale price variable cost O i = variable cost salvage value (a conservative measure) In general, U i = 3~4O i

18 µ Demand Density Estimation D i0 and D i follow a bivariate normal distrn Estimate Estimate µ i0, µ i, σ i0, σ i and ρ i (correlation) y ij sales estimate of product i by member j Actual deviation is well correlated with predicted standard deviation

19 Estimate Assume that ρ i is the same for all products within 5 major product categories Estimate ρ I as the correlation between total and initial demand in the previous season Let k be the fraction of season sales in period 1 => µ = ku i0 i Let δ be the correlation coeff.. between D i0 2 )=> (1 ρ i ) (Proposition 1) (D i - D i0 )=> σ i0 = σ i ρi δi 2 (1 δ i ) i0 and

20 Proposition 1 σio = σi ρi δi 2 (1 ρi ) 2 (1 δi ) σ i2 - std of (D i - D i0 ) (For equations and explanation, see Proposition 1 on page 95 of the Fisher and Raman paper) Q-Q Q plot shows that the student s s t distribution with t=2 may be better approximation.

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

23 Bounding Results (See Figures 6 and 7 on page 98 of the Fisher and Raman paper.)

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 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