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

Reducing the Cost of Demand Uncertainty through Accurate Response to Early Sales Fashion Industry ¾ Long lead time ¾ Unpredictable demand ¾ Complex Supply Chain ¾ Enormous inventory loss z 25% of retail sales ¾ Lost Sales Quick Response ¾ Lead time reduction through z z z Efforts in IT, Logistics improvement Reorganization of production process ¾ Complications z z Production planning (how much and when) Need a method to incorporate observed demand information This Paper ¾ Provides a model for response-based production planning ¾ A two-stage stochastic program ¾ Use relaxations to obtain feasible solutions and bounds ¾ Implementation at Sport Obermeyer Time Line Full Price season Mid Feb LT=1 May Sept.1 Sept.1 Production period Dec. 31 spring Reorders (filled from avail. Inv) Place Orders Jan.1 Markdown season Notation ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ n xi0 xi (For notation descriptions, see page Di0 90 of the Fisher and Raman paper) Di Oi Ui K ci(xi,Di) = Oi(xi–Di)+ + Oi (Di -xi)+ Notation (2) ¾ fi(Di0,Di) ¾ gi(Di0) ¾ hi(Di|Di0) ¾ f(D0,D) ¾ ¾ g( D 0 ) h(D|D0) 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 (P) n ∑x i =1 i x0 ≥ 0 n ≤ K + ∑ xi 0 i =1 Z(x0) is convex Z 1 ( x, D0 ) = ED|D0 c( x, D) (P2) Z 2 ( x0 , D0 ) = min Z 1 ( x, D0 ) x ≥ x0 n ∑x i =1 i n ≤ K + ∑ xi 0 ci(xi,Di) is convex in xi ⇒ c(x, D) convex in x ⇒ Z1(x, D0) convex in x ⇒ Z2 convex in x0 i =1 However, no closed form expression for x in D0, so convex opt. is not tractable Approximation to P “Replace the capacity constraint in 2nd period with a lower limit on total 1st period production” x0* - optimal value of x0 W ( L) = min ED0 min ED|D0 c( x, D) x 0≥0 n x ≥ x0 n n ∑ xi 0 = L∑ xi ≤ K + ∑ xi 0 i =1 i =1 Define L * = ∑ x i*0 i =1 W ( L) = min Z ( x0 ) n ∑x i =1 i0 x0 ≥ 0 n = L where Z ( x0 ) = ∑ EDi 0 min EDi|Di 0 ci ( xi , Di ) i =1 i =1 (P) n (P) xi ≥ xi 0 Solve (P) for a Given L 1. Given hi(Di|Di0), xi* (Di0) = argmin ci(xi,Di) (opt. newsboy solution) W ( L) = min Z ( x0 ) x0 ≥ 0 n n i =1 i =1 ∑ xi 0 = L where Z ( x0 ) = ∑ EDi 0 min EDi|Di 0 ci ( xi , Di ) xi = max(xi* (Di0), xi0) solves min EDi|Di0 ci(xi,Di) Then Z(x0) can be computed using numerical (P) 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* xi ≥ xi 0 Choosing L ¾ ¾ ¾ 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)) Lower bounds on W(L) Relaxing the constraint x >= x0 , we get: (P2) W 2 ( L) = ED0 min ED|D0 c( x, D) x ≥0 n ∑x i =1 ¾ ¾ ¾ ⇒ i ≤K+L 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* Minimum Production Quantities Let Sj defines the sets of products that satisfies a min. initial production level Mj0, j=1,…, m 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 P2 Minimum Production Quantities ¾ The modified problem can be solved as before with 2 changes: z z z 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: Zj* has the same form as (P), thus can be solved similarly Sport Opermeyer Full Price season Mid Feb May Sept.1 LT=1 Place Orders Sept. (2yrs ago) Commit 40% Forecast Jan.1 Dec. 31 spring Reorders (filled from avail. Inv) Nov. Oct. (1yrs ago) (1yrs ago) Design Markdown season Sept.1 Production period Sport Obermeyer Application ¾ L = 0.4D ¾ Ui = wholesale price – variable cost ¾ Oi = variable cost – salvage value (a conservative measure) ¾ In general, Ui = 3~4Oi µ Demand Density Estimation ¾Di0 and Di follow a bivariate normal distrn ¾Estimate 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 Estimate that ρi is the same for all products within 5 major product categories ¾Estimate ρI as the correlation between total and initial demand in the previous season ¾Assume ¾Let k be the  i => µl i 0 = ku fraction of season sales in period 1 δ be the correlation coeff. between Di0 and (Di - Di0 )=> σl = σl ⎡⎢ ρ − δ (1 − ρl ) ⎤⎥ (Proposition 1) ¾Let 2 i0 i ⎢ ⎣ i i i 2 i (1 − δ ) ⎥⎦ Proposition 1 ⎡ (1 − ρi2 ) ⎤ σ io = σ i ⎢ ρi − δ i 2 ⎥ (1 − δ i ) ⎥⎦ ⎢⎣ σi2 - std of (Di - Di0 ) Q-Q plot shows that the student’s t distribution with t=2 may be better approximation. Summary ¾ Provides a model for response-based production planning ¾ A two-stage stochastic program ¾ Use relaxations to obtain feasible solutions and bounds ¾ Results at Sport Obermeyer Critique ¾ ¾ ¾ ¾ ¾ ¾ 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