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. "Reducing the Cost of Demand
Uncertainty Through Accurate Response to Early Sales." Operations Research 44, no. 1 (1996): 87-99.
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)
(For notation descriptions, see page
90 of the Fisher and Raman paper)
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
(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 P2
Minimum Production Quantities
¾
The modified problem can be solved as before with 2 changes:
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:
(See equations on page 92, left hand
column, of Fisher and Raman paper)
z
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.
(For equations and
explanation, see
Proposition 1 on page
95 of the Fisher and
Raman paper)
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.)
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
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