Robustness Analysis of GEMTEX the Newsboy Problem P.L.Douillet - - PowerPoint PPT Presentation

GEMTEX P.L.Douillet iesm 2005

G.E.M.T.E.X (Roubaix, France)

Robustness Analysis of the Newsboy Problem

pierre.douillet@ensait.fr besoa.rabenasolo@ensait.fr

GEMTEX P.L.Douillet iesm 2005

robustness is a key concern

probability distributions are often used to express a limited knowledge too often, side assumptions are introduced that are not founded on that actual knowledge, but

• nly on computational facilities

the robustness of the conclusions drawn

must be checked !

GEMTEX P.L.Douillet iesm 2005

y c r min y, G y, rcy

the newsboy paradigm

Scarf’s notations : order quantity, : demand cdf, : unit cost, : unit selling price satisfied demand non sold units are discarded actual gain

GEMTEX P.L.Douillet iesm 2005

G y, E G y,

• cyr

y 0 d y

• y d

G0G µ, µ y 1 c r

a well known formula

naive solution expected gain analytical solution

GEMTEX P.L.Douillet iesm 2005

• y d

left, right

l 1 1

y 0 d

r 1

• y d

G0G y, 1 yl ry : y : G y, G0

the cost of uncertainties

define and by and and obtain thus

GEMTEX P.L.Douillet iesm 2005

µ 1000 c12 r20 /µ 0, .1, .2, .4

a comparative study

using different models normal lognormal triangular “two Diracs (Scarf’s model)” and the following parameters fixed , , variable (namely )

GEMTEX P.L.Douillet iesm 2005

µ

(r-c) µ µ y

/µ

normal model

additive independence (consumers ?) necessitates small values of

GEMTEX P.L.Douillet iesm 2005

µ

(r-c) µ µ y

lognormal model

multiplicative independence (atmospherics ?) special shape of the maximum locus

GEMTEX P.L.Douillet iesm 2005

µ

(r-c) µ µ y

triangular model

positive values, three parameters, easy to use have you a knowledge against that model ?

GEMTEX P.L.Douillet iesm 2005

µ

2

• 2
• 12 /

µ 1 3

• 2 1

36 2 2 2

• 2

12 32 M3 1 1080 3 92

several formulae

• ,

skewness ,

GEMTEX P.L.Douillet iesm 2005

(r-c) µ µ y

when c/r < 1/2

GEMTEX P.L.Douillet iesm 2005

• discussion about hypotheses

does model a lack of knowledge due e.g. to their cost or model the intrinsic wild behavior of the market ? is an average over all the many parallel independent worlds or is induced from an assumed ergodic property of historical data ? can be ever measured, even afterwards, when the demand overflows the inventory ?

GEMTEX P.L.Douillet iesm 2005

G0Gµ r 1 rl 1 rl

yµ y 1 lr yµ Gµ G µ, Pr >µ lE |<µ r

the naive and obstinate merchant

when , holds when , verifies since , and the quantity is a measure of the dispersion of the demand

GEMTEX P.L.Douillet iesm 2005

1 rl

/ / 3/4 1/ 2 < 1/ 2 1/ 6 .. 8 2/27

the "inter-mean" interval

distribution exact approx uniform 0.433 normal 0.399 lognormal < 0.399 triangular 0.408 .. 0.419 general 0.5 ?

GEMTEX P.L.Douillet iesm 2005

if c r 1 2 µ2 <1 then yrobustµ r/2c c rc

• therwise

yrobust0 d 1 Dirac() Dirac()

• µ,

y Grobust maxy min|µ, G y,

recalling the Scarf's bound

Scarf’s functions

• ver all the

that shares the same , the worst distribution against a given order quantity is a Scarf’s function thus is

GEMTEX P.L.Douillet iesm 2005

µ, µ, µ 1000 600 300 c/r5/9 cr10

comparison

Scarf’s max-min using fixed max-min using fixed , , ,

GEMTEX P.L.Douillet iesm 2005

8000 3292 G 0.73 0.56

• 1

θ

graphical proof of Scarf's theorem

GEMTEX P.L.Douillet iesm 2005

3250 G

• 1

0.7 0.56 θ

using the same method

GEMTEX P.L.Douillet iesm 2005

1 rl