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

G.E.M.T.E.X (Roubaix, France) Robustness Analysis of GEMTEX the Newsboy Problem P.L.Douillet iesm 2005 pierre.douillet@ensait.fr besoa.rabenasolo@ensait.fr

� 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 GEMTEX only on computational facilities � the robustness of the conclusions drawn P.L.Douillet must be checked ! iesm 2005 robustness is a key concern

� Scarf’s notations : order quantity, : demand cdf, y � � : unit cost, : unit selling price c r GEMTEX satisfied demand � � � min y , � P.L.Douillet � non sold units are discarded iesm 2005 actual gain G y , � � r � � � cy the newsboy paradigm

� naive solution G 0 � G µ, µ � expected gain G y , � � E G y , � � GEMTEX � cy � r � y 0 � d � � � y � y d � � � P.L.Douillet iesm 2005 � y � � 1 � c � analytical solution r a well known formula

� left , � right � define y d � � and by � � � � 1 � r � 1 y 0 � d � � y � d � � and � 1 � � � � � � l � GEMTEX � and obtain P.L.Douillet G 0 � G y , � � 1 � � y � � l � � � r � y iesm 2005 � � : � y : G y , � � G 0 � thus the cost of uncertainties

using different models � normal � lognormal � triangular GEMTEX � “two Diracs (Scarf’s model)” P.L.Douillet and the following parameters iesm 2005 � fixed µ � 1000 c � 12 r � 20 , , � variable � /µ (namely 0, .1, .2, .4 ) a comparative study

(r-c) µ GEMTEX P.L.Douillet 0 y µ µ iesm 2005 � additive independence (consumers ?) � necessitates small values of � /µ normal model

(r-c) µ GEMTEX P.L.Douillet 0 y µ µ iesm 2005 � multiplicative independence (atmospherics ?) � special shape of the maximum locus lognormal model

(r-c) µ GEMTEX P.L.Douillet 0 y µ µ iesm 2005 � positive values, three parameters, easy to use � have you a knowledge against that model ? triangular model

� � � 2 � � � � � � � � � � � � � � � 2 � � � � � � � � � 0 µ GEMTEX � �� 1 � 2 � � � / � � � µ � 1 � 2 � 1 P.L.Douillet , � � � 2 � � � � 2 � � � � 2 � � � � � � 3 36 iesm 2005 12 � 3 � � 2 M 3 � � skewness 2 , � 1 1080 � 3 � 9 � � 2 � � several formulae

(r-c) µ GEMTEX P.L.Douillet iesm 2005 y µ when c/r < 1/2

� 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 � GEMTEX independent worlds or is induced from an � assumed ergodic property of historical data ? P.L.Douillet � can be ever measured, even afterwards, when iesm 2005 � the demand overflows the inventory ? discussion about hypotheses

y � 1 � � � l � �� r � when y � µ , holds G µ � G µ, � y � µ � when , verifies G 0 � G µ � r � 1 � � � r � � l GEMTEX � l � E � | � <µ � since , and � � Pr � >µ � r � � P.L.Douillet the quantity iesm 2005 � � � 1 � � � r � � l is a measure of the dispersion of the demand the naive and obstinate merchant

� � � 1 � � � r � � l � / � � / � distribution exact approx uniform 0.433 3/4 normal 0.399 1/ 2 � GEMTEX lognormal < 0.399 < 1/ 2 � P.L.Douillet triangular 0.408 .. 0.419 1/ 6 .. 8 2/27 general � 0.5 ? iesm 2005 the "inter-mean" interval

� Scarf’s functions d � � � 1 � � Dirac( � � � ) � � Dirac( � � � ) � over all the that shares the same µ, � , the � worst distribution against a given order quantity is a Scarf’s function y GEMTEX � thus G robust � max y min � |µ, � G y , � is P.L.Douillet 1 � � 2 <1 then y robust � µ � � r /2 � c if c iesm 2005 r µ 2 c r � c otherwise y robust � 0 recalling the Scarf's bound

� Scarf’s max-min using fixed µ, � � max-min using fixed µ, � GEMTEX µ � 1000 � � 600 � � 300 , , P.L.Douillet c / r � 5/9 c � r � 10 , iesm 2005 comparison

8000 G 3292 GEMTEX P.L.Douillet 0 iesm 2005 -1 0.56 0.73 θ graphical proof of Scarf's theorem

G 3250 GEMTEX P.L.Douillet 0 iesm 2005 -1 0.56 0.7 θ using the same method

GEMTEX � � � 1 � � � r � � l P.L.Douillet iesm 2005