Using the Intermeans Parameter to Model the Dispersion of Demand - PowerPoint PPT Presentation

✬ ✩ CESA 2006 Using the Intermeans Parameter to Model the Dispersion of Demand pierre.douillet@ensait.fr besoa.rabenasolo@ensait.fr École Nationale Supérieure des Arts et Industries Textiles Roubaix, France ✫ ✪

✬ ✩ Douillet-Rabenasolo CESA 2006 ⇒ • the newsboy paradigm . . . . . . . . . . . . . 3 introduction cost of a given choice cost of uncertainties example : lognormal Φ , given µ • robust solutions . . . . . . . . . . . . . . . . 8 • the intermeans parameter . . . . . . . . . . . 13 • sampling properties . . . . . . . . . . . . . . 18 • conclusion . . . . . . . . . . . . . . . . . . . . 21 ✫ ✪ Ensait - Roubaix 2

✬ ✩ Douillet-Rabenasolo CESA 2006 the newsboy paradigm introduction • (Scarf’s notations) y : order quantity, Φ ( ξ ) : demand cdf, c : unit cost, r : unit selling price • unsold units are discarded • the satisfied demand is : ξ + = min ( y, ξ ) G . • naive solution : � = G ( µ, µ ) ... but actual gain (ex post) : G ( y, ξ ) = r ξ + − c y ✫ ✪ Ensait - Roubaix 3

✬ ✩ Douillet-Rabenasolo CESA 2006 cost of a given choice � ∞ • define θ y . dΦ ( ξ ) together with: = y . . ξ a ξ b = E ( ξ | y < ξ ) ; = E ( ξ | ξ < y ) y y • define G ( y, Φ) . = E ( G ( y, ξ )) ξ , obtain: � G − G ( y, Φ) = � � � � ξ a y − ξ b θ y y − y ( r − c ) + (1 − θ y ) c y • and conclude: ∀ Φ ∀ y : G ( y, Φ) ≤ � G ✫ ✪ Ensait - Roubaix 4

✬ ✩ Douillet-Rabenasolo CESA 2006 cost of uncertainties • knowing Φ , we have y ∗ = arg max y G ( y, Φ) • exact analytical solution : θ y ∗ = θ ∗ = 1 − Φ ( y ∗ ) = c/r • the corresponding (least) miss to gain can be rewritten as: � � � ξ a ∗ − ξ b G − G ( y ∗ , Φ) = θ ∗ (1 − θ ∗ ) r ∗ ✫ ✪ Ensait - Roubaix 5

✬ ✩ Douillet-Rabenasolo CESA 2006 example : lognormal Φ , given µ (r-c) µ µ y µ • positive values, but assume multiplicative independence • curve σ �→ ( y ∗ , G ∗ ) , assuming c/r < 1 / 2 ✫ ✪ Ensait - Roubaix 6

✬ ✩ Douillet-Rabenasolo CESA 2006 √ • the newsboy paradigm . . . . . . . . . . . . . 3 ⇒ • robust solutions . . . . . . . . . . . . . . . . 8 knowledge versus facilities basic questions Scarf’s theorem graphical proof • the intermeans parameter . . . . . . . . . . . 13 • sampling properties . . . . . . . . . . . . . . 18 • conclusion . . . . . . . . . . . . . . . . . . . . 21 ✫ ✪ Ensait - Roubaix 7

✬ ✩ Douillet-Rabenasolo CESA 2006 robust solutions knowledge versus facilities • probability distributions can be used to impersonate our actual knowledge about the real world... or about the limits of our knowledge • but, too often, side assumptions are introduced that does not come from the actual framework, but only from computing easiness ✫ ✪ Ensait - Roubaix 8

✬ ✩ Douillet-Rabenasolo CESA 2006 basic questions • does Φ model a lack of knowledge due e.g. to their cost or model the intrinsic wild behavior of the markets ? • is Φ guessed from many parallel independent worlds or induced from historical data (questionable ergodicity) ? • can ξ be ever measured, even afterwards, when the demand overflows the inventory ? • robust solution against a family F of distributions : G ∗ . = G ( y ∗ , F ) . = max Φ ∈F G ( y, Φ) min ✫ ✪ y Ensait - Roubaix 9

✬ ✩ Douillet-Rabenasolo CESA 2006 Scarf’s theorem • when playing against F ( µ, σ ) , the worst case is a Dirac two points distribution where ξ is either ξ a or ξ b • when ( c/r ) − 1 < 1 + σ 2 /µ 2 , then better buy nothing • otherwise, the robust decision is:  �  y ∗ = µ + σ ( r/ 2 − c ) / c ( r − c )    �   G ∗ = µ ( r − c ) − σ c ( r − c ) ✫ ✪ Ensait - Roubaix 10

✬ ✩ Douillet-Rabenasolo CESA 2006 graphical proof playing against the F ( µ, σ, Dirac ) family is: 8000 E(G) • chose an y , i.e. a curve • wait for answer G ( y, ξ ) 3292 • E ( G ) depends on θ 0 0 0.56 0.73 θ all curves are going through the same point θ = c/r ✫ ✪ Ensait - Roubaix 11

✬ ✩ Douillet-Rabenasolo CESA 2006 √ • the newsboy paradigm . . . . . . . . . . . . . 3 √ • robust solutions . . . . . . . . . . . . . . . . 8 ⇒ • the intermeans parameter . . . . . . . . . . . 13 cost of mean a measure of dispersion comparison δ versus σ a surprising result • sampling properties . . . . . . . . . . . . . . 18 • conclusion . . . . . . . . . . . . . . . . . . . . 21 ✫ ✪ Ensait - Roubaix 12

✬ ✩ Douillet-Rabenasolo CESA 2006 the intermeans parameter cost of mean � � • we have � ξ a µ − ξ b G − G ( µ, Φ) = θ µ (1 − θ µ ) × r µ • this kind of factorization applies only to y ∗ from θ ∗ = c/r and to µ from the obvious ∀ y : θ y ξ a y + (1 − θ y ) ξ b y = µ G − G ∗ ≤ δ r , independent of c/r , where • thus � � � δ . ξ a µ − ξ b = θ µ (1 − θ µ ) µ ✫ ✪ Ensait - Roubaix 13

✬ ✩ Douillet-Rabenasolo CESA 2006 a measure of dispersion • from now on, all θ, ξ a , ξ b are relative to µ � ξ a − ξ b � δ . = θ (1 − θ ) • this δ has some similarities with the interquartile range • properties : ξ b < µ − δ < µ + δ < ξ a and x y = δ 2 x y δ δ b a µ ξ µ−δ µ+δ ξ ✫ ✪ Ensait - Roubaix 14

✬ ✩ Douillet-Rabenasolo CESA 2006 comparison δ versus σ Φ δ/σ (exact) δ/σ (approx) 0 . 25 gap √ uniform 3 / 4 ≈ 0 . 433 √ √ triangular 1 / 6 · · · 8 2 / 27 . 408 · · · . 419 √ normal 1 / 2 π ≈ 0 . 399 exp 1/e ≈ 0 . 368 � Dirac θ (1 − θ ) 0 · · · 0 . 5 θ = 7% , 93% √ lognormal ≤ 1 / 2 π 0 . · · · 0 . 399 σ/µ ≈ 2 ✫ in all realistic situations : 0 . 25 σ ≤ δ ≤ 0 . 50 σ ✪ Ensait - Roubaix 15

✬ ✩ Douillet-Rabenasolo CESA 2006 a surprising result • when playing against F ( µ, δ ) , the worst case is not necessarily a two points distribution • when playing against G F ( µ, δ, Dirac ) , all curves are going through 3250 the same point θ = c/r • and now 0 G ∗ Dirac = µ 0 5/9 0.7 θ ✫ ✪ Ensait - Roubaix 16

✬ ✩ Douillet-Rabenasolo CESA 2006 √ • the newsboy paradigm . . . . . . . . . . . . . 3 √ • robust solutions . . . . . . . . . . . . . . . . 8 √ • the intermeans parameter . . . . . . . . . . . 13 ⇒ • sampling properties . . . . . . . . . . . . . . 18 sampling properties of variance sampling properties of δ experimental behavior • conclusion . . . . . . . . . . . . . . . . . . . . 21 ✫ ✪ Ensait - Roubaix 17

✬ ✩ Douillet-Rabenasolo CESA 2006 sampling properties sampling properties of variance • obtain sample ω by n independent drawings from Φ • put s 2 . = var ( ω ) and define S 2 . = s 2 × n/ ( n − 1) . Then: � n − 1 σ 4 � � S 2 � � S 2 � = 1 M 4 − n − 3 = σ 2 E ; var n � S 2 � � M 4 � n × var 2 = σ 4 − 1 + E 2 n − 1 ✫ ✪ Ensait - Roubaix 18

✬ ✩ Douillet-Rabenasolo CESA 2006 sampling properties of δ • d . = δ ( ω ) is well defined, even if x n = m d ( ω ) = d ( ω \ { x n } ) × ( n − 1) /n • define D = d × bias _ factor so that E ( D ) = δ (Φ) : � S 2 � n × var n × var Φ D/d E 2 ( D ) E 2 n 1 2 Dirac’s idem θ (1 − θ ) − 4 + n − 1 n − 1 2 / 3 n 1 4 2 unif. 3 + n − 26 / 45 + · · · 5 + n − 2 / 3 n − 1 • bias factor depends on Φ ✫ ✪ Ensait - Roubaix 19

✬ ✩ Douillet-Rabenasolo CESA 2006 experimental behavior • exact bias factors have not been found for other Φ • experiments : n = 4 , 7 , 10 , 13 and each time N = 1600 • using m ( d ) ≈ E ( d ) and S 2 ( d ) ≈ var ( d ) leads to: � S 2 � n × var n × var n × var Φ E 2 ( d ) E 2 ( S ) E 2 4 2 unif. ≈ 0 . 4 ≈ 0 . 3 5 + n − 1 2 gauss ≈ 0 . 6 ≈ 0 . 5 2 + n − 1 2 exp ≈ 1 . 5 ≈ 1 . 7 8 + n − 1 log ≈ 1 . 5 ≈ 1 . 7 ≈ 10 ✫ ✪ Ensait - Roubaix 20

✬ ✩ Douillet-Rabenasolo CESA 2006 conclusion • the best decision for the newsboy problem depends heavily on the choice of the dispersion measure • the well known "Scarf’s rule" follows when assuming an exact knowledge of µ, σ • but another strategy follows when assuming an exact knowledge of µ, δ • this happens while 0 . 3 ≤ δ/σ ≤ 0 . 5 for all relevant Φ and ✫ ✪ while estimator d doesn’t behave worse than estimator S Ensait - Roubaix 21

✬ ✩ Douillet-Rabenasolo CESA 2006 • an exact identification of reduced parameters concerning the demand models seems to be questionable • extraction of knowledge from history cannot be model-free ( ξ a is beyond any experience) • a description using larger families of pdf such as F ( µ ± ∆ µ, σ ± ∆ σ ) or F ( µ ± ∆ µ, δ ± ∆ δ ) seems a better way for a robust description ✫ ✪ Ensait - Roubaix 22