ONLINE SUPPLEMENT Optimal Markdown Pricing and Inventory Allocation for Retail Chains with Inventory Dependent Demand

Transcription

1 Submittd to Maufacturig & Srvic Opratios Maagmt mauscript MSOM 5-4R2 ONLINE SUPPLEMENT Optimal Markdow Pricig ad Ivtory Allocatio for Rtail Chais with Ivtory Dpdt Dmad Stph A Smith Dpartmt of Opratios Maagmt & Iformatio Systms, Lavy School of Busiss, Sata Clara Uivrsity, Sata Clara, Califoria 95053, ssmith@scudu Nardra Agrawal Dpartmt of Opratios Maagmt & Iformatio Systms, Lavy School of Busiss, Sata Clara Uivrsity, Sata Clara, Califoria 95053, agrawal@scudu Appdix A: Solutio for th Optimal Pric Trajctory Cosidr th followig modifid form of th optimal cotrol problm (2 that icluds a ivtory holdig cost trm max t { p(td(t, I(t, p(t c I I(t dt + c I, t ad I = I 0 D(t, I(t, p(tdt 0, Th optimality coditios for th Hamiltoia H = (p λd c I I H I subjct to I (t = D(t, I(t, p(t ar = p λ] D I c I = λ ad H = p λ] D p p + D = 0, with th boudary coditio λ(t = c + θ, whr θi(t = 0 Usig th dmad fuctio i ( ad substitutig I = D, w obtai λ = D D I / D p + c I = c y (II + c I ad p + D/ D y(i p = p c = λ Thus it follows that p = λ, ad itgratig th first xprssio abov givs p(t = c l (y(i(t + c It + costat

2 2 Smith ad Agrawal: Th costat is obtaid by valuatig th Hamiltoia coditios at tim t = t ad subtractig th rsultig quatio, which givs p(t = p(t + c I (t t + c ( y(i l y(i This shows that th pric trajctory ca b xprssd as p(t = P (I(t + c I (t t, whr P (I = p(t + c ( y(i l y(i With c I = 0, th optimal trajctory i (3 rsults Th trmial pric p = p(t = P (I is obtaid from th boudary coditio at t = t Wh y(0 = 0, it follows that I > 0 Th boudary coditio implis that θ = 0 wh I > 0 Thus, th boudary coditio bcoms λ(t = c, which togthr with p c = λ, implis that p = λ(t + c / = c ( + / Th trm c I (t t ca b icorporatd ito th sasoality factor k(t to obtai k (t = k(t c I (t t /c Th dmad rat with th optimal pric trajctory ca thrfor b writt as D(t, I(t, p(t = k (ty(i p/c This allows th optimal ivtory trajctory I(t to b dtrmid dirctly by itgratio I(t = I 0 y(i p/c t k (tdt Th cost c I lowrs th pric for smallr t valus sic th trm c I (t t is gativ Thr is also th variabl cost trm to b valuatd i th profit calculatio Sic I(t dclis i proportio to k (t, this trm ca b valuatd as follows t { t c I I(tdt = c I I 0 y(i K, whr K = p/c t k (tdt t For th cas of a costat pric p 0, (8 still holds ad th variabl cost trm c I I(tdt is addd to th objctiv fuctio Th optimal pric p 0 ca still b dtrmid by a o dimsioal sarch, but thr is o closd form xprssio for th total sals durig, t ] Appdix B: Proof of Thorm ad Lmma Proof of Thorm : (A To simplify otatio i this drivatio, w will supprss th dpdc o x ad us I for I (x Substitutig for I 0 (x usig (2, w hav th followig trms i L that dpd o I = I (x ] c ( µi + I α K ] + α I α K + α I + I α K ]l( + I α K Thrfor, th FONC L = 0 yilds I (x L I (x = ( µ( + αi α K + α α I α K

3 Smith ad Agrawal: 3 + α α + αi K ]l( + I α K + α I + I α K (α I α 2 + I α K K ] = + αi α K µ + α ] α l{ + I K = 0, (A2 aftr caclig trms ad combiig th multiplicativ factors always positiv, w hav th quivalt FONC show with th dpdc o x µ + α α l{ + I (xk (x = 0, Sic th first multiplicativ factor is which implis that α I (x = K (x α (µ ] α To simplify otatio, dfi th paramtr ψ For th stor with idx x, w thrfor hav that ad from (2 that ψ = ψ(µ = α (µ ] I (x = K (x α ψ(µ α I 0 (x = K (x α ψ(µ + K (x + α α ψ(µ α = K (x α ψ(µ + ψ(µ α ] It ca b vrifid that th scod ordr coditio is also satisfid for α < sic 2 L { α 2 = α(α I I (x 2 K µ + α α l( + I K +( + αi α K α (α I α K < 0 + I α K Suppos that thr is a variabl ivtory cost c I pr uit, ad w us th optimal pric trajctory (3 at ach stor x Dfi K (x, t = t k(x, t c I (t t/c dt with K (x = K (x, t p/c I α r Thus, if thr is a variabl cost c I ad K (x = p/c I α r t icluds th additioal pric trajctory trm i (3 Also, a trm K (x, tdt of holdig ivtory, th factors K (x ar rplacd by K (x, which { { c I I 0 (x y(i (xk (x = c I I (x + K I α (x I α (xk (x corrspodig to (A is subtractd from th objctiv fuctio + αi α K whr th last trm ca b writt as + αi α This mas that (A2 bcoms ] µ c I + α ] α l{ + I K + αc I I α K = 0, K ] K K K K

4 4 Smith ad Agrawal: Th proof is basd o showig that + αi α (xk (x is a costat that is idpdt of x This is cosistt with th FONC abov, providd that K (x/k (x is idpdt of x A sufficit coditio for this to b tru is that th ratio G (t = K (x, t/k (x corrspodig to G(t i th proof of Lmma is idpdt of x That is, all stors hav th sam sasoal variatio Howvr, this solutio to th FONC may ot b uiqu, as it was i th cas with o variabl ivtory cost If th ratio K (x/k (x is idpdt of x, th thr is a costat ψ such that th FONC for optimal ivtory allocatio ar satisfid by I (x = K (x α ψ ad I 0 (x = K (x α ψ + (ψ α ] I this cas, th paramtr ψ ca still b dtrmid from th total ivtory costrait as i (9, but it caot b rlatd to µ as i (6 i this cas A sufficit coditio for th ivtory allocatio rsult to hold is that all stors hav th sam sasoal variatio, i, K(x, t is sparabl Th form of th optimal ivtory allocatio dos ot appar to xtd to th costat pric cas, howvr Proof of Lmma : If K(x, t = K(x, t G(t, w itgrat th rlatioship (4 for stor x to xprss th optimal ivtory trajctory at ach tim t as I(t, x = I 0 (x I (x α K (xg(t It ca b s from (3 that th optimal pric for stor x at tim t satisfis P (x, I(x, t = p + αc ( I(x, t l (A3 I (x From (2 usig Thorm to substitut for I 0 (x ad I (x, w hav I(t, x I (x = I 0(x I (x I (x α K (xg(t = + ψ α ψ α G(t Thus, th optimal pric trajctory (A3 i this cas is P (x, I(x, t = p + c α l ( + ψ α G(t], which is idpdt of x Th scod part of th Lmma follows sic I(t, x/i 0 (x = (I(t, x/i (x (I (x/i 0 (x = G(tψ α /( + ψ α QED Appdix C: Proof of Thorm 2 Assig th Lagrag multiplir c µ to th ivtory costrait as bfor ad form th Lagragia L for (23 Takig th partial drivativ of th Lagragia, th followig FONC for I 0 (x is obtaid L I 0 (x = µc + p 0 (x + (c p 0 (xi α 0 (xi α (x = 0, aftr substitutig from (22 This FONC ca b rwritt i trms of th fractio of usold ivtory U (x α = p 0(x µc p 0 (x c for all x, whr U (x = I (x/i 0 (x (A4

5 Smith ad Agrawal: 5 Th partial drivativ with rspct to pric yilds L p 0 (x = I 0(x I (x + p 0 (x µc ]K (xf (p 0 (xi 0 (x α = 0, aftr substitutig th xprssio (A4 abov This ca b rarragd to obtai K (xi 0 (x α U (x = p 0 (x µc ]f (p 0 (x Rarragig th xprssio (22 for I (x α, w obtai K (xi 0 (x α = U (x α α]f(p 0 (x (A5 Equatig th right had sids of th two xprssios abov givs a FONC for p 0 (x Sic U (x dpds o x oly through p 0 (x, th right had sid of this FONC ca b xprssd i trms of valus p 0 ad U that ar idpdt of x Also, (A4 givs th fractio of ivtory U (x = U that is salvagd at all stors udr th optimal allocatio, ad (A5 shows that I 0 (x is proportioal to K (x α as follows ( I 0 (x = ρk (x α ( αf(p0 α for all x, whr ρ =, U α ad ρ is idpdt of x Th quatio for ρ allows us to writ U as a fuctio of p 0 as follows ] ( α U = ρ f(p α 0 α For a giv I ad X, ρ ca b obtaid from th ivtory costrait I = x X I 0 (x = ρ x X Appdix D: Proof of Lmma 2 To simplify otatio, lt K = K(t I α r p/c ad dfi K (x α I, or ρ = x X K QED α (x U( = th total dig ivtory wh stors stockd W wish to dtrmi th sig of U ( bcaus th total umbr of uits sold icrass if ad oly if U( dcrass If stors ar stockd, th rlatioship for ivtory ad sals i (5 for ay stor that is stockd ca b writt as I = U( ( U( + y K ( U( or I = U( + y K Takig th drivativ with rspct to, w hav ( ( U( U( U ( 0 = U ( + y K + y Now, lt u = U(/ ad solv for U ( to gt U( 2 ] K U ( = uy (u y(u]k + K y (u Sic y(0 = 0 ad y (u 0 for all u, w s that U ( > 0 if ad oly if y(u y(0 < uy (u For u > 0, this last coditio holds if y(i is covx ad th sig is rvrsd if y(i is cocav Sic th total umbr of uits sold is I U(, w s that th total umbr sold icrass with if y(i is cocav ad dcrass with if y(i is covx QED

6 6 Smith ad Agrawal: Appdix E: Proof of Thorm 3 W focus o rvu maximizatio sic th ivtory costs ar suk costs Th total ivtory I will b allocatd qually to a subst of stors For th optimal pric trajctory cas with xpotial pric ssitivity, th total rvu ca b obtaid from (6 as follows R(I 0 = c I 0 + I 0 I + I0 l I ( y(i y(i ] di If ivtory I is allocatd qually to idtical stors, this ca b writt as ( I R = c I + I U( + I / { ( ( ] U( l(y(i l y di U(/ This itgral ca b simplifid by usig th chag of variabl z = I ad substitutig th fuctioal form y(i = (I/I r α to obtai c I + I U( + α I U( l(z l(u(]dz This xprssio ca b itgratd to obtai ( I R = c I + α I U(] + αi ( ] l I (A6 U( Diffrtiatig with rspct to, w hav = c ( αu ( + αi ] U ( = c { U( U ( α + α I U( Sic < I /U( always holds, this shows that th drivativ of total rvu with rspct to has th opposit sig of U ( ad th sam sig as th drivativ of th total umbr sold For costat pricig, w kow that th total rvu is Thus w wish to show that { I ] R(I, = p 0 I ( + c I ( = I p 0 + (c p 0 I ( R(I, = (c p 0 I ( + (c p 0 I ( > 0 Sic (c p 0 < 0, w d to show that I (/I ( < From (8 ad th fact that ach stor rcivs I 0 /, w s that This holds as log as p 0 Appdix F: Proof of Thorm 4 I ( I ( = I / I / ( αk f(p 0 < is chos so that dmad K f(p 0 I / ad α < QED This proof is oly for th optimal pric trajctory ad xpotial pric ssitivity Lt = th umbr of idtical stors that ar stockd, U( = th total usold ivtory, ]

7 Smith ad Agrawal: 7 ad us th total rvu quatio R ( I i (A6 With th fixd cost F, w wat to optimiz R ( I F with rspct to, subjct to th dig ivtory costrait i (5 For idtical stors, (5 ca b writt as Now, w wat to optimiz yilds th FONC I = U( + R ( I ( U( α K, whr K = K(t p/c I α r, or I = U( + α U( α K (A7 F with rspct to, subjct to (A7 Th partial drivativ { ( I R F = c { U ( + α U( I U(] F = 0 Diffrtiatig th costrait (A7 shows that th followig must hold This ca b rarragd to obtai 0 = U ( + ( α α U( α K + αu ( α U( α K, U ( = ( α (U(/α K + α (U(/ α K = ( α (U(/α K + αi U(]/U( Substitutig this ito th FONC, w obtai ( α c U( ( α K = F, (A8 which ca b solvd for U(/ to obtai ( /α U( F /c = K ( α Usig (A7 to substitut i (A8 yilds I U( = F /c α, which givs a formula for th amout sold at ach stor Combiig this with th amout of ivtory usold abov implis that th optimal ivtory at ach stor is ( /α I F = /c + F /c K ( α α QED It ca also b vrifid that th ivtory allocatio i Thorm 4 is cosistt with th optimal ivtory allocatios for oidtical stors i Thorm by sttig ψ = K α( α ( F /c ( α /α This is also aalogous to (3 for th cas i which I is a dcisio variabl ad µ = c/c ad ψ ar costats Compariso to th Cas with No Ivtory Effct It is itrstig to compar th abov rsults i th idtical stor cas to th cas α = 0 Ths formulas ca b drivd dirctly from a w optimizatio, or by simply takig th limit i th xprssios

8 8 Smith ad Agrawal: abov Lttig α approach 0, w s that th uits sold pr stor gos to F /c It ca vrifid that this is th optimal sals pr stor wh thr is o ivtory ffct, providd that th dmad pr stor is sufficitly high to mak this a achivabl sals targt Th Usold Uits i (27 should approach 0 as α approachs 0, which is th optimal rsult wh thr is o ivtory ffct if But w ot that K F /c K ( α < or quivaltly if th Sold Uits = F /c α < K This will happ if ad oly is th umbr of uits dmadd at th pric p, wh thr is o ivtory ffct Sic th optimal trajctory with th ivtory ffct uss prics that ar at last this larg, th iquality must hold for ay st of paramtr valus that allows th Sold Uits = F /c α is larg ough, th iqualitis abov will b rvrsd stors at all ad simply salvag all uits to b achivd Clarly, if F This would ma that it is optimal to stock o Appdix G: Two Proofs Rlatd to Optimizig th St of Stors Stockd Extdig th Optimal Sarch to Iclud Bouds o th Ivtoris Lt us cosidr oly th uppr boud for simplicity, sic both bouds ar hadld i a similar way First fid th optimal X without th bouds Th dfi X max = Max{x K (x α (ψ + ψ α I max, rcallig that th idics x ar arragd i dcrasig ordr of K (x Lt I 0 (x = I max for all x X max Th roptimiz th X usig th rvisd ivtory costrait X max x X K (x α (ψ + ψ α = I x X max I max This may caus som additioal ivtory allocatios to xcd I max If so, rpat th procss util o ivtory allocatios violat th uppr boud costrait For x X max, th profit quatio (7 still holds, but ow w must us ψ = ψ(x, whr ψ(x is obtaid from K (x α (ψ(x + ψ(x α = I max for all x X max This ψ(x also dtrmis th rvisd dig ivtory I (x = K (x α ψ(x, which th allows th pric trajctory to b dtrmid from (A3 Bcaus of this, th optimal pric trajctoris will d to b dtrmid sparatly for ach stor with a costraid ivtory lvl As a rsult, th pric trajctoris will o logr b sychroizd i tim bcaus som I (x hav b modifid Showig that thr is a Uiqu Local Maximum i th Cotiuous Cas If w trat th stor idics x as a cotiuous variabl, it ca b show by calculus that a local optimum must b uiqu With discrt idics x, this proof is also sufficit to imply that thr ar o sparatd local maxima To prov this, lt th idx x b dfid as a cotiuous prctil rakig of K (x,

9 Smith ad Agrawal: 9 ad cosidr th st of stors {x X, whr X must hold This implis that th Lagragia for optimizig th total t profit with rspct to {I 0 (x ad X ca b writt as ] L = R(I 0 (xdx + c µ I I 0 (xdx XF + ξ( X, x X which is also subjct to (2 Optimizatio with rspct to I 0 (x lads to th allocatio rsults i Thorm x X I additio, th Kuh-Tuckr coditios giv th FONCs for X R(I 0 (X c µi 0 (X F ξ = 0, ad ξ( X = 0 Ths FONCs ca b solvd as two sparat cass a] ad b] Cas a For X =, fid th uiqu valu of ψ such that I 0 (xdx = ψ + ψ α ] K (x α dx = I x If R(I 0 ( c µi 0 ( F usig this valu of ψ ad th corrspodig µ, th X = must hold This complts th solutio, bcaus µ is dtrmid from ψ ad ξ > 0 is dtrmid from th FONC with X = Cas b Othrwis, w hav X <, which implis that ξ = 0 W thrfor d to solv th two FONCs for µ ad X simultaously To simplify otatio, lt κ(x = K (x α dx, x X x which allows th ivtory costrait quatio to b writt as Y (X, µ = ψ(µ + ψ(µ α ]κ(x I = 0 To obtai th scod quatio, w us µ = + α l ( + ψα which simplifis to R(I 0 (X c µi 0 (X F = c I 0 (X α to obtai ψ(µ α ψ(µ + ψ(µ F = 0, α Y 2 (X, µ = c K (X α α ψ(µα F = 0 Ths two quatios ca b solvd simultaously for X ad ψ = ψ(µ sic thr is a o to o rlatioship btw µ ad ψ To udrstad th bhavior of th solutio abov, dfi th two implicit fuctios X (ψ ad X 2 (ψ such that Y (X (ψ, ψ 0 ad Y 2 (X 2 (ψ, ψ 0 ca b vrifid that bcaus K (X is dcrasig i X X (ψ = Y (X, ψ/ ψ Y (X, ψ/ X = ( + αψα κ(x (ψ + ψ α K (X α X 2(ψ = Y 2(X 2, ψ/ ψ Y 2 (X 2, ψ/ X 2 = K (X By takig th total drivativ of th two FONCs, it α αψ α < 0 ad, K (X α X ψ α > 0, Thus, as ψ is adjustd, thr ca b at most o valu X such that X = X (µ = X 2 (µ Existc follows bcaus 0 X < holds as wll This shows that thr is at most o solutio of th FONC i th cotiuous cas, bcaus ach X dtrmis a uiqu valu of ψ This uiquss rsult applis to th sarch o X i th discrt cas as wll, i that th discrt cas ca b approximatd by th cotiuous cas But i th discrt cas, thr may b tis btw th optimal umbr of stors to stock ad a ighborig valu or +