Online Supplement: Advance Selling in a Supply Chain under Uncertain Supply and Demand

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1 Onlin Supplmnt Avanc Slling in a Supply Cain unr Uncrtain Supply an Dman. Proos o Analytical sults Proo o Lmma. Using a = minl 0 ; x g; w can rwrit () as ollows (x ; w ; x ; w ) = a +(m0 w )a +( +" x w )x Tus, i m0 w < l 0 ; tn x = m0 w is t uniqu optimal orr quantity tat maximizs ; otrwis, any quantity x ( l 0 ) incluing m0 w is optimal. Proo o Lmma. W n to consir only t cas wn w m 0 bcaus otrwis x AB = 0 rom Lmma. Lt us rst consir t cass (i) an (ii) in wic m 0 > 0. Obsrv rom () tat M is linar in w wn w m 0 l 0, an quaratic in w wit its maximum at w = m0 wn m 0 l 0 < w m 0 It is asy to sow tat M is unimoal an tat its maximum is attain at w AB = m 0 l 0 i m 0 l 0 > m0 (i.., l0 < m0 ), an at wab = m0 otrwis. In t cas (iii) wn m 0 0, or any w ; x AB = 0 rom Lmma. By substituting w AB in ac cas into x AB in Lmma, w obtain x AB By substituting (w AB ; x AB ) into () an (), w obtain t xprssions or M an ; rspctivly. Proo o Lmma 3. (a) W can rwrit (5) as ollows E ~"; ~ [ (x ; w )] = x + ( w )x g F ( x )g + x 0 () + ( w )()gf () Using t Libniz intgral rul, w ~"; ~ [ (x ;w = x + ( w )g F ( x )g Sinc F ( x ) 0; E ~"; ~ [ (x ; w )] is unimoal an maximiz at x A (w ) = w (b) Sinc M (w ) in (6) is concav in w or any ~ ; E ~ [ M (w )] is also concav in w, nc a n uniqu w A xists. As illustrat in t blow gur, i ~ 05; arg min ~ oi w ; w w = 05; n w otrwis, arg min ~ oi w ; w w = ~ > 05 Tus, i ~ 05 8 ~ ; w A = 05; otrwis, w w A > 05 By substituting wa an xa (wa ) into (5) an (6), w obtain t rsults or EA an E A M mar. W can sow tat " 075 is su cint to guarant positiv pric in quilibrium unr avanc slling as ollows. (Tis conition is also su cint or positiv pric unr ynamic slling bcaus x AB x A ) In quilibrium, xa (wa ) = wa, so p A = + ~" min ~ ; x A (wa )g = + ~" min ~ ; wa g Dpning on t valu o wa ; tr ar tr cass (i) I wa <, tn min ~ ; wa g = wa 8 ~ ; an minp A j~" ["; "]; ~ [; ]g = " + +wa wa, tn min min ~ ; wa gj ~ [; ]g = wa, an minp A j~" ["; "]; ~ (ii) I [; ]g = " + +wa (iii) I wa >, tn min ~ ; wa g = ~ 8 ~, an minp A j~" ["; "]; ~ [; ]g = + " In (i) an (ii), minp A j~" ["; "]; ~ [; ]g > 0 i an only i " +wa +w, wr 075 bcaus w A 05. In (iii), minpa j~" ["; "]; ~ [; ]g > 0 i an only i ", wr < wa = +wa 075 Tror, wn " 075, p A > 0 or all ~" an ~

2 Figur Manuacturr s Ex-Post Pro t M unr Avanc Slling Proo o Torm. (a) W can rwrit E A M givn in (6) as E A M = max w w min ~ ; w F ( ~ ) = wa min ~ ; wa F ( ~ ) (8) W can also rwrit t manuacturr s x-ant pro t E B M unr rgular slling as E B M = " " B M( ~ ; ~")G(~")F ( ~ ) max w min ~ ; w F ( w ~ ); (9) wr t inquality is u to Jnsn s inquality an t proprty tat B M prsnt in Corollary is incrasing an convx in ~". Finally, obsrv tat t intgran o (9) is gratr tan or qual to tat o (8) or ~, nc E B M EA M EB M = EA M wn on o t ollowing two i conitions ar satis (i) Pr ~ = = Pr +~" ~ i = bcaus B M (~ ; ~") is linar in ~", an w tat maximizs t intgran o (9) is qual to w A = (s t gur abov), or (ii) Pr [~" = 0] = Pr +~" ~ i = bcaus E B M = EA M = 8 rom Lmma 3(i) an Corollary (i). (b) Suppos Pr +~" ~ i = Tis implis tat 05( +") 05, so E A = 6 by Lmma i 3(i). From Corollary (i), E B = E (+~") 6 in tis cas. By Jnsn s inquality, E B EA ; wr t quality ols wn Pr[~" = 0] = Wn Pr +~" ~ i <, Tabl sows tat E B E A can b itr positiv or ngativ. Proo o Torm. (a) T rsult ollows irctly rom Torm (a) an Lmma. (b) Suppos Pr +~" ~ i = Tn, rom Torm (b), E B EA. Sinc EAB E B 8; ~ ; ~", E AB E A, wr t quality ols wn EAB = E B = EA. Wn Pr +~" ~ i < ; Tabl 3 sows tat E AB E A can b itr positiv or ngativ. n Proo o Proposition. From t proo o Lmma 3, M (w ) = min ~ o w ; w w in (6) is concav in ~ or any givn w. By t wll-nown proprty o t scon-orr stocastic ominanc, E ~ u( ~ i ) E ~ u( ~ i ) or any concav unction u. Tus, E ~ [ M (w )] E ~ [ M (w )] or any givn w D n w 0 arg maxe ~ [ M (w )] an w 00 arg maxe ~ [ M (w )] Tn, E ~ [ A M ] = w w E ~ [ M (w 0 )] E ~ [ M (w 00)] E ~ [ M (w 00)] = E ~ [ A M ] T rst inquality ollows rom t

3 optimality o w 0 an t scon inquality ollows rom t scon-orr stocastic ominanc o ~ ovr ~ Proo o Proposition. (a) From Corollary, B M is incrasing an concav in ~ or ~ < +~" an it is constant or ~ +~". Tus, B M is non-crasing an concav in ~. By ollowing t sam argumnt as in t proo o Proposition, w can sow t sir rsult. Similarly, w can obtain t rsults or w B an xb by sowing tat wb is non-incrasing an convx in ~ an tat x B is non-crasing an concav in ~ (b) From Corollary, B M is incrasing an convx in ~" or ~" ~ an it is linarly incrasing or ~" > ~. Tus, B M is incrasing an an convx in ~". By ollowing t sam argumnt as in t proo o Proposition, w can sow t sir rsult. Similarly, w can sow tat w B is incrasing an convx in ~" an tat x B is non-crasing an concav in ~"; an obtain t sir rsults.. Computation unr Spci c Probability Distributions Tis sction prsnts t clos-orm xprssions o t quilibrium outcoms unr avanc or rgular slling, an xplains t procur to comput t quilibrium outcoms unr ynamic slling wn t spci c istributions o ~ an ~" ar us as scrib in 6. Unr t avanc slling stratgy, a A (w ) pns on t ranom yil ~ in t ollowing mannr i w ~ + r; a A (w ) = w an i r ~ w, aa (w ) = ~ Tus, w can xprss E ~ [ M (w )] in (6) as E ~ [ M (w )] = E ~ [w a A (w )] = +r w a w r + a r w r ; wr a min + r; max r; w. By noting tat a can ta on tr i rnt valus pning 8 on w, E ~ [ M (w )] can b rwrittn as w i 0 w ( + r) >< E ~ [ M (w )] = w 6r [w ( + r + p r)g] [w ( + r p r)g] i ( + r) < w < ( r) > w ( w ) i ( r) w Lmma 3 as sown tat i ( r) 05, w A = 05 I ( r) < 05, w can sow w A = 3 ( r) + 3p ( + r + r ) ( + r) + ( w 0 ) (0) T proo o t abov rsult is as ollows. Lt g (w ) w, g (w ) w 6r [w ( + r + p r)g] [w ( + r p r)g], an g 3 (w ) w ( w ) Not tat g is a cubic unction o w an its omain [ ( + r); ( r)] is contain in [ ( + r + p r); ( + r p r)]. Sinc 3 g w 3 < 0 an (+r p r) > 0; g is unimoal wit its maximum at a largr root o g (w ) w = 0; wic is qual to w 0 givn in (0). (Not tat g (w ) w = 0 is a quaratic quation o w.) Also, g 3 is concav in w wit its maximum at 05. Sinc g w w = (+r) = g w w = (+r) = > 0, 3

4 w A os not xist in t rst intrval o w. Suppos ( r) 05 Tn, g w w = ( r) = + ( r) 0; nc wa = 05 [ ( r); ]. Nxt, suppos ( r) < 05. Tn, g w w > 0; g w = (+r) w < 0 an g 3 w = ( r) w < 0; nc = ( r) wa = w0 is optimal. In tis cas, w A > 05 bcaus g w w = =05 6r ( r)g i > 0 By substituting w = w A into xa (w ), w can obtain t clos-orm xprssions or x A (wa ) an similarly or EA M an E A. For t rgular slling stratgy, using t x-post quilibrium outcoms givn in Corollary, w comput t x-ant quilibrium outcoms. Using Corollary, w can writ t x-ant xpct wolsal pric as E ~;~" w B = Pr(~" = ) Pr( ~ b ) + + i b r [ + ] r +Pr(~" = ) = r + 3r + ( + ) b + ( ) c g b + c ( r) ; Pr( ~ c ) + c wr b min + r; max r; + an c min + r; max r; Similarly, w can xprss otr x-ant quilibrium outcoms suc as E B M, ExB, an EB. For ac o six possibl cass o aving i rnt pairs o b or c ; w av riv t clos-orm xprssion o t quilibrium outcoms, wic ar availabl upon rqust. For t ynamic slling stratgy, w av prsnt in Lmma t clos-orm xprssions o t quilibrium outcoms in t scon-prio gam. As w sow blow, t trmination o t rtailr s quilibrium pr-boo quantity in t rst prio involvs t analysis o 39 cass. Tus, insta o ning t clos-orm xprssions or ac cas, w vis an cint procur to comput t quilibrium outcoms in t rst prio. First, consir t rtailr s problm in t rst prio. T conitions provi in Lmma can b r-writtn as ollows ~ l 0 ~m0 as ~ +~"+x trsol numbrs min + r; max r; ++x r, ~ l 0 > 0 as ~ > x ; an ~m 0 > 0 as x < +~". D n ; min + r; max r; +x, an min + r; max r; x ; an t inicator unction I(y) = i y is tru an I(y) = 0, otrwis. Tn E ~;~" [ (x ; w )] in (3) can b rwrittn as E ~;~" [ (x ; w )] = +r ( x w ) x r + r ( w ) r + Pr(~" = )I x < + + Pr(~" = )I x < 6 6 ( + x ) r + i ( x ) r +r ( x ) r + i ( x ) r, an I x < ; E~;~" [ (x ; w )] tas on +r Dpning on t valus o ;,, I x < + i rnt unctional orms o x. Not tat ; or is itr a constant or a linar unction o x ; an I x < + or I x < i rnt combination o ;,, I x < + is itr 0 or. Hnc, in ac intrval o x aving a, an I x <, E~;~" [ (x ; w )] is at most a cubic unction o x. T pr-boo orr quantity x AB (w ) tat maximizs E ~;~" [ (x ; w )] or any givn w is itr a bounary point btwn any two intrvals or an intrior point at wic t rst orr conition is satis. Tr ar potntially 39 caniats or x AB (w ) wic consist o 9 bounary points btwn two intrvals o x an 30 intrior optimal points witin any intrval o i [ ] r

5 x. T bounary points ar 0; +, ; (+r) (+) (at wic + r = ++x (at wic r = ++x ), (+r) ( ) (at wic + r = +x ), ), ( r) (+) ( r) ( ) (at wic r = +x ), ( + r) (at wic + r = x ), an ( r) (at wic r = x ). To n t intrior optimal points, w rst simpliy t xprssion o E ~;~" [ (x ; w )] or ac o t ollowing 30 cass = I x < = 0; so E~;~" [ (x ; w )] can av 3 i rnt xprs- (i) I x +, I x < + sions wn = + r; r or x ; (ii) I x < +, I x < + = an I x < = 0; so E~;~" [ (x ; w )] can av 9 i rnt xprssions wn = + r; r or x, an = + r; r or ++x ; (iii) I x <, I x < + = I x < = ; so E~;~" [ (x ; w )] can av 8 i rnt xprssions wn = + r; r or x, = + r; r or ++x, an = + r; r or +x (not 8 cass xist insta o 7 cass bcaus ). For ac o t abov 30 cass, w can asily obtain an intrior optimal point rom t rst orr conition (wic w omit r). By comparing E ~;~" [ (x ; w )] at ts 39 caniats, w can n t rtailr s bst rspons x AB (w ) or any givn w Nxt, consir t manuacturr s problm in t rst prio. Similar to E ~;~" [ (x ; w )], w can rwrit E ~;~" [ M (x ; w )] in () as E ~;~" [ M (x ; w )] = +r (w x ) r + r (w ) r + Pr(~" = )I x < + +r 8 ( + x ) r + i ( + ) ( x ) r + Pr(~" = )I x < +r 8 ( x ) r + ( ) ( x ) r W can cintly comput t pr-boo wolsal pric w AB tat maximizs E ~;~" M (x AB (w ); w ) as ollows. W rst comput t rtailr s pr-boo quantity x AB (w ) as a unction o w an intiy bounary points btwn any two ajacnt intrvals o w at wic x AB (w ) switcs rom on o t 39 caniat points to anotr. In ac intrval o w, E ~;~" M (x AB (w ); w ) is a continuous unction, nc its local maximum is attain at itr a bounary point or an intrior point at wic t rst orr conition is satis. By comparing local maxima, w can intiy a global optimal point w AB. 3. Invntory Holbac Consir t Staclbrg gam tat tas plac bor t supply an man uncrtaintis ar rsolv. In tis gam, t manuacturr rst sts is pr-boo wolsal pric w an tn t rtailr trmins r pr-boo orr quantity x. In aition, atr obsrving t raliz man an supply, t rtailr as an option o witoling t part o t pr-boo orr quantity tat s as rciv. W us suprscript AH to not quilibrium outcoms in tis gam. For gnral probability istributions o ~ an ~", u to t complxity o t mol, t problm is intractabl. Tus, w assum t sam istributions as in t prvious sction, an compar i 5

6 quilibrium outcoms in tis gam wit tos unr avanc or rgular slling. Suppos tat ~ is uniormly istribut btwn r an + r, wr r (0; ], an tat ~" = wit probability 0.5 an ~" = wit probability 0.5, wr (0; ]. Lt Q not t quantity tat t rtailr slls to t mart. Atr obsrving ~", t rtailr trmins Q ( x ) to maximiz r x-post pro t ( + ~" Q)Q w x Tus, t optimal Q AH quals +~" Clarly, t rtailr woul not pr-boo mor tan +, wic is QAH wn ~" = ; i.., x +. T x-ant xpct pro t ( o t rtailr in quation () o t bas mol is tn moi into w x + 05( + x )x + 05( x )x i x E ~" [ (x ; w )] = w x + 05( + x )x + 05( ) i x >. Tis can b intrprt as ollows. I t rtailr as rciv x rom r pr-boo orr, it is optimal or r to sll t ntir quantity tat s as rciv rom r pr-boo orr; i.., Q AH = x Otrwis, wn t man turns out to b low (i.., ~" = ), it is optimal or t rtailr to witol x (> 0) an sll only Q AH = ; an wn t man is ig (i.., ~" = ), it is optimal to sll Q AH = x ( ( ; + ]) From t abov quation or E ~" [ (x ; w )], givn t pr-boo ( wolsal pric w ; w obtain t ollowing optimal pr-boo quantity w x AH i x or quivalntly w (w ) = + w i x > or quivalntly w < Not tat + w > w i an only i w <. Compar wit x A (w ) = w or all w in our bas mol, x AH (w ) implis tat, givn t pr-boo wolsal pric w, t rtailr pr-boos mor wit t olbac option tan in t bas mol. In anticipation o t rtailr s bst rspons x AH (w ); t manuacturr trmins is prboo wolsal pric w to maximiz 8 is nxpct pro t < w min ~ o ; w E M (w ) = w a AH i w (w ) = n w min ~ o ; + w i w <. In t ollowing, w will rst analyz (Intrval ) w an (Intrval ) w < sparatly, an tn combin t rsults. W will sow tat E M (w ) is itr unimoal or bimoal in w. (Intrval ) In tis intrval, t rtailr os not witol t invntory. Tus, w can us t rsults o t bas mol wit no olbac option prsnt in t prvious sction. Sinc E M (w ) is unimoal, it is optimal or t manuacturr to st w = max05; g i ( w = maxw 0 ; g otrwis, wr w 0 is givn in (0). r) 05; an (Intrval ) In tis intrval, t rtailr witols t invntory atr obsrving raliz ~ an ~". Dpning on t ranom yil ~, a AH (w ) can b xprss as ollows i + w ~ + r; a AH (w ) = + in (6) as w an i r ~ w, aah (w ) = ~ Tus, w can xprss E ~ [ M (w )] E ~ [ M (w )] = E ~ [w a AH (w )] = +r + w w w r + w r w r ; wr w min + r; max r; + w. By noting tat w can ta on tr i rnt valus pning on w, E ~ [ M (w )] can b rwrittn as 6

7 8 w i 0 w + ( + r) >< w + r w ( + r + p r) E ~ [ M (w )] = w + ( + r p + i ( + r) < w < + ( r) r) > + (w ) + ( + ) i + ( r) w By ollowing t sam mto as prsnt in Sction o tis onlin supplmnt, w can sow tat E M (w ) is unimoal in tis intrval an tat t optimal wolsal pric in tis intrval is i ( r) + ; w = min + ; g; otrwis, w = minw 00 ; g, wr w 00 = + 3 ( + r) + 6p ( + r + r ) ( + )( + r) + ( + ) (Intrvals & ) Bcaus E M (w ) is unimoal in ac o Intrval an Intrval, E M (w ) is unimoal or bimoal in w across t two intrvals. Tror, a global optimum w AH can b oun by comparing t maximum valu o E M (w ) in ac o t two intrvals. By substituting w AH into x AH (w ), w can obtain x AH (w AH ) an t rsulting xpct pro ts o bot rms. Using t mto scrib abov, w av comput t quilibrium outcoms o 80 scnarios by varying rom 0 to an varying an r rom 0 to 09 wit an incrmnt o 0 W tn compar t quilibrium outcoms wit tos unr avanc or rgular slling. To illustrat, w prsnt t rsults o 7 scnarios in Tabls A an A. From our numrical xprimnts, w raw t ollowing obsrvations (a) In all 80 scnarios, E AH M EA M an xah x A. Howvr, w av obsrv bot wah w A > 0 an wah w A < 0; an bot EAH > EA an EAH < EA. (b) In all 80 scnarios, E B M EAH M. Howvr, w av obsrv bot wb w AH > 0 an w B w AH < 0; bot x B > xah an x B < xah ; an bot E B > EAH an EB < EAH. Obsrvation (a) sows tat t rtailr pr-boos a largr quantity wit t olbac option. Howvr, tis os not ncssarily bn t t rtailr bcaus t manuacturr can anticipat t rtailr s bst rspons an trmin is wolsal pric accoringly. As a rsult, it is only t manuacturr wo will always bn t rom tis aitional option. Obsrvation (b) con rms tat our rsults obtain in t bas mol continu to ol in t olbac mol t manuacturr is always bttr o unr rgular slling tan unr avanc slling, wras t rtailr may prr itr stratgy. T comparison btwn avanc slling wit t olbac option an ynamic slling is similar to t comparison btwn avanc slling witout tis option an ynamic slling. Tabl A3 summarizs t comparativ statics o tis gam. It sows tat t cts o capacity an supply uncrtainty r on t quilibrium outcoms ar t sam as tos unr avanc slling witout t olbac option. Wil man uncrtainty os not a ct t quilibrium outcoms witout t olbac option, it os a ct t quilibrium outcoms o tis gam. T cts o on t xpct pro ts o bot rms ar t sam as tos unr rgular slling man uncrtainty bn ts t manuacturr but os not always bn t t rtailr. In aition, w av obsrv tat t rtailr s pr-booing quantity is incrasing in man uncrtainty, but t 7

8 manuacturr s wolsal pric is not monotonic in. Tabl A. Equilibrium Outcoms unr Avanc Slling E cts o t Holbac Option. Paramtrs Equilibrium Outcom Comparisons r E AH M EA M EAH E A wah w A x AH x A Tabl A. Equilibrium Outcoms gular Slling vs. Avanc Slling wit t Holbac Option Paramtrs Equilibrium Outcom Comparisons r E B M EAH M E B EAH Ew B w AH Ex B x AH Tabl A3. Comparativ Statics o t Gam unr Avanc Slling wit Holbac " # " r # " # x AH w AH E AH E AH M