Submitted to Mnufctuing & Sevice Oetions Mngement mnuscit Electonic Comnion fo Otiml Design of Co-Poductive Sevices: Intection nd Wok Alloction Guillume Roels UCLA Andeson School of Mngement, 110 Westwood Plz, Los Angeles, CA 90095, USA, goels@ndesonucledu 1
e-comnion to Roels: Otiml Design of Co-Poductive Sevices ec1 Poofs of Sttements Define :=, :=, nd e,,,:=e 1, 1,, Accodingl, fφ,e,,, c c =fφ,e,,, c c nd oblem 1 is equivlent to: m fφ,e,,, c c =v φ,,, 0 EC1 The fist-ode otimlit conditions e: f e e c =0 nd f e e c =0 EC Thoughout the oofs, we conside oblem EC1 insted of 1 With slight buse of nottion, we omit the ue bs Lemm EC1 When >m{1,}, the objective function in EC1 is stictl concve in, Poof Recll tht we omit the ue bs to simlif nottion, ie, e,,,= +1 1 We hve: e = e 1 1 0 e = e 1 1 1 0 e = e 1 1 G 0 e = e 1 1 1 + 1 1 e = e 1 1 1 1 1 + 1 e = e 1 1 1 1 1, EC3 EC4 EC5 EC6 EC7 EC8 in which G ln + 1 ln +1 ln +1 B Jensen s inequlit, G is nonnegtive since ξlnξ is conve Accodingl, e e e e = e 1 3 1 EC9 e e e e = e 1 3 1 EC10 e e e = e 1 4 1 EC11
ec e-comnion to Roels: Otiml Design of Co-Poductive Sevices to When >m{1,}, e 0 Hence, f e e +f ee e <0 The deteminnt of the Hessin is equl fee e +f e e fee e +f e e fe e +f ee e e = f ee f e e e +e e e e e +f e e e e = f e e 1 4 f ee e+1 f e >0 EC1 Hence, the Hessin is ositive semi-definite Poof of Lemm 1 Aling the imlicit function theoem to EC omitting ue bs ields: Hence, [ ] [ ] 1[ ] fee e = +f e e f e e +f ee e e fee e e +f e e f e e +f ee e e f ee e +f ee f ee e e +f e e 1 = f ee e +f ee f ee e +f ee fe e +f ee e e [ fee e +f ][ ] ee f e e f ee e e fee e e +f e e f e e f ee e e f ee e +f e e f ee e e +f e e fee e +e e +f e e fee e +f e e fe e +f ee e e = e fee e f e e e e e +f e f ee e e e e e +fee e e e e fee e f e e e e e +f e f ee e e e e e +f ee e e e = e fee e f e e e e e +f ee e e e e fee e f e e e e e +f ee e e e We hve: e = e 1 1 1 1 ln + 1 G EC13 1 e = e 1 1 1 ln + 1 G EC14 Using EC3, EC4, EC9, EC11 togethe with EC13 nd EC14 nd then lugging EC5 ield: fee e +e e +f e e fee e +f e e fe e +f ee e e = f ee e f e e e e e e +e e f e e e e e e +e e e e e = f ee e f e e 31 4 fee 3 1 4 1 G = f e e 3 31 4 1 Gf eee+f e EC15
e-comnion to Roels: Otiml Design of Co-Poductive Sevices ec3 Combining EC5, EC1, nd EC15 ields fee e +e +e e +f ee fee e +f ee fe e +f ee e e = e 3 3 1 1 Gf e 4 f ee e+1 f e f e e 3 31 4 1 Gf eee+f e = e 3 3 1 Gf e 1 4 0 BLemmEC1,f ee e +f e e f ee e +f e e fe e +f ee e e 0Hence,e +e +e 0 Theefoe, fom EC1 nd omitting the ue bs, we obtin tht v φ =f φee +e +e 0 Poof of Lemm Aling the imlicit function theoem to EC omitting ue bs nd using EC6, EC7, EC8, nd EC11 ield: We hve: e +e +e fee e +f ee fee e +f ee fe e +f ee e e = e fee e f e e e e e +f e f ee e e e e e +fee e e e e fee e f e e e e e +f e f ee e e e e e +fee e e e +e fee f e e e +e e e e e +f e e e e = e f e e e e e e f e e e e e +e f e e e e = fee 1 3 1 e e +e e = 1 e1 0 EC16 e = e 1 1 1 1 + EC17 e = e 1 1 1 1 1 EC18 Plugging EC17 nd EC18 ields f e e 1 3 1 e e +e = f e e3 31 3 1, which is nonnegtive if nd onl if Theefoe, fom EC1 nd omitting the ue bs, we obtin tht v φ=f φe e +e +e 0 if nd onl if Lemm EC lnzz+1/z 1 fo ll z 0
ec4 e-comnion to Roels: Otiml Design of Co-Poductive Sevices Poof Let gz lnzz+1/z 1 We fist show tht gz is seudo-concve Let z be such tht g z =lnz/z 1 z+1/[zz 1]=0 Then, g z = z+1 z z 1 +4 1 zz 1 4 lnz z 1 3 z=z z +1 = z z 1 +4 1 z z 1 z +1 z z 1 = 1 z <0 B Theoem 345 in Cmbini nd Mtein 009, gz is seudo-concve Net, obseve tht z=1 is sttion oint: g lnzz z +1 1+lnz z 1=lim =lim z 1 zz 1 z 1 z 1 +z 1z =lim /z z 1 4z 1+z =0 Since gz is seudo-concve, z =1 is thus globl mimum Since g1= lim z 1 lnzz+ 1/z 1= lim z 1 lnz+1+1/z=0, we conclude tht gz 0 fo ll z 0 Lemm EC3 lnz/z 1 z+1/z fo ll z 0 Poof Let gz lnz/z 1 z+1/z We fist show tht gz is seudo-concve Tking the deivtive, we obtin tht g z= z lnz+z 13z 1/z z 1 Hence, g z 0 if nd onl if z lnz+z 13z 1 0 Define hz z lnz+z 13z 1 Tking the deivtive, we obtin tht h z= 4zlnz+4z 1 It is tivil to check tht h z is concve nd ttins its mimum t z =1 Becuse h 1=0, we thus obtin tht h z 0 fo ll z 0 Since h1=0, we theefoe find tht hz 0 if nd onl if z 1, ie, g z 0 if nd onl if z 1 B Theoem 37 in Cmbini nd Mtein 009, gz is seudo-concve nd ttins its mimum t z=1 Since lim z 1 gz= =0, we theefoe conclude tht gz 0 fo ll z 0 Lemm EC4 Fo n θ 1 nd ζ 0, the function Hζ,θ 1 θlnζθ ζ θ ζ θ +1 ζ θ 1 is incesing in θ if nd onl if ζ 1, nd it is incesing in ζ ove 0,1 nd ove 1, when eithe θ 0 o ζ [ζ,ζ], in which ζ =1/150 nd ζ =150 Poof Fist, ling L Hositl s ule shows tht Hζ, θ is continuous t θ = 0 when ζ 1 nd Hζ,0 = lnζ /lnζ In contst, Hζ,θ is discontinuous t ζ = 1 nd lim ζ 1 Hζ,θ = nd lim ζ 1 Hζ,θ = B Lemm EC, the deivtive Hζ,θ/ θ = ζ θ lnζ1 θ lnζ θ ζ θ +1/ζ θ 1/ζ θ 1 is nonnegtive if nd onl if ζ 1 Fothesecondt,wehve Hζ,θ/ ζ =θζ θ 1 θ 1 θlnζ θ ζ θ +1/ζ θ 1/ζ θ 1 Suose tht θ 0 Then, b Lemm EC, Hζ,θ/ ζ θζ θ 1 θ 1 θ/ζ θ 1 0 Net,suosethtθ>0Suosetht Hζ,θ/ ζ =0,ie, Gζ,θ θ 1 θlnζ θ ζθ +1 = ζ θ 1 0 At tht oint, Gζ,θ/ ζ =θ1 θζ θ 1 lnζθ ζθ +1 /ζ θ 1, which, given tht θ >0, is ζ θ 1 ζ θ
e-comnion to Roels: Otiml Design of Co-Poductive Sevices ec5 nonnegtive if nd onl if ζ 1 b Lemm EC3 Hence when θ>0, Hζ,θ/ ζ cosses zeo t most twice s ζ inceses, once fom below when ζ <1 nd once fom bove when ζ >1 Hence if θ>0, Hζ,θ is incesing in ζ ove ζ,1 nd ove 1,ζ fo some ζ,ζ When ζ =1/150 o ζ =150, it cn then be checked tht Hζ,θ/ ζ >0 fo ll θ Lemm EC5 Unde Assumtion, h, if nd onl if φ,, φ,, Poof Lemm EC4 shows tht, when [ c /c +ζc,c /c +ζc ], the function h, is decesing in evewhee ecet t = c /c +c whee the function jums fom to + Accodingl, h, cosses t most twice on tht intevl, once fom bove when <c /c +c, nd once fom below when >c /c +c Hence, between these two cossing oints h, if nd onl if >c /c +c In ddition, h, is decesing in if nd onl if > c /c +c b Lemm EC4, so tht, if h, if nd onl if > c /c +c, then h, if nd onl if >c /c +c fo ll leq Poof of Lemm 3 Fom EC1 nd omitting the ue bs, obseve tht v = f e e +e +e +f ee e e +e +e Becuse e =e e e 1 + 1 e1 ln 1 e1, nd becuse, using EC17 nd EC18 togethe with EC3, EC4, nd EC16, e = e e e 1 + 1 1 e 1 nd e =e e e 1 1 1 e 1, we obtin e +e +e = e e 1 e +e +e + 1 e1 ln 1 e1 1 + 1 1 e 1 Fom the fist-ode otimlit conditions EC, we obtin tht / = c 1 /c Hence, = ln Accodingl, e +e +e = e e 1 e +e +e + 1 e1 ln 1 = e e 1 e +e +e e1 ln = e e 1 e +e +e 1 e1 Hence, e1 1 e1 + ln 1 e 1 1 ln +1 v = f eee+f e e 1 e e +e +e 1 f e ln + +1 1 e1 1
ec6 e-comnion to Roels: Otiml Design of Co-Poductive Sevices The second tem is nonnegtive if nd onl if b Lemm EC5 given tht / = c 1 /c nd becuse f e >0, nd so is the fist tem when f ee e+f e 0 b Lemm 1 nd EC16 Poof of Poosition 1 Since the fesible sets e oducts of chins, the define lttice b Emle 3b nd d in Tokis 1998 B Lemm 1, v φ 0, nd b Lemms nd 3, v φ 0 nd v 0 if nd onl if Using the fist-ode otimlit conditions EC, we obtin tht φ,, φ,, if nd onl if c /c +c B Cooll 61 in Tokis 1998, we thus obtin tht v φ,, is suemodul on Φ A [ c c +c,] R,] nd v φ,, is suemodul on Φ A [, c c +c ] R,] Given tht Fφ;u is stochsticll incesing in u, the esult follows b Theoem 3101 in Tokis 1998 Poof of Lemm 4 Conside oblem EC1 Thoughout the oof, we omit the ue bs Plugging EC3 nd EC4 into EC, we obtin tht / =c 1 /c Tking the deivtive of the ight-hnd side with esect to φ,, nd ields the desied esult Poof of Poosition The oof follows fom combining Lemm 4 with Poosition 1 Poof of Poosition A-1 Without loss of genelit, set gφ = 1 nd bφ = b Define v,,,=e,,, b, in which e,,,= +1 1 Using EC, it cn be veified tht the otiml solution to 1 must stisf in which ζ :=c 1 /c =ζ 1 nd = 1 b c 1 b ζ b b +1, EC19 1 Without loss of genelit, suose tht Conside the coss-deivtives: v = bb +1 b 1 1 1, v = b 1 +1 b b ln +1 ln ln +1 +1 +1 ln ln, v = b1 1 +1 b b z lnz lnz +1 z +1 z lnz Hence, v 0 if nd onl if b When, v 0 b Jensen s inequlit becuse ξlnξ is conve When b, b z lnz lnz +1 z +1 z lnz 1 z lnz lnz +1 z +1 z lnz 0, nd theefoe v 0 As esult when b, v,, is suemodul if The esult then follows fom Tokis 1998, Theoem 8 The cse in which is teted simill
e-comnion to Roels: Otiml Design of Co-Poductive Sevices ec7 b Fom EC19, we obtin tht 0 if nd onl if h:= ζ 1 +1 b is nondecesing Tking the deivtive nd denoting z /, we obtin h = b z +1 b b b z lnz z +1 lnz +1 Becuse the function ξlnξ is conve nd equl to zeo when ξ=1, b b z lnz z lnz h b z +1 b = b z +1 b z lnz, nd the ight-hnd side is nonnegtive wheneve z 1, ie,, showing i In the following, suose tht >, ie, z>1 Let m:= b b z lnz z +1 lnz +1 Thus, h 0 if nd onl if m 0 Thus, in ode to estblish the esult, we need to show tht m 0 wheneve =0, ie, wheneve m=0 Tking the deivtive, we obtin: m = z m=0 lnz b + b b b = z lnz 1 b +b lnz b z +1 lnz lnz +1 EC0 Let n:=b +b lnz1 /z +1 Thus, when 0, m 0 when =0 if nd onl if n 0 when =0 Obseve tht nb<0 Moeove, n = =0 lnz 1 z b+b lnz <0, z +1 z +1 when z>1 nd <b Hence, thee eists <b such tht n 0 when =0 if nd onl if Suose fist tht = 0 In tht cse, function EC0 is nonositive fo ll, nd theefoe m 0 wheneve m=0 Suose net tht <0 Function EC0 is thus ositive fo ll,0 nd nonositive othewise Hence, m cosses zeo otentill thee times, fom bove on, ], fom below on,0, nd fom bove on [0,b Suose fo contdiction tht this is the cse, nd let 1,, nd 3 be those thee cossing oints with < 1 < <0 3 < b In tht cse, m 0 on, 1 ] [, 3 ] nd m 0 on [ 1, ] [ 3,b] Becuse m0 = 0, m must be conve t zeo Howeve becuse < 0, n0 < 0 nd m is stictl concve t zeo As esult, m cosses zeo t most once, nd the cossing must occu fom bove on,] Tht is, thee eists some ˆ such tht m 0 if nd onl if ˆ, o equivlentl such tht 0 if nd onl if ˆ The cse when > 0 cn be teted simill, theefoe estblishing ii
ec8 e-comnion to Roels: Otiml Design of Co-Poductive Sevices Simil to 1, v 0 if nd onl if b Conside the following coss-deivtives: v = b 1 +1 b b + 0, v = b +1 b b 1 1 1 + 1 b Hence, v 0 nd v 0 when b As esult when b, v,, is suemodul The esult then follows fom Tokis 1998, Theoem 8 b Fom EC19, we obtin tht 0 if nd onl if h:= ζ +1 b 1 1 is nondecesing We hve h = ζ b b +1 1 ζ 1 b ζ When, ζ 1 1, nd h 0 Let m := b/1 ζ 1 ζ b/ ; tht is, h hs the sme sign s m When m=0, then =0 nd theefoe, Refeences m = b m=0 = ζ b 1 + ζ 1 ζ = 1 b 1 b + b 1+ b ζ 1 + b1 As esult when m=0, m 0 if nd onl if b / Theefoe, h cosses zeo t most twice, fist fom below nd then fom bove Cmbini, A, L Mtein 009 Genelized Conveit nd Otimiztion Singe-Velg, Belin, Gemn