# Promised Lead-Time Contracts Under Asymmetric Information

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5 92 Opeations Reseach 56(4), pp , 28 INFORMS optimal fo this classical seial system. Othe papes povide simple poofs, efficient computational methods, and extensions to the infinite-hoizon case (see, fo example, Fedeguen and Zipkin 1984, Chen and Zheng 1994). Fo a stationay seial system, Gallego and Öze (23) show that a myopic base-stock policy is optimal. They show how to allocate costs to fims (stages) in a cetain way to obtain optimal base-stock levels. Next, we summaize this method. Let x e jt and y e jt be fim j s echelon inventoy position 6 befoe and afte odeing, espectively, in peiod t, whee j = s epesents the supplie and j = the etaile. Also, let L y e t = 1 c y e t + l E [ h h s y e t Dl+1 + p + h y e t Dl+1 ] (1) be the etaile s expected inventoy cost in peiod t whee is the discount facto. Define y m as the smallest minimize of L. This minimize is the etaile s optimal base-stock level. Next, define the implicit penalty cost IP m y = L miny m y L y m (2) The supplie s expected cost chaged in peiod t is then L s y e st = 1 c sy e st + L Eh s y e st DL+1 + IP m y e st DL+1 (3) The supplie s optimal echelon base-stock level ys m is the smallest minimize of L s. Unde this echelon base-stock policy, fim j odes a sufficient amount in each peiod to aise its echelon inventoy position xjt e to the echelon base-stock level ym j fo j s. An equivalent optimal policy, known as an installation base-stock policy, is given by installation base-stock levels Y s = maxys m y m and Y = minys mym. Let x jt and y jt be fim j s installation inventoy position 7 befoe and afte odeing in peiod t, espectively. Fim j odes a sufficient amount in each peiod to aise its installation inventoy position x jt to the installation base-stock level Y j fo j s (see, fo example, Axsäte and Rosling 1993, Chen and Zheng 1994 fo a discussion on the equivalence of installation and echelon base-stock policies fo a seial system). 5. Local Contol with Pomised Lead-Time Contact Unde local contol, the supplie offes a pomised leadtime contact to the etaile. If the etaile accepts, the fims establish an inventoy isk-shaing ageement fo T peiods. Next, the fims choose thei espective optimal stocking levels. The emainde of the sequence of events is as descibed in 3.1. We use backwad induction to chaacteize the optimal decisions; i.e., we solve fo the optimal inventoy-stocking decisions, followed by the contacting decision The Supplie sand the Retaile s Inventoy Poblem Fo a given pomised lead-time contact (K), each fim minimizes its expected inventoy cost ove the next T peiods. The pe-peiod lump-sum payment K is independent of the ode quantity and inventoy on hand; hence, it has no effect on the inventoy eplenishment policy. Note also that the etaile is guaanteed on-time shipment of all odes unde a pomised lead-time contact. Hence, each fim independently solves a stationay, peiodic-eview inventoy contol poblem with no upsteam supply estictions. The following dynamic pogamming ecusion minimizes the cost of managing each fim s inventoy ove a finite hoizon with T t peiods emaining until temination: J jt x jt = min y jt x jt G j y jt + EJ jt+1 y jt D t fo all t 1T, J jt+1 8 fo j s, whee G s y st = 1 c s y st + Eh s y st D L p s D L+1 y st + and G y t = 1 c y t + Eh y t D l p D l+1+ y t + Fo these stationay poblems, a myopic base-stock policy is optimal (Veinott 1965). The optimal base-stock levels fo the supplie and etaile ae the minimizes of G s and G, espectively: ( ) Y 1 ps 1 c s = F s L+1 and h s + p s ( ) Y p = F 1 p 1 c l+1+ h + p To have a easonable solution, we assume that p j 1 c j fo fim j. Hence, with pomised lead time, fim j odes up to an optimal base-stock level Yj if its installation inventoy position x jt is below this level at the beginning of peiod t. The expected discounted inventoy cost ove T t peiods equals the sum of the discounted single-peiod costs, i.e., T J s x st Y s = k t G s Y s = G s whee k=t G s c s + E { h s Y s DL+1 + +p s D L+1 Y s +}

10 Opeations Reseach 56(4), pp , 28 INFORMS 97 diffe in thei poduction and pocessing lead times, cost of boowing fom an altenative souce, and the likelihood of the etaile s poviding a low vesus a high sevice level. In each scenaio, we make a combination of selections fom the following sets: L l p s Thoughout this section, an all-o-nothing pomised lead time is optimal because demand is nomally distibuted (Poposition 12) The Supplie s Cost Unde the Pomised Lead-Time Contact In Figue 2, we plot the supplie s expected inventoy cost as a function of lead times (L l), the supplie s shotage cost p s, and unde both full and asymmetic infomation. We highlight fou obsevations. Fist, the supplie s cost unde full infomation is always lowe than unde asymmetic infomation. Compae, fo example, the cost cuve unde full and asymmetic infomation when L l = 4 2. Second, the supplie s expected cost inceases with he cost of altenative soucing, i.e., he penalty cost p s. Fo example, unde asymmetic infomation when = 8 and L l = 2 4, the supplie s expected cost is 66.37, 67.62, and when p s = 19, 49, and 99, espectively. Thid, the supplie s expected cost is highe with lowe. Fo example, unde asymmetic infomation when p s = 49 and L l = 2 4, the supplie s expected cost is 76.15, 71.89, and when = 2,.5, and.8, espectively. This obsevation indicates that contacting with a low-sevice etaile is moe pofitable than contacting with a high-sevice etaile. Ou fouth and final obsevation is the decease in the supplie s expected inventoy cost when L inceases fo a constant L+l. A supplie located close to the etaile incus lowe expected inventoy cost unde a pomised lead-time contact. In othe wods, a supplie that is futhe away fom the etaile faces moe inventoy isk. The following poposition poves this obsevation. Poposition 13. Fo constant system lead time L + l, nomally distibuted demand, and two possible etaile sevice levels, the supplie s expected inventoy cost unde asymmetic infomation inceases as L deceases The Supplie sstocking Level: Local vs. Cental In the est of this section, we compae the inventoy levels unde local contol to those chosen by a cental decision make. In Figue 3, we plot the pecent incease in the supplie s inventoy level unde local contol fom the inventoy level allocated unde cental contol, i.e., Ys Y s/y s 1%. This pecent can be intepeted as ove- (espectively, unde-) investment in inventoy due to local contol when the pecent is positive (espectively, negative). The pomised lead time is zeo fo this figue. Conside, fo example, the scenaio in which p = 17, L = 2, and l = 4. When p s = 49, the pecentage incease is 4.7%. In othe wods, the supplie caies 4.7% moe inventoy as compaed to a system-optimal inventoy allocation. Note that the pecent incease is also plotted as a function of the etaile s penalty cost, the shotage cost at the supplie, and the lead times. The following thee obsevations ae woth noting. Figue 2. Supplie s expected inventoy cost. 9 λ =.2 λ =.5 λ =.8 λ =.2 λ =.5 λ =.8 λ =.2 λ =.5 λ =.8 75 Asymmetic info., L = 2, I = Asymmetic info., L = 3, I = 3 Asymmetic info., L = 4, I = 2 Full infomation, L = 2, I = 4 Full infomation, L = 3, I = 3 Full infomation, L = 4, I = p s = 19 p s = 49 p s = 99

12 Opeations Reseach 56(4), pp , 28 INFORMS 99 the etaile s location than the etaile would cay independently when the holding cost at the supplie location is sufficiently high. Such compaisons show when and how much the supplie and the etaile unde- o oveinvest in inventoy as compaed to an integated system managed by a cental decision make. The models can be used to quantify this diffeence in inventoy investment and how this diffeence changes with the system paametes, such as lead times. We also chaacteize conditions unde which the supplie should assume all o none of the inventoy isk. In paticula, the optimal pomised lead-time contact stipulates eithe build-to-stock ( = ) o build-to-ode ( = L + 1) sevice to the etaile unde eithe full o asymmetic infomation when the single-peiod cost functions ae concave in the pomised lead time. A supplie woking with multiple independent etailes can conside offeing these two contacts and constuct a potfolio of etailes with pomised lead times zeo and L + 1. We leave this topic to futue eseach. We assume that the supplie boows inventoy fom an altenative souce to satisfy etaile odes on time. Altenatively, if the supplie simply puchases emegency units fom an outside souce o expedites emegency units though ovetime esouces, the supplie pays a one-time unit emegency cost p s above he nomal poduction cost. This scenaio esults in the supplie facing the equivalent of a lost-sales inventoy contol poblem: sales ae lost fo the supplie s nomal poduction pocess. Fo a peiodiceview poblem unde lost sales and positive lead times, Nahmias (1979) and efeences theein offe a two-tem myopic heuistic, which can be used to extend the pomised lead-time esults to this altenate case. The concept of isk shaing though pomised lead-time contacts is a fetile avenue fo futue eseach. Some issues to exploe include nonstationaities in cost o demand paametes, the impact of possible contact enegotiation, altenative infomation asymmeties, and the situation whee the etaile takes the lead in contact development. 1. Electonic Companion An electonic companion to this pape is available as pat of the online vesion that can be found at infoms.og/. Appendix. Poofs Poof of Poposition 1. To pove pat (a), ecall that is the mean demand fo a single peiod. The etaile s minimum expected inventoy cost can be witten as G p = p l h + p l+1+ p /h +p uf l+1+ u du (9) Hence, G p [ = p h + p l+1+ p /h +p l+ p /h +p uf l+1+ u du ] uf l+ u du We divide this expession by h + p to obtain G p [ l+1+ = uf h + p l+1+ u du l+ ] uf l+ u du whee = p /h + p. To show that G p, we show G p /h + p. To do so, we fist show that G p /h + p is concave in. Note that 2 / 2 G p /h + p = f l+1+ Fl+1+ 1 f l+ Fl+ 1/f l+1+fl+1+ 1 f l+fl+ 1. Because F has a log-concave density, F l+ is less than o equal to F l+1+ in dispesive ode. Hence, fom Equation (2.B.7) in Shaked and Shanthikuma (1994), we have f l+1+ Fl+1+ 1 f l+fl+ 1 fo all in ( 1). Hence, 2 / 2 G p /h + p. Note that G p /h + p is continuous in 1 and equals zeo fo 1. Togethe with concavity, this implies G p /h + p. To pove pat (b), we note that G p p = l l+1+ uf l+1+ udu 1 F 1 l+1+ Fo =, /p G p = l + 1 +, and fo = 1 G p /p =. Because 2 G p /p = 1 /f l+1+ F 1 l+1+ < we have G p /p >. To pove (c), we note that G p p [ = l+1+ uf l+1+ u du l+ ] uf l+ u du 1 F 1 1 l+1+ Fl+ Fo =, G p /p =, and fo = 1, G p /p =. To pove G p /p >, we show that it is deceasing in. To do so, note that 2 G p p [ fl+ Fl+ 1 = 1 f l+1+f 1 f l+ Fl+ 1f l+1+fl+1+ 1 l+1+ ]

16 Opeations Reseach 56(4), pp , 28 INFORMS 913 To pove pat (b), fom Equation (3), we have y L sy = 1 c s + L h s + L+1 y EIPm y D L+1 (12) Fom Equation (2), we have y IPm y = y L y y y m y>y m By the definition of y m, the above function is continuous ove all x. We define the fist deivative of L s y in two egions. When y y m, then y u ym fo all u. Hence, fom Equation (12), we have y L sy = 1 c s + L h s + L+1 y L y uf L+1 u du = 1 c s + L h s + L+1 1 c l p + h s + L+l+1 p + h F L+l+2 y (13) which is inceasing in y. When y>y m,wehave/yipm y u = fo u< y y m. Hence, fom Equation (12), we have y L sy = 1 c s + L h s + L+1 y y m y L y uf L+1 u du = 1 c s + L h s + L+1 1 c l p + h s 1 F L+1 y y m + L+l+1 p + h y y m F l+1 y uf L+1 u du (14) which is inceasing in y ove the egion y y m because 2 /y 2 L s y = L+l+1 p + h f y y m l+1 y u f L+1 u du >. Because it is also inceasing in the othe egion, /yl s y is inceasing in y fo all y. Because p P, wehave/yl s = 1 c s + L h s + L+1 1 c l p + h s, and lim y /yl s y = 1 c s + L h s >. Hence, /yl s y cosses the zeo line only once, and ys m is defined as the unique point whee it cosses zeo. When p H, wehave/yl s y m. Theefoe, ys m y m and it is obtained by setting Equation (13) equal to zeo; i.e., /yl s ys m =. Solving fo ym s yields the closed-fom solution in pat (b). Note that ys m y m. When p >H,wehave/yL s y m<. Hence, ym s y m and it is the y that sets Equation (14) equal to zeo. Note also that ys m >y m. The installation base-stock levels ae obtained fom thei definition; i.e., Y = minys mym and Y s = maxys m ym. Poof of Poposition 9. To pove pat (a), note that Y = y m when p >H. We set Y = Fl+1 1l p + h s 1 c / l p + h = Fl+1+ 1 p 1 c /p + h = Y. Applying F l+1 to both sides and eaanging tems yields the theshold B. Note that when h s B, Y Y p. Note also that B is inceasing in because F l+1+ x F l+1+ x fo any x and. Pat (b) follows immediately fom Poposition 8(b) and the definition of Ys. Poof of Poposition 1. The poof follows fom concavity and Poposition 2(a) and 2(b). Poof of Poposition 11. Fist, ecall fom Poposition 5(c) that 1 a = f 1. Hence, 1 a = L + 1 because G s + G p 1 is concave. Second, conside the two tivial cases. Note that L a 2 a N a fom Poposition 5(a). Hence, if 1 a =, then i a = fo all i. Similaly, if N a = L + 1, then i a = L + 1 fo all i. Pats (a) and (b) and the expession fo Ki a follow fom Poposition 4(a). Next, conside 1 a = L+1 and N a L+1. Note that each tem in the objective function (6) is concave in, that is, W i i G s + G p i + i 1 G p i G p i 1 is concave in. Let i max be the smallest maximize. Next, we show that i a i 1 a a i+1 fo i 2 N 1. Assume fo a contadiction that i 1 a >a i > i+1 a max. We conside thee cases: i <i a, i max >i a, and i max = i a max.ifi <i a, then /W i < fo all i aa i 1. Theefoe, the supplie can incease i a to i 1 a and educe he expected cost without violating the monotonicity constaint in (6), contadicting the optimality of i a. If i max >i a, then /W i > fo all i+1 a a i. Theefoe, the supplie could decease i a to i+1 a and educe he expected cost without violating the monotonicity constaint in (6), contadicting the optimality of i a max.ifi = i a, then i i+1 a a i 1 yields a lowe expected cost fo the supplie without violating monotonicity, again contadicting the optimality of i a. Theefoe, fo all i 2N 1, we must have i a i 1 a a i+1. Simila aguments lead to N a N a 1. The above esult implies that thee exists a finite stictly inceasing sequence k 1 k 2 k n (whee k n N ) fo some n such that 1 a = 2 a = =a k 1 = L + 1; k a 1 +1 = = k a 2 k2 and k a n 1 +1 = 2 a = = k a n 1 = N a = k n. In othe wods, thee ae n segments. Next, we show that in fact n 2, i.e., thee can be at most two such segments. Conside the segment k j fo some j 2N 1. Note that W kj kj k j k=k j 1 +1 kg s k j +G p k kj +

17 914 Opeations Reseach 56(4), pp , 28 INFORMS k 1 G p k kj G p k 1 kj is also concave in kj. Let k max j be the maximize of W kj kj. Following the same aguments as above, one can show that kj kj 1 kj+1. Hence, the n segments can be educed to n 1 segments, that is, thee exists a new finite stictly inceasing sequence k 1 k 2 k n 1. This agument can be applied iteatively until the sequence of numbes is educed to two numbes, and hence two segments. Theefoe, thee exists an m such that i a = 1 a = L + 1 fo i m and i a = N a fo i>m. The expession fo Ki a follows fom Poposition 4(a). Poof of Poposition 12. Let G n be a geneal fom of the optimal expected cost function, whee n is the numbe of peiods of uncetainty, i.e., n = L+1 fo the supplie o n = l fo the etaile. Then, fom (9), G n = pn h + p Fn 1p/h+p uf n u du fo geneic holding and penalty costs h and p. To pove pat (i), when F is Nomally distibuted, the optimal base-stock level is Y = Fn 1 p/h + p = n + z n, whee z = 1 p/h + p. The esulting optimal cost is G n = h + pz n. Hence, G n depends on only though n, which is concave in fo both n = L + 1 and n = l Note also that we have G p i G p i 1= h +p i p i /h +p i h + p i 1 p i 1 /h + p i 1 l The tem in backets is the diffeence between optimal inventoy cost functions with l = = and penalty costs p i and p i 1, espectively. Fom Poposition 1(b), this tem is positive. Hence, G p i G p i 1 is concave in. To pove pat (ii), we define n G n = G n G n 1. To show that G n is concave in n, we show n G n> n+1 G n. By algeba, this inequality holds when n+1 p/h+p + uf n+1 u du 2 n 1 p/h+p n p/h+p uf n u du uf n 1 u du > (15) Letting = p/h + p, we note that the left side of (15) equals zeo fo 1. The second deivative of the left side of (15) with espect to equals 1/f n+1 Fn+1 1 2/ f n Fn 1 + 1/f n 1 Fn 1 1, which must be negative fo (15) to be tue fo all 1. That is, f n Fn 1 1 must be concave in n. Next, we show that G p i G p i 1 is concave in. To do so, we show that /p G p 2G p 1 + G p 2. Fom (9), the above is equivalent to [ l+1+ p /h +p uf l+1+ u du 2 + l+ p /h +p l+ 1 p /h +p uf l+ u du ] ( ) h uf l+ 1 u du + h + p [ ( ) ( F 1 p l+1+ 2F 1 p l+ h + p h + p ( + F 1 p l+ 1 h + p ) )] (16) The limit of the left side of (16) as = p /h + p goes to one equals l l + + l + 1 =. Hence, (16) holds as long as the deivative of the left side with espect to is negative. This deivative equals 1 1/f l+1+ Fl /f l+ Fl /f l+ 1Fl+ 1 1, which is negative when f n Fn 1 1 is concave in n. Poof of Poposition 13. The poof is defeed to the online companion that can be found at infoms.og/. Endnotes 1. Recent infomation technologies such as compliance management systems and RFID have been cited as enables fo accounting of sevice- and quality-elated activities acoss the supply chain (Lee and Öze 25). 2. In 5, we show that the etaile follows a stationay base-stock policy, so the etaile odes in each peiod to ecove the pevious peiod s custome demand. As a esult, the supplie obseves the same demand steam as the etaile. 3. We dop the subscipt t fom the definition of D n+1 fo bevity. 4. This altenative souce may be a finished-goods inventoy caied eithe by anothe fim o fo anothe etaile. It may also epesent poduction o pocessing esouces lent by anothe opeation. To ou knowledge, this altenative souce concept appeaed fist in Lee et al. (2). See also Gaves and Willems (2), who assume guaanteed sevice between evey stage in a supply chain. 5. We note the diffeence between the centalized (fistbest) solution concept used in supply chain contacting liteatue and the cental contol concepts used in multiechelon inventoy liteatue. In the liteatue, these nomenclatues ae aely discussed togethe. This is due to the fact that most papes eithe study coodination issues in supply chain contacting fo a single-peiod poblem (as in Cachon 23), o they study cental and local contol in multiechelon inventoy poblems unde full infomation (as in Axsäte and Rosling 1993). 6. Inventoy at fim j and downsteam plus the pipeline inventoy to fim j minus custome backodes. 7. Inventoy on hand at fim j plus its pipeline inventoy minus the backode due to downsteam location s ode (o custome demand fo j = ). 8. Using a simila agument in Veinott (1965), the inventoy contol poblem with linea salvage value fo each fim is conveted into an equivalent poblem with zeo salvage. 9. We use the tem deceasing and inceasing in the weak sense, so deceasing means noninceasing.

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