MULTIPERIOD QUANTITY-FLEXIBILITY CONTRACTS
6.2. Model and Problem Formulation
In this section, we design a one-period, two-stage quantity-flexibility supply contract between a buyer and a supplier. The contract is an agreement between a buyer and a supplier. The contract makes it possible for the buyer to have an option to increase a certain percentage of its initial orders in a later stage.
Specially, with limited information about its customer demand and market price, the buyer signs a quantity-flexibility contract with the supplier that details the terms of supply: the purchase quantity q and the unit price p. The contract allows the buyer to argument the initial purchase quantity by up to an amount (;q in a later stage at a price pc such that pc > p. In addition to the contract, the buyer has an option to purchase the same product from a spot market at the market price. The decision and information dynamics are illustrated in Figure 6.1.
At stage 1, with the knowledge of unit price p, the contract-unit price pc of the future optional purchase, the distribution of the spot-market price, and the customer demand, the buyer makes a decision of initial purchase quantity q.
The buyer is also aware that the information of the customer demand and the spot-market price will be updated at stage 2. At that time, the uncertainty of customer demand is reduced.
At stage 2, it is possible for the buyer to make a final adjustment in responding to the new information obtained between stage 1 and stage 2. The buyer can purchase additional product qc^ such that qc < ^q, at the contract price pc- Moreover, the buyer can purchase the same product from a spot market at the market price. We further assume that the spot-market price can be modeled as a random variable Ps taking value in the interval \psi^ Psh] with psh > Psi > 0- The decision at stage 2 is to choose the purchase quantity qs from the spot market at the prevailing market price ps and qdqc < ^q) on-contract at price Pc. Note that the degree of quantity flexibility is determined by the flexibility bound (; and the initial-purchase quantity g jointly.
Finally, after stage 2, the customer demand realizes. The buyer is assumed to lose revenue r for each unit of unsatisfied demand, and excess inventory is assumed to have a salvage value of s. To avoid trivial cases, we assume throughout this chapter that
r > max{pshjPc} and s < mm{psi,p}, (6.1) The above sequence of events is displayed in Figure 6.1
We use D to denote customer demand and / to represent the information observed between stage 1 and stage 2. We assume that D and / are random variables, not necessarily independent. Let
6(*5 •) = the joint distribution function of D and / ;
^(•, •) = the joint density function of D and / ; A() = the marginal distribution function of / ;
A() = the marginal density function of / ;
ip{'\i) = the conditional density function of D given I = i\
^(•|z) = the conditional distribution function of D given I = i, The optimal profit is defined as
TTI = max Hi (^)
= max < -pq + E max Il2{q, qs, qc, I, Ps)\\ , (6.2)
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= E (r-{DA{q + qs + Qc))
+s-{q + qs + qc- D)+ - pcQc - Paqs j ir,Ps
(6.3) In (6.2), pq, represents the ordering cost incurred at stage 1. The second term of (6.2),Il2{q,qs,qc,I-iPs), corresponds to the random profit received by the buyer at stage 2 given I and Pg. Therefore, the buyer's problem is to determine the optimal purchase decisions, denoted hy{q*,q*,q*),for maximizing the total ex- pected profit. Clearly, q* and q* depend on q, I, and Pg. To highlight the above dependence, we sometimes write these contingent decisions as g*(g, I, Pg) and q*{q, I, Pg), respectively. To solve the problem, we first determine the optimal ql{q, i-iPs) and g*(g, i.,Pa) for given q,I = i and Pg — Ps—that is, first solve
max 'U.2{q,qs,qc,hVs)- (6.4)
0<qs<oo 0<9c<59
With the notation defined above, given (7, Pg) = {i,pg), equation (6.3) can be written as
n2(g,gs,gc,i,Ps)
fQ+ls+qc /"oo
'q+qs+q, rq+qs+Qc
rq+qs+qc /"oo / z •'il){z\i)dz + r • {q + qg + qc) '4){z\i)dz
Jo Jq+qs+qc rq+qs+Qc
+s / [{q + qs + qc) - z] • 'ip{z\i)dz - Pcqc - Psqs- Jo
rq+qs+qc
= -{r-s) [iq + qs + qc) - z] • i^{z\i)dz Jo
+r-{q + qc + qs)-Pcqc-Psqs- (6.5) If the unit-order cost at stage 1 and the contractual unit-order cost are larger
than the unit-order costs of the spot market at stage 2—that is, Psl <Psh<P< Pc,
then for any observed market price, the best strategy is to purchase all re- quired product from the spot market—that is, g* = 0 and q* = 0. To find out g*(0, i,Ps), in view of (6.5), it suffices to find the value of qg that maximizes the function
rqs roo rqs
r z •'ip{z\i)dz + rqg il){z\i)dz + s [qg - z] • ilj{z\i)dz - pgqg.
Jo Jqs Jo
This is a newsvendor problem, and its solution is
q;(o,i^p,)==^-^(^L-Pi\?j
As a result, the model described above reduces to a classic newsvendor model. If Psi < P < Psh < Pc. then for any observed market price, g* = 0. Consequently, this case is the same as the case Psi < P < Pc < Psh with ? = 0. Similarly, if p < psi < Psh < Pc. then g* = 0, and it is the case p < Psi < Pc < Psh with <;^ = 0. In summary, based onp < pc, it suffices to consider the following cases:
P<Pc<Psl< Psh] P<Psl<Pc< Psh] Psl<P<Pc< Psh^ (6.6) REMARK 6.1 Note that if the spot-market price is very large—that is, psi —>
oo and p^h ^-^ oo—then the spot market is prohibitively expensive or nonexis- tent. Thus, the model reduces to a pure contract model.
R E M A R K 6.2 Here the spot-market price is realized at stage 2. If we were to use information I to update both the demand D and the spot-marker price P5, an extension of the following analysis could be easily carried out.
In the next section, we take up the buyer's problem at stage 2.