MULTIPERIOD QUANTITY-FLEXIBILITY CONTRACTS
6.3. Contingent Order Quantity at Stage 2
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.
flexible bound. That is,
^" -I r-pc r — s I -q qtiQ^hPs) ^ -I n'-Vs
r — s
-[ +
i]-{l-\-<;)q
Before giving the proof, let us explain the theorem in words. Statement (i) says that when the contract price pc is higher than the prevailing market price p^, then the buyer purchases nothing on the contract at stage 2. Instead, the buyer purchases the product from the spot market. The purchase quantity is deter- mined by the difference of the critical fractile of the updated demand distribution and the amount purchased at stage 1. The critical fractile is determined by the demand distribution, the sales price r, the salvage value s, and the spot-market price ps. When the market price ps is higher than the contractual price pc, then the buyer purchases on the contract first and considers purchasing from the spot market only after exhausting the quantity flexibility provided in the contract.
Note that the buyer can purchase qq at most. Therefore, the marginal purchase price can be the contract price pc or the spot-market price ps- The buyer first exhausts its option to purchase on the contract with the contract price pc as the marginal purchasing price in the critical fractile calculation. Otherwise, in addition to exhausting the purchase option in the contract, the buyer purchases a desired additional amount from the spot market with the spot-market price ps as the marginal price in the critical fractile calculation.
REMARK 6.3 When q — 0—that is, when there is no flexibility at stage 2—
the contract price Pc does not impact the decision maker. So the optimal order quantity at the spot market is given by
qtiQ^hPs) ^- r -Ps
r — s
M
A J
-q(6.7) Note that, for this special case with the assumption in which Pg has a geometric distribution, Gumani and Tang [12] also obtain (6.7).
Proof of Theorem 6.1 Let us first consider (i). Note that max Yi2{q,qs,qc,hPs)
0<qs<oo
( rq+qs+Qc
= max <r z * ip{z\i)dz
0<qc<^q ^ ^ ^
POO
+r ^ (q + qs + Qc) I IIJ[Z\I)6Z
rq+qs+Qc ^
+s [q + Qs + qc- z]- ^p{z\i)6z - pcqc - Psqs > • (6.8) It follows from simple calculations that g^ (g, i,Ps) given by (i) of the theorem maximizes
-{r - s) / {t- z) • tp(z\i)dz + (r - ps)t + psq Jo
on the interval [g, +oo). If Ps < Pc, then for any qs > 0 and qc > 0,
•'q+qs+qc />oo
z •'tp{z\i)dz + r • {q-\-Qs + qc) ip{z\i)6^
J q+qs+qc r q+qs+qc
+s [q + qs + qc- z]- ip(z\i)dz - pcqc - Psqs
Jo rq+qs+qc roo
<r I z • ip{z\i)6z + r • {q + qs + qc) '4j(z\i)dz
Jo Jq+qs+qc
J
rq+qs+qc [q + qs + qc- z]- i/j{z\i)dz - ps • (qs + qc) 0< max < —(r — s) / {t — z) - ip{z\i)dz + {r — ps)t-\-psq} .
q<t<oo [ J o J
Consequently, (0, g*(g, i,Ps)) also maximizes the following function in (q^qs),
-q+qs+qc roo
z • tl;{z\i)dz + r • [q + qs + qc) / ip{z\i)d:.
Jq+qs+qc rq+qs+qc
+s {q + qs + qc- z) • ip(z\i)dz - pcqc - Psqs
on the region [0, <^q] x [0, oo). Therefore, the proof of (i) is completed. Jo Now we consider (ii). Using (6.8), it follows from simple calculations that
•I fr-Pc
[(1+0^1 A
^"r — s z Vg maximizes
-1 r-ps
-{r - s) / {t- z) • ip{z\i)dz + (r - Pc)t + Pcq Jo
on the interval [q, (1 + <;)q\, and [(1 + <;)q] V ^
- ( r - s) I (t- z) • '4)(z\i)dz + [r - ps)t Jo
r — s I maximizes
+Ps • [(1 + O d A ^- 1 /
1
\fr- V '/^
- P . s
— s
\
'^ I
) V q
on the interval ((1 + ^)q^ oo). If ps > Pc^ then for any given QS > 0, qc> 0, 5 > 0, we have
PsQs + PcQc < Ps • {QS + S)+PC' (QC - ^ ) - This implies that
rq-tqs-tqc re / zi){z\i)dz + r ' (q + Qs + qc) /
Jo JQ'\
rq+qs+qc roo / z • i;{z\i)dz + r ' {q + qs + qc) / ij{z\i)6z
Jo Jo+Q.^^Qr r I zyj[z\i)az + r ' [q + qs -i- qc) I ^p{z\i)dz
'o Jq+qs+qc rq+qs+qc
+s {q + Qs + qc- z) ' i^{z\i)dz - pciqc - ^) - Ps{qs + ^) Jo
"-q+qs+qc
< r
^0 Jq+qs+qc rq+qs+qc
+ 5 / [q + qs + qc- z]' ij{z\i)dz - ps • {qs + ^c).
Jo
Consequently, {ql(q^ hPs)i qtiq^ hPs)) also maximizes the function
rq+qs+qc roo r z ' '\l){z\i)dz + r • (g + ^5 + gc) / 'i\){^z\i)dz
Jo Jq+qs+qc rq+qs+qc
+8 {q + qs + qc- z) ' il;{z\i)dz - pcqc ~ Psqs Jo
of {qciqs) on the region [0, <;g] x [0, oo). Therefore, the proof of (ii) is
completed. D With an assumption that the demand D is conditionally stochastically monotone
with respect to signal / , we provide an explicit expression of the optimal pur- chase quantity with respect to i. Without loss of generality, we assume D to be conditionally stochastically increasing with respect to / . For the case of a conditionally stochastically decreasing with respect to / , it is possible for us to redefine the signal / so that the case of the conditionally stochastically decreas- ing can be translated to the case of a conditionally stochastically increasing.
T H E O R E M 6.2 Let the demand D be conditionally stochastically increasing with respect to L Then for an observed market price ps^ there exist i{q^Pc)>
^{q^Pc)y Kq-iPs)* ^^d i{q^Ps) defined by the relations
^
^ -11
1
- 1 /
1
( r-pc
\ r — s
^r-ps
\^ r — s
i{Q,Pc) = g , *
'iiq^Ps) = g , *
1 /
I
11
\
^r-pc\
\^ r — s \ f r-ps
\ r — s
^{QJPC)] = ( i + ^)g,
KQ^PS)] = (i + 0^>
such that
(i) ifVs < Pc, then
Q*c{q^hPs) = 0, qtiqJ^Ps) = (ii) ifpc < Ps, then
0,
^ - 1 [ Lz££
' r—s i)-q, ifi>i(q,ps)\
(tciq^hPs)
q^siq^hPs) =
0, ifi<i(qjPc),
^ " ^ ( T ^ I ^ ) - ^ , if~i{q.Pc)<i<i(q.Pc).
if i > i{q,Pc),
^q, 0,
^ - 1 ( IzPs ' r—s
if i < i{q^Ps), i)-{l + q)q, ifi>i{q,ps).
Proof Statements (i) and (ii) follow directly from the corresponding results (i) and (ii) in Theorem 6.1, respectively, when D is conditionally stochastically
increasing with respect to / . D
REMARK 6.4 Statements (i) and (ii) indicate that when the conditional de- mand distribution D given I — i has a monotonicity structure, the optimal purchase quantity at stage 2 has the same monotone structure with respect to the observed information i.
REMARK 6.5 When <; = 0—that is, when there is no flexibility at stage 2, then, for any observed market price, if the conditional distribution of D given / = 2 is increasing in i, the optimal spot-market purchase is
qsiq^hPs) =
0, if i <iiq,Ps),
^ " ' ( T E ^ I O " ^ '
if^>^(^'P^)-
(6.9) Note that for this special case with the assumption in which Ps has a geometric distribution, Gumani and Tang [12] also obtain (6.9).
REMARK 6.6 Note that (6.1) implies
r > m.ax{E[Ps],Pc} and s < inm{E[Ps],p}. (6.10)
Regarding Theorem 6.1, since its proof is based on the classical newsboy prob- lem, it can be easily shown that if Pc < ^ < Ps^ then Qsiq^hPs) = 0, and if Ps ^ s < P^ then Qsiq^hPs) = 0. These are the cases that do not occur under (6.1), but occur under (6.10). Going along the lines of the proof of Theorem 6.2, we can show that Theorem 6.2 holds also for these cases.