MULTIPERIOD QUANTITY-FLEXIBILITY CONTRACTS
6.4. Optimal Purchase Quantity at Stage 1
6.4.1 The Case of Worthless Information Revision
The case of worthless-information revision is that the information / ob- served between stage 1 and stage 2 cannot further reduce the demand uncer- tainty. Mathematically, the random variables / and D are independent. Hence,
^(.|i) = ^(.) and e(-, •) = A(-) • *(•)• From Theorem 6.1, ql{q,i,Ps) and g*(g, i,Ps) are independent of z. Therefore, in this subsection, we denote them as ql[q,Ps) and q*s{q,Ps), respectively.
T H E O R E M 6.4 (WORTHLESS-INFORMATION REVISION). In addition to Assumptions 6.1 and 6.2, we also assume that ^(-li) = ^(O and 9(-, •) =
A ( - ) • * ( • ) •
(A) Ifp < inm{psijPcjPsh}> then the optimal order quantities are given by
= ^ — • p
r — s
(fciq'.Psi) = qliq'^Psi) = o
the optimal expected total order quantity is given by ^"^ {{r — p)/{r — s))] and the optimal expected profit is
{T — s) j z • h{z)dz, Jo
(B) Ifp > inin{psijPc^Psh}^ then we have the following three subcases, (B.l) When [—p + (Spsi + (1 ~ P)Pc] ^ 0, the optimal order quantities are given by
* ^ ^-if-p+ppsi + {i-(^y q*c{q*^Psi) = 0,
q*s(q\psi) = ^ . r — s ql{q*^Psh) = q*s{q\psh) = 0\
{l-f3)(r-s) -I (r-psi
the optimal expected total order quantity is given by
/? . ^ - 1 (^!:Il££i^ + (1 _ ^) . ^ -iir-psi\ , ... ^s ^^,_i / - p + fe/ + ( l - / 3 ) r {1-(5)(T-S)
and the optimal expected profit is
(3 / zhilj{z)dz + (1-P) / zip{z)dz \ . (B.2) When [-p + /Spsi + (1 - P)pc] < 0 and
I-P + PPsl + (1 - P)Pc + (1 + 0 ( 1 - /^)fe/. - Pc)] > 0,
^/lefT the optimal order quantities are given by
* ^ _ l _ ^ - i / ( I - m-^<^){r-Pc)-p-^PPsi + (1 - /?)Pc QciQ^^Psl) = 0,
^ r — s QciQ^Psh) = <iq\
ql{q.\psh) = 0;
the optimal expected total order quantity is given by
. ^ _ . ^ , p - l ( { l - m + ^){r -Pc)-P + (Spsl + (1 - (3)Pc\
^ ^^' V ( l - « ( l + 0 ( r - s ) ;
+/?.^-W^-^^'r — s
ancf the optimal expected profit is
( rq*+q*s{Q*,Psi) /•(i+<r)9* ) {r-s)lp z- ip{z)dz + {l- (3) z- il){z)6z \ .
(B.3) When [-p + (3psi + (1 - (3)pc] < 0 and
[-P + fe/ + (1 - (5)pc + (1 + c^)(l - (3)(psh - Pc)] < 0, the optimal order quantities are given by
g* - 0,
ql(q\Psi) = 0,
qciQ^Psh) = 0,
?:(«•, m) = * - ' ( ^ ) ;
the optimal expected total order quantity is given by
and the optimal expected profit is
(3 z- ip{z)dz + (1 - /?) / z- tp{z)dz > .
R E M A R K 6.10 When <^ = 0, the results of (B.l) and (B.2) are the same.
Furthermore, if p > f3psi + (1 - ^)Psh^ then p > (3psi + (1 - (3)pc. Therefore, when <; — Q, from Theorem 6.4 (B.2) and (B.3) we have that \ip < (3psi + (1 — P)Psh^ then the optimal order quantities are given by
^ il-P)ir-s) '
the optimal expected total order quantity is given by
(6.20) and the optimal expected profit is
{
rQ*+Q*siQ*^Psi) /•<?* "1(3 z- ip{z)dz + {l- (5) Z' il){z)dz \ . (6.21) lfp> jSpsi + (1 — (3)Psh-> then the optimal order quantities are given by
r — s the optimal expected total order quantity is given by
p. ^-1 ("j^) +(!-/?)• ^-' (''-;rzf) 5
(6.22)
and the optimal expected profit is
{
pqliq* .Pal) rql{q*,Psh) l/? / z- ^{z)(\z + {1-P) z- ijiz)dz \ . (6.23) These results are also obtained by Gumani and Tang [12] when ^(•) is a normal distribution.
REMARK 6.11 When 13 = 0, the spot-market price is definitely higher than the unit price at stage 1. From Theorem 6.4 (B.l), we get that the optimal order quantity is
g* = ^ ^ ' r — s the optimal expected total order quantity is
and the optimal expected profit is
{r-s) / zil;(z)6z. (6.25) Jo
This is the same as Theorem 4 (b) of Brown and Lee [5] with ^(•) being a normal distribution.
Proof of Theorem 6.4 Here we give only a proof of (B. 1) and (B .2), since the other results in the theorem can be established similarly. Since p > psi in Case B, then in view of p < pc and (6.6), we have
Psi <P<Pc< Psh- (6.26) Thus,
T — S J \ T — S J \T — S
It suffices to show that when (3psi + (1 "~ (^)Pc > P-> (f given in (B.l) is a maximizer of the function
Hi[q] = -pq+E [n2(g, q*M Ps). q*ciq. Ps), / , Ps)] ;
and when (3psi + (1 - f3)Pc < P and [-p + (3psi + (1 - P)pc + {1+ <;){!- P)(Psh — Pc)] > 0, q* given in (B.2) is a maximizer of n i ( g ) .
First we look at the proof of (B. 1). The proof is divided into three subcases.
CaseB.1.1: [q > ^-\{r - Psi)/{r - s))]
By Theorem 6.1,
QciQ^Psi) "= q*s{q,Psi) = QciQ^Psh) = q*s{q^Psh) = o.
Then
n i ( q ) = -pq + s {q- z)- i;{z)dz + r z- ip{z)dz + r g [ l - ^{q)].
Jo Jo This implies that
dU,{q)
dq = -p + s-'^{q)+r[l-'^{q)]
= r — p — {r — s) • "^(q)
< r-p-(r -psi)
< 0.
Hence, Ui{q) is decreasing in [^ ^{{r — psi)/{r — s))^ CXD).
CaseB.1.2: [^-'{{r - pc)/ir - s)) < q < ^''{{r - psi)/{r - s))]
It follows from Theorem 6.1 that
niW
-pq+Vi{r,s,Psi) + pPsiq+ a-(3)\s [\q-z)-i;iz)dz L Jo
-\-r z • ip{z)dz + rg • (1 - ^ ( g ) ) Jo
where
V/(r,s,ps/)
= / 3 | - ( r - 5 ) + {r-psi)-'^-
'^-H(r-Psl)/ir-s))
r — s ip{z)dz
r — s Therefore,
d n i ( g ) dg
< -p + Ppsi + {1 - P)pc.
This implies that Hi (g) is increasing on the interval [^ ^{{f~Pc)/{f~s)).,q*]
and decreasing on the interval [g*, ^ ~ ^ ( ( r — Psi)/{r — s))].
CaseB.1.3: [q < "^'^ir - Pc)/{r - s))]
Proceeding as in Case B.1.2, we can show that Ili(q) is increasing on the interval [0, ^ ~ ^ ( ( r - p c ) / ( r - s))].
Combining Cases 1-3 completes the proof for (B.l).
Finally, we look at (B.2). Similarly, the proof is also divided into several cases.
Case B.2.1: [q < ^ - ^ ( ( r - Psh)/{r - s)) and (1 + q)q < ^ " ^ ( ( r - Psh)/ir - s))]
Ili{q) can be written as
n i ( ^ ) = -pq + Vi{r,s,Psi)-\- /3psiq
+ ( l - / ? ) | - ( r - . ) -
Jo
r-Psh
- 1 / ^ - Psh r — s Pc^q
^{z)6z
-Psh * •1 Psh
( l + <^)9
Consequently, by [ - p + /3p,/ + (1 - (5)pc + (1 + <;)(! - (3){psh - Pc)\ > 0, 6Ui(q)
dq = -p + PPsl + (1 - /3) l-Pc^ + Pshi'^ + ^)] > 0.
So Ui{q) is increasing for q satisfying q < ^ ^{{r — Psh)/{f — s)) and {1 + c;)q <^'\{r - psh)/(r - s)).
CaseB.2.2: [q < ^-H{r-psh)/{r-s)), {l+<;)q > ^-\{r-psh)/{r-s)) and (1 + <;)q < ^ - ^ ( ( r - Pc)/{r - s))]
Under this case, ITi {q) can be written as n i ( g ) = -pq+Vi{r,s,Psi) + i3psiq
+ ( 1 - /?) I - (r - s) y^ l{l + <;)q-z]- ijiz)dz + r • (l + q)q-pc<^q}.
Consequently, d n i ( g )
dg -P + PPsl
+ (1 - /?) [ - ( r - 5)(1 + <^) • ^ ( ( 1 + q)q) + r • (1 + <?) - pc^]
In the following, if (l + ( r ) * - i ( ( r - p 5 / i ) / ( r - s ) ) > * - i ( ( r - p c ) / ( r - s ) ) , we go to Cases B.2.3 and B.2.5-B.2.7. If (1 + ^)'^~^{{r - Vsh)/{r - s)) <
^~^((r - PC)I{T - s)), we go to Cases B.2.4-B.2.7.
Case B.2.3: [q < ^ - i ( ( r - p , / , ) / ( r - 5 ) ) a n d (1 + 0 ? > ^''\{r-Vc)/{r- s))]
We have ni(9)
+ ( l - ^ ) | - ( r - . ) -
/•*-i((r-pc)/(r-s))
+r •^" 1 n'-Pc r — s
^
Pc
1 f ^1^) - ;
\r-sj \
U-i f !LZ^)
L \r-s J
• '0(2;)d2;
-Q | .
J
Then
dni(g)
dg = -p + ppsl + (1 - /3)pc < 0.
Case B.2.4: [ ^ - i ( ( r - Psh)/{r - s)) < q < ^-\{r - pc)/{r - s)), and (l + < ^ ) g < ^ - i ( ( r - p , ) / ( r - s ) ) ]
We have
niW
+(1 - /^) { - (^ - ^) y^ [(1+^)'? - ^1 • ^w^^
+ r ( l + (;)g-pc<rg Consequently,
dni(g) dg
= -P + PPsl
+ (1 - /?) [-(r - s)(l + <;) • ^ ( ( 1 + <;)q) + r • (1 + <;) - p^^].
Case B.2.5: [^-^((r - Psh)/{r - s)) < q < ^-\{r - pc)/{r - s)), and {l-h<^)q>^-H{r-Pc)/ir-s))]
We have n i ( g )
= -pq + Vi{r, s.psi) + (3psiq
+ ( l - / 3 ) { - ( r - s ) .
^^-H{r-Pc)/ir-s))
1 r-pc
4 - 7 . . ^ -
r — s
^^
Pc
1 fr-pc
r — s ip(z)dz
*i^)-]}-
Then
d n i ( g )
dq = -p-\-(3psi + {l-(3)pc<0.
Case B.2.6: [ ^ - i ( ( r - Pc)/(r - s)) < q < ^ - ^ ( ( r - p , 0 / ( ^ - s))]
We have
n i ( g ) = -pq + Vi{r,s,Psi)-}-l3psiq
+ (1-P)l -{r-s) [q- z] •ip{z)6z + rq\.
Consequently, d n i ( g )
d^ - -p + l3psi + {l-(3)l-(r-s)-^{q)-i-r]
< -p + (5psi + (l-P)Pc
< 0.
CaseB.2.7: [q > ^'Hir - Psi)/ir - s))]
We have
n i ( g ) = -Pq + P\ - {r- s) [q- z]-'(p{z)dz-\-rq\
+(1-P)l -{r-s) [q-z]-2p(z)dz-^rqy Consequently,
d n i ( ^ )
dq = -p + Pl-{r - s) • ^ ( g ) + r] + (1 - l3)[-{r - s) • ^{q) + r]
< -p + (3psi + {l-(3)pc
< 0.
According to (1 + c,)^-^((r - Psh)l{r - s)) > ^ - ^ ( ( r - pc)/ir - s)) or (1 + q)^-\ir - psh)/(r - s)) < ^-\ir - Pc)/(r - s)), (B.2) follows from CasesB.2.1-B.2.3andB.2.5-B.2.7 or Cases B.2.1-B.2.2andB.2.4-B.2.7,
respectively. D We now provide intuitive insights into the various results obtained in Theorem
6,4. Case A addresses the situation when the initial unit-order cost is less than the lowest possible market price. In this case, if the observed information is useless, then the buyer gains nothing by delaying his purchase to stage 2. Thus, the entire purchase is made at stage 1, and nothing is purchased at stage 2.
Indeed, in this case, the contract is of no value.
In Case B, we have (6.26). Clearly, QciQ^^Psi) = 0 in this case.
In (B. 1), the expected relevant price at stage 2 is clearly Ppsi + (1 — P)Pc^ and it is higher than the initial price p. Therefore, the buyer will buy a sufficiently large quantity q* at the initial price p so that he would not need to buy any quantity at all when the market price is high. Moreover, q* will not be too large to prohibit the buyer from taking advantage of buying in the market when the spot price is low.
We now consider (B.2) and (B.3). Note that since <? > 0, the condition
P > PPsl + (1 - P)Psh + (1 - P)^ • {Psh - Pc) (6.28) in (B.3) implies p > (3psi + (1 — (^)Pc- Thus, in both cases (B.2) and (B.3),
f^Psi + (1 ~ l^)Pc is lower than the initial price p. In contrast to (B.l), it seems reasonable, therefore, to reduce or completely postpone the purchase to stage 2 in (B.2) and (B.3). The (B.3) condition (6.28), however, also implies p >
(3psi+(1—(^)Psh' This says that the expected market price at stage 2 is lower than the initial price p, which argues for a complete postponement of the purchase.
Consequently, the initial purchase quantity is zero, and the entire respective newsvendor quantity is bought from the market depending on the prevailing market price at stage 2.
This leaves us with (B.2), where we still have p > Ppsi + (1 — P)Pc^ but we do not have (6.28). In other words, the high market price psh is not low enough for (6.28) to hold and thus argues perhaps for a reduction in the initial purchase amount rather than a complete postponement. Let us therefore consider an initial purchase of one unit at stage 1 and <; unit at stage 2. Clearly, the purchase of q unit at stage 2 will take place at psi when the market price is low and at pc when the market price is high. Thus, the per unit expected cost of a reduced purchase at stage 1 followed by an additional purchase up to the contracted amount is
P + /3<iPsi + (1 - (3)<;pc