35
Æ 7
7
Systems Engineering — Theory & Practice
: 1000-6788(2015)07-1666-12
: F273.7
:
Vol.35, No.7
July, 2015
2015
A
CVaR Æ
Æ
!"#$%&'()*+,-. /01
234 50167894 :;)*+, +<
= >?:;@ A +B -.# CDEF@)*+,G
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bcXYdefgh#>?-.ijk
" VWlm )*+, R &'(-.
,
,
(
,
430074)
,
(ORP)
.
,
(SRP)
,
SRP
,
,
, ORP
.
,
,
.
.
;
;
;
;
Research on two different supply chain rebates and penalty
contracts with CVaR criterion
LI Jian-bin, YU Niu, LIU Zhi-xue
(School of Management, Huanzhong University of Science and Technology, Wuhan 430074, China)
Abstract We investigate two different rebates and penalty supply chain contracts when the retailer is
risk-averse under demand uncertainty. One is based on the retailer’s order quantity (ORP) and the other is
based on the real market sales (SRP). The results show that, if the rebate or penalty size is too large, then
the retailer under the ORP contract will inflate his order quantity so that it cannot coordinate the supply
chain, which is totally different from the SRP contract. And if the retailer’s risk averse level satisfies some
certain conditions, the supply chain coordination could be also achieved by designing appropriate contract
parameters for the two rebates and penalty contracts. Finally, we use numerical examples to verify the
effectiveness of the different types of contracts.
Keywords risk-averse; CVaR; rebate and penalty; coordination; supply chain contract
0
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). , , ,
Æ . Cachon[3] , !
. Tayor [4] "# , Æ $, "%& . Chiu [5] Tayor[2] ' , Æ , ( Æ
: 2014-07-29
: !" # (71171088); $!) "
*#+!" (NCET-13-0228); # $,"!%!
# (2014YQ002); %-" $!"# # (2014BDF112)
: &'. (1980–), (, ) ! / " , $$, 0%&1', # * &$: % +$ '& , ( -, )'( -, 2%( ( 3, Email: jbli@mail.hust.edu.cn; ) ) &': (. (1985–), (, % - *" , 0% # *&, # * &$: % +$ '& , ( -, E-mail:
%()*+
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,-./
nyu@mail.hust.edu.cn.
[1]
&'., 3: /)* CVaR +,4'&,*0$56*-#*
7
1667
++ + . Wong [6] ,7 .$, -18,. / . 0-, [7] ,7
/$-., /0+ .0. 2$3 [8] 45
-.6/, /0 + 71920:;1, <3
8911 . 2 32 [9] 2=: 3.0, 1 1
>3. 45 [10] ;<Æ
, /4 Æ 4?, 1 8911 .
5 67=5 , 45, 6> Æ
. @6
7 , 8 6?6> .$, / /Æ
Æ (ORP, orders rebate and penalty), (b) A 7
: (a) A (SRP, sales rebate and penalty). Æ26Æ, 6?6> 7
78 (8), 878, @78B (
) (rebate), %/8, @ C%78B
() (penalty); 52685, 9D:9Æ ORP 9. E2
@;:F6, ORP 66?6> , SRP <69:2 ? . :=6, @6; ORP Æ <
>A [11−13] 8:F. < >A 6 6?B ;, C 8B>A (?; ); @6; ORP
: = D5, A2<=GEH:I, 8 C (?; ), : ;8B<@ (J;
).
Æ@6F676K =<; .0 [14−17] . Chiu [18] >L
=?=<? @< —— =89M (mean variance), <!'/0 + . Æ Chiu [18] 6, @6/G@,N<HA>?/ CVaR @<?
=<>OI@. Rockafellar [19−20] CVaR B?@J. KL, CC732@
CVaR @</ÆD:-. Chen [21] CVaR ? @<MPQA6/
N+18Æ .0. BA [22] =QA6/;<A
, =<>OBO 1
[23]
=<1 . P4R ;< .$ , 1/
“BC” CEQCRDCC A
;F S TS, < CVaR ? @<, F
=<D? 8DTE=EU. PFH [24] CVaR ? @<, 45
=<>O/ >GE?Æ N+ 6/, VF=<>O? .
G5M 32 [25−26] WH/ CVaR @< , UXI0HF & &EB
< J , VIK @6;; Æ
. J CVaR @<Y@0L
M- +/, @6/ CVaR @<? =<>O?, D/ ORP , SRP .0, < =<D? T. , =<5, ÆW SRP 6, ORP , 8 :N> ( ÆX8M8), K< L GC
1 . >O=<, ORP , SRP =%=<5 . =<>OCEZ3,., /0+O Y<, =1 :; ; V
6 M >O=<, @ . 15, LSHF
+.
1
1.1
0123
4G56
;< =<>O , =<5.$HP, 8[
$@ c, w ID , J( p IJNZ-\$QO
35
9:2. P/ D 6, ]Q[R< (CDF) ,S^_?R<F F (·) , f (·), <3 F (·)
6TU`K\R<, JR< F (·). PR5, 79X8 s S.
S], P/ s < c < w < p. , 1 :; , /
/Æ
: (a) A Æ (orders rebate and penalty,
ORP), (b) A 7 Æ
(sales rebate and penalty, SRP). P/ , /Æ 8 T , 7> T , B
u (u > 0), T , 3B
u. T.8UL:
, Y< (w, u, T ). M?V
(a) VÆ;Æ, ; Æ
8 , < a (b);
(b) 6>KW=<I @? MN, 7
Q, <;
1668
−1
;
(c) AL^b80W, <2E (P;Æ) @cX ;
(d) Pd, 7 D 9;
(e) ALÆX9YÆ
, Æ +, MONXP$.
Q Æ, ReS R< Πsc(Q),
Πsc (Q) = p min(Q, D) − cQ + s(Q − D)+
(1)
x+ = max{0, x}. T5, 8.0K9MPQA.0, 1 Q∗0 = arg max
E(Πsc (Q)) =
Q
p−c
F −1 ( p−s ) , :;1>. Y ORP , SRP ; Y<F
O(w, u, T ) , S(w, u, T ), < O(w, u, T ) , ÆR<F:
Π̄r (Q; w, u, T ) = p min(Q, D) − wQ + s(Q − D)+ + u(Q − T )
Π̄m (w, u, T ; Q) = (w − c)Q − u(Q − T )
(2)
(3)
(2) , I0 p min(Q, D) ZZ$@ wQ, JU SI0 s(Q −
D) Æ
I0 u(Q − T ). T5, Q > T , u(Q − T ) 8?, @9;, Q < T
, u(Q − T ) 8U, @V; . (3) , (w − c)Q ZZNX
9;I0). VC, Re
$@ u(Q − T )(88U, F
W S(w, u, T ) , ÆR<:
+
Π̃r (Q; w, u, T ) = p min(Q, D) − wQ + s(Q − D)+ + u[min(Q, D) − T ]
Π̃m (w, u, T ; Q) = (w − c)Q − u[min(Q, D) − T ]
(4)
(5)
Æ ORP 6, SRP 66>7 , 22:
u(Q − T ) Æ u(min(Q, D) − T ) :F. MFC, Q < D , 7 min(Q, D) = Q, !
ORP Æ SRP :6V. a=?=<? @< —— ,.=< 8 (CVaR) Sf.
1.2 O P
: CVaR
@ Æ, ReQRg[=<? @< CVaR X. P/ π(Q, D) 618 μ ,
D R<, < qη (π(μ, D)) F π(μ, D) η <:
7 89:
qη (π(μ, D)) = Inf{z|Pr(π(μ, D) ≤ z) ≥ η}
(6)
=<? @<[7 η <=, S:;<,? % η <=,
2YB. !Z, Y@0LMM1 CVaR G@,N<HA9;>?/. η-CVaR ]3X:
CVaR
T?6
1
CVaRη (π(μ, D)) = max v + E[min(π(μ, D) − v, 0)]
v∈R
η
(7)
E FPÆLE, η ∈ (0, 1] =<>OCE, J[ 2=<>O?, 8C , F 2C
>O=<. MFC, η = 1 F 26=<5.
6> Agrawal [27], Gan [16−17] ;<=<>O# X, :\ _\Z3 \ ,.: (i) , X98 PÆ :%]KNh; (ii) =
&'., 3: /)* CVaR +,4'&,*0$56*-#*
7
1669
<>O ? @< ( CVaR ) PÆ:;1>; (iii) 1>3. ,
,. (i) 6$?V Æ,., 5 X9` “ ]” , 4 ^9`
Nh, , H “1aU” ?V8 b. _c+ , ReP/X9N
hF + Æ PÆ, F/ κ , κ F, VXRd7+, , 4 κ + κ ≤ E[Π (Q )]. ,. (ii) 6T5, =
?=<? , 1 21e PÆ:;1>. 15, 1 :;
1>, _\1 , , 4 r
r
2
2.1
m
m
sc
∗
0
Q∗0 .
;<=>W?@ABX ORP C SRP DE
FGHI
O7JKLY
MN
V 9;
ReS;<=<5
ORP
ORP
.
(2)
PÆR<:
π̄r (Q; w, u, T ) = E[Π̄r (Q; w, u, T )] = pS(Q) − wQ + sI(Q) + u(Q − T )
Q
S(q) = min(Q, D) = Q − 0 F (x)dx
, I(Q) = (Q − D)+ = Q − S(Q)
.
ORP
,
.
F PÆ
Re
i0
Æ]918 1 Y< (w̄∗ , ū∗ , T̄ ∗ ) Z3 (i) ū∗ < w̄∗ − s; (ii) ū(w̄∗ )
E[Πsc (Q∗0 )] − κr , ORP :\ .
V (8) Q jf<<k , 9:
OP
QR
FP
= w̄∗ − c; (iii) κm ≤ ū∗ T̄ ∗ ≤
dπ̄r (Q; w, u, T )
= p(1 − F (Q)) − w + sF (Q) + u = 0,
dQ
0 < F (·) < 1,
0 < p+u−w
u < w − s.
p−s < 1,
2
d π̄r (Q; w, u, T )
= −(p − s)f (Q) < 0,
dQ2
Q̄∗ = F −1 ( p+u−w
ORP
p−s ),
9: F (Q) = p+u−w
p−s , V
`8D1
(8)
<
4
ZC
19
1>. :1 SRP ∗
∗
∗
∗
= p−c
, < Q̄ = Q0 , 4 p+u−w
p−s
p−s , 9 r̄ = w̄ − c. p−c
Q̄∗ = F −1 ( p−s
) , Æ %]KNh, 4:
π̄r (Q∗0 ; w̄∗ , ū∗ , T̄ ∗ ) = E[Πsc (Q∗0 )] − ū∗ T̄ ∗ ≥ κr ,
π̄m (w̄∗ , ū∗ , T̄ ∗ ; Q∗0 ) = E[Πsc (Q∗0 )] − π̄r (Q∗0 ; w, u, T ) = ū∗ T̄ ∗ ≥ κm .
U .9: κm ≤ ū∗T̄ ∗ ≤ E[Πsc (Q∗0 )] − κr . FÆ.
i0 1 6=<5, ORP 1 :; _\Z3,.. X8> (s ≥ w̄∗ − ū∗) NX 8N> (ū∗ ≥ w̄∗ − s) , ORP 6 “ ]” , Ll:7C ; <3 C7, C7, 41PR]9
7, 26 , V6TLf[ “mg+”[28] . C!, 1:; ,
R,. ū∗ ≤ w̄∗ − s 6_:, ^ 9a:/: 9a, 7N +C6>7
, ZKPn18, B%18$@; F9a, +Æ\], I80W,
1+C:. ,. (ii) ū∗ = w̄∗ − c NF ORP , , 4 Q̄∗ = Q∗0 . ^, :; , ū∗ Q w̄∗ . !Z, G5 8 T̄ ∗ , ū∗ w̄∗ − c,
4d7 (w̄∗ − ū∗ − c = 0), V8 T̄ ∗ > Æ . ,.
(iii) T̄ ∗ 8+a^, !Æ => ]KNh, X9
“X_”. o, T̄ ∗ C>, C>, C . p` 1 M_`q ORP `q.
1 T̄ ∗ = 0 , M ORP :1 :; , <_ \ Z3 π̄m (w̄∗ , ū∗ , T̄ ∗ ; Q∗0 ) = 0, 4
ORP :1:; .
(7)
` 1 X6IT, T̄ ∗ = 0 , ORP @a3$ , !5: Q̄ > 0, `@^9 ū∗ > 0. P/ ORP :\1:; , ST
QR
IJNZ-\$QO
35
, 1:; , aZ3,. ū = w̄ − c, V!d7
w̄ − ū − c = 0, PÆ2 , @^9 . +
+ (Nh) ?, T56 L; 8 . C!, :1
1670
∗
∗
∗
∗
:; .
2.2
FGHIO7JKLY
MN
SRP
>S, V (4) 9 SRP PÆR<:
π̃r (Q; w, u, T ) = E[Π̃r (Q; w, u, T )] = (p + u)S(Q) − wQ + sI(Q) − uT
(9)
pi0 2 SRP 1:; ,..
ũ
2 Y< (w̃∗ , ũ∗ , T̃ ∗ ) Z3 (i) ũ(w̃∗ ) = (w̃ −c)(p−s)
; (ii) κm + p−s
E[Πsc (Q∗0 )] ≤ ũ∗ T̃ ∗ ≤
c−s
ũ
)E[Πsc (Q∗0 )] − κr , SRP :\ .
(1 + p−s
V (9) F Q jf<,Ejf<9:
OP
QR
∗
∗
∗
T5
dπ̃r (Q; w, u, T )
= p + u − w − (p + u − s)F (Q),
dQ
d2 π̃r (Q; w, u, T )
= −(p + u − s)f (Q).
dQ2
`Vj,.^8D1 Q̃ = F (
) 19 PÆ
1>. k Q̃ = Q , 9: ũ(w̃ ) =
, 4Z3!,. SRP . ! ÆPÆ2 %]KNh, 4:
⎧
d2 π̃r (Q;w,u,T )
dQ2
∗
∗
< 0,
∗
0
∗
∗
(w̃ −c)(p−s)
c−s
−1 p+u−w
p+u−s
p + ũ∗ − s
⎪
∗
∗
∗
∗
⎪
E[πsc (Q∗0 )] − ũ∗ T̃ ∗ ≥ κr ,
⎨ π̃r (Q0 ; w̃ , ũ , T̃ ) =
p−s
⎪
ũ∗
⎪
⎩ π̃m (w̃∗ , ũ∗ , T̃ ∗ ; Q∗0 ) = −
E[Πsc (Q∗0 )] + ũ∗ T̃ ∗ ≥ κm .
p−s
ũ
ũ
U .9: κm + p−s
E[Πsc (Q∗0 )] ≤ ũ∗ T̃ ∗ ≤ (1 + p−s
)E[Πsc (Q∗0 )] − κr . FÆ.
i0 2 , w̃∗ ∈ (c, p), _8Z3,. (i), (ii) 8 ũ∗ ,8
∗
T̃ , 19 SRP :\ . SRP 6Xa. Æ + T, 7
1 , :PÆbS (
T̃ ∗ V; ũ∗ ÆPn]9 8bS (w̃∗ − s)
,) ÆPÆI0 ( > T̃ ∗ ^9) cE (trade-off). 8i0, SRP
1:; , 8 ũ∗ Q w̃∗ > >, 3 ũ∗ = (w̃ −c)(p−s)
> w̃∗ − c,
c−s
∗
!d7U, Va8 T̃ +Æ . !
Z, Vi0 1 ^ ORP ū(w̄∗ ) = w̄∗ − c, ^ ũ(w̃∗ ) > ū(w̄∗ ). Td, :; , w, SRP 8> ORP 8. 9`
1, Re
M.
2 T̃ ∗ = 0 , M SRP :1 :; , <_ \ Z3 π̃m (w̃∗ , ũ∗ , T̃ ∗ ; Q∗0 ) < 0, 4
ORP :1:; .
(7)
ũ
Vi0 2 ^ T̃ ∗ = 0 , π̃m (w̃∗ , ũ∗, T̃ ∗; Q∗0 ) = − p−s
E[Πsc (Q∗0 )] < 0, !PÆ ,
T5 L?V8 , . T +, P/
+
Q∗w , u ∈ (0, w − s), Re
.
∗
∗
∗
ST
QR
∗
ST
QR VÆ6Q^
3 Q∗w < Q̃∗ < Q̄∗ < Q∗0 .
p−c
Q∗0 = F −1( p−s
), ORP ∗
−1 p+u−w
Q̃ = F ( p+u−s ), ∗
−1 p−w
Qw = F ( p−s ), T, T5 Q∗w < Q̃∗ < Q̄∗ < Q∗0 .
V`^, ORP SRP 1 => 1 .
MFC, 1:; , Q̃∗ = Q̄∗ = Q∗0 . Td, Æ
, = +C , 1 :; :. FZ, V Q̃∗ < Q̄∗ ^,
1 > SRP 1 , T, ORP ,
∗
−1 p+u−w
Q̄ = F ( p−s ), SRP
&'., 3: /)* CVaR +,4'&,*0$56*-#*
7
Æ d, 6? ,. u ∈ (0, w − s).
3
3.1
(
)
1671
+, V! u _\Z3
UV CVaR WXYX ORP C SRP DE
FGHIO7Z_LY
MN[\
CVaR-ORP
V, ReS 6=<>O ORP . 6>=<? @< CVaR X (7) 6=<5PÆR< (8), 9;!=<>O/ 3.0:
1
max CVaRη [π̄r (Q; w, u, T )] = max max ĝ(Q, v) := v + E[min(π̄r (Q; w, u, T ) − v, 0)]
Q
Q v∈R
η
1
ĝ(Q, v) = v + E[min(π̄r (Q; w, u, T ) − v, 0)]
η
Q
1
= v−
[v − (s + u − w)Q − (p − s)x + uT ]dF (x)
η
0
+∞
+
+
[v − (p + u − w)Q + uT ] dF (x) .
(10)
Q
i0 3 CVaR =<? @< ORP 1:; :Z3,..
3 CVaR =< ? @< , Y< (ŵ∗ , û∗ , T̂ ∗ ) Z3 (i) û∗ < η1 (p − s) + ŵ∗ − p;
OP
∗
1
∗
η [(1 − η)p + η ŵ
û(ŵ ) =
CVaR-ORP
QR
− c]; (iii)
κm + ( η1
:\ .
^
ĝ(Q, v) = v −
1
η
0
v jf<9:
`
∂ĝ(v, Q)
∂v
3:
≤ û T̂ ≤
Q̂∗ ,
1
η
E[Πsc (Q∗0 )] − κr
+∞
Q
1
η
+ ( η1
− 1)(p − c)Q∗0
(ii)
,
[v − (p + u − w)Q + uT ]+ dF (x),
Sb Q, v∗ (Q) 5, Jh?`R< ĝ(Q, v∗(Q)),
v < (s + u − w)Q − uT , ! ĝ(v, Q) = v, v Ujf<
(s + u − w)Q − uT ≤ v ≤ (p + u − w)Q − uT ,
ĝ(v, Q) = v −
<3
∗
[v − (s + u − w)Q − (p − s)x + uT ]dF (x) −
∗
1:
2:
∗
Q
=<>O/ 1
U Q̂ . ;< \` :
`
`
− 1)(p − c)Q∗0
v−(s+u−w)Q+uT
p−s
:
∂ĝ(v,Q)
∂v
= 1 > 0;
[v − (s + u − w)Q + uT − (p − s)x]dF (x),
0
1
v − (s + u − w)Q + uT
∂ĝ(v, Q)
=1− F
,
∂v
η
p−s
= 1,
v=(s+u−w)Q−uT
∂ĝ(v, Q)
∂v
v=(p+u−w)Q−rT
1
= 1 − F (Q).
η
v > (p + u − w)Q − uT ,
ĝ(v, Q) = v −
1
η
Q
0
1
[v − (s + u − w)Q + uT − (p − s)x]dF (x) − [v − (pu − w)Q + uT ](1 − F (Q)),
η
v jf<9:
∂ĝ(v, Q)
1
1
= 1 − [F (Q) + 1 − F (Q)] = 1 − < 0.
∂v
η
η
+` 1∼3, 6>R< ĝ(v, Q) , ^ Q > 0, v∗ (Q) ∈ [(s+u−uT, (P +u−w)Q−uT ].
v∗(Q) 8, `q:
] = 0, 9
1) e 1 − η1 ≤ 0, 4 Q ≥ F −1 (η), < 1 − η1 F [ v−(s+u−w)Q+uT
p−s
v ∗ (Q) = (p − s)F −1 (η) + (s + u − w)Q − uT.
IJNZ-\$QO
1672
@ v∗(Q) 8h?`R<, 9:
1
ĝ(v (Q), Q) = v (Q) −
η
∗
v∗ (Q)−(s+u−w)Q+uT
p−s
∗
= (p − s)F
−1
35
[v ∗ (Q) − (s + u − w)Q + uT − (p − s)x]dF (x)
0
1
(η) + (s + u − w)Q − uT −
η
F −1 (η)
[(p − s)F −1 (η) − (p − s)x]dF (x)
0
= (p − s)F −1 (η) + (s + u − w)Q − uT − (p − s)F −1 (η) +
p−s
= (s + u − w)Q +
η
Q jf<9:
T5 ĝ(v (Q), Q) 6
∗
2)
e 1−
1
η
F −1 (η)
p−s
η
F −1 (η)
xdF (x)
0
xdF (x),
0
dĝ(v ∗ (Q), Q)
= s + u − w < 0,
dQ
\ZR<, ! 8c8i.
> 0, < v (Q) = (p + u − w)Q − uT , h?`R<9:
Q
∗
ĝ(v ∗ (Q), Q) = (p + u − w)Q − uT −
1
η
Q
[(p − s)Q − (p − s)x]dF (x)
0
1
1
= (p + u − w)Q − uT − (p − s)QF (Q) + (p − s)
η
η
Q Fjf<,Ejf<9:
Q
xdF (x),
0
p−s
p−s
dĝ(v ∗ (Q), Q)
= (p + u − w) −
[F (Q) + Qf (Q)] +
Qf (Q)
dQ
η
η
p−s
= p+u−w−
F (Q),
η
d2 ĝ(v ∗ (Q), Q)
p−s
f (Q) < 0.
= −
dQ2
η
k dĝ(v dQ(Q),Q) = 0, 9 û∗ < η1 (p − s) + w − p(0 < F (·) < 1) , Q̂∗ = F −1(η p+u−w
p−s ), 4 =< ? @<
∗
Q̂ , 19:;1>. MFC, η = 1 , Q̂∗ = F −1 ( p+u−w
CVaR , 8D1
p−s )
f@6 =<51 . !, CVaR-ORP 1:; , 1 p−c
1 , 4Z3 Q̂∗ = Q∗0 , 4 η p+u−w
= p−s
, 9 û(ŵ∗ ) = η1 [(1 − η)p + η ŵ∗ − c].
p−s
jC, ! ÆP
Æ<Z3: ⎧
∗
1
⎪
⎪
− 1 (p − c)Q∗0 − û∗ T̂ ∗ ≥ κr ,
⎨ π̂r (Q∗0 ; ŵ∗ , û∗ , T̂ ∗ ) = E[Πsc (Q∗0 )] +
η
1
⎪
⎪
− 1 (p − c)Q∗0 + û∗ T̂ ∗ ≥ κm .
⎩ π̂m (ŵ∗ , û∗ , T̂ ∗ ; Q∗0 ) = −
η
U .9: κm + ( η1 − 1)(p − c)Q∗0 ≤ û∗T̂ ∗ ≤ E[Πsc (Q∗0 )] − κr + ( η1 − 1)(p − c)Q∗0 . i09F.
i0 3 , =?=<? @< CVaR 5, Z3,. (i)∼(iii) , ORP a5 =<5
. Æ=<5 ORP T, =<>O (Q̂∗ = F −1(η p+u−w
p−s ))
, <3=<>OCE η C , C>O=<, 1 C , :; û∗ 8C>.
3V,. û∗ < η1 (p − s) + ŵ∗ − p ^, Y< û∗ 8a^Æ p 66?, ga ,
p C>, û∗ 8a^C>. MFC, η = 1 , 8,.a3 û∗ < ŵ∗ − s, Æ=<5RV. F
9a, û(ŵ∗ ) = η1 [(1 − η)p + ηŵ∗ − c] > ū∗, 4 6=<>O, , 82L. 0L, Re<, =? CVaR @<5,
a56$5.
4 T̂ ∗ = 0 , M CVaR @< ORP :1:; , <_\Z3 π̂m (ŵ∗ , û∗ ,
T̂ ∗ ; Q∗0 ) < 0, 4 CVaR-ORP :1:; .
(7)
Æ 6=<5 (d ) `qT, (dU) %. T, M 6=<>O, @:kl7bS=<, Ta5 :; .
ST
QR
&'., 3: /)* CVaR +,4'&,*0$56*-#*
7
C!, CVaR-ORP 1:; _\Z3 T̂
C
3.2
FGHIIO7Z_LY
.
CVaR-SRP
MN[\
∗
> 0,
1673
<3 T̂ ∗ C>, C>, ? Re 6=<>O SRP . 9 CVaR-ORP Q, >S9;
CVaR @< 3.0:
1
max CVaRη [π̃r (Q; w, u, T )] = max max ǧ(Q, v) := v + E[min(π̃r (Q; w, u, T ) − v, 0)]
Q
Q v∈R
η
1
ǧ(Q, v) = v + E[min(π̃r (Q; w, u, T ) − v, 0)]
η
Q
1
= v−
[v + (w − s)Q − (p + u − s)x + uT ]dF (x)
η
0
+∞
+
[v − (p + u − w)Q + uT ] dF (x) .
+
(11)
Q
;< CVaR @< SRP .0, Re
i0.
p−c
4 e CVaR @< =<>O? η ∈ ( p−s , 1], 3 Y< (w̌∗ , ǔ∗ , Ť ∗ ) Z3
OP
(p−s)[(1−η)p+η w̌ ∗ −c]
,
η(p−s)−(p−c)
∗
(1−η)(p−c)(w−c)Q0 −[η(p−w)−(p−c)]E[Πsc (Q∗
0 )]
η(p−s)−(p−c)
(i) ǔ(w̌∗ ) =
(ii) κm +
≤ ǔ∗ Ť ∗ ≤
∗
(w−s)[ηE[Πsc (Q∗
0 )]+(1−η)(p−c)Q0 ]
η(p−s)−(p−c)
− κr
, CVaR-SRP :\ .
F9 i0 3, S9; CVaR @< 1 : Q̌∗ = F −1(η p+u−w
p+u−s ). k
(p−s)[(1−η)p+η
w̌
−c]
p+u−w
p−c
p−c
Q̌∗ = Q∗0 , 9 η p+u−s = p−s . V u > 0, ` p−s < η ≤ 1 , 9 ǔ(w̌∗ ) =
.
j
η(p−s)−(p−c)
C, ! ÆPÆZ3:
QR
∗
⎧
(w − s)[ηE[Πsc (Q∗0 )] + (1 − η)(p − c)Q∗0 ]
∗
∗
∗
∗
⎪
⎪
− ǔ∗ Ť ∗ ≥ κr ,
⎨ π̌r (Q0 ; w̌ , ǔ , Ť ) =
η(p − s) − (p − c)
⎪
(w − s)[ηE[Πsc (Q∗0 )] + (1 − η)(p − c)Q∗0 ]
⎪
⎩ π̌m (w̌∗ , ǔ∗ , Ť ∗ ; Q∗0 ) = E[Πsc (Q∗0 )] −
+ ǔ∗ Ť ∗ ≥ κm .
η(p − s) − (p − c)
U .9:
(1 − η)(p − c)(w − c)Q∗0 − [η(p − w) − (p − c)]E[Πsc (Q∗0 )]
η(p − s) − (p − c)
∗
∗
(Q
(w
−
s)[ηE[Π
sc
0 )] + (1 − η)(p − c)Q0 ]
≤ ǔ∗ Ť ∗ ≤
− κr .
η(p − s) − (p − c)
κm +
i09F.
p−c
i0 4 X6T5. M =<>OCEN (η < p−s
), : 1 :; _ \ Z3
∗
ǔ < 0, T56Æ7 Æ. T, M >O=<, CVaR @< SRP p−c
. η ∈ ( p−s
, 1] , Z3,. (i), (ii) CVaR-SRP . !Z, Re<9
; .
5 Ť ∗ = 0 , M CVaR @< SRP :1:; , <_\Z3 π̌m (w̌∗ , ǔ∗ ,
Ť ∗ ; Q∗0 ) < 0, 4 CVaR-SRP :1:; .
(7)
VC, 6=<>O, Æ=<5 SRP 9, :; U,
L/8 , ! .
∗
∗
∗
∗
∗
6 e η ∈ ( p−c
p−s , 1], w ∈ (c, p), < ǔ > û > ū , ǔ > ũ .
p−c
η ∈ ( p−s
, 1] , w ∈ (c, p),
ST
QR
ST
QR
ǔ∗ − û∗ =
(p − s)[(p − c) − η(p − w)] p − c − η(p − w)
−
η(p − s) − (p − c)
η
IJNZ-\$QO
1674
35
p−s
1
−
η(p − s) − (p − c) η
η(p − s) − η(p − s) + (p − c)
[(p − c) − η(p − w)]
η[η(p − s) − (p − c)]
p−c
> 0.
[(p − c) − η(p − w)]
η[η(p − s) − (p − c)]
(p − s)[(1 − η)p + ηw − c] (w − c)(p − s)
−
η(p − s) − (p − c)
c−s
p−s
[p − c − η(p − w)](c − s) − (w − c)[η(p − s) − (p − c)]
(c − s)[η(p − s) − (p − c)]
p−s
[(p − c)(w − s) − η(p − c)(w − s)]
(c − s)[η(p − s) − (p − c)]
(1 − η)(p − c)(w − s)(p − s)
> 0.
.
(c − s)[η(p − s) − (p − c)]
= [(p − c) − η(p − w)]
=
=
ǔ∗ − ũ∗ =
=
=
=
FÆ
V =<>O SRP Q̌∗ ≤ Q̃∗, 1:
; , 88Z3 ǔ∗ > ũ∗. V, =<>O ORP , CVaR-SRP 4?2>, 4 ǔ∗ > û∗ .
`, 1:; , CVaR-SRP 8>=<5
8, 3> CVaR-ORP 8.
4
]^a_
eH@6;/0hÆ
+, 6/Y<: p = 150, c = 80,
1
s = 10, P/ U[0, 1000] =b[. VÆ6Q, -9 Q = 500. HFÆ
, ] 1(a) ( w = 120) , Æ
8 u a 3 ,
a3`q. o, 6
=<5<6=<>O, Æ
= +C (> ), <31
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:; , /ÆS 8 u LQ w ( ] (b) ;F). j
o, , 6=<>O, Æ 6=<5` T, 19
:; , : 4?, 4Æ 8.
@6;Y<,. , h`q Æ
=1:; , ] 2(a)∼2(d)
F. w, /0+OÆ 8 u 8 T , 1
:; . ]o, /0 , T :AEa3, 19:; ∗
0
u
Q
500
250
批发价契约下的订货量
ORP契约下的订货量
SRP 契约下的订货量
CVaR-ORP契约下的订货量
CVaR-SRP契约下的订货量
450
ORP契约下的返利与惩罚值
SRP 契约下的返利与惩罚值
CVaR-ORP 契约下的返利与惩罚值
CVaR-SRP契约下的返利与惩罚值
200
400
150
350
100
300
50
250
200
10
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c
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20
25
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110
120
130
140
150
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16000
16000
零售商的保留利润
ORP契约下零售商的利润
制造商的保留利润
ORP契约下制造商的利润
14000
12000
10000
10000
8000
8000
6000
6000
4000
4000
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300
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2(a) ORP
零售商的保留利润
SRP 契约下零售商的利润
制造商的保留利润
SRP 契约下制造商的利润
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12000
2000
250
1675
400
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450
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2000
550
500
16000
3
600
650
2(b) SRP
3
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700
750
800
850
900
950
T
16000
零售商的保留利润
CVaR-ORP契约下零售商的利润
制造商的保留利润
CVaR-SRP契约下零售商的利润
14000
12000
零售商的保留利润
CVaR-SRP契约下零售商的利润
制造商的的保留利润
CVaR-SRP契约下制造商的利润
14000
12000
10000
10000
8000
8000
6000
6000
4000
4000
`
2000
650
700
750
2(c) CVaR-ORP
fglkqofnrii
800
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850
900
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2000
460
w
470
480
490
2(d) CVaR-SRP
fglkqofnrii
500
510
520
530
T
100
u
800
0
CVaR-ORP契约下的返利与惩罚值
CVaR-SRP契约下的返利与惩罚值
600
Ŧ
CVaR-ORP契约下的批发价
CVaR-ORP契约下的批发价
400
Ŧ
200
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1676
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`q, M=?=<I@ (gI@ [30] ), ZL ? FZ, @61/G@,N<HA
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}~
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tru &s *jk*0ps4hno'&,h&
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NlNZ
mn $v tp 3 '&,*0$56*-#* IJNZq
uxi yo( jklwvxpqrq4'&,zywjzs
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