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v
: salvage value per unit.
h
: shortage cost per unit.
Q
:orderquantity.
X
: random demand. Its probability density function is
f
(
x
)andcumulative
distribution function is
F
(
x
).
Y
: the fraction of good units, i.e., the amount of good units received is
YQ
.
It is independent of the demand, and its probability density function is
g
(
y
)and
mean is
μ
.
It is reasonable to assume that
s
≥
c
≥
v
≥
0and
h
≥
0. Then for any
X
=
x
and
Y
=
y
, the retailer's realized profit is
π
(
Q,x,y
)=
sx
−
cyQ
+
v
(
yQ
−
x
)
,
x
≤
yQ,
(1)
syQ
−
cyQ
−
h
(
x
−
yQ
)
,x>yQ.
Suppose the retailer is loss-averse and we use the following piecewise-linear loss
aversion utility function:
U
(
π
)=
π,
0
,
λπ, π <
0
,
π
≥
(2)
where
λ
1 is the retailer's loss aversion coecient. Note that if
λ
=1,the
retailer is risk-neutral. The above function has been used in the literature on the
inventory management because of its simplicity (see e.g., [12], [15]). Then the
retailer's expected utility is
E
[
U
(
π
(
Q,X,Y
))] =
1
0
≥
∞
U
(
π
(
Q,x,y
))
f
(
x
)
g
(
y
)
dxdy.
(3)
0
To calculate (3), we divide the region of integration into four subregions, as
shown in Fig. 1. It is shown that the retailer's profit
π
(
Q,x,y
) is negative in
S
1
and
S
4
while positive in
S
2
and
S
3
. Thus the precise expression for (3) is as
follows:
E
[
U
(
π
(
Q,X,Y
))] =
λ
π
(
Q,x,y
)
f
(
x
)
g
(
y
)
dxdy
+
π
(
Q,x,y
)
f
(
x
)
g
(
y
)
dxdy
S
1
∪S
4
S
2
∪S
3
1)
1
0
(
c−v
)
yQ
s−v
=(
λ
−
[(
s
−
v
)
x
−
(
c
−
v
)
yQ
]
f
(
x
)
g
(
y
)
dxdy
0
1)
1
0
∞
+(
λ
−
[(
s
−
c
+
h
)
yQ
−
hx
]
f
(
x
)
g
(
y
)
dxdy
(1+
s−c
h
)
yQ
+
1
0
yQ
[(
s
−
v
)
x
−
(
c
−
v
)
yQ
]
f
(
x
)
g
(
y
)
dxdy
0
+
1
0
∞
[(
s
−
c
+
h
)
yQ
−
hx
]
f
(
x
)
g
(
y
)
dxdy.
yQ
(4)
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