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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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