
56 Fashion Retail Supply Chain Management
Define:
s =
r + τ − c
r − v
.
We consider only the case when τ<c −v (or else consumer welfare is so significant
that the fashion retailer will have to order an infinite amount of the product). We can
simplify U
R,0
∗
in the following:
U
R,0
∗
= (r − v)µ
0
− (c − v −τ)(µ
0
+
d
0
+ δ
−1
(s)) − (r − v)
d
0
+ δ
−1
(
s
)
= (r − c)µ
0
− (r − v)
d
0
+ δφ(
−1
(s)).
Similarly, at Time 1, we define U
R,1
(q|µ
1
) =EP
R,1
(q|µ
1
) +CW
1
(q), and derive the
optimal ordering quantity as follows:
q
∗
R,1
(µ
1
) = arg{max
q
U
R,1
(q|µ
1
)}
= arg
max
q
(r − v)µ
1
− (c − v)q −(r − v)
d
1
+ δ
q − µ
1
d
1
+ δ
+ τq
.
It is straightforward to find that
d
2
U
R,1
(q|µ
1
)
dq
2
< 0 and hence