fill
.
rate =
E
sales
E
D
=
E
D
−
E
backorder
E
D
= 1
−
E
backorder
µ
1
.
How to choose base stock level
S
to meet the in-stock probability requirement?
Let
D
∼
F
.
Given the instock probability, say 97.5%, find
S
such that
P
(
D
≤
S
) =
F
(
S
) = 0
.
975
. Hence,
we need to find inverse,
S
=
F
−
1
(0
.
975)
. If
D
∼
(
µ, σ
2
)
, then
S
=
µ
+ 1
.
96
·
σ
; in Excel,
S
=
norm
.
inv
(
0
.
975
,
mu
,
sigma
)
.
How to choose base stock level
S
to hit the fill rate target?
First, from the fill rate require-
ment
fill
.
rate =
E
sales
/
E
D = (
µ
−
E
back
.
order)
/µ
1
, we find the expected backorder. Then use
E
back
.
order =
σ
ℓ
+1
L(z)
to find
z
; from
z
=
S
−
µ
ℓ
+1
σ
ℓ
+1
we find base stock level
S
.
5.2
EOQ Model
Your annual demand for food is stable at rate
D
. The fixed ordering cost of grocery shopping is
K
, regardless of quantity you buy; the unit inventory holding cost is
h
. If every time you buy
x
amount of food, it will last
x/D
year. Hence, you need to buy
D/x
times per year, and your
total
ordering cost
is
KD/x
. In each cycle, the average inventory is
x/
2
, and hence the
total hold cost
is
hx/
2
. The main trade-off in determining optimal quantity
x
∗
is to balance the ordering cost and
holding cost. Formally, you have a nonlinear programming problem
min
x
≥
0
V
(
x
) =
KD
x
+
hx
2
.
The objective function is
convex
(i.e.,
V
′′
(
x
)
≥
0
), hence the necessary and sufficient condition for
the optimal solution is
V
′
(
x
) =
−
KD
x
2
+
h
2
= 0
,
22

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which yields the
economic order quantity
(EOQ)
x
∗
=
√
2
KD
h
.
23

6
Week 6
6.1
Risk-Pooling
The idea of risk-pooling is based on the correlation of random variables. Consider random vari-
ables
X
,
Y
,
Z
. The
covariance
of
X
and
Y
is
Cov(
X, Y
) =
E
[(
X
−
µ
X
)(
Y
−
µ
Y
)] =
E
(
XY
)
−
µ
X
µ
Y
.
The
correlation coefficient
of
X
and
Y
is
ρ
=
Cov(
X,Y
)
σ
X
·
σ
Y
, with
−
1
≤
ρ
≤
1
. In particular,
ρ <
0
indicates
negative correlation,
ρ >
1
suggests positive correlation, and
ρ
= 0
means no correlation.
4
In particular,
Cov(
X, Y
) =
ρ
·
σ
X
·
σ
Y
,
Cov(
X, X
) =
σ
2
X
, and
V ar
(
X
+
Y
) =
V ar
(
X
) +
V ar
(
Y
) + 2
Cov
(
X, Y
)
.
Suppose two demands
X
and
Y
have the same mean
µ
and standard deviation
σ
. When they
are managed
independently
, the total standard deviation is
2
σ
. When they are managed
jointly
, we
have
pooling demand
Z
=
X
+
Y
, with
µ
Z
=
E
Z
=
E
(
X
+
Y
) =
µ
+
µ
= 2
µ
σ
2
Z
=
V ar
(
Z
) =
V ar
(
X
+
Y
) =
σ
2
+
σ
2
+ 2
ρσσ
= 2(1 +
ρ
)
σ
2
σ
Z
=
√
2(1 +
ρ
)
·
σ.
(18)
cv
Z
=
µ
Z
σ
Z
=
√
2(1 +
ρ
)
·
σ
2
µ
=
√
(1 +
ρ
)
2
·
σ
µ
From Eq. (18), we see that as long two demands are not perfectly positively correlated (i.e.,
ρ <
1
),
pooling always reduces risk (relative to unpooled demands):
√
2(1 +
ρ
)
·
σ
≤
√
2
×
2
·
σ
= 2
σ.
6.2
Optimization
In a changing business world, firms often need to made decisions with
uncertain
outcomes. For
example, to plan a promotion, a furniture company needs to stock chairs before customer demand
and selling price are fully known. To save cost, a manufacturer needs to buy components overseas,
and run production long before customers place their orders. In each case, the decision is made
when the future is uncertain—bet
before
tossing the coin.