Supplement
C Special Inventory Models
1. Noninstantaneous Replenishment
1. Maximum cycle inventory
a. Maximum cycle inventory is:
Imax = Q
pp−d( ) = Qp−d
p
where
=p
production rate
=d
demand rate
=Q
lot size
b. Cycle inventory is no longer
2Q
, as with the basic EOQ method, instead, it is what?
2. Total cost
a. Total annual cost = Annual holding cost + Annual ordering or setup cost
( ) ( ) ( ) ( )
S
Q
D
H
p
dpQ
S
Q
D
H
I
C+
−
=+= 22
max
3. Economic Production Lot Size (ELS): optimal lot size
a. Derived by calculus:
dp
p
H
DS
ELS −
=2
b. Why does the ELS result in a larger lot size than the EOQ?
4. Application C.1: Noninstantaneous Replenishment to show the difference between the
ELS model and the EOQ model.
A domestic automobile manufacturer schedules 12 two-person teams to assemble 4.6 liter
DOHC V-8 engines per work day. Each team can assemble 5 engines per day. The
automobile final assembly line creates an annual demand for the DOHC engine at 10,080
units per year. The engine and automobile assembly plants operate 6 days per week, 48 weeks
per year. The engine assembly line also produces SOHC V-8 engines. The cost to switch the
production line from one type of engine to the other is $100,000. It costs $2000 to store one
DOHC V-8 for one year.
a. What is the economic lot size?
=
−
=dp
p
H
DS
ELS 2
=_______engines
b. How long is the production run?
=
p
Q
=_______production days
c. What is the average quantity in inventory
324
60
3560
2
555,1
22
max =
−
=
−
=p
dpQ
I
engines
d. What is the total annual cost?
( ) ( ) ( ) ( )
=
+
−
=+= S
Q
D
H
p
dpQ
S
Q
D
H
I
C22
max
2. Quantity Discounts
1. Quantity discounts
2. Total annual cost = Annual holding costs + Annual ordering costs + Annual cost of materials
( ) ( )
PDS
Q
D
H
Q
C++= 2
where P = per unit price
a. Why do EOQs not necessarily produce the best lot size?
•
•
3. Two-step solution procedure to find the best lot size
a. Step 1.
b. Step 2.
• Application C.2: Finding Q with Quantity Discounts
A supplier’s price schedule is:
Order Quantity
Price per Unit
0–99
$50
100 or more
45
If ordering cost is $16 per order, annual holding cost is 20 percent of the purchase price,
and annual demand is 1,800 items, what is the best order quantity?
Step 1:
== H
DS
EOQ 2
00.45
units (infeasible)
== H
DS
EOQ 2
00.50
units (feasible)
Step 2:
( ) ( )
=++= PDS
Q
D
H
Q
C2
76
( ) ( )
=++= PDS
Q
D
H
Q
C2
100
The best order quantity is _____ units.
3. One-Period Decisions
1. Dilemma facing many retailers is how to handle seasonal goods. (why?)
2. Newsboy problem.
a. Step 1:
b. Step 2:
c. Step 3:
d. Step 4:
3. Solved Problem 3: One-Period Decisions
For one item, p = $10 and l= $5. The probability distribution for the season’s demand is:
Demand
Demand
(D)
Probability
10
0.2
20
0.3
30
0.3
40
0.1
50
0.1
Complete the following payoff matrix, as well as the column on the right showing expected
payoff. What is the best choice for Q?
D
Expected
Q
10
20
30
40
50
Payoff
10
$100
$100
$100
$100
$100
$100
20
50
200
200
200
200
170
30
0
____
300
____
300
____
40
–50
100
250
400
400
175
50
–100
50
200
350
500
140
Payoff if Q = 30 and D = 20:
( )
=−− DQlpD
Payoff if Q = 30 and D = 40:
=pD
Expected payoff if Q = 30:
The best choice for Q is: