Inventory Control:
Part 2 - Lot-Size Inventories
1
Types of Inventories

By Function
- Lot-Size (Cycle or Replenishment)
- Instantaneous (Purchase)
- Non-Instantaneous (Produce)
- Safety (Fluctuation or Buffer)
- Anticipation (Seasonal)
- Transportation (Pipeline)
- Hedge (Beyond Scope of Class)
2
Lot-Size Stocks:
Instantaneous Receipts
3
Lot-Size Stocks:
Instantaneous

Let
Q = Order Quantity
A = Usage (Forecast Demand)
S = Order (Setup) Cost per Order
c = Purchase Price per Item
i = Cost as % of Purchase
H = Holding Cost per Unit = ic

TC = Order + Holding = S(A/Q) + ic(Q/2)
4
Lot-Size Stocks:
Instantaneous, Example
Joe the plumber has gone into the designer
plunger business. He buys basic plungers from a
well-known supplier in Washington D.C. and
customizes them. He maintains a huge raw
materials plunger inventory in Defiance, Ohio.
Demand averages about 1,000 items per month;
holding costs per month are 50% of purchase
costs; and order costs are $30. Joe buys
plungers for $12. How much and how often
should Joe order? Determine total relevant costs.
5
Lot-Size Stocks:
Instantaneous, Example

A
S
c
i
=
=
=
=
1000 Plungers per Month
$ 30 per Order
$ 12 per Item
50% of Purchase Costs
H = (.50)($12) = $ 6 per Unit per Month

TC = Order + Holding = (30000/Q) + (6)(Q/2)
6
Lot Size Stocks:
Instantaneous, Example
Q
50
100
150
200
Order
$600
300
200
150
Holding
$150
300
450
600
Total (TC)
$750
600
650
750
Best Q or Q* is Apparently 100
7
Costs
Lot Size Stocks:
Instantaneous, Example
800.0
700.0
600.0
500.0
400.0
300.0
200.0
100.0
0.0
Order
Holding
TRC
50.0
100.0
150.0
200.0
Q
8
Lot Size Stocks:
Instantaneous
Let Holding
(ic)Q/2
Q2 ic
Q*
= Order Cost at Best Answer
= S(A/Q)
= 2AS
= (2AS/ic)0.5 = EOQ
(1)
= (2ASic)0.5
(2)
N*
= A/Q* = # of Orders
(3)
T*
= 1/N* = Reorder Time
(4)
TC*
9
Cycle Stocks:
Instantaneous Example

A
S
c
i
H

Q*(EOQ) = (2AS/ic)0.5 = (2x1000x30/6)0.5 = 100
TC*
= (2ASic)0.5 = (2x1000x30x6)0.5 = $600
N*
= A/Q*
= 1000/100
= 10


=
=
=
=
=
1000 Items per Month
$30 per Order
$12 per Item
50% of Purchase Costs
(.50)($12) = $6 per Unit per Month
10
Lot-Size Stocks:
Instantaneous, Price Discounts
A distributor buys an average of 1,600 Snortoff
Vodka bottles a year. Bottles cost $0.98 if orders
are at least 800 bottles; otherwise bottles cost
$1.00. Order costs are $5.00 and holding costs
are 10% of purchase per year. Determine the
economic order quantity.
11
Lot-Size Stocks:
Price (or Quantity) Discounts

Let x = Price Break Point

If

Example Problem
If Q < 800, Regular Price c1 = $1.00
Q  800, Discounted Price c2 = $0.98
Q < x, We Have Regular Price c1
Q  x, We Have Discounted Price c2
Also: A = 1600 per Year, S = $5, i = 10%
12
Lot-Size Stocks:
Price Discounts

TC
TC
= Order + Holding + Purchase
= S(A/Q) + ic(Q/2) + cA
(1)

Q*
= (2AS/ic)0.5
(2)

TC* = (2ASic)0.5 + cA
(3)
13
Lot-Size Stocks,
Price Discount Rules
1.
Compute Q* Using Equation (2) and c2. If
Answer is  x, Stop. You Have Answer.
2.
Calculate TC* Using Equation (3) and c1.
Calculate TCx Using Equation (1), c2,
and Q = x.
3.
If TCx  TC*, Q* = x.
4.
If TC* < TCx, Calculate Q* from Equation
(2) Using c1.
14
Lot-Size Stocks,
Quantity Discount Example
= (2AS/ic)0.5
= [(2x1600x5)/(.10x.98)]0.5 = 404
404 < 800, So Go On!
(1)
Q*
(2)
TC* = (2ASic)0.5 + cA
= [(2x1600x5x.1x1)]0.5+(1x1600)=$1640
TCx = S(A/Q) + ic(Q/2) + cA
= (5)(1600)/800) + (.1x.98)x(800/2) +
(.98)(1600) = $1617
(3) $1617 < $1640, So Q* = x = 800
15
Lot-Size Stocks:
Non-Instantaneous Receipts
16
Lot-Size Stocks:
Non-Instantaneous

Let
Q = Run Size
A = Forecast Demand (or Usage Rate = d)
S = Setup Cost per Order
H = Holding Cost per Item
p = Production/Delivery Rate
Tp= Time Machine On
IMAX is Maximum Inventory

TC = Setup + Holding
17
Lot-Size Stocks:
Non-Instantaneous

Suppose p = 100 per Hour, d = 50 per Hour,
Tp = 2 Hours

What is Q? What is IMAX?

Note that Q = pTp (100x2) or Tp (Time On) = Q/p

Also, IMAX = (p - d)Tp (50x2)
= (p - d)(Q/p) = [1-(d/p)]Q
18
Lot-Size Stocks:
Non-Instantaneous

TC = Setup + Holding

TC = S(A/Q) + H(IMAX/2)
TC = S(A/Q) + (1-(d/p)) (HQ/2)
(1)

Q* = [(2AS/H) (p/(p-d))]0.5
(2)

TC* = [2ASH (1-(d/p))]0.5
(3)

N*
(4)
= (A/Q*)
19
Incorporating Q* (Or EOQ)
into MRP



We Can Use EOQ as Lot Size in MRP
Creates Excessive Inventory Due to “Lumpy”
Demand
Let Q* = EOQ = 250
Week
1
2
3
Net Requirements
100
50
150
Planned Order Receipts
250
Ending Inventory
(Available)
150
4
5
6
7
8
9
10
Total
75
200
55
80
150
30
890
110
1360
250
100
200
250
200
125
175
250
120
40
140
20
Period Order Quantity (POQ)




POQ = Q* / A = T*
A is Often in Weeks
POQ Normally Reduces Inventory and
Number of Orders When Compared with
EOQ Ordering
E.g. POQ = 250 / 89 = 2.81  3 Weeks
21
POQ Example
Week
1
2
3
Net Requirements
100
50
150
Planned Order Receipts
300
Ending Inventory
(Available)
200
4
5
6
7
8
9
10
Total
75
200
55
80
150
30
890
30
0
870
330
150
0
0
255
260
55
0
180
POQ Reduces Inventory and Number of Orders.
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