# Deterministic Inventory Management

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```					 Order Quantities when
Demand is Approximately
Level
Chapter 5
Inventory Management
Dr. Ron Tibben-Lembke
Inventory Costs

Costs associated with inventory:
   Cost of the products
   Cost of ordering
   Cost of hanging onto it
   Cost of having too much / disposal
   Cost of not having enough (shortage)
Shrinkage Costs

   How much is stolen?
   2% for discount, dept. stores, hardware,
convenience, sporting goods
   3% for toys & hobbies
   1.5% for all else
   Where does the missing stuff go?
   Employees: 44.5%
   Shoplifters: 32.7%
   Administrative / paperwork error: 17.5%
   Vendor fraud: 5.1%
Inventory Holding Costs

Category               % of Value
Housing (building) cost        6%
Material handling              3%
Labor cost                     3%
Opportunity/investment         11%
Pilferage/scrap/obsolescence   3%
Total Holding Cost              26%
ABC Analysis
   Divides on-hand inventory into 3 classes
   A class, B class, C class
   Basis is usually annual \$ volume
   \$ volume = Annual demand x Unit cost
   Policies based on ABC analysis
 Develop class A suppliers more
 Give tighter physical control of A items

 Forecast A items more carefully
Classifying Items
as ABC
% Annual \$ Usage
100
80
60
40
A
20              B               C
0
0           50          100      150
% of Inventory Items
ABC Classification Solution

Stock #   Vol.     Cost    \$ Vol.     %   ABC
206      26,000   \$ 36 \$936,000
105         200    600 120,000
019       2,000     55 110,000
144      20,000      4   80,000
207       7,000     10   70,000
Total                    1,316,000
ABC Classification Solution

Stock #   Vol.     Cost    \$ Vol.    %      ABC
206      26,000   \$ 36 \$936,000     71.1   A
105         200    600 120,000       9.1   A
019       2,000     55 110,000       8.4   B
144      20,000      4   80,000      6.1   B
207       7,000     10   70,000      5.3   C
Total                    1,316,000 100.0
Economic Order Quantity

Assumptions
   Demand rate is known and constant
   Shortages are not allowed
   Costs:
 A - setup cost per order
 v - unit cost

 r - holding cost per unit time
EOQ
Inventory
Level

Q*             Decrease Due to
Optimal        Constant Demand
Order
Quantity

Time
EOQ
Inventory
Level

Q*             Instantaneous
Optimal        Receipt of Optimal
Order          Order Quantity
Quantity

Time
EOQ

Inventory
Level
Q*

Reorder
Point
(ROP)

Time

EOQ

Inventory
Level
Q*
Average
Inventory Q/2
Reorder
Point
(ROP)

Time

Total Costs

   Average Inventory = Q/2
   Annual Holding costs = rv * Q/2
   # Orders per year = D / Q
   Annual Ordering Costs = A * D/Q
   Annual Total Costs = Holding + Ordering
Q      D
TC (Q)  vr *  A *
2      Q
How Much to Order?

Annual Cost

Holding Cost
= H * Q/2

Order Quantity
How Much to Order?

Annual Cost

Ordering Cost
= A * D/Q

Holding Cost
= H * Q/2

Order Quantity
How Much to Order?
Total Cost
Annual Cost          = Holding + Ordering

Order Quantity
How Much to Order?
Total Cost
Annual Cost               = Holding + Ordering

Optimal Q       Order Quantity
Optimal Quantity

Q      D
Total Costs =         vr *  A *
2      Q
Optimal Quantity

Q      D
Total Costs =         vr *  A *
2      Q
Take derivative       vr     D
with respect to Q =       A* 2
2      Q
Optimal Quantity

Q      D
Total Costs =         vr *  A *
2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
2      Q
Optimal Quantity

Q      D
Total Costs =         vr *  A *
2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
2      Q
Solve for Q:

vr DA
 2
2 Q
Optimal Quantity

Q      D
Total Costs =         vr *  A *
2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
2      Q
Solve for Q:

vr DA               2 AS
 2           Q 
2
2 Q                  vr
Optimal Quantity

Q      D
Total Costs =         vr *  A *
2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
2      Q
Solve for Q:

vr DA               2 AS             2 AS
 2           Q 
2
Q
2 Q                  vr               vr
Sensitivity

   Suppose we do not order optimal EOQ, but
order Q instead, and Q is p percent larger
   Q = (1+p) * EOQ
   Percentage Cost Penalty given by:
 p2 
PCP  50     
1 p 
     
   EOQ = 100, Q = 150, so p = 0.5
50*(0.25/1.5) = 8.33 a 8.33% cost increase
Figure 5.3 Sensitivity
Percentage Cost Penalty using Q different from the EOQ

30

25

20

15
PCP

10

5

0
-0.6      -0.4        -0.2           0        0.2         0.4   0.6
-5
p
A Question:

   If the EOQ is based on so many
horrible assumptions that are never
really true, why is it the most
commonly used ordering policy?
Benefits of EOQ

   Profit function is very shallow
   Even if conditions don’t hold
perfectly, profits are close to optimal
   Estimated parameters will not throw
you off very far
Tabular Aid 5.1

   For A = \$3.20 and r = 0.24%
   Calculate Dv =total \$ usage (or sales)
   Find where Dv fits in the table
   Use that number of months of supply
   D = 200, v = \$16, Dv=\$3,200
   From table, buy 1 month’s worth
   Q = D/12 = 200/12 = 16.7 = 17
How do you get a table?

   Decide which T values you want to
consider: 1 month, etc.
   Use same v and r values for whole table
   For each neighboring set of T’s, put them
into

288 A
Dv 
T1T2 r
How do you get a table?

   For example, A = \$3.20, r = 0.24
   To find the breakpoint between 0.25 and 0.5
   Dv = 288 * 3.2 / (0.25 * 0.5 * 0.24)
   = 921.6 / 0.03 = 30,720
   So if Dv is less than this, use 0.25, more
than that, use 0.5
   Find 0.5 and 0.75 breakpoint:
   Dv = 288 * 3.2/(0.5 * 0.75 * 0.24) = 10,2240
   Some simple calculations to get set up
   No thinking to figure out lot sizes
   Every product with the same ordering cost
and holding cost rate can use it
   Real benefit - simplified ordering
   Every product ordered every 1 or 2 weeks, or
every 1, 2, 3, 4, 6, 12 months
   Order multiple products on same schedule:
   Get volume discounts from suppliers
   Save on shipping costs
   Savings outweigh small increase from non-EOQ orders
Uncoordinated Orders

Time
Simultaneous Orders

Time
Same T = number months supply allows firm to order at
same time, saving freight and ordering expenses
Adjusted some T’s, changed order times
Offset Orders

Same T = number months supply allows firm to control
maximum inventory level by coordinating replenishments
With different T, no consistency
Quantity Discounts

   How does this all change if price
changes depending on order size?
   Explicitly consider price:

Q
vr
Discount Example

D = 10,000   A = \$20     r = 20%

Price    Quantity     EOQ
v = 5.00     Q < 500      633
4.50     501-999      666
3.90     Q >= 1000    716
Discount Pricing

Total Cost

Price 1      Price 2           Price 3

X 633
X 666
X 716

500              1,000      Order Size
Discount Pricing

Total Cost

Price 1      Price 2           Price 3

X 633
X 666
X 716

500              1,000      Order Size
Discount Example

Order 666 at a time:
Hold 666/2 * 4.50 * 0.2= \$299.70
Order 10,000/666 * 20 = \$300.00
Mat’l 10,000*4.50 = \$45,000.00 45,599.70

Order 1,000 at a time:
Hold 1,000/2 * 3.90 * 0.2=\$390.00
Order 10,000/1,000 * 20 = \$200.00
Mat’l 10,000*3.90 = \$39,000.00 39,590.00
Discount Model

1.Compute EOQ for each price
2.Is EOQ ‘realizeable’? (is Q in range?)
If EOQ is too large, use lowest
possible value. If too small, ignore.
3.Compute total cost for this quantity
4.Select quantity/price with lowest total
cost.

   Use same order size Q  2 DA
vr

   Order before inventory depleted
   R = DL where:
 D = annual demand rate
 L = lead time in years

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