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Dr. Ron Lembke
Economic Order Quantity
Assumptions
Demand rate is known and constant
No order lead time
Shortages are not allowed
Costs:
S - setup cost per order
H - holding cost per unit time
EOQ
Inventory
Level
Q*
Optimal
Order
Quantity
Decrease Due to
Constant Demand
Instantaneous
Average
Receipt of Optimal
Inventory Q/2
Order Quantity
Time
Total Costs
Average Inventory = Q/2
Annual Holding costs = H * Q/2
# Orders per year = D / Q
Annual Ordering Costs = S * D/Q
Annual Total Costs = Holding + Ordering
Q
D
TC (Q) H * S *
2
Q
How Much to Order?
Total Cost
Annual Cost
= Holding + Ordering
Ordering Cost
= S * D/Q
Holding Cost
= H * Q/2
Optimal Q
Order Quantity
Optimal Quantity
Total Costs =
Q
D
H* S*
2
Q
Take derivative
with respect to Q =
H
D
S* 2 0
2
Q
Set equal
to zero
Solve for Q:
H DS
2
2 Q
2 DS
Q
H
2
2 DS
Q
H
EOQ
Inventory
Level
Q*
Reorder
Point
(R)
Time
Lead Time
Adding Lead Time
2 DS
Q
H
Order before inventory depleted
Use same order size
R dL
Where:
d = demand rate (per day)
L = lead time (in days)
both in same time period (wks, months, etc.)
A Question:
2 DS
Q
H
If the EOQ is based on so many horrible assumptions
that are never really true, why is it the most commonly
used ordering policy?
Cost curve very flat around optimal Q, so a small
change in Q means small increase in Total Costs
If overestimate D by 10%, and S by 10%, and H by 20%,
they pretty much cancel each other out
Have to overestimate all in the wrong direction before Q
affected
Sensitivity
Suppose we do not order optimal Q*, but order Q instead.
Percentage profit loss given by:
TC (Q) 1 Q * Q
TC (Q*) 2 Q Q *
Should order 100, order 150 (50% over):
0.5*(0.66 + 1.5) =1.08 an 8%cost increase
Quantity Discounts- Price Break
How does this all change if price changes
depending on order size?
Holding cost as function of cost:
H=i*C
Explicitly consider price:
2 DS
Q
i C
Discount Example
D = 10,000
S = $20
Price
C = 5.00
4.50
3.90
Quantity
Q < 500
500-999
Q >= 1000
Must Include
Cost of Goods:
2 DS
Q
i C
i = 20%
EOQ
633
667
716
Q
D
TC (Q) iC S DC
2
Q
Discount Pricing
Total Cost
Price 1
Price 2
Price 3
X 633
X 667
X 716
500
1,000
Order Size
C=$5
C=$4.5
C=$3.9
Discount Example
Order 667 at a time:
Hold 667/2 * 4.50 * 0.2= $300.15
Order 10,000/667 * 20 =
$299.85
Mat’l 10,000*4.50 =
$45,000.00
45,600.00
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 next cheapest price
2. Is EOQ feasible? (is EOQ in range?)
If EOQ is too small, use lowest possible Q to get
price.
3. Compute total cost for this quantity
4. Repeat until EOQ is feasible or too big.
5. Select quantity/price with lowest total cost.
Summary
Economic Order Quantity
Perfectly balances ordering and holding costs
Very robust, errors in input quantities have small impact
on correctness of results
Discount Model
Start with EOQ calculations, using H= iC
Compute EOQ for each price,
Determine feasible quantity
Compute Total Costs:
Holding, Ordering, and Cost of Goods.
Dr. Ron Lembke
Random Demand
Don’t know how many we will sell
Sales will differ by period
Average always remains the same
Standard deviation remains constant
How would our policies change?
How would our order quantity change?
EOQ balances ordering vs holding, and is unchanged
How would our reorder point change?
That’s a good question
Constant Demand vs Random
•Steady demand
•Always buy Q
•Reorder at R=dL
•Sell dL during LT
•Inv = Q after arrival
•Random demand
•Always buy Q
•Reorder at R=dL + ?
•Sell ? during LT
•Inv = ? after arrival
Inv
Q
R
L
Q
Q
Q
R
L
Random Demand
Reorder when on-hand inventory is equal to the
amount you expect to sell during LT, plus an extra
amount of safety stock
Assume daily demand has a normal distribution
If we want to satisfy all of the demand 95% of the time,
how many standard deviations above the mean should
the inventory level be?
Just considers a probability of running out, not the
number of units we’ll be short.
Demand over the Lead Time
R = Expected Demand over LT + Safety Stock
R d L z L
d = Average demand per day
L = Lead Time in days
L = st deviation of demand over Lead Time
z from normal table, e.g. z.95 = 1.65
Random Example
What should our reorder point be?
Demand averages 50 units per day, L = 5 days
Total demand over LT has standard deviation of 100
want to satisfy all demand 90% of the time
To satisfy 90% of the demand, z = 1.28
R d L z L
R 50* 5 1.28*100
R 250 128 378
Random Demand
•Random demand
•Always buy Q
•Reorder at R=dL + SS
Inv
R
•Sometimes use SS
•Sometimes don’t
•On average use 0 SS
St Dev of Daily Demand
What if we only know the average daily demand, and
the standard deviation of daily demand?
Lead time = 4 days,
daily demand = 10,
Daily demand has standard deviation = 5,
What should our reorder point be, if z = 3?
St Dev Over LT
If the average each day is 10, and the lead time is 4
days, then the average demand over the lead time must
be 40.
d * L 10* 4 40
What is the standard deviation of demand over the
lead time?
Std. Dev. ≠ 5 * 4
St Dev Over Lead Time
We can add up variances, not standard deviations
Standard deviation of demand over LT =
L Ldays day
45 10
R d * L z L d * L z Ldays day
Demand Per Day
R dL z * L * D
Or, same
thing:
L L * D
R d L z L
L = Lead time in days
d = average demand per day
D = st deviation of demand per day
z from normal table, e.g. z.95 = 1.65
Random Demand
Fixed Order Quantity
Demand per day averages 40 with standard deviation
15, lead time is 5 days, service level of 90%
L = 5 days
d = 40
R d L z * L * D
z = 1.30,
R 200 43.6 243.6
D = 15
R 40 * 5 1.30 * 5 *15
Fixed-Time Period Model
Place an order every, say, week.
Time period is fixed, order quantity will vary
Order enough so amount on hand plus on order gets
up to a target amount
Q = S – Inv
Order “up to” policies
S d L T z * L T * D
Service Level Criteria
Type I: specify probability that you do not run out
during the lead time
Chance that 100% of customers go home happy
Type II: (Fill Rate) proportion of demands met from
stock
100% chance that this many go home happy, on average
Two Types of Service
CycleDemand
1
180
2
75
3
235
4
140
5
180
6
200
7
150
8
90
9
160
10
40
Sum
1,450
Stock-Outs
0
0
45
0
0
10
0
0
0
0
55
Type I:
8 of 10 periods
80% service
Type II:
1,395 / 1,450 =
96%
Summary
Fixed Order Quantity – always order same
Random demand – reorder point needs to change
Standard Deviation over the LT is given
Standard Deviation per day is given
Fixed Time Period
Always order once a month, e.g.
Amount on hand plus on order will add up to S
Different service metrics