Transcript Inventory Discount Model

Ordering and Carrying Costs
Annual Cost
The Total-Cost Curve is U-Shaped
Q
D
TC  H  S
2
Q
Ordering Costs
QO (optimal order quantity) Order Quantity
(Q)
Cost
Total Cost
Adding Purchasing cost
doesn’t change EOQ
TC with PD
TC without PD
PD
0
EOQ
Quantity
D=
Q
100
200
300
400
500
600
700
800
900
1000
1100
1200
9600 H=
Ordering Carring
7200
800
3600
1600
2400
2400
1800
3200
1440
4000
1200
4800
1029
5600
900
6400
800
7200
720
8000
655
8800
600
9600
16 S=
75
12000
10000
8000
Ordering
6000
Carring
4000
2000
0
0
500
1000
1500
D=
Q
100
200
300
400
500
600
700
800
900
1000
1100
1200
9600 H=
16 S=
Ordering Carring
Total
7200
800
8000
3600
1600
5200
2400
2400
4800
1800
3200
5000
1440
4000
5440
1200
4800
6000
1029
5600
6629
900
6400
7300
800
7200
8000
720
8000
8720
655
8800
9455
600
9600
10200
75
12000
10000
8000
Ordering
6000
Carring
Total
4000
2000
0
0
500
1000
1500
D=
Q
100
200
300
400
500
600
700
800
900
1000
1100
1200
9600 H=
Total
Purchasing
8000
9600
5200
9600
4800
9600
5000
9600
5440
9600
6000
9600
6629
9600
7300
9600
8000
9600
8720
9600
9455
9600
10200
9600
16 S=
75 P=
1
12000
10000
8000
Total
6000
Purchasing
4000
2000
0
0
500
1000
1500
D=
Q
100
200
300
400
500
600
700
800
900
1000
1100
1200
9600 H=
16 S=
Total
PurchasingGrandTotal
8000
9600
17600
5200
9600
14800
4800
9600
14400
5000
9600
14600
5440
9600
15040
6000
9600
15600
6629
9600
16229
7300
9600
16900
8000
9600
17600
8720
9600
18320
9455
9600
19055
10200
9600
19800
75 P=
1
25000
20000
15000
Total
Purchasing
GrandTotal
10000
5000
0
0
500
1000
1500
Quantity Discount
By quantity discount, we mean the price per unit
decreases as order quantity increases.
When quantity discounts are offered, there is a
separate, U-shaped, total cost curve for each unit
price.
When unit price decreases, the total cost curve
drops.
A different total cost curve is applied to each price.
If we have quantity discount, then we should weigh
the benefit of price discount against the increase in
inventory cost.
Example
Demand for a product is 816 units / year ==> D = 816
Ordering cost is $12 / order ==> S = 12
Carrying cost is $4 / unit / year ==> H = 4
Price schedule is as follows
Quantity (Q)
1-49
50-79
80-99
100 or more
Price (P)
20
18
17
16
What is the best quantity that we could order to minimize
our total annual cost?
Total Cost Including Purchasing Cost
Cost
p1
p2
p3
p4
0
EOQ
Quantity
Total Cost with different Purchase Price
Smaller unit prices will raise total cost curve less
than larger unit prices.
For each price, there is a separate U-shaped total
cost curve for total cost.
Note that no single curve is applied to the entire
range of quantities.
Each curve is applied to a portion of the range.
Quantity Discount
Large quantity purchases
Price Discount - purchasing cost
Fewer orders - Ordering costs
More inventory - inventory cost
Our objective is to minimize the total annual costs
TC = SD/Q + HQ/2 + PD
In our initial model we assumed price is fixed.
Therefore we did not include PD in the model.
Total Cost With Price Discount
Cost
p1
p2
0
EOQ
p3
p4
Quantity
Total Cost Including Purchasing Cost
Cost
p1
p2
p3
0
EOQ
p4
Quantity
Total Cost Including Purchasing Cost
The applicable or feasible total cost is initially on the
curve with the highest unit price and then drops down
curve by curve at the price breaks.
Price breaks are the minimum quantities needed to
obtain the discounts.
If carrying cost is stated in terms of cost / unit of
product / year, there is a single EOQ which is the
same for all cost curves.
Solution Procedure
1- Compute EOQ without price considerations. This EOQ is
the same for all prices.
2- But this EOQ is feasible for only one price. Identify the
corresponding price and quantity.
3-If EOQ is feasible for the lowest price ==>it is the solution. If
it is not, then calculate:
a) TC for EOQ and corresponding feasible price.
Note that TC is…
TC = HQ/2 + SD/Q +PD
b) calculate TC for all Qs of price break after the
above prices.
Compare their TC to find the best Q ==>it is the solution.
Total Cost Including Purchasing Cost
Cost
p1
p2
p4
p3
0
EOQ
Q
Quantity
Total Cost Including Purchasing Cost
Cost
p1
p2
0
EOQ
p4
p3
Q
Quantity
Total Cost Including Purchasing Cost
Cost
p1
p3
p2
0
EOQ
Quantity
p4
Example
Demand for a product is 816 units / year ==> D = 816
Ordering cost is $12 / order ==> S = 12
Carrying cost is $4 / unit / year ==> H = 4
Price schedule is as follows
Quantity (Q)
1-49
50-79
80-99
100 or more
Price (P)
20
18
17
16
What is the best quantity that we could order to minimize
our total annual cost?
Example
2SD
EOQ 
H
2(12)(816)
EOQ 
4
EOQ  70
(Q)
1-49
50-79
80-99
100 or more
(P)
20
18
17
16
Q=70 is in the 50-79 range. Therefore, the corresponding
price is $18.
Obviously, we do not consider P=20, but what about
P=17 or P=16?
Total Cost Including Purchasing Cost
Cost
p1
p2
p3
0
EOQ
p4
Quantity
Example
Is Q = 70 and P = 18 better or
Q = 80 and P = 17 or
Q = 100 and P = 16
TC = HQ/2 + SD/Q + PD
TC ( Q = 70 , P = 18) = 4(70)/2 +12(816)/70 + 18(816)
TC = 14968
TC ( Q = 80 , P = 17) = 4(80)/2 +12(816)/80 + 17(816)
TC = 14154
TC ( Q = 100 , P = 16) = 4(100)/2 +12(816)/100 + 16(816)
TC = 13354
Example
Demand for a product is 25 tones / day and there are 200
working days / year. ==> D = 25(200) = 5000.
Ordering cost is $48 / order ==> S = 48
Carrying cost is $2 / unit / year ==> H = 2
Price schedule is as follows:
Quantity (Q)
600-...
400-599
0-399
Price (P)
8
9
10
What is the best quantity that we could order to minimize
our total annual cost?
Example
2SD
EOQ 
H
2(48)(5000)
EOQ 
2
EOQ  490
Q
600-...
400-599
0-399
P
8
9
10
Q=490 is in the 400-599 range. Therefore, the corresponding
price is $9.
Obviously, we do not consider P=10 but what about P=8?
Example
Is Q = 490 and P = 9 better or
Q = 600 and P = 8
We should compare their TC
TC = HQ/2 + SD/Q + PD
TC ( Q = 490 , P = 9) = 2(490)/2 + 48(5000)/490 + 9(5000)
TC = 490 + 489.8 + 45000 = 45979.8
TC ( Q = 600 , P = 8) = 2(600)/2 + 48(5000)/600 + 8(5000)
TC = 41000
Assignment 12b
Problem 2: A small manufacturing firm uses approximately 3400
pounds of chemical dye per year. Currently the firm purchases
300 pounds per order and pays $3 per pound. The supplier has
just announced that orders of 1000 pounds and more will be
filled at a price of $2 per pound. The ordering cost is $100 and
inventory carrying cost is 51 cents per unit per year.
a) Determine the order size that will minimizes the total cost.
b) If the supplier offered a discount at 1500 pounds instead of 1000
pounds, what order size will minimize total cost?