The Total-Cost Curve is U-Shaped

Ordering Costs

QO Order Quantity (Q)

An

nu

al C

os

t

(optimal order quantity)

SQ

DH

QTC

2

Ordering and Carrying Costs

Co

st

EOQ

TC with PD

TC without PD

PD

0 Quantity

Adding Purchasing costdoesn’t change EOQ

Total Cost

D= 9600 H= 16 S= 75Q Ordering Carring

100 7200 800200 3600 1600300 2400 2400400 1800 3200500 1440 4000600 1200 4800700 1029 5600800 900 6400900 800 7200

1000 720 80001100 655 88001200 600 9600

0

2000

4000

6000

8000

10000

12000

0 500 1000 1500

Ordering

Carring

D= 9600 H= 16 S= 75Q Ordering Carring Total

100 7200 800 8000200 3600 1600 5200300 2400 2400 4800400 1800 3200 5000500 1440 4000 5440600 1200 4800 6000700 1029 5600 6629800 900 6400 7300900 800 7200 8000

1000 720 8000 87201100 655 8800 94551200 600 9600 10200

0

2000

4000

6000

8000

10000

12000

0 500 1000 1500

Ordering

Carring

Total

D= 9600 H= 16 S= 75 P= 1Q Total Purchasing

100 8000 9600200 5200 9600300 4800 9600400 5000 9600500 5440 9600600 6000 9600700 6629 9600800 7300 9600900 8000 9600

1000 8720 96001100 9455 96001200 10200 9600

0

2000

4000

6000

8000

10000

12000

0 500 1000 1500

Total

Purchasing

D= 9600 H= 16 S= 75 P= 1Q Total PurchasingGrandTotal

100 8000 9600 17600200 5200 9600 14800300 4800 9600 14400400 5000 9600 14600500 5440 9600 15040600 6000 9600 15600700 6629 9600 16229800 7300 9600 16900900 8000 9600 17600

1000 8720 9600 183201100 9455 9600 190551200 10200 9600 19800

0

5000

10000

15000

20000

25000

0 500 1000 1500

Total

Purchasing

GrandTotal

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 = 816Ordering cost is $12 / order ==> S = 12Carrying cost is $4 / unit / year ==> H = 4Price schedule is as follows

Quantity (Q) Price (P)1-49 2050-79 1880-99 17100 or more 16

What is the best quantity that we could order to minimize our total annual cost?

Co

st

0 Quantity

Total Cost Including Purchasing Cost

p1

p2

p3

p4

EOQ

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 purchasesPrice 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.

Co

st

EOQ0 Quantity

Total Cost With Price Discount

p1

p2p3

p4

Co

st

EOQ0 Quantity

Total Cost Including Purchasing Cost

p1

p2

p3p4

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.

Co

st

EOQ0 Quantity

Total Cost Including Purchasing Cost

p1

p2

p3

p4

Q

Total Cost Including Purchasing Cost C

ost

EOQ0 Quantity

p1

p2 p3p4

Q

Total Cost Including Purchasing Cost C

ost

0 Quantity

p1

p2

p3

p4

EOQ

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) Price (P)1-49 2050-79 1880-99 17100 or more 16

What is the best quantity that we could order to minimize our total annual cost?

Example

H

SDEOQ

2

4

)816)(12(2EOQ

70EOQQ=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?

(Q) (P)1-49 2050-79 1880-99 17100 or more 16

Co

st

EOQ0 Quantity

Total Cost Including Purchasing Cost

p1

p2

p3p4

Example

Is Q = 70 and P = 18 better orQ = 80 and P = 17 orQ = 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 = 48Carrying cost is $2 / unit / year ==> H = 2Price schedule is as follows:

Quantity (Q) Price (P)600-... 8400-599 90-399 10

What is the best quantity that we could order to minimize our total annual cost?

Example

H

SDEOQ

2

2

)5000)(48(2EOQ

490EOQ

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?

Q P600-... 8400-599 90-399 10

Example

Is Q = 490 and P = 9 better orQ = 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

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?

Assignment 12b