EOQ and ROP

12.7 A computer training school stocks workbooks with the following characteristics:
Demand D = 19,500 units/year
Ordering cost S = $25/order
Holding cost H = $4/unit/year
a) Calculate the EOQ for the workbooks.
b) What are the annual holding costs for the workbooks?
c) What are the annual ordering costs?

a)
Q* = 2 D S/H

Q* = (2 * 19,500 * 25)/4 = 494 workbooks (per order)

b) Annual holdings costs = [Q/2]H = [494/2]*4 = $988/year

c) Annual ordering costs = [D/Q]S = [19500/494]*25 = $987/year

12.8. If D = 8,000 per month, S = $45 per order, and H = $2 per unit per month,
a) What is the economic order quantity?
b) How does your answer change if the holding cost doubles?
c) What if the holding cost drops in half?

a)

Q* = 2 D S/H

Q = 2*8,000*45
2 = 600 units per order

b)
If H' = 2*H --> Q' = 600/ 2 = 424.2 units per order

c)
If H'' = H/2 --> Q'' = 600/ 1/2 = 848.5 units per order

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12.10. Matthew sells beds and assorted supplies. His best-selling bed has an annual
demand of 400 units. Ordering cost is $40; holding cost is $5 per unit per year.
a) To minimize the total cost, how many units should be ordered each time an order is
placed?
b) If the holding cost per unit was $6 instead of $5, what would be the optimal order
quantity?

D = 400 beds/year
H = $5/bed/year
S = $40/order

a)
Q* = 2 D S/H

Q = 2*400*40
5 = 80 beds/order

b)
H' = $6/bed/year

Q = 2*400*40
6 = 73 beds/order

2
12.11. Southeastern Bell stocks a certain switch connector at its central warehouse for
supplying field service offices. The yearly demand for these connectors is 15,000 units.
Southeastern estimates its annual holding cost for this item to be $25 per unit. The cost
to place and process an order from the supplier is $75. The company operates 300 days
per year, and the lead time to receive an order from the supplier is 2 working days.
a) Find the economic order quantity.
b) Find the annual holding costs.
c) Find the annual ordering costs.
d) What is the reorder point?

D =Annual demand: 15,000 connectors

H = Holding cost per connector per year: $25
S = Order cost per order: $75
L = Lead time: 2 days
Working days per year: 300

a)
Q* = 2 D S/H

Q = 2*15,000*75
25 = 300 connectors per order

b) Annual holdings costs = [Q/2]H = [300/2]*25 = $3750/year

c) Annual ordering costs = [D/Q]S = [15,000/300]*75 = $3750/year

d)
d = 15000/300 = 50 connectors/day
ROP = d*L = 50*2 = 100 connectors

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12.12. Lead time for one of your fastest-moving products is 21 days. Demand during
this period averages 100 units per day.
a) What would be an appropriate reorder point?
b) How does your answer change if demand during lead time doubles?
c) How does your answer change if demand during lead time drops in half?

L = 21 days
d = 100 units/day

a) ROP = d*L = 21*100 = 2100 units

b) d' = 200 units/day --> ROP' = 200*21= 4200 units
c) d'' = 50 units/day --> ROP'' = 50*21 = 1050 units

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12.14. Thomas’s fastest-moving inventory item has a demand of 6,000 units per year.
The cost of each unit is $100 and the inventory carrying cost is $10 per unit per year.
The average ordering cost is $30 per order. It takes about 5 days for an order to arrive,
and the demand for 1 week is 120 units. (This is a corporate operation, and there are 250
working days per year.)
a) What is the EOQ?
b) What is the average inventory if the EOQ is used?
c) What is the optimal number of orders per year?
d) What is the optimal number of days in between any two orders?
e) What is the annual cost of ordering and holding inventory?
f) What is the total annual inventory cost, including the cost of the 6,000 units?

D =Annual demand: 6,000 units

p = $100
H = Holding cost per unit per year: $10
S = Order cost per order: $30
L = Lead time: 5 days
Working days per year: 250

a)
Q* = 2 D S/H

Q = 2*6,000*30
10 = 190 units per order

b) Average inventory = Q/2= 190/2 = 95 units

c) Number of orders N = D/Q=6000/190 = 31,6 orders per year

d) Time between orders TBO = Q/d = 190 /24 = 7.9 days

where d = 6000 units/250 days = 24 units/day (daily demand)

e) Annual holdings costs = [Q/2]H = [190/2]*10 = $950/year

Annual ordering costs = [D/Q]S = [6000/190]*30 = $947.37/year

f)
Total costs = D * p + [Q/2]H +[D/Q]S =6000 *100 + 950 +947.37 = $601,897.37/year

5
12.15. Joe’s machine shop uses 2,500 brackets during the course of a year. These
brackets are purchased from a supplier. The following information is known about the
brackets:
D =Annual demand: 2,500
H = Holding cost per bracket per year: $1.50
S = Order cost per order: $18.75
L = Lead time: 2 days
Working days per year: 250
a) Given the above information, what would be the economic order quantity (EOQ)?
b) Given the EOQ, what would be the average inventory? What would be the annual
inventory holding cost?
c) Given the EOQ, how many orders would be made each year? What would be the
annual order cost?
d) Given the EOQ, what is the total annual cost of managing the inventory?
e) What is the time between orders?
f) What is the reorder point (ROP)?

a)
Q* = 2 D S/H

Q = 2*2500*18.75
1.50 = 250 brackets per order

b) Average inventory = Q/2= 250/2 = 125 brackets

Annual holdings costs = [Q/2]H = [250/2]*1.50 = $187.50/year

c) Number of orders N = D/Q=2500/250 = 10 orders per year

Annual ordering costs = [D/Q]S = [2500/250]*18.75 = $187.50/year

d) Total costs = [Q/2]H +[D/Q]S = [2500/250]*18.75 + [250/2]*1.50 = $187.50 +

$187.50 = $375/year

e) Time between orders TBO = Q/d = 250 /10 = 25 days

where d = 2500 brackets/250 days = 10 brackets/day (daily demand)

f) ROP = d*L = 10*2 = 20 brackets

6
12.16. Abey sells 1,200 units of a certain spare part that costs $25 for each order, with
an annual holding cost of $24.
a) Calculate the total cost for order sizes of 25, 40, 50, 60, and 100.
b) Identify the economic order quantity and consider the implications for making an
error in calculating economic order quantity.

Demand D = 1,200 units/year

Ordering cost S = $25/order
Holding cost H = $24/unit/year

Total relevant cost = order cost + holding cost = DS + QH

Q 2

For Q = 25: 1,200 * 25 + 25 * 24 = $1,500/year

25 2

For Q = 40: 1,200 * 25 + 40 * 24 = $1,230/year

40 2

For Q = 50: 1,200 * 25 + 50 * 24 = $1,200/year

50 2

For Q = 60: 1,200 * 25 + 60 * 24 = $1,220/year

60 2

For Q = 100: 1,200 * 25 + 100 * 24 = $1,500/year

100 2

Small variations will not have a significant effect on total costs.

Q* = 2 D S/H

Q* = (2 * 1, 200 * 25)/24 = 50 units/order

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12.17. M. Cotteleer Electronics supplies microcomputer circuitry to a company that
incorporates microprocessors into refrigerators and other home appliances. One of the
components has an annual demand of 250 units, and this is constant throughout the
year. Carrying cost is estimated to be $1 per unit per year, and the ordering cost is $20
per order.
a) To minimize cost, how many units should be ordered each time an order is placed?
b) How many orders per year are needed with the optimal policy?
c) What is the average inventory if costs are minimized?
d) Suppose that the ordering cost is not $20, and Cotteleer has been ordering 150 units
each time an order is placed. For this order policy (of Q = 150) to be optimal, determine
what the ordering cost would have to be.

D = 250 microprocessors/year
H = $1/microprocessor/year
S = $20/order

a)
Q* = 2 D S/H

Q = 2*250*20
1 = 100 microprocessors per order

b) N = D/Q = 250/100 = 2.5 orders/year

c) Average inventory = Q/2 = 100/2 = 50 microprocessors

d)
D = 250 microprocessors/year
H = $1/microprocessor/year
S= ?
Q = 150 microprocessors/order

2*250*S = 150 --> S = $45/order

1

8
xx (old). Annual demand for the notebook binders at Duncan’s Stationery Shop is
10,000 units. Dana Duncan operates her business 300 days per year and finds that
deliveries from her supplier generally take 5 working days.
a) Calculate the reorder point for the notebook binders that she stocks.
b) Why is this number important to Duncan?
D = 10,000 binders/year
L = 5 days
a) d = 10000/300 = 33.3 binders/day
ROP = demand during lead time = d*L = 5*33.3 = 167 binders
b) This number is important because it tells the shop manager when it is time to
reorder. It helps the company keep enough units in inventory to prevent stockouts and
backorders while new supplies arrive.

xx (old). A law office has traditionally ordered ink refills 60 units at a time. The firm
estimates that carrying cost is 40% of the $10 unit cost and that annual demand is about
240 units per year. The assumptions of the basic EOQ model are thought to apply.
a) For what value of ordering cost would its action be optimal?
b) If the true ordering cost turns out to be much greater than your answer to (a), what is
the impact on the firm’s ordering policy?

Q = 60 refills/order
i = 40 % of purchase price per year in holding costs, where H = i*P
p = $10/refill
D = 240 refills/year
S=?

H = i * p = 0.4*10 = $4/refill/year
a) This problem reverses the unknown of a standard EOQ problem to solve for S

Q* = 2 D S/H

2*240*S = 60 --> S = $30/order

4

b) If the true ordering cost (S) is much greater than $30, then the firm is ordering too
little at a time. The order size (Q) should increase.

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Stochastic demand and safety stock

12.46. A Hotel distributes a mean of 1,000 bath towels per day to guests at the pool and
in their rooms. This demand is normally distributed with a standard deviation of 100
towels per day, based on occupancy. The laundry firm that has the linens contract
requires a 2-day lead time. The hotel expects a 98% service level to satisfy high guest
expectations.
a) What is the safety stock?
b) What is the ROP?

d = 1000 towels/day (average)

sigma = 100 towels/day
L = 2 days
Service level = 98%

a)
Safety stock = z * sigma during lead time

Sigma during lead time = 100* 2 = 141.42 towels

F(z) .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

z = 2.054

SS = Z * σ dLT = 2.054*141.42 = 291 clean towels

ROP = d*L + safety stock = 1000*2 + 291 = 2291 clean towels

10
xx. (old) A gourmet coffee shop is open 200 days a year and sells an average of 75
pounds of Kona coffee beans a day. (Demand can be assumed to be distributed
normally, with a standard deviation of 15 pounds per day.) After ordering (fixed cost =
$16 per order), beans are always shipped from Hawaii within exactly 4 days. Per-pound
annual holding costs for the beans are $3.
a) What is the economic order quantity (EOQ) for Kona coffee beans?
b) What are the total annual holding costs of stock for Kona coffee beans?
c) What are the total annual ordering costs for Kona coffee beans?
d) Assume that management has specified that no more than a 1% risk during stockout
is acceptable. What should the reorder point (ROP) be?
e) What is the safety stock needed to attain a 1% risk of stockout during lead time?
f) What is the annual holding cost of maintaining the level of safety stock needed to
support a 1% risk?
g) If management specified that a 2% risk of stockout during lead time would be
acceptable, would the safety stock holding costs decrease or increase?

d = 75 pounds/day (average)
Sigma = 15 pounds/day
S = $16 per order
H = $3/pound/year
L = 4 days

a)
D = 75 * 200 = 15,000 pounds/year

Q* = 2 D S/H

Q = 2*15,000*16
3 = 400 pounds/order

b) Annual holdings costs = [Q/2]H = [400/2]*3 = $600/year

c) Annual ordering costs = [D/Q]S = [15000/400]*16 = $600/year

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e) stockout risk = 1% Service level = 100% - stockout risk = 99%

Safety stock = z * sigma during lead time

F(z) .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

z = 2.33

Sigma during lead time = 15 * 4 = 30 pounds

SS = Z * σ dLT = 2.33*30 = 70 pounds

d) ROP = d * L + safety stock = 75*4 + 7 = 370 pounds

f) Annual safety stock holding cost = SS * H = 70 * 3 = $210/year

g) The lower we make our target service level, the less SS we need:

If the risk increases to a 2%, the service level decreases (98%) and Z decreases (2.054)
too. In consequence, the safety stock will decrease.

SS = Z * σ dLT = = 2.054*30 = 61.6 pounds

12
Discounts

12.22. Bell Computers purchases integrated chips at $350 per chip. The holding cost is
$35 per unit per year, the ordering cost is $120 per order, and sales are steady, at 400
per month. The company’s supplier, Rich Blue Chip Manufacturing, Inc., decides to
offer price concessions in order to attract larger orders. The price structure is shown
below.

Rich Blue Chip’s Price Structure

1–99 units $350
100–199 units $325
200 or more units $300

a) What is the optimal order quantity and the minimum annual cost for Bell Computers
to order, purchase, and hold these integrated chips?
b) Bell Computers wishes to use a 10% holding cost rather than the fixed $35 holding
cost in (a). What is the optimal order quantity, and what is the optimal annual cost?

a)
D = 400 chips/month --> D = 400 *12 = 4800 chips/year
H = $35/chip/year
S = $120/order

EOQ = (2 * 4,800 * 120)/35

= 181.4 chips ~181 chips (rounded)

Since Q < 200, the unit price will be $325. We have to study the case Q = 200 (at a
price of $300)

TC = p * D + S* D/Q + H * Q/2

Total cost (Q=181) = Cost of goods + Ordering cost + Carrying cost =

=$325 *4,800 + 4,800*120/181 + 35*181/2 = = $1,566,350/year

Total cost (Q = 200)

=$300 *4,800 + 4,800*120/200 + 35*200/2 = = $1,446,380/year

Bell Computers should order 200 units for a minimum total cost of $1,446,380.

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b)

D = 4800 chips/year
i = 10% of purchase price per year in holding costs, where H = i * P
S = $120/order

Q300 = 2DS/H

= (2 * 4, 800 * 120)/(0.1 * 300)

= 196 chips (not valid) < 200. 196 units cannot be bought at $300.

Q325 = (2 * 4, 800 * 120)/(0.1 * 325)

= 188 chips (valid) (100, 199)

We will have to compare this value with the case Q = 300 (at a price of $300)

Q350 = (2 * 4, 800 * 120)/(0.1 * 350)

= 181 (infeasible) > 100. 181 units would not be bought at $350.

Total cost (Q=188) = Cost of goods + Ordering cost + Carrying cost =

=$325 *4,800 + 4,800*120/188 + 0.1*325*188/2 = $1,566,119/year

Total cost (Q = 200)

=$300 *4,800 + 4,800*120/200 + 0.1*300*200/2 = $1,445,880/year

The minimum order quantity is 200 units yet again because the overall cost of
$1,445,880 is less than ordering 188 units, which has an overall cost of $1,566,119.

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12.23. A company has an annual demand for an airport metal detector of 1,400 units.
The cost of a typical detector to Wang is $400. Carrying cost is estimated to be 20% of
the unit cost, and the ordering cost is $25 per order. If the owner orders in quantities of
300 or more, he can get a 5% discount on the cost of the detectors. Should he take the
quantity discount?

D = 1,400 detectors/year
i = 20% of purchase price per year in holding costs, where H = i * P
S = $25/order

Regular price = $400

Price with discount = 0.95* 400 = $380

EOQ (with discount) = (2 * 1,400 * 25)/(0.2 * 380)

= 30.3 detectors (< 300 invalid)

EOQ (regular price) = (2 * 1,400 * 25)/(0.2 * 400)

= 29.6 ~ 30 detectors (< 300 valid)

Because this economic order quantity is below the discounted price, we must compare it
to an order quantity of 300 units.

TC = p * D + H * Q/2 + S* D/Q

Total cost (no discount) = Cost of goods + Ordering cost + Carrying cost =
=$400 *1,400 + 1,400*25/30 + 30*400*0.2/2 = = $562,367/year

Total cost for the discount (Q = 300 units)

= $380*1,400 + 1,400 *25/ 300 + 300 *380 *0.2 / 2 = $543,517/year

The optimal strategy is to order 300 units at an annual total cost of $543,517.

15
12.24. Lisa, the manager of a hotel, is disturbed by the amount of silverware she is
losing every week. She decides she needs to order some more silverware, but wants to
take advantage of any quantity discounts her vendor will offer.
For a small order (2,000 or fewer pieces), her vendor quotes a price of $1.80/piece.
If she orders 2,001–5,000 pieces, the price drops to $1.60/piece.
5,001–10,000 pieces brings the price to $1.40/piece,
and 10,001 and above reduces the price to $1.25.
Lisa’s order costs are $200 per order, her annual holding costs are 5%, and the annual
demand is 45,000 pieces.
For the best option:
a) What is the optimal order quantity?
b) What is the annual holding cost?
c) What is the annual ordering (setup) cost?
d) What are the annual costs of the silverware itself with an optimal order quantity?
e) What is the total annual cost, including ordering, holding, and purchasing the
silverware?

D = 45,000 pieces
S = $200 per order
i = 5% of purchase price per year in holding costs, where H = i * p

a)
Q1.25 = 2DS/H

(2 * 45, 000 * 200)/(0.05 * 1.25)

=

= 16,971> 10,000 Since the value is valid and it takes advantage of the greatest discount,
this is the optimal solution. (Besides, Q 1.40 , Q1.60 and Q 1.80 are infeasible values.)

b, c, d, e)

TC = PD + HQ/2 + SD/Q =
= $1.25 * 45, 000 + (0.05 * $1.25 *16971)/2 + ($200 * 45, 000)/16971 =
= $56,250 + $530.34 + $539.32 = $57,310.66 per year

16
12.25. Rocky Mountain Tire Center sells 20,000 go-cart tires per year. The ordering
cost for each order is $40, and the holding cost is 20% of the purchase price of the tires
per year. The purchase price is $20 per tire if fewer than 500 tires are ordered,
$18 per tire if 500 or more—but fewer than 1,000—tires are ordered,
and $17 per tire if 1,000 or more tires are ordered.
a) How many tires should Rocky Mountain order each time it places an order?
b) What is the total cost of this policy?

D = 20,000/year
i = 20 percent of purchase price per year in holding costs, where H = i*P
S = $40/order

p = $20/tire if fewer than 500 are ordered;

$18/tire if between 500 and 999 are ordered; and
$17/tire if 1,000 or more are ordered

Q17 = 2DS/H

(2 * 20, 000 * 40)/(0.2 * 17)

=

= 686 (not valid) < 1000

Q18 = (2 * 20, 000 * 40)/(0.2 * 18)

= 666.7 (valid) (500, 1000)

Q20 = (2 * 20, 000 * 40)/(0.2 * 20)

= 632.5 (not valid) > 500

We compare the cost of ordering 667 with the cost of ordering 1,000.

TC667 = p*D + HQ/2 + SD/Q

= $18 * 20, 000 + (.2 * $18 * 667)/2 + ($40 * 20, 000)/667
= $360,000 + $1,200 + $1,200 = $362,400 per year

17
TC1,000 = p*D + HQ/2 + SD/Q
= $17 * 20, 000 + (.2 * $17 * 1, 000)/2 + ($40 * 20, 000) /1, 000
= $340,000 + $1,700 + $800 = $342,500 per year

Rocky Mountain should order 1,000 tires each time.

xx.(old) Cesar Rego Computers, a chain of computer hardware and software retail
outlets, supplies customers with memory and storage devices. It currently faces the
following ordering decision relating to purchases of DVDs:
D = 36,000 disks
S = $25
H = $0.45
Purchase price = $0.85
Discount price = $0.82
Quantity needed to qualify for the discount = 6,000 disks
Should the discount be taken?

Q= (2 * 36, 000 * 25)/0.45

Q = 2,000 disks/order (< 6,000).

The total costs for Q = 2,000 disks per order:

Holding cost = Q/2 × H = 1,000 × 0.45 = $450/period
Ordering cost = D/Q × S = 36,000/2,000 × 25 = $450/period
Purchase cost = D × P = 36,000 × 0.85 = $30,600/period
Total cost = $31,500/period

The total costs for Q = 6,000 disks per order:

Holding cost = Q/2 × H = 3,000 × 0.45 = $1,350/period
Ordering cost = D/Q × S = 36,000/6,000 × 25 = $150/period
Purchase cost = D × P = 36,000 × 0.82 = $29,520/period
Total cost = $31,020/period

The quantity discount will save $480 on this item. The company should also consider
aspects such as available space, obsolescence and deterioration. However, 6,000 units
only represent two months of inventory.

18