CHAPTER 6: INVENTORY ANALYSIS
6.3 Solutions to the Problem Set
Problem 6.1
The data in the question is: flow rate R = 50,000 parts/yr, fixed setup cost S = $800, purchasing cost C =
$4/part, and cost of capital r = 20%/yr. Thus, the annual unit holding cost is H = rC = $0.8/yr. The
economic order quantity tells us to purchase each time
2RS
2  50,000  800 = 10,000 units.

H
0.8
a) Q =
b) Order R/Q = 5 times per year.
Problem 6.2
BIM Computers: Assume 8 working hours per day.
 We know Q = 4 wks supply = 1,600 units, R = 400units/wk = 20,000 units/yr, purchase cost
per unit C = $1250-20% = $1,000. Thus, holding cost H = rC = 20%/year * $1,000 =
$200/yr. Switch over or setup cost S = $2,000 + 1/2hr*$1,500/day*1day/8hr = $2,093.75.
Thus, # of setups per year = R/Q = 20,000 units/yr / 1600 units/setup = 12.5 setups/yr. Thus,
 Annual setup cost = (R/Q) S = 12.5setups/yr * $2,093.75/setup = $26,172/yr.
 Annual Purchasing Cost = RC = 20,000 units/yr * $1,000/unit = $ 20 M/yr.
 Annual Holding Cost = (Q/2)*H = 800*$200/yr = $160,000/yr.
 Thus, total annual production and inventory cost = $20,186,172.


2 RS
2  20000  2093.75
= 647 units.

H
200
EOQ =




#setups = R/Q = 20,000 /647 = 30.91. Thus, annual setup cost = 30.91setups/yr *
$2,093.75/setup = $64,711/yr.
annual holding cost = (Q/2) * H = 323.5 * $200/yr = $64,711/yr (notice that at
optimal EOQ annual holding cost equal setup costs)
annual purchasing cost remains $20M/yr
the resulting annual savings equals $186,172 - $129,422 = $56,750.
Problem 6.3
Victor's data: flow unit = one dress, flow rate R = 30 units/wk, purchase cost C = $150/unit, order lead
time L = 2 weeks, fixed order cost S = $225, cost of capital r = 20%/yr. Victor currently orders ten weeks
supply at a time, hence Q = 10wks * 30 units/wk = 300 units.
a. Costs for Victor's current inventory management:
 Annual variable ordering (purchasing) cost = RC = $150/unit * 30 units/wk * 52 wks/yr
= $234,000/yr.
 Annual fixed ordering (setups) cost = (# of orders/yr) S = (R/Q) S = (30*52/yr/300) $225
= $1,170/yr.
49
Chapter 6


Annual holding cost = H (Q/2) = (rC) (Q/2) = $30/yr * 150 = $4,500/yr.
Total annual costs = $239,670.
b. To minimize costs, Victor should order in batches of
Q* = EOQ =


2RS
2  30  52  225
= 153 units.

H
30
Thus, he should place an order for 153 units two weeks before he expects to run out. That is,
whenever current inventory drops to RL = 30 units/wk * 2 wks = 60 units, which is the reorder point.
His annual cost will be
RC +
2RSH  2  30  52  225  30 + $234,000 = $4,589 + $234,000 = $238,589.
c. Inventory turns = R/I, where average inventory I = Q/2 with cycle stock only.


Current policy: turns = R/(Q/2)=2*R/Q = 2*30units/wk / 300units = 2/10week = 52*2/10 per
year = 10.4 times per year.
Proposed policy: Q is roughly halved, so turns roughly double to 20.4 times per year.
Problem 6.4
The retailer: Current Fixed Costs S1 = $1000. Current optimal lot size Q1 = 400. New, desired lot size Q2
= 50. We must find the fixed cost S2 at which Q2 is optimal. Since Q1 is optimal for S1, we have
Q1 = 400 =
2RS1
2  R 1000
. So, R/H = 160000/2000 = 80.

H
H
Now,
Q2 = 50 =
2 RS 2
,
H
or S2 = 502 /(2*80) = 15.625. So ABC should try to reduce the fixed costs to $15.625.
Problem 6.5
Major Airlines: This question illustrates the basic tradeoff between fixed and variable costs in a service
industry; thus the concepts of EOQ discussed in class in the context of inventory management are much
more generic.
The process view here is illuminating and it goes as follows: flow unit = one flight attendant (FA). The
process transforms an input (= "un-trained" FA) into an output (= "quitted" FA). The sequence of
activities is: undergo training for 6 weeks, go on vacation for one week, wait in a buffer of "trained, but
not assigned FA" until being assigned, serve as a FA on flights, and finally quit the job.
Untrained
FA
R
Training
T = 6 wks
Vacation
1 wks
Pool of
trained FAs
? wks
Serve on flights
2 years
Quitted
FA
Chapter 6
The question asks for the tradeoff between training costs (higher class size is preferred) versus 'holding
costs' in the buffer (smaller class size -> fewer attendants waiting in buffer is preferred).
 (a)
 Flow rate R = 1000 every two years = 500 attendants per year = 10 per week.
 Fixed costs of training involves hiring ten instructors and support personnel for 6 weeks.
Thus, fixed costs of training S = 10 * ($220+$80) * 6 weeks = $18,000 per training session.
 Annual holding cost is the cost incurred to hold one flow unit (FA) in the buffer for one year:
H = $500 per month * 12 = $6,000 / person / year.
 Thus, Economic Class Size (EOQ) = 54.77 or 55 per class. Thus, we should run R/Q = 500 /
55 = 9.09 classes per year
 Per person variable cost of training is the stipend paid for 6 weeks of training + stipend for a
week of vacation = $500/mo,person * 7 wk * 12 mo/yr / 50 wk/yr = $840 per person. Notice
that the annual variable cost is constant $840/person * 500 person/yr = $420,000/yr
regardless of the class size.
 Total Annual Cost = Fixed Costs of Training + Variable Costs of Training + Holding Costs =
($18,000 * 9.09) + ($840 * 500) + (55/2)($6000) = $748,636.36 per year.
 Time Between starting consecutive classes (say, T) = Q/R = 5.5 weeks. Thus, we will have
two classes overlap for a 1/2 week (and thus we need two sets of trainers and training class
rooms). The inventory-time diagram looks as follows (assuming for simplicity that we start
the training process at time 0):
I (in training)
110
55
Class 1
0
Class 2
Class 3
6
t (weeks)
5.5
I (on vacation)
Class 1
Class 2
Class 3
55
0
6 7
I (in buffer)
Class 1
t (weeks)
Class 2
Class 3
55
0
6 7
t (weeks)
Chapter 6

(b): This part of the question illustrates the following: Often, in reality, people wish to adopt policies
that are simple (e.g., starting training every 6 weeks is simpler than trying to track the exact days to
start training when subsequent trainings start every 5.5 weeks. But what is the implication of
deviation from the optimal? In this case, quite small. This is because the optimal cost structure near
the (optimal) EOQ is quite flat. Thus any solution close to optimality will suffice.
 If time between classes (T = Q/R ) has to be 6 weeks, then Q = T R = 6 wks * 10
attendants/wk = 60 attendants.
 Total Cost of this policy = ($18,000)(500/60) + ($840)(500) + (60/2)($6000) = $750,000 per
year.
Problem 6.6
Fixed cost of filling an ATM m/c, S = $100.
To estimate demand, observe that the average size of each transaction = $80. With 150 transactions per
week, annual demand R is estimated to be = 150*52*80 = 624,000.
With cost of money of 10%, unit holding cost, H = $0.10 / year
Then, the economic quantity to place in the ATM machine is given by the EOQ formula:
Q
2RS
2  624000  100

 $35,327
H
0.1
The number of times the ATM needs to be filled = R/Q = 624000/35327 = 17.66 per year.
Problem 6.7
The annual demand, R = 150,000 lbs/yr. The purchase price per lb is $1.50. However the shipping cost
exhibits a quantity discount model. The holding cost per year is then 15% of the sum of the purchase and
shipping cost. The administrative costs of placing an order = $50/order.
(a) In addition, rental cost of the forklift truck adds to the fixed cost giving a total fixed cost, S =
50+350 = $400/order. We can use a spreadsheet model as shown in Table TN 6.1. The optimal
order quantity = 22,000 lbs with an annual cost of $249,916.77.
(b) If GC buys a forklift and builds a new ramp, then the per-transaction fixed cost will simply be the
administrative cost of $50 per order. The economic order quantity and annual operating costs of
this option is shown in Table TN 6.2. The economic order quantity is 15000 lbs. with an annual
operating cost = $246,833.75. The annual savings = 249,916.77 - 246,833.75 = $3,083.02. The
net present value of cost savings (over 5 years) with cost of capital of 15% = $10,334.76.
Assuming a useful life of 5 years for the forklift and ramp, an investment of less than $10,334
generates a positive NPV.
Problem 6.8
Changeover time = 4hrs giving a fixed cost, S = 4 * 250 = $1,000.
Annual demand, R = 1000/mo * 12 = 12,000 units / yr.
Chapter 6
Unit cost, C = 100
Holding cost = $25 / unit / yr.
a) The optimal production batch size is
Q
2 RS
2 12000 1000

 $980
H
25
b) To reduce batch size by a factor of 4, the setup cost needs o be reduced by a factor of (4)2 = 16.
That is, S should reduce to 1000/16 = $62.5. This reduction can be achieved by reducing the
changeover time or the cost per unit time during changeovers.
Problem 6.9
a) From the EOQ formula, observe that the order quantity is proportional to the square root of
annual demand (R). Since cycle inventory is half of the order quantity, it too is proportional to R.
Since HP motors has a higher R, the cycle inventory for HP motors is also higher.
b) Average time spent by a motor T = Icycle / R. Since Icycle is proportional to
R , T is proportional
to 1/ R . Therefore time spent by a HP motor is less than the time spent by an LP motor.
Problem 6.10
Each retail outlet faces an annual demand, R = 4000/wk * 50 = 200,000 per year. The unit cost of the
item, C = $200 / unit. The fixed order cost, S = $900. The unit holding cost per year, H = 20 % * 200 =
$40 / unit / year.
a) The optimal order quantity for each outlet
Q
2 RS
2  200000  900

 $3000
H
40
with a cycle inventory of 1500 units. The total cycle inventory across all four outlets equals 6000
units.
b) With centralization of purchasing the fixed order cost, S = $1800. The centralized order quantity
is then,
Q
2RS
2  800000 1800

 $8485
H
40
and a cycle inventory of 4242.5 units.
Chapter 6
Table TN 6.1: Problem 6.7 (with use of rental forklift)
S
R
(h+r)
purchase
price
50 per order
150000 lbs / yr
15.00% % per year
1.5 per lb.
Order Shipping Total
size (Q) cost per variable
pound
6000
6500
7000
7500
8000
8500
9000
9500
10000
10500
11000
11500
12000
12500
13000
13500
14000
14500
15000
15500
16000
16500
17000
17500
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.13
0.13
0.13
0.13
0.13
0.13
holding Number Annual Average
Annual
Annual
Total
cost
of orders order cost Cycle
holding procurement Annual costs
cost =
pound
(R/Q) (SxR/Q) Inventory
cost
cost (CxR)
TC
purchase per year
(Q/2)
(HxQ/2)
cost +
(H)
shipping
(C)
1.67 0.2505
25.00 1250.00
3000
751.5
250500 252,501.50
1.67 0.2505
23.08 1153.85
3250
814.125
250500 252,467.97
1.67 0.2505
21.43 1071.43
3500
876.75
250500 252,448.18
1.67 0.2505
20.00 1000.00
3750
939.375
250500 252,439.38
1.67 0.2505
18.75
937.50
4000
1002
250500 252,439.50
1.67 0.2505
17.65
882.35
4250 1064.625
250500 252,446.98
1.67 0.2505
16.67
833.33
4500
1127.25
250500 252,460.58
1.67 0.2505
15.79
789.47
4750 1189.875
250500 252,479.35
1.65 0.2475
15.00
750.00
5000
1237.5
247500 249,487.50
1.65 0.2475
14.29
714.29
5250 1299.375
247500 249,513.66
1.65 0.2475
13.64
681.82
5500
1361.25
247500 249,543.07
1.65 0.2475
13.04
652.17
5750 1423.125
247500 249,575.30
1.65 0.2475
12.50
625.00
6000
1485
247500 249,610.00
1.65 0.2475
12.00
600.00
6250 1546.875
247500 249,646.88
1.65 0.2475
11.54
576.92
6500
1608.75
247500 249,685.67
1.65 0.2475
11.11
555.56
6750 1670.625
247500 249,726.18
1.65 0.2475
10.71
535.71
7000
1732.5
247500 249,768.21
1.65 0.2475
10.34
517.24
7250 1794.375
247500 249,811.62
1.63 0.2445
10.00
500.00
7500
1833.75
244500 246,833.75
1.63 0.2445
9.68
483.87
7750 1894.875
244500 246,878.75
1.63 0.2445
9.38
468.75
8000
1956
244500 246,924.75
1.63 0.2445
9.09
454.55
8250 2017.125
244500 246,971.67
1.63 0.2445
8.82
441.18
8500
2078.25
244500 247,019.43
1.63 0.2445
8.57
428.57
8750 2139.375
244500 247,067.95
Chapter 6
Table TN 6.2: Problem 6.7 (without use of rental forklift)
S
R
(h+r)
purchase
price
50 per order
150000 lbs / yr
15.00% % per year
1.5 per lb.
Order Shipping Total
size (Q) cost per variable
pound
6000
6500
7000
7500
8000
8500
9000
9500
10000
10500
11000
11500
12000
12500
13000
13500
14000
14500
15000
15500
16000
16500
17000
17500
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.13
0.13
0.13
0.13
0.13
0.13
holding Number Annual Average
Annual
Annual
Total
cost
of orders order cost Cycle
holding procurement Annual costs
cost =
pound
(R/Q) (SxR/Q) Inventory
cost
cost (CxR)
TC
purchase per year
(Q/2)
(HxQ/2)
cost +
(H)
shipping
(C)
1.67 0.2505
25.00 1250.00
3000
751.5
250500 252,501.50
1.67 0.2505
23.08 1153.85
3250
814.125
250500 252,467.97
1.67 0.2505
21.43 1071.43
3500
876.75
250500 252,448.18
1.67 0.2505
20.00 1000.00
3750
939.375
250500 252,439.38
1.67 0.2505
18.75
937.50
4000
1002
250500 252,439.50
1.67 0.2505
17.65
882.35
4250 1064.625
250500 252,446.98
1.67 0.2505
16.67
833.33
4500
1127.25
250500 252,460.58
1.67 0.2505
15.79
789.47
4750 1189.875
250500 252,479.35
1.65 0.2475
15.00
750.00
5000
1237.5
247500 249,487.50
1.65 0.2475
14.29
714.29
5250 1299.375
247500 249,513.66
1.65 0.2475
13.64
681.82
5500
1361.25
247500 249,543.07
1.65 0.2475
13.04
652.17
5750 1423.125
247500 249,575.30
1.65 0.2475
12.50
625.00
6000
1485
247500 249,610.00
1.65 0.2475
12.00
600.00
6250 1546.875
247500 249,646.88
1.65 0.2475
11.54
576.92
6500
1608.75
247500 249,685.67
1.65 0.2475
11.11
555.56
6750 1670.625
247500 249,726.18
1.65 0.2475
10.71
535.71
7000
1732.5
247500 249,768.21
1.65 0.2475
10.34
517.24
7250 1794.375
247500 249,811.62
1.63 0.2445
10.00
500.00
7500
1833.75
244500 246,833.75
1.63 0.2445
9.68
483.87
7750 1894.875
244500 246,878.75
1.63 0.2445
9.38
468.75
8000
1956
244500 246,924.75
1.63 0.2445
9.09
454.55
8250 2017.125
244500 246,971.67
1.63 0.2445
8.82
441.18
8500
2078.25
244500 247,019.43
1.63 0.2445
8.57
428.57
8750 2139.375
244500 247,067.95