Problemsheet #1
IENG431, Fall, 2011
1. The companies First United Joiners (F) and Best Furniture (B) are in the same industry. The capacity
of company F is four times as much as the capacity of B. Their yearly demands of raw material are
400,000 and 100,000 tons, respectively. Company F buys 80,000 tons five times in a year. Her order
cost is $55,000. Company B buys 25,000 tons four times in a year. The order cost is $11,000. Both
companies apply the policy that the newly purchased raw material arrives when the stock is
exhausted. Assume that the price of the raw material and the interest rate used to determine the
inventory holding cost, are the same. Compare the inventory holding unit cost of the two companies.
The First United Joiners Company buys 80,000 tons and the next transport arrives when the stock
decreased to zero. Thus the average inventory level is 40,000 tons. Let p and i be the unit price of the raw
material and the interest rate, respectively. Then the yearly inventory holding cost is 40000pi. The
company must pay the order cost five times what is $275,000. Thus the total cost of buying and storing of
the raw material is 275000+40000pi. Hence the unit cost of 1 ton is
275000  40000 pi
 0.6875  0.1 pi.
400000
In a similar way the Best Furniture Company has an average inventory level of 12,500 tons and her total
order cost in a year is $44,000. Thus the unit cost is
44000  1250 pi
 0.44  0.125 pi.
100000
Company F has a better policy if and only if
0.6875  0.1 pi  0.44  0.125 pi  0.2475  0.025 pi  9.9  pi.
That means that F has a better policy if the unit price of the raw material multiplied by the interest rate is
greater than 9 dollars and 90 cents. It is because the relative inventory level of F is smaller than the
relative inventory level of B. These two values are 0.1 and 0.125, respectively. The smaller value is the
advantage of the higher number of purchases.
2. You are the inventory manager of your company. Compare two offers for an integrated circuit. Your
company needs 12,000 pieces a year. In both offers the price of the IC is $60. In offer 1 there is no
order cost, but the minimal order quantity is 1000. In offer 2 the order cost is $100. On the other
hand there is now lower bound on the purchased quantity. Which offer is better at interest rates 10
and 20 percent, respectively? Which interest rate is the breakeven point?
As there is no order cost in the case of offer 1, thus it is worth to order as many times as possible
and keep the inventory level low. As the minimal order quantity is 1000, the best option is to order 12
times a year 1000 pieces. Thus the average inventory level is then 500. Let i be the interest rate. The
inventory holding cost in one year is 500×60×i=30,000i. In the case of the second offer the Harris formula
can be applied. The inventory holding unit cost is h=60i. Hence
EOQ 
2  100  12000
1
 200 .
60i
i
The average inventory level is the half of the EOQ. Thus the yearly inventory holding cost is
EOQ
1
h  100 60i  6000 i .
2
i
The average yearly number of purchases is the yearly demand divided by the EOQ. Hence the average
yearly order cost is
100
12000
 6000 i .
EOQ
(See P218 for further details why the yearly inventory holding and yearly order costs are equal to each
other.) Offer 1 is better than offer 2 if and only if
30000i  12000 i  i 
2
4
i
 0.16.
5
25
Thus the breakeven point is 16 percent. At 10 percent the first offer is better and at 20 percent the second
one.
3. The Clean Water Company fills 5 litre plastic bottles with drinking water. The company buys
desalinated water and cleans it further on. The price of the raw material is ¢10 per bottle. The bottles
are blown up from so-called preforms. It price is ¢20. The process cost is ¢12 per bottle. The
production line is served by two workers. They work in 8-hour shifts. The work force cost is $12 per
hour per worker. It is possible to take overtime. During the first 2 hours of overtime salary cost is $24
per hour per worker. In the third and fourth hours of the overtime the work force cost is $48 per hour
per worker. (The workers earn better during overtime, of course. Higher salary implies higher taxes.)
It is not possible to take more than 4 hours overtime a day. The production capacity of the filling line
in an 8-hour shift is 2400 bottles. Assume that the production quantity on a day is q. Give the variable
cost and the variable unit cost as the functions of q.
Notice that the two workers must be paid for a whole shift. Thus the work force cost is at least
2×8×12=192 dollars. The quantity q is expressed as the number of filled bottles. The total process cost
(raw material, process and preform) of a bottle is ¢42. Hence if the production quantity is not greater than
2400 then the production cost is 192+0.42q dollars. The capacity of the filling line is 2400/8=300 bottles
per hour. Thus at most 600 bottles can be produced in the first two hours of the overtime. If only this part
of the overtime is taken then 2400 < q ≤ 3000. The required overtime is
q  2400
.
300
If this quantity is multiplied by 48 then the work force cost in the overtime is obtained in dollars. The
process cost is still ¢42 per bottle for the whole quantity. Thus the total cost in the region 2400 < q ≤ 3000
is
192  0.42q 
q  2400
48.
300
The last case is when 3000 < q ≤ 3600. Until the end of the tenth working hour the workforce cost is
$288. The process cost is unchanged. The additional part of the work force cost can be determined
similarly to the previous case. Thus the total cost is:
288  0.42q 
q  3000
96.
300
The variable unit cost is obtained if the variable cost is divided by the production quantity:
192
if 0 < q ≤ 2400
 0.42
q
192
48 2400  48
192
if 2400 < q ≤ 3000
 0.42 

 0.58 
q
300
300q
q
288
96 960
672
if 3000 < q ≤ 3600.
 0.42 

 0.74 
q
300
q
q