Problems

1. An automobile manufacturer uses about 60,000 pairs of bumpers (front bumper and rear bumper) per year, which it orders from a supplier. The bumpers are used at a reasonably steady rate during the 240 working days per year. It costs $3.00 to keep one pair of bumpers in inventory for one month, and it costs $25.00 to place an order. A pair of bumpers costs $150.00.

1. Which inventory model should be used here: the instantaneous model, the noninstantaneous model, or the single-period model? How do you know?
2. Write the annual carrying cost function.
3. Write the annual ordering cost function.
4. Write the annual total cost function.
5. What is the EOQ?
6. What is the significance of the EOQ?
7. What is the total annual expense of ordering the EOQ every time?
8. How many orders will be placed per year?
9. What is the total annual expense of ordering 600 pairs of bumpers every time? How much is saved per year by ordering the EOQ?

2. The Acme Bumper Co. manufactures bumpers for automobiles for one of the big three auto companies. About 60,000 pairs of bumpers (front bumper and rear bumper) are ordered by the auto company per year, at a price of $150.00 per pair. Pairs of bumpers are produced at a rate of about 400 per working day, and the company operates 240 days per year. The company manufactures other products, and it must set up the manufacturing system for a production order for pairs of bumpers, which costs $250.00. It costs $2.50 to store one pair of bumpers for one month.

1. Which inventory model is appropriate here: the instantaneous model, the noninstantaneous model, or the single-period model? How do you know?
2. Write the annual carrying cost function.
3. Write the annual setup cost function.
4. Write the annual total cost function.
5. What is the EOQ?
6. What is the significance of the EOQ?
7. What is the total annual expense of producing the EOQ every time?
8. How many production runs will be required per year?
9. What is the total annual expense of manufacturing 3,000 pairs of bumpers per production run? How much is saved per year by producing the EOQ every time?

3. Use the data from Problem 1 to solve this problem, in addition to the data below. The supplier offers a system of quantity discounts, as follows:

Order Quantity Cost per Pair Discount
1 to 299 pairs $150.00 0%
300 to 499 pairs $150.00 4%
500 or more pairs $150.00 10%

1. Which inventory model is appropriate here: the instantaneous model, the quantity discount model, or the single-period model? How do you know9
2. Write the annual total cost function for orders for less than 300 pairs.
3. Write the annual total cost function for orders for 300 to 499 pairs.
4. Write the total cost function for orders for at least 500 pairs.
5. Draw the graph of the three total cost functions on a piece of rectangular coordinate paper.
6. What is the EOQ?
7. What is the total annual expense, including the purchasing cost of ordering the EOQ every time?

1. Which reorder-point model is appropriate here, fixed lead time and variable demand, variable lead time and fixed demand, or variable demand and variable lead time?
2. What is the mean usage during lead time?
3. What is the standard deviation of the usage during lead time?
4. What is the reorder point? Draw a diagram like the one in Figure 13-11 in your textbook.
5. What is the significance of the reorder point?
6. What is the safety stock?

5. The purchasing agent has ascertained that the employees use about 50 pencils per day, with very little variation. Pencils are ordered from a supplier whose shipments are irregular. They are normally distributed with a mean of 10 working days and a variance of 4. He has established a 95 percent service level for pencils.

1. Which reorder-point model is appropriate here: variable demand and fixed lead tirne, fixed demand and variable lead time, or variable demand and variable lead time?
2. What is the mean usage during lead time?
3. What is the standard deviation of usage during lead time?
4. What is the reorder point? Draw a diagram in the style of Figure 13-13 in your textbook.
5. What is the significance of the reorder point?
6. What is the safety stock?

6. The purchasing agent has ascertained that the usage of letterhead paper has an approximately normal distribution with a mean of 200 sheets per working day and a standard deviation of 13 sheets. Letterhead paper is ordered from a printer whose deliveries are variable, with a mean of 25 working days and a variance of 4 working days. The purchasing agent has established a 99 percent service level for letterhead paper.

1. Which reorder-point model is appropriate here: variable demand and fixed lead time, fixed demand and variable lead time, or variable demand and variable lead time?
2. What is the mean usage during lead time?
3. What is the standard deviation of usage during lead time?
4. What is the reorder point? Draw a diagram in the style of Figure 13-13 in your textbook.
5. What is the significance of the reorder point?
6. What is the safety stock?

7. (One step beyond; based on statistics.) The Whirlwind Appliance Store sells freezers for home use. These are large, upright units which resemble a refrigerator. They differ in operation in that the entire interior is kept at a freezing temperature. The store orders the freezers from the factory, which is very reliable and will deliver the order within one week (7 calendar days) almost invariably. Here are the volumes of sales for the past 50 weeks: 2, 1, 4, 5, 2, 0, 3, 6, 7, 3, 4, 2, 2, 4, 3, 5, 5, 1, 3, 6, 4, 5, 1, 0, 1, 4, 3, 5, 7, 6, 3, 5, 4, 1, 5, 5, 4, 5, 6, 4, 3, 2, 5, 4, 1, 5, 2, 5, 4. Future sales are expected to be similar. The store manager wishes to establish a 95 percent service level.

1. What is the frequency distribution for this data? Does it appear to be approximately normal?
2. What are the mean sales of freezers per week?
3. What is the empirical probability distribution?
4. What is the cumulative probability distribution?
5. What is the reorder point? (Hint: use the cumulative probability distribution.)
6. What is the safety stock?

8. An appliance store knows that the distribution of monthly sales of 50" TV sets is approximately normal with a mean of 6 sets and a standard deviation of 1.5 sets. The sets cost $2,750 each, and are priced at $4,000. The sets are ordered from the manufacturer on the 10th of the month (or the preceding business day), and there is a lead time of seven business days. There are 20 business days in a typical month. The manager wishes to maintain a 90 percent service level. The store is open six days a week and Sunday afternoons. On March 31, the store had five 50" TV sets on hand. Here is data regarding sales for the next three months:

Month Sales
April 4 sets
May 8 sets
June 7 sets

1. Which inventory model is appropriate here: the fixed quantity and variable order interval model, the variable quantity and fixed order interval model, or the single period model?
2. How many sets should be ordered on April 10?
3. What will be the ending inventory on April 30?
4. How many sets should be ordered on May 10?
5. What will be the ending inventory on May 31?
6. How many sets should be ordered on June 10?

9. Use Table 13-3 in your textbook to answer these questions.

1. What is the expected number of units short per cycle in Problem 4?
2. What is the implication of the expected number of units short in Problem 4?
3. What is the expected number of units short in Problem 5?
4. What is the significance of the expected number of units short in Problem 5?

10. Here is a list of 15 items, containing the annual volume of sales and the unit price, An A-B-C inventory system is to be established on the basis of the annual dollar value of each item. Classification A will contain about 60% of the dollar value; B will contain about 30%; and C will contain about 10%. Determine the items which belong in each category.

1. Determine the annual dollar value for each item and the total annual dollar value.
2. Which items should be assigned to the A category? To the B category? To the C category?

11. A delicatessen sells potato salad from trays in the cooler cabinet; the salesperson packs the salad into cardboard containers according to the customers' orders. The supervisor prepares one batch of salad in the morning and disposes of any leftover salad at closing time. Potato salad sells for $1.49 per pound, and costs $0.80 to prepare. Sales of potato salad are believed to be normally distributed with a mean of 60 pounds per day and a variance of 256.

1. Which inventory model is appropriate here: the single-period model, the instantaneous model, or the quantity discount model? How do you know?
2. What is the shortage cost, <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image95::/sites/dl/free/0072443901/24520/Image95.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image95 (0.0K)</a>Image95 ?
3. What is the excess cost, <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image96::/sites/dl/free/0072443901/24520/Image96.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image96 (0.0K)</a>Image96 ?
4. How large a batch should the supervisor mix up in the morning?

12. (One step beyond) Kerey White operates a booth which sells specialty merchandise at the State Fair. This year she is planning to sell t-shirts which are inscribed:

FAMILY FUN
THE STATE FAIR
FOR ALL AGES

The price of the t-shirts will be $25.00.each. Ms. White buys plain white t-shirts from a factory at a cost of $864 per gross (144 shirts), with a 3% discount for payment within 10 days, which she always takes. Ms. White inscribes the design on the t-shirts, herself, and estimates that her labor is worth $4.00 per shirt. Unsold T-shirts are useless, but can be sold to a scrap dealer for $0.16 per pound; there are eight t-shirts in a pound. The number of t-shirts which will be sold at the State Fair is, of course, unknown. But, based on previous experience, Ms. White estimates that the mean sales are 3 gross, and that the probability of selling more than 3 gross decreases as the quantity increases.

1. Which inventory model should be used here-the instantaneous model, the quantity discount model, or the single period model?
2. Which probability distribution should be used here?
3. What is the shortage cost, C_s?
4. What is the excess cost, C_e?
5. What is the service level?
6. How many t-shirts should Ms. White prepare (in gross)?
7. Estimate Ms. White's profits for your answer in f.

13. The owner of Smith's Speedy Service Station calculates that he sells about 15,000 gallons of regular unleaded gas per year. He sells gas from self-service pumps and wishes to maintain a service level of 99% per year. Although the price of gasoline fluctuates with the seasons, he estimates that it averages about $1.00 per gallon. It costs $.25 to store one gallon for one year. It costs $25.00 to place and receive one order. The lead time for an order is 1 day and the standard deviation of sales is about 5 gallons per day.

1. What average number of gallons short will be consistent with the specified service level?
2. What is the EOQ?
3. What is the average number of gallons short per order cycle?
4. What lead time service level is necessary for the 99% service level?

Solutions

1. a. Use the instantaneous model.
b. The annual carrying cost = HQ/2 = 3(12)Q/2 = 18Q.
c. The annual ordering cost = DS/Q = 60000(25)/Q = 1500000/Q.
d. The annual total cost = 18Q + 1500000/Q.
e. <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image97::/sites/dl/free/0072443901/24520/Image97.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image97 (1.0K)</a>Image97<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image98::/sites/dl/free/0072443901/24520/Image98.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image98 (1.0K)</a>Image98 » 289 pairs of bumpers.
f. Every time the company places an order, it should order 289 pairs of bumpers.
g. The annual total expense = (Q/2)H + (D/Q)S = (289/2)36 + (60,000/289)25 = $10,392.00 per year.
h. The number of orders = D/Q = 60000/289 » 208 per year.
i. The annual total expense = 18(600) + 1500000/600 = $13,300. The annual savings are $13,300 - $10,392 = $2,908.00.

2. a. Use the noninstantaneous model.
b. The annual carrying cost = (HQ/2)[(P-U)/P] = (2.50(12)Q/2)[(400 - 60000/240)]/400 = 5.625Q.
c. The annual setup cost = DS/Q = 60000(250)/Q = 15000000/Q.
d. The annual total cost = 5.625Q + 15000000/Q.
e. EOQ = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image99::/sites/dl/free/0072443901/24520/Image99.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image99 (1.0K)</a>Image99<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image100::/sites/dl/free/0072443901/24520/Image100.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image100 (1.0K)</a>Image100 = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image101::/sites/dl/free/0072443901/24520/Image101.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image101 (1.0K)</a>Image101 x = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image102::/sites/dl/free/0072443901/24520/Image102.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image102 (1.0K)</a>Image102 1,633 pairs of bumpers.
f. The company should place a production order for 1,633 pairs of bumpers every time.
g. The annual total cost = 5.625EOQ + 15000000/EOQ = 5.625(1633) + 15000000/1633 » $18,371.00 per year.
h. The number of production runs = D/EOQ = 60000/1633 = 36.74 runs per year.
i. The annual total cost = 5.625(3000) + 15000000/3000 = $21,875.00. The annual savings are $21,875.00 - $18,371.00 = $3,504.00.

3. a. Use the quantity discount model.
b. The annual total cost = 18Q + 1500000/Q + 150D.
c. The net price = 150(1 - d) = 150(1 - .04) = $144.00. The annual total cost = 18Q + 1500000/Q + 144D.
d. The net price = 150(1 - d) = 150(1 - .10) = $135.00. The annual total cost = 18Q + 1500000/Q + 135D.
e.
f. The EOQ is the minimum feasible point, which is 500 pairs of bumpers.
g. The annual total cost = 18(500) + 1500000/500 + 135(500) = $79,500.00

4. a. Use the fixed lead time and variable demand model.
b. The mean usage during lead time = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image103::/sites/dl/free/0072443901/24520/Image103.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image103 (0.0K)</a>Image103 (LT) = 30(5) = 150 floppy disks.
c. The standard deviation = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image104::/sites/dl/free/0072443901/24520/Image104.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image104 (1.0K)</a>Image104 floppy disks.
d. ROP = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image103::/sites/dl/free/0072443901/24520/Image103.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image103 (0.0K)</a>Image103 (LT) + z(standard deviations) = 150 + 2.58(8.94) = 173.08 floppy disks

e. When the quantity of floppy disks on hand falls to 173.08 disks (or fewer) place an order for the economic order quantity.
f. The safety stock = ROP - mean usage = 173.08 - 150 = 23.08 floppy disks.

5. a. Use the fixed demand and variable lead time model.
b. The mean usage during lead time = d <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image105::/sites/dl/free/0072443901/24520/Image105.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image105 (0.0K)</a>Image105 = 50(10) = 500 pencils.
c. The standard deviation = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image106::/sites/dl/free/0072443901/24520/Image106.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image106 (1.0K)</a>Image106 pencils.
d. ROP = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image103::/sites/dl/free/0072443901/24520/Image103.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image103 (0.0K)</a>Image103 (LT) + z(standard deviations) = 500 + 1.65(100) = 665 pencils.
e. When the quantity of pencils on hand falls to 665 (or fewer), place an order for the economic order quantity
f. The safety stock = ROP - the mean usage = 665 - 500 = 165 pencils.

6. a. Use the variable demand and variable lead time model.
b. The mean usage during lead time = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image103::/sites/dl/free/0072443901/24520/Image103.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image103 (0.0K)</a>Image103 (LT) = 200(25) = 5,000 sheets.
c. The standard deviation = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image107::/sites/dl/free/0072443901/24520/Image107.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image107 (1.0K)</a>Image107 = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image108::/sites/dl/free/0072443901/24520/Image108.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image108 (1.0K)</a>Image108 = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image109::/sites/dl/free/0072443901/24520/Image109.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image109 (1.0K)</a>Image109 = 405.25 Sheets.
d. ROP = The mean usage + z(standard deviations) = 5000 + 2.33(405.25) = 5,944.23 sheets.
e. When the quantity of letterhead paper on hand falls to 5,944.23 sheets (or fewer) place an order for the Economic Order Quantity.
f. The safety stock = ROP - the mean usage = 5,944.23 - 5,000 = 944.23 sheets.

7. a,b,c

The frequency distribution is highly skew negative and is not approximately normal.
d. <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image110::/sites/dl/free/0072443901/24520/Image110.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image110 (1.0K)</a>Image110 freezers per week.
e. The first cumulative probability which exceeds .95 is .96, and the reorder point is 6 freezers.
f. The safety stock = 6 - 3.6 = 2.4 freezers.

8. a. Use the fixed order interval and variable <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image111::/sites/dl/free/0072443901/24520/Image111.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image111 (1.0K)</a>Image111 quantity model.
b. The expected demand = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image103::/sites/dl/free/0072443901/24520/Image103.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image103 (0.0K)</a>Image103 (OI + LT) = 6 sets.
The safety stock = sets.
The order quantity = the expected demand + the safety stock - the quantity on hand = 6 + 1.92 - 5 » 3 sets.
c. The ending inventory on April 30 will be 5 + 3 - 4 = 4 sets.
d. The order quantity = the expected demand + the safety stock - the quantity on hand= 6 + 1.92 - 4 » 4 sets.
e. The ending inventory on May 31 will be 4 + 4 - 8 = 0 sets.
f. The order quantity = the expected demand + the safety stock - the quantity on hand= 6 + 1.92 - 0 » 8 sets.

9. a. E(z) = E(2.58) = .002
E(n) = E(z)(standard deviations) = .002(8.94) = .02 floppy disks.
b. The mean number which will be short per order cycle is .02 floppy disks. In other words, no shortage should be experienced in most order
cycles, but very rarely there may be a shortage of one disk.
c. E(z) = E(1.65) » .020.
E(n) = E(z)(standard deviations) = .020(100) = 2 pencils.
d. The mean number which will be short per order cycle is 2 pencils. In other words, sometimes there will be a shortage of around 2 pencils (e.g., 1, 2, 3, or 4, but probably not more, pencils).

10. a.

* (800 x $26.00)
b. Rearrange the items in the order of decreasing annual dollar value. Calculate the cumulative totals and the cumulative percents. Here are the first few items:

*34.35% = 100(127,800)/372,050. Classification A will contain items o, c, m, and g. B will contain items d, n, a, l, e, and h. C will contain items f, i, j, b, and k.

11. a. Use the single period model.
b. C_s = Revenue per unit - cost per unit = $1.49 -$0.80 = $0.69 per pound.
c. C_e = Cost per unit = salvage value per unit = $0.80 - 0 = $0.80.
d. The service level = C_s/(C_s + C_e) = .69/(.69 + .80) = .46.
e. The value of z for the service level = -0.04. The batch size = the mean demand + z(standard deviations) = 60 - 0.04(16) = 59.36 pounds of potato salad.

12. a. The single-period model, since the t-shirts are salable only at one event.
b. The Poisson distribution displays decreasing probabilities as the value of the random variable increases.
c. <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image112::/sites/dl/free/0072443901/24520/Image112.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image112 (0.0K)</a>Image112 = revenue per unit - cost per unit = 25(144) - 864(.97) = 3600 - 838.08 = $2761.92.
d. <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image113::/sites/dl/free/0072443901/24520/Image113.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image113 (0.0K)</a>Image113 = cost per unit - salvage value per unit = 864(.97) - .02(144) = 838.08 - 2.88 = $835.20.
e. The service level = <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image114::/sites/dl/free/0072443901/24520/Image114.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image114 (1.0K)</a>Image114 .
f. Use the cumulative Poisson distribution with m = 3, in Table C to obtain these values:

Because P(x) = .815 is the lowest probability to exceed .7678, Ms. White should prepare 4 gross of t-shirts.
g. Profits = revenues - expenses = 4(3600) - 4(838.08) = 4(2761.92) = $11,047.68. It is improper to count Ms. White's labor as an expense.

13. a. <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image115::/sites/dl/free/0072443901/24520/Image115.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image115 (1.0K)</a>Image115 gal.
b. <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image116::/sites/dl/free/0072443901/24520/Image116.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image116 (1.0K)</a>Image116 gal.
c. <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image117::/sites/dl/free/0072443901/24520/Image117.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image117 (1.0K)</a>Image117 gal. per cycle.
d. Find E(z).
<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image118::/sites/dl/free/0072443901/24520/Image118.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image118 (1.0K)</a>Image118 (Table 13-4 in your textbook)
<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image119::/sites/dl/free/0072443901/24520/Image119.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image119 (1.0K)</a>Image119
Refer to Table 13-3 in your textbook. For E(z) = .35, z <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image120::/sites/dl/free/0072443901/24520/Image120.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image120 (0.0K)</a>Image120 .08 and the service level <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image120::/sites/dl/free/0072443901/24520/Image120.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image120 (0.0K)</a>Image120 .53.