884 C H A P T E R 1 6 Probabilistic Inventory Models

From (3), Walton should order q* calendars, where q* is the smallest number for which

P(D # q*) $ }23

}. As a function of q, P(D # q) increases only when q 5 100, 150, 200,

250, or 300. Also note that P(D # 100) 5 .30, P(D # 150) 5 .50, and P(D # 200) 5

.80. Since P(D # 200) is greater than or equal to }23

}, q* 5 200 calendars should be ordered.

R E M A R K S 1 In terms of marginal analysis, the probability of selling the 200th calendar that is ordered is P(D $ 200) 5 .50. This implies that the 200th calendar has a 1 2 .50 5 .50 chance of being un-sold. Thus, the 200th calendar will increase Walton’s expected costs by .50(22.50) 1 .50(1.25) 52$0.625. Hence, the 200th calendar should be ordered. On the other hand, the probability that the201st calendar will be sold is P(D $ 201) 5 .20, and the probability that the 201st calendar willnot be sold is 1 2 .20 5 .80. Therefore, the 201st calendar will increase expected costs by.20(22.50) 1 .80(1.25) 5 $0.50. Thus, the 201st calendar will increase expected costs and shouldnot be ordered.2 In Example 1, co and cu could easily have been determined without recourse to (2) and (2.1).For example, being one more unit over actual demand increases Walton’s costs by 2 2 0.75 5 $1.25.Thus, co 5 $1.25. Similarly, being one more unit under actual demand will cost Walton 4.50 2 2.00 5$2.50 in profit. Hence, cu 5 $2.50. If we are able to determine co and cu without using Equations(2) and (2.1), we should do so. In more difficult problems, however, they can be very useful (seeExamples 2 and 3).

P R O B L E M SGroup A

TA B L E 3

Computation of Total Cost If d $ q 1 1

Cost

Buy q calendars at $2/calendar 2q

Sell d calendars at $4.50/calendar 24.50q

Total cost 22.50q

1 In August 2003, a car dealer is trying to determine howmany 2004 models should be ordered. Each car costs thedealer $10,000. The demand for the dealer’s 2004 modelshas the probability distribution shown in Table 4. Each caris sold for $15,000. If the demand for 2004 cars exceeds thenumber of cars ordered in August, the dealer must reorderat a cost of $12,000 per car. If the demand for 2004 carsfalls short, the dealer may dispose of excess cars in an end-of-model-year sale for $9,000 per car. How many 2004models should be ordered in August?

2 Each day, a news vendor must determine how manyNew York Herald Wonderfuls to order. She pays 15¢ for eachpaper and sells each for 30¢. Any leftover papers are a totalloss. From past experience, she believes that the number ofpapers she can sell each day is governed by the probabilitydistribution shown in Table 5. How many papers should sheorder each day?

3 If cu is fixed, will an increase in co increase or decreasethe optimal order quantity?

TA B L E 4

No. of CarsDemanded Probability

20 .30

25 .15

30 .15

35 .20

40 .20

TA B L E 5

No. of PapersDemanded Probability

50 .30

70 .15

90 .25

110 .10

130 .20

1 6 . 3 The News Vendor Problem: Discrete Demand 885

4 If co is fixed, will an increase in cu increase or decreasethe optimal order quantity?

5 The power at Ice Station Lion is supplied via solar cells.Once a year, a plane flies in and sells solar cells to the icestation at a price of $20 per cell. Because of uncertaintyabout future power needs, the ice station can only guess thenumber of cells that will be required during the coming year(see probability distribution in Table 6). If the ice stationruns out of solar cells, a special order must be placed at acost of $30 per cell.

a Assuming that the news vendor problem is relevant,how many cells should be ordered from the plane?

b In part (a), what type of cost is being ignored?

6 The daily demand for substitute teachers in the LosAngeles teaching system follows the distribution given inTable 7. Los Angeles wants to know how many teachers tokeep in the substitute teacher pool. Whether or not thesubstitute teacher is needed, it costs $30 per day to keep asubstitute teacher in the pool. If not enough substitute teachersare available on a given day, regular teachers are used to coverclasses at a cost of $54 per regular teacher. How many teachersshould Los Angeles have in the substitute teacher pool?†

Group B

7 Every four years, Blockbuster Publishers revises itstextbooks. It has been three years since the best-sellingbook, The Joy of OR, has been revised. At present, 2,000

copies of the book are in stock, and Blockbuster mustdetermine how many copies of the book should be printedfor the next year. The sales department believes that salesduring the next year are governed by the distribution inTable 8. Each copy of Joy sold during the next year bringsthe publisher $35 in revenues. Any copies left at the end ofthe next year cannot be sold at full price but can be sold for$5 to Bonds Ennoble and Gitano’s bookstores. The cost ofa printing of the book is $50,000 plus $15 per book printed.How many copies of Joy should be printed? Would theanswer change if 4,000 copies were currently in stock?

8 Vivian and Wayne are planning on going to Lamazenatural childbirth classes. Lamaze classes meet once a weekfor five weeks. Each class gives 20% of the knowledge neededfor “natural” childbirth. If Vivian and Wayne finish theirclasses before the birth of their child, they will forget duringeach week 5% of what they have learned in class. To maximizetheir expected knowledge at the time of childbirth, duringwhich week of pregnancy should they begin classes? Assumethat the number of weeks from conception to childbirthfollows the probability distribution given in Table 9.

9‡ Some universities allow an employee to put an amount

q into an account at the beginning of each year, to be usedfor child-care expenses. The amount q is not subject tofederal income tax. Assume that all other income is taxedby the federal government at a 40% rate. If child-careexpenses for the year (call them d ) are less than q, theemployee in effect loses q 2 d dollars in before-tax income.If child-care expenses exceed q, the employee must pay theexcess out of his or her own pocket but may credit 25% ofthat as a savings on his or her state income tax.

Suppose Professor Muffy Rabbit believes that there is anequal chance that her child-care expenses for the comingyear will be $3,000, $4,000, $5,000, $6,000, or $7,000. Atthe beginning of the year, how much money should sheplace in the child-care account?

TA B L E 6

No. of Cells Probability

50 .20

60 .15

70 .30

80 .10

90 .15

100 .10

TA B L E 7

Number Needed Probability

200 .03

275 .03

350 .03

400 .05

450 .40

500 .30

550 .06

600 .07

650 .03

†Based on Bruno (1970).

TA B L E 8

Copies Demanded Probability

5,000 .30

6,000 .20

7,000 .40

8,000 .10

TA B L E 9

Week of Birth Probability

36 .05

37 .15

39 .20

40 .30

41 .15

42 .10

43 .05

‡Based on Rosenfeld (1986).