1.
A newsstand receives its weekly order of “Weekly News” magazine on Monday & cannot reorder. Each copy costs Rs 15 and sells for Rs 20. Unsold copies may be returned the following week for Rs 4. When the news stand runs out of copies and cannot supply a customer wanting, it estimates its “good-will” loss at Rs 6 in future profits, figuring that the customer will take his business elsewhere for a couple of weeks, on the average. Demand distribution is given below: --------------------------------------------------------------------------------Demand (No. of Copies) 10 15 20 25 --------------------------------------------------------------------------------Probability .28 .35 .17 .20 --------------------------------------------------------------------------------(a) Construct a payoff table and use it to determine the optimal number of copies to stock and the expected profit. (b) Computed Expected Value of Perfect Information (EVPI). Enter Details Marginal Profit Marginal Loss Opportunity Loss
5 11
Sr no
6
1 2 3 4
Probabilit y
Produce Deman d
0.28 0.35 0.17 0.2
10 15 20 25
1
Produce
Total Expected Profit
2.
1 10
2 15
3 20
4 25
50 20 -10 -40
-5 75 45 15
-60 20 100 70
-115 -35 45 125
35.5
50 75 100 125 Max Profit
15 11.3
Max(Profit)
21.2
-11.8
35.5
EPPI 82.2 5
EVPI 46.7 5
Michael, Nancy, & Associates (MNA) produce color printers. The demand for their printers could be light, medium, or high with the following probabilities.
Probability
Light Demand 0.4
Medium Demand 0.3
High Demand 0.3
The company has three production alternatives for the coming period. The payoffs (in millions of dollars) associated with the three alternatives are shown below.
Alternative 1 Alternative 2 Alternative 3
Light Demand 18 25 3
Medium Demand 28 17 40
High Demand 20 -5 16
a. Compute the expected value of the three alternatives. Which alternative would you select, based on the expected values? b. Find the optimal strategy using EOL criteria
Prob 0.4 Light Demand
Payoff Table Alternative Alternative Alternative 1 2 3 18 25 3
Medium 0.3 demand 28 17 40 0.3 Heavy Demand 20 -5 16 E(X) 21.6 13.6 18 Select Alternative 1 based on Expected Value Opportunity Loss Table Alternative Alternative Alternative Prob 1 2 3 0.4 Light Demand 7 0 22 Medium 0.3 demand 12 23 0 0.3 Heavy Demand 0 25 4 E(Loss) 6.4 14.4 10 Select Alternative 1 based on Expected Opportunity Loss
3.
Technico Ltd has installed a machine costing Rs 4 Lacs and is in the process of deciding on appropriate number of spare parts required for repairs. The spare parts cost Rs 4000 each but are available only if they are ordered now. In case the machine fails and no spares are available, the cost to the company of mending the part is Rs 18000. The plant has estimated life of 8 years and the probability distribution of failures during this time, based on experience of similar machines is as follows. No of 0 1 2 3 4 5 6+ failures Probabilit .1 .2 .3 .2 .1 .1 0 y Ignoring any discounting for time value for money, determine following: a) The cost table and the optimal choice on the basis of expected cost principle. b) The regret table and the optimal choice on the basis of expected regret criterion c) Find EVPI Enter Details Marginal Profit Marginal Loss Opportunity Loss
Sr no
4000 4000 1800 0
Probabilit y
Produce Deman d
1
0.1
0
2
0.2
1
3
0.3
2
4
0.2
3
5 6
0.1 0.1
4 5
0
0 1800 0 3600 0 5400 0 7200 0 9000
1
2
3
-4000
-8000
12000
-4000 2200 0 4000 0 5800 0 7600
-8000
12000
-8000 2600 0 4400 0 6200
12000 12000 30000 48000
4 1600 0 1600 0 1600 0 1600 0 1600 0 3400
5
Max(Profit)
-20000
0
-20000
-4000
-20000
-8000
-20000
-12000
-20000 -20000
-16000 -20000
0 1
Produc e Deman d
0.1
0
0.2
1
0.3
2
0.2
3
0.1
4
0.1
5 Produc 1 e Total Expected Profit
4.
0 Max Profit
4140 0
2920 0
2060 0
1780 0
17400
-20000
0
1
2
3
4
5
0 1400 0 2800 0 4200 0 5600 0 7000 0
4000
8000
1200 0
0
4000
8000
1600 0 1200 0
14000
0
4000
8000
2000 0 1600 0 1200 0
28000
14000
4000
8000
42000
28000 42000
0 1400 0
4000
56000
0 1400 0 2800 0
3220 0
20000
11400
8200
8600
-9200
8200
2000 0 1600 0 2800 0 4200 0 5600 0 7000 0 Min Loss
1080 0
-17400
EVPI
Max(Profit)
0
3
EPPI
8200
EPPI 3460 0
EVPI 2640 0
As a fund-raiser for a student organization, some students have decided to sell individual pizzas out side the Union on Friday. Each pizza will sell for $1.75 and costs the organization $0.75. Historical sales indicated that between 55 and 60 dozen pizzas be sold with the probability distribution given below: Dozen of pizzas 55 56 57 58 59 60 Probability 0.15 0.20 0.10 0.35 0.15 0.05 To maximize the profit contribution, how many pizzas should be ordered? Assume pizzas must be ordered by dozen. What is the expected value of perfect information in this problem? What is the maximum amount the organization would be willing to pay for perfect information?
probabilit y
0.15 0.2 0.1 0.35 0.15 0.05 1
5.
0
Produce
Total Expected Profit
Probabilit y
0
stock-> deman d 55 56 57 58 59 60
55
56
57
58
59
60
max
660 660 660 660 660 660
651 672 672 672 672 672
642 663 684 684 684 684
633 654 675 696 696 696
624 645 666 687 708 708
615 636 657 678 699 718
660 672 684 696 708 718
668.8 5
673. 5
676.0 5
671.2 5
663. 2
687. 5
11.45
660 EVPI
Center City motors Sales has recently incorporated. Its chief asset is franchise to sell automobiles of a major American manufacturer. CCMS’s general manager is planning the staffing of the leadership’s garage facilities. From information provided by the manufacturer and from the nearby dealerships, he has estimated the number of annual mechanic hours that the garage will be likely to need. Hours 10000 12000 14000 16000
Probability 0.2 0.3 0.4 0.1 The manager plans to pay each mechanics $9.00 per hour and to charge customer $16.00. The mechanics will work for 40 hour week and get annual 2 week vacation. Determine how many mechanics Center City should hire? How much should Center City pay to get perfect information about the number of mechanics it needs? probabili ty
stock> deman d
0.2
5
0.3
6
0.4
7
0.1
8
1 11712
6.
6
7
6640 0 6640 0 6640 0 6640 0
4768 0 7968 0 7968 0 7968 0
2896 0 6096 0 9296 0 9296 0
6640 0
7328 0
7056 0
8
Max
10240
66400
42240
79680
74240 10624 0
92960 10624 0
55040
84992
EVPI
Emily Scott, head of a small business consulting firm, must decide how many M.B.A.s to hire as fulltime consultants for the next year. (Emily has decided that she will not bother with any part-time employees.) Emily knows from experience that the probability distribution on the number of consulting jobs her firm will get each year is as follows: Consulting jobs 24 27 30 33 Probability 0.3 0.2 0.4 0.1 Emily also knows that each M.B.A. hired will be able to handle exactly three consulting jobs per year. The salary of each M.B.A. is $60,000. Each consulting job is worth $30,000 to Emily’s firm. Each consulting job that the firm is awarded but cannot complete costs the firm $10,000 in future business lost. (a) How many M.B.A.s should Emily hire? (b) What is the expected value of perfect information to Emily? probabilit y
Ans 9or10 EVPI 54
0.3 0.2 0.4 0.1 1
7.
5
stock-> deman d 24 27 30 33
8
9
10
11
Max
(in thousands) 240 210 180 150
180 270 240 210
120 210 300 270
60 150 240 330
240 270 300 330
201
225
225
177
279
The Writer’s Workbench operates a chain of word-processing franchises in college towns. For an hourly fee of $8.00, Writer’s Workbench provides access to a personal computer, word-processing software, and a printer to students who need to prepare papers for their classes. Paper is provided at no additional cost. The firm estimates that its hourly variable cost per machine (principally due to paper, ribbons, electricity, and wear and tear on the computers and printers) is about 85¢. Deborah Rubin is considering opening a Writer’s Workbench franchise in Ames, Iowa. A preliminary market survey has resulted in the following probability distribution of the number of machines demanded per hour during the hours she plans to operate: Number of machines 22 23 24 25 26 27
Probability 0.12 0.16 0.22 0.27 0.18 0.05 If she wishes to maximize her profit contribution, how many machines should Deborah plan to have? What is the hourly expected value of perfect information in this situation? Even if Deborah could obtain a perfectly accurate forecast of the demand for each and every hour, why wouldn’t she be willing to pay up to the EVPI for that information in this situation?
Ans 26 EV PI 1.8
probabili ty
stock> deman d
0.12
22
157.3
0.16
23
157.3
0.22
24
157.3
0.27
25
157.3
0.18
26
157.3
0.05 1
27
22
23
24
25
157.3
156.4 5 164.4 5 164.4 5 164.4 5 164.4 5 164.4 5
155. 6 163. 6 171. 6 171. 6 171. 6 171. 6
154.7 5 162.7 5 170.7 5 178.7 5 178.7 5 178.7 5
157.3
163.4 9
168. 4
171.5 5
26
27
185.9
153.0 5 161.0 5 169.0 5 177.0 5 185.0 5 193.0 5
172.5 4
172.0 9
153.9 161.9 169.9 177.9 185.9
max
157.3 164.4 5 171.6 178.7 5 185.9 193.0 5 174.3 2
