Applied Operations Research
6500:662
Lecture Notes
(required)
Dr. Jayprakash Patankar
Department of Management
College of Business Administration
Fall 1998
Section 1 Decision Theory
1. Set up a pay off matrix
1 2 j
1
2
i xij
Xij is payoff if alternative i is chosen and event j occurs.
2. Types of environments
Certainty
Uncertainty
Risk
Conflict
3. Select a criterion:
Maximax
Maximin
Equally Likely (Laplace)
Criterion of Realism (Hurwicz)
Minimax
EMV
# 1.
A revolutionary new process has been developed for the manufacture of golf balls. These golf balls are packaged in sets of three, and the manufacturer will sell a package of golf balls to a sporting goods wholesaler for $1.50. However, in order to become a distributor of these revolutionary new golf balls, the wholesaler must agree to one of two promotional actions:
1. Sell the package of golf balls for $2.50, agreeing to refund the entire purchase price if one or more of the golf balls is found to be defective.
2. Sell the package of golf balls for $2.00 agreeing to refund $0.50 for each defective golf ball.
Determine the pay-off table for the wholesaler.
# 2.
A student visiting another university for a football game discovers it is their homecoming weekend and that many coeds are wearing large mums to honor the event. The next year he considers selling mums outside the stadium the day of the homecoming game at his own school. A little marketing research convinces him that one percent of those attending buy a mum. At past games, attendance has been 10,000, 15,000, or 20,000 people. He can buy mums for one dollar fifty cents each and will se them for two dollars.
# 3.
You are president of an American corporation which manufacturer aircraft There has been some talk in Washington about (1) going to Mars, going to Mars, and (3) going to Mars with the Russians. In your judgment, the probability of these states is 0.5, 0.4, and 0.1, respectively. You must make plans in the light of these possible states. Your alternatives are (a) continue to make airplanes only, which would provide you with an expected $10 million profit, regardless of the Mars decision (b) to design a payload system for the Mars shot (if the program falls through, you would lose $30 million; if it does not fall through, you will make a profit of $40 million, regardless of Russian participation or not); or (c) to design a payload booster system. If the program falls through, you will lose $100 million. If it goes with Russian participation, it is likely that we will use Russia's booster system, and your profits will be $20 million. If we go alone, your profits will be $80 million.
#1 Payoff Table
0 1 2 3
Plan I 1 -1.5 -1.5 -1.5
Plan II .5 0 -.5 -1
Do Nothing 0 0 0 0
Using Maximax criterion, he will select Plan I ($1)
Using Maximin criterion, he will select Do nothing (0)
Criterion of Realism
Let a be coeff. of realism.
Exp. payoff = a (max payoff) + (1-a)(min payoff)
For a = .6
Exp. payoff (I)= .6(1)+.4(-1.5) = 0*
Exp. payoff (II)= .6(.5)+.4(-1) =-.1
Exp. payoff (Do nothing)= .6(0)+.4(0) = 0*
Minimax: Select strategy that minimizes the maximum opportunity loss.
Opportunity Loss: It is the difference between the payoff for the best alternative and the alternative chosen by the decision maker.
O. L. T.
0 1 2 3 Row Max
I 0 1.5 1.5 1.5 1.5
II .5 0 .5 1.0 1.0
DoN 1 0 0 0 0*
Optimal strategy is Do nothing.
A jeweler must buy diamonds in large lots "as is" when buying in Amsterdam. Lots can be classified as poor, good, and excellent depending on the percent of perfect diamonds in the lot. A jeweler has the opportunity to buy a lot of 500 diamonds for $150,000. He can sell perfect stones for $500 and all others for $100. He takes a sample of size 2 from the lot and finds one is perfect. Prior probabilities, states of nature and the opportunity loss table are given below. (Classification based on percent of perfect stones, Prior probabilities are .3, .5, and .2, respectively.
Poor(q=.2) / Good
(q=.4) / Excellent
(q=.7)
Buy
/ 60,000 / 20,000 / 0Don’t buy / 0 / 0 / 40,000
For the data in the above exercise, calculate the expected value of sample information.
O. L. T.
P / G / EProbability / .3 / .5 / .2
Buy / 60,000 / 20,000 / 0
Don’t buy / 0 / 0 / 40,000
EOL(Buy) = .3(60000)+.5(20000)+.2(0)
= 28000
EOL(DB) = .3(0)+.5(0)+.2(40000)
= 8000 *
Don't Buy.
EVPI = 8000
n=2 x=1 x = # of perfect stones
Lot / Prior / Cond. / Joint / Posterior0 / -
.2 / P / .3 / .32 / .096 / .2286
.4 / G / .5 / .48 / .240 / .5714
.7 / E / .2 / .42 / .084 / .2000
1.0 / .42 / 1.0000
EOL(Buy)
= .2286(60000)+.5714(20000)+.2(0)
= 25144
EOL(DB)
= .2286(0)+.5714(0)+.2(40000)
= 8000 *
Don't Buy
(EVPI|x=1, n=2) = 8000
You are in the pottery business and buy bowls in lots of 100 "as is" for $1800 from a broker. Some bowls have surface irregularities and can only be sold for $10, while perfect bowls can be sold for $30. You classify lot as "Good," "Fair", and "Poor" based on the percent of irregular bowls, 20%, 50% and 70%, respectively. The prior probabilities are .1, .2, and .7, respectively.
a. Calculate the payoff table.
b. Will you buy the lot before having sampled?
c. If you take a sample of size one, and find a perfect bowl,
will you buy the lot?
d. What is the expected value of perfect information?
For the data in the above exercise, calculate the expected value of sample information. (sample size is one).
O. L. T.
G / F / PProbability / .1 / .2 / .7
Buy / 0 / 0 / 200
Don’t buy / 800 / 200 / 0
EOL(Buy) = .7x200 = 140
EOL(DB) = .1x800+.2x200+.7x0 = 20 *
Opt. Decision: Don't Buy
EVPI = 120
n=1 x=1 x = # of perfect bowls
Lot / Prior / Cond. / Joint / PosteriorG / .1 / .8 / .08 / .08/.39=.205
F / .2 / .5 / .10 / .1/.39 =.256
P / .7 / .3 / .21 / .21/.39=.539
1.0 / .39 / 1.0
EOL(Buy)
= .205(0)+.256(0)+.539(200) =107.8 *
EOL(DB)
= .205(800)+.256(200)+.539(0)=215.2
Opt. Decision: Buy the Lot.
(EVPI|x=1) = 107.8
EVPI = EPPI - Max EMG
EPPI = .3x440+.6x312+.1x284 = 347.6
EVPI = 347.6 - 305.6 =
Posterior Analysis
Payoff Table
G / F / PProbability / .1 / .2 / .7
Buy / 800 / 200 / -200
Don’t buy / 0 / 0 / 0
Good & Buy:
Payoff: 80x30+20x10-1800 = 800
F & Buy:
Payoff: 50x30+50x10-1800 = 200
P & Buy:
Payoff: 30x30+70x10-1800 = -20
E(Buy)= .1x800+.2x200+.7(-200) = -20
E(DB) = 0
Optimal Decision: Don't Buy.
n=1 Let x = 0
Lot / Prior / Cond / JOINT / pOSTERIORG / .1 / .2 / .02 / .0328
F / .2 / .5 / .10 / .1639
p / .7 / .7 / .49 / 1.0000
.61 / 1.0000
EOL(Buy)
= .0328(0)+.1639(0)+.8033(200)
= 160.66
EOL(DB)
= .0328(800)+.1639(200)+.8033(0)
= 59.02
(EVPI|x=0) = 59.02
EVSI = EVPI - EVPI
before sample after sample
= 120 - [59.02x.61 + 107.8x.39]
= 41.96
A hot dog entrepreneur, named Frank, sells his products at afternoon ball games. The decision he is continually faced with is how many how dogs to buy for tomorrow's game. Through long experience he has simplified his problem to the bare essentials. If it rains, he will sell 7 gross. But the fair-weather fans turn out in fair weather and eat about four more gross. Frank's revenue per gross of hot dogs sold is $10.
Frank's first problem is to decide whether to buy 7 gross of hot dogs today or 11 gross. If he orders 11 gross, there's nothing he can do tomorrow except hope it doesn't rain (since the unneeded hot dogs go to waste). On the other hand, he can hedge today by ordering 7 gross and then tomorrow morning, depending on weather conditions, he can order four more gross (at additional cost, of course). The cost per gross of hot dogs is $4 for a regular order, but $7 for a rush order. Frank has determined the probabilities of rain as follow:
probability of rain in the morning is .7
probability of rain in P.M. if rain in A.M. is .9
probability of rain in P.M. if clear in A.M. is .1
Use a decision tree to help Frank solve his problem.
Jim Sellers is thinking about producing a new type of electric razor for men. If the market were favorable, he would get a return of $100,000, but if the market for this new type of razor were unfavorable, he would lose $60,000. Since Ron Bush is a good friend of Jim Sellers, Jim is considering the possibility of using Bush Marketing Research to gather additional information about the market for the razor. Bush has suggested that Jim either use a Survey or a pilot study to test the market. The survey would be a sophisticated questionnaire administered to a test market. It will cost $5,000. Another alternative is to actually run a pilot study. This would involve producing a limited number of the new razors and actually trying to sell them in two cities that are typical of American cities. The pilot study is more accurate, but is also more expensive. It will cost $20.000. Ron Rush has suggested that it would be a good idea for Jim to conduct either the survey or the pilot before Jim makes the decision concerning whether or not to produce the new razor. But Jim is not sure if the value of the survey or the pilot is worth the cost.
Jim estimates that the probability of a successful market without performing a survey or pilot study is .5. Furthermore, the probability of a favorable survey result given a favorable market for razors is .7. and the probability of a favorable survey result given an unsuccessful market for razors is .2. In addition, the probability of an unfavorable pilot study given an unfavorable market is .9, and the probability of an unsuccessful pilot study result given a favorable market for razors is .2.
(a) Draw the decision tree for this problem without the probability values.
(b) Compute the revised probabilities needed to complete the decision, and place these values in the decision tree.
(c) What is the best decision for Jim? Use expected monetary value as the decision criterion.