Problem Set #3
Integer Programming & Decision Analysis
QMETH 501—Professor Hillier
due Wednesday, March 10
1. An Electrical Utility Startup Problem.
A model faced by an electrical utility each day is that of deciding which generators to start up. The utility in
question has five generators with the characteristics shown in the following table. If any megawatts are
produced by a generator, the generator must be started up and the fixed startup cost is incurred. A total of 6500
MW must be generated today.
Fixed Startup Cost
Cost per Megawatt Generated
Maximum Capacity (MW)
A
$3000
$5
2100
B
$2000
$4
1800
Generator
C
$2500
$6
2500
D
$1500
$6
1500
E
$1000
$7
3000
a. Formulate the algebraic model for this problem— be sure to define all decision variables, write the objective
function as a function of the decision variables, and write all constraints (including nonnegativity) as functions
of the decision variables. (Type the algebraic model in your spreadsheet.)
b. On the same worksheet, build a linear integer programming spreadsheet model, and solve it using Solver.
2. Who Wants to be a Millionaire.
You are a contestant on “Who Wants to be a Millionaire”. You have already answered the $250,000 question,
and now must decide if you would like to answer the $500,000 question. You can choose to walk away at this
point with $250,000 in winnings, or you may decide to answer the $500,000 question. If you answer the
$500,000 question correctly, you can then choose to walk away with $500,000 in winnings, or go on and try to
answer the $1,000,000 question. If you answer the $1,000,000 question correctly the game is over, and you win
$1,000,000. If you answer either the $500,000 or the $1,000,000 question incorrectly, the game is over
immediately and you take home “only” $32,000.
A feature of the game “Who Wants to be a Millionaire” is that you have three “lifelines”—namely “50-50”,
“ask the audience”, and “phone a friend”. At this point (after answering the $250,000 question), you have
already used two of them, but you have the “phone a friend” lifeline remaining. With this option, you may
phone a friend to get their advice on the correct answer to a question before answering it. You may use this
option only once (i.e., you can use it on either the $500,000 question or the $1,000,000 question, but not both).
Since your friends are all smarter than you are, “phone a friend” significantly improves your odds for answering
a question correctly. Without “phone a friend”, if you choose to answer the $500,000 question you have a 65%
chance of answering correctly, and if you choose to answer the $1,000,000 question you have a 50% chance of
answering correctly (the questions get progressively harder). With “phone a friend”, you have an 80% chance of
answering the $500,000 question correctly, and a 65% chance of answering the $1,000,000 question correctly.
a. Use TreePlan to construct and solve a decision tree to decide what to do. Should you answer the $500,000
question? Should you use “phone a friend”? Should you save “phone a friend” for the $1,000,000 question?
Should you even answer the $1,000,000 question if you get to it? Is that your final answer? State (in words
below the tree) what is the best course of action, assuming your goal is to maximize your expected winnings.
b. Use the equivalent lottery method discussed in class to determine your utility function (in particular, your utility
values for all of the possible payoffs in the game). Re-solve the tree, replacing the expected payoffs with your
utility values, to maximize your expected utility. Does the best course of action change? Include on the
spreadsheet a description of how you obtained your utility values (e.g., what equivalent lotteries did you
consider?).
Problem Set #3 continues on the next page
Problem Set #3 continued
3. UW Toys and the Professor Hillier Action Figure.
UW Toys has developed a brand new product—a Professor Hillier Action Figure (PHAF). They are now
deciding how to market the doll.
One option is to immediately ramp up production and launch an ad campaign throughout the state of
Washington. This option would cost $600,000. Based on past experience, new action figures either take off and
do well, or fail miserably. Hence, they predict one of two possible outcomes— total sales of 650,000 units or
total sales of only 40,000 units. The net profit per unit sold is $2. They currently think there is about a 60%
chance that the product will do well (sell 650,000 units) and a 40% chance that it will do poorly (sell 40,000
units).
Another option is to test market the product in Yakima. This would require less capital for the production run,
and a much smaller ad campaign. Again, they predict one of two possible outcomes for Yakima. The product
will either do well (sell 6000 units) or do poorly (sell 600 units). The cost for this option is estimated to be
$10,000. The net profit per unit sold is $2 for the test market as well. University Toys has often test marketed
toys in the Yakima market. Products that sell well statewide have also sold well in Yakima 70% of the time.
Products that sell poorly statewide also sell poorly in Yakima 80% of the time.
There is a complication with the Yakima test market option, however. A rival toy manufacturer is rumored to be
considering the development of a Dean Gupta Action Figure (DGAF). After doing the test market in Yakima, if
University Toys decides to go ahead and ramp up production and market throughout the state, the cost of doing
so would still be $600,000. However, the sales prospects depend upon whether DGAF has been introduced into
the market or not. If DGAF has not entered the market, then the sales prospects will be the same as before (i.e.,
650,000 units if PHAF does well, or 40,000 units if PHAF does poorly, on top of any units sold in the test
market). However, if DGAF has been introduced, the increased competition will diminish sales PHAF. In
particular, they expect to sell 500,000 if PHAF does well, and 30,000 units if it does poorly (on top of any units
sold in the test market). Note that the probability of PHAF doing well or doing poorly is not affected by DGAF,
just the final sales totals of each possibility. The probability that DGAF will enter the market before the end of
the test market in Yakima is 20%. On the other hand, if UW Toys markets PHAF immediately, they are
guaranteed to beat DGAF to market (thus making DGAF a nonfactor).
a. Use TreePlan to develop a decision tree to help UW Toys decide the best course of action and the expected
payoff.
b. Now suppose UW Toys is uncertain of the probability that DGAF will enter the market before the end of the
test market in Yakima would be completed (if it were done). How does the expected payoff vary as the
probability that the DGAF would enter the market changes? Generate a data table that shows the expected
payoff as the probability that DGAF enters varies from 0% to 100% (at 10% increments). How does the best
course of action change over this range?