Download Homework5

Math 442/542
Homework 5
(problems 1 and 2 are due Friday, March 11)
1.
A company will be producing the same new product at two different
factories, and then the product must be shipped to two warehouses. Factory
1 can send an unlimited amount by rail to warehouse 1 only, whereas factory
2 can send an unlimited amount by rail to warehouse 2 only. However,
independent truckers can be used to ship up to 50 units from each factory to
a distribution center, from which up to 50 units can be shipped to each
warehouse. The shipping cost per unit for each alternative is shown in the
following table, along with the amounts to be produced at the factories and
the amounts needed at the warehouses.
To
From
Factory 1
Factory 2
Distr. center
Allocation
Unit shipping cost
Distribution
Warehouse
center
1
2
3
7
4
9
2
4
60
90
Output
80
70
a) Formulate the network representation of this problem as a minimum
cost flow problem.
b) Formulate the linear programming model for this problem.
2.
Reconsider the minimum cost flow problem formulated in Problem 1.
Ignore the upper bounds on the arcs when solving the following
subproblems.
a) Obtain an initial BF solution by solving the feasible spanning tree that
corresponds to using just the two rail lines plus factory 1 shipping to
warehouse 2 via the distribution center.
b)
Use the network simplex method to solve this problem.
3.
Consider the following parlor game between two players. It begins
when a referee flips a coin, notes whether it comes up heads or tails, and
then shows this result to player 1 only. Player 1 may then (1) pass and
thereby pay $5 to player 2 or (2) bet. If player 1 passes, the game is
terminated. However, if he bets, the game continues, in which case player 2
may then either (1) pass and thereby pay $5 to player 1 or (2) call. If player
2 calls, the referee then shows him the coin; if it came up heads, player 2
pays $10 to player 1; if it came up tails, player 2 receives $10 from player 1.
a) Give the pure strategies for each player. (Hint: Player 1 will
have four pure strategies, each one specifying how he would
respond to each of the two results the referee can show him;
player 2 will have two pure strategies, each one specifying how
he will respond if player 1 bets.)
b) Develop the payoff the table for this game, using expected
values for the entries when necessary. Then identify and
eliminate any dominated strategies.
c) Show that none of the entries in the resulting payoff table are a
saddle point. Then explain why any fixed choice of a pure
strategy for each of the two players must be an unstable
solution, so mixed strategies should be used instead.
d) Give the LP formulations (one for each player) for this
problem.
4. This question is worth 4 points. Give yourself one of the following scores:
0, 1, 2, 3, 4 (as you know, A is 4, B is 3, etc.). You will receive this score
provided your score is not the highest score selected among all of your
classmates'. If your number is the highest (or tied for the highest) one
recorded, you will receive 0 points. (Briefly explain your reasoning).
Note: Your final grade will be based on the score
you get from this problem.☺