MATH 313: Operations Research Exam II Due Date: Tuesday, Dec. 1st, 2015, by 6:00 pm 1. (10 pts) Find all possible minimum spanning trees respectively of the two diagrams given below: 2. (10 pts) S is a warehouse, and T is a store. Find all possible shortest paths from S to T . 3. (10 pts) Depart from A, and reach the destination H. B to G are all middle stops. The capacity of vehicles can pass through on each road is given. Find the maximum number of vehicles that can reach H. 4. (14 pts) For the following payo table. Strategy 1 1 4 Player 1 2 -3 3 6 4 7 Player 2 2 3 -4 -5 -4 -9 7 -8 3 -9 4 6 -2 -9 5 Use the minimax criterion to nd the best strategy for each player. Does this game have a saddle point? Is it a stable game? 5. (16 pts) Consider the game having the following payo table. Player 2 1 2 3 4 1 2 2 3 -2 Player 1 2 4 3 2 5 Strategy Use the graphical procedure to determine the value of the game and the optimal mixed strategy for each player according to the minimax criterion. 6. (16 pts) You have the chance to invest your money in either a 7.5% bond that sells at face value or an aggressive growth stock that pays only 1% dividend. If ination occurs, the interest rate will go up to 8%, in which case the principal value of the bond will go down by 10%, and the stock value will go down by 20%. If recession materializes, the interest rate will go down to 6%. In this case, the principal value of the bond is expected to go up by 5%, and the stock value will increase by 20%. If the economy remains unchanged, the stock value will go up by 8%, and the bond principal value will remain the same. Economists estimate a 20% chance of ination and 15% of recession. You are basing your investment decision on next year's economic conditions. (a) Represent the problem as a decision tree. (b) Would you invest in stocks or bonds? 7. (24 pts) Farmer McCoy can plant either corn or soybeans. The probabilities that the next harvest prices will go up, stay the same, or go down are .25, .30, and .45, respectively. If the prices go up, the corn crop will net $30,000 and the soybeans will net $10,000. If the prices remain unchanged, McCoy will (barely) break even. But if the prices go down, the corn and soybeans crops will sustain losses of $35,000 and $5000, respectively. The farmer has the additional option of using the land as a grazing range, in which case a payo of $7500 is guaranteed. The farmer has also secured additional information from a broker regarding the degree of stability of future commodity prices. The broker's assessment of \"favorable\" and \"unfavorable\" is described by the following conditional probabilities: The symbols a1 and a2 represent the \"favorable\" and \"unfavorable\" assessments, and s1 , s2 , and s3 represent the \"up\