Professor Jackson                           ______________________

Due Mon. Dec. 8         Math  177                   Name

                      Practice Final 

Extra Credit = 16 points, 2 points each  

1. An oil refinery has available three different processes to produce gasoline.  Each process produces varying amounts of three grades of gasoline (regular, premium, and super).  These amounts, in hundreds of gallons per hour of operation, are given in the table below, along with the cost in dollars, of an hour's operation of each of the processes.  Each week the refinery must produce at least 5000 gallons of regular, 2500 gallons of premium, and 3000 gallons of super.  a) Formulate an LP problem to find the production schedule that satisfies the demand at a minimum cost.  b) Find the optimal production schedule using either the two-phase simplex method or the dual simplex algorithm.

2. Eli Daisy Company uses chemicals 1 and 2 to produce two drugs.  Drug 1 must be at least 70% chemical 1, and drug 2 must be at least 60% chemical 2.  Up to 40 oz of drug 1 can be sold at $6 per oz; up to 30 oz of drug 2 can be sold at $5 per oz.  Up to 45 oz of chemical 1 can be purchased at $6 per oz, and up to 40 oz of chemical 2 can be purchased at $4 per oz.  Formulate an LP problem that can be used to maximize Eli Daisy's profit. 

3. A shop is responsible for making and delivering 225 differentials each month for the next four months.  Manufacture of a differential requires 2 man-hours of labor and 3 units of A.  Each month the shop has available 400 man-hours of labor at $8/hr and 150 man-hours of overtime at $10/hr, and an unlimited supply of A.  However, the cost of a unit of A increases from month to month, as given in the table below.  Any differentials above 225 made in one month can be stored for later delivery, at a cost of $2/month/unit.  Formulate an LP problem for finding a production schedule, which minimizes the overall cost (including storage).

4. Solve the following LP problem using the basic simplex method. 

Maximize 2x2 + x3

Subject to x1 + x2 — 2x3 £ 7

           -3x1 + x2 + 2x3 £ 3

              x1,x2,x3 ³ 0

5. a) State the dual of the LP problem below.

   b) Find an optimal solution of the following LP problem and

   its dual.

Minimize 10x1 + 4x2

Subject to 3x1 + 2x2 >= 72

           6x1 + 2x2 >= 84

           3x1 + 6x2 >= 60

             x1,x2 >= 0

6. In solving the LP problem below we have the following initial and final tableaus.  a) Suppose b1 is changed from 28 to 28 + D.  Determine the range on D so that x4 and x2 remain as basic variables in the optimal solution.  b) What is the new optimal solution if b2 is changed from 50 to 60? 

Maximize z = 11x1 + 4x2 + x3 + 15x4

Subject to   3x1 + x2 + 2x3 + 4x4 <= 28

             8x1 + 2x2 — x3 + 7x4 <= 50

                x1,x2,x3,x4 >= 0

    | x1   x2   x3   x4   x5   x6  |       initial tableau

 x5 |  3    1    2    4    1    0  |  28

 x6 |  8    2   -1    7    0    1  |  50

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    | -11  -4   -1   -15   0    0  |   0

    | x1   x2   x3   x4   x5   x6  |           final tableau

 x4 | -2    0    5    1    2   -1  |   6

 x2 | 11    1  -18    0   -7    4  |   4

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    |  3    0    2    0    2    1  | 106

7. In spring, in preparation for the summer trade, a shop makes outdoor tables.  Three types of tables can be made.  The amounts of wood and labor required, and the selling price, for each type of table, are given below.  For the project, the shop has available up to 2000 hr. of labor at $8/hr.  The wood is purchased from a mill, and costs $5/unit for the first 2500 units, and $4.50/unit for any units above 2500.  Assuming that all tables made now will be sold in the coming summer, formulate an integer programming problem to determine how the shop can maximize its profit. 

 8. Western Oil Company has oil fields in Alaska and California.  The Alaska field can produce up to 900,000 barrels per day and the California field can produce up to 500,000 barrels per day.  Oil from the fields is either sent to a refinery in Texas or a refinery in Oklahoma.  It costs $1000 to refine 100,000 barrels of oil at the Texas refinery and $1100 at the Oklahoma refinery.  Refined oil is shipped to customers in Illinois and Ohio.  Chicago customers require 800,000 barrels per day of refined oil; St. Louis customers require 500,000 barrels per day of refined oil.  The costs of shipping 100,000 barrels of oil (refined or unrefined) between cities are given in the table below.  Formulate a balanced transportation model for minimizing the cost of refining and shipping the oil to the customers.

 9. Player 1 selects a number from {1,2,3}, and player 2 selects a number from {1,2}.  If the sum of the selected numbers is even, player 1 wins that amount from player 2; if the sum is odd, player 2 wins that amount from player 1. 

a)  Determine the payoff matrix for this game. 

b)  Determine the security level of the mixed strategy (1/3,1/3,1/3) for player 1. 

c) Use any method (graphing or linear programming) to find the best possible mixed strategy for player 2. 

                                                                      The  End