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Posted on 24 November 2009
The displays the optimal programs as seen in the next results table.
| The L.P. Input Tableau |
| |
|
X1 |
X2 |
X3 |
| Z0 |
0 |
600 |
600 |
2400 |
| Z1 |
24000 |
-600 |
-100 |
-500 |
| Z2 |
6500 |
-100 |
-100 |
-125 |
| Z3 |
24000 |
-100 |
-600 |
-500 |
| Z4 |
100000 |
-500 |
-500 |
-4000 |
| The L.P. Output Tableau |
| |
|
Y1 |
Y3 |
Y4 |
| Z |
72000 |
-0.52 |
-0.52 |
-0.47 |
| X1 |
20 |
-0 |
0 |
0 |
| Y2 |
0 |
0.15 |
0.15 |
-0.01 |
| X2 |
20 |
0 |
-0 |
0 |
| X3 |
20 |
0 |
0 |
-0 |
3. Solve the zero sum two person game on P.114 of Intriligator, Michael D., Mathematical Optimization and Economic Theory. Englewood Cliffs, N.J.: Prentice Hall, 1971.
The payoff matrix is:
4. Solve the zero sum two person game on P.761 of Chiang, Alpha C., Fundamental Methods of Mathematical Optimization,. 2nd ed. New York: McGraw-Hill, 1974.
The payoff matrix is:
5. The linear programming problem on P. 152 of Loomba, N. Paul. Linear Programming: An Introductory Analysis. New York: McGraw-Hill, 1964, exhibits degeneracy. Try solving it by hand with the primal simplex method, and then check your answer with the online l.p. solver.
| max P = 22 x1 + 30 x2 + 25 x3 |
| 2 x1 + 2 x2 + 0 x3 <= 100 |
| 2 x1 + 1 x2 + 1 x3 <= 100 |
| 1 x1 + 2 x2 + 2 x3 <= 100 |
| x1 >= 0, x2 >= 0, x3 >= 0 |
6. Look through the constructed example of "cycling" on P. 190 of Hadley, G. Linear Programming. Reading, Mass: Addison-Wesley, 1962. Then use the dual simplex method to solve the problem.
| max P = .75 x1 - 20 x2 + 0.5 x3 - 6 x4 |
| .25 x1 - 8 x2 - 1 x3 + 9 x4 <= 0 |
| 0.5 x1 - 12 x2 - 0.5 x3 + 3 x4 <= 0 |
| 0 x1 + 0 x2 + 1 x3 + 0 x4 <= 1 |
| x1 >= 0, x2 >= 0, x3 >= 0, x4 >= 0 |
7. Dorfman, Robert, Paul Samuelson, and Robert Solow. Linear Programming and Economic Analysis. New York: McGraw-Hill, 1958. In this classical treatise, the authors claim that "much of standard economic analysis is linear programming." They provide the example (85-92) of a firm with 4 production process that convert a limited quantity of raw materials into a product. The variables, x1, x2, x3 , and x4, represent the level of each process's activity. The objective function, P, represents each production process's profitability. Thus "1 unit" of production process 1 generates $60 of profit from 100 tons of raw materials. The coefficients of the technology matrix, A, capture the efficiency of each production process. The first column of A means: (in a given week) one unit of production process 1 can process 100 tons of raw materials using 7% of input 1 (stills) and 3% of input 2 (retorts).
| max P = 60 x1 + 60 x2 + 60 x3 + 60 x4 |
| 100 x1 + 100 x2 + 100 x3 + 100 x4 <= 1,500 |
| 7 x1 + 5 x2 + 3 x3 + 2 x4 <= 100 |
| 3 x1 + 5 x2 + 10 x3 + 15 x4 <= 100 |
| x1 >= 0, x2 >= 0, x3 >= 0, x4 >= 0 |
The three constraints represent the availability of inputs: raw materials (1,500 tons), stills (100%), and retorts (100%).
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