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Operations Research Problems
1. P.17 of Hillier, Fredrick S. and Lieberman, Gerald J., Operations Research, 2nd edition. San Francisco: Holden-Day, 1974. Wyndor Glass Co. Problem.
max Z = 3 * X1 + 5 * X2
2. The following problem has 3 variables and 4 <= constraints whose planes intersect in Euclidean space E3 at the optimal vertex – the optimal primal program.
The following three dimensional graphs show how the four restraining planes intersect at the point (20,20,20).
The online linear programming solver displays the optimal programs as seen in the next results table.
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.
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.
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).
The three constraints represent the availability of inputs: raw materials (1,500 tons), stills (100%), and retorts (100%).
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