By Stephen P. Bradley
Publication through Bradley, Stephen P., Hax, Arnoldo C., Magnanti, Thomas L.
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Extra info for Applied Mathematical Programming
Construct the coefficient matrix, define each of the variables, and explain briefly the significance of each equation or inequality in the linear-programming model for optimal operation, during March and April, of the hydroelectric power system described as follows: The system consists of two dams and their associated reservoirs and power plants on a river. The important flows of power and water are shown in the accompanying diagram. In the following table, all quantities measuring water are in units of 103 acre-feet (KAF).
Thus the maximum value for z is obtained when x3 = x4 = 0. To summarize this observation, we state the: Optimality Criterion. Suppose that, in a maximization problem, every nonbasic variable has a nonpositive coefficient in the objective function of a canonical form. Then the basic feasible solution given by the canonical form maximizes the objective function over the feasible region. Unbounded Objective Value Next consider the example just discussed but with a new objective function: Maximize z = 0x1 + 0x2 + 3x3 − x4 + 20, (Objective 2) subject to: x1 − 3x3 + 3x4 = 6, x2 − 8x3 + 4x4 = 4, xj ≥ 0 (1) (2) ( j = 1, 2, 3, 4).
However, since b0 , b1 , and b2 cannot, in general, be chosen so that the actual prices Pi are exactly equal to the forecast prices Pˆi for all observations, the agent would like to minimize the absolute value of the residuals Ri = Pi − Pˆi. Formulate mathematical programs to find the ‘‘best’’ values of b0 , b1 , and b2 by minimizing each of the following criteria: 6 (Pi − Pˆi )2 , a) i=1 Least squares Acknowledgments 37 6 |Pi − Pˆi |, b) Linear absolute residual i=1 c) Max |Pi − Pˆi |, 1≤i≤6 Maximum absolute residual (Hint: (b) and (c) can be formulated as linear programs.
Applied Mathematical Programming by Stephen P. Bradley