HW8 - 2 x 1 x 2 = 5(c Show that the duals obtained in...

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8th Homework for IEOR162 Linear Programming Semester: Fall 2010 Instructor: Dr. Sarah Drewes Due to: Tuesday, November 23, 2010, (at the beginning of class) P1 Consider the following simplex tableau for a maximization problem: z x 1 x 2 x 3 x 4 x 5 rhs 1 δ 2 0 0 0 10 0 -1 1 1 0 0 4 0 α -4 0 1 0 1 0 γ 3 0 0 1 β The parameters α, β, δ, γ are unknown parameters. For each of the following statements (a)-(c) find conditions on the parameters that will make the statement true. (a) The current solution is optimal and there are multiple optimal solutions. (b) The problem is unbounded. (c) The current solution is feasible, but not optimal. P2 Given the primal problem max z = x 1 + 2 x 2 s.t. 3 x 1 + x 2 6 2 x 1 + x 2 = 5 x 1 , x 2 0 (1) (a) Use the rules presented in the lecture to directly find the dual of this problem (without transformation into normal form). (b) Now transform the LP (1) into normal max form. Take the dual of the transposed LP. Let y 0 2 and y 00 2 denote the variables for the two inequality constraints introduced for
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Unformatted text preview: 2 x 1 + x 2 = 5. (c) Show that the duals obtained in part (a) and (b) are equivalent by substituting y 2 = y 2-y 00 2 in the dual derived in part (b). 1 P3 Consider the following auto production model that determines the number of trucks ( x 1 ) and the number of cars ( x 2 ) produced daily. The profit of each truck is 3, the profit of each car is 2 (in hundreds of dollars). The paint shop can produce 60 cars per day (if only cars) or 40 trucks per day(if only trucks), the body shop can produce 50 cars per day (if only cars) or 50 trucks per day(if only trucks). Thus, the LP maximizing the daily profit is given by: max z = 3 x 1 + 2 x 2 s.t. 1 40 x 1 + 1 60 x 2 ≤ 1 1 50 x 1 + 1 50 x 2 ≤ 1 x 1 , x 2 ≥ (2) Formulate the dual problem of (2) and give an economic interpretation. 2...
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  • Fall '07
  • Zhang
  • English-language films, Dual problem, current solution, Multiple Optimal Solutions, Dr. Sarah Drewes, IEOR162 Linear Programming

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