hw6_aut2013 - Introduction to Optimization MS&E 111/ENGR...

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Introduction to Optimization MS&E 111/ENGR 62, Fall 2013-2014, Stanford University Instructor: Ashish Goel Homework 6. Given 11/22/13. Due 12/04/13 at 6:00PM (at the Huang HW Cabinet) Problem 1 Provide numerical examples (objective function and constraints) for each of the following cases. Say whether the primal and dual are unbounded and whether each has an optimal solution. (a) Primal feasible, dual feasible. (Example answer: Both primal and dual have optimal solutions. And optimal primal solution = optimal dual solution). (b) Primal feasible, dual infeasible. (c) Primal infeasible, dual feasible. (d) Primal infeasible, dual infeasible. Problem 2 ARC Consultants need to know how long it will take to complete a particular job. The job consists of n different tasks. Each task k takes t k hours to complete (regardless of how many people work on it). Additionally, certain tasks cannot be performed until other tasks have been completed. This information is represented in a n × n matrix P where the ( i, j )th element of the matrix is a 1 if task j must be performed before task i and a 0 otherwise. (Note that all elements on the diagonal must be zero as a task cannot be

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