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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 15.053 – Optimization Methods in Management Science (Spring 2007) Recitation 5, March 8 th and March 9 th , 2007 Problem 1: Taking the Dual Let the following LP be the primal. Convert it into the dual (Use the SOB rule). 12 4 34 3 4 1234 3 4 Maximize 2 3 7 s.t. 3 2 2 5 2 2 4 4 2 5 3 3 , 0, uris, 0 xx x x x xxxx x x +− −+ −≤ ++ − = ++−≥ ≥≤ Problem 2: Duality Facts: Short Answer &True/False Part A: If a constraint of the primal problem is NOT tight (binding) at the optimal solution, then the corresponding dual variable (shadow price) is zero. Part B: If a constraint in the dual variable is binding, then the corresponding primal variable is 0. Part C: If a primal variable is 0, what does it say about the dual problem? Part D: The dual of the dual always has an objective function that is greater than the dual. Part E: If a LP is feasible, then so is it dual and the optimal costs are equal. Part F: Given a maximization primal problem every dual simplex pivot increases the value of the dual problem’s objective function.
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Part G: If the dual is unique then the primal BFS is not degenerate. Part H: If the dual is degenerate than the primal has multiple optimal solutions. Part I: The dual simplex method is particularly useful given a dual feasible solution to the original problem and a new cost vector c. Part J: Solving the dual allows us to derive the upper bound on the primal LP. Problem 3 – Connecting Excel Output, Calculations and Graphical Analysis Gemstone Tool company produces wrenches and pliers. They are both made from steel, and the process involves molding on a molding machine, and assembling on an assembly machine. The amount of steel used and the daily availability of steel are shown in the following table. The next two lines contain the machine requirements for each product as well as the availability of machine time. The final two lines contain the daily demand and their per unit contribution to earnings. Wrenches Pliers Availability Steel (lbs.) 1.5 1.0 27,000 lbs./day Molding Machine (hrs) 1.0 1.0 21,000 hours /day Assembly Machine (hrs) .3 0.5 9,000 hours / day Demand Limit (tools/day) 15,000 16,000 Contribution to earnings ($/1000 units) $130 $100
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This note was uploaded on 01/18/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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