quiz2solutions

quiz2solutions - the entering and departing variables....

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640:354:01, 02/12/07 Quiz #2, with solutions 1. Consider the following LP in standard form: Maximize 2 x 1 + 3 x 2 + x 3 , subject to x 1 + 10 x 2 + x 3 4 x 2 + x 3 2 x 1 + 5 x 2 + x 3 3 x 1 ,x 2 ,x 3 0 Write the corresponding canonical form. Write the initial simplex tableau, and the cor- responding dictionary. Solution: The canonical form is obtained by introducing slack variables: Maximize 2 x 1 + 3 x 2 + x 3 , subject to x 1 +10 x 2 + x 3 + x 4 = 4 x 2 + x 3 + x 5 = 2 x 1 +5 x 2 + x 3 + x 6 = 3 x 1 , x 2 , x 3 , x 4 , x 5 , x 6 0 The simplex tableau is: x 1 x 2 x 3 x 4 x 5 x 6 x 4 1 10 1 1 0 0 4 x 5 0 1 1 0 1 0 2 x 6 1 5 1 0 0 1 3 z -2 -3 -1 0 0 0 0 The corresponding dictionary is: x 4 = 4 - x 1 - 10 x 2 - x 3 x 5 = 2 - x 2 - x 3 - x 5 x 6 = 3 - x 1 - 5 x 2 - x 3 z = 2 x 1 + 3 x 2 + x 3
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2. Suppose we have a linear program in canonical form. True or false? (If false, justify!) (a) Given an initial basic feasible solution, the simplex method will lead to an optimal solution in finitely many steps, provided that such a solution exists, no matter how we choose
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Unformatted text preview: the entering and departing variables. False. Unless we use Blands rule, cycling may occur, thus resulting in non-termination of the method. (b) The value of z strictly increases from one tableau to the next, whenever the simplex method is followed. False. The value of z remains the same if the value of the departing variable is zero (that is, if degeneracy occurs). (c) Every row of every tableau, including the objective row, corresponds to an equation which is satised by all feasible solutions. True. 2...
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quiz2solutions - the entering and departing variables....

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