are155-homework-6

# are155-homework-6 - University of California Davis...

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Unformatted text preview: University of California, Davis Department of Agricultural and Resource Economics “We are what we repeatedly do. Excellence then is not an act, but a habit.” Aristotle Copyright c 2010by Quirino Paris. ARE 155 Winter 2010 Prof. Quirino Paris HOMEWORK #6 Due Tuesday, February 16 1. Solve the following LP problem using the appropriate algorithm max Z = 3 x 1 + x 2 subject to x 1 − 2 x 2 ≥ − 4 3 x 1 + 4 x 2 ≤ 12 2 x 1 − 4 x 2 ≤ 4 x 1 ≥ , x 2 ≥ B) Exhibit the complete primal and dual solutions. Recalculate the optimal value of the primal and the dual objective functions. Exhibit the optimal primal feasible basis and its inverse. C) Verify that the product of the optimal basis and its inverse results in the identity matrix. D) Write down the dual problem and identify the optimal dual feasible basis using the information contained in the last row of the optimal tableau. E) Verify that the dual variables can be obtained also by multiplying the row vector of revenue coeﬃcients associated with basic activities by the inverse of the optimal basis (Place the row vector of the revenue coeﬃcients in front of the inverse matrix). F) Verify that the primal basic variables can be obtained by multiplying the inverse of the optimal basis by the column vector of right-hand-side (RHS) coeﬃcients (Place the column vector of RHS coeﬃcients after the inverse matrix). G) Verify that, in the optimal tableau, the product of any primal slack variable and its corresponding dual variable is always equal to zero. Verify also that the prod- uct of any dual slack variable and its corresponding primal variable is always equal to zero. This property is called “COMPLEMENTARY SLACKNESS CONDITION.” Interpret these results from an economic viewpoint (see chapter 9, pages 140- 141). 2. Solve the following LP problem by the appropriate algorithm max Z = − (3 / 4) x 1 − x 2 1 subject to x 1 + x 2 ≥ 1 2 x 1 + 3 x 2 ≥ 2 x j ≥ , j = 1 , 2 ....
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