assn7 - gramming problem using only linear constraints(b...

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ORIE 3300/5300 ASSIGNMENT 7 Fall 2010 Individual work. Due: 3 pm, Friday October 29. 1. Read Section 4.2 (on a multiperiod production model) in the AMPL book. Do question 4-3. 2. Consider the linear program maximize c T x subject to Ax = b x 0 , where c = 2 7 4 1 - 1 A = 0 1 1 - 1 1 1 2 1 1 0 b = 3 5 . (a) Is B = [2 , 3] a basis? (b) Write down the matrices A B and A N , and the vectors c B and c N . (c) Calculate A - 1 B . (d) Solve the equations A B u = b for u and A T B y = c B for y . (e) Use your answers to write down the tableau corresponding to B . (f) What is an optimal solution to the problem? 3. Consider again the problem in Q2. Suppose the current basis is B = [5 , 4]. Calculate the current values of the basic variables, ˆ x B . Perform one iteration of the revised simplex method from this basic feasible so- lution, using the least index rule for the entering and leaving variables. OVER 1
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4. Suppose you are given a linear program with variables x 1 , x 2 , . . . , x n . (a) Suppose feasible solutions must satisfy the extra condition max { 3 x 1 , x 2 } ≤ 6 . How could you incorporate this condition into your linear pro-
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Unformatted text preview: gramming problem, using only linear constraints? (b) Suppose instead feasible solutions must satisfy the extra condition | x 3 + 2 x 4 | ≤ 5 . How could you incorporate this condition into your linear pro-gramming problem, using only linear constraints? (c) Suppose instead you wish to change your objective to maximizing the function min { x 1 , 4 x 2 } . How could you model the new prob-lem as a linear programming problem? ( Hint: introduce a new variable as the objective; then add suitable constraints restricting its value.) (d) Suppose instead feasible solutions must satisfy the extra condition max { 3 x 1 , x 2 } ≥ 6 . Do you think it is possible, in general, to incorporate this condition using only linear constraints? ( Hint : what geometric properties do feasible regions of linear programming problems have?) 2...
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