hw2sol - Homework 2 Solution Exercise 1, page 89: Consider...

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Homework 2 Solution Exercise 1, page 89 : Consider the following LP maximize z =2 x 1 +3 x 2 subject to x 1 x 2 6 3 x 1 +2 x 2 6 x 1 ,x 2 0 . (a) Express the problem as a standard LP. (b) Determine all the basic solutions of the problem and classify them as feasible or infeasible. (c) Use direct substitution in the objective function to determine the optimal basic feasible solution. (d) Verify graphically that the solution in (c) is indeed optimal. (e) Show how the infeasible basic solutions are represented on the graphical solu- tion space. Solution to Exercise 1, page 89 : (a) By introducing two slack variables, we transform the given LP problem to the standard form: maximize z x 1 x 2 subject to x 1 x 2 + x 3 =6 3 x 1 x 2 + x 4 x 1 2 3 4 0 . (b) In this problem, each basic solution has 2 basic and 2 nonbasic variables. The nonbasic variables are those whose values are set to 0, while the basic are those value is determined by solving the equality constraints. The number of basic solutions is equal to the number of combinations when choosing 2 out of 4, which is 6. They are obtained as follows 1. Choosing x 1 and x 2 as basic variables (means that x 3 = x 4 = 0), the equalities reduce to x 1 x 2 , 3 x 1 x 2 . 1
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By solving these equations, we obtain x 1 = 6 7 and x 2 = 12 7 (point B in Figure 1). This basic solution is feasible, and by substituting the solution in the objective, we have the corresponding objective value z = 48 7 .
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hw2sol - Homework 2 Solution Exercise 1, page 89: Consider...

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