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Problem 11

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{array}{rr} \text { Minimize } & C=2 x+5 y \\ \text { subject to } & x+2 y \geq 4 \\ & 3 x+2 y \geq 6 \\ & x \geq 0, y \geq 0 \end{array} $$

Expert verified

The optimal solution of the primal problem is \( x = 2, y = 0 \) with a minimal value of \( C = 4 \). The dual problem is given by:
Objective:
Maximize:
\( W = 4w_1 + 6w_2 \)
Subject to:
\( w_1 + 3w_2 \leq 2 \)
\( 2w_1 + 2w_2 \leq 5 \)
\( w_1 \geq 0, w_2 \geq 0 \)

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Chapter 6

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{ccrcrc|c} x & y & z & u & v & P & \text { Constant } \\ \hline 1 & 0 & \frac{3}{5} & 0 & \frac{1}{5} & 0 & 30 \\ 0 & 1 & -\frac{19}{5} & 1 & -\frac{3}{5} & 0 & 10 \\ \hline 0 & 0 & \frac{26}{5} & 0 & 0 & 1 & 60 \end{array} $$

Chapter 6

Solve each linear programming problem by the method of corners. $$ \begin{array}{l} \text { Maximize } P=4 x+2 y \\ \text { subject to } \quad x+y \leq 8 \\ \quad 2 x+y \leq 10 \\ x \geq 0, y \geq 0 \end{array} $$

Chapter 6

Consider the linear programming problem $$ \begin{array}{lr} \text { Maximize } & P=2 x+7 y \\ \text { subject to } & 2 x+y \geq 8 \\ x+y & \geq 6 \\ x & \geq 0, y \geq 0 \end{array} $$ a. Sketch the feasible set \(S\). b. Find the corner points of \(S\). c. Find the values of \(P\) at the corner points of \(S\) found in part (b). d. Show that the linear programming problem has no (optimal) solution. Does this contradict Theorem \(1 ?\)

Chapter 6

A financier plans to invest up to $$\$ 500,000$$ in two projects. Project A yields a return of \(10 \%\) on the investment whereas project \(\bar{B}\) yields a return of \(15 \%\) on the investment. Because the investment in project \(\mathrm{B}\) is riskier than the investment in project \(\mathrm{A}\), the financier has decided that the investment in project \(\mathrm{B}\) should not exceed \(40 \%\) of the total investment. How much should she invest in each project in order to maximize the return on her investment? What is the maximum return?

Chapter 6

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=x+4 y-2 z \\ \text { subject to } & 3 x+y-z \leq 80 \\ & 2 x+y-z \leq 40 \\ & -x+y+z \leq 80 \\ x & \geq 0, y \geq 0, z \geq 0 \end{array} $$

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