Chapter 3: Chapter 3

You are given a linear programming problem. a. Use the method of corners to solve the problem. b. Find the range of values that the coefficient of \(x\) can assume without changing the optimal solution. c. Find the range of values that resource 1 (requirement 1) can assume. d. Find the shadow price for resource 1 (requirement 1). e. Identify the binding and nonbinding constraints. $$ \begin{array}{ll} \text { Maximize } & P=3 x+4 y \\ \text { subject to } & 2 x+3 y \leq 12 \\ & 2 x+y \leq 8 \\ & x \geq 0, y \geq 0 \end{array} $$

A veterinarian has been asked to prepare a diet for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than \(8 \mathrm{oz}\) and must contain at least 29 units of nutrient I and 20 units of nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: brand \(\mathrm{A}\) and brand \(\mathrm{B}\). Each ounce of brand A contains 3 units of nutrient \(I\) and 4 units of nutrient II. Each ounce of brand B contains 5 units of nutrient \(\mathrm{I}\) and 2 units of nutrient II. Brand A costs 3 cents/ounce and brand B costs 4 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. An optimal solution of a linear programming problem is a feasible solution, but a feasible solution of a linear programming problem need not be an optimal solution.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Suppose you are given the following linear programming problem: Maximize $P=a x+b y\(, where \)a>0\( and \)b>0\(, on the feasible set \)S$ shown in the accompanying figure. Explain, without using Theorem 1 , why the optimal solution of the linear programming problem cannot occur at the point \(Q\) unless the problem has infinitely many solutions lying along the line segment joining the vertices \(A\) and \(B\). Hint: Let \(A\left(x_{1}, y_{1}\right)\) and $B\left(x_{2}, y_{2}\right)\(. Then \)Q(\bar{x}, \bar{y})\(, where \)\bar{x}=x_{1}+$ \(\left(x_{2}-x_{1}\right) t\) and \(\bar{y}=y_{1}+\left(y_{2}-y_{1}\right) t\) with \(0

Consider the linear programming problem $$ \begin{array}{rr} \text { Maximize } & P=2 x+7 y \\ \text { subject to } & 2 x+y \geq 8 \\ & x+y \geq 6 \\ x & \geq 0, y \geq 0 \end{array} $$ 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 ?

Consider the linear programming problem $$ \begin{aligned} \text { Minimize } & C=-2 x+5 y \\ \text { subject to } & x+y \leq 3 \\ & 2 x+y \leq 4 \\ & 5 x+8 y \geq 40 \\ & x \geq 0, y \geq 0 \end{aligned} $$ a. Sketch the feasible set. b. Find the solution(s) of the linear programming problem, if it exists.

Find the graphical solution of each inequality. $$ y \geq-1 $$

You are given a linear programming problem. a. Use the method of corners to solve the problem. b. Find the range of values that the coefficient of \(x\) can assume without changing the optimal solution. c. Find the range of values that resource 1 (requirement 1) can assume. d. Find the shadow price for resource 1 (requirement 1). e. Identify the binding and nonbinding constraints. $$ \begin{array}{ll} \text { Maximize } & P=2 x+5 y \\ \text { subject to } & x+3 y \leq 15 \\ & 4 x+y \leq 16 \\ & x \geq 0, y \geq 0 \end{array} $$

Find the maximum and/or minimum value(s) of the objective function on the feasible set \(S\). $$ Z=3 x+2 y $$

You are given a linear programming problem. a. Use the method of corners to solve the problem. b. Find the range of values that the coefficient of \(x\) can assume without changing the optimal solution. c. Find the range of values that resource 1 (requirement 1) can assume. d. Find the shadow price for resource 1 (requirement 1). e. Identify the binding and nonbinding constraints. $$ \begin{array}{cc} \text { Minimize } & C=2 x+5 y \\ \text { subject to } & x+2 y \geq 4 \\ & x+y \geq 3 \\ & x \geq 0, y \geq 0 \end{array} $$