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Problem 10

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. $$ \begin{array}{ll} \text { Maximize } & P=4 x+5 y \\ \text { subject to } & x+y \leq 30 \\ & x+2 y \leq 40 \\ x & \leq 25 \\ & x \geq 0, y \geq 0 \end{array} $$

Expert verified

The optimal solution for the given linear programming problem occurs at point C, where \(x = 20\) and \(y = 10\), resulting in a maximum value of the objective function \(P = 180\). The coefficient of \(x\) can assume a range of values in \((0, \frac{45}{25})\) without changing the optimal solution. Resource 1 can take a range of values in \([30, 30)\). The shadow price for resource 1 is 0, implying no impact on the objective function. Binding constraints include \(x + y \leq 30\) and \(x + 2y \leq 40\), while nonbinding constraints are \(x \leq 25\), \(x \geq 0\), and \(y \geq 0\).

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Chapter 3

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{l} 3 x-6 y \leq 12 \\ -x+2 y \leq 4 \\ x \geq 0, y \geq 0 \end{array} $$

Chapter 3

WATER SuPPLY The water-supply manager for a Midwest city needs to supply the city with at least 10 million gal of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gallons of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is \(\$ 300\) and that for pipeline water is \(\$ 500\), how much water should the manager get from each source to minimize daily water costs for the city?

Chapter 3

Solve each linear programming problem by the method of corners. $$ \begin{aligned} \text { Minimize } & C=2 x+10 y \\ \text { subject to } & 5 x+2 y \geq 40 \\ & x+2 y \geq 20 \\ & y \geq 3, x \geq 0 \end{aligned} $$

Chapter 3

Solve each linear programming problem by the method of corners. $$ \begin{aligned} \text { Maximize } & P=2 x+5 y \\ \text { subject to } & 2 x+y \leq 16 \\ & 2 x+3 y \leq 24 \\ y & \leq 6 \\ & x \geq 0, y & \geq 0 \end{aligned} $$

Chapter 3

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{l} 2 x-y \geq 4 \\ 4 x-2 y<-2 \end{array} $$

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