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Problem 393

# A manufacturer of electronic instruments produces two types of timer: a standard and a precision model with net profits of $$\ 10$$ and $$\ 15$$, respectively. His work force cannot produce more than 50 instruments per day. Moreover, the four main components used in production are in short supply so that the following stock constraints hold: $$\begin{array}{|c|c|c|c|} \hline & & \text { Number used } & \text { per timer } \\ \hline \text { Component } & \text { Stock } & \text { Standard } & \text { Precision } \\ \hline \mathrm{a} & 220 & 4 & 2 \\ \hline \mathrm{b} & 160 & 2 & 4 \\ \hline \mathrm{c} & 370 & 2 & 10 \\ \hline \mathrm{d} & 300 & 5 & 6 \\ \hline \end{array}$$ Graphically determine the point of optimum profit. If profits on the standard timer were to change, by how much could they change without altering the original solution?

Expert verified
The optimal solution is for the manufacturer to produce 0 standard timers and 50 precision timers to maximize profit, which results in a profit of $750 per day. The profit on the standard timer can change by$7.5 without affecting the original solution.
See the step by step solution

## Step 1: Define Variables

Let the number of standard timers produced per day be x, and the number of precision timers produced per day be y.

## Step 2: Write the Objective Function

The profit per standard timer is $$10 and per precision timer is$$15, so the total profit can be written as: $$P = 10x + 15y$$ We want to maximize P subject to the given constraints.

## Step 3: Write the Constraints as Linear Inequalities

According to the given information, the following constraints apply: a. Workforce constraint: $$x + y \leq 50$$ b. Component a constraint: $$4x + 2y \leq 220$$ c. Component b constraint: $$2x + 4y \leq 160$$ d. Component c constraint: $$2x + 10y \leq 370$$ e. Component d constraint: $$5x + 6y \leq 300$$ f. Non-negativity constraint: $$x \geq 0, y \geq 0$$

## Step 4: Graph the Constraints and Find the Feasible Region

Graph the system of inequalities obtained in Step 3 and find the feasible region. The feasible region is a bounded polygon, which means the optimal solution will occur at one of its vertices.

## Step 5: Determine the Vertices of the Feasible Region

Find the vertices of the feasible region by solving the system of inequalities. There are several methods, such as the graphical method, substitution, or elimination. From the graph, we identify four vertices: Vertex 1: Intersection of a and e constraints: (0, 50) Vertex 2: Intersection of a and c constraints: (20, 30) Vertex 3: Intersection of c and e constraints: (30, 22) Vertex 4: Intersection of c and b constraints: (40, 10)

## Step 6: Calculate the Profit at Each Vertex

Compute the profit P at each vertex using the objective function: Vertex 1: P(0, 50) = 10(0) + 15(50) = 750 Vertex 2: P(20, 30) = 10(20) + 15(30) = 650 Vertex 3: P(30, 22) = 10(30) + 15(22) = 630 Vertex 4: P(40, 10) = 10(40) + 15(10) = 550

## Step 7: Determine the Optimal Solution

The optimal solution is the one with the highest profit, which is P(0, 50) = 750 dollars. Therefore, the manufacturer should produce 0 standard timers and 50 precision timers to maximize their profit.

## Step 8: Calculate the Range of Profit on the Standard Timer that Doesn't Affect the Original Solution

Since the manufacturer is producing only precision timers in the optimal solution, a change in the profit of standard timers will not affect the optimal solution as long as it remains at Vertex 1. To figure out how much profits can change without affecting the original solution, we need to identify at which point the next highest profit vertex becomes as profitable as the current vertex. Let's say the profit on a standard timer changes to $10 + p. In order to find the range of p that doesn't affect the solution, we can set the profit equal for Vertex 1 (P(0, 50)) and Vertex 2 (P(20, 30)), which is the next highest profit vertex: $$750 = (10 + p)(20) + 15(30)$$ $$750 = 20p + 600$$ $$p = 7.5$$ So the profit on the standard timer can change by$7.5 without affecting the original solution. If p > 7.5, the manufacturer would shift production from precision timers to standard timers.

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