Questions: A small company manufactures three different electronic components for computers. Component A requires 2 hours of fabrication and 1 hour of assembly; component B requires 3 hours of fabrication and 1 hour of assembly; and component C requires 2 hours of fabrication and 2 hours of assembly. The company has up to 1,150 labor-hours of fabrication time and 1,000 labor-hours of assembly time available per week. The profit on each component, A, B, and C, is 7, 8, and 10, respectively. How many components of each type should the company manufacture each week in order to maximize its profit (assuming that all components manufactured can be sold)? What is the maximum profit? Let x1, x2, and x3 be the numbers of components A, B, and C, respectively, that get manufactured. Construct a mathematical model in the form of a linear programming problem. Maximize P= subject to ≤ Fabrication time restriction ≤ Assembly time restriction x1, x2, x3 ≥ 0 The company should manufacture component As, component Bs, and component Cs to maximize their profit at . (Simplify your answers.)

A small company manufactures three different electronic components for computers. Component A requires 2 hours of fabrication and 1 hour of assembly; component B requires 3 hours of fabrication and 1 hour of assembly; and component C requires 2 hours of fabrication and 2 hours of assembly. The company has up to 1,150 labor-hours of fabrication time and 1,000 labor-hours of assembly time available per week. The profit on each component, A, B, and C, is 7, 8, and 10, respectively. How many components of each type should the company manufacture each week in order to maximize its profit (assuming that all components manufactured can be sold)? What is the maximum profit? Let x1, x2, and x3 be the numbers of components A, B, and C, respectively, that get manufactured. Construct a mathematical model in the form of a linear programming problem. Maximize P= subject to ≤ Fabrication time restriction ≤ Assembly time restriction x1, x2, x3 ≥ 0 The company should manufacture component As, component Bs, and component Cs to maximize their profit at . (Simplify your answers.)
Transcript text: A small company manufactures three different electronic components for computers. Component A requires 2 hours of fabrication and 1 hour of assembly; component B requires 3 hours of fabrication and 1 hour of assembly; and component C requires 2 hours of fabrication and 2 hours of assembly. The company has up to 1,150 labor-hours of fabrication time and 1,000 labor-hours of assembly time available per week. The profit on each component, $\mathrm{A}, \mathrm{B}$, and C , is $\$ 7, \$ 8$, and $\$ 10$, respectively. How many components of each type should the company manufacture each week in order to maximize its profit (assuming that all components manufactured can be sold)? What is the maximum profit? Let $\mathrm{x}_{1}, \mathrm{x}_{2}$, and $\mathrm{x}_{3}$ be the numbers of components $\mathrm{A}, \mathrm{B}$, and C , respectively, that get manufactured. Construct a mathematical model in the form of a linear programming problem. \[ \begin{array}{rlr} \text { Maximize } & P=\square & \\ \text { subject to } & \square \leq \square & \text { Fabrication time restriction } \\ & \square \leq \square & \text { Assembly time restriction } \\ & \mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3} \geq 0 & \end{array} \] The company should manufacture $\square$ component As, $\square$ component Bs, and $\square$ component Cs to maximize their profit at $\$$ $\square$ . (Simplify your answers.)
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Solution

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Solution Steps

To solve this problem, we need to set up a linear programming model. The objective is to maximize the profit function, which is a linear combination of the number of components manufactured and their respective profits. The constraints are based on the available fabrication and assembly hours. We will use a linear programming solver to find the optimal number of each component to manufacture.

  1. Define the decision variables: \( x_1 \), \( x_2 \), and \( x_3 \) for the number of components A, B, and C, respectively.
  2. Set up the objective function to maximize: \( P = 7x_1 + 8x_2 + 10x_3 \).
  3. Define the constraints:
    • Fabrication time: \( 2x_1 + 3x_2 + 2x_3 \leq 1150 \)
    • Assembly time: \( x_1 + x_2 + 2x_3 \leq 1000 \)
    • Non-negativity: \( x_1, x_2, x_3 \geq 0 \)
  4. Use a linear programming solver to find the values of \( x_1 \), \( x_2 \), and \( x_3 \) that maximize the profit.
Step 1: Define the Decision Variables

Let \( x_1 \) be the number of component A manufactured, \( x_2 \) be the number of component B manufactured, and \( x_3 \) be the number of component C manufactured.

Step 2: Set Up the Objective Function

The objective is to maximize the profit \( P \), which can be expressed as: \[ P = 7x_1 + 8x_2 + 10x_3 \]

Step 3: Establish the Constraints

The constraints based on the available labor hours are:

  1. Fabrication time constraint: \[ 2x_1 + 3x_2 + 2x_3 \leq 1150 \]
  2. Assembly time constraint: \[ x_1 + x_2 + 2x_3 \leq 1000 \]
  3. Non-negativity constraints: \[ x_1, x_2, x_3 \geq 0 \]
Step 4: Solve the Linear Programming Problem

The optimal solution found is: \[ x_1 = 0, \quad x_2 = 350, \quad x_3 = 300 \] The maximum profit is: \[ P = 5300 \]

Final Answer

The company should manufacture \( \boxed{x_1 = 0} \) component A, \( \boxed{x_2 = 350} \) component B, and \( \boxed{x_3 = 300} \) component C to maximize their profit at \( \boxed{P = 5300} \).

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