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A manufacturing plant makes two types of inflatable boats: two-person boats and four-person boats. Each two-person boat requires 1 labor-hour from the cutting department and 2 labor- hour from the assembly department. Each four-person boat requires 2 labor-hours from the cutting department and 2 labor-hours from the assembly department. The maximum labor- hours available per month in the cutting department and the assembly department are 600 and 700, respectively. The company makes a profit of $25 on each two-person boat and $40 on each four-person boat. (A) Write the linear programming that maximizes the profit. (B) Solve part (A) graphically.

A manufacturing plant makes two types of inflatable boats: two-person boats and four-person boats. Each two-person boat requires 1 labor-hour from the cutting department and 2 labor- hour from the assembly department. Each four-person boat requires 2 labor-hours from the cutting department and 2 labor-hours from the assembly department. The maximum labor- hours available per month in the cutting department and the assembly department are 600 and 700, respectively. The company makes a profit of $25 on each two-person boat and $40 on each four-person boat. (A) Write the linear programming that maximizes the profit. (B) Solve part (A) graphically.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Title: Optimizing Profit in Manufacturing Introduction: In this problem, a manufacturing plant produces two types of inflatable boats: two-person boats and four-person boats. The challenge is to determine the optimal production strategy to maximize profit given specific labor constraints. Problem Details: 1. **Production Requirements:** - Each two-person boat requires: - 1 labor-hour from the cutting department - 2 labor-hours from the assembly department - Each four-person boat requires: - 2 labor-hours from the cutting department - 2 labor-hours from the assembly department 2. **Labor Availability:** - Cutting department: 600 labor-hours per month - Assembly department: 700 labor-hours per month 3. **Profit:** - $25 profit per two-person boat - $40 profit per four-person boat Tasks: (A) Write the linear programming model that maximizes profit. (B) Solve the linear programming model graphically. --- **Task (A): Linear Programming Model** Objective Function: Maximize \( P = 25x + 40y \) Where: - \( x \) = number of two-person boats - \( y \) = number of four-person boats Subject to the constraints: 1. \( 1x + 2y \leq 600 \) (Cutting department limit) 2. \( 2x + 2y \leq 700 \) (Assembly department limit) 3. \( x \geq 0, y \geq 0 \) (Non-negativity constraints) **Task (B): Graphical Solution** To solve graphically, plot the constraints on a graph with \( x \) and \( y \) axes representing two-person and four-person boats, respectively. Identify the feasible area and locate the corner points. Calculate the profit at each corner point to find the maximum profit.
Transcribed Image Text:Title: Optimizing Profit in Manufacturing Introduction: In this problem, a manufacturing plant produces two types of inflatable boats: two-person boats and four-person boats. The challenge is to determine the optimal production strategy to maximize profit given specific labor constraints. Problem Details: 1. **Production Requirements:** - Each two-person boat requires: - 1 labor-hour from the cutting department - 2 labor-hours from the assembly department - Each four-person boat requires: - 2 labor-hours from the cutting department - 2 labor-hours from the assembly department 2. **Labor Availability:** - Cutting department: 600 labor-hours per month - Assembly department: 700 labor-hours per month 3. **Profit:** - $25 profit per two-person boat - $40 profit per four-person boat Tasks: (A) Write the linear programming model that maximizes profit. (B) Solve the linear programming model graphically. --- **Task (A): Linear Programming Model** Objective Function: Maximize \( P = 25x + 40y \) Where: - \( x \) = number of two-person boats - \( y \) = number of four-person boats Subject to the constraints: 1. \( 1x + 2y \leq 600 \) (Cutting department limit) 2. \( 2x + 2y \leq 700 \) (Assembly department limit) 3. \( x \geq 0, y \geq 0 \) (Non-negativity constraints) **Task (B): Graphical Solution** To solve graphically, plot the constraints on a graph with \( x \) and \( y \) axes representing two-person and four-person boats, respectively. Identify the feasible area and locate the corner points. Calculate the profit at each corner point to find the maximum profit.
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