Mat540 Quantitative Methods Week 6 Homework Chapter 2 711704

Mat540quantitative Methodsweek 6 Homeworkchapter 21 A Company Produce

Formulate linear programming models for multiple production and resource allocation problems, then solve them using graphical analysis. Problems include production of two products on two assembly lines, furniture manufacturing with constraints on labor and materials, drug formulation with antibiotic requirements, clothing production balancing wool and labor, and an LP model with complex constraints. Additionally, solve a specific LP problem graphically.

Paper For Above instruction

Introduction

Linear programming (LP) is a mathematical technique used for optimizing a linear objective function, subject to linear equality and inequality constraints. It has broad applications in industry for resource allocation, production scheduling, and cost minimization or profit maximization. This paper formulates and analyzes multiple LP problems that pertain to production operations, emphasizing graphical solutions where applicable. Each problem illustrates different real-world manufacturing and formulation challenges, spanning assembly lines, furniture production, pharmaceutical composition, clothing manufacturing, and general LP problem-solving.

Problem 1: Production on Two Assembly Lines

The first problem involves producing two products processed on two assembly lines with limited available hours. Let x₁ and x₂ denote the number of units produced of products 1 and 2, respectively. The constraints are based on processing hours on each line, and the goal is to maximize profit.

  • Assembly line 1 has 100 hours; product 1 requires 10 hours, product 2 requires 14 hours.
  • Assembly line 2 has 42 hours; product 1 requires 7 hours, product 2 requires 3 hours.
  • Profit per unit: $6 for product 1, $4 for product 2.

Formulation:

Maximize Z = 6x₁ + 4x₂


subject to:


10x₁ + 14x₂ ≤ 100 (Line 1 hours)


7x₁ + 3x₂ ≤ 42  (Line 2 hours)


x₁, x₂ ≥ 0

The graphical analysis involves plotting the constraints and identifying the feasible region, then evaluating the profit function at each corner point to find the optimal solution.

Problem 2: Furniture Manufacturing Resource Allocation

Richly, the Pinewood Furniture Company produces chairs and tables. Variables x and y represent the number of chairs and tables produced daily, respectively. Constraints are based on available labor and wood, with demand limitations.

  • Available resources: 80 hours of labor, 36 board-feet of wood.
  • Demand for chairs capped at 6 units/day.
  • Resources required:
  • Chairs: 8 hours labor, 2 ft2 wood.
  • Tables: 10 hours labor, 6 ft2 wood.
  • Profit per unit: $400 for chairs, $100 for tables.

Formulation:

Maximize Z = 400x + 100y


subject to:


8x + 10y ≤ 80 (Labor)


2x + 6y ≤ 36 (Wood)


x ≤ 6 (Demand for chairs)


x, y ≥ 0

The graphical solution involves plotting these inequalities, locating the feasible region, and determining the combination of chairs and tables that provides maximum profit.

Problem 3: Resource Usage at Optimal Production

By solving the LP in Problem 2, we identify the optimal number of chairs and tables. The unused resources can be calculated by substituting these optimal quantities into resource constraints, providing insight into resource efficiency.

Problem 4: Pharmaceutical Ingredient Formulation

The company formulates a drug from two ingredients, each contributing to three antibiotics in varying proportions. The goal is to minimize ingredient costs while meeting antibiotic production constraints.

  • Cost per gram: $80 (ingredient 1), $50 (ingredient 2).
  • Antibiotic contributions per gram:
  • Antibiotic 1: 3 (ingredient 1), 1 (ingredient 2), total 6 units required.
  • Antibiotic 2: 0 (ingredient 1), 6 (ingredient 2), at least 4 units required.
  • Antibiotic 3: 2 (ingredient 1), 6 (ingredient 2), at least 12 units required.

Formulation:

Minimize Cost = 80x + 50y


subject to:


3x + 0y ≥ 6 (Antibiotic 1 requirement)


0x + 6y ≥ 4 (Antibiotic 2 requirement)


2x + 6y ≥ 12 (Antibiotic 3 requirement)


x, y ≥ 0

The LP involves solving for x and y to meet the constraints at minimum cost, typically using graphical analysis for approximate solutions or linear programming software for exact solutions.

Problem 5: Clothing Production Balancing Wool and Labor

The clothing manufacturer produces coats and slacks, constrained by wool and labor. Variables x and y denote coats and slacks produced daily.

  • Resources available: 150 sq. yards of wool, 200 hours of labor.
  • Resources per product:
  • Coat: 3 yards wool, 10 hours labor.
  • Slacks: 5 yards wool, 4 hours labor.
  • Profit per unit: $50 (coats), $40 (slacks).

Formulation:

Maximize Z = 50x + 40y


subject to:


3x + 5y ≤ 150 (Wool)


10x + 4y ≤ 200 (Labor)


x ≤ 6 (Demand for coats, assuming a cap; interpret as constraints or as an additional condition)


x, y ≥ 0

Graphical analysis involves plotting the feasible region and selecting the point that maximizes profit within the constraints.

Problem 6: Graphical LP Solution

Maximize Z = 5x₁ + 8x₂, subject to the inequalities:

  • 4x₁ + 5x₂ ≤ x₁ + 4x₂ ≤ 40
  • x₁ ≤ 8, x₂ ≤ 8, x₁, x₂ ≥ 0

The solution involves plotting the constraints in the x₁-x₂ plane, identifying the feasible region, and evaluating the objective function at the vertices to find the maximum value.

Conclusion

These problems demonstrate the application of linear programming in various production and resource allocation scenarios. Graphical solutions provide valuable intuition and approximate solutions for two-variable problems, while more complex models may require simplex or software-based methods. The key is to accurately formulate the constraints and objective function reflective of real-world limitations and goals.

References

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