Formulating And Solving Linear Programming Problems Graphica

Formulating and Solving Linear Programming Problems Graphically

In this discussion, we examine multiple linear programming (LP) problems related to production, resource allocation, and cost minimization. The core task in each case is to formulate the LP model—defining decision variables, objective functions, and constraints—and then solve these models graphically to determine optimal production levels, resource utilization, and minimal costs.

First, we explore a manufacturing scenario involving two products processed on two assembly lines with limited available hours. We formulate the LP to maximize profit subject to processing time constraints per line. Next, we analyze a furniture company's goal of maximizing profit by deciding how many chairs and tables to produce, considering resource limitations and product demand. We then evaluate resource usage—labor and wood—based on optimal production quantities.

Subsequently, we address a pharmaceutical formulation problem, determining the quantities of two ingredients to meet antibiotic requirements at minimal cost. Additionally, we consider a clothing manufacturer’s decision on producing coats and slacks to maximize profit under resource constraints. Lastly, a basic LP problem is presented to maximize a linear objective function given a set of linear inequalities.

Paper For Above instruction

The process of formulating and solving linear programming models is fundamental in operational research and decision-making contexts. These models are pivotal in optimizing resource utilization, maximizing profits, or minimizing costs subject to various constraints. The subsequent analysis covers multiple illustrative problems demonstrating the application of LP methodologies, primarily focusing on graphical solutions.

Problem 1: Production on Assembly Lines

A company produces two products on two assembly lines with limited available hours. Assembly line 1 has 100 hours, and line 2 has 42 hours. Product 1 requires 10 hours on line 1 and 7 hours on line 2, whereas product 2 requires 14 hours on line 1 and 3 hours on line 2. The profit per unit is $6 for product 1 and $4 for product 2. The goal is to determine the number of units of each product to produce to maximize profit, subject to processing time constraints.

Let x₁ be the units of product 1 and x₂ the units of product 2. The LP model is:

• Maximize Z = 6x₁ + 4x₂
• Subject to:
  • 10x₁ + 14x₂ ≤ 100 (Line 1 hours)
  • 7x₁ + 3x₂ ≤ 42 (Line 2 hours)
  • x₁, x₂ ≥ 0

Graphical analysis involves plotting the constraints and identifying the feasible region. The corner points are evaluated to find the optimal solution. Solving these constraints yields approximate values where profit is maximized—an exercise often completed using graphing software or manual plotting.

Problem 2: Furniture Production Optimization

The Pinewood Furniture Company produces chairs and tables from labor and wood. Availability is 80 hours of labor and 36 board-feet of wood daily. Chairs require 8 hours of labor and 2 board-feet, while tables require 10 hours and 6 board-feet. Demand limits chairs to 6 units per day. Profits are $400 per chair and $100 per table. The LP formulation is:

• Maximize Z = 400x₁ + 100x₂
• Subject to:
  • 8x₁ + 10x₂ ≤ 80 (labor hours)
  • 2x₁ + 6x₂ ≤ 36 (wood)
  • x₁ ≤ 6 (demand limit)
  • x₁, x₂ ≥ 0

Using graphical methods, plotting the constraints enables the identification of the feasible region. The optimal point, found at a vertex of this region, signifies the number of chairs and tables with maximum profit. Unused resources are computed by substituting the optimal values into the constraints.

Problem 3: Antibiotic Mixture Cost Minimization

A pharmaceutical company formulates a drug from two ingredients, each containing three antibiotics. Ingredient 1 costs $80 per gram; ingredient 2 costs $50 per gram. The goal is to meet antibiotic 1, 2, and 3 minimum units—6, 4, and 12 respectively—while minimizing costs.

Decision variables: x₁ and x₂ (grams of ingredients 1 and 2). The LP model is:

• Minimize Z = 80x₁ + 50x₂
• Subject to:
  • 3x₁ + x₂ ≥ 6 (antibiotic 1)
  • 1x₁ + 6x₂ ≥ 4 (antibiotic 2)
  • 2x₁ + 6x₂ ≥ 12 (antibiotic 3)
  • x₁, x₂ ≥ 0

Graphical analysis involves plotting the inequalities and determining the feasible region at the intersection points that minimally cost based on the objective function.

Problem 4: Clothing Production

A clothier produces coats and slacks with resource constraints of 150 square yards of wool and 200 labor hours. Coats require 3 yards of wool and 10 hours, slacks require 5 yards and 4 hours. The profits are $50 for coats and $40 for slacks. The LP is:

• Maximize Z = 50x₁ + 40x₂
• Subject to:
  • 3x₁ + 5x₂ ≤ 150 (wool)
  • 10x₁ + 4x₂ ≤ 200 (labor hours)
  • x₁, x₂ ≥ 0

Graphing these constraints identifies the optimal quantities, and the unused resources are computed by substituting the optimal points into the respective constraints.

Problem 5: Linear Programming with Multiple Constraints

This problem involves maximizing Z = 5x₁ + 8x₂ with constraints:

• 4x₁ + 5x₂ ≤ 50
• 2x₁ + 4x₂ ≤ 40
• x₁ ≤ 8
• x₂ ≤ 8
• x₁, x₂ ≥ 0

Graphical solution proceeds by plotting constraints and locating the highest value of the objective at the feasible vertices, ensuring optimal resource allocation within bounds.

Conclusion

These examples demonstrate the process of formulating LP models tailored to specific operational scenarios and solving them graphically. While straightforward for two-variable problems, such graphical solutions become complex with higher dimensions, necessitating computational tools. Nonetheless, understanding the geometric interpretation enhances comprehension of LP solutions and their practical implications in resource management and profit maximization.

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