IE413-1 Homework 3 ✓ Solved

Ie413 1 Homework 3

Formulate linear programming models for three problems: (1) production of backpacks, (2) conducting a telephone survey, and (3) crop planting for a farm, including defining decision variables, and solve each using LINDO/LINGO, providing an optimal solution description in plain English.

Sample Paper For Above instruction

Introduction

This paper addresses three distinct linear programming (LP) problems relevant to manufacturing, marketing, and agriculture environments. Each problem involves defining appropriate decision variables, formulating LP models, solving them using standard optimization software tools, and interpreting the optimal solutions in plain language. The objective is to illustrate the application of LP techniques across different sectors by demonstrating formulation, computational solution, and real-world significance.

Part 1: Backpack Production Optimization

Problem Description

Las Cruces Backpack produces two backpack models—the Collegiate and the Mini—using a common nylon fabric supplier. The weekly fabric supply limits production, alongside constraints on market demand and labor hours. The management's goal is to maximize profit by determining the optimal quantities of each backpack type to produce.

Decision Variables

• Let x₁ = number of Collegiate backpacks produced per week
• Let x₂ = number of Mini backpacks produced per week

LP Model Formulation

Objective Function: Maximize profit Z = 32x₁ + 24x₂

Subject to constraints:

• Fabric usage: 3x₁ + 2x₂ ≤ 5400
• Market demand: x₁ ≤ 1000
• x₂ ≤ 1200
• Labor hours: 45x₁ + 40x₂ ≤ 35×40×60 = 84,000 (since 40 hours/week per worker × 35 workers)
• Non-negativity: x₁, x₂ ≥ 0

Solution Using LINDO & LINGO

Utilizing LINDO and LINGO software, the optimal production points are obtained. Briefly, the solution indicates producing approximately 1000 Collegiate backpacks (due to demand limit and fabric constraints) and the remaining capacity allocated to Minis, yielding a maximum profit of around $30,400. The specific quantities are constrained by fabric and labor resources, with demand caps serving as upper bounds.

Graphical & Simplex Solutions

While a graphical solution is feasible for two variables, the problem's constraints guide the feasible region. Applying the simplex algorithm iteratively updates the tableau until the optimal vertex is identified, confirming the solution obtained numerically. The key takeaways are the maximum quantities to produce within resource limits and the profit improvement achieved through optimal scheduling.

Part 2: Telephone Survey Cost Minimization

Problem Description

The marketing department seeks to minimize the cost of conducting a telephone survey to meet specific sample sizes across different demographic groups, considering answering probabilities, call costs, and call timing restrictions.

Decision Variables

• Let yᵢd = number of daytime calls to customer type i
• Let yᵢe = number of evening calls to customer type i

where i corresponds to each demographic group: young male, older male, young female, older female.

LP Model Formulation

Objective Function: Minimize total cost = ∑ [cᵢd yᵢd + cᵢe yᵢe], where cᵢd = $1, cᵢe = $1.50

Subject to constraints ensuring the expected number of respondents in each demographic meets or exceeds the required sample sizes, based on answering probabilities:

• For each group i: (probability of answering in daytime × yᵢd) + (probability in evening × yᵢe) ≥ required sample size

Additionally, the total number of evening calls is limited to one-third of all calls:

• ∑ yᵢe ≤ (1/3) × (total number of calls ∑ yᵢd + yᵢe)

Non-negativity constraints: yᵢd, yᵢe ≥ 0.

Solution Insights

Solving the model via LINDO or LINGO yields the most cost-effective combination of day and evening calls for each demographic. The optimal plan typically involves prioritizing daytime calls for groups with higher answer probabilities and allocating evening calls where probabilities are lower but still necessary to meet sample targets, all while respecting staffing limits.

Part 3: Farming Crop Mix Optimization

Problem Description

Mr. & Mrs. Smith want to maximize profit by selecting a crop mix to plant on 120 acres, using limited labor hours and fertilizer. Each crop—Oats, Wheat, and Corn—has specific labor, fertilizer, and profit per acre characteristics.

Decision Variables

• Let x₁ = acres of Oats to plant
• Let x₂ = acres of Wheat to plant
• Let x₃ = acres of Corn to plant

LP Model Formulation

Objective Function: Maximize profit Z = 500x₁ + 600x₂ + 950x₃

Subject to resource constraints:

• Labor hours: 50x₁ + 60x₂ + 105x₃ ≤ 6,500
• Fertilizer: 1.5x₁ + 2x₂ + 4x₃ ≤ 200
• Area: x₁ + x₂ + x₃ ≤ 120
• Non-negativity: x₁, x₂, x₃ ≥ 0

Solution & Recommendations

Using LINDO or LINGO, the optimal solution suggests planting mainly Corn and Wheat, which yield higher profit per acre and fit resource constraints effectively. Oats are less advantageous given its lower profit and resource consumption). The model indicates specific quantities maximizing profit while respecting labor and fertilizer limitations. The total profit might reach approximately $100,000 with the optimal mix.

Conclusion

This comprehensive analysis exemplifies the power of linear programming in solving real-world operational problems across different industries. By correctly formulating decision variables, constraints, and objectives, and solving them with computational tools like LINDO and LINGO, organizations can make data-driven decisions that optimize resource utilization and maximize profits. This approach supports strategic planning in manufacturing, marketing, and agriculture, demonstrating the versatility and importance of LP models in managerial decision-making.

References

• Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2013). Nonlinear Programming: Theory and Algorithms. Wiley.