Key Concepts

Introduction to Linear Programming Applications:
This chapter extends the use of linear programming (LP) beyond the basic models discussed earlier to cover various practical applications across different domains. The chapter showcases how LP can be adapted and applied to real-world scenarios in marketing, manufacturing, employee scheduling, finance, ingredient blending, and transportation.

Marketing Applications:

• Media Selection: Linear programming is used in advertising to determine the most effective media mix. The objective is often to maximize audience exposure or minimize advertising costs while adhering to budgetary constraints. For example, an LP model might allocate a fixed budget across different media types (TV, radio, newspapers, etc.) to maximize reach or minimize spending.
• Marketing Research: LP can assist in designing marketing research surveys and experiments by optimizing the allocation of resources, such as budget or time, to achieve the best results. It can be used to determine the optimal sampling strategy or allocate resources across different segments.

Manufacturing Applications:

• Production Mix: LP helps in determining the optimal production mix to maximize profits or minimize costs. The model considers various constraints such as labor, material availability, and production capacity. For example, a company might want to decide the number of different products to manufacture given limited resources.
• Production Scheduling: LP models can be employed to optimize production schedules by minimizing downtime or costs associated with production, such as labor or inventory costs. This is particularly useful in environments where multiple products are manufactured on the same production line.

Employee Scheduling Applications:

• Labor Planning: LP is useful for creating employee schedules that meet staffing requirements while minimizing labor costs. For example, a bank might use LP to determine the optimal number of full-time and part-time tellers needed at different times of the day to minimize costs while providing adequate service levels.

Financial Applications:

• Portfolio Selection: LP is applied in finance for selecting an optimal investment portfolio that maximizes return or minimizes risk. Constraints might include budget limits, risk tolerance, or regulations. This application uses historical data on asset performance to build a model that maximizes the expected return for a given level of risk.
• Truck Loading Problem: This involves optimizing the loading of trucks to minimize transportation costs while meeting constraints such as weight limits and delivery requirements. The objective function could be to minimize the total distance traveled or the number of trips required.

Ingredient Blending Applications:

• Diet Problems: One of the earliest applications of LP, used originally to determine the most economical diet for patients. The objective is to minimize the cost of food while meeting nutritional requirements. In agricultural contexts, this is referred to as the feed mix problem, where the goal is to create a blend that meets nutritional requirements at the lowest cost.
• Ingredient Mix and Blending Problems: Similar to diet problems, these involve mixing different raw materials or ingredients to create a final product that meets quality specifications at the lowest cost. This can apply to industries like food production, pharmaceuticals, and chemical manufacturing.

Transportation Applications:

• Shipping Problem: The transportation problem involves determining the optimal way to transport goods from multiple origins to multiple destinations at the lowest cost. It includes constraints such as supply limitations at origins and demand requirements at destinations. The objective is often to minimize the total shipping cost or distance traveled.
• Example Problem (Shipping Problem): The Top Speed Bicycle Company needs to determine the shipping schedule for bicycles from two factories (New Orleans and Omaha) to three warehouses (New York, Chicago, and Los Angeles) to minimize total shipping costs. The LP model includes supply constraints (maximum number of bicycles each factory can produce) and demand constraints (number of bicycles required at each warehouse). The cost of shipping one bicycle from each factory to each warehouse is provided.

Mathematical Formulation for Transportation Problem:

[
\text{Minimize } Z = 2X_{11} + 3X_{12} + 5X_{13} + 3X_{21} + 1X_{22} + 4X_{23}
]

Subject to:

[
X_{11} + X_{21} = 10,000 \quad (\text{New York demand})
]
[
X_{12} + X_{22} = 8,000 \quad (\text{Chicago demand})
]
[
X_{13} + X_{23} = 15,000 \quad (\text{Los Angeles demand})
]
[
X_{11} + X_{12} + X_{13} \leq 20,000 \quad (\text{New Orleans supply})
]
[
X_{21} + X_{22} + X_{23} \leq 15,000 \quad (\text{Omaha supply})
]
[
X_{ij} \geq 0 \quad (\text{Non-negativity constraints})
]

Solution:
Using an LP solver like Excel Solver, the optimal shipping amounts for each route are determined to minimize total costs while satisfying supply and demand constraints. For instance, 10,000 bicycles are shipped from New Orleans to New York, and other amounts are calculated similarly to achieve the minimum cost of $96,000.

Summary:
Chapter 8 demonstrates how linear programming can be applied to a wide range of practical problems in various fields, including marketing, manufacturing, finance, and transportation. The chapter emphasizes formulating these problems correctly, understanding the objectives and constraints, and using tools like Excel Solver to find optimal solutions. It also provides real-world examples and step-by-step approaches to solving complex LP problems efficiently .