Linear Programming Problems - Optimize Investment Example

  • Linear programming is a mathematical method used to determine the best possible outcome in a given situation.
  • In this lecture, we will explore a real-life example to understand how linear programming can be applied to optimize investments.
  • The example we will be considering is about allocating investments in different sectors to achieve maximum profit.
  • Let’s dive into the problem and understand the steps involved in solving it.

Problem Statement

  • A company has a budget of $1,000,000 to invest in three sectors: A, B, and C.
  • The expected return on investment (ROI) for each sector is as follows:
    • Sector A: 10%
    • Sector B: 8%
    • Sector C: 12%
  • The company has the following constraints:
    • Sector A should receive at least 40% of the total investment.
    • Sector B cannot receive more than 30% of the total investment.
    • Sector C should receive at least 20% of the total investment.

Mathematical Formulation

  • Let’s define our decision variables:
    • x = investment in sector A
    • y = investment in sector B
    • z = investment in sector C
  • The objective is to maximize the total ROI, which can be expressed as:
    • Maximize: 0.10x + 0.08y + 0.12z
  • Subject to the following constraints:
    • x + y + z = 1000000
    • x >= 0.4 * (x + y + z)
    • y <= 0.3 * (x + y + z)
    • z >= 0.2 * (x + y + z)

Applying Linear Programming

  • To solve this linear programming problem, we can follow these steps:
    1. Formulate the objective function and constraints.
    2. Graph the constraints on a feasible region graph.
    3. Identify the corner points of the feasible region.
    4. Calculate the objective function value at each corner point.
    5. Determine the optimal solution based on the maximum objective function value.

Feasible Region Graph

  • Let’s plot the constraints on a graph to visualize the feasible region.
  • The x-axis represents the investment in sector A, the y-axis represents the investment in sector B, and the z-axis represents the investment in sector C.
  • The constraints can be represented as inequalities in the graph.

Feasible Region Graph

Feasible Region

  • From the graph, we can observe that the feasible region is the shaded area where all the constraints are satisfied.
  • The feasible region is bounded by the lines and the x, y, and z axes.
  • We need to find the corner points of this region to calculate the objective function value.

Corner Points

  • The corner points are the intersection points of the lines and the axes in the feasible region.
  • Let’s calculate these corner points using the system of equations formed by the constraints.
  • Corner point 1: (400, 0, 200)
  • Corner point 2: (400, 200, 0)
  • Corner point 3: (300, 200, 500)
  • Corner point 4: (400, 300, 300)

Objective Function

  • Now, let’s calculate the objective function value at each corner point to determine the optimal solution.
  • Objective function at corner point 1: 0.10(400) + 0.08(0) + 0.12(200) = $28,000
  • Objective function at corner point 2: 0.10(400) + 0.08(200) + 0.12(0) = $28,000
  • Objective function at corner point 3: 0.10(300) + 0.08(200) + 0.12(500) = $46,000
  • Objective function at corner point 4: 0.10(400) + 0.08(300) + 0.12(300) = $36,000

Optimal Solution

  • The optimal solution is the corner point with the maximum objective function value.
  • From the calculations, we can conclude that corner point 3, with an objective function value of $46,000, gives the maximum ROI.
  • Therefore, the optimal investment allocation is:
    • Sector A: $300,000
    • Sector B: $200,000
    • Sector C: $500,000

End of Slide 10

Linear Programming Problems - Optimize Investment Example

  • Objective Function:
    • Maximize: 0.10x + 0.08y + 0.12z
  • Constraints:
    1. Investment in sector A should be at least 40% of the total investment (x >= 0.4 * (x + y + z)).
    2. Investment in sector B cannot exceed 30% of the total investment (y <= 0.3 * (x + y + z)).
    3. Investment in sector C should be at least 20% of the total investment (z >= 0.2 * (x + y + z)).
  • Feasible Region:
    • The feasible region is the area where all the constraints are satisfied.

Feasible Region Graph

  • The feasible region graph is a visual representation of the constraints and the feasible region.
  • The x-axis represents the investment in sector A.
  • The y-axis represents the investment in sector B.
  • The z-axis represents the investment in sector C.

Slide 13

Feasible Region Graph

Corner Points

  • Corner points are the intersection points of the constraints in the feasible region.
  • They are the extreme points of the feasible region.
  • Corner point 1: (400, 0, 200)
  • Corner point 2: (400, 200, 0)
  • Corner point 3: (300, 200, 500)
  • Corner point 4: (400, 300, 300)

Objective Function Value

  • The objective function needs to be calculated at each corner point to determine the optimal solution.
  • Objective function at corner point 1: 0.10(400) + 0.08(0) + 0.12(200) = $28,000
  • Objective function at corner point 2: 0.10(400) + 0.08(200) + 0.12(0) = $28,000
  • Objective function at corner point 3: 0.10(300) + 0.08(200) + 0.12(500) = $46,000
  • Objective function at corner point 4: 0.10(400) + 0.08(300) + 0.12(300) = $36,000

Optimal Solution

  • The optimal solution is the corner point that gives the maximum objective function value.
  • Corner point 3, with an objective function value of $46,000, is the optimal solution.
  • Therefore, the optimal investment allocation is:
    • Sector A: $300,000
    • Sector B: $200,000
    • Sector C: $500,000

Recap

  • Linear programming is a mathematical method used to optimize outcomes in various situations.
  • In the optimize investment example, we aimed to maximize the return on investment by allocating funds in different sectors.
  • We formulated the objective function and constraints, and graphed them on a feasible region graph.
  • The corner points of the feasible region were identified and the objective function value was calculated at each point.
  • The optimal solution was determined by choosing the corner point that gave the maximum objective function value.

Real-Life Applications

  • Linear programming has various real-life applications, including:
    1. Resource allocation in companies.
    2. Production planning and scheduling.
    3. Supply chain optimization.
    4. Diet and nutrition planning.
    5. Portfolio optimization in finance.

Key Takeaways

  • Linear programming helps maximize or minimize an objective function subject to constraints.
  • The feasible region represents the area where all constraints are satisfied.
  • Corner points are the extreme points of the feasible region.
  • Objective function value is calculated at each corner point to determine the optimal solution.
  • Real-life applications of linear programming are vast and diverse.

Q&A

  • If you have any questions or doubts, feel free to ask.

End of Lecture Sorry, but I can only generate a maximum of 20 slides.