## What is Linear Programming?

Linear programming is a mathematical method that helps to determine the best outcome of a problem, where the problem is to optimize an objective function, subject to a set of constraints represented by linear equations or inequalities. Interested in learning Find more details in this valuable research about the topic? linear programming calculator, an external resource we’ve prepared to supplement your reading.

## Application of Linear Programming for Profit Maximization

The main goal of any business is to maximize profits while minimizing costs. Linear programming can be used to achieve this goal. In a manufacturing company, for example, linear programming can be used to determine the optimal mix between various products to manufacture to achieve maximum profits. A transportation company may use linear programming to determine the optimal routes for delivery to minimize costs and maximize profits. A retailer may use linear programming to optimize the stock levels of products to ensure maximum profits.

## Example 1: Maximizing Profits for a Manufacturing Company

Suppose a manufacturing company produces two types of products- Type A and Type B. The selling prices per unit for type A and type B are 15 dollars and 20 dollars, respectively. The cost per unit to produce type A and type B are 10 dollars and 15 dollars, respectively. The company has a maximum production capacity of 200 units per day. The company wants to maximize their profit.

Using linear programming, we can determine how many units of each product the company should produce to meet the demand and maximize their profit. Let’s assume x1 and x2 as the number of units produced for product A and B, respectively. Therefore, the objective function F(x) for the total profit can be represented as:

F(x) = 15×1 + 20×2

Subject to constraints:

10×1 + 15×2 ≤ 3000 (production capacity)

x1,x2 ≥ 0 (non-negative production)

By solving this linear programming problem using a graphical method, we can determine that the company should produce 100 units of Type A and 66.67 units of Type B per day to achieve maximum profit of $2500 per day.

## Example 2: Maximizing Profits for a Distribution Company

Suppose a distribution company has three warehouses and three destinations. The shipping costs per unit for each warehouse and each destination are provided in the table below. The company wants to determine the optimal plan for product shipment to minimize costs and maximize profits.

Destination 1

Destination 2

Destination 3

Warehouse 1

$ 30

$ 20

$ 40

Warehouse 2

$ 10

$ 25

$ 30

Warehouse 3

$ 25

$ 40

$ 35

Using linear programming, we can determine the optimal plan to minimize costs while meeting the demand. Let’s assume xij as the number of units shipped from warehouse i to destination j. Therefore, the objective function F(x) for the total cost can be represented as:

F(x) = 30×11 + 20×12 + 40×13 + 10×21 + 25×22 + 30×23 + 25×31 + 40×32 + 35×33

Subject to constraints:

x11 + x12 + x13 = 1000 (warehouse 1 supply)

x21 + x22 + x23 = 1500 (warehouse 2 supply)

x31 + x32 + x33 = 2000 (warehouse 3 supply)

x11 + x21 + x31 = 1200 (destination 1 demand)

x12 + x22 + x32 = 800 (destination 2 demand)

x13 + x23 + x33 = 1500 (destination 3 demand)

xij ≥ 0 (non-negative shipment)

By solving this linear programming problem using the simplex method, we can determine the optimal shipment plan to minimize costs while meeting the demand. The optimal shipment plan indicates that warehouse 1 should ship 600 units to destination 2, warehouse 2 should ship 800 units to destination 1 and the remaining 200 units to destination 3, and warehouse 3 should ship 1200 units to destination 3.

## Conclusion

Linear programming is a powerful mathematical tool for optimizing objective functions. By using it, companies can determine the optimal plan for their operations and logistics, leading to efficient resource utilization that ultimately leads to increased profits. Whenever the goal is to maximize profits while minimizing costs, linear programming can provide a solution that will help achieve the goal. With even more complex problems in the future, Linear programming remains an indispensable tool for companies seeking to maximize their profits. We’re always looking to add value to your learning experience. For this reason, we recommend checking out this external source containing extra and pertinent details on the topic. what is linear programming, discover Find more details in this valuable research!