Linear programming, also abbreviated as LP, is a simple method we use to depict complicated real-world relationships by using a linear function. Therefore, we obtain the best outcome by minimizing or maximizing the objective function.

This is a follow up to the post about the optimization model. We will therefore be looking at some examples of linear programming. In other words, giving more explanation on the topic.

**Linear Programming Formula**

A linear programming problem will consist of decision variables, an objective function, constraints, and non-negative restrictions.

- The decision variables, x, and y decide the output of the LP problem and represent the final solution.
- The objective function, Z, is the linear function that needs to be optimized (maximized or minimized) to get the solution.
- The constraints are the restrictions that are imposed on the decision variables to limit their value.

Below is the general formula of a linear programming problem

Objective Function: Z = ax + by

Constraints: cx + dy ≤ e, fx + gy ≤ h. The inequalities can also be “≥”

Non-negative restrictions: x ≥ 0, y ≥ 0

**How to Solve Linear Programming Problems?**

The most important part of solving a linear programming problem is to first formulate the problem using the given data.

The steps to solve linear programming problems are given below:

Step 1: Identify the decision variables.

Step 2: Formulate the objective function. Check whether the function needs to be minimized or maximized.

Step 3: Write down the constraints.

Step 4: Ensure that the decision variables are greater than or equal to 0. (Non-negative restraint)

Step 5: Solve the linear programming problem. Use either the simplex or graphical method or Microsoft excel.

**EXAMPLE**

A company makes two products (X and Y) using two machines (A and B). To produce each unit of X, it requires 1 hour processing time on machine A and 1 hour 30 minutes processing time on machine B. And to produce each unit of Y, it requires 2 hours of processing time on machine A and 1 hour of processing time on machine B.

Also, the forecasted available processing time on machine A is 40 hours and on machine B is 35 hours.

The profit is N4 per product X N7 per product Y. Company policy is to maximize the combined sum of the units of X and the units of Y per production.

Formulate the problem of deciding how much of each product to make as a linear program.

**SOLUTION**

We need to follow the steps stated earlier in order to solve this problem.

Firstly, we are to identify the decision variables. The decision variables in this example are products X and Y.

Secondly, let us formulate the objective function. The objective function Z is to maximize profit on the quantity of X and Y.

Thus, Z=4X+7Y

Thirdly, write down the constraints.

In this example, the constraints are:

Machine A available processing time is 40 hours and for

Machine B available processing time is 35 hours

Then, formulate the problem:

Z=4X+7Y (Profit)

X+2Y<=40 (Machine A constraints)

1.5X+Y<=35 (Machine B constraints)

X,Y>=0 (Non-negativity)

The answer using the solver tool in excel is:

Therefore, the company needs to produce 15 units of product X and 12.5 units of Y to maximize profit.

Thanks for reading.