Linear programming is a tool used to help make quantitative business decisions based on constraints and whether the goal is to maximize or minimize the linear function. When performing linear programming the starting process is similar depending on whether you are maximizing or minimizing. This process of either minimizing or maximizing is known as optimization. Usually, when looking into linear programming for business purposes you are given information pertaining to how the business operates and what the goal of the problem is.
It is important to understand the goal of a linear programming problem before the objective function is developed, as the objective function will lead the problem going forward.
The first step in linear programming is describing the objective function. “The objective function is a mathematical equation that describes the production output target that corresponds to the maximization of profits with respect to production. It then uses the correlation of variables to determine the value of the final outcome. In other words, it’s a formula businesses use to achieve profitability and production goals.” (myaccountingcourse.com)
The next step is finding the constraints. Constraints are restrictions in the function that the feasible area must fall within. Constraints are usually limited by factors such as work hours available or materials on hand. Constraints can also be minimums such as a factory being unable to make less than zero of a product.
Next, we must graph the constraints to find our feasible regions. These feasible regions are overlapping areas of a graph that contain solutions to our objective function. The difference here is whether we are maximizing or minimizing. Usually, the goal is to either maximize profit or minimize cost. When we maximize we are trying to find the highest possible value our objective function can give that fits within the bounds of the constraints. Likewise, when minimizing we are trying to find the best possible solution that falls within the constraints, this will be the solution that gives us the lowest numbers. The vertices of the constraints give us useful values that we can use to solve our problem especially those that intersect all constraints. Generally, for a minimizing problem, our feasible area can be unbound which means feasible solutions can go on beyond a minimum value infinitely, therefore, it is important to find these intersecting points as we are aiming for the lowest feasible result. Maximizing problems are usually constrained into a much smaller area and (in the case of production) usually not unbound into the negative so it is important to find the highest possible solution in the feasible area.(Mario 2020)
Once we have graphed out or linear program we can plug in the coordinates of the vertices into our objective function which will give us feasible solutions.
References
What is objective function?, Myaccountingclass.com, https://www.myaccountingcourse.com/accounting-dictionary/objective-function
Linear Programming (Optimization) 2 Examples Minimize & Maximize, https://www.youtube.com/watch?v=Y7e7DCsDUMY&ab_channel=Mario%27sMathTutoring