|1. Introduction||2. Use||3. Methods|
|4. Objective Function||5. Constraints||6. Examples|
In managerial accounting, linear programming refers to the application of various mathematical techniques to determine an optimum solution.
A common example of the use of linear programming is to find the optimum mix of products or services that shall lead to maximum profits (i.e. objective function) while taking into consideration any shortage of resources (i.e. constraints).
As we learned in the tutorial on single limiting factor analysis, financial problems involving only one limiting factor can be solved simply by ranking the alternatives according to their contribution per unit of limiting factor.
Unfortunately, such approach cannot be applied to problems involving multiple limiting factors. This is where the use of linear programming techniques such as the graphical method become necessary in order to find the best solution.
Linear programming can be used to solve financial problems involving multiple limiting factors and multiple alternatives. However, where the number of alternatives ( e.g. types of products) is greater than 2, only a specific method of linear programming (known as the simplex method) can be used to determine the optimum solution.
In the following sections, we will learn how to apply linear programming to problems involving only 2 alternatives. Simplex Method shall be covered later in a separate article.
There are 2 methods of solving multiple limiting factor problems involving 2 alternatives:
a) Graphical Method
b) Equation Method
Objective Function is an equation that defines what you want to achieve by solving the financial problem.
Objective of solving a linear programming problem could for example be to maximize contribution by the production and sale of optimum quantities of products. Objective function simply presents such objectives in the form of mathematical equations.
Constraints are any limitations that prevent an organization from maximizing its profits.
Constraints have to be 'programmed' into a linear programming problem in the form of mathematical expressions so that the optimum solution is within feasible limits.
Some examples of constraints are as follows:
- Limiting factor constraints
These are mathematical expressions of the scarce resources (e.g. land, labor, machine hours, etc.) that prevent a business from maximizing its sales.
- Demand constraints
These constraints quantify the maximum demand of products or services.
- Minimum Supply constraints
These constraints quantify the minimum supply of products or services (e.g. due to contractual commitments).
- Non-negativity constraints
If the objective of a business is to minimize cost, solving the linear programming problem without defining non-negativity constraints (e.g. the number of units of product X and Y should not be lower than zero) would suggest an optimum solution of producing negative infinite units of both products.
Non-negativity constraints simply ensure that the optimum solution always returns positive.