Linear programming

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Linear programming is a method by which an objective function is optimized, either by maximizing or minimizing, where the variables are raised to the power of 1. This, taking into account different restrictions given.

Linear programming, then, is a process by which a linear function will be maximized. That is, an equation of the first degree, where the variables are raised to the power of 1.

We must remember that this type of equation is a mathematical equality that can have one or more unknowns. Thus, it has the following basic form, where a and b are the constants, while x and y are the variables.

ax + b = y

Now, through linear programming, this function could be optimized, finding the maximum or minimum value of y. This, taking into account that x is subject to certain restrictions. Perhaps it is greater than 0 and less than 20, for example.

Elements of linear programming

The main elements of linear programming are the following:

  • Objective function: It is that function that is optimized, either by maximizing or minimizing its result.
  • Restrictions: These are the conditions that must be met when optimizing the objective function. It can be algebraic equations or inequalities.

Linear programming exercise

Let's see, to finish, a linear programming exercise.

Suppose that we have the following function, which expresses the benefit that a person obtains when acquiring certain products, being the utility U and the products, x and y.

U = 4x + 7y

Likewise, the individual faces a budgetary restriction, with his budget being 70 monetary units (cu), and the prices of products x and y are 6 and 14 cu, respectively.

70≥6x + 14y

In this case, if we graph the functions, we will realize that the greatest utility occurs when the person buys only the good x (11 units), thus having a utility of 44 (4 × 11 + 0x7). Instead, if you buy 9 units of x and 1 of y, for example, your profit would be 42 (9 × 4 + 1 × 7). Meanwhile, if you spend everything on good y, you could only buy 5, which would give you a profit of 35 (4 × 0 + 5 × 7).

It is worth mentioning that, in the graph above, the gray line is one of the indifference curves.

At this point, we must also remember that the goods x and y can take only integer values.

The case presented may be that of two goods that satisfy the same need, for example, hunger.However, one of them, good x, while offering a little less utility, is less expensive, priced at CU6, while good y costs more than double CU14.

To maximize the objective function, you can use online tools that allow you to enter the linear equation and the respective restrictions, automatically giving the result.

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