Many problems in business, science and engineering involve two or more simultane

Question

Many problems in business, science and engineering involve two or more simultaneous equations, we call these equations together a system of equations. One very common application in business is the break even analysis. If a system of equations contains a cost equation, and a revenue equation, the solution for the system represents the break even point. Think about other common business applications that can be explored using system of equations. Provide at least one business concept, and explain how it can be solved using systems of equations

Explanation / Answer

Linear Programming:

Linear Programming (LP) is a mathematical procedure for determining optimal allocation of scarce resources. LP is a procedure that has found practical application in almost all facets of business, from advertising to production planning. Transportation, distribution, and aggregate production planning problems are the most typical objects of LP analysis. In the petroleum industry, for example a data processing manager at a large oil company recently estimated that from 5 to 10 percent of the firm's computer time was devoted to the processing of LP and LP-like models.

Linear programming deals with a class of programming problems where both the objective function to be optimized is linear and all relations among the variables corresponding to resources are linear. Rarely has a new mathematical technique found such a wide range of practical business, commerce, and industrial applications and simultaneously received so thorough a theoretical development, in such a short period of time. Today, this theory is being successfully applied to problems of capital budgeting, design of diets, conservation of resources, games of strategy, economic growth prediction, and transportation systems. In very recent times, linear programming theory has also helped resolve and unify many outstanding applications.

Any LP problem consists of an objective function and a set of constraints. In most cases, constraints come from the environment in which you work to achieve your objective. When you want to achieve the desirable objective, you will realize that the environment is setting some constraints (i.e., the difficulties, restrictions) in fulfilling your desire or objective. This is why religions such as Buddhism, among others, prescribe living an abstemious life. No desire, no pain. Can you take this advice with respect to your business objective?

What is a function: A function is a thing that does something. For example, a coffee grinding machine is a function that transform the coffee beans into powder. The (objective) function maps and translates the input domain (called the feasible region) into output range, with the two end-values called the maximum and the minimum values.

When you formulate a decision-making problem as a linear program, you must check the following conditions:

For example, the following problem is not an LP: Max X, subject to X < 1. This very simple problem has no solution.

As always, one must be careful in categorizing an optimization problem as an LP problem. Here is a question for you. Is the following problem an LP problem?

Max X2
subject to:
X1 + X2 < 0
X12 - 4 < 0

Although the second constraint looks "as if" it is a nonlinear constraint, this constraint can equivalently be written as:
X1 > -2, and X2 < 2.
Therefore, the above problem is indeed an LP problem.

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