The solution to a mathematical equation can be feasible or infeasible. There can be mathematical models with boundaries as well. Such models are termed as optimization models where the solution resides within a set of values. Usually such models are expressed with a set of constraints. For example, the classical functions of pricing the supply and demand for products, both these functions together create a fixed value for the price.

Here is a sample mathematical model

Objective: Maximize Profits from selling two products P1 and P2 at Price $3 and $4 respectively.

Constraint: Labor hours are limited to 40 per week. Product 1 takes 2 hours to produce and product 2 takes 3 hours to produce. (Per unit)

Maximize 3P1 + 4P2 — (1)

SUBJECT TO

2P1 + 3P2 < 40 — (2)

If you carefully notice the system of equations above, the first equation constantly increases for any value of P1 and P2. But the increase is restricted by equation number 2 which enforces a boundary on the system. Hence the solution set returns feasible values.

Equations can be both deterministic as well as stochastic. Stochastic systems are systems that do not have fixed values such as USD 3 as cost of product or labor hours as 2 hours per product. The expected values can be specified as a probability distribution. An Example of a probability distribution is the arrival rate of automobiles in a junction. One cannot determine the exact rate as the source would be dependant upon a lot of factors.

Simulation is an extended technique of analyzing variations of input and output using expected values for a large number of trials. Many contrasting system conditions can be specified and the simulation can be run for a large number of trials.

Math models are common place and are used to describe physical phenomena, astronomical phenomena and population growth. They are also used in production planning, manufacturing etc. In synopsis a mathematical model can create unbounded values or bounded values. A system with boundaries can be used to study extreme objectives such as profit maximization, time minimization etc.,