How a mathematical model is formulated?

We can visualize the formulation of mathematical problem by solving the following problem.

Problem: Jackson is a college student, who earns money by typing letter and menu scripts in his spare times. He has a given amount of spare time available in a given period and each page of a project utilizes a specified amount of that time. Jackson earns a given profit per page. There is practically an unlimited demand for his work. Jackson wants to earn as much money as possible.

Solution: since Jackson wants to earn as much as possible, his objective is to maximize profit. Total earnings are determined multiplying the profit per page and typing number of pages.

By Letting,

P= total profit,

R= Profit per page,

Q= number of Pages

Jackson’s objective is to maximize profit can be stated as follows:

P=QR---------------------- (i)

This type of mathematical expression is called objective function or goal of the problem.

Here, Total profit is restricted by Jackson’s available time. The demand for his work will equal to the time utilized per page multiplied by quantity of pages. This demand must not exceed his available time.

By Letting,

t= Time utilized per page,

T= Jackson’s available time.

The relationship can be described with the following mathematical expression:

tQ, ≤ T ------------------ (ii)

The symbol less than equal to (≤) indicates that the total time required must be less than or equal to the available time period. This type of expression is known as constraints.

Another restriction is that Jackson cannot type a negative number of pages, i.e.,

Q ≥ 0 ---------------- (iii)

The above mathematical expression states that the quantity of pages must be greater than or equal to zero (0). This type of expression is known as non-negative function.

Jackson’s problem is to determine the quantity of pages (Q) that will maximize his profit (P) per period from the typing service. This problem also recommended quantity must not require more than his available time.

By accumulating, the equation number (i), (ii), (iii), Jackson’s problem can be represented with the following mathematical model:

Maximize P= QR----------- (i)

Subject to,       tq ≤T--------------- (ii)

Q ≥ O ------------------- (iii)