Mathematical Problem

Problem:

Nabila is a university student. She produces playing tools for children with clay. She would like to earn as much as possible. She uses her spare time to produce these tools. She has a specified amount of time for this work. Nabila earns a given amount of profit per product. There is practically an unlimited demand of her work. Show the above description with mathematical expression.

Solution:

Since Nabila wants to earn as much as possible, her objective is to maximize profit. Total earnings are determined multiplying the profit per playing tool and number of playing tools produced.

By Letting,

P= total profit,

R= Profit per playing tool.

Q= Number of playing tools produced.

Nabil’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 Nabila’s available time. The demand for her work will equal to the time utilized per playing tool multiplied by quantity of playing tools. This demand must not exceed her available time.

By Letting,

t= Time utilized per playing tool,

T= Nabila’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 Nabila cannot produce a negative number playing tool, 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.

Nabila’s problem is to determine the quantity of pages (Q) that will maximize her profit (P) per period from the production 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)

Role of Quantitative Techniques (QT) in Industry and Business.

Quantitative technique especially operation research technique has gained increasing importance since World War II in the technology of business administration. This technique greatly helps in tackling the integrated and complex problems of the modern business and industry. Quantitative techniques for decision making are infecting examples of the use of scientific management. However the roles of quantitative techniques are explained below.

1.     They provide a tool for scientific analysis:

These techniques provide the executives with a more precise description of a cause. They replace the intuitive and subjective approach. The use of these techniques has transformed the conventional techniques of operational and investment problems in business and industry. Quantitative techniques thus encourage and enforce disciplined thinking about organization’s problems.

2.     They provide solutions for various business problems:

The quantitative techniques are being used in the field of production, procurement, marketing, and such other fields. Problems like- how best can the manager and executives allotted the available resources to various departments. So that in a given time the profit are maximized or costs are minimized planning decision business and industry largely governed by the picture of anticipated demands and quantitative techniques help to forecast about demand. So, quantitative techniques are very important.

3.     They enable proper deployment of resources:

Quantitative techniques render valuable help in proper deployment of resources. For example- Programming- Evaluation- Review- Technique (PERT) requires various related data to identify critical path. In the same way when it require supply data and determine the probability of completing an event or project itself by specified data.

4.     They help in minimizing waiting and servicing costs:

The waiting line and/ or queuing theory help the management in minimizing the total waiting and servicing costs. This technique also analyses the feasibility of adding facilities and thereby helps the business people to take the correct and profitable decision.

5.     They assist in choosing an optimum strategy:

Game theory is specially used to determine the optimum strategy in a competitive situation and enable the businessman to maximize profits or minimize losses by adopting optimum strategy.

6.     They help in resources allocation:

They render great help in optimum resource allocation by the help of linear programming. Linear programming techniques are popularly used by modern management in resource allocation and selecting production mix.

7.     They facilitates the process of decision making:

The decision theory enables the businessman to select the best courses of action when information is given in probabilistic form.

8.     Inventory problem:

These techniques enable the management to decide when to buy and how much to buy.

9.     Statistical techniques:

Statistical techniques are also of great help to business man in more than one way. Some of the statistical techniques are considerable importance in sales forecasting whereas other facilitates from comparison between the various phenomena. In statistics there are various techniques such as quality control technique, sampling theory to decision making, various significant tests to judge the reliability etc. Similarly regression analysis, variance analysis, time series analysis, index number etc are useful tools of statistical analysis from where business get a great help and right decision is being taken.

Limitation of Quantitative Techniques (QT)