Operations Research

 Operations Research

Operation Research is also known as-

i.                    Quantitative techniques for decision making.

ii.                  Management Science.

iii.                Quantitative techniques.

iv.                Operation management.

Meaning of Operation Research:

Before coming to the answer, we should know – what is Operation? And what is Research?

Operation: An operation is a set of acts required for the accomplishment of some desired outcomes or objectives. 

Research: Research is a systematic quest to discover the hidden truth.

Operation Research is a systematic and scientific method of providing executive departments with a quantitative basis for decision regarding the operation under their control.

  --Morse and Kimball.

Operation research is a scientific approach to problem solving for executive management.

 --H. M. Wagner

On the other hand, Operation research is the application of scientific method by interdisciplinary teams to problem involving the control of organized systems so as to provide solutions which best to serve the purpose of the organization as a whole.

So we can say that operation research or quantitative technique may be defined as those techniques which provide the decision makers with a systematic and powerful means of analysis and help based on quantitative data in exploring policies for achieving predetermined goals.

Important Operation Research Techniques

(Tools of Operation Research)

Followings are some important operations research techniques:

1.      Linear Programming:

These techniques are used in finding a solution for optimizing a given objective, such as profit maximization or cost minimization under certain constraints. Linear Programming techniques solve product mix and distribution problems of business and industry. It is a technique used to allocate scare resources in an optimum manner in problems of scheduling product mix and so on.

2.      Queuing theory:

Waiting line or queuing line deals with mathematical study of queues. The queues are formed whenever the current demand for service exceeds the current capacity to provide that service. Waiting line technique concerned itself with the random arrival of customers at a service station where the facility is limited. Mainly with the help of it we can find the optimum capacity to be installed which will lead to a short of economic balance between cost of service and cost of waiting.

3.      Game theory:

Game theory is used to determine the optimum strategy in competitive situation. Simply as possible competitive situation is that of two persons are involved, one person wins exactly and other person losses.

4.      Inventory Control Planning:

Inventory control planning aims at optimizing inventory levels. Inventory may be defined as the useful idle resource which has economic value, such as- raw material, spare parts, finished products etc. Mainly it helps to get answer of two questions- How much to buy? And when to buy?

5.      Decision Theory:

Decision theory concerned with making sound decision under conditions of certainty, risk and uncertainty. From the various alternatives decision theory helps to select the best one.

6.      Network Analysis:

Network analysis involves the determination of an optimum sequence of performing certain operations concerning some jobs. In order to minimize the overall time and/ or cost, it uses various models-

         
         i.        Programming evaluation and review techniques (PERT);
         ii.     Critical path method (CPM);
         iii.    Gantt Chart etc.

What is Model?

A model is a simplest representation of real objects or situations.

Here, the representation includes only essential and relevant features. For example: A scale model rail road is a physical replica of general appearance and the operating characteristics of the real of the real things. There are 3 types of model. These are-

1.      Iconic Model;

2.      Analog model; and

3.      Mathematical model.

1.      Iconic Model: A physical replica of the real situation or object is called iconic model. For example, Toys, Photograph of rail road etc.

2.      Analog model: A physical form that doesn’t look like the real object or situation is called analog model. For example, An organization’s chart, Graphical representation etc.

3.      Mathematical Model: A system of symbols and mathematical expression that represents the real situation is called mathematical model. For example,

P=QR, Where,

P= Total Profit,

Q= Quantity Produced

R= Profit per Unit.

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)

 

Historical background of Operation Research

No science has been even been born on a single day. Operation research is not exception of it. Its root is as old as science and society. Those are roots of operation research extend to early 1800’s. But completely disclosed in 1885 When F. W. Taylor emphasized the application of scientific analysis to the method of production.

After then, in 1914, F. W. lanchester used mathematical equation in the field of forecasting outcomes of military battles. In 1917, A. K. Erlong used queuing theory in the field of production of waiting time for callers using an automatic telephone exchange. In 1924, W. Shewhard used the theory of probability and statistical inference in case of production quality control charge and in 1930 H. C. Levinson used mathematical expression in the field of marketing relationship.

The name operation research was evolved in the year 1940. During Second World War, a team of scientist (Blackett’s Circus) in UK applied scientist method to study military operations to win the world and the techniques thus developed was named as operation research.

Later on, the economic crisis of UK required radical improvement. This resulted in an industrial revolution and the operation research techniques so far developed in defense problems to provide a more vigorous and scientific approach to the problems. After a decade of second world war, operation research techniques was rapidly developed in the field of industrial, academic and government organizations.

After the Second World War, due to high development technology, people used machinery instead of manpower as controller. The new revolution began around 1970’s when electric computers were available in market.

In business sector, from 1950, the operation research techniques were used highly. Not only these, even in America it was introduced as an academic subject. Since then, this subject has been gaining ever increasing importance to the students of mathematics, statistics, commerce, economics etc.

Nowadays, we cannot think any business without operation research techniques.

Battle’s Circus

Blackett’s Circus was one of the most publicized operation research groups on Great Britain. During Second World War Professor P. M. S Blackett of the University of Mancherter was the director of this group. The group was formed by eleven members. They were-

1. 3 Physiologists.
2. 2 Mathematical physicists.
3. 1 Astro physiologist.
4. 1 Army officer.
5. 1 Surveyor.
6. 1 General physicist.
7. 2 Mathematicians.

The group was formed for the following reasons:

1. For increasing the early coming radar systems;
2. In empty aircraft gunnery;
3. In empty submarine war-fare;
4. In civilian defense;
5. For determining convoy size; and
6. For conducting bombing raids upon Germany.

Management Science Process

Quantitative decision making is not a substitute for competent management. Rather it is methodology that can significantly improve the executive decision making. So, every business students should learn about it. Followings are the graphical representation and description of management science process:

Management Science Process

1. Define the problem:

The First step of the management science process is to define the problem. Quantitative decision making approach is problem oriented. So we should define the managerial problem clearly and concisely. The problem must be stated precisely and that should be suitable for analysis. Many operation research studies were failed simply because the problem was poorly defined. So at first we should emphasize more on it.

2. Formulate a quantitative model:

In the second step, we need to formulate a model. In formulating model, we should consider controllable and uncontrollable inputs. A model is a simplified representation of real objects or situations. The representation inputs only essential, relevant and important features.

3. Gather relevant quantitative data:

Organization would gather data from past accounting records, sales, financial, inventory, production and engineering records and reports. Published documents such as government statistical summaries may be important sources of data. Managers and operating personnel can provide information about markets, financial conditions, productivity and other factors that are unavailable elsewhere.

4. Analyze and solve the quantitative model:

In this step we have to analyze collected data and solve the quantitative model. In most cases, there is tremendous volume of available data and a considerable amount of time is required to collect and organize the information. Furthermore, data are usually not in form of suitable form for decision making purposes. More effort then, is necessary for processing and analyzing the data. As a result many organizations have designed and implemented formal system for collecting, analyzing and reporting relevant and timely information. Such a structure id referred to MIS (management Information System).

5. Implement the solution:

It is the last phase of decision making. The quantitative technique analysis process is not complete, until the model’s solution information is reported to the decision maker and results are implemented. Such data constitute only one of the inputs considered by the manager when a final decision is being made. So it shows the success or failure of the process. For this reason it is called vital step among all.

Mathematical problem

Problem:

Nabila is a university student. She produces playing tools for 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)



Linear programming

Problem:

Zenith Inc. manufactures two types of kitchen utensils:

1.      Knives.

2.      Forks.

Both must be pressed and polished. The shop manager estimates that there will be a maximum 70 hours available next week in pressing machine center and 100 hours in polishing machine center. However each case of knives requires an estimated 12 minutes (.20 hour) of pressing and 30 minutes (.50 hour) of polishing while in case of forks requires 24 minutes (.40hour) of pressing and 15 minutes (.25 hour) of polishing. The company can sell as many knives as it produces at the prevailing market price of Tk 12 per case. Forks can be sold for Tk 9 per case. Cost of production per case knives is Tk 4 and forks Tk 3.

Zenith wants to determine how many cases of knives and forks the company should produce to maximize profit.

Solution (Equation type expression):

Zenith’s problem is to determine the quantity of knives and forks that will maximize profit and selected quantities cannot be used more than available pressing and polishing time.

Objective: To maximize profit.

Zenith’s total profit = The contribution from knives + the contribution from forks.

Since,

Each case of knives can be sold @ Tk 12

Each case of knives’ production cost@ Tk 4

So, contribution from each case of knives = Tk (12-4) = Tk 8

Since,

Each case of forks can be sold @ Tk 9

Each case of fork’s production cost@ Tk 3

So, contribution from each case of forks = Tk (9-3) = Tk 6

This Tk 8 and Tk 6 per case knives and per case forks contribution multiplied by number of cases gives total profit from knives and forks respectively.

By letting,

X= Number of knives zenith will produce in next week.

Y= Number of forks zenith will produce in next week.

Z= Zenith’s want of total profit.

Now we can represent Zenith’s total profit,
            Z= 8X + 6Y ------------- (I)

The company wants to choose the level of decision variables (X and Y) that maximizes total profit (Z). The objective can be expressed as,
             Maximize Z = 8X + 6Y ----------(II)

Restrictions:

Available pressing and polishing capacity will limit knives and forks Zenith can produce. Since each case of knives uses .20 hour of pressing time i.e., .20X is the total time required to press X cases knives, similarly .40Y is the total time required to press Y cases of forks.

Consequently, .20X + .40Y gives the total time required to press X cases of knives and Y cases of forks.

Zenith can select any product combination doesn’t require more than 70 hours available pressing time. The mathematical representation of this condition will be-
            .20X + .40Y ≤ 70 (hour) ----------- (III)

Another system constraint deals with polishing operation management of the business/ company knows that each case of knives uses .50 hour and each case of forks uses .25 hour polishing hour.

Since, there are only 100 hours of polishing time. So we can represent the above description as following:
             .50X + .20Y ≤ 100 (hour) ------------ (IV)

It is physically impossible for zenith to produce negative number of knives and forks. Therefore, management must ensure that decision variables X and Y have values greater than or equal to 0 (zero).

             Symbolically, X ≥ 0 and Y ≥ 0
             In abbreviated form X, Y ≥ 0 ---------- (V)

Complete Formulation:

By collecting objective function (II), system constraints (III) and (IV) and non-negativity condition (V), zenith’s management can be represented the machine shop problem with the following mathematical equation/ function:

               Maximize, Z = 8X + 6Y

Subject to,
.20X + .40Y ≤ 70 (System Constraints)
.50X + .25Y ≤ 100 (System Constraints)
X, Y ≥ 0 (Non-negative function)

Solution (Numerical Expression):

Let us, first consider the inequality into equation we have, 

.20X + .40Y = 70 ------------- (I)

And .50X + .25Y = 100 -------------- (II)

For the equation number (I), when X=0 then

.20 x 0 + .40Y = 70

Or, Y = 175             [X, Y= 0, 175]

When Y=0 then

.20X   + .40 x 0 = 70

Or, X = 350            [X, Y= 350, 0]

For the equation number (II), when X=0 then

.50 x 0 + .25Y = 100

Or, Y = 400             [X, Y= 0, 400]

When Y=0 then

.50X   + .25 x 0 = 70

Or, X = 200            [X, Y= 200, 0]

Now equation number (I) multiplied by 5 and equation number (II) multiplied by 2. Then deduct (II) from (I), we get-

X + 2Y = 350

X + .50Y = 200

--------------------

1.50Y = 150

So, Y = 100

Putting the value of Y, in equation (I) we get,

.20X + .40 x 100= 70

Or, X = 150

So, X, Y = 150, 100

Now all the values of X and y are ---

For equation Number (I)

When X = 0,   then    X, Y = 0, 175

When Y = 0    then X, Y = 350, 0

For equation Number (II)

When X = 0,   then    X, Y = 0, 400

When Y = 0    then X, Y = 200, 0

And X, Y = 150, 100

Putting the above values on graph, we get---

Click on the picture to see the original size.

 

By putting values of variables on X and Y axis and using shadow, we get the crossing point of two lines at (150, 100). So it would be the target point where zenith may get highest profit or lowest profit.

Now, Z = 8X + 6Y

 = 8 x 1500 + 6x 100

 = 1800

Competitive analysis:

Corner point	Total profit
0, 175	Z = (8 x 0) + (6 x 175)= 1050
0, 0	Z = (8 x 0) + (6 x 0)= 0
200, 0	Z = (8 x 200) + (6 x 0)= 1600
150, 100	Z = (8 x 150) + (6 x 100)= 1800

Comment:

Under the above solution, we suggest the production manager to produce 150 cases of knives and 100 cases of forks to maximize profit or minimize costs.


 

Problem 1( For Assignment): Rahim Factory manufactures two articles- share and towel. To manufacture share a certain machine has to be worked for 1.5 hours and in addition a craftsman has to work for two hours. To manufacture the towel the machine has to work for 2.5 hours, in addition the craftsman has to work for 1.5 hours. In a week, the factory can avail 80 hours of machine and 70 hours of craftsman. The profit on each share is Tk 50 that on each towel is TK 40.

If all the articles produced can be sold away, find how many of each kind i.e., share and towel should be produced to earn maximum profit per week. Formulate the model using linear programming model and solve it.

Problem 2 (For Assignment): Mrs Saleha has learnt from a nutrition book that her family needs at least 330 gm of protein and 45 mg of iron per day. These nutrients can be obtained from meat and vegetables. Each pound of meat costs on an average of $1.6 and contains average of 150 gm. and 15 mg of iron, while each pound of vegetable costs 50 cents ($1/2) and has 10 gm. of protein and 5 mg of iron. Mrs. Saleha wants to determine the quantities of food that meet the nutritional requirements at least costs.

Simplex Method

Simplex Method: A methodology designed to systematically solve large scale linear programming problems. This method is an algebraic approach based equality relationship. Yet linear programs typically involve inequality. To use the simplex method, the decision makers first must convert each inequalities restrictions into equality through adding slack variables of deducting surplus variables. Simplex method was first introduced by G.B Dantzig and his associates in 1947 in USA among the department of airforce.

Slack Variables: In a linear programming model when a structural constraint is in the “less than or equal to” form a non-negative variable is added to the left hand side to convert inequality into equation, then these variables are called as slack variables.

For example: 3X + 5Y ≤ 100

Or, 3X + S₁ + 5Y + S₂ = 100.

Here S₁ and S₂ are slack variables.

Surplus variables: In a linear programming model when a structural constraint is in the “Greater  than or equal to” form a non-negative variable is deducted to the left hand side to convert inequality into equation, then these variables are called as slack variables.

For example: 3X + 5Y ≤ 100

Or, 3X - S₁ + 5Y - S₂ = 100.

Here S₁ and S₂ are Surplus variables.

Key column: In a simplex table the column which contains the most positive number in the objective row is called Key column or pivot column.

Key Row: In a simplex table the raw which contains the lowest displacement ratio is called key row or pivot row.

Key number: In a simplex table the number which lies at the intersection of the key row and key column is called key number.

Problem 3 (Slack Variables): Dell Inc. manufactures two sizes of baseball- little league and major league. The company earns a profit of $2 of little league baseball and $3 per of major league baseball. Each product is assembled and packaged. There is a maximum of 1,800 hours available between the assembling and packaging department during a given time period. It takes 9 minutes to assemble a box of little league baseball and 15 minutes to assemble a box of major league baseball. A box of little league requires eleven minutes packaging whereas 5 minutes to major league. Dell Inc. seeks 2 combination of little league baseball and major league baseball that will maximize total profit within the available assembling and packaging time.

Requirement: Formulate this problem with simplex model and solve it.