# Operations Research – Part 1

###### Operations Research – An Overview

The subject of Operations Research is one of the most fascinating subjects world has ever known, or at least in that world through my eyes. It’s this very subject that helps financial decision makers take informed prudent decisions that later goes on to achieve the objectives of the enterprise.

The concept of Operations Research has been around for more than 100 years but the origin of the modern day Operations Research traces back at the Bawdsey Research Station in the UK in 1937. The idea was conceived and conceptualized as a means to analyze and improve the working of the UK’s early warning radar system, Chain Home (CH). The rise of Operation Research was during the II World war era where more than 1,000 men and women were engaged in Operations Research in the UK. United States also used the subject of OR for many of its Military programs including the Manhattan Project(USA’s 2nd World War Nuclear Program). Since then OR has expanded its reach and is currently used widely in the field of management where it’s classified under the larger umbrella of Management Sciences.

###### Understanding Optimality!

The subject of Operations research revolves around the concept of Optimality. In Fact the objective of Operation Research is to find the optimum values for a certain set of variables such that their combined net result is the most desirable.   Let us understand Optimality with an example.

A person is at a restaurant, where the buffet spread is quiet wide. Although he is at a position to have all of the dishes served, a plate full every dish can’t let him taste all the dishes. Therefore, there should be a certain quantity he shall consume so that he can taste all the dishes. But then, consuming the same quantity of all the dishes may not bring him joy. Therefore he should consume say x units of favorite dishes and y units of other dishes. In short he should optimize the consumption of dishes so that he tastes all the dishes and the joy from eating out is maximum.

###### Optimization under the Business Environment

Similar to our earlier example, in our real life business environment we need to optimize the values of many parameters, so that the joint effect of these parameters either decreases cost or increases revenue. Let us understand this very concept with the help of an example.

A company makes 2 Products A and B. A unit of A requires 1 unit of Resource R1 and 3 units of Resource R2. A unit of B requires 1 unit of Resource R1 and 2 units of Resource R2. A unit of A pays out a profit of Rs. 6 a unit and B pays off Rs. 5 a unit. 5 units of R1 and 12 units of R2 is available to the company. How much number of units of A and B should the company make so that its profit is maximized?

In the above example, the company needs to optimize the quantities of A and B produced so that its profits are maximized, and it does not strain the company of its resource restrictions.

###### Formulating an OR problem

The first stage to solving an OR problem is to express it as a Mathematical model. Let us try and express the above example as a set of Mathematical equations.
Let X_1be the units of Product A made and X_2 be the units of Product B made.
Here, the company wants to maximize its profit, i.e. the net profit from the production and sale of both the products should be the maximum. We know the per unit profit from each product. Therefore, total profit Z from sale of X_1units of A and X_2units of B is represented by the formulate:

$Z\; =\; 6X_1\; +\; 5X_2\;$

Our choice of $X_1\;$ and $X_2\;$ is limited by the resource constraints for resources R1 and R2. Product A and Product B needs equal no. of units of resource R1 and the maximum units of R1 available is 5 units. Representing this as an equation we get:

$X_1\; +\; X_2\; \leq\; 5\;$ (Meaning the sum of $X_1$ and $X_2$ can go up to a maximum value of 5)

Similarly, with the next resource, we have:
$3 \times X_1 + 2 \times X_2 \leq 12$

We also know that X_1 and X_2 can only take positive values and 0. –ve values for $X_1$ and $X_2$ is absurd. We can represent this as:
$X_1 ,X_2\geq0$

Summing it all up, we need to solve for optimal quantities of $X_1$ and $X_2$, such that:
The value of Z under the equation $Z = 6X_1 + 5X_2$ is maximized, subject to the constraints:

$X_1+ X_2 \leq 5$

$3 \times X_1 + 2 \times X_2\leq12$

And,
$X_1 ,X_2\geq0$

###### Understanding the parts of an OR problem

We can write our earlier example as :

 Maximize: $Z = 6X_1 + 5X_2$ Objective Function Subject to: $X_1 + X_2 \leq 5$ Constraints Subject to: $3X_1 + 2X_2\leq 12$ Constraints And, $X_1 ,X_2 \geq 0$

$X_1 \& X_2$ are called Decision Variables. The equation that represents the variable Z, being the profit company intends to maximize is called the “Objective Function” of the formulation. Equations that represents the resource constraints entity faces are called “Constraints”.

###### Understanding Linearity from our example:

From our Mathematical representation of the example problem, its quiet clear that all equations are liner i.e. of the $y = mx + c$ type. Neither Objective function, nor the constrains violate linearity property.

Formulating and solving for the decision variables, from an OR problem that can be represented as a set of Linear Equations, forming the objective function and constrains are the basic contents of the topic Linear Programming.