Operation Research

According to Churchman, Aackoff, and Aruoff, Operations research is defined as: the application of scientific methods, techniques and tools to operation of a system with optimum solutions to the problems, where 'optimum' refers to the best possible alternative.

Types of Operation Research

The OR models are broadly classified into the following types:

Physical models

These models include all forms of diagrams, graphs, and charts. There are two types of physical models.

Iconic models are primarily images of objects or systems, represented on a smaller scale.

Analogue models are small physical systems having characteristics similar to the objects they represent, such as toys.

Mathematical or symbolic models

These models employ a set of mathematical symbols to represent the decision variable of the system. Some e.g. of mathematical models are allocation, sequencing, and replacement models.

By nature of environment

These models can be further classified as follows:

Deterministic models - These are the models in which everything is defined and the results are certain, such as an EOQ model.

Probabilistic models - These are the models in which the input and output variables follow a defined probability distribution, such as the games theory.

By the extent of generality

These models can be further classified as follows:

General models These are the models which you can apply to any problem. For example, linear programming.

Specific models - These are the models that you can apply only under specific conditions. For example, you can use the sales response curve or equation in the marketing function.

Phases of Operations Research

The scientific method in OR study generally involves these three phases;

Judgment phase

This phase includes the following activities:

Determination of the operations

Establishment of objectives and values related to the operations

Determination of suitable measures of effectiveness

Formulation of problems relative to the objectives

Research phase

This phase utilises the following methodologies:

Operation and data collection.

Formulation of hypothesis and model

Observation and experimentation to test the hypothesis on the basis of additional data

Analysis of the available information and verification of the hypothesis using pre-established measure of effectiveness

Prediction of various results and consideration of alternative methods

Action phase

This phase involves making recommendations for the decision process. The recommendations can be made by those who identify and present the problem or by anyone who influences the operation in which the problem has occurred.

Linear Programming Problem

Linear Programming (LP) is a mathematical technique designed to help managers in their planning and decision-making. It is usually used in an organisation that is trying to make the most effective use of its resources. Resources typically include machinery, manpower, money, time, warehouse space, and raw materials.

Graphical Method of solving Linear Programming Problem

The method of solving an LPP on the basis of the above analysis is known as the graphical method. The working rule for the method is as follows.

Step 1: Formulate the problem in terms of a series of mathematical equations representing objective function and constraints of LPP.

Step 2: Plot each of the constraints equation graphically. Replace the inequality constraint equation to form a linear equation. Plot the equations on the planar graph with each axis representing respective variables.

Step 3: Identify the convex polygon region relevant to the problem. The area which satisfies all the constraints simultaneously will be the feasible region. This is determined by the inequality constraints.

Step 4: Determine the vertices of the polygon and find the values of the given objective function Z at each of these vertices. Identify the greatest and the least of these values. These are respectively the maximum and minimum value of Z.

Step 5: Identify the values of (x1, x2) which correspond to the desired extreme value of Z. This is an optimal solution of the problem.

Degeneracy in transportation problems

A basic solution to an m-origin, n destination transportation problem can have at the most m+n-1 positive basic variables (non-zero), otherwise the basic solution degenerates.

The degeneracy can develop in two ways:

Case 1 - The degeneracy develops while determining an initial assignment

To resolve degeneracy, you must augment the positive variables by as many zero-valued variables as is necessary to complete the required m + n 1 basic variable. These zero-valued variables are selected in such a manner that the resulting m + n 1 variable constitutes a basic solution. The selected zero valued variables are designated by allocating an extremely small positive value to each one of them. The s are kept in the transportation table until temporary degeneracy is removed or until the optimum solution is attained, whichever occurs first. At that point, we set each = 0.

Case 2 - The degeneracy develops at the iteration stage. This happens when the selection of the entering variable results in the simultaneous drive to zero of two or more current basic variables.

To resolve degeneracy, the positive variables are augmented by as many zero-valued variables as it is necessary to complete m+n-1 basic variables. These zero-valued variables are selected from among those current basic variables, which are simultaneously driven to zero. The rest of the procedure is same as in case 1.

MODI method of finding solution through optimality test

A feasible solution has to be found always. Rather than determining a first approximation by a direct application of the simplex method, it is more efficient to work with the transportation table. The transportation algorithm is the simplex method specialized to the format of table involving the following steps:

I. Finding an initial basic feasible solution

II. Testing the solution for optimality

III. Improving the solution, when it is not optimal

IV. Repeating steps (ii) and (iii) until the optimal solution is obtained

The solution to transportation problem is obtained in two stages

In the first stage, we find the basic feasible solution using any of the following methods:

North-west corner rule

Matrix minima method or least cost method

Vogels approximation method

In the second stage, we test the basic feasible solution for its optimality by MODI method.

After evaluating an initial basic feasible solution to a transportation problem, the next question is how to get the optimum solution. The basic techniques are illustrated as follows:

1. Determine the net evaluations for the nonbasic variables (empty cells)

2. Determine the entering variable

3. Determine the leaving variable

4. Compute a better basic feasible solution

5. Repeat steps (1) to (4) until an optimum solution has been obtained.

Hungarian Method Algorithm

The following steps are adopted to solve an Assignment Problem using the Hungarian method algorithm.

Step 1: Prepare row ruled matrix by selecting the minimum values for each row and subtract it from the other elements of the row.

Step 2: Prepare column-reduced matrix by subtracting minimum value of the column from the other values of that column.

Step 3: Assign zero row-wise if there is only one zero in the row and cross (X) or cancel other zeros in that column.

Step 4: Assign column wise if there is only one zero in that column and cross other zeros in that row.

Step 5: Repeat steps 3 & 4 till all zeros are either assigned or crossed. If the no. of assignments is equal to no. of rows present, you have arrived at an optimal solution, if not, proceed to step 6.

Step 6: Mark () the unassigned rows. Look for crossed zero in that row. Mark the column containing the crossed zero. Look for assigned zero in that column. Mark the row containing assigned zero. Repeat this process till all the makings are done.

Step 7: Draw a straight line through unmarked rows and marked column. The no. of straight line drawn will be equal to the number of assignments made.

Step 8: Examine the uncovered elements. Select the minimum.

Subtract it from the uncovered elements.

Add it at the point of intersection of lines.

Leave the rest as is.

Prepare a new table.

Step 9: Repeat steps 3 to 7 till optimum assignment is obtained.

Step 10: Repeat steps 5 to 7 till number of allocations = number of rows.

The assignment algorithm applies the concept of opportunity costs. The cost of any kind of action or decision consists of the opportunities that are sacrificed in taking that action.

Monte Carlo Simulation

The technique of Monte-Carlo involves the selection of random observations within the simulation model.

The principle of this technique is replacement of actual statistical universe by another universe described by some assumed probability distribution and then sampling from this theoretical population by means of random numbers.

The Monte-Carlo method is a simulation technique in which statistical distribution functions are created by using a series of random numbers. This approach has the ability to develop many months or years of data in a matter of few minutes on a digital computer.

The method is generally used to solve the problems that cannot be adequately represented by mathematical models or where solution of the model is not possible by analytical method.

The steps involved in Monte-Carlo simulation procedure are:

Domiance

In a rectangular game, the pay-off matrix of player A is pay-off in one specific row (rth row) exceeding the corresponding pay-off in another specific row (sth row).

This means that whatever course of action is adopted by player B, for A, the course of action Ar yields greater gains than the course of action As.

Therefore, Ar is a better strategy than As irrespective of Bs strategy. Hence, you can say that Ar dominates As.

Alternatively, if each pay-off in a specific column (pth column) is less than the corresponding pay-off in another specific column (qth column), it means strategy Bp offers minor loss than strategy Bq irrespective of As strategy. Hence, you can say that Bp dominates Bq.

Therefore, you can say that:

a) In the pay-off matrix, if each pay-off in rth row is greater than (or equal to) the corresponding pay-off in the sth row, Ar dominates As.

b) In the pay-off matrix, if each pay-off in pth column is less than (or equal to) the corresponding pay-off in the qth column, Bp dominates Bq.

At times, a convex combination of two or more courses of action may dominate another course of action.

Constituents of a Queuing System

The constituents of a queuing system include arrival pattern, service facility and queue discipline.

Arrival pattern: It is the average rate at which the customers arrive.

Service facility: Examining the number of customers served at a time and the statistical pattern of time taken for service at the service facility.

Queue discipline: The common method of choosing a customer for service amongst those waiting for service is First Come First Serve.

Difference between PERT and CPM

PERT

CPM

PERT was developed in connection with a Research and Development (R&D) work.

CPM was developed in connection with a construction project, which consisted of routine tasks.

It is an event-oriented network as in the analysis of a network, emphasis is given on the important stages of completion of a task.

It is suitable for establishing a trade-off for optimum balancing between schedule time and cost of the project.

It is normally used for projects involving activities of non-repetitive nature in which time estimates are uncertain.

It is used for projects involving activities of repetitive nature.

Research Phase

Action Phase

Judgement Phase

Step 1

Define the problem

Step 2

Construct the approximate model

Step 3

Prepare the model for Experimentation

Step 4

Step 5

Using Step 1 to 3, experiment with the model

Step 7

Summarise and examine the results abtained in step 4

Step 6

Evaluate the results of the stimulation

Formulate proposals for advice to management

Models

Physical

Iconic

Analog

Mathematical

Models by nature of environment

Models by extent of generality

Deterministic

Probabilistic

General

Specific