Operations research (OR) is an analytical method of problem-solving and decision-making that is useful in the management of organizations. In operations research, problems are broken down into basic components and then solved in defined steps by mathematical analysis. Analytical methods used in OR include mathematical logic, simulation, network analysis, queuing theory , and game theory .The process can be broadly broken down into three steps. 1. A set of potential solutions to a problem is developed. (This set may be large.) 2. The alternatives derived in the first step are analyzed and reduced to a small set of solutions most likely to prove workable. 3. The alternatives derived in the second step are subjected to simulated implementation and, if possible, tested out in real-world situations. In this final step, psychology and management science often play important roles

1. OPERATIONS RESEARCH
VISHNURAJ C.R
ANJANA K.S
T4 MBA
RBS

2. Meaning of OR
OR is the systematic and method oriented study of the basic structure, functions and
relationships of an organization. OR provides quantitative measure for decision making.it
provides techniques for taking wise decisions and arriving at optimal solutions.it is a study of
optimization techniques.
Definition of OR
According to P.H Morse,” OR is a scientific method of providing executive departments with
a quantitative basis for decisions under their control.”
Features/Characteristics/Nature of OR
✓ Systematic orientation
✓ Interdisciplinary team approach
✓ Scientific approach
✓ Decision making
✓ Optimization objective
✓ Mathematical models and quantitative solution
✓ Bad answers to the problem
✓ Use of computers
Importance of OR
✓ It provides a tool for scientific analyzer
✓ It provides solutions for various business problems
✓ It enables proper development of resources
✓ It helps in minimizing waiting and servicing costs
✓ It assists in choosing an optimum strategy
✓ It facilitates the process of decision making
Transportation problem
These are particular class of allocation problems. The main objective of this problem is to
transport various amounts of a single homogeneous commodity that are stored at several origins
to a number of destinations.
Feasible solution
A feasible solution to a transportation problem is a set of non- negative individual allocation
which satisfy the row and column sum restrictions.
Basic feasible solution
A feasible solution to a m*n transportation problem is said to be a basic feasible solution. If
the total no: of allocations is exactly equal to m+n-1.
Optimal solution

3. A feasible solution is said to be optimal, if it minimizes to total transportation cost.
Initial basic feasible solution
Initial feasible solutions are those which satisfy the required condition that is the allocation
made in every row taken together is equal to the availability shown that row similarly for each
column. Total allocation should be equal to the requirement in that column. Initial solution can
be optimal by:
1.North West corner rule 2. Least/Lowest Cost entry method 3. Vogel’s approximation
method
Degeneracy in transportation problem
Degeneracy occurs whenever the no: of individual allocations is less than m+n-1[m-row, n-
column]. here we allocate delta to one or more empty cells, so that the total no: of allocations
is m+n-1
Unbalanced transportation problem
A transportation problem is said to be an unbalanced transportation problem, if the sum of all
available amounts is not equal to the sum of all requirements.
Maximization in transportation problem
A transportation problem in which the objective is to maximize, can be solved by converting
the given maximization problem into minimization problem. For this select highest value and
subtract all other values from this highest value, then the given problem becomes minimization
problem.
Assignment problem
It is a special case of transportation problem. Its objective is to assign, the no: of origins to the
equal no: of destinations or tasks at a minimum cost.
Maximization in assignment problems
The objective of some assignment problem is to maximize the effectiveness like maximizing
profit, such problems can be converted into minimization problems, for this, subtract all the
elements from the highest element of that matrix.
Unbalanced assignment problems
An assignment problem is said to be unbalanced problem, whenever a no: of task or job is not
equal to no: of facilities or persons. Here, we add dummy rows or columns to the given matrix
to make it a square matrix. The cost in these dummy rows and columns are taken to be zero.
Prohibited/Restricted assignments
In some assignment problems, it may not be possible to assign a particular task to a particular
facility due to space or tasks or other restrictions.in such situations we can assign a very big

4. cost to the corresponding cells, so that it will be automatically excluded in minimizing process
of assignment.
Diff: b/w Transportation and Assignment problem
o Transportation problem is one of the sub classes of linear programming problems,
where assignment problem is a special case of transportation problem
o The main objective of transportation problem is to transport various quantum of a
commodity that are initially stored at various origins to different destinations in such a
way that the transportation cost is minimum. The objective of assignment problem is to
assign a no: of origins to the equal no: of destinations.
o In T.P no: of origins and no: of destinations need not be equal, so that the no: of rows
and no: of columns need not be equal.in assignment problems the no: of persons and
no: of tasks are equal so that the no: of rows and no: of columns are equal.
o TP are said to be said to be unbalanced if the total demand and total supply are not equal
while A.P are unbalanced when no: of rows are not equal to no: of columns.
o In T.P, a positive quantity is allocated from a source(origin)to a destination.in A.P a
source(job)is assigned to a destination (a man)
Queuing Theory (Waiting line theory)
Queuing theory is a quantitative technique, which consist in constructing mathematical models
of various types of queuing systems. These models can be used for making predictions about
how the system can adjust with the demands on it. Queuing theory deals with analysis of queues
and queuing behavior. A queue is formed when the demand for a service exceeds the capacity
to provide that service.
Objectives/Aim of queuing theory
Customers wait for service. The time thus lose by them is expensive. The cost associated with
waiting in queue are known as waiting time cost. Similarly, if there are not customers service
station will be ideal. Cost associated with service or the facilities are known as “Service cost”.
Its main objective is to achieve an economic balance between these 2 types of cost.ie; study of
theory helps in minimizing the total waiting and service cost.
Characteristics or Elements of queuing system
*Input process/Arrivals –The input describes the way in which the customers arrive and join
the system. Arrivals may occur at regular or random fashion.
*Service mechanism-it concerns with the service time and service facilities. Service can either
be fixed or distributed in accordance with some probability distribution.
*Queue discipline-if any of the service facilities is free, the incoming customer is taken into
service immediately. If, however all the services facilities are busy, the customers in the queue
may be handled in a no: of ways as service becomes free.

5. *Output process-after availing the service, the customer departs from the system.
Game theory
Game theory is a theory of conflict and it is a mathematical theory which deals with competitive
situations.it is a type of decision theory which is concerned with the decision making in
situations where two or more rational opponents are involved under conditions of competition
and conflicting interest.
Game is defined as an activity b/w two or more person’s accordance with a set of rules at the
end of which each person receives some benefit or satisfaction or suffers loss.
Strategy
The strategy of a player is the predetermined rule by which a player decides his course of action
during the gain.ie; a strategy for a given player is a set of rules or programs that specify which
of the available courses of action, he should select at each play. There are 2 types: pure strategy
& mixed strategy.
Pure strategy-a pure strategy decision always to choose a particular course of action.it is a
predetermined course of action. The player knows it in advance.
Mixed strategy-a player is said to adopt in a single strategy when he does not adopt in a single
strategy all the time. But would play different strategies each at a certain time. A mixed strategy
is a decision to choose a course of action for each action for each play in accordance with
some particular probability distribution.
Saddle point
A saddle point of a payoff matrix is that positioning in the payoff matrix where the maximin
coincides with the minimax. Payoff at the saddle point is the value of the game.in a game
having a saddle point optimum strategy of maximizing player is always to choose the row
containing saddle point and for minimizing player to choose the column containing saddle
point.
Zero sum game
In a game, if the algebraic sum of the outcomes of all the players together is zero, the game is
called zero sum game, otherwise it is called non-zero sum game.
Two person zero game (rectangular game)
Two person zero sum game is the simplest of game model. There will be two persons in the
conflict and sum of the pay-offs of both players is zero.ie; a gain of one is the expense of the
other.
Fair game
A game is said to be fair if the value of the game is zero.

6. Value of the game
The value of the game is the maximum guaranteed gain to the maximizing player, if both the
players use their best strategies.it is the expected pay-off of a play when all the players of the
game follow their optimal strategies.
##Solution to a pure strategic game
Solution of mixed strategy problems
Mixed strategy problems can be solved by
1.Probability method 2. Graphic method 3. Linear programing method
1.Probability method-this method in applied when there is no saddle point and the payoff
matrix has two rows and two columns only.
Principle of dominance:
The principle of dominance states that if the strategy of a player dominates over another
strategy in all conditions. Then the later strategy can be ignored because it will not be affected
by the solution in anyway.
2.Graphic method-if pay-off matrix is of order 2*n or m*2, graphic method can be applied.
3.Linear programing method
A LPP includes a set of simultaneous linear equations or in equations which represent
restrictions related to the limited resources and a linear function which expresses objective
function representing total profit or cost. The objective of LPP is to maximize profit or to
minimize cost as the case may be, subject to a no: of limitations known as constraints.
Linear programming may be defined as a method of determining an optimum program of
interdependent activities in the view of available resources.
Solution to LPP
LPP can be solved by (i)Graphical method (ii)Simplex method
(i)Graphical Method-LPP involving two variables can be solved by graphical method. This
method is simple and easy to apply. A layman can easily apply this method to solve a LPP.
(ii)Simplex Method-Simplex method is a LPP technique in which we start with a certain
solution which is feasible. We improve this solution in a no: of consecutive stages which is
feasible until we arrive at an optimal solution. For arriving at the solution of LPP by this
method, the constraints and the objective function are presented in a table known as Simplex
method.
Slack variables

7. If a constraint has a sign less than or equal to, then in order to make it an equality we have to
add some variables to the LHS.The variables which are added to LHS of the constraints to
convert them into equalities are called Slack variables.
Surplus variables
If a constraint has a sign greater than or equal to, then in order to make it an equality we have
to subtract some variables to the LHS.The variables which are subtracted from LHS of the
constraints to convert them into equalities are called Surplus variables.
Net Evaluation-It is the net profit or loss, if one unit of variable in the respective column is
introduced.
Minimum Ratio-It is the lowest non-negative ratio in the replacing ratio column.
Key column
Key row-the row which relates to the minimum ratio is the key row.
Key number-key number is that number of the simplex table which lies both in the key row
and key column.
Iteration-Iteration means step by step process, followed in simplex method to move from one
basic feasible solution to another.
Artificial variable-artificial variables are fictitious variables. They are incorporated only for
computational purposes. They have no physical meaning.
Artificial variable technique-constraints of some LPP may have greater than or equal to.in such
problems we introduce artificial variables, we are able to get starting basic feasible solution.
This technique of LPP in which artificial variables are used for solving is known as Artificial
variable technique.it can be solved by *Big M method *Two phase method.
Duality &Sensitivity analysis
Every LPP is associated which another LPP called Dual. Original problem is called primal.
Dual problems may be defined as mirror image problems of primal problems.
Sequencing
A sequencing problem is a problem determining an appropriate order for a series of jobs to be
done on a finite no: of service facilities so as to minimize the total time taken for finishing all
the jobs. The selection of an appropriate order is called sequencing.
Replacement theory
The replacement theory deals with situations that arise when some items such as machines,
men, light, bulbs etc. require replacement due to their deteriorating efficiency, failure or
breakdown.

8. Types of replacement problems
(i)Replacement of major or capital items that deteriorate with time (when money value is not
counted)
(ii)Replacement of major or capital items that deteriorate with time (when money value is
counted)
(iii)Replacement of items that fail completely: -This type of replacement can be done in two
ways:
(a)Individual replacement policy-according to this policy the failed item is replaced by the
new one.
(b)Group replacement policy-in general, a system contains a large no: of identical cost items
such that the probability of their failure increases with time or age.in such cases there is a setup
cost for replacement which is independent of the number replaced. Here in such systems it may
be advantageous to replace all items at some fixed interval. Such policy is called Group
replacement policy.
Network analysis
A project is composed of a no: of jobs, activities or task that are related to each other and all of
these should be completed in order to complete the project.an activity of a project can start only
at the completion of many other activities. A network is a combination of activities and events
of a project.
Activity –an activity is a task associated with a project.it is the work to be undertaken to
materialize a specific event.
Dummy activity-usually a job or a task requires time and cost but there are certain activities
which do not take time or resources. They are known as dummy activities.
Event-events represents instants in time when certain activities have been started or
completed.an event or node in a network diagram is a junction of 2 or more arrows representing
activities.
Network diagram- a graph drawn connecting the various activities and events of a project is a
network diagram. They are of 2 types: Event oriented diagrams & Activity oriented diagrams.
Event oriented diagrams-they are also known as PERT network diagram. Here emphasis is
given to the events of the project.
Activity oriented diagrams-they are also known as CPM Network diagram. Here emphasis is
given to activities of the project.
Earliest and Latest Event times.
Earliest event times: The earliest event time or occurrence time is the earliest at which on event
can occur.

9. Latest event times: the latest event time is the latest time by which an event must occur to keep
the project on schedule.
Start & finish times of an activity.
Earliest start time(EST)it is the earliest time by which it can commence.
Earliest finish time(EFT)if the activity proceeds at its early time and takes its estimated
duration for completion then it will have an earliest finish time of an activity is the earliest time
by which it can be finished.
EFT=EST+Activity duration
Latest finish time(LFT)-The latest finish time is an activity is the latest time by which an
activity can be finished without delaying the completion of the project.
Latest start time(LST)-LST of an activity is the latest time by which an activity can be started
without delaying the completion of the project.
LST=LFT-activity duration
Slack and float
Slack is a term associated with events it denotes the flexibility range within which an event
can occur. The term float is associated with activity times. Float denotes the range within which
activity, start time or its finish time may fluctuate without affecting the completion of the
project.
Types of floats:
(i)Total float: it is the time spend by which the starting or finishing of an activity can be delayed
without delaying the completing of the project.
Total float=LFT-EFT or Total float=LST-EST
(ii)Free float: it is that position of positive total float that can be used by an activity without
delaying any succeeding activity.
Free float=EST of successor-EFT of the present activity.
(iii)Independent float: the independent float is defined as the excess of minimum available
time over the required activity duration.
I.F=EST for subsequent activity –LFT for proceeding Activity-Duration
(iv)Interfering float-it is just another name given to the head event slack especially in CPM
networks.it is the difference between total float and free float.
(v)Critical path-the path which is the longest on the basis of final duration is called critical
path.
(vi)Critical activity-activity lying on the critical path are critical activities.ie; an activity whose
float is zero is called critical activity.
Critical path method(CPM)

10. CPM is a network technique.it was originally discovered for applications to industrial situations
like construction, manufacturing, maintenance etc. It is always activity oriented.
Program Evaluation and Review Technique(PERT)
PERT is a management technique in which we try to exercise logical disciplines in planning
and controlling projects. The main objective of PERT analysis is to find out whether a job could
be finished on a given date.ie; to minimize the total time for a project.
Time estimates:
Here 3 estimates are made:
Optimistic time estimates-this is the shortest possible time in which an activity can be
completed under ideal conditions.
Pessimistic time estimates-it is the maximum time that would be required to complete the
activity.
Most likely time estimates-it is the time which the activity will take most frequently. If
performed a no: of times. This time estimates lies b/w the optimistic and pessimistic time
estimates.
Diff: b/w CPM & PERT
o In CPM, emphasis is given on activities. In PERT emphasis is given on events.
o In CPM, Time estimates for completion of activities are with a fair degree of accuracy.
In PERT, Time estimates are not so accurate and defined.
o CPM is a deterministic model with a well-known activity time based on experience
whereas PERT is a probabilistic model with uncertainty in activity duration
o CPM is used for non-repetitive jobs, and PERT is used for repetitive jobs.
o CPM is mainly used for construction and business problems. PERT is used for planning
and scheduling research programs.
Decision theory
A decision is a process of choosing an alternative courses of action when no: of alternatives
exists. Decision making is an everyday activity in life. Right decisions will have salutary effect
and wrong one may prove to be disastrous. Decision may be tactical or strategic. Tactical
decisions are those which affects the business in the short run. Strategic decisions are those
which have far reaching effect during the course of business.
Payoff and regret tables:
Payoff table is in a matrix form.it list the acts and events.it considers the economics of the
problem by calculating payoff value for each act and event combination. Similarly regret table,
is a matrix showing opportunity loss for each act under a state of nature.

11. Types of decision making situations
(i)Decision making under certainty
(ii)Decision making under uncertainty
Maximax criterion-The term maxi-max is an abbreviation of the phrase maximum of
maxima.
Mini-max Criterion-Mini max is just opposite to maximax. The term minimax is an
abbreviation of the phrase minimum of maximum.
Maxi-min criterion-The maxi min criterion of decision making stands for choice
between alternative courses of action.
Laplace criterion-The act having maximum expected pay-off is selected.
Hurwicz alpha criterion-This method is a combination of maximum criterion and a
maxi-max criterion. Here the decision maker’s degree of optimism is represented by
alpha.
(iii)Decision making under risk-The most popular methods used are:
Expected Monetary Value (EMV)criterion-when the probabilities can be assigned to the
various states of nature, it is possible to calculate the expected payoff for each course of action.
These expected payoffs are known as EMV.
Expected Opportunity Loss(EOL)criterion-when the probabilities for various states of nature
are known, it is possible to calculate the expected losses for each course of action.
Expected Value of Perfect Information(EVPI)-it gives an upper bound of the amount which
the decision maker can spend for obtaining perfect information.
EVPI=Expected Value with Perfect Information-optimum EMV.
Decision trees
Decision tree is one of the devices of representing a diagrammatic representation of sequential
and multidimensional aspects of a particular decision problem for systematic analysis and
evaluation. Under this method, the decision problem, alternative courses of action, states of
nature and the likely outcomes of alternatives are diagrammatically or graphically represented
as if they are branches and sub-branches of horizontal tree.