2. SYLLABUS
UNIT -I
Development - Definition- Characteristics and Phases -
Types of models - operation Research models -
Applications.
ALLOCATION: Linear Programming Problem Formulation
- Graphical solution -Simplex method -Artificial variables
techniques -Two-phase method, Big-M method - Duality
Principle.
3. UNIT – II
TRANSPORTATION PROBLEM - Formulation - Optimal
solution, unbalanced transportation problem -
Degeneracy.
Assignment problem - Formulation - Optimal solution -
Variants of Assignment Problem-Traveling Salesman
problem.
4. UNIT - III
SEQUENCING - Introduction - Flow -Shop sequencing -
n jobs through two machines -n jobs through three
machines - Job shop sequencing - two jobs through 'm'
machines.
REPLACEMENT: Introduction - Replacement of items
that deteriorate with time -when money value Is not
counted and counted - Replacement of items that fail
completely, group replacement.
5. UNIT-I
LINEAR PROGRAMMING PROBLEM,SIMPLEX
METHOD
Introduction
We take several decisions in our life. Most of
these decisions are taken by common sense.
But the decisions taken by mere common sense
may mislead or confuse us.
Therefore it has become necessary for managers and
engineers to believe in the science that provides the
evidential support and scientific base.
6. Operations Research is one such science that provides
better solutions to the managers, engineers and any
practitioners with better solutions.
This science came in to existence during world war II.
Though it was first employed for military operations, its
applications are extended to any field on the earth in
some form or other.
Thus OR is considered as the science that deals with
decision making.
7. ORIGIN and DEVELOPMENT :
The term “Operations Research” was first coined by
Mc Closky and Trefthen in 1940 in a small town Bowdsey
of UK.
The name “Operations Research” was given to this
subject because it has started with the research of
military operations. During world war –II , the military
commands of UK and USA engaged several teams of
scientists to discover tactical and strategic military
operations.
8. Their mission was to formulate specific proposals and to
arrive to the decisions that can optimally utilize the
scarce resources to acquire maximum possible level of
effective results
In simple words it is yielding greatest results with little
efforts
Thus it has gained popularity and was called “an art of
winning the war without actually fighting it”
9. Following the end of the war the success and
encouraging results of British teams have attracted
industrial managers to apply these methods to solve their
complex problems.
The first method in this direction was simplex method
(LPP)developed in1947 by D.B.Dantzig
USA.
Since then several scientists have been developing this
science in the interest of making operations to yield high
profits or least costs.
10. Now this science has become universally applicable to
any area such as transportation, hospital management,
agriculture, libraries, city planning, financial institutions,
constructional management and so on.
In India many industries have been realizing the
advantages by implementing the OR methodology.
A few to quote in this regard are Delhi Cloth Mills, Indian
Airlines and Railways, Hindustan Lever Ltd.(HLL), Tata
Iron & Steel Co.(TISCO),Fertilizer Corporation of
India(FCI),LIC etc.
11. Definition:
Operations Research is concerned with scientifically
deciding how to best design and operate man-machine
systems usually requiring the allocation of scarce
resources
- OR society, America
Operations Research is the systematic application of
quantitative methods, techniques and tools to the
analysis of problems involving the operation of systems
-Daellenbach and George
12. STEPS (PHASES) OF OR METHOD:
Step 1: Identification of problem
Step 2: Collection of relevant data
Step 3: Formulation of data into mathematical form
Step 4: Selecting and applying right OR application model
Step 5: Analysis and calculation
Step 6: Decision Making
13. Application Models of Operations Research:
1.Allocation Models
2.Inventory Models
3.Competitive Models
4.Network Models
5.Sequencing Models
6.Waiting Line Models
7.Replacement Models
8.Dynamic Programming Model
9.Markov Chain Models
10.Monte- Carlo Simulation
14. Significance and Applications of OR in Industrial
Problems:
1.Design of aircraft and aerospace structure for minimum
weight
2.Finding the optimal trajectories of space vehicles
3.Design of civil engineering structures like
frames,foundations,bribges,towers,chimneys and dams
for minimum cost
4.Optimum design of linkages, cams, gears, machine tools
and other mechanical components
15. 5.Design of material handling equipment like conveyer
trucks and cranes for minimum cost
6.Design of pumps ,turbines, and heat transfer
equipment for maximum efficiency
7.Optimum design of electrical machinery like
motors,generators,and transformers
8.Shortest route taken by salesman visiting different
cities
9.Optimum Production, planning and control
10.Planning the best strategy to obtain maximum profit in
the presence of competitor
17. Linear Programming Problem:
If the relevant data is translated in to appropriate
mathematical or OR model then that is called as “
Linear Programming Problem “ in which the
variables are linearly related. This process of
translation is called as formulation
18. The formulation of relevant data in Linear Programming
Problem is carried out in the following four steps.
Step 1 : Selection of variables
Step 2 : Setting the objective function
Step 3 : Identification of constraint set
Step 4 : Writing the condition of variables
19. STEP 1: Selection of Variables:
In the given data firstly the variables are to be identified.
These are also called Decision Variables which are often
seen in the following ways
(a) Number of different types of products to be
manufactured per day or week
(b)Number of different types of products to be sold in a
marketing problem
20. (c) Number of resources such as men /machine /
material required to do certain job in an allocation
problem
(d) Number of units of certain model or type to be bought
at lowest cost
(e) Number of ingredients to be mixed or used
(f) Number of units to be ordered
(g) Number of unit advertised
(h) Number of fertilizers used in a agriculture problem
(i) Number of tablets to purchase in medical problem
21. STEP 2: Setting objective function:
This is to set the goal in the problem. There are only two
types of objective functions in OR
These are maximization of profits and minimization of
costs
Maximize Z = Px1 + Qx2
STEP 3: Identifying the constraint set :
A constraints set is a set of limitations in achieving the
objective
22. Constrains are two types
1. Inequality type (≤ or ≥)
2. Equality type (=)
Inequality type constraints is classified availability type
constraint (≤) and requirement type constraint (≥)
23. Availability type constraint(≤)
Ex:1. Suppose you are cooking some curry dish and you
have to put some salt in it, say 10 gm. You will be
constrained to put less than or equal to 10 gm only, but
excess is not allowed.
2. Suppose you have only Rs.100 in your wallet and
went to a restaurant. You will be constrained to order
worth less than or equal to 100 only. Hence x ≤ 100,
where x is rupees spent.
24. 3. You have a machine on which you can can utilize 8
hours a day. Thus you can have a maximum of 48 hours
a weak(6 days). Now number of hours you can utilize on
the machine, say x ≤ 48.
Requirement type constraint (≥)
Ex: 1. you have a production unit at which you have an
order of 50 units minimum. Now the number of units to be
produced is , x ≥ 50.
25. 2. Suppose you are selling a product, then you will say it
should be greater than or equal to its manufacturing and
transport costs.
Exact constraint (=)
Ex: 1. you have to order the lens power for your spectacles.
Here you will not permit lesser or
greater than your specified power.
2. When you purchase the shoe, you neither allow ‘less
than’ nor ‘greater than’ your feet size.
26. STEP 4: Writing the condition of variables:
Here the conditions of the decision variables will be
predetermined. The conditions often found in LPP are in
two kinds.These are non-negative and unrestricted. A
non negative conditions is represented by greater than or
equal to zero
(≥ 0).
27. GRAPHICAL SOLUTION OF LPP:
Graphical Solutions are easier to understand and
reproduce. Also a pictorial view is always a better
representation. Thus graphical solutions have gained
prominence in Operations Research.
28. Graphical Solution Procedure:
Assume the constraints as equations and find any two
points for each equation, so that the equation can be
represented as a straight line on graph.
Similarly draw all the constraint lines
Shade the appropriate areas as given by the
constraints. If the constraint is ≤ type shade the area
below the line. If the constraint is ≥ type shade the area
above the line. If the constraint is = type do not shade
any area,and the line itself is the region
29. Identify the feasible region by locating the area satisfying
all constraint that is common with all the constraint areas.
This region will have the optimal solutions. If there is no
common area possible then the solution is infeasible
30. SIMPLEX METHOD
It was originally proposed by D.B.Dantzig in 1948.It
starts at a basic level of the problem.
At each step it projects the improvement in the objective
function over its previous step. Thus the solution
becomes optimum when no further improvement is
possible on the objective function
31. Simplex Algorithm:
Step 1: Formulation of LPP
Step 2: Convert the Constraints into equality
form
Step 3: Find the IBPS (Initial Basic Feasible
Solution)
Step 4: Construct the initial Simplex Table as
given below
32. Step 5 : Find out going and incoming variables
Step 6 : Re-write next table
Step 7 : Check whether all the values of Zj-Cj
are positive or not. If all are positive
the optimal solution is reached. If not
repeat the iteration till all the values of
Zj-Cj become positive
33. TRANSPOTRATION PROBLEM:
Transportation problem is another case of application to
Linear Programming Problems, where some physical
transportation of resources to be made from one place to
another to meet certain set of requirements within the
given availability.
34. The places from where the resources are to be transferred
are referred as Origins. The other side of this
transportation to where the resources are transported are
called Destinations such as market centres,godowns etc.
These will have certain requirements or demands
36. Transportation Algorithm:
Step 1 : Formulate the Problem in the form of
Matrix
Step 2 : Standardizes the TP
Step 3 : Check the rim condition
i.e., total supply= total demand.
Step 4 : Obtain the IBFS by the below methods
a)North West Corner Method (NWCM)
b) Vogel’s Approximation Method (VAM)
or Penalty method
c)Least Cost Entry Method (LCEM)
37. Step 5: Test the degeneracy n(c) ≥ r + c -1
Step 6: Optimality Test:
Obtain the final Solution by
(i) Stepping stone method
(ii)Modified Distribution Method
Step 7: Update the solution
38. ASSIGNMENT PROBLEM
In most of the cases the department will have
the specialists for operating certain critical or
sophisticated machinery , equipment while in some
other cases anybody can operate any machine.
Whatever the type of step may it be ,it is
essential for a manager to see that maximum work
is to be deliver from his man power resources
which are precious and scarce. This can be done
when and only when right job is given to right man
39. Generally though any body can do any job, all the men will
not have the same efficiency and knowledge on all the
jobs. The same job one may do fast while the other may
do it slowly or a person may do one job fast and other
slow. So, assigning the right job to the right person who is
doing efficiently is called as “Assignment”.
40. The Assignment Algorithm:
Step 1: Balance Checking
no. of rows(r) = no. of columns(c)
Step 2: Standarard form:
The minimisation case is supposed to be
standard form.
Step 3: Opportunity cost table:
locate the smallest element in each row
and column and subtract from all the
elements
41. Step 4: Assignment:
Enrectangle the zero in each row and
column and strike the other zeros.
Here while enrectangle preference will
be given to the zero which is not having
extra zeros.
Step 5: Obtain the final solution
