Operations Research

Lecture Notes

By

Prof A K Saxena

Professor and Head

Dept of CSIT

G G Vishwavidyalaya,

Bilaspur-India

Operations Research

Some important tips before start of course material to

students

• Mostly we followed Book by S D Sharma, as prescribed

for this syllabus

• At places, we use some internet links not necessarily

mentioned there at.

• We acknowledge all such resources.

• As the course is mostly mathematical in nature, we will be

solving problems in class room. The problems will involve

a lot of mathematics, calculations although simple but will

be so time consuming to express on computers, we leave it

up to students to ask in details any particular topic or

problem in class or during contact hours.

• So ready to take off !!!!

History of Operations Research

The term Operation Research has its origin during

the Second World War. The military management

of England called a team of scientists to study the

strategic and tactical problems which could raise

in air and land defence of the country. As the

resources were limited and those need to be fully

but properly utilized. The team did not involve

actually in military operations like fight or

attending war but the team kept off the war but

studying and suggesting various operations related

to war.

What is Operations Research?

Several definitions have been given

• Operations research (abbreviated as OR hereafter) is a scientific

method of providing executive departments with a

quantitative basis for decisions regarding the operations

under their control: Morse and Kimbal (1944)• 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.

• Operational Research (OR) is the use of advanced analytical techniques to

improve decision making. It is sometimes known as Operations Research,

Management Science or Industrial Engineering. People with skills in OR hold

jobs in decision support, business analytics, marketing analysis and logistics

planning – as well as jobs with OR in the title.

•As such a number of definitions can be found in literature, you

can express the term OR with the spirit mentioned in the literature.

Meaning of Operations Research?

As stated early, the OR does not mean to get involved

in the operations but suggestion for better execution of

operations. Suggesting strategy how the operations

can be improved and get better results. The genesis of

OR is in finding better ways to solve a problem. Thus

it is analytical not purely hard core action oriented.

As we explore several options for the analysis of

operations, we search and re-search the effects of

operations. If one solution offers some result, try

second solution and see and compare with previous

and so on unless we satisfy ourselves.

Therefore research term sounds to indicate that there

would be enough thinking on the outcome of several

results. Hence Operation Research.

Meaning of Operations Research?

A simple example of OR

Given different routs to reach from source A to

destination B. Also on these routes there can be

various ways to travel. For simplicity, we assume we

have travelling modes x,y,z each having different

travelling time and cost incurred on travel.

I have a limited money or budget and a limited time

also to reach destination B. Now all options can reach

me A to B but they will not be fit for me. I want a

solution which I can use so that I can afford journey

both in terms of cost and time.

OR can be used here.

Expression of problems in

Operations Research?

A typical example of OR problem

Most of the problems in OR are of the following form

• Given an objective function also called fitness

function which depends on certain variables or

parameters. The objective function has to be

optimized, i.e. maximized or minimized.

•A set of constraints given which should be satisfied

while solving the problem

•Some conditions on the nature of parameters so as to

ignore those parameters at all which do not fulfill

these conditions e.g. x2 =4 will give x=+2 and x=-2 but

we do not want to consider x=-2 so we have to declare

x>=0 at the start of the problem.

Operations Research

Introduction to LPP

Equations and Inequalities/Constraints

We are familiar with terms equations such as

2x+ 3y -3z = 12

is an equation as we have an equality sign relating left

hand side with right hand side

But in OR we will mostly deal with types of following

2x + 3y <= 14

Or

2x + 3y >= 14

These types of representations will be called as constraints

or inequalities in OR

LPP stands for linear Programming Problem which means

finding optimum solutions (minimum or maximum)

represented by a function of variables or parameters called

objective function, denoted by z

Subject to a set of linear equations or inequalities called

constraints.

Operations Research

Introduction to LPP

In an LPP, the equations or inequalities are of the linear form

like a11 x1 + a12 x2 + …+ a1n xn = b1or

a11 x1 + a12 x2 + …+ a1n xn >= b1or

In general

where aij, bi, and cj are given constants. aij are coefficients of

decision variables x’s, bi are constraints values and cj are

coefficients associated to objective function z

Operations Research

Introduction to LPP

A complete solution to an LPP comprises of following steps

1.Formulating problem to a proper mathematical form

2.Solving the problem graphically/algebraically

In our discussion onward, we will first learn how to

formulate given problem in the standard form, then learn

first its graphical solution then algebraic solution. In

algebraic solution, we will apply simplex method.

Operations Research

Introduction to LPP

An example problem in formulated form

Max Max z= z= 55xx11 + 7+ 7xx22

s.t. s.t. xx11 << 66

22xx11 + 3+ 3xx22 << 1919

xx11 + + xx22 << 88

xx11 >> 0 and 0 and xx22 >> 00

ObjectiveObjectiveFunctionFunction

““RegularRegular””ConstraintsConstraints

NonNon--negativitynegativityConstraintsConstraints

Operations Research

Introduction to LPP

Exercise on formulating an LPPA toy company manufactures two types of dolls, A and B.

Each doll of type B takes twice as long to produce as one of

type A, and the company would have time to make a

maximum of 2000 per day. The supply of plastic is sufficient

to produce 1500 dolls per day of A and B combined. Each B

type doll requires fancy dress of which there are only 600

per day available. If the company makes a profit of Rs 3

and Rs 5 on doll A and B respectively, then how many dolls

of A and B should be produced per day in order to maximize

the total profit. Formulate this problem.

Operations Research

Introduction to LPP

Formulation of LPPIn this problem (and these types of problems) start from last. We are to maximize the

profit. So the objective function z will be

Maximize z

Then two products are dolls of A and B types. So decision variables will be x1 and

x2. Now let x1 denotes the number of dolls of type A required for maximize profit z

and x2 be dolls for B type. Profit on each doll of A is Rs 3 and that for each doll B is

Rs 5 so

Max z= 3x1 + 5x2

As x2 takes twice time than x1 and total time allowed per day can produce 2000

dolls so x1 + 2x2 <=2000

Fancy dress material is available for B type 600 dolls so

X2<=600

Also plastics availability is enough to produce 1500 dolls for A and B both so

x1+x2<=1500

As x1, x2 are numbers of dolls to be produced per day so x1,x2 >=0

Writing all steps together

Max z= 3x1 + 5x2 (Objective Function)

s.t.

x1 + 2x2 <=2000 (Time constraint)

x1+ x2 <=1500 (Plastics availability constraint)

X2<=600 (Fancy dress material constraint)

and x1,x2 >=0 (non negativity problem)

Operations Research

Introduction to LPP

Formulation of LPPThere are similar type of other formulation problems in

LPP. The easy way to do is

1.First read the LPP to find the term Profit/Cost/Time or

similar term to maximize/minimize/optimize

2.Usually the profit/time etc. associated with each of the

product will determine the decision variables viz. x1,x2,…

3.Read each sentence carefully, a constraint ( in rare case

an equality) with some numeric value is given.

4.Convert all such sentences to constraints/inequalities.

5.Write the objective function first like Max(Min) z=….

6.Then write subject to (s.t.) and all inequalities in

following lines below s.t.

7.Do not forget to write non negativity conditions x1>=0,

x2>=0, etc…

Operations Research

Introduction to LPP

Formulation of LPPThere are similar type of other formulation problems in

LPP. The easy way to do is

1.First read the LPP to find the term Profit/Cost/Time or

similar term to maximize/minimize/optimize

2.Usually the profit/time etc. associated with each of the

product will determine the decision variables viz. x1,x2,…

3.Read each sentence carefully, a constraint ( in rare case

an equality) with some numeric value is given.

4.Convert all such sentences to constraints/inequalities.

5.Write the objective function first like Max(Min) z=….

6.Then write subject to (s.t.) and all inequalities in

following lines below s.t.

7.Do not forget to write non negativity conditions x1>=0,

x2>=0, etc…

Operations Research

Introduction to LPP

Solution of LPPWe will see how LPP can be solved after these are

formulated. There can be two type of solutions to discuss

1.Graphical solution

2. Algebraic mainly simplex method

First we shall discuss graphical method to solve LPP.

We adopt an easy approach here by taking a rough sketch

of graphs manually but in principle correct.

Operations Research

Introduction to LPP

Graphical Solution of LPPThe concept of graph and linear equations.

In a graph, we have two axes, axis of x and axis of y.

+

+ (-,+) IInd (+,+) Ist

+

y O

- (-,-) IIIrd (+,-) IVth

-

-

..---------- - x ++++++++++…

The axes can be divided in four quadrants. Any point (x,y)

lies in one of the quadrants. The origin O is the point

having (0,0) coordinates. Any point in four quadrants will

be (x,y), (-x,y),(-x,-y) and (x,-y) in first, second, third and

fourth quadrant respectively.

Operations Research

Introduction to LPP

Graphical Solution of LPPThe Suppose an LPP is given in the formulated form.

Max(min) z = c1 x1 +c2 x2 +…cnxns.t.

a11 x1 +a12 x2 +…..a1n xn (<=>)b1a21 x1 +a22 x2 +…..a2n xn (<=>)b2……………………………………………………………

am1 x1 +am2 x2 +…..amn xn (<=>)bmwith xi’s >=0

1.Consider all constraints as equations

2.Plot all lines (equations) on the graph

3.Indicate point of intersection of every two lines intersecting

each other or the point of intersection of a line with axis as the

case may be. If you are not using the graph accurately the solve

the two lines algebraically to know point of intersection.

4.Shade the region of every line which is towards the axis (<=) or

away from axis (>=).

Operations Research

Introduction to LPP

Graphical Solution of LPP5. As we have xi’s >=0, all valid regions will lie in the first region going

towards origin (<=) or towards infinity (>=)

6. After all lines (constraints ) are plotted and shaded, the common

region, shaded and surrounded by all lines will give the feasible region.

7. Now plot objective function line z at the origin and move it parallel

away from first quadrant in the +infinity direction.

8.Keep the line z sliding in the feasible region. A point will be reached

which is the extreme point in the feasible region. In most of the cases of

maximum, this is the farthest point from origin and for cases of

minimum, this point is the closest to origin. This point is called the point

of optimum solution of z.

9.Find out the value of z at this point. The point is the solution point with

the value of z as calculated there.

10.For a quick solution, take all intersection points and shade the

common region called feasible region. Find out the coordinates of every

corner point in the feasible region. Calculate z at each of these points,

and finalize the point with maximum(minimum) value of z as the solution

point with value of z as calculated there at.

Operations Research

Introduction to LPP

Plotting of linesSuppose a line (or inequality) is given as follows

x1 + 4 x2 (< = >) 4

Then first for plotting purpose write it as

x1 /4 + x2 /1 (< = >) 1 (i.e. x/a + y/b =1 form)

Now plotting becomes easier

2-

1-

| | | |

1 2 3 4

(0,0)

The slanted line represents x1 + 4 x2 (< = >) 4 or x1 /4 + x2 /1 (< = >) 1 .

The line cuts intercepts 4 from axis x1 and +1 from axis x2. This is why we

brought the line in the form x1 /4 + x2 /1 (< = >) 1

Operations Research

Introduction to LPP

Plotting of linesIf we have lines (or inequality) of the form

x1 - 4 x2 (< = >) 4

Then first for plotting purpose write it as

x1 /4 + x2 /-1 (< = >) 1 (i.e. x/a + y/b =1 form)

Now plotting becomes as follows

2-

1-

| | | |

1 2 3 4

-1- (0,0)

-2-

The slanted line represents x1 - 4 x2 (< = >) 4 or x1 /4 + x2 /-1 (< = >) 1 .

The line cuts intercepts 4 from axis x1 and -1 from axis x2. This is why we

brought the line in the form x1 /4 + x2 /-1 (< = >) 1

Operations Research

Introduction to LPP

Plotting of linesSimilarly we can plot lines of other two types lying in second (-,+)

and third (-,-) quadrants.

The graphical solution to several LPP problems have been

practiced in class room. Ply try solved and unsolved problems.

Operations Research

Introduction to LPP

General form of LPPThe LPP can be in general one of the following forms

Max(min) z = c1 x1 +c2 x2 +…cnxns.t. the m constraints

a11 x1 +a12 x2 +…..a1n xn (<=>)b1a21 x1 +a22 x2 +…..a2n xn (<=>)b2……………………………………………………………

am1 x1 +am2 x2 +…..amn xn (<=>)bmwith xi’s >=0

Slack and Surplus Variables

Slack variable: If a constraint has <= sign x1 + x2 <=20 then to

make it equality, we need to add some non negative term s to the

left hand side of the constraint. Thus we have x1 + x2 + s =20

then s is called a slack variable.

Surplus variable: If a constraint has >= sign x1 + x2 >=20 then

to make it equality, we need to subtract some non negative term

s from the constraint. Thus we have x1 + x2 - s =20 then s is

called a surplus variable.

Operations Research

Introduction to LPP

Standard form of LPPThe general form of LPP is given previously. The standard form

has following characteristics

Objective function should have only Maximum and NOT Min. So

even if we have Min z = c1 x1 +c2 x2 +…cnxn, we will convert to Max

form by Max z = -Min(z) or we can have Max z = -c1 x1 -c2 x2 -…-

cnxn, and can write z’ =-z so Min z = Max z’

Convert all constraints to equalities using slack or surplus

variables so that we have

a11 x1 +a12 x2 +…..a1n xn + xn+1 = b1a21 x1 +a22 x2 +…..a2n xn + 0xn+1 + xn+2 =b2 -(2)

……………………………………………………………

am1 x1 +am2 x2 +…..amn xn + 0xn+1 + 0xn+2 + xm+n =bm……………………………………………………………

with all xi’s >=0 -(3)

The objective function will become now

Max z = c1 x1 +c2 x2 +…cnxn +0xn+1+0xn+2 +…0xn+m -(1)

If any x is unrestricted in sign, convert it to x’ – x” where x’ and

x” are >=0

Operations Research

Introduction to LPP

Matrix form of LPPThe general and standard forms of LPP are given previously. The

standard form can be converted to the Matrix form as follows

Max z = CXT (transpose of X)

subject to AX = b, b >=0 and x >=0

Where x = (x1,x2,…xn, xn+1,…xn+m)

c =(c1,c2,..cn,0,…0)

b= (b1,b2,…,bm)

And Matrix A =

a11 a12 ..a1n 1 0 0 … 1

a21 a22 ..a2n 0 1 0 … 0………………………..

am1 am2 ..amn 0 0 0 … 0

Students are advised to attempt all problems related to the

concepts so far and ask the doubts if any.

Operations Research

Introduction to LPP

Important Definitions of LPPSee the standard form of LPP and equations 1,2,3

Solution of LPP: Any set of variables x = (x1,x2,…xn, xn+1,…xn+m) is called

solution of LPP if it satisfies (2) only.

Feasible Solution of LPP: Any set of variables x = (x1,x2,…xn, xn+1,…xn+m)

is called feasible solution of LPP if it satisfies (2) as well as (3).

Basic solutions and Basic Variable: A solution to (2) is a basic solution

if it is obtained by setting n out of m+n variables equal to 0 and then

solving for remaining m variables with the determinant of coefficients of

these m variables is non zero.

Usually we call those variables as basic variables which are used to get

identity matrix in solving LPP using simplex method.

Basic Feasible solutions: A basic solution to (2) is a basic feasible

solution if it also satisfies (3).

Optimum Basic Feasible solutions: A basic feasible solution which also

satisfies (1) is called a Basic Feasible solutions.

Unbounded Solution: If the value of objective function z can be increased or decreased infinitely then such a solution is called an unbounded solution.

After these definitions, we are ready to start solution of LPP using simplex method.

Students must try some numeric problems based on the lectures

completed so far.

Operations Research

Transportation Problems

We often face the situation when we have to transport material from

various places to various other places and want to minimize the

time or cost of transporting these material (e.g. TV, computers etc)

Transportation Problem(TP) is to transport various quantities of a

single item stored at various origins to different locations or

destinations such that the cost of transportation is minimum.

Mathematical Formulation

Suppose there are m origins, ith origin storing ai units of a certain

product, there are n destinations and destination j requiring bj units

of same item, the cost of transporting each of m origins to each of n

destinations is known and given. Let C be the cost of shipping or

transporting ith source to jth destination

It is assumed that ΣΣΣΣai = ΣΣΣΣbj (i=1,…m; j=1,2,..m) such TP are called

balanced TP. If ΣΣΣΣai ≠ ΣΣΣΣbj then those TP are unbalanced TP. In case of

unbalanced TP we add one row or column with zero valued Cij in that

row or column with the value of ai or bi = of that row as column equal

to the difference of (ΣΣΣΣai - ΣΣΣΣbj )

Operations Research

Transportation Problems

Mathematical Formulation

So ΣΣΣΣai = ΣΣΣΣbj the problem is now to determine non negative values of

xij satisfying the availability constraints

for i=1,2,..,m

And the requirement constraint

for j=1,2,..,n

And minimize the total cost of transportation

(Objective function)

∑=

=

n

j

iij ax1

∑=

=

m

i

jij bx1

∑∑==

=

n

j

ijij

m

i

cxz11

Operations Research

Transportation ProblemsTabular Representation of TP (Given)

Warehouse/

Factories

W1 W2 .. Wn Factory

Capacities

F1 C11 C12 .. c1n a1

F2 C21 C22 C2n a2

… .. .. .. …

Fm Cm1 Cm2 .. Cmn am

Requirements b1 b2 .. bn ΣΣΣΣa = ΣΣΣΣb

Tabular Representation of TP (To find out)

Warehouse/

Factories

W1 W2 .. Wn Factory

Capacities

F1 x11 x12 .. x1n a1

F2 x21 x22 x2n a2

… .. .. .. …

Fm xm1 xm2 .. xmn am

Operations Research

Transportation Problems

To find out the solution of TP

i.Find out initial basic feasible solution (ibfs)

ii.Find out the optimal solution

First following methods are used to find out ibfs

a.North west corner method

b.Row minima method

c.Column minima method

d.Lowest cost entry method or matrix minima method

e.Vogel’s approximation method

All above ibfs methods are easy and have been discussed thoroughly

with examples in class rooms.

The optimal solution is obtained after ibfs is obtained by any of the

five methods. The optimal solution will give the best value. The

examples have been done in class room.

Students are advised to contact in case of any doubt/problem about

these exercises.

Operations Research

Transportation Problems

Degeneracy in Transportation Problems

The basic feasible solution to an m-origin and n-destination TP can

have maximum m+n-1 positive basic variables. If the number of

basic variables is exactly m+n-1 then BFS is a non-degenerate. If

however basic variables are less than m+n-1 then BFS are

degenerate.

Operations Research

Transportation Problems

Unbalanced TP: As stated earlier that when Σai ≠ Σbj then such TP are said to be unbalanced TP

e.g. given a TP, here total requirement (30) ≠ total capacity (34) so

this problem is unbalanced

Because there is a lack of 4 requirement so add 4 to requirement by

an extra column as follows. Now the TP becomes balanced. Solve it

as usual.

Mills M1 M2 M3 M4 M5 Capacity

F1 4 2 3 2 6 8

F2 5 4 5 2 1 12

F3 6 5 4 7 3 14

Requirement 4 4 6 8 8 30≠ 34

Mills M1 M2 M3 M4 M5 Mf Capacity

F1 4 2 3 2 6 0 8

F2 5 4 5 2 1 0 12

F3 6 5 4 7 3 0 14

Requirement 4 4 6 8 8 4 34 =34

Operations Research

Assignment Problems

Basic

We know that all fingers of our either hand are not equal. Same is

the case with efficiency of each individual. One person is better in

one job than another whereas other person is better than first in

some other job. If we can assign suitable job to each individual then

total efficiency will increase.

Suppose there are n jobs to be performed by n persons and each of

the persons can do each of the job but with different efficiency. Let

Cij be the cost of performing jth job by ith person, the problem is to

assign each job to each person such that no job is performed by

more than one person or no person is assigned more than one job,

and the total cost of performing all jobs is minimum.

Operations Research

Assignment Problems

Tabular form of assignment problem

Jobs 1 2 .. j .. n

Persons P1 C11 C12 .. C1j .. C1n

P2 C21 C22 .. C2j .. C2n

: : : : : : :

i Ci1 Ci2 : Cij : Cin

: : : : : : :

n Cn1 Cn2 : Cnj : Cnn

Operations Research

Assignment Problems

Mathematical Formulation of

Assignment Problem

Minimize the total cost z = i=1,2,..n,j=1,2,..n

Subject to restrictions of the form

xij = 1 if ith person is assigned jth job

0 if not

(one job is done by ith person, i-1,2,..,n)

(one person should be assigned jth job, j=1,2,…n)

Remember that size of the assignment table is n by n which is

balanced. For unbalanced problems, we will adopt similar action as

in TP, we will discuss later here.

∑∑==

n

j

ijij

n

i

xc11

∑=

=

n

j

ijx1

1

∑=

=

n

i

ijx1

1

Operations Research

Assignment Problems

Solution of Assignment problemsGiven a table of size n X n (balanced)

(i)Simple method of single zero assignments first row wise then

column wise or vice versa

a.Find the row having only single zero. Other zeros in that row are

either assigned or crossed. Mark this zero as assigned, cut all other

zeros in the column of that row assignment.

b.Complete all rows in manner as given a

c.When all row assignments have been made, repeat the process for

column assignment. Here cross all zeros of the row where column

assignment has been made

d.Complete all columns in manner as given a

e.Check if each row and each column has one and only assignment

f.Note the position of each assignment and sum the original costs at

these locations. This will give the minimum cost.

g.If any row or column still has one or more zeros left unassigned or

uncrossed then above method will not be applicable. Then apply

minimum lines drawing method as explained in classroom with

many examples.

Operations Research

Assignment Problems

Solution of Assignment problemsGiven a table of size n X m (unbalanced)

First check if rows m < column n, then add additional rows with all

zero values to make m=n

If columns n < rows n, then add additional column to make m=n

The problem now becomes balanced

Apply the same procedure explained before to solve this balanced

problem

Try to solve as many exercises as possible of both types

Contact in case of any doubt, problem.

Operations Research

Replacement Problems

BasicsWe know that almost every object needs replacement after a certain period.

We go on repairing our vehicle again and again but after a time, repair

becomes more expensive than to buy a new vehicle. Due to following factors

replacement becomes necessary

a.The old item has become worse, works badly or requires expensive

maintenance for its operation or running.

b.Old item has failed due to accident or otherwise and can not work.

c.A better or more efficient item has come in the market.

The question is to take a decision whether to continue with the old item or

buy a new one (i.e. replace old item). More important question or the main

question of replacement problem is to decide when to replace old item?

Two types of replacement problems will be mainly dealt here

i.When the cost value, scrap value, operational costs etc are given and to find

out the time when to replace existing item. There is no increase or decrease in

the value of money. The cost of a machine is Rs 40000/ today will remain

same even after 10 years on today.

ii.When the cost value, scrap value, operational costs etc are given and to find

out the time when to replace existing item. There is an increase or decrease in

the value of money. The cost of a machine is Rs 40000/ today will becomes

20% after every passing years as seen on today.

Both types of problems have been explained with examples in class. Students

are advised to contact in case of any doubt or problem.

Operations Research

Project Management CPM/PERT

BasicsPlease elaborate the followingA project is defined as a combination of activities which must be executed in

such a manner or order before the whole project or task can be completed.

Steps in Project CPM/PERT techniques

1.Planning: Divide the main project into small projects which will further be

divided into small activities.

2.Scheduling: Prepare a time table to show start and finish of every activity.

3.Allocation of resources: To allocate a physical resource like equipment,

money, space to project.

4.Controlling : The overall activities must be controlled so that the project

may complete in time. Also find out critical path.

We have discussed in class room various networks, find out Earliest time,

Latest time, critical path, float, slack time. Do other exercises and contact in

case of doubt/problem.