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.