MTH601 Operation Research Quiz by VUBuddy

Lecture # 01

Operation Research:

What is Operation Research?

Operation Research (O.R) is an art of wining wars without actual fighting. (Arthur Clark) O.R is a scientific approach to the problems. (H.M. Wagner)

Linear Programming:

Linear programming is a mathematical technique for determining the optimal solution and obtaining a particular objective when there are alternative uses of resources. The objective may be cost minimization or profit maximization. The word linear means that the relationships are represented by straight line ax+by = c The word programming that is concerned with optimal allocation of limited resources.

Linear Function:

A linear function contains terms of which is composed of only a single variable raised to the power one. Linear functions are those whose graph is a straight line. e.g. 3x+2y = 7 (linear) 3
/+2y = 7 (Non-linear)

Objective Function:

It is a linear function of decision variables. z =
+
 is the most typical form of objective functions are maximize f(x) or minimize f(x).

۔minimum   اور ںmaximum     ں  decide    وہDecision Variable

Constraints:

These are the linear equation arising out of practical limitations. The mathematical forms of constraints are f(x) ≤ b or f(x)  b or f(x) = b Feasible Solution: A non-negative solution which satisfies all the constraints is known as feasible solution. The region comprising all feasible solutions is referred to as feasible region.

By-Pin2 · By-Pin2 · Optimal Solution:

The solution where the objective function is maximize or minimize is known as optimal solution.

General linear programming problem:

Consider the following optimize

n n z c x c x c x subjectto

n n a x a x a x which is b

n n a x a x a x which is b

m m mn n m a x a x a x which is b

n x x x  In summation form the above problem can be written as

n j j i z c x

1,2,...., n ij j i j a x b i m

1,2,...., j and x j n

j s ij s i s c a b are decision variables.

Graphical Solution of Linear Programming problem:

Q2. Solution:

x  2

x = 2 At (0,0) 0  4

By-Pin2 · By-Pin2 · Corner Point:

A point of solution region where two of its boundary line are intersect is called corner point or vertex of solution region.

Example:

Maximize z x x x x

x  x 

x x 

Solution: Given that 1 x x

x x

At (5,0) x x

At (0, 5) x x or x

And x 

(2,0) x And x

(0,5) x 

At (0,0) for 1 x x

0 5 true

For x 

0 2 false

Solution away from origin

For x 