Introduction to Linear programming problem

1. OPERATIONS RESEARCH

2. • Operations Research (Management Science) is a scientific approach to
decision making that seeks to best design and operate a system, usually
under conditions requiring the allocation of scarce resources.”
• OR is scientific technique
• It is problem solving technique
• It is for executives to take decision for organizations based on some
constraints

3. • Finding optimal trajectory in missile launching
• Design of structures like bridges, dams etc with minimum cost
• Shortest route taken by salesmen visiting various cities
• Optimal production planning, controlling and scheduling
• Inventory management
• Transportation, loading and unloading
• Work force planning
APPLICATIONS OF OR

4. Step 1. Formulate the Problem
It involves description of the objective, identification
of the decision variables and constraints of the system
Step 2. Formulate a Mathematical Model of the
Problem
Mathematical model consist of set of equations that
describe the system.
Step 3. Deriving the solution from the model
Determine the value of decision variables that
optimizes the given objective function.
THE METHODOLOGY OF OR

5. Step 4. Select a Suitable Alternative
Given a model and a set of alternatives, the analyst chooses the alternative (if
there is one) that best meets the organization's objectives.
Sometimes the set of alternatives is subject to certain restrictions and
constraints. In many situations, the best alternative may be impossible or too
costly to determine.
Step 5. Validation of the model:
Model is valid if it can give reasonable prediction of performance of the
system
Step 6:Controlling the solution
Specify the solution procedure and operating procedure for implementation
Step 7. Implement and Evaluate Recommendation
Tested results of he model should be implemented to work.
CONT..

6. • The number of assumptions made should be as few as possible.
• Model should be simple and consistent.
• Number of variables utilized should be small.
• It should be easy and economical to construct.
FEATURES OF OR

7. Linear programming is an optimization technique for finding an optimal value
of a function, called objective function, of several independent variables. The
variables are subjected to constraints.
Steps in formulation of linear Programming problem:
Step1:
Write down the decision variables
Step2:
Formulate the objective function to be optimized as a linear function of
decision variables.
Step 3:
Formulate the constraints as linear inequalities or equations in terms of
decision variables.
Step 4:
Add non negative constraints for decision variables.
LINEAR PROGRAMMING
PROBLEM

8. To find n decision variables x1,x2,x3……xn to maximize or minimize the objective
function
Z= C1x1+C2x2+C3x3……+Cnxn
Satisfying m constraints
a11x1+a12x2+a13x3+ …+a1nxn(<= = >=)b1
a21x1+a22x2+a23x3+ …+a2nxn(<= = >=)b2
.
.
.
am1x1+am2x2+am3x3+ …+amnxn(<= = >=)bm
Non negativity constraints x1>=0,x2 >=0,x3 >=0……xn >=0
Z is objective function, C1,C2,C3,……Cn are coefficients in the objective function.
a11, a21,am1 are co-efficient of constraints.
b1 , b2 , bm are resources.
MATHEMATICAL FORMULATION
OF LPP