Definition: Operation Research is the representation of real world system by mathematical models with the use of quantitative methods for solving models
with a view to optimizing.
Characteristics of OR:
1. It is system oriented.
2. The use of interdisciplinary teams.
3. Application of scientific methods.
4. Uncovering of new problems.
5. Use of Computer
6. Quantitative solution
The cost to the company, if decision A is taken is X, if decision B is taken is Y.
Application of OR:
OR model consist of -
-Decision Variables
-Constraints
-An Objective function
-A solution to the model
Decision Variables:
These are the unknowns to be determined by the solution to the model.
Constraints: Any restrictions on the values of decision variables that are expressed mathematically by means of inequality are often called constraints.
The coefficients of the constraints in the objective function are called the model parameters.
Objective Function:
An objective function is a function of decision variables that the decision maker wants to maximize (profits) or minimize( costs).
Solution:
Any specification of the values for the decision variables is called a solution.
Feasible Solution:
A solution of the model is feasible if it satisfies all constraints.
Optimal solution: A solution of the model is optimal if in addition to being feasible, it yields the best (max/min) value of the objective function.
l,w – decision Variables
max lw – objective function
l>=0, w>=0 – constraints
Formulating mathematical models:
The following methods are used-
1. Linear Programming (LP)
- Solving Linear programming problems
- Graphical methods
- Simplex method
- Duality Theory sensitivity analysis
2. Integer Programming
3. Decision analysis & Games
4. Goal Programming
Conditions of LP:
It is used for optimizing problems that satisfies the following condition-
1. There is a well-defined objective function to be optimized which can be expressed as a linear function of decision variables.
2. There are constraints on the attainment of the objective and they are capable of being expressed as linear inequalities in terms of variables.
3. There are alternative causes of action
4. The decision variables are interrelated and nonnegative.
Redundant Restriction:
The restriction which has no effect on feasible region or solution space is called redundant restriction.
Basic and Nonbasic Variables:
In a set of m╳n equations (m<n) if m variables have a unique solution, then they are basic variables and their solution is referred to as basic solution.
In a set of m╳n equations, the (n-m) variables that are set to zero are known as non-basic variables and their solution is non-basic solution.
Disadvantages of LP:
1. Calculation is lengthy.
2. It may give misleading conclusion unless the problem is correctly constructed.
Assumptions/Properties of LP:
In LP models, the objective and the constraints are all linear. The linearity must satisfy three basic properties.
1. Proportionality:
This property requires the combination of each variables in both the objective function and the constraints to be directly proportional to the value of the variable.
Z= bx1
z∝x1 =>z=kx1
2. Additivity:
This property requires the total combination of all the variables in the objective function and in the constraints to be the direct sum of the individual combination of each variable.
Z=6x1+x2
3. Certainty: All the objective and constraints coefficients of the LP model are deterministic. This means they are known constants.
Steps to formulate LP:
Step-1: We need to study the given situation to find the key decision to be made.
Step-2: We identify the involved variable and designate them by symbols xj(i=1,2,…n)
Step-3: We express the possible alternatives
Step-4: Identifying the objective function and expressing it in linear function of the variables.
Step-5: We express the influencing factors such as limitation or constraints as linear inequalities in terms of the variables of the problem.
Simplex method:
The simplex method is an iterative procedure which gives the solution to a LPP in a finite number of steps.
This is simple iterative procedure to find an optimum basic feasible solution from an initial basic feasible solution by a finite no. of iteration.
These iterations consist of finding a new basic feasible solution which improves the value of objective function from that of previous solution. The process is continued until an optimum basic feasible solution is obtained or there is an indication of an unbounded solution.
Slack variables:
When the constraints are inequations connected by the sign ≤, then in each equation an extra variable is added to the left hand side of the equation to convert it into an equation. These variables are known as slack variables. For example,
X1-2x2+x3≤5
=>x1-2x2+x3+s1=5
The variable s1 is known as slack variable which is non-negative.
Similarly, Surplus Variable;
Characteristics of OR:
1. It is system oriented.
2. The use of interdisciplinary teams.
3. Application of scientific methods.
4. Uncovering of new problems.
5. Use of Computer
6. Quantitative solution
The cost to the company, if decision A is taken is X, if decision B is taken is Y.
Application of OR:
OR model consist of -
-Decision Variables
-Constraints
-An Objective function
-A solution to the model
Decision Variables:
These are the unknowns to be determined by the solution to the model.
Constraints: Any restrictions on the values of decision variables that are expressed mathematically by means of inequality are often called constraints.
The coefficients of the constraints in the objective function are called the model parameters.
Objective Function:
An objective function is a function of decision variables that the decision maker wants to maximize (profits) or minimize( costs).
Solution:
Any specification of the values for the decision variables is called a solution.
Feasible Solution:
A solution of the model is feasible if it satisfies all constraints.
Optimal solution: A solution of the model is optimal if in addition to being feasible, it yields the best (max/min) value of the objective function.
l,w – decision Variables
max lw – objective function
l>=0, w>=0 – constraints
Formulating mathematical models:
The following methods are used-
1. Linear Programming (LP)
- Solving Linear programming problems
- Graphical methods
- Simplex method
- Duality Theory sensitivity analysis
2. Integer Programming
3. Decision analysis & Games
4. Goal Programming
Conditions of LP:
It is used for optimizing problems that satisfies the following condition-
1. There is a well-defined objective function to be optimized which can be expressed as a linear function of decision variables.
2. There are constraints on the attainment of the objective and they are capable of being expressed as linear inequalities in terms of variables.
3. There are alternative causes of action
4. The decision variables are interrelated and nonnegative.
Redundant Restriction:
The restriction which has no effect on feasible region or solution space is called redundant restriction.
Basic and Nonbasic Variables:
In a set of m╳n equations (m<n) if m variables have a unique solution, then they are basic variables and their solution is referred to as basic solution.
In a set of m╳n equations, the (n-m) variables that are set to zero are known as non-basic variables and their solution is non-basic solution.
Disadvantages of LP:
1. Calculation is lengthy.
2. It may give misleading conclusion unless the problem is correctly constructed.
Assumptions/Properties of LP:
In LP models, the objective and the constraints are all linear. The linearity must satisfy three basic properties.
1. Proportionality:
This property requires the combination of each variables in both the objective function and the constraints to be directly proportional to the value of the variable.
Z= bx1
z∝x1 =>z=kx1
2. Additivity:
This property requires the total combination of all the variables in the objective function and in the constraints to be the direct sum of the individual combination of each variable.
Z=6x1+x2
3. Certainty: All the objective and constraints coefficients of the LP model are deterministic. This means they are known constants.
Steps to formulate LP:
Step-1: We need to study the given situation to find the key decision to be made.
Step-2: We identify the involved variable and designate them by symbols xj(i=1,2,…n)
Step-3: We express the possible alternatives
Step-4: Identifying the objective function and expressing it in linear function of the variables.
Step-5: We express the influencing factors such as limitation or constraints as linear inequalities in terms of the variables of the problem.
Simplex method:
The simplex method is an iterative procedure which gives the solution to a LPP in a finite number of steps.
This is simple iterative procedure to find an optimum basic feasible solution from an initial basic feasible solution by a finite no. of iteration.
These iterations consist of finding a new basic feasible solution which improves the value of objective function from that of previous solution. The process is continued until an optimum basic feasible solution is obtained or there is an indication of an unbounded solution.
Slack variables:
When the constraints are inequations connected by the sign ≤, then in each equation an extra variable is added to the left hand side of the equation to convert it into an equation. These variables are known as slack variables. For example,
X1-2x2+x3≤5
=>x1-2x2+x3+s1=5
The variable s1 is known as slack variable which is non-negative.
Similarly, Surplus Variable;
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