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Lesson Plan Math Class XII Ch-12 | Linear Programming
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LESSON PLAN MATHEMATICS CH-12
LINEAR PROGRAMMING PROBLEMS
NAME OF THE TEACHER | DINESH KUMAR | ||||
CLASS | 10+2 | CHAPTER | 12 | SUBJECT | MATHEMATICS |
TOPIC | LINEAR PROGRAMMING | DURATION : 05 Class Meetings |
- Knowledge of Cartesian coordinate system to represent the graph
- Knowledge of representing linear inequalities on the graph and method to find their solution (Chapter 6 class XI)
- Introduction of Linear Programming
- Related terminology such as constraints, objective function, feasible region and optimization.
- Different types of linear programming problems (L.P.).
- Mathematical Formulation of L.P. problems.
- Graphical method of solution for problems in two variables.
- Feasible or infeasible region (bounded or unbounded), Feasible and infeasible solution.
Introduction :
The term Linear means that all inequations, equations and the functions to be maximized or minimized are linear. The term ‘programming’ means planning and it refers to a particular plan amongst several alternatives for maximizing profit or minimizing cost etc. In this chapter we discuss linear programming problems with two variables only.
Constraints:
The inequations or equations in the variables of a LPP which describe the conditions under which the optimization (maximization or minimization) is to be accomplished are called constraints.
Objective Function :
Linear function of the form z = ax + by which is to be maximized or minimized is called objective function. Here a , b are the constants and x , y are the variables.
Feasible Region :
The common solution region of all the inequations or equations related to the particular problem is called a Feasible Region.
Feasible solution:
Every point in the feasible region is called the feasible solution.
Infeasible Solution :
All points outside the feasible region are called infeasible solution.
Optimal Feasible Solution :
A feasible solution of a LPP is said to be an optimal feasible solution if it optimizes(maximize or minimize ) the objective function.
Convex Set :
A set is a convex set, if every point on the line segment joining any two points in it lies in it. Set of all feasible solutions of a LPP is a convex set.
Mathematical Formulations of Linear Programming Problems :
In general there is not any set procedure to formulate LPP. However the following steps will help in the formation of LPP.
a) In every LPP certain decisions are to be made. These decisions are represented by certain decision variables like x and y.
b) Identify the objective function and express it as a linear function of the decision variables x and y.
c) Now find the type of optimization that is maximization or minimization.
d) Identify the set of constraints, in terms of decision variables and express them as the linear inequations or equations.
Formulate the given LPP in mathematical form.
a) Convert all the inequations into equations and draw their graphs.
b) Find the region identified by each inequality. For this choose a check point (0,0) for all graphical lines which are not passing through the origin. If any graph passing through the origin then for that use check point either on the x-axis or on the y-axis.
c) Now find the feasible region determined by all the constraints.
d) Find the coordinates of the vertices(corner points) of the convex polygon of the feasible region.
e) Calculate the value of the objective function at every corner point of the convex polygon and then find the optimal solution of the LPP.
Corner – Point Method of Solving LPP
If feasible region is unbounded then follow the following steps
a) Calculate the maximum and minimum value of the objective function at the corner points of feasible region.
b) Let maximum value is M or minimum value is m
c) Now draw the graph of ax + by = M and find the open half plane ax + by >M. If this open half plane has no point common with the unbounded feasible region, then M is the maximum value of Z. Otherwise Z has no maximum value.
d) Draw the graph of ax + by = m and find the open half plane ax + by < m, If this open half plane has no point common with the unbounded feasible region, then m is the minimum value of Z. Otherwise Z has no minimum value.
Teacher should explain each type of problem in front of the students and help the students in solving more related problems.
- Terms constraints, objective function, feasible region and solution,
- Optimal feasible solution and bounded convex plane.
- Students should know the method finding the feasible region and optimal solution of the constraints by the graphical method and corner point method.
- Review questions given by the teacher.
- Students can prepare the presentation or model on Linear Programming Problems.
- Solve NCERT problems with examples.
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