Linear Programming: Class 12 Mathematics NCERT Chapter 12
Key Features of NCERT Material for Class 12 Maths Chapter 12 – Linear Programming
In the last chapter 11, you learned about 3d geometry. In this chapter, you all will get to know about Linear programming.
Quick revision notes
Linear Programming Problem: A linear programming problem is one in which we need to discover optimal worth (maximum or minimum) of a linear function of a few factors (called objective function) subject to specific conditions that the factors are non-negative and fulfilling by a lot of linear inequalities with factors, are at times called division factors.
Terms identified with Linear Programming
Objective Function: A linear function z = px + qy (p and q are constants) which must be expanded or limited, is called an objective function.
Constraints: The linear inequalities or conditions or limitations on the factors of the linear programming problem are called constraints. The conditions x ≥ 0, y ≥ 0 are called non-negative limitations.
Optimal Value: The maximum or minimum estimation of an objective function is known as its optimal worth.
Advancement Problem: A problem, which tries to amplify or limit a linear function subject to specific constraints as controlled by a lot of linear inequalities, is called a streamlining problem.
Feasible Region: The normal region dictated by all the constraints including non-negative constraints x,y>0 of a linear programming problem is known as the feasible region for the problem. The region other than the feasible region is called an infeasible region. The feasible region is consistently an arched polygon.
Feasible Solutions: Points inside and on the limit of the feasible region speak to possible arrangements of the constraints. Any point outside the feasible region is called an infeasible arrangement.
Optimal Feasible Solution: Any point in the feasible region that gives the optimal estimation of the objective function is known as the optimal feasible arrangement.
Bounded and Unbounded Region: A feasible region of an arrangement of linear inequalities is supposed to be bounded, on the off chance that it very well may be encased inside a circle. Else, it is called unbounded.
Fundamental Theorems for Solving Linear Programming
Hypothesis 1: Let R be the feasible region for a linear programming problem and let z = hatchet + by be the objective function. At the point when z has an optimal worth (maximum or minimum), where the factors x and y are dependent upon constraints depicted by linear inequalities. This optimal worth must happen at a corner point (vertex) of the feasible region.
Note: A corner purpose of a feasible region is a point in the region which is the convergence of two limit lines.
Hypothesis 2: Let R be the feasible region for a linear programming problem and let z = hatchet + by be the objective function. In the event that R is bounded, at that point z has both a maximum and a minimum incentive on R and each of these repeats at a corner purpose of JR.
Note that: Maximum or a minimum may not exist, if the feasible region is unbounded.
Corner Point Method: The corner point technique says that, on the off chance that a maximum or minimum worth exists, at that point it will happen at a corner purpose of the feasible region.
Steps for Applying Corner Point Method
Locate the feasible region of the linear programming problem and decide its corner focuses either by investigation or by settling the two conditions of the lines converging by then.
Assess the objective function z = hatchet + by at each corner point. Leave M and m alone, separately mean the biggest and the littlest estimations of these focuses.
On the off chance that the feasible region is bounded, at that point M and m individually are the maximum and minimum estimations of the objective function at corner focuses.
On the off chance that the feasible region is unbounded ,, at that point
(a) M is the maximum estimation of objective function z, if the open half plane controlled by hatchet + by > M shares no point practically speaking with the feasible region. Something else, z has no maximum worth.
(b) m is the minimum estimation of z, if the open half plane dictated by hatchet + by < m shares no point for all intents and purpose with the feasible region. Something else, z has no minimum worth.
In the event that two corner purposes of the feasible region are both optimal arrangements of a similar kind, i.e both produce a similar maximum or minimum,
at that point any point on the line section joining these two focuses is additionally an optimal arrangement of a similar sort.