This chapter focuses on linear programming, a method used to optimize certain objectives within given constraints, which is applicable in various fields like economics and management.
Linear Programming – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - II, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Linear Programming chapter of Mathematics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Objective Function: Z = ax + by
Z represents the objective function, a linear function to be maximised or minimised, where 'a' and 'b' are constants, and 'x' and 'y' are decision variables.
Inequality Constraints: ax + by ≤ c
Represents a constraint where 'c' is the upper limit of resources or capabilities. Ensures solutions are within feasible limits.
Non-negativity Constraints: x ≥ 0, y ≥ 0
Ensures that the decision variables x and y cannot take negative values, reflecting real-world constraints.
Maximisation Problem: Max Z = cx + dy
Where 'c' and 'd' are coefficients representing profit contributions of each decision variable (x and y), this setup is used to achieve maximum profit.
Minimisation Problem: Min Z = cx + dy
This involves determining the minimum cost or resource usage, similar to maximisation but focusing on reducing outputs.
Feasible Region: defined by inequalities
The region where all constraints overlap represents feasible solutions to the linear programming problem.
Corner Point Method Steps: 1. Identify vertices 2. Evaluate Z at vertices
This method states that optimal solutions occur at corner points of the feasible region, which must be evaluated for value.
Theorem 1: Optimal value occurs at vertices
The maximum or minimum of the objective function exists at one or more vertex points of the feasible region.
Theorem 2: Bounded Feasible Region
If the feasible region is bounded, both maximum and minimum values will be found at corner points.
Unbounded Region Effect: Infinite solutions possible
In an unbounded feasible region, maximum or minimum values may not exist; evaluation near boundaries is required.
Equations
Investment Constraint: 2500x + 500y ≤ 50000
Represents a limit on resources available for investment in decision variables x and y.
Storage Constraint: x + y ≤ 60
Limits the total number of items stored, ensuring the total of decision variables does not exceed capacity.
Profit Calculation: P = 250x + 75y
P calculates total profit from the sale of items derived from variables x and y.
Slope of line for Limitations: y = mx + b
Helps determine intersection points of inequalities, forming boundaries of the feasible region.
Graph Method: Plot inequalities
Graphically represents constraints and finds feasible regions by shading appropriate areas.
Intersection Point Calculation: Solve linear equations
To find feasible solutions, simultaneous equations representing constraints must be solved.
Evaluating Vertices: Z = aX + bY at vertex points
Determine the value of the objective function at each corner of the feasible region.
Tabulating Corners: List (x, y, Z)
Create a table to assess which vertex offers the maximum or minimum outcomes.
Maximize Z given constraints
A linear programming task focused on increasing output subject to defined restrictions.
Minimize Z given constraints
A task focused on achieving the lowest possible output within the set limits.
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