03_Ex 5.3

Exercise Questions

Questions	Links
Q1. Maximize $f(x, y)=2 x+5 y$	5.3 Q-1
Q2. Maximize $f(x, y)=x+y$	5.3 Q-2
Q3. Maximize $z=2 x+3 y$	5.3 Q-3
Q4. Maximize $z=2 x+y$	5.3 Q-4
Q5. Maximize the function is defined $f(x, y)=2 x+3 y$	5.3 Q-5
Q6. Maximize $z=3 x+y$ …	5.3 Q-6
Q7. Each unit of food $X$ costs Rs. 25 and…	5.3 Q-7
Q8. Dealer wishes to purchase a number of fans and…	5.3 Q-8
Q9. A machine can produce product A by…	5.3 Q-9

Overview

This exercise focuses on the optimization of linear functions under given constraints, a key problem in linear programming. The objective is to find the maximum or minimum value of a function given a set of inequalities, often representing physical, economic, or social constraints. Understanding how to solve these problems is essential for a wide range of applications such as resource allocation, production planning, and operations research.

Key Concepts

Linear Programming: A method to achieve the best outcome (maximum or minimum) in a mathematical model whose requirements are represented by linear relationships.
Feasible Region: The region of the graph that satisfies all constraints, typically forming a polygon with vertices representing potential optimal solutions.
Corner Point Theorem: The optimal solution of a linear programming problem occurs at one of the corner points (vertices) of the feasible region.

Important Formulas

Objective Function: The function that is being maximized or minimized, e.g., $f(x, y) = 2x + 5y$ .
Constraints: Inequalities that limit the values of $x$ and $y$ , e.g., $2y - x \leq 8$ and $x - y \leq 4$ .
Corner Points: The points where the constraints intersect and are tested for optimality by evaluating the objective function.

Tips and Tricks

Graphing the Constraints: Always graph the constraints to visualize the feasible region. Use intersection points to find the corner points.
Evaluating at Corner Points: Once the corner points are identified, evaluate the objective function at each point to find the maximum or minimum value.
Check for Validity: Ensure all points satisfy the constraints before calculating the objective function.

Summary

This exercise illustrates the process of maximizing a linear function subject to various constraints. The steps involve:

Graphing the constraints and identifying the feasible region.
Identifying the corner points of the feasible region.
Evaluating the objective function at each corner point to find the maximum value.
Analyzing the results to determine the optimal solution.

Reference

By Sir Shahzad Sair: