Summary

1. Key Terminology in Linear Programming

• Decision Variables: The unknown quantities to be determined, typically denoted by $x$ and $y$.
• Objective Function: The linear function $Z = ax + by$ which needs to be maximized or minimized.
• Constraints: The linear inequalities or equations representing the restrictions on the resources, such as $cx + dy \le e$, $cx + dy \ge e$, or $cx + dy = e$.
• Non-negativity Restrictions: The conditions $x \ge 0, y \ge 0$ ensuring that physical quantities cannot be negative.
• Feasible Region: The common region determined by all the constraints, including non-negativity constraints. Every point inside or on its boundary is a feasible solution.
• Optimal Solution: Any feasible point that gives the maximum or minimum value of the objective function.

2. Fundamental Theorems of LPP

• Theorem 1 (Corner Point Theorem): If an optimal value (maximum or minimum) of the objective function exists for a linear programming problem, then it occurs at a corner point (vertex) of the feasible region.
• Theorem 2: If the feasible region is bounded, then the objective function $Z = ax + by$ has both a maximum and a minimum value, and these occur at corner points of the feasible region.
• Important Note: If the objective function has the same optimal value at two adjacent corner points, then every point on the line segment joining them also gives the same optimal value.

3. The Corner Point Method (Steps to Solve)

1. Graph the Constraints: Convert inequalities into equations, draw the corresponding lines, and identify the correct half-planes.
2. Locate the Feasible Region: Find the common region satisfying all the constraints.
3. Find Corner Points: Determine the coordinates of all vertices of the feasible region, either directly from the graph or by solving equations of intersecting boundary lines.
4. Evaluate the Objective Function: Substitute each corner point into $Z = ax + by$.
5. Identify the Optimum: The largest value gives the maximum, and the smallest value gives the minimum, provided the optimum exists.

4. Rules for Unbounded Regions

If the feasible region is unbounded, a maximum or minimum may or may not exist.

• For Maximization: Suppose the largest value obtained at the corner points is $M$. Consider the open half-plane $ax + by > M$.
• If this half-plane has no common point with the feasible region, then $M$ is the true maximum.
• If it has common points, then the problem has no maximum value.
• For Minimization: Suppose the smallest value obtained at the corner points is $m$. Consider the open half-plane $ax + by < m$.
• If this half-plane has no common point with the feasible region, then $m$ is the true minimum.
• If it has common points, then the problem has no minimum value.

5. Special Cases

• Multiple Optimal Solutions: If the maximum or minimum value occurs at two adjacent corner points, then it occurs at every point on the line segment joining them.
• Infeasible Problem: If the constraints contradict each other, the feasible region is empty and no feasible solution exists.
• Redundant Constraint: A constraint whose removal does not change the feasible region.

6. Important Exam Tips for Board Exams

• Formulation Marks: Read word problems carefully. Define your variables clearly, write the objective function separately, and list all constraints neatly.
• Graph Presentation: Use a proper scale, label the axes, write the equations of boundary lines, and shade the feasible region clearly.
• Corner Point Table: Always prepare a neat table showing each corner point and the corresponding value of $Z$.
• Final Conclusion: End with a complete sentence, such as: "Therefore, the maximum profit is Rs 1500 when 10 units of A and 20 units of B are produced."

7. Important Exam Tips for JEE Main & Advanced

• Fast Intersections: Make a rough but correct sketch first, then solve only the equations that actually form the boundary of the feasible region.
• Modulus Constraints: Inequalities like $|x| + |y| \le a$ represent a square tilted by $45^\circ$, centered at the origin, with vertices $(a,0)$, $(0,a)$, $(-a,0)$, and $(0,-a)$.
• Parameter Matching ($px + qy$): If an objective function $Z = px + qy$ has the same optimal value at two corner points, evaluate $Z$ at both points and equate the results to find the relation between $p$ and $q$.
• Unbounded Regions: Do not assume that an unbounded feasible region means no optimum exists. For many maximization problems with positive coefficients, the maximum may fail to exist, but you must verify this using the objective function and the direction in which it increases.