Description

LINEAR PROGRAMMING — Complete Study Guide (007)
Advanced Level Mathematics — Form Five

A comprehensive, exam-ready resource covering everything from LP formulation and graphical solutions to transportation problems — all in one clearly organized document.

📘 What’s Inside
1. Introduction to Linear Programming

Clear definition: optimizing an objective subject to linear constraints

Key components table: objective function, decision variables, constraints, feasible region

Real-life applications

2. Formulation of LP Problems

5-step formulation procedure — from identifying variables to writing the complete model

Standard form of an LP model

3 fully worked formulation examples:

Production problem (maximizing profit)

Diet problem (minimizing cost)

Blending problem (coffee mixing with percentages)

3. Graphical Method for Two Variables

5-step graphical procedure

6 fully worked examples:

Maximization problem (solving simultaneous equations for the corner point)

Minimization problem (identifying the binding constraint)

Alternative minimization

Maximization with special cases

Multiple optimal solutions — recognizing when an edge gives the same value

Unbounded region — identifying when no finite maximum exists

No feasible solution — recognizing inconsistent constraints

4. Transportation Problems

Definition and key components diagram (sources → destinations)

Mathematical formulation: decision variables, objective, supply and demand constraints

2 fully worked formulation examples:

Simple 2-warehouse, 3-store problem

Balanced 3-factory, 4-warehouse problem with cost matrix

Network representation

5. Minimum Cost Method (NW Corner Rule)

5-step procedure explained

Fully worked example allocating step by step through the cost matrix

Total cost calculation

6. Optimality Check — MODI Method

6-step MODI procedure

Fully worked example:

Calculating u and v values

Finding improvement indices for unoccupied cells

Identifying a negative index (−5)

Tracing the closed loop

Determining θ and forming the new improved solution

Recalculating cost — showing exactly how much is saved

7. Chapter Summary

Key concepts table (objective function, decision variables, constraints, feasible region, corner point, optimal solution)

Graphical method summary

Special cases table (unique solution, multiple optimal solutions, unbounded, no solution)

8. Practice Questions

Three sections: Formulation (production and animal feed problems), Graphical Solutions (maximize and minimize), and Transportation (NW corner method with a 3×4 cost matrix)

✅ Why This Guide Works
Every LP type demonstrated — maximization, minimization, unbounded, no solution, and multiple optimal solutions

Simultaneous equations shown in full — no skipped steps when finding corner points

MODI method fully worked — from initial solution to improved solution with the exact cost saving calculated

Exam-focused — formulation templates, step-by-step procedures, and practice questions by topic

Complete coverage — from writing an LP model to optimizing a transportation problem in a single document

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