Operations Research and Linear Programming: Simplex and Graphical Methods Explained

Operations Research

1. Operations Research
Operations Research (OR) is a scientific and mathematical approach used to help managers make the best possible decisions under given constraints and limited resources.

Scope

• Production – production planning and product mix.
• Marketing – advertising and sales decisions.
• Finance – investment and budgeting.
• HR – manpower planning and scheduling.
• Transportation – reducing transportation cost.
• Inventory – managing stock and resources.
• Project Management – planning and scheduling.

Limitations

• Depends on accurate data.
• Mathematical models may be complex.
• Models are based on assumptions that may not reflect reality.
• Human behaviour is difficult to express mathematically.
• Requires skilled people and technology.
• The mathematically optimal solution may not always be practical.

Linear Programming

4. Linear Programming
Linear Programming is a mathematical technique used to find the optimum allocation of limited resources among competing activities, subject to certain constraints. It is generally used to:

• Maximize → Profit / Revenue / Production
• Minimize → Cost / Time

Basic Terms of LPP

• Decision Variables – Unknown quantities that we need to determine.
  Example: x = units of Product A, y = units of Product B
• Objective Function – Shows what we want to maximize or minimize.
  Example: Maximize Z = 40x + 30y
• Constraints – Limitations on resources.
  Example: 2x + y ≤ 100
• Non-negativity – Variables cannot be negative.
  Example: x ≥ 0, y ≥ 0
• Feasible Solution – A solution satisfying all constraints.
• Optimal Solution – The feasible solution giving the maximum or minimum objective value.

Applications of LPP in Business

LPP helps businesses in:

• Product mix – deciding how much of each product to produce.
• Production planning – optimum use of labour, machines and materials.
• Resource allocation – allocating limited resources.
• Advertising – allocating advertising budget.
• Investment – selecting investment combinations.
• Workforce planning – allocating employees.
• Transportation – reducing distribution cost.

Simplex Method

7. Simplex Method
The Simplex Method is an iterative method used to find the optimal solution of an LPP by moving from one feasible solution to another until the optimum solution is reached.

Simplex Steps

• 1. Convert to standard form – Add slack variables.
• 2. Prepare initial simplex table
• 3. Select entering variable – For the common Z − ... = 0 table, choose the most negative value in the Z-row.
• 4. Ratio test – Ratio = RHS / positive coefficient. Choose the smallest positive ratio.
• 5. Select pivot element – Intersection of the entering-variable column and the leaving-variable row.
• 6. Perform pivoting – Make pivot = 1 and other values in that column = 0.
• 7. Repeat – Continue until there are no negative values in the Z-row.
• 8. Read final solution – This is your optimal solution.

Simplex Exam Trick

If your table is written like: Z − 3x − 2y = 0, then:

• Entering variable: Most negative number
• Leaving variable: Smallest positive ratio
• Optimal: No negative number in the Z-row

Slack Variable

For a constraint 2x + y ≤ 100, add a slack variable: 2x + y + s₁ = 100.

The slack variable represents unused resources.

For two constraints: x + y + s₁ = 4 and 2x + y + s₂ = 5.

Graphical Method

The Graphical Method is used mainly when there are two variables.

Steps:

• 1. Formulate LPP
• 2. Convert inequalities into equations
• 3. Find intercepts
• 4. Draw lines on graph
• 5. Identify feasible region
• 6. Find corner points
• 7. Calculate Z at each corner point
• 8. Highest Z = Maximum, Lowest Z = Minimum

REMEMBER: Graphical method = Corner points → Calculate Z → Choose best value.