1. Meaning of a Mathematical Model
A mathematical model of a linear programming problem is the complete set of equations and inequalities that describe the real situation. It translates words into symbols so that the problem can be analyzed and solved systematically.
The model includes the objective function, the constraints, and the non-negativity conditions.
2. Components of a Mathematical Model
Every LPP model has a few essential parts. Writing them clearly helps understand the structure of the problem.
2.1. Decision Variables
These are symbols used to represent the unknown quantities we want to determine. Usually written as \( x, y, z \), etc.
Example: Let \( x \) and \( y \) be the units of two products produced in a workshop.
2.2. Objective Function
This is the expression to be maximized or minimized. It shows the total profit, total cost, or any quantity of interest.
Example: Maximize profit \( Z = 5x + 7y \).
2.3. Constraints
These are the restrictions or limits that must be satisfied. They come from limited resources such as time, money, or material.
Example constraint:
\( 3x + 2y \le 20 \)
This might mean the total machine-time cannot exceed 20 hours.
2.4. Non-Negativity Conditions
Since negative production or negative usage does not make sense, we always include:
\( x \ge 0, \quad y \ge 0 \)
3. General Form of an LPP
The mathematical model of any linear programming problem usually looks like this:
\( \text{Maximize/Minimize } Z = c_1x_1 + c_2x_2 + \dots + c_nx_n \)
Subject to:
\( a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n \le b_1 \)
\( a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n \le b_2 \)
\( \vdots \)
\( x_1, x_2, \dots, x_n \ge 0 \)
4. Steps to Form a Mathematical Model
To convert a real situation into an LPP model, follow these steps:
- Step 1: Identify the decision variables.
- Step 2: Write the objective function using these variables.
- Step 3: Identify all limitations and express them as constraints.
- Step 4: Add the non-negativity conditions.
4.1. Example of Model Formation
A workshop makes two products, A and B.
- Profit per unit of A: 40
- Profit per unit of B: 50
- Machine time: A requires 2 hours, B requires 4 hours
- Total machine time available: 40 hours
Step 1: Let \( x \) = units of A, \( y \) = units of B.
Step 2: Objective function:
\( Z = 40x + 50y \)
Step 3: Constraint:
\( 2x + 4y \le 40 \)
Step 4: Non-negativity:
\( x \ge 0, \quad y \ge 0 \)