The model components of linear programming are decision variables, an objective function, and constraints.
Linear programming is a mathematical technique used to optimize a linear objective function subject to a set of linear constraints. These components work together to define and solve optimization problems. Let's break down each component:
1. Decision Variables
- Definition: These are the variables that the decision-maker controls. They represent the quantities that can be changed to achieve the optimal solution.
- Example: If a company is trying to maximize profit from producing two products, X and Y, the decision variables would be the quantity of product X to produce and the quantity of product Y to produce. These are typically denoted as x and y.
2. Objective Function
- Definition: This is a mathematical expression that defines the goal of the problem. It's a linear function of the decision variables that needs to be maximized (e.g., profit) or minimized (e.g., cost).
- Example: Continuing with the previous example, if the profit from product X is $5 per unit and the profit from product Y is $8 per unit, the objective function to maximize profit (Z) would be: Maximize Z = 5x + 8y.
3. Constraints
- Definition: These are linear inequalities or equalities that limit the values of the decision variables. They represent restrictions or limitations on the resources available or the requirements that must be met.
- Examples:
- Resource Constraint: If the production of product X requires 2 hours of machine time and the production of product Y requires 3 hours of machine time, and there are only 120 hours of machine time available, the constraint would be: 2x + 3y ≤ 120.
- Demand Constraint: If the company must produce at least 10 units of product X to meet demand, the constraint would be: x ≥ 10.
- Non-negativity Constraint: Decision variables usually represent physical quantities and cannot be negative. Therefore, constraints like x ≥ 0 and y ≥ 0 are common.
In summary, a linear programming model consists of identifying the decision variables, formulating a linear objective function to be optimized, and defining linear constraints that restrict the possible values of the decision variables. Solving this model involves finding the values of the decision variables that satisfy all the constraints and result in the optimal value of the objective function.
