نموذج البرمجة الخطية

Introduction to Linear Programming Models

• Linear programming (LP) is a quantitative method for optimizing a single objective function while considering various constraints.
• LP is used in various sectors, including production, distribution, finance, etc.
• It aims to maximize profits or minimize costs.

Components of a Linear Programming Model

• Objective Function: The mathematical expression representing the quantity to be maximized or minimized.
• Constraints: Restrictions on the use of resources or limitations placed on the decision variables. These limitations are expressed as linear inequalities.
• Decision Variables: The unknowns that need to be determined to optimize the objective function.
• Non-negativity constraints: These constraints stipulate that decision variables cannot be negative, which is a practical requirement for most real-world applications.

Steps for Formulating a Linear Programming Model

• Define the Decision Variables: Clearly identify the variables to be determined.
• State the Objective Function: Formulate the mathematical expression to either maximize or minimize the desired outcome.
• Formulate the Constraints: Express the limitations and restrictions on the resources or conditions as linear inequalities.
• Specify Non-negativity Constraints: Ensure that all decision variables are non-negative.

Example of a linear programming problem

• A company produces two products (A and B). Each unit of product A requires 30 cubic feet of wood and 5 labor hours, and yields a profit of 50$. Each unit of product B requires 20 cubic feet of wood and 10 labor hours, and yields a profit of 75$.
• Resources: 400 cubic feet of wood and 120 labor hours are available.
• Variables:
  • x1: number of units of product A
  • x2: number of units of product B
• Objective function: maximize profit (z) = 50x1 + 75x2
• Constraints:
  • 30x1 + 20x2 ≤ 400 (wood constraint)
  • 5x1 + 10x2 ≤ 120 (labor constraint)
• Non-negativity constraints: x1, x2 ≥ 0