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

تقدم هذه الاختبار نظرة عامة على نماذج البرمجة الخطية وأهم مكوناتها. ستتعلم كيفية صياغة نموذج البرمجة الخطية من خلال التعرف على الدوال الهدف والقيود والمتغيرات. هذا الاختبار مكتوب لتوضيح الطريقة المستخدمة في تحسين الأهداف المختلفة في القطاعات المختلفة.