Linear programming is a mathematical method for finding the best solution under certain constraints. Imagine you are a factory manager who needs to decide how many units of product A and product B to produce. You want to maximize profit, but you have limited materials and time. Linear programming helps you find the optimal production plan.
Linear programming has three essential elements. First, the objective function - this is what you want to maximize or minimize, like maximizing profit from selling cookies. Second, constraints - these are the limitations you face, such as limited flour or baking time. Third, decision variables - these are the quantities you need to determine, like how many chocolate cookies and oat cookies to make.
Let's look at a concrete example. Suppose you run a bakery and want to maximize profit from making chocolate cookies and oat cookies. Each chocolate cookie uses 10 grams of flour and sells for 2 yuan profit. Each oat cookie uses 8 grams of flour and sells for 1.5 yuan profit. You have 100 grams of flour and 12 hours of time available. The yellow area shows all possible production combinations that satisfy your constraints.
To find the optimal solution, we check the profit at each corner point of the feasible region. At point A (0,0), profit is 0. At point B (0,12), profit is 18 yuan. At point C (2,10), profit is 19 yuan. At point D (10,0), profit is 20 yuan. The maximum profit of 20 yuan occurs at point D, meaning we should produce 10 chocolate cookies and 0 oat cookies.
In summary, linear programming is a powerful mathematical tool that helps us find the best solution under given constraints. It consists of three key elements: objective function, constraints, and decision variables. The optimal solution always lies at a corner point of the feasible region. Linear programming is widely used in production planning, resource allocation, and many other real-world problems, making it an essential tool in operations research and management science.