Berikut adalah artikel tentang resep lengkap tentang: Pemrograman Linear, Perumusan Model dan Solusi Grafis.
Linear Programming: Formulation of Model and Graphical Solution - A Complete Recipe
Linear programming (LP) is a powerful mathematical method used to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. This comprehensive guide will walk you through the entire process, from problem formulation to graphical solution. Weβll provide a step-by-step recipe, ensuring you can successfully tackle your own linear programming problems.
What is Linear Programming?
Linear programming is a technique used to optimize an objective function, subject to a set of constraints. The objective function and constraints are all expressed as linear equations or inequalities. Think of it as finding the best solution within a defined set of limitations. This is widely applicable in various fields, including:
- Business: Optimizing production schedules, inventory management, resource allocation.
- Engineering: Designing efficient systems, optimizing network flows.
- Finance: Portfolio optimization, risk management.
Formulating the Linear Programming Model: A Step-by-Step Guide
To successfully solve a linear programming problem, a clear and accurate model is crucial. Here's how to formulate it:
1. Define the Decision Variables
Identify the key factors you can control. These become your decision variables, often represented by 'x' and 'y' (or more, depending on the complexity). For example:
- Problem: A bakery wants to maximize profit from selling cakes and cookies.
- Decision Variables:
x= number of cakes,y= number of cookies
2. Define the Objective Function
This is what you want to maximize or minimize. It's a linear equation expressing the relationship between the decision variables and the desired outcome. For our bakery example:
- Objective: Maximize profit (Z). Assume each cake yields $5 profit and each cookie yields $2 profit.
- Objective Function:
Z = 5x + 2y
3. Define the Constraints
These are the limitations on your decision variables. They represent real-world restrictions, such as resource availability, production capacity, etc. Constraints are expressed as linear inequalities. For our bakery example:
- Constraint 1 (Oven Time): Each cake requires 20 minutes in the oven, each cookie requires 5 minutes. The oven is available for 400 minutes per day. This translates to:
20x + 5y β€ 400 - Constraint 2 (Ingredient Availability): The bakery has limited amounts of flour and sugar. These can be expressed as additional inequalities.
- Non-negativity Constraints: You cannot produce a negative number of cakes or cookies. Thus:
x β₯ 0,y β₯ 0
4. Complete Model Representation
Putting it all together, the complete linear programming model for our bakery example is:
Maximize: Z = 5x + 2y
Subject to:
20x + 5y β€ 400
[Other ingredient constraints]
x β₯ 0
y β₯ 0
Graphical Solution: Visualizing the Optimal Solution
For problems with two decision variables (like our example), a graphical solution is straightforward:
1. Plot the Constraints
Treat each inequality as an equation and plot it on a graph. Shade the feasible regionβthe area where all constraints are satisfied.
2. Identify the Corner Points
The optimal solution always lies at one of the corner points of the feasible region. Find the coordinates of these points.
3. Evaluate the Objective Function
Substitute the coordinates of each corner point into the objective function. The point that yields the highest (or lowest, depending on your objective) value is the optimal solution.
4. Interpret the Results
The optimal solution gives you the values of the decision variables (number of cakes and cookies) that maximize profit, along with the maximum profit itself.
Conclusion
Linear programming provides a powerful framework for optimizing decision-making under constraints. By systematically following the steps of model formulation and employing a graphical solution (for smaller problems), you can efficiently find the best possible outcome for various real-world scenarios. Remember to meticulously define your variables, objective, and constraints for accurate and meaningful results. This "recipe" will serve as a valuable tool in tackling a wide array of linear programming challenges.
