5.11 Linear Programming
3 min read•june 18, 2024
Linear programming helps solve real-world problems. It's all about finding the best solution within given . This powerful tool is used in business, economics, and engineering to maximize profits or minimize costs.
The process involves setting up an , graphing constraints, and finding the . By identifying and constraints, we can visualize the problem and determine the best possible outcome.
Linear Programming
Objective function formulation
Top images from around the web for Objective function formulation
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
linear programming - Canonical form simplex method - Mathematics Stack Exchange View original
Is this image relevant?
optimization - Linear Programming Formulation of Traveling Salesman (TSP) in Wikipedia ... View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
linear programming - Canonical form simplex method - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
Top images from around the web for Objective function formulation
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
linear programming - Canonical form simplex method - Mathematics Stack Exchange View original
Is this image relevant?
optimization - Linear Programming Formulation of Traveling Salesman (TSP) in Wikipedia ... View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
linear programming - Canonical form simplex method - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
- Linear equation represents goal of problem, either maximizing or minimizing quantity
- involve maximizing profit, revenue, or production (widget manufacturing)
- involve minimizing cost, time, or resource usage (transportation logistics)
- Identify decision variables in problem
- Decision variables are unknowns that need to be determined to optimize objective function
- Represented by symbols such as , , or other letters (product quantities)
- Write objective function in terms of decision variables
- Combine decision variables with respective coefficients (unit profits)
- If represents units of Product A and represents units of Product B, with profit per unit of A as 15, objective function to maximize profit would be
Graphing linear inequality systems
- Constraints are limitations or restrictions on decision variables
- Represented as linear inequalities (production time limits)
- If company has limited hours available for production, constraint could be , where and are decision variables and 120 is total available hours
- Graph each constraint as straight line on coordinate plane
- Convert inequality to equation by replacing inequality sign with equals sign
- Plot line on graph
- Determine which side of line satisfies inequality and shade that region ()
- Feasible region is area on graph where all constraints are satisfied simultaneously
- Intersection of all shaded areas corresponding to constraints
- Any point within feasible region represents possible solution to linear programming problem (production plan)
- is crucial for determining valid solutions
Optimal solution determination
- are vertices of feasible region, where constraint lines intersect
- Also called
- Represent potential optimal solutions to linear programming problem
- Find coordinates of each corner point by solving system of equations formed by intersecting constraint lines
- Use substitution or elimination methods to solve for and values
- Evaluate objective function at each corner point
- Substitute and values of each corner point into objective function
- Calculate value of objective function at each corner point (total profit)
- Compare values of objective function at corner points
- For maximization problem, corner point with highest objective function value is optimal solution
- For minimization problem, corner point with lowest objective function value is optimal solution
- If objective function has same value at multiple corner points, there are multiple optimal solutions ()
Optimization and Solution Methods
- Linear programming is used for optimization problems in various fields
- Objective is to find the best solution among all feasible alternatives
- is an efficient algorithm for solving linear programming problems
- Systematically moves from one feasible solution to another, improving the objective function value
- form the basis of constraints and objective functions in linear programming
Key Terms to Review (21)
≤: The symbol '≤' represents a relationship in mathematics indicating that one value is less than or equal to another. This symbol is crucial for expressing inequalities and understanding the boundaries of solutions in various mathematical contexts, helping to determine feasible solutions, especially in optimization problems.
≥: The symbol '≥' represents the concept of 'greater than or equal to' in mathematics, establishing a relationship between two values. This symbol is crucial in expressing linear inequalities, where one side of the inequality can either exceed or be equal to the other side. It helps define boundaries in mathematical expressions and is foundational in various applications like optimization and systems of inequalities.
Alternative optima: Alternative optima refer to different solutions in a linear programming problem that yield the same maximum or minimum objective function value. This concept is significant because it shows that there may be multiple ways to achieve the same outcome while satisfying the constraints of the problem, leading to various feasible solutions.
Constraints: Constraints are conditions or limitations placed on a mathematical problem that define the feasible region in which a solution must lie. They serve as boundaries that restrict the possible values of decision variables in optimization problems. In the context of linear programming, constraints help to model real-world limitations, ensuring that solutions are practical and applicable.
Corner points: Corner points are specific points on the boundary of a feasible region in linear programming where the optimal solution to a linear optimization problem can be found. These points are significant because, according to the Fundamental Theorem of Linear Programming, if there is an optimal solution, at least one of the corner points will provide it. Evaluating these corner points is essential to determine which one yields the best value for the objective function.
Decision variables: Decision variables are the unknown values in a mathematical model that decision-makers will determine to optimize an objective function. They represent the choices available to the decision-maker and are essential in formulating linear programming problems. By adjusting these variables, one can explore different scenarios to identify the best possible outcomes under given constraints.
Elimination method: The elimination method is a technique used to solve systems of linear equations by removing one variable, allowing for the straightforward solution of the remaining variable. This method involves adding or subtracting equations to eliminate one of the variables, simplifying the process of finding the values of both variables in a system. It is particularly useful when dealing with larger systems where substitution may become cumbersome.
Extreme points: Extreme points are the vertices of a feasible region in linear programming, where the optimal solutions to a linear objective function may be found. They play a critical role in identifying the best possible outcomes for given constraints, as linear programming solutions are often located at these vertices rather than within the interior of the feasible region. The concept emphasizes how solutions can vary significantly depending on which extreme point is chosen.
Feasibility: Feasibility refers to the condition of being possible or practical within a given context, especially concerning the constraints outlined by linear inequalities. In scenarios involving two-variable inequalities, feasibility indicates whether a certain set of values satisfies all the conditions imposed by those inequalities. This concept is crucial for determining valid solutions in optimization problems where certain criteria must be met.
Feasible region: The feasible region is the set of all possible solutions that satisfy a given set of constraints in a mathematical context. This concept is crucial when dealing with inequalities and optimization problems, as it visually represents the area where all constraints overlap. The feasible region is often graphed in two-dimensional space, showing the combinations of variables that meet all specified conditions.
Graphical method: The graphical method is a visual technique used to solve linear programming problems by plotting the constraints on a graph and identifying the feasible region where all constraints overlap. This method allows for the determination of optimal solutions by examining the vertices of the feasible region, as linear programming assumes that the maximum or minimum values occur at these points.
Linear equations: Linear equations are mathematical statements that express the equality of two linear expressions, typically written in the form $ax + by = c$, where 'a' and 'b' are coefficients, 'x' and 'y' are variables, and 'c' is a constant. These equations represent straight lines when graphed on a coordinate plane, and they are fundamental in various applications, particularly in optimization and resource allocation.
Linear Inequality: A linear inequality is a mathematical expression that shows the relationship between two values where one value is not equal to the other, typically expressed in the form of $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$. These inequalities can represent ranges of values rather than single points, and they are used to model situations where constraints exist, making them essential in understanding how to evaluate and compare different scenarios.
Max Z: Max Z represents the objective function value that is maximized in a linear programming problem. This term plays a central role in optimization, where the goal is to find the highest possible value of the objective function given certain constraints. In practical terms, max Z often relates to maximizing profits, productivity, or efficiency within specified limits.
Maximization problems: Maximization problems are mathematical challenges that involve finding the largest possible value of a particular function, often subject to certain constraints. These problems are crucial in decision-making processes where optimal solutions are needed, such as maximizing profit, minimizing cost, or achieving the best possible outcome given limited resources. They often utilize techniques like linear programming to identify the maximum point within feasible regions defined by constraints.
Minimization problems: Minimization problems are mathematical optimization challenges where the goal is to find the lowest value of a particular function subject to a set of constraints. These problems often involve maximizing efficiency or reducing costs in various scenarios, such as resource allocation or transportation. They can be formulated and solved using linear programming techniques, which provide systematic methods to identify optimal solutions within defined boundaries.
Objective Function: An objective function is a mathematical expression that defines the goal of an optimization problem, typically representing the quantity to be maximized or minimized. This function plays a central role in determining the best outcome based on constraints and resources available. By evaluating different variable values, the objective function helps identify optimal solutions within feasible regions defined by linear inequalities.
Optimal solution: An optimal solution is the best possible outcome in a given mathematical model, particularly in linear programming, where it maximizes or minimizes an objective function while satisfying a set of constraints. Achieving an optimal solution means finding the most efficient use of resources or the best decision among a set of alternatives, ensuring that all limitations are respected. This concept is central to decision-making processes in various fields such as economics, engineering, and operations research.
Optimization: Optimization is the process of finding the best solution or outcome from a set of possible choices, often subject to certain constraints. It plays a vital role in decision-making where the goal is to maximize or minimize a specific function, whether it be cost, time, efficiency, or resources. This concept is applied in various fields to analyze and improve systems, ensuring that limited resources are used effectively.
Simplex method: The Simplex method is an algorithm used for solving linear programming problems, which involves maximizing or minimizing a linear objective function subject to linear constraints. This method efficiently navigates the vertices of the feasible region defined by the constraints, moving towards the optimal solution. It is widely utilized in various fields like economics, engineering, and logistics due to its effectiveness in handling complex optimization problems.
Substitution method: The substitution method is a technique used to solve systems of linear equations by expressing one variable in terms of the other and substituting that expression into the second equation. This approach allows for finding the values of both variables in a straightforward manner, making it easier to analyze relationships within the equations. It is particularly useful when dealing with two-variable systems, simplifying the process of finding intersection points in graphs and determining feasible regions in optimization problems.