Linear Programming: Mathematical Optimization

A mathematical procedure for finding the maximum or minimum value of a linear objective function subject to linear constraints.

Linear Programming (LP) is a mathematical method for determining the best outcome in a mathematical model whose requirements are represented by linear relationships. It is used extensively in various fields to maximize or minimize an objective function subject to a set of linear constraints.

Historical Context

Linear programming was developed during World War II to plan expenditures and returns in order to reduce costs to the military and increase losses to the enemy. George Dantzig is credited with the development of the simplex algorithm in 1947, which has been a primary solution method for LP problems ever since.

Types/Categories

1. Standard Form

A linear programming problem is in standard form if it seeks to maximize an objective function.

2. Canonical Form

In canonical form, the goal is to either maximize or minimize an objective function with inequality constraints.

3. Integer Linear Programming (ILP)

Constraints and objectives are the same as LP, but all variables are constrained to be integers.

4. Mixed-Integer Linear Programming (MILP)

Some, but not necessarily all, of the variables are constrained to be integers.

Key Events

• 1947: George Dantzig introduces the simplex algorithm.
• 1950s-60s: Development of interior-point methods.
• 1984: Narendra Karmarkar introduces a new polynomial-time algorithm for LP.

Detailed Explanation

Mathematical Formulation

A typical linear programming problem can be expressed as follows:

Objective function:

$$ \text{Maximize } Z = c_1x_1 + c_2x_2 + \ldots + c_nx_n $$

Subject to:

$$ a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n \leq b_1 $$

$$ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n \leq b_2 $$

$$ \vdots $$

$$ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n \leq b_m $$

$$ x_1, x_2, \ldots, x_n \geq 0 $$

Where:

• \( Z \) is the objective function to be maximized.
• \( c_i \) are coefficients of the objective function.
• \( a_{ij} \) are coefficients of the constraints.
• \( b_j \) are the constraint boundaries.
• \( x_i \) are the decision variables.

Simplex Algorithm

The Simplex method solves linear programming problems by moving from vertex to vertex along the edges of the feasible region until the optimum is reached.

Graphical Solution Method

For problems with two variables, graphical methods can be employed. The feasible region is plotted and the objective function is represented as a line. The line is moved to find the maximum or minimum value within the feasible region.

    graph TD;
	    A[Start] --> B[Identify Constraints]
	    B --> C[Plot Feasible Region]
	    C --> D[Plot Objective Function]
	    D --> E[Shift Line for Optimization]
	    E --> F[Find Optimal Solution]
	    F --> G[End]

Importance and Applicability

Linear programming is vital in various fields such as:

• Economics: For optimal allocation of resources.
• Engineering: For design optimization.
• Military: For logistics and resource allocation.
• Finance: For investment portfolio selection.
• Manufacturing: For production planning.

Examples

Example 1: Diet Problem

Minimize cost while meeting nutritional requirements with available foods as constraints.

Example 2: Transportation Problem

Minimize transportation cost from suppliers to consumers while satisfying demand and supply constraints.

Considerations

Sensitivity Analysis

Examines the effect of changes in the coefficients of the objective function or constraints on the optimal solution.

Infeasibility and Unboundedness

• Infeasibility: No solution satisfies all constraints.
• Unboundedness: The objective function can be increased indefinitely.

Related Terms with Definitions

• Feasible Region: The set of all possible points that satisfy the problem’s constraints.
• Objective Function: The function to be maximized or minimized.
• Constraint: A condition that the solution must satisfy.

Comparisons

• Linear Programming vs Integer Programming: Integer programming adds the restriction that some or all variables must take integer values.
• Linear Programming vs Nonlinear Programming: Nonlinear programming deals with at least one nonlinear constraint or a nonlinear objective function.

Interesting Facts

• Linear programming has been applied to model global climate changes and assist in urban planning.
• The world’s largest linear programming model had millions of constraints and variables.

Inspirational Stories

George Dantzig’s development of the simplex method not only revolutionized operations research but also had significant impacts on economics and engineering.

Famous Quotes

“The scientist explores what is, the engineer creates what never has been.” - Theodore von Karman

Proverbs and Clichés

• “Necessity is the mother of invention.”

Expressions

• “Optimal solution”
• “Feasibility study”

Jargon and Slang

• Basis: A set of linearly independent columns.
• Slack: The difference between the left-hand side and the right-hand side of a constraint.

FAQs

What is Linear Programming used for?

It is used to determine the optimal allocation of resources to achieve a desired objective.

Is Linear Programming still relevant today?

Yes, it is widely used in logistics, finance, manufacturing, and many other industries.

Can Linear Programming solve non-linear problems?

No, for non-linear problems, non-linear programming methods are used.

References

1. Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press.
2. Karmarkar, N. (1984). “A New Polynomial-Time Algorithm for Linear Programming.” Combinatorica, 4(4), 373-395.

Summary

Linear Programming is a fundamental optimization technique used to achieve the best outcome in mathematical models. With applications spanning multiple fields, LP remains a critical tool for resource allocation and decision-making processes. The advent of computational methods like the simplex algorithm has enabled large-scale application of LP, making it an indispensable asset in modern problem-solving.