Linear programming

Linear programming (LP) is a mathematical optimization technique used to determine the best possible outcome from a set of linear relationships subject to specified constraints. It seeks to maximize or minimize a linear objective function while satisfying a system of linear equations or inequalities representing resource limitations, operational requirements, or other restrictions. Linear programming is one of the most widely used methods in Operations research, economics, engineering, logistics, manufacturing, finance, and management science.

A linear programming model consists of decision variables, an objective function, and one or more constraints. The decision variables represent the quantities to be determined, the objective function defines the quantity to be maximized or minimized, and the constraints represent limitations on available resources or operational conditions. Because both the objective function and the constraints are linear, linear programming problems can be solved efficiently using specialized optimization algorithms.

Linear programming is applied to a wide range of decision-making problems, including production planning, transportation, inventory management, portfolio optimization, scheduling, workforce allocation, energy distribution, telecommunications, and supply chain management. It provides decision-makers with mathematically optimal solutions that improve resource utilization, reduce costs, and increase operational efficiency.

Since its development during the mid-twentieth century, linear programming has become one of the fundamental tools of quantitative decision-making and mathematical optimization.

History

The foundations of linear programming were established during the 1930s and 1940s through research in mathematical economics and military logistics.

The field advanced significantly after the development of the simplex algorithm by the American mathematician George Dantzig in 1947. The simplex method enabled large-scale optimization problems to be solved efficiently and contributed to the rapid adoption of linear programming across government, industry, and scientific research.

Subsequent advances in computing technology and optimization algorithms have enabled modern software to solve linear programming problems involving millions of decision variables and constraints.

Objectives

The principal objectives of linear programming are to:

maximize profits or revenues;
minimize costs;
optimize resource allocation;
improve operational efficiency;
support decision-making;
satisfy operational constraints.

These objectives contribute to more efficient planning and management.

Components

A linear programming model generally consists of:

decision variables;
objective function;
linear constraints;
non-negativity restrictions.

These components define the optimization problem.

Mathematical formulation

A general linear programming problem may be expressed as:

Maximize or minimize:

$Z = c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n}$

subject to:

$a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} \leq b_{1}$

$a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} \leq b_{2}$

$⋮$

$a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} \leq b_{m}$

and

$x_{1}, x_{2}, \dots, x_{n} \geq 0$

where:

$x_{i}$ are the decision variables;
$c_{i}$ are the objective function coefficients;
$a_{i j}$ are the constraint coefficients;
$b_{i}$ are the available resources or limits.

Example

Suppose a manufacturer produces two products.

Let:

$x$ = number of units of Product A;
$y$ = number of units of Product B.

The profit is:

US$40 per unit of Product A;
US$30 per unit of Product B.

The objective is to maximize:

$Z = 40 x + 30 y$

subject to:

$2 x + y \leq 100$

$x + 3 y \leq 90$

$x \geq 0, y \geq 0$

The linear programming model determines the production quantities of Product A and Product B that maximize total profit while satisfying both resource constraints.

Assumptions

Classical linear programming relies on several assumptions.

These include:

linear relationships;
proportionality;
additivity;
divisibility of decision variables;
certainty of model parameters.

These assumptions simplify mathematical optimization.

Solution methods

Several algorithms are commonly used to solve linear programming problems.

These include:

simplex algorithm;
revised simplex algorithm;
dual simplex algorithm;
interior-point methods.

Modern optimization software automatically selects the most appropriate algorithm for a given problem.

Relationship with mathematical optimization

Linear programming is one of the principal branches of Mathematical optimization.

It provides efficient methods for solving optimization problems in which both the objective function and the constraints are linear.

Many advanced optimization techniques have evolved from linear programming.

Relationship with operations research

Linear programming is one of the foundational methods of Operations research.

It is commonly combined with:

transportation models;
assignment problems;
network flow models;
inventory models;
scheduling models.

These techniques support complex organizational decision-making.

Relationship with transportation models

The Transportation model is a specialized form of linear programming.

Because transportation problems possess a distinctive mathematical structure, they can often be solved more efficiently using specialized transportation algorithms than by applying general linear programming techniques.

Relationship with supply chain management

Linear programming is widely used in Supply chain management to:

optimize production;
allocate inventory;
minimize transportation costs;
improve distribution planning;
manage warehouse operations.

These applications improve operational efficiency across supply networks.

Applications

Linear programming is widely applied in:

manufacturing;
logistics;
transportation;
finance;
agriculture;
telecommunications;
energy systems;
production planning;
military logistics.

It is useful wherever limited resources must be allocated efficiently.

Advantages

Linear programming provides several important benefits.

It identifies mathematically optimal solutions, improves resource utilization, reduces operational costs, supports objective decision-making, handles large optimization problems efficiently, and can be applied across numerous industries.

Modern optimization software enables extremely large linear programming models to be solved rapidly.

Limitations

Despite its usefulness, linear programming relies on assumptions that may not fully represent real-world conditions. Many practical problems involve nonlinear relationships, uncertainty, discrete decision variables, or multiple conflicting objectives that require more advanced optimization methods.

Accurate results also depend upon the quality of the input data and the validity of the underlying mathematical model.

Importance

Linear programming is one of the most important techniques in mathematical optimization and operations research. By providing efficient methods for allocating scarce resources and optimizing complex decision-making problems, it has become an indispensable analytical tool in business, engineering, economics, logistics, finance, and public policy.