Linear programming is a mathematical way to find the best outcome when you have a goal and limited resources, with relationships that can be written as straight-line equations or inequalities.
Despite its name, the "programming" in linear programming does not come from computer programming. In fact, it was invented before computers were capable of handling even a small LP. Here, “programming” means planning. You describe the decisions, the goal, and the rules; a linear programming solver then finds the best feasible values for those decisions.
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Linear Programming Meaning in Simple Terms
Suppose a factory makes several products. Each product earns a different margin and consumes labor, materials, and machine time. The factory cannot exceed its available resources, but it wants to choose a product mix that earns the highest total margin.
Linear programming turns that business question into a model. Every allowed plan is a feasible solution. The plan with the best objective value is an optimal solution, subject to the assumptions and data in the model.
Most linear programming problems contain five connected parts:
Decision variables: The quantities you can choose, such as production volume, shipment flow, or budget allocation.
Objective function: The measure to maximize or minimize, such as profit, cost, distance, waste, or emissions.
Linear constraints: The limits or requirements that every solution must meet.
Feasible region: All combinations of variable values that satisfy those constraints.
Optimal solution: The feasible combination with the best objective value.
“Linear” means each variable has a proportional, additive effect. If one extra unit uses two labor hours, then ten use twenty hours. The model does not contain products of variables, powers such as x², or other nonlinear functions.
A Short Linear Programming Example
Consider a workshop that makes desks and shelves. Let x be the number of desks and y the number of shelves produced during a planning period.
Each desk contributes $60 and each shelf contributes $40.
A desk uses four carpentry hours and a shelf uses two; 80 hours are available.
A desk uses two finishing hours and a shelf uses three; 60 hours are available.
The model is:
The feasible region contains every nonnegative pair (x, y) that respects both capacity limits. Here, the best corner point is x = 15 and y = 10, with an objective value of $1,300. It uses all 80 carpentry hours and all 60 finishing hours.
That result also illustrates sensitivity. If the shop adds finishing capacity but no carpentry capacity, the best product mix may change; the value of extra capacity can be analyzed from the LP solution.
The example treats production quantities as continuous. If the products must be made only in whole units, the model should restrict the variables to integers, which makes it an integer or mixed-integer linear program. Rounding a linear program's solution to force integer answers is not a reliable fix. Small rounding errors can accumulate, leading to plans that overcommit or undercommit real capacity.
Linear Programming vs. Linear Optimization and Integer Programming
Linear programming and linear optimization normally mean the same thing. Both describe optimizing a linear objective subject to linear equality or inequality constraints with continuous decision variables.
The important distinction is between continuous and discrete decisions, which can be combined to form the much more common MILP (often just called a MIP).
Model type | Variable values | Typical decision |
|---|---|---|
Linear programming (LP) | Continuous values, including fractions | How many tons to ship or how much budget to allocate |
Integer linear programming (ILP) | Whole numbers | How many trucks or machines to use |
Mixed-integer linear programming (MILP or MIP) | A mix of continuous and integer values | Which facility to open and how much it should ship |
Nonlinear programming (NLP) | Continuous values | Decisions with nonlinear physics, returns, or risk |
Mixed-integer nonlinear programming (MINLP) | A mix that includes integer values | Nonlinear relationships plus discrete choices |
A high-level workforce capacity plan may use continuous hours and fit an LP, while assigning named employees to individual shifts usually requires binary variables and a MIP. Routing, packing, and many scheduling problems are also commonly modeled with integer decisions.
The word “linear” describes the model, not the size of the business problem. Large-scale linear programs can include detailed networks, many time periods, and extensive operational rules while remaining linear.
How Linear Programming Problems Are Solved
A two-variable LP can be drawn on a coordinate plane. Each linear equality or inequality creates a boundary, and their overlap forms the feasible region. For a bounded LP with an optimal solution, at least one optimum occurs at a corner point, so a graphical method can compare those corners.
That method stops being practical once a model has more than two or three decision variables. Real models need algorithms that work in high-dimensional spaces.
The simplex method
The simplex algorithm moves among corner points in search of an improved objective. George Dantzig developed the method in 1947, and it remains a core LP algorithm today. The INFORMS oral history with Dantzig describes its origin in planning problems once handled by hand.
Primal simplex and dual simplex use related paths through the problem. Dual simplex is especially useful when a model is changed and re-solved, because an existing solution basis can often provide a strong start.
Interior-point methods
An interior-point, or barrier, method travels through the interior of the feasible region instead of stepping from corner to corner. It can be effective for large, difficult, sparse linear programs solved from scratch.
Neither simplex nor barrier wins on every model. Modern solvers inspect the model and can choose among algorithms, run alternatives concurrently, or let the user select a method after testing.
Presolve and solution recovery
Before the main algorithm starts, presolve looks for safe ways to simplify the model. It may remove redundant rows, fix variables, or tighten bounds, then it maps the final solution back to the original variables.
These reductions can make a major difference, but solver performance still depends on formulation, scaling, data, hardware, and settings. Gurobi’s LP algorithm guidance explains the simplex and barrier tradeoffs in more detail.
Where Linear Programming Is Used in Real Life
Linear programming is a foundational tool in operations research because many resource-allocation problems have additive costs and capacities. Common applications include:
Supply chain and logistics: Allocate product across plants, warehouses, and customers while meeting demand at minimum transportation and handling cost.
Production planning: Choose output levels given raw materials, labor, equipment capacity, inventory limits, and demand.
Energy systems: Dispatch generation or plan flows through a network while meeting demand and operating limits.
Finance and budget allocation: Allocate capital across eligible uses under exposure, policy, and funding constraints when the objective and risk treatment are linear.
Marketing and media: Divide a budget across channels while meeting reach, mix, and spend rules.
Workforce planning: Set aggregate staffing levels or hours; use integer variables when individual shifts and assignments must be selected.
LP also supports scenario analysis. Teams can change demand, capacity, costs, or policy limits and re-solve the model to see how the recommended plan and objective value respond.
The model’s answer is only as useful as its formulation and inputs. A mathematically optimal plan can still be operationally poor if an important constraint is missing, a coefficient is wrong, or a supposedly linear relationship behaves differently at the relevant scale.
Try a real model, not just a toy problem. Browse Gurobi’s functional code examples and use the Optimizer documentation to build in Python, C, C++, Java, .NET, MATLAB, or R.
When Is Linear Programming the Right Technique?
LP is a strong candidate when four conditions mostly hold:
Proportionality: Each unit of an activity has a constant contribution and resource use.
Additivity: Total impact is the sum of the individual impacts.
Divisibility: Fractional values make sense for the decision variables.
Defined inputs: The model has usable estimates for costs, capacity, demand, and other coefficients.
If whole-item choices matter, add integer variables. If relationships are genuinely nonlinear, consider a quadratic or nonlinear model rather than forcing a misleading linear approximation. A piecewise-linear approximation can sometimes preserve useful accuracy while keeping a model tractable, but it should be validated against the real process.
Before selecting a method, write down the decisions you control, the outcome you care about, and the rules the plan must obey. This simple exercise often reveals whether the problem is an LP, a MIP, another optimization class, or not yet ready to model.
Why Use a Linear Programming Solver?
Manual solving is valuable for learning, but it does not scale to thousands or millions of variables, constraints, and nonzero coefficients. A solver automates presolve, algorithm selection, numerical checks, and solution recovery. It also reports whether a model is optimal, infeasible, unbounded, or stopped early.
Gurobi Optimizer supports LP as well as mixed-integer, quadratic, quadratically constrained, and nonlinear model classes. Its APIs let teams embed a model in an application, change data, re-optimize, inspect solution attributes, and apply the plan in an operational workflow.
A solver cannot decide whether the model represents the business correctly. Teams should validate constraints with subject-matter experts, test edge cases, compare recommendations with known outcomes, and monitor the deployed decision process.
Ready to test your own use case? Start a free 30-day evaluation, which currently includes access to benchmarking, model tuning, technical guidance, and support.
Linear Programming Meaning: Frequently Asked Questions
What is linear programming in simple terms?
Linear programming is a method for choosing the best values for a set of decisions when the goal and limits are linear. It can maximize an outcome such as profit or minimize an outcome such as cost while respecting every stated constraint.
What is the difference between linear programming and linear optimization?
None. “Linear programming” and “linear optimization” are two names for optimizing a linear objective function subject to linear constraints with continuous variables.
What are the main components of a linear programming problem?
The main components are decision variables, a linear objective function, and linear constraints. Together, the constraints and variable bounds define the feasible region; the solver searches that region for an optimal solution.
What is linear programming used for in real life?
Organizations use LP for production planning, network flows, inventory and transportation allocation, energy dispatch, blending, finance, and budget decisions. Problems that require indivisible choices, such as selecting facilities or assigning people to shifts, usually add integer variables.
What is the simplex method, and how does it relate to linear programming?
The simplex method is an algorithm for solving LPs. It moves between corner points in search of a better objective until it finds an optimum or identifies another outcome, such as unboundedness.
Can linear programming be solved by hand, or do you need software?
Small problems with two variables can be solved by hand with a graph or simple algebra. Real-world problems normally require solver software because the number of variables, constraints, and possible corner points grows far beyond what a person can evaluate manually.
What’s the difference between linear programming and integer programming?
LP allows decision variables to take fractional values. Integer programming requires some or all variables to be whole numbers, often to represent yes/no choices, counts, assignments, routes, or schedules; those restrictions usually make the problem harder to solve.
Turn the definition into a decision model. Talk with a Gurobi optimization expert about your objective, constraints, and data, or start a free trial and benchmark your own model.
