Linear Programming Explained: How Businesses Find the Best Plan

Linear programming is a mathematical method for finding the best outcome when a business has limited resources and its objective and constraints can be expressed as linear equations or inequalities. For example, a company can use it to decide how much of two products to make with a fixed amount of labor and materials. The result is the best plan according to the model—not necessarily the best decision if important real-world factors were left out.

What is quantitative analysis in business?

Quantitative analysis uses numerical information and mathematical methods to help evaluate business decisions. Depending on the question, it might involve forecasting demand, comparing investment options, analyzing risk, or allocating limited resources.

Linear programming is one tool within that broader field. It is useful when a decision has a clear numerical objective, several possible choices, and limits that can be written as linear relationships. For a broader introduction, see the advantages of quantitative analysis.

What is linear programming in quantitative analysis?

A linear programming model has three main parts:

• Decision variables: The quantities to choose, such as how much of each product to make.
• Objective function: A linear expression to maximize or minimize, such as total contribution to profit.
• Constraints: Linear equations or inequalities that limit the choices, such as available labor hours. Variables also commonly have nonnegativity constraints.

A feasible solution satisfies every constraint. An optimal solution is a feasible solution with the best objective value. There can be more than one optimal solution; a model can also be infeasible or have an unbounded objective.

“Linear” means the model assumes each unit adds a constant amount to the objective and resource use. It does not directly represent effects such as a variable multiplied by itself or by another variable.

What are quantitative analysis examples in business?

Consider a business that can make products A and B. Let x be the number of units of A and y the number of units of B. Each unit of A contributes $3 toward profit, and each unit of B contributes $2.

The business has 40 labor hours and enough material for 30 total units. A uses two labor hours per unit, while B uses one. Its model is:

• Maximize contribution: 3x + 2y
• Labor constraint: 2x + y ≤ 40
• Material constraint: x + y ≤ 30
• Nonnegativity: x ≥ 0 and y ≥ 0

The best plan is x = 10 and y = 20. It uses all 40 labor hours and all the material available for 30 units, producing a contribution of $70. For comparison, making 20 units of A or 30 units of B produces $60 in each case; those are the other nonzero corner points of the feasible region.

Businesses can build similar models to allocate an advertising budget, choose a production mix, or minimize transportation costs. The actual model depends on the decision and reliable estimates of resource use and returns.

How does sensitivity analysis help with business decisions?

Sensitivity analysis asks what happens when a model input changes. A manager might test whether the same plan remains best if a product’s contribution changes, or how much an extra hour of labor is worth.

In the example, suppose labor increases from 40 to 44 hours while all other assumptions stay the same. The new optimal plan is 14 units of A and 16 of B, with a contribution of 74 dollars. The additional four labor hours increase the objective by $4; this does not mean every further hour will always be worth $1.

Sensitivity analysis is not a substitute for checking uncertain inputs. For decisions where outcomes or probabilities are unclear, see how to make decisions under uncertainty.

What is the best quantitative analysis software?

The best tool depends on the size of the problem and what you need to explain. A spreadsheet with an optimization add-in can be a practical starting point for a small model; programming libraries and dedicated optimization software can handle larger or more complex models.

Whichever tool you use, check that the variables, objective, and constraints match the real decision. Review the solution for feasibility and interpret it in business terms rather than treating a solver’s output as a decision by itself.

Standard linear programming permits fractional values. If the business must choose whole units or yes-or-no options, an integer programming model may be more appropriate. Linear models can also miss nonlinear costs, changing resource use, or factors that are difficult to quantify.

What does a quantitative business analysis course outline cover?

A course outline can help you see whether a program goes beyond the mechanics of solving equations. Look for modeling, interpretation of solutions, sensitivity analysis, and applications alongside broader decision-making methods.

The following table summarizes UPI Study’s Quantitative Analysis course, including its format, credit recommendation, and price.

Which colleges list Quantitative Analysis on an equivalency chart?

These colleges show Quantitative Analysis (Math 250) on their equivalency charts with UPI Study. Open a college for its full chart.

See the rest of the equivalency on the Quantitative Analysis course page, or open a college above for its full chart. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Frequently Asked Questions about Business Math

Final Thoughts on Business Math

To practice, choose a small business decision and write down its variables, objective, and constraints before opening a solver. Then change one assumption and explain whether the recommended plan changes.