What it does
Linear programming finds the best value of a linear objective, such as profit or cost, when the choices are limited by linear constraints: hours, materials, budgets, demand. Integer programming adds the rule that some choices must be whole numbers, or yes/no. This solver takes a model typed as plain text and gives:
- the optimal solution, or a clear message that the model is infeasible (no point meets every constraint) or unbounded (the objective improves for ever);
- for linear programs, the shadow price of every constraint and the reduced cost of every variable;
- branch and bound for
int(whole number) andbin(0 or 1) variables; - a picture of the feasible region when the model has two variables, with the constraint lines, the objective line through the optimum and, for integer models, the allowed whole-number points;
- the simplex tableaux, step by step, for small linear models, which is useful for checking homework done by hand.
How to write a model
maximize profit: 40 tables + 30 chairs
subject to
wood: 4 tables + 2 chairs <= 60
labour: 2 tables + 3 chairs <= 54
demand: tables <= 12- Start with
maximizeorminimize, an optional name and a colon, then the objective. - Write one constraint per line with
<=,>=or=. Variables and numbers may be on both sides, and2 <= x + y <= 8is split into two constraints. A name in front (wood:) labels the constraint in the results; otherwise they are called c1, c2 and so on. - Every variable is 0 or more unless you say otherwise under
bounds:x <= 10,-5 <= y <= 5,z >= -3orz free. int x, ymakes variables whole numbers;bin a, bmakes them 0 or 1.#starts a comment.
What the numbers mean
Shadow price: how much the best objective changes if a constraint’s right-hand side goes up by one unit. In the workshop example the wood constraint has a shadow price of 7.5: one more board metre of wood would add 7.5 to the best profit, so it is worth paying up to 7.5 for it. A constraint with slack left over has a shadow price of 0. Shadow prices hold for small changes; a large change can move the optimum to a different corner.
Reduced cost: how much the objective changes for each unit a variable is pushed up from the bound where it sits. It is 0 for variables between their bounds. A negative reduced cost in a maximisation tells you how much the variable’s objective coefficient would have to rise before it is worth using.
Slack: how far the left side is from the right side. A slack of 0 means the constraint is binding.
Worked example
Choose the first example. The solver answers Maximum profit = 720 with 9 tables and 12 chairs. Wood (60 of 60) and labour (54 of 54) are binding, with shadow prices 7.5 and 5; the demand limit of 12 tables has 3 to spare and a shadow price of 0. The picture shows the shaded feasible region, and the dashed profit line touches it at the corner (9, 12).
The second example has whole items only and slightly different limits. Branch and bound needs 9 nodes and finds a profit of 700 at 10 tables and 10 chairs, while the linear relaxation promises a little more.
The method
Linear models are solved by the two-phase simplex method on a full tableau: phase 1 finds a feasible corner by minimising the sum of artificial variables (or proves there is none), and phase 2 moves from corner to corner improving the objective. The default pivot rule picks the largest improvement and switches to Bland’s rule if many steps in a row make no progress, which rules out cycling; you can also choose Bland’s rule from the start. Variables with lower bounds are shifted, variables with only an upper bound are measured down from it, and free variables are split into two parts. Integer models use branch and bound, depth first, on the most fractional variable, and stop after 20,000 nodes.
What it does not do
It is made for teaching and for small and medium models (up to 300 variables and 300 constraints). It uses a dense tableau, not the sparse factorisations of industrial solvers, so very large or badly scaled models may be slow or lose accuracy. It does not give ranging (how far a coefficient can move before the basis changes), and it does not read MPS files.
Frequently asked questions
Why does my model say infeasible? Check the signs and right-hand sides, and remember the default lower bound of 0. Deleting constraints one at a time shows which ones clash.
Is my model sent anywhere? No. Everything is worked out in your browser and nothing is stored.
Where the numbers come from
- Methods: the two-phase simplex method with Bland's anti-cycling rule and branch and bound, as described in standard operations research textbooks; the solver was written for fibub.
- Checked against SciPy's linprog and milp (HiGHS) on 1,800 random models (optimal, infeasible and unbounded; free, shifted and upper-bounded variables), against brute-force search for small integer models, and the shadow prices and reduced costs against the optimality (KKT) conditions and HiGHS's marginals.
- The examples are made up for this tool.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.