LP Toolkit: Simplex, Graphical, Integer, and Duality

Linear programming is one family of tools, not a single button. Solver360 splits the workflow into solvers that share the same mathematics so you can move from a sketch to a dual check without rewriting the model.

Start with the right view

| Situation | Use | |-----------|-----| | Exactly two decision variables; you want a picture | Graphical Method | | Any size LP; you need tableaus and pivots | Simplex Method | | Some variables must be whole numbers | Integer Programming | | You need the dual LP and shadow-price proof | Duality & Shadow Prices |

A shared example

📐 Formula

Maximize Z = 3x₁ + 5x₂ subject to x₁ ≤ 4, 2x₂ ≤ 12 x₁, x₂ ≥ 0

On the Simplex solver you get x₁ = 4, x₂ = 6, Z = 42, with shadow prices (3, 2.5).

The same instance on Duality builds

📐 Formula

Minimize W = 4y₁ + 12y₂ subject to y₁ ≥ 3, 2y₂ ≥ 5 y ≥ 0

and recovers y = (3, 2.5), W = 42** — strong duality and the shadow-price interpretation in one place.

With two variables you can also open the graphical solver to see the corner (4, 6) on the feasible region.

If production must be integer units, pass the same data to Integer Programming; the LP relaxation is what Simplex already solved.

Minimization and ≥ constraints

Two-phase Simplex (and the duality converter) handle ≥ and = rows. A standard diet-style model

📐 Formula

Minimize Z = 3x₁ + 2x₂ subject to x₁ + x₂ ≥ 4, 2x₁ + x₂ ≥ 6

has optimum (2, 2), Z = 10. Load it from the worked examples on Simplex or the min example on Duality.

Practical tips

Use share links and JSON export on Simplex to hand a problem to a classmate or paste into notes.
Shadow prices are local: they apply inside the allowable RHS range before the basis changes.
Integer optima can differ from the LP relaxation — always re-solve with Branch and Bound when integrality is required.

Key takeaways

Graphical for intuition (2 variables); Simplex for general LPs; Duality to explain resource value; Integer when fractions are forbidden.
Strong duality says optimal primal and dual objectives match; dual variables are the shadow prices of the matching constraints.
Deep-link into the solvers above whenever you want every tableau, plot, or dual side by side.