# Linear Programming Simplex Solver (Two-Phase)

Two-phase simplex for 2–6 variables and 1–8 ≤/≥/= constraints: Bland's rule, per-iteration pivot log, optimal/unbounded/infeasible status, and substitution checks.

> Canonical page: https://elysiatools.com/en/tools/linear-programming-simplex

- **Category:** Math & Numbers

- **Keywords:** linear programming, simplex method, two-phase simplex, optimization, objective function, constraints, bland's rule, operations research, duality, feasible region

## Overview

The Linear Programming Simplex Solver solves linear optimization models with 2 to 6 decision variables and 1 to 8 linear constraints using the two-phase simplex algorithm. It supports maximization and minimization goals across standard inequality (<=, >=) and equality (=) constraints, incorporating Bland's rule to avoid cycling while outputting complete per-iteration pivot logs, solution statuses, and substitution verification checks.

## Inputs

- **Objective coefficients (e.g. 3, 5)** (text): Coefficients of the objective c·x, one per variable (2–6 numbers).
- **Goal** (select)
- **Constraints (one per line: coefficients, comparator, rhs)** (textarea): One constraint per line: coefficients (comma/space separated), then <=, >= or =, then the right-hand side. 1–8 lines.
- **Decimal Places** (number)

## When to use

- When solving small-scale linear programming optimization problems for operations research coursework or resource planning.
- When requiring step-by-step simplex iteration logs showing entering and leaving basic variables across Phase 1 and Phase 2.
- When verifying mathematical solutions, feasibility, or boundedness for linear models with mixed inequality and equality constraints.

## How it works

- Enter objective function coefficients for 2 to 6 decision variables and select your goal as either Maximize or Minimize.
- Input 1 to 8 linear constraints, defining coefficients, relational operators (<=, >=, or =), and right-hand side constants on separate lines.
- The solver introduces slack, surplus, or artificial variables, executing Phase 1 if necessary to find an initial basic feasible solution and Phase 2 to reach optimality using Bland's pivot selection rule.
- Review the resulting optimal variable values, objective value, status report (Optimal, Infeasible, or Unbounded), and direct substitution checks.

## Use cases

- Product-mix optimization to maximize manufacturing profit given labor, material, and machine hour limits.
- Diet and blending cost-minimization problems requiring nutritional or raw material lower bounds.
- Operations research study and homework verification comparing manual tableau pivot steps against automated pivot logs.

## Frequently asked questions

### What is the role of the two-phase method in this solver?

Phase 1 introduces artificial variables to determine an initial basic feasible solution for models with >= or = constraints, and Phase 2 optimizes the primary objective function.

### How does the solver prevent cycling during degenerate pivots?

It applies Bland's rule, selecting the lowest-indexed eligible candidate variable for entering and leaving the basis.

### What problem sizes does this tool support?

It handles linear programs with 2 to 6 decision variables and 1 to 8 linear constraints.

### How does the solver handle negative right-hand side values?

Constraints with negative right-hand side values are normalized by multiplying both sides by -1 and reversing the inequality sign before constructing the tableau.

### What solution statuses can the solver return?

The solver reports whether the model is OPTIMAL, INFEASIBLE (no solution satisfies all constraints), or UNBOUNDED (objective can increase or decrease infinitely).

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