Linear Programming Simplex Solver (Two-Phase)

Linear Programming Simplex Solver (Two-Phase) | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: text, select, textarea, number
Output type: text
Sample coverage: 4
API ready: Yes

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.

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

1Enter objective function coefficients for 2 to 6 decision variables and select your goal as either Maximize or Minimize.
2Input 1 to 8 linear constraints, defining coefficients, relational operators (<=, >=, or =), and right-hand side constants on separate lines.
3The 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.
4Review 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.

Examples

1. Manufacturing Product-Mix Optimization

Operations Planner

Background

A small workshop manufactures two products (x1 and x2) with different profit margins and shared production limits.

Problem

Determine the optimal production quantity of each unit to maximize total profit without exceeding department hours.

How to use

Set objective coefficients to 3, 5 with goal Maximize, then supply machine and labor constraints as separate lines.

Objective: 3, 5
Goal: max
Constraints:
1, 0 <= 4
0, 2 <= 12
3, 2 <= 18
Decimal Places: 4

Outcome

Returns an optimal solution of x1 = 2, x2 = 6 with a maximum profit value of 36.

2. Diet Cost Minimization

Nutrition Student

Background

A student needs to solve a classic blending problem to meet minimum daily nutrient requirements at minimal expense.

Problem

Find the lowest-cost food combination subject to multiple greater-than-or-equal nutritional requirements.

How to use

Set objective coefficients to 2, 3 with goal Minimize, and enter the nutrient lower bounds using >= constraints.

FAQ

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).

Objective: 2, 3
Goal: min
Constraints:
1, 1 >= 4
1, 3 >= 6
Decimal Places: 4

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Manufacturing Product-Mix Optimization

2. Diet Cost Minimization

FAQ

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