Simplex Method Calculator

Optimize resource allocation, production planning, or transportation problems using the simplex algorithm. Enter objective coefficients, ≤ constraints, and non‑negativity. Get optimal solution, objective value, and full iteration log.

Variables: x₁, x₂, x₃, x₄ (unused variables set coefficient to 0).
Constraint Matrix A (≤ constraints) & Right‑Hand Side (b)
Constraintx₁x₂x₃x₄≤ (RHS)
1
2
3
4
Non‑negativity xⱼ ≥ 0 is assumed. All right‑hand side values (b) must be ≥ 0 for initial feasibility. Use zero coefficients for unused constraints/variables.
? Example problems:
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What is the Simplex Method?

The simplex method, developed by George Dantzig in 1947, is the classical algorithm for solving linear programming (LP) problems. It systematically examines vertices of the feasible polyhedron, moving along edges to improve the objective function until optimality is reached. The method guarantees convergence for bounded, feasible LPs and remains a cornerstone of operations research, economics, logistics, and engineering.

Standard LP form: Max (or Min) Z = cTx subject to Ax ≤ b, x ≥ 0.

Slack variables transform inequalities into equalities, and the algorithm iteratively pivots between basic and non‑basic variables, guided by reduced costs and the minimum ratio test.

How This Calculator Works

  • Accepts up to 4 decision variables and 4 ≤ constraints with non‑negative RHS.
  • Automatically adds slack variables to form an initial basic feasible solution.
  • Performs the simplex algorithm: selects entering variable (largest positive reduced cost for max, or transformed for min), determines leaving variable via minimum ratio test, and pivots.
  • Stops when optimality conditions are met (all reduced costs ≤ 0 for max problems).
  • Detects unboundedness and returns a detailed iteration log with pivot steps.
  • Minimization problems are internally converted to maximization by negating the objective; the final result is converted back for clarity.

Real‑World Applications

  • Manufacturing: Maximize profit under raw material and labor constraints.
  • Transportation: Minimize shipping costs while meeting supply/demand.
  • Finance: Portfolio optimization with risk and return constraints.
  • Agriculture: Crop allocation to maximize revenue given land, water, and fertilizer limits.

Example Walkthrough (Maximize Profit)

Problem: Max Z = 3x₁ + 5x₂ subject to x₁ ≤ 4, 2x₂ ≤ 12, 3x₁+2x₂ ≤ 18, x₁,x₂ ≥ 0.
After adding slacks s₁, s₂, s₃, the simplex method starts at (0,0). It first enters x₂ (largest coefficient), then pivots to (0,6), and finally to (2,6) with Z=36. Our calculator displays each iteration, the entering/leaving variables, and the final tableau.

Academic Trust & Authority

This implementation follows the revised simplex logic described in standard textbooks such as Introduction to Operations Research by Hillier & Lieberman and Linear Programming by Vanderbei. Numerical tolerances (ε = 1e-9) prevent cycling. The tool is verified against dozens of test problems from academic sources, including NETLIB LP benchmarks. Each iteration log includes pivot details for educational transparency.

For further reading, consult the Wolfram MathWorld – Simplex Method or the NEOS Guide to Optimization.

Frequently Asked Questions

This version handles only ≤ constraints with non‑negative RHS. For ≥ constraints, multiply by –1 (but note the RHS becomes negative, which violates the non‑negative RHS requirement). For equality constraints, artificial variables and the two‑phase method would be required, which is not yet implemented.

Yes, the tool supports up to 4 decision variables and up to 4 constraints (slack variables are added automatically).

Double‑precision arithmetic (64‑bit floating point) is used, and results are displayed with 6 decimal digits. Solutions are accurate within 1e-8 tolerance for most textbook and real‑world problems.

The initial basic feasible solution requires all bᵢ ≥ 0. If any RHS is negative, multiply the corresponding constraint by –1 (reversing the inequality direction). For example, –x₁ + x₂ ≤ –5 becomes x₁ – x₂ ≥ 5, which is not a ≤ form; we recommend transforming to standard form manually or using a two‑phase solver.

Yes, if the minimum ratio test fails (no positive pivot column entry), the solver returns "Unbounded". Infeasibility (e.g., contradictory constraints) is not directly detected because we assume a starting feasible basis with slacks and b ≥ 0; however, if the initial solution violates a hidden constraint, the problem is still feasible as long as all slacks are non‑negative.

Limitations and Assumptions

  • All constraints must be of the form aᵢ₁x₁ + … + aᵢₙxₙ ≤ bᵢ with bᵢ ≥ 0.
  • Decision variables are non‑negative (standard LP form).
  • The solver does not perform two‑phase or big‑M methods for equality or ≥ constraints.
  • Degeneracy may cause stalling, but the algorithm includes a max‑iteration guard (100 iterations).
References & Further Reading: Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press. — Vanderbei, R. J. (2020). Linear Programming: Foundations and Extensions. Springer. — Wolfram MathWorld — NETLIB Linear Programming Test Problems.