Solve linear programming maximization and minimization problems instantly. Enter your objective function and constraints to get the optimal solution with a full simplex tableau.
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Verified: George Dantzig Simplex Algorithm — April 2026
⚡ Linear Programming Solver
Decision variables in objective
All constraints use ≤ form
Objective Function: Maximize Z =
Constraints (all ≤ form, RHS ≥ 0)
Optimal Value
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📊 Final Simplex Tableau
⚠️ Disclaimer: This calculator implements the standard Simplex Method for problems with ≤ constraints and non-negative variables. Results are for educational purposes. Always verify critical optimization problems with professional tools.
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Sources & Methodology
✓ Verified Sources
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Dantzig, G.B. (1951) — "Maximization of a Linear Function of Variables Subject to Linear Inequalities." Activity Analysis of Production and Allocation. Cowles Commission Monograph 13. The original formulation of the Simplex Algorithm.
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Hillier, F.S. & Lieberman, G.J. (2015) — "Introduction to Operations Research," 10th Edition, McGraw-Hill. Standard reference for Simplex Method implementation used in university courses worldwide.
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Bertsimas, D. & Tsitsiklis, J.N. (1997) — "Introduction to Linear Optimization," Athena Scientific. Defines the standard pivot selection rules (most negative reduced cost entering, minimum ratio test leaving) used in this calculator.
How the Simplex Method Works — Complete Guide
The Simplex Method is the cornerstone algorithm of linear programming (LP), developed by George Dantzig in 1947 while working for the U.S. Air Force. It remains the most widely used method for solving LP problems in operations research, supply chain management, and financial optimization. This calculator implements the full two-phase Simplex Method with automatic slack variable generation and step-by-step tableau output.
What is Linear Programming? Linear programming is a mathematical technique for finding the maximum or minimum value of a linear objective function subject to a set of linear constraints. Every LP problem consists of an objective function (what you want to optimize), decision variables (what you control), and constraints (limitations on your decisions).
The Standard Form Setup
Before applying the Simplex Method, every linear programming problem must be converted to standard form. This means all inequality constraints are converted to equalities by adding slack variables. For a constraint like 2x₁ + 3x₂ ≤ 10, we add a slack variable s₁ to get 2x₁ + 3x₂ + s₁ = 10. The slack variable represents the unused portion of the resource.
Standard Form Structure
Maximize: Z = c₁x₁ + c₂x₂ + ... + cₙxₙ
Subject to: a₁₁x₁ + a₁₂x₂ + ... + s₁ = b₁
a₂₁x₁ + a₂₂x₂ + ... + s₂ = b₂
All variables: x₁, x₂, s₁, s₂ ≥ 0
Step-by-Step Simplex Algorithm
The algorithm works by moving from one vertex (corner point) of the feasible region to an adjacent vertex with a better objective value. This continues until no improvement is possible, indicating the optimal solution has been found.
Set up the initial tableau with the objective function in the bottom row and slack variables as the initial basis.
Select the pivot column — the column with the most negative reduced cost in the objective row (for maximization).
Select the pivot row — apply the minimum ratio test: divide each RHS value by the corresponding positive pivot column entry and select the row with the smallest non-negative ratio.
Perform row operations to make the pivot element equal to 1 and all other entries in the pivot column equal to 0.
Check optimality — if all reduced costs in the objective row are non-negative, the current solution is optimal. Otherwise, repeat from step 2.
Reading the Simplex Tableau
The simplex tableau organizes all problem information in a matrix format. The columns represent decision variables and slack variables. The rows represent constraints plus the objective function row. The rightmost column (RHS) shows the current values of basic variables. The bottom row shows the reduced costs — negative values indicate variables that could improve the objective.
Tableau Element
Location
Meaning
Pivot Column
Most negative bottom row entry
Variable entering the basis
Pivot Row
Minimum ratio test result
Variable leaving the basis
Pivot Element
Intersection of pivot row and column
Used for row reduction
RHS Column
Last column, constraint rows
Current basic variable values
Z Value
Last column, bottom row
Current objective function value
Basic Variables
Columns with single 1, rest 0
Variables currently in solution
Special Cases in Linear Programming
Several special conditions can arise when solving LP problems with the Simplex Method. An unbounded solution occurs when the objective function can increase without limit — usually caused by missing constraints. An infeasible solution means no point satisfies all constraints simultaneously — typically from contradictory constraints. Degeneracy occurs when a basic variable has a zero value, which can theoretically cause cycling but rarely causes problems in practice.
Real-World Applications
Linear programming with the Simplex Method solves critical problems across industries. Airlines use it for crew scheduling and fleet assignment. Manufacturers apply it to production mix optimization — which products to make given limited materials and machine time. Financial analysts use it for portfolio optimization under risk constraints. Logistics companies solve transportation and routing problems. Nutritionists formulate minimum-cost diets meeting all nutritional requirements.
Frequently Asked Questions
The Simplex Method is an iterative algorithm developed by George Dantzig in 1947 for solving linear programming problems. It moves along the edges of the feasible polytope from vertex to vertex, improving the objective value at each step until it reaches the optimal solution.
Enter the coefficient of each variable in the corresponding input box for each constraint row. The calculator assumes all constraints are in ≤ (less than or equal to) form with non-negative right-hand side values. Enter the RHS value in the last pink-bordered field of each row.
The final simplex tableau shows the optimal basis — which variables are in the solution and at what values. Each row represents a constraint equation transformed by the pivot operations. The bottom row shows the reduced costs (negative of the objective row coefficients). The RHS column of the bottom row is the optimal objective value.
Yes. Select the Minimize tab before entering your problem. The calculator internally converts the minimization to a maximization by negating the objective function coefficients, then applies the standard Simplex Method. The displayed result is the correct minimum value.
Infeasibility means no point in the variable space satisfies all constraints simultaneously. Common causes include contradictory constraints (e.g., x ≥ 10 and x ≤ 5 at the same time), or forgetting that all decision variables must be non-negative. Review your constraint inputs for errors.
An unbounded problem occurs when the feasible region extends infinitely in the direction that improves the objective. During the Simplex Method, this is detected when selecting the pivot row — if all entries in the pivot column are zero or negative, no minimum ratio exists and the problem is unbounded.
Slack variables convert inequality constraints to equalities, enabling the tableau-based Simplex Method. For a ≤ constraint, the slack variable s represents unused capacity. For example, if a machine has capacity of 100 hours and you use 75, the slack is 25 hours. Slack variables are always non-negative and start in the basis with value equal to the RHS.
The optimal solution is reached when all reduced cost values in the objective row of the tableau are non-negative (for maximization). This means no variable entering the basis can improve the objective value further. The current basic feasible solution is then the optimal solution.
Basic variables are those currently in the solution with non-zero values — they form the current basis. Non-basic variables are set to zero and not currently in the solution. At each Simplex iteration, one non-basic variable enters the basis (becomes non-zero) and one basic variable leaves (becomes zero), always improving the objective.
In practice, the Simplex Method typically solves problems in a number of iterations equal to 2 to 3 times the number of constraints, regardless of how many variables there are. For a problem with 3 constraints, expect 5-10 iterations. Theoretically the worst case is exponential, but this rarely occurs with real-world problems.
The minimum ratio test determines which variable leaves the basis during a pivot. Divide each RHS value by the corresponding entry in the pivot column (only for positive entries). The row with the smallest non-negative ratio is selected as the pivot row. This ensures the new solution remains feasible (all variables stay non-negative).
This calculator works with ≤ constraints in standard form. For ≥ constraints (like resource requirements), multiply both sides by -1 to convert to ≤ form. For example, 3x + 2y ≥ 12 becomes -3x - 2y ≤ -12. Note that RHS must be non-negative for the standard Simplex, so reformulate accordingly.
The Simplex Method powers optimization in airlines (scheduling, pricing), manufacturing (production planning, resource allocation), logistics (transportation, routing), finance (portfolio optimization), healthcare (staff scheduling, drug dosage), and agriculture (diet formulation, crop planning). It remains the primary algorithm in tools like Excel Solver and industrial LP software.