Linear Programming Simplex Solver

What Is a Linear Program?

A linear program is an optimization problem with three defining characteristics:

• Linear objective function: The quantity to maximize or minimize is a linear combination of decision variables (e.g., z = 3x₁ + 5x₂).
• Linear constraints: All restrictions are linear inequalities or equalities (e.g., 2x₁ + x₂ ≤ 100, x₁ + 3x₂ ≥ 50).
• Decision variables: The unknowns you're solving for, typically required to be non-negative (x ≥ 0).

Example: Maximize profit z = 40x₁ + 30x₂ (where x₁ and x₂ are units of two products) subject to labor constraint 2x₁ + x₂ ≤ 80, machine constraint x₁ + 2x₂ ≤ 60, and non-negativity x₁, x₂ ≥ 0.

Linear programs have elegant mathematical properties: the feasible region (all points satisfying constraints) is convex, and optimal solutions (when they exist) occur at vertices (corner points) of this region. The simplex method exploits this structure by searching vertices systematically.

Decision Variables, Objective Function, and Constraints

Decision variables (x₁, x₂, ..., xₙ):

• Quantities you control: units to produce, hours to schedule, dollars to invest, routes to select.
• Example: x₁ = units of Product A, x₂ = units of Product B.

Objective function (z):

• A linear function you want to maximize (profit, revenue, output) or minimize (cost, time, waste).
• Form: z = c₁x₁ + c₂x₂ + ... + cₙxₙ, where cᵢ are known coefficients.
• Example: Maximize profit z = 40x₁ + 30x₂ (Product A gives $40 profit/unit, Product B gives $30/unit).

Constraints:

• Linear inequalities (≤, ≥) or equalities (=) representing resource limits, capacity bounds, minimum requirements, or balance conditions.
• Example: 2x₁ + x₂ ≤ 80 (total labor-hours available), x₁ + 2x₂ ≤ 60 (machine-hours available).
• Each constraint restricts the feasible region—the set of all variable combinations that satisfy all constraints.

Non-negativity:

• Most LP models require xᵢ ≥ 0 (you can't produce negative units or schedule negative hours).
• Some models allow variables to be unrestricted in sign, but standard form assumes non-negativity.

Feasible Region and Optimal Solution

The feasible region is the set of all points (combinations of decision variables) that satisfy every constraint. Graphically (in 2D), it's the overlapping area where all constraint lines/half-planes intersect. Algebraically, it's the solution set of the system of inequalities.

Key properties:

• Convexity: The feasible region is a convex polyhedron (polygon in 2D, polytope in higher dimensions). This means a line segment connecting any two feasible points lies entirely within the region.
• Vertices (corner points): Extreme points where constraint boundaries intersect. For example, the point (20, 30) might be where the labor and machine constraints cross.
• Fundamental theorem of linear programming: If an optimal solution exists and is unique, it occurs at a vertex. If multiple optimal solutions exist, at least one is at a vertex.

The optimal solution is the feasible point that maximizes (or minimizes) the objective function. The simplex method finds this by evaluating vertices systematically, moving from one to a better neighbor until no improvement is possible.

Special cases: (1) Infeasible: No point satisfies all constraints (e.g., x ≤ 5 and x ≥ 10 simultaneously). (2) Unbounded: The objective can increase (for maximization) or decrease (for minimization) indefinitely without violating constraints (e.g., maximize x with only x ≥ 0 as constraint).

The Simplex Method (High-Level Overview)

The simplex method is an iterative algorithm that moves from vertex to vertex of the feasible region, improving the objective at each step, until reaching optimality.

Conceptual steps:

1. Start at a feasible vertex: Often the origin (all variables = 0) after adding slack variables to convert inequalities to equalities.
2. Check optimality: Examine the objective row (reduced costs). For maximization, if all reduced costs are ≤ 0, the current solution is optimal.
3. Choose entering variable: If not optimal, select a variable with positive reduced cost (for maximization) to bring into the basis (make non-zero).
4. Choose leaving variable: Determine which current basic variable should become zero to maintain feasibility (using minimum ratio test).
5. Pivot: Perform row operations (Gaussian elimination) to update the tableau, making the entering variable basic and the leaving variable non-basic.
6. Repeat: Return to step 2 until optimal or detect unboundedness/infeasibility.

The simplex method typically requires several iterations (pivots), but each iteration strictly improves the objective (or maintains it in degenerate cases). The algorithm is guaranteed to terminate in finite steps for non-degenerate problems.

Why it works: Because optimal solutions are at vertices, and the simplex method explores vertices in a way that never revisits the same vertex (under non-degeneracy), it efficiently finds the optimum without exhaustive search.

Mode 1: Standard Maximization Problem

1. Select "Maximize" as the objective type from the dropdown.
2. Set Number of Variables: Enter how many decision variables your problem has (typically 2-10).
3. Enter Objective Coefficients: Type the profit/revenue coefficients for each variable (e.g., for z = 40x₁ + 30x₂, enter 40 for x₁ and 30 for x₂).
4. Add and Configure Constraints:
   • Click "+ Add" to create each constraint.
   • Enter coefficients for each variable in that constraint.
   • Select the constraint type: ≤ (less than or equal), = (equal), or ≥ (greater than or equal).
   • Enter the right-hand-side (RHS) value (resource limit, requirement, etc.).
5. Optional Settings:
   • Show Tableau: Check to see simplex tableaus at each iteration (educational).
   • Sensitivity Analysis: Check to compute shadow prices and reduced costs.
   • Graphical View: For 2-variable problems, check to see the feasible region plot.
6. Click "Solve".
7. Review Results:
   • Solution status: Optimal, Unbounded, or Infeasible.
   • Optimal objective value (Z*).
   • Optimal variable values (x₁, x₂, ...).
   • Shadow prices (dual values) showing value of relaxing constraints by one unit.
   • Active constraints (binding at optimum).

Use this mode for: Production planning (maximize profit), resource allocation (maximize output), portfolio selection (maximize return), or any homework problem asking for maximum value.

Mode 2: Minimization Problems

1. Select "Minimize" from the objective dropdown.
2. Enter cost coefficients in the objective function (e.g., for minimize cost = 5x₁ + 3x₂, enter 5 and 3).
3. Configure constraints the same way as Mode 1 (typically ≥ constraints for meeting minimum requirements, plus ≤ for capacity limits).
4. Click "Solve".
5. Review Results: Minimum cost, variable values that achieve it, and sensitivity information.

Use this mode for: Diet problems (minimize cost while meeting nutrition requirements), transportation problems (minimize shipping cost), workforce scheduling (minimize labor cost), or any homework problem seeking minimum value.

Mode 3: Educational / Tableau View

1. Enter your LP problem as in Mode 1 or 2.
2. Check "Show Tableau" to display simplex tableaus.
3. Click "Solve".
4. Explore Iterations: If the UI supports step-by-step mode, use "Next" or "Previous" buttons to step through each pivot:
   • Observe which variable enters the basis (becomes non-zero).
   • Observe which variable leaves the basis (becomes zero).
   • See how the objective value improves with each pivot.
   • Watch the tableau transform until optimality is reached (all reduced costs ≤ 0 for maximization).

Use this mode for: Learning how simplex works internally, verifying manual tableau calculations for homework, preparing for exams that require showing simplex steps.

General Tips for Using the Solver

• Keep units consistent: If objective is in dollars, RHS should be in matching units (hours, units, etc.).
• Check for typos: A negative RHS or swapped ≤/≥ can make problems infeasible when they shouldn't be.
• Understand results: "Optimal" means a best solution was found. "Unbounded" means you can improve indefinitely (often indicates a modeling error). "Infeasible" means constraints contradict each other.
• Use sensitivity analysis: Shadow prices tell you how much the objective would change if you had one more unit of a resource—useful for "what-if" planning.
• Iterate scenarios: Try changing objective coefficients or RHS values to see how the optimal solution shifts. This builds intuition about LP sensitivity.
• Remember scope: This calculator is for learning and preliminary planning. For critical business decisions, validate with domain expertise and professional optimization software.

Standard Form of a Linear Program

A linear programming problem is typically written in standard form for maximization:

Where:

• cᵢ are objective function coefficients (profit/revenue per unit of xᵢ)
• aᵢⱼ are constraint coefficients (resources consumed by xⱼ in constraint i)
• bᵢ are right-hand-side values (available resources)
• xⱼ ≥ 0 enforces non-negativity

To solve with simplex, we convert inequalities to equalities by adding slack variables. For example, 2x₁ + x₂ ≤ 80 becomes 2x₁ + x₂ + s₁ = 80, where s₁ ≥ 0 represents unused resource. For ≥ constraints, we subtract surplus variables and may add artificial variables in Phase I of two-phase simplex.

Simplex Concept (Without Full Tableau Derivation)

A basic feasible solution (BFS) corresponds to a vertex of the feasible region. At each BFS:

• We select a subset of variables (including slacks) to be basic (non-zero).
• Remaining variables are non-basic (set to zero).
• The number of basic variables equals the number of constraints.

The simplex tableau organizes coefficients in matrix form:

• Rows represent constraints (equalities after adding slacks).
• Columns represent variables (original + slacks).
• The bottom row (objective row) shows reduced costs.

Pivot operations use Gaussian elimination to:

• Bring an entering variable into the basis (change from 0 to positive).
• Remove a leaving variable from the basis (change from positive to 0).
• Update the tableau to reflect the new BFS.

Optimality, Unboundedness, and Infeasibility

Optimality condition (maximization):

• When all reduced costs (coefficients in the objective row) are ≤ 0, no variable can improve the objective.
• The current BFS is optimal.
• For minimization, optimality requires all reduced costs ≥ 0.

Unboundedness:

• Occurs when an entering variable with positive reduced cost (for max) can increase indefinitely without making any basic variable negative.
• No leaving variable is found in the minimum ratio test.
• The objective can be improved without bound.
• In practice, unboundedness usually indicates a modeling error (missing constraint).

Infeasibility:

• No point satisfies all constraints simultaneously.
• Detected in Phase I of two-phase simplex when artificial variables cannot be driven to zero.
• Example: Constraints x ≤ 5 and x ≥ 10 cannot both be true.

Worked Example 1: Simple Two-Variable Maximization

Worked Example 2: Comparing Solutions with Sensitivity

1. Operations Research Homework: Checking Manual Solutions

Scenario: You're in an OR course and manually solved a simplex problem by hand, going through several tableaus. You want to verify your final answer (optimal variable values and z*) and check that you didn't make an arithmetic error during pivoting.

How this tool helps: Enter the objective and constraints, enable "Show Tableau," solve, and compare the calculator's tableaus step-by-step to your handwritten work. If your final answer matches, you're confident in your solution. If it differs, review where your pivot diverged. This saves hours of double-checking and helps you learn from mistakes.

2. Production Planning Case Study: Maximizing Profit

Scenario: A manufacturing company produces three products (A, B, C) with different profit margins. Each product consumes labor, machine hours, and raw materials in different amounts. Management wants to know: What production mix maximizes total profit given limited resources?

How this tool helps: Formulate the LP: decision variables = units of A, B, C; objective = profit per unit (say, 10A + 15B + 12C); constraints = labor ≤ 500 hours, machine ≤ 300 hours, material ≤ 1000 kg, each with appropriate coefficients. Solve. The calculator returns optimal units to produce, maximum profit, and shadow prices showing which resources are bottlenecks. This informs capacity planning and resource investment priorities.

3. Workforce Scheduling: Minimizing Labor Cost

Scenario: A call center needs to schedule agents across different shifts (morning, afternoon, night) to meet minimum coverage requirements for each shift while minimizing total labor cost. Each shift has different wage rates.

How this tool helps: Formulate as minimization LP: decision variables = number of agents assigned to each shift; objective = total cost (sum of agents × wage per shift); constraints = each shift must meet minimum coverage (≥ constraints). The solver returns the least-cost staffing plan that meets all coverage requirements, helping managers balance service quality and budget.

4. Diet Problem: Meeting Nutritional Requirements at Minimum Cost

Scenario: A nutritionist wants to design a meal plan that meets daily requirements for calories, protein, vitamins, and minerals using available foods (bread, milk, chicken, vegetables, etc.), each with different costs and nutritional content. The goal is to minimize total cost while meeting all nutrition constraints.

How this tool helps: Set up LP: decision variables = servings of each food; objective = total cost; constraints = calories ≥ 2000, protein ≥ 50g, vitamin C ≥ 60mg, etc. The solver finds the cheapest combination of foods that satisfies all nutritional needs. While real diet planning involves more complexity, this provides a conceptual starting point and educational insight into resource allocation.

5. Portfolio Selection: Maximizing Return Under Risk Constraints

Scenario: An investor has a budget to allocate across several stocks and bonds. Each asset has an expected return and contributes to portfolio risk. The investor wants to maximize expected return while keeping total risk below a threshold and maintaining diversification (e.g., no more than 30% in any single asset).

How this tool helps: Formulate LP: decision variables = dollars invested in each asset; objective = sum of (return rate × investment); constraints = total budget ≤ $100k, risk measure ≤ threshold, individual asset caps. Solve to find optimal allocation. (Note: Real portfolio optimization often uses quadratic programming for variance, but linear approximations are educational and useful for preliminary analysis.)

6. Scenario Testing and Sensitivity Analysis for Exam Prep

Scenario: You're preparing for an operations research exam where you'll need to explain how optimal solutions change when constraints or objective coefficients change. You want to build intuition by testing many small variations quickly.

How this tool helps: Enter a base problem, solve, note shadow prices. Then increase one RHS constraint value by 1 unit, re-solve, observe how z* changes (should match shadow price). Change an objective coefficient, re-solve, see if the optimal variable mix shifts. Repeat for multiple scenarios. This rapid iteration builds deep understanding of LP sensitivity and duality, essential for exam success.

7. Transportation and Logistics Optimization (Simplified)

Scenario: A company has multiple warehouses and multiple retail stores. Each warehouse has a supply of goods, each store has a demand. Shipping costs vary by warehouse-store pair. The goal is to minimize total shipping cost while meeting all store demands and not exceeding warehouse supplies.

How this tool helps: Formulate as minimization LP: decision variables = units shipped from each warehouse to each store; objective = sum of (cost per unit × quantity shipped); constraints = supply limits (≤) at warehouses, demand requirements (≥) at stores. The solver returns the least-cost shipping plan. (For larger problems, specialized transportation simplex algorithms exist, but this tool handles small to moderate cases for homework and conceptual learning.)

8. Resource Blending: Mixing Materials to Meet Specifications

Scenario: A chemical plant blends raw materials (crude oils, additives) to produce gasoline with specific octane, sulfur content, and vapor pressure specifications. Each raw material has different properties and costs. The goal is to minimize blending cost while meeting all quality specs.

How this tool helps: Formulate LP: decision variables = volume of each raw material; objective = total cost; constraints = octane blend ≥ 87, sulfur ≤ 0.05%, vapor pressure within range, etc. Solve to find optimal blend recipe. Real blending problems are complex, but LP provides a foundational model for learning and preliminary planning.

1. Mixing Up Maximize vs Minimize

Mistake: Entering a cost function but selecting "Maximize," or entering a profit function but selecting "Minimize."

Why it matters: This completely inverts the solution. You'll get the worst possible outcome instead of the best.

How to avoid: Always double-check: "Am I maximizing profit/revenue/output or minimizing cost/time/waste?" Match the objective type accordingly. Re-read the problem statement carefully before clicking Solve.

2. Inconsistent Units Across Constraints and Objective

Mistake: Using hours in one constraint, minutes in another, and dollars per hour in the objective without converting to a common unit.

Why it matters: Coefficients become meaningless, and the optimal solution will be numerically incorrect and uninterpretable.

How to avoid: Before entering numbers, define units clearly (e.g., all time in hours, all cost in dollars, all quantities in units). Convert all inputs to match. For example, if labor is measured in minutes, convert to hours if other constraints use hours.

3. Mis-Entering Coefficients or Swapping Inequality Signs

Mistake: Typing 2x₁ + 3x₂ ≤ 100 when the problem says 3x₁ + 2x₂ ≤ 100, or using ≥ when the constraint is actually ≤ (or vice versa).

Why it matters: This changes the feasible region completely. You might get "infeasible" when the problem should have a solution, or get a solution that violates the real constraints.

How to avoid: After entering all coefficients, review each constraint against the problem statement. Read carefully: "at most" means ≤, "at least" means ≥, "exactly" means =. Double-check signs and numbers before solving.

4. Forgetting Non-Negativity or Logical Bounds

Mistake: Not enforcing xᵢ ≥ 0 when the variable represents a quantity that cannot logically be negative (units produced, hours scheduled, etc.).

Why it matters: The solver might return negative variable values, which make no sense in context. For example, "produce -5 chairs" is nonsensical.

How to avoid: Standard form assumes non-negativity by default. If you have variables that can be negative (e.g., profit/loss differences), model them carefully with auxiliary variables or note explicitly. For typical problems, always confirm xᵢ ≥ 0 is enforced.

5. Over-Interpreting Fractional Solutions in Integer Contexts

Mistake: The LP solver returns x₁ = 12.7, x₂ = 8.3 (fractional), but the real problem requires integer values (e.g., number of employees or machines). You round to nearest integers and assume that's optimal.

Why it matters: Rounding LP solutions does NOT guarantee optimality or even feasibility for integer problems. The true integer optimum might be far from the rounded LP solution.

How to avoid: Recognize that LP is a relaxation of integer programming (IP). If your problem requires integers, use IP solvers or acknowledge that LP provides an upper bound (for max) or lower bound (for min) and a heuristic starting point. For homework, clarify whether fractional solutions are acceptable or if integrality is required.

6. Assuming the Model Captures Everything Important

Mistake: Building an LP model with a few constraints and assuming the optimal solution is the absolute best decision for the real system, ignoring factors not in the model (quality, risk, worker morale, market demand variability, etc.).

Why it matters: LP is a simplification. Real decisions involve complexities beyond linear constraints and objectives. Over-reliance on LP results without considering broader context can lead to poor operational outcomes.

How to avoid: Treat LP solutions as decision-support insights, not absolute mandates. Validate results with domain expertise, sensitivity analysis, and scenario testing. Combine quantitative optimization with qualitative judgment. Use LP for preliminary planning and learning, not as the sole basis for high-stakes decisions.

7. Ignoring Infeasibility or Unboundedness Warnings

Mistake: The solver returns "Infeasible" or "Unbounded," and you assume the tool is broken or give up without investigating the model.

Why it matters: Infeasibility and unboundedness are usually signs of modeling errors (contradictory constraints, missing constraints, incorrect signs).

How to avoid: If you get "Infeasible," check for contradictory constraints (e.g., x ≤ 10 and x ≥ 15). If "Unbounded," check if you forgot a constraint that limits growth (e.g., maximizing profit with no resource limits). Review your formulation carefully. These messages are diagnostic tools, not failures.

8. Not Utilizing Sensitivity Analysis and Shadow Prices

Mistake: Getting the optimal solution and stopping there, without exploring how the solution would change if constraints or objective coefficients shifted slightly.

Why it matters: Sensitivity analysis reveals which constraints are binding (bottlenecks), which resources are most valuable (via shadow prices), and how robust the solution is to parameter changes. This information is critical for strategic planning and resource allocation.

How to avoid: Always enable "Sensitivity Analysis" if available. Study shadow prices: they indicate the marginal value of relaxing each constraint by one unit. Use this to prioritize investments (e.g., add capacity where shadow price is highest). Test "what-if" scenarios by changing RHS or objective coefficients and re-solving.

9. Misunderstanding the Meaning of Shadow Prices

Mistake: Thinking a shadow price of $10 per hour means you should always pay $10 to acquire one more hour, regardless of how many hours you add.

Why it matters: Shadow prices are marginal values, valid only for small changes around the current solution. They hold within a "validity range." Beyond that range, the basis changes and the shadow price changes.

How to avoid: Interpret shadow prices as "the value of one additional unit, assuming nothing else changes drastically." For large changes, re-solve the LP with the new RHS value. Use shadow prices for directional guidance, not absolute pricing for bulk purchases.

10. Presenting Results Without Context or Units

Mistake: Reporting "The optimal solution is x₁ = 50, x₂ = 20, z = 3300" without explaining what x₁, x₂, and z represent, or omitting units.

Why it matters: Stakeholders need to understand what the numbers mean in real-world terms. "Produce 50 chairs and 20 tables for a profit of $3,300" is actionable; "x₁ = 50" is not.

How to avoid: Always translate LP results back into the original context. State clearly: "Optimal solution: produce 50 units of Product A and 20 units of Product B, yielding a maximum profit of $3,300 per week." Include units (dollars, hours, units) and timeframes (per day, per week). This makes results understandable and actionable for decision-makers.

1. Use LP as a Baseline for More Complex Models

Start with a simple LP formulation to understand problem structure, identify key variables and constraints, and estimate rough optimal values. Then extend to integer programming (if integrality matters), add nonlinear terms (if relationships aren't linear), or incorporate stochastic elements (if uncertainty is critical). LP provides a tractable foundation before adding complexity.

2. Master Sensitivity Analysis and "What-If" Thinking

Don't just solve once—explore how the optimal solution changes when parameters shift. Increase RHS values by 10%, decrease objective coefficients by 5%, swap a ≤ for ≥, etc. Observe which changes have large impacts and which don't. This builds intuition about LP robustness and helps you identify critical parameters that need precise estimation vs. parameters where rough guesses suffice.

3. Interpret Shadow Prices (Dual Values) Strategically

Shadow prices reveal the marginal value of resources. Use them to prioritize capacity investments: if labor has a shadow price of $20/hour and materials have $5/unit, investing in more labor capacity yields higher returns (within validity ranges). In exams and case studies, explaining shadow prices demonstrates deep understanding of LP duality and economic interpretation. Shadow prices also connect to the dual LP, where constraints become variables and vice versa—learning duality theory is a powerful advanced topic.

4. Recognize Binding vs Non-Binding Constraints and Bottlenecks

After solving, identify which constraints are active (slack = 0) and which have slack > 0. Active constraints are bottlenecks limiting your objective. Non-binding constraints are not currently restrictive. Focus improvement efforts (adding capacity, renegotiating limits) on binding constraints—they're where you'll get the most value. This insight guides operational decisions and strategic planning.

5. Understand When Integer Programming Is Needed

If decision variables must be whole numbers (number of machines, employees, projects), LP gives a fractional relaxation—an upper bound for maximization, lower bound for minimization. Use LP to estimate the best-case scenario quickly, then apply IP solvers (branch-and-bound, cutting planes) for exact integer solutions. Knowing when to stop at LP (when fractions are acceptable or close enough) vs. when to move to IP (when integrality is critical) is key for efficient problem-solving.

6. Combine LP with Scenario Analysis for Risk Management

Real-world parameters (demand, costs, capacities) are uncertain. Solve LP for best-case, worst-case, and expected-case scenarios to understand the range of possible outcomes. Identify solutions that are robust across scenarios (not optimal in any single scenario, but good in all). This approach, called robust optimization or stochastic programming, uses LP as a building block for handling uncertainty.

7. Learn the Dual Problem and Strong Duality Theorem

Every LP has a "dual" LP where constraints become variables and variables become constraints. The primal LP (original problem) and dual LP have equal optimal objective values (strong duality). Understanding duality provides deeper insights into shadow prices, economic interpretation, and alternative solution methods. Studying duality is essential for advanced OR courses and provides elegant theoretical connections.

8. Recognize Special LP Structures (Network Flow, Transportation)

Some LP problems have special structures that allow faster specialized algorithms. Network flow problems (shortest path, max flow, min cost flow) and transportation problems can be solved with network simplex, which is much faster than general simplex. If your problem fits one of these structures, formulate it accordingly and use specialized solvers. This is powerful for large-scale logistics and network optimization.

9. Practice Formulation Skills with Diverse Problem Types

The hardest part of LP is often formulation—translating a messy real-world problem into objective function and constraints. Practice on varied problems: production, blending, scheduling, assignment, diet, portfolio, network design, etc. The more problem types you've seen, the faster you'll recognize patterns and formulate correctly. Use textbooks, OR course problem sets, and case competitions for practice.

10. Use LP as a Communication and Decision-Support Tool

LP is not just for finding optimal solutions—it's a framework for structured thinking about trade-offs, constraints, and objectives. Use LP models to facilitate discussions with stakeholders: "If we invest in more capacity here (change RHS), profit increases by this much (shadow price)." Visualize feasible regions (in 2D) to explain why certain solutions aren't possible. LP models make complex trade-offs transparent and help align teams around data-driven decisions. This communication value is as important as the numerical solution itself.