LP Linear Programming Calculator

Calculator Form

Optimization goal

First variable name

Second variable name

Objective coefficient for x

Objective coefficient for y

Decimal places

Lower bound for x

Upper bound for x

Lower bound for y

Upper bound for y

Feasibility tolerance

RHS change for shadow estimate

Force x and y to be non-negative

Constraints

Name	x coefficient	y coefficient	Sign	RHS

Example Data Table

Item	Expression	Meaning
Objective	Maximize Z = 40x + 30y	Profit from two products
Constraint 1	2x + y ≤ 100	Labor limit
Constraint 2	x + y ≤ 80	Material limit
Constraint 3	x ≤ 40	Maximum units for product x
Non-negative	x ≥ 0, y ≥ 0	No negative production
Expected optimum	x = 20, y = 60, Z = 2600	Best feasible corner

Formula Used

Objective: Z = c₁x + c₂y. The calculator either maximizes or minimizes this value.

Constraint: a_ix + b_iy ≤, ≥, or = r_i. Every accepted point must satisfy every constraint.

Corner intersection: D = a₁b₂ - a₂b₁. If D is not zero, x = (r₁b₂ - r₂b₁) / D and y = (a₁r₂ - a₂r₁) / D.

Slack: for ≤ constraints, slack = RHS - LHS. For ≥ constraints, surplus = LHS - RHS.

Shadow estimate: estimated effect = changed optimal Z minus original optimal Z, divided by the RHS change.

How to Use This Calculator

Choose whether the objective should be maximized or minimized.
Enter names for the two decision variables.
Add the two objective coefficients.
Enter each constraint with coefficients, sign, and RHS value.
Keep non-negative variables checked for standard production models.
Add lower or upper bounds if a variable has a practical limit.
Press the calculate button and review the result above the form.
Download CSV or PDF reports for records and sharing.

Linear Programming Planning Guide

An LP linear programming calculator helps turn a practical planning question into a structured mathematical model. It works with two decision variables. It then searches the feasible region created by the constraints. The tool is useful when resources are limited and choices compete.

Objective Function

Linear programming starts with an objective function. The objective can be profit, cost, time, weight, yield, or score. You choose maximize when bigger is better. You choose minimize when smaller is better. Each variable receives a coefficient. The calculator multiplies each coefficient by its related variable.

Constraints and Corners

Constraints define the limits. They can represent labor hours, machine capacity, material stock, demand, budget, storage space, or policy rules. Each constraint creates a boundary line. The feasible solution must satisfy every boundary and every selected variable bound. In two variable models, the best solution normally appears at a corner point.

Calculation Method

This calculator uses the graphical corner point method. It builds boundary equations from every constraint. Then it tests all intersections. It keeps only points that meet every rule. Each feasible point receives an objective value. The highest or lowest value becomes the recommended optimum. Slack or surplus values show unused resources and over satisfied limits.

Advanced Options

The advanced options help refine a model. Lower and upper bounds restrict each variable. Non negative settings prevent unrealistic negative production. Precision settings control the displayed decimals. A small RHS change is also tested. This creates a simple shadow estimate. It shows how the objective may change when a limit is adjusted slightly.

Model Setup

Use the example table before entering your own data. Start with a clear business question. Define the two variables. Add the objective coefficients. Then write each limit as a linear constraint. Keep units consistent. Do not mix hours with days or dollars with cents unless converted.

Review Results

Results should be checked with judgment. Linear programming assumes straight line relationships. It also assumes divisible quantities unless integer rules are handled later. If the solution suggests fractional units, round carefully and recheck feasibility. The downloadable reports make review easier. They help compare scenarios, document assumptions, and share the result with a team.

Binding lines deserve extra attention. They often control the final answer. Review them before changing real schedules or budgets in practice carefully first.

FAQs

1. What does this calculator solve?

It solves two variable linear programming models. It can maximize or minimize a linear objective while checking linear constraints, bounds, slack, feasible corner points, and a simple shadow estimate.

2. Can it handle more than two variables?

No. This version uses the graphical corner point method for two variables. Larger models need simplex, interior point, or dedicated optimization software.

3. Why are corner points important?

For a bounded linear model, the optimum occurs at a feasible corner point. The calculator tests intersections of boundary lines and keeps only points satisfying every rule.

4. What is slack?

Slack is unused capacity in a less than or equal constraint. If a labor limit is 100 and the solution uses 92, the slack is 8.

5. What is surplus?

Surplus applies to greater than or equal constraints. It shows how much the left side exceeds the required minimum at the selected solution.

6. What does binding mean?

A binding constraint has almost zero slack or surplus. It directly limits the solution and may strongly affect the objective value.

7. Why might a model be unbounded?

A model can be unbounded when the objective can improve without limit. Add realistic resource limits, demand caps, or upper bounds to control the feasible region.

8. Should I round the final answer?

Round only after checking feasibility. If variables represent whole items, round carefully and recalculate the objective and constraints manually.