Linear Programming Calculator (Two Variables): Maximise or Minimise by the Corner Point Method

The corner point method, done for you.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose maximise or minimise and enter the objective.

Enter the constraints.

The calculator lists the corner points and the best answer.

Formulas used

  • Z = c_1x + c_2y
  • a_ix + b_iy \le r_i
  • x \ge 0,\ y \ge 0

Worked example

With the default numbers in the calculator above:

  1. Best value of the objective: 7.5
  2. Value of x at the best point: 1.5
  3. Value of y at the best point: 1.5
  4. Corner points of the feasible region and the objective at each: (1.5, 1.5) gives 7.5; (0, 2) gives 4; (2, 0) gives 6; (0, 0) gives 0

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Goal Minimise; Objective: number in front of x 1; Objective: number in front of y 1; Number of constraints 2; Constraint 1: number in front of x 1; Constraint 1: number in front of y 2; Constraint 1: sign greater than or equal to; Constraint 1: right-hand side 4; Constraint 2: number in front of x 3.

  1. Best value of the objective: 2.8
  2. Value of x at the best point: 1.6
  3. Value of y at the best point: 1.2
  4. Corner points of the feasible region and the objective at each: (1.6, 1.2) gives 2.8; (4, 0) gives 4; (0, 6) gives 6

Common questions

What is the corner point method?

The best value of a linear objective is always at a corner of the feasible region, so you find the corners and compare the objective at each.

What if there is no answer?

If the constraints cannot all be met the problem is infeasible. If the region has no limit in the direction that improves the objective it is unbounded.

Can there be more than one best point?

Yes. If the objective is parallel to an edge, every point on that edge is equally good.

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Further reading: Any operations research or discrete mathematics text (linear programming).

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