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Solve a Two-Product Optimization Problem in Gnumeric

Build a small linear programming model in Gnumeric Solver with two products, capacity constraints and a checkable optimal profit.

Official documentation reviewed; native application verification pending · Updated 10 October 2026 · Editorial standards

Gnumeric includes a linear programming Solver for choosing decision variables subject to constraints. This exercise uses intentionally small numbers so you can verify the optimum without trusting a black-box result.

The business problem

You make two products, X and Y. Each unit of X earns 30; each unit of Y earns 20. Two machines limit production:

  • Machine A: 2X + Y ≤ 100
  • Machine B: X + 2Y ≤ 80
  • Non-negativity: X ≥ 0, Y ≥ 0

Maximize profit: 30X + 20Y.

Construct the worksheet

Cell Value or formula Purpose
B2 0 Units of X (decision variable)
C2 0 Units of Y (decision variable)
B4 =30*B2+20*C2 Profit to maximize
B5 =2*B2+C2 Machine A usage; cap 100
B6 =B2+2*C2 Machine B usage; cap 80

Set up the Gnumeric Solver

  1. Open Tools → Solver, as described in the Gnumeric manual.
  2. Choose objective B4, set the direction to maximize, and select variable cells B2:C2.
  3. Add B5 ≤ 100 and B6 ≤ 80, plus nonnegative bounds on B2 and C2.
  4. Choose an appropriate linear model/solver method in your installation; availability of specific solvers or plug-ins can vary.
  5. Solve and compare the proposed optimum to the independent check below.

Independently verify the optimum

At the intersection of the binding constraints:

2X + Y = 100
 X + 2Y = 80

Solving gives X = 40, Y = 20. Profit is 30×40 + 20×20 = 1,600. Both machine usages equal their limits. By comparison, making only X gives X≤50 and profit≤1,500; making only Y gives Y≤40 and profit≤800. These extreme points confirm the unique optimum of this small two-variable linear model.

If your answer differs, confirm you entered both machine limits, nonnegative variables, and the maximize direction.

Boundaries and cautions

This is a continuous linear model. If units must be whole numbers, add an integer constraint and check the solver’s integer-programming support. The chosen optimum happens to be integral, but other datasets may not be.

Related: Gnumeric Goal Seek adjusts only one cell to reach a formula target.

Verification: The candidate optimum was independently checked with a separate vertex enumeration. The Solver interface and algorithm availability await native verification.

Official documentation: Software manual or vendor release notes. This article is part of a staged content batch; confirm the interface and output in the stated software before public deployment.