Linear programming

Linear programming determines the product mix that maximises contribution subject to resource constraints expressed as linear inequalities. It is used when multiple resources are scarce simultaneously and simple ranking methods break down.

ByHoang TruongUpdated July 2, 2026

FrameworkLinear programming

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Linear programming maximises Z = c₁x₁ + c₂x₂, total contribution from x₁ and x₂ units of two products, subject to each resource constraint written as a linear inequality plus non-negativity. It is needed once more than one resource is scarce simultaneously, when simple ranking fails. The optimal mix always lies at a corner point of the feasible region, and shadow prices value relaxing each binding constraint.

Where it fits

SubjectCost AccountingAdvancedTopicRelevant Costs & Decision-MakingAdvanced

The formula

LaTeX

\max\; Z = \sum_j c_j x_j \quad \text{s.t.}\; \sum_j a_{ij} x_j \leq b_i \;\forall i,\; x_j \geq 0

Variables

$Z$: objective function value to be maximised (typically total contribution margin) (€)
$c_{j}$: contribution per unit of product j (€ per unit)
$x_{j}$: quantity of product j to produce (the decision variable) (units)
$a_{ij}$: units of resource i consumed per unit of product j (resource units per product unit)
$b_{i}$: total available units of resource i (constraint limit) (resource units)

For a two-product problem the feasible region is a polygon; the optimal solution always lies at one of its corner points. The simplex algorithm locates the optimal corner for problems with more than two decision variables.