Chapter 4: Linear Programming

Study Guide Highlights

Purpose and model assumptions

Linear programming provides a structured way to choose between feasible production combinations when more than one resource or requirement constrains the decision. The chapter sets out the assumptions that make the model workable, including stable relationships between activity and resource use. A mathematical optimum depends on those assumptions and therefore represents a model of the decision rather than a complete description of business reality.

Variables, objectives and constraints

The model uses variables to represent the quantities being chosen, an objective function to express what the organisation wants to maximise or minimise, and constraints to represent limits or minimum requirements. Clear definitions and consistent units matter throughout. Non-negativity is also part of the model because production quantities cannot be meaningfully represented by negative values.

Feasible regions and vertices

Each constraint removes combinations that cannot be achieved, leaving a feasible region containing the combinations that satisfy all restrictions. The important candidate solutions lie at the region's vertices, and a graph can show which constraints meet at the optimum. A graph does not have to be decorative; its value is in making the feasible set and relevant intersections visible.

Exact intersections and practical output

Where the optimal point occurs at the intersection of binding constraints, simultaneous equations can be used to identify its exact coordinates. The chapter also considers what happens when a mathematical solution produces fractional units that are not practical. The underlying objective remains to compare feasible alternatives consistently, while recognising that whole-unit production may require a practical adjustment rather than blind acceptance of a fractional result.