MATH2070 Chap.1 Linear Programming Models and Geometry

Linear Programming Models and Geometry

Linear programming begins with translation. A decision context becomes variables with units, a linear objective and linear constraints. The feasible region is the intersection of every requirement, so feasibility must be checked before objective values matter. In two variables, objective contours and corner comparisons reveal finite optima, alternative optima, infeasible systems and unbounded improvement.

The chapter closes by interpreting slack and returning the selected coordinates to the original decision units. Capacity statements, minimum requirements and exact balances create different inequality directions, and a dimensional check helps ensure coefficients sit beside the correct variables. Boundary intersections become candidates only after substitution into every original constraint.

Parallel objective contours can make a complete edge optimal, while an open feasible region is unbounded in value only when it contains an improving ray. Continuous answers also need care when the decisions are indivisible: rounding is not guaranteed to preserve feasibility or optimality.

A complete modelling argument therefore connects prose, geometry, algebra and interpretation rather than presenting a graph or coordinate pair in isolation.

What this chapter covers

• 01

  Decision variables and units

• 02

  Linear objectives

• 03

  Constraint direction

• 04

  Feasible convex regions

• 05

  Corner point comparison

• 06

  Infeasible and unbounded cases

Worked example · free

Optimise a studio production plan

Key terms

Decision variable

A controllable quantity whose value is selected by the optimisation model.

Objective function

The linear quantity that the model maximises or minimises.

Feasible region

The set of points satisfying every constraint simultaneously.

Slack

The unused amount below an upper-bound resource constraint.

Binding constraint

A requirement satisfied at equality by the selected solution. It has zero slack and helps identify the active boundary that determines a vertex, although binding alone does not prove that the vertex is optimal.

Improving ray

A feasible direction along which the objective can continue to improve without limit. Its existence establishes objective unboundedness, unlike an unbounded feasible region whose open directions may all worsen the objective. It is checked by comparing a feasible direction vector with the linear objective coefficients. The sign of that comparison determines whether movement increases or decreases the objective. A direction must also remain inside every constraint for all non-negative step lengths; otherwise it is not a feasible ray. Graphically, draw the objective normal and compare it with the open edges of the feasible region. Algebraically, test the direction against the homogeneous part of the constraints. This distinction prevents an open drawing from being labelled unbounded in value when the objective actually points back toward a finite boundary. State the direction and its objective change explicitly so the diagnosis can be checked without relying on the sketch.

Linear Programming Models and Geometry FAQ

Exam move

Begin with formulation-only drills. Cover the solution and turn each sentence into a variable definition, objective term or labelled constraint. Write units beside every coefficient and test an obviously large activity level against each inequality; this reveals a reversed capacity or minimum more reliably than memorising direction words.

Include equality, upper-bound and lower-bound examples, and state non-negativity only when its economic meaning is defensible. On a second pass, sketch the feasible geometry without solving every intersection. Mark axis intercepts, shade one test point, draw an objective contour and indicate its improving direction.

Predict whether the model looks infeasible, bounded, unbounded or capable of multiple optima before doing algebra. Then compute only the intersections that can be vertices and substitute them into all original constraints, rejecting points that satisfy rearranged lines but not the model. For numerical practice, build a vertex table containing coordinates, feasibility checks, objective values and contextual interpretation.

Keep exact fractions until the final comparison so rounding cannot reverse a close decision. If whole items are required, treat the continuous answer as a bound and examine feasible integer neighbours rather than rounding both coordinates automatically. Record binding constraints and slack because those observations prepare the later dual interpretation.

Use an error ledger with separate columns for modelling, algebra and conclusion. A modelling error requires a new translation drill; an algebra error requires slower substitution; an interpretation error requires writing units and nouns in the final sentence.

Once a week, solve a fresh two-variable problem under a short time limit, then explain aloud why the optimum occurs at a vertex and which assumptions would change the model. Finish by checking that the reported activity plan, resource use and objective value all refer to the same time period.

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MATH1061 · MATH1961