ETF2480 Chap.1 Linear Programming Model Anatomy
Linear Programming Model Anatomy
Linear programming begins with translation. A decision variable names a controllable quantity and its unit; the objective states what the organisation wants to maximise or minimise; constraints represent capacity, demand, policy or physical balance; and domains state signs, bounds or integrality.
Matrix form then aligns one variable with each column and one restriction with each row, allowing the same mathematics to scale from a small hand model to R or Python. The central exam skill is not merely recognising an LP. It is explaining why each coefficient and inequality represents the business situation, checking the returned status, and reporting the solution with its units and assumptions.
What this chapter covers
- 01
Decision variables and units
- 02
Linear objectives and coefficient meaning
- 03
Capacity, demand and balance constraints
- 04
Non-negativity and variable domains
- 05
Matrix form cTx and Ax <= b
- 06
Equivalent max, min and inequality transformations
Worked example · free
Build a product-mix model from capacity data
- +1Let p and b be poster and brochure batches, measured in weekly batches.
- +2Maximise 28p + 44b subject to 2p + b <= 50, p + 2b <= 52, and p,b >= 0.
- +2Solving the two binding equations gives p = 16 and b = 18; both capacity checks hold exactly.
- +1The maximum contribution is 28(16) + 44(18) = $1,240, conditional on divisibility and constant coefficients.
Key terms
- Decision variable
- A controllable quantity whose value the optimisation model selects, stated with a unit and any index sets.
- Objective function
- The expression that ranks feasible decisions by profit, cost, distance, service or another stated goal.
- Constraint
- A mathematical restriction representing a resource, requirement, balance, policy or logical relationship.
- Feasibility
- The property of satisfying every constraint and variable-domain restriction at the same time.
- Matrix form
- A compact representation in which columns correspond to variables and rows correspond to constraints.
Linear Programming Model Anatomy FAQ
How do I choose decision variables?
Identify the choices management directly controls, then define each with a noun, time period, unit and index when needed. Avoid variables that are merely outputs of the decision unless they simplify the model and are linked by a defining equation.
How can I check an inequality direction?
Translate the row back into plain language and substitute an obviously impossible plan. At most, capacity and upper bounds usually use a less-than-or-equal direction. At least, service and minimum requirements usually use a greater-than-or-equal direction. Equality is reserved for exact balance or definition.
Why is matrix order important?
A solver associates each objective and constraint coefficient with a column position, not with your intended noun. If column order changes without changing every vector and row, the code represents a different problem even though its dimensions remain valid.
What is the final formulation audit?
Confirm that every variable has a unit and domain, every coefficient matches the declared column order, and every constraint translates back to a defensible business sentence. Then test simple feasible and impossible plans before solving.
Exam move
Practise formulation before software. Take a short scenario and underline controllable nouns, objective phrases, capacities, minimum requirements and quantities that must balance. Define variables with units, then write one labelled row at a time and explain its direction. Perform a dimensional audit: a profit coefficient times units should produce dollars, while a labour coefficient times units should produce hours.
Next, convert the formulation into matrix form and verify that each column belongs to the same variable throughout. After solving, inspect status, substitute the returned values into every row and recompute the objective independently. Finish with a short management statement that names the chosen quantities, objective value, binding limits and one assumption.
This sequence mirrors the marking roles published for Assignment I and prevents a correct solver command from masking a wrong model. Build a personal error log organised by variable, objective, row and domain. For every missed coefficient, rewrite the sentence that should have generated it and record the units on both sides.
Alternate maximisation and minimisation cases so the objective verb is never inferred from a familiar diagram. Include at least one equality-balance case, one minimum-service case and one capacity ceiling in each practice set. Before accepting a formulation, try a zero plan, a one-unit plan and a deliberately excessive plan; explain which rows each should satisfy or violate.
In code, print the variable order beside the objective vector and matrix dimensions before calling the solver. If a returned solution surprises you, diagnose the algebra first instead of assuming the algorithm failed. Finally, rerun one scenario after changing a single capacity and predict the direction of the response beforehand.
That prediction forces the coefficients to retain their managerial meaning rather than becoming anonymous entries in a matrix.
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