QBUS2310 Chap.1 Optimisation Decisions and Linear Models
Optimisation Decisions and Linear Models
Management science turns a decision into a mathematical model. The essential move is to separate choices from given information, state one measurable objective, and translate every operational limit into a constraint. A solution is useful only when it is feasible, interpretable, and connected to the original decision.
This chapter establishes the vocabulary used throughout the unit and shows why prescriptive analytics goes beyond prediction. Reviewing these six terms first shows whether a candidate plan is genuinely feasible before its objective value is trusted. Decision Variable is a controllable quantity whose value the optimisation model chooses.
Parameter is given information that describes costs, capacities, requirements, or other fixed inputs. Objective Function is the measurable expression that the model maximises or minimises. Feasible Region is the set of all variable values that satisfy every constraint and sign restriction. Optimal Value is the best objective value attained by a feasible solution.
Unbounded Model is a model in which feasible objective improvement continues without a finite limit. The formulation above is verified in two steps: A complete formulation names units for x and y, gives a linear objective, lists each capacity inequality, and includes nonnegativity. Feasibility is checked first by substitution. Only feasible plans may be compared by objective value.
What this chapter covers
- 01
Prescriptive analytics and decision quality
- 02
Decision variables and parameters
- 03
Objectives and constraint systems
- 04
Feasible and optimal solutions
- 05
Infeasibility and unboundedness
- 06
Linear objective and constraint tests
- 07
Sign restrictions and free variables
- 08
Model assumptions and validation
Optimisation Decisions and Linear Models worked example
- +1Let x and y denote nonnegative production quantities. These are choices, while capacities and unit contributions are parameters.
- +1Write the contribution objective as a linear expression in x and y. State whether it is maximised or minimised.
- +1Translate each capacity into a left-hand resource use no greater than the available amount, then add sign restrictions.
- +1Substitute the proposed plan into every constraint before comparing objective values. A high objective never repairs infeasibility.
Key terms
- Decision Variable
- A controllable quantity whose value the optimisation model chooses.
- Parameter
- Given information that describes costs, capacities, requirements, or other fixed inputs.
- Objective Function
- The measurable expression that the model maximises or minimises.
- Feasible Region
- The set of all variable values that satisfy every constraint and sign restriction.
- Optimal Value
- The best objective value attained by a feasible solution.
- Unbounded Model
- A model in which feasible objective improvement continues without a finite limit.
Optimisation Decisions and Linear Models FAQ
Why does prescriptive analytics and decision quality matter?
Decision Variable: A controllable quantity whose value the optimisation model chooses. Parameter: Given information that describes costs, capacities, requirements, or other fixed inputs. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How can I check feasible and optimal solutions?
Parameter: Given information that describes costs, capacities, requirements, or other fixed inputs. Objective Function: The measurable expression that the model maximises or minimises. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
What separates decision variable from parameter?
Objective Function: The measurable expression that the model maximises or minimises. Feasible Region: The set of all variable values that satisfy every constraint and sign restriction. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Which error is most likely around linear objective and constraint tests?
Feasible Region: The set of all variable values that satisfy every constraint and sign restriction. Optimal Value: The best objective value attained by a feasible solution. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How should I practise model assumptions and validation?
Optimal Value: The best objective value attained by a feasible solution. Unbounded Model: A model in which feasible objective improvement continues without a finite limit. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Exam move
Practise the four-part modelling sentence: choice, goal, limits, signs. For any story, underline controllable quantities, circle numerical data, and write units beside each variable before doing algebra. Rehearse the chapter method in this order: Let x and y denote nonnegative production quantities. These are choices, while capacities and unit contributions are parameters.
Write the contribution objective as a linear expression in x and y. State whether it is maximised or minimised. Translate each capacity into a left-hand resource use no greater than the available amount, then add sign restrictions. Substitute the proposed plan into every constraint before comparing objective values. A high objective never repairs infeasibility.
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