QBUS2310 Chap.4 Building Linear Optimisation Models
Building Linear Optimisation Models
The hardest modelling step is not solving an optimisation model but deciding what every symbol means. Different applications share conservation, capacity, coverage, and balance patterns. This chapter compares those patterns across production, diet, transport, workforce, inventory, and financing settings, with special attention to indexing and timing.
These paired terms let a reader check that every balance equation accounts for what enters, leaves, or is stored. Flow Balance is an equation matching what enters, leaves, is produced, or is consumed by an entity. Coverage Constraint is a requirement that enough scheduled resources serve a time period or demand group. Inventory Balance is an equation linking opening stock, production, demand, and closing stock over time.
Transportation Model is a network allocation model that sends supply to demand locations at minimum cost. Diet Model is a cost-minimisation model meeting lower bounds on nutrients or other requirements. Cash Flow is money entering or leaving a period, with timing that must be represented explicitly.
The shipping plan above is checked against every balance in turn: Ship three and one from the first plant, then zero and five from the second. Every supply and demand balance holds, all flows are nonnegative, and direct substitution gives cost 17.
What this chapter covers
- 01
Manufacturing resource models
- 02
Diet lower-bound models
- 03
Transportation balances
- 04
Cyclic workforce coverage
- 05
Inventory flow conservation
- 06
Regular and overtime capacity
- 07
Cash-flow timing
- 08
Assumptions and units
Building Linear Optimisation Models worked example
- +1Define x_ij as units shipped from plant i to outlet j. Use one row balance for each plant and one column balance for each outlet.
- +1Let a=x_11. The balances imply flows a, 4-a, 3-a, and 2+a, with zero at most a at most three.
- +1Substitute into cost: a+4(4-a)+3(3-a)+2(2+a)=29-4a.
- +1The cost falls as a rises, so a=3. The optimal flow matrix is [[3,1],[0,5]] and total cost is 17.
Key terms
- Flow Balance
- An equation matching what enters, leaves, is produced, or is consumed by an entity.
- Coverage Constraint
- A requirement that enough scheduled resources serve a time period or demand group.
- Inventory Balance
- An equation linking opening stock, production, demand, and closing stock over time.
- Transportation Model
- A network allocation model that sends supply to demand locations at minimum cost.
- Diet Model
- A cost-minimisation model meeting lower bounds on nutrients or other requirements.
- Cash Flow
- Money entering or leaving a period, with timing that must be represented explicitly.
Building Linear Optimisation Models FAQ
Why does manufacturing resource models matter?
Flow Balance: An equation matching what enters, leaves, is produced, or is consumed by an entity. Coverage Constraint: A requirement that enough scheduled resources serve a time period or demand group. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How can I check cyclic workforce coverage?
Coverage Constraint: A requirement that enough scheduled resources serve a time period or demand group. Inventory Balance: An equation linking opening stock, production, demand, and closing stock over time. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
What separates flow balance from coverage constraint?
Inventory Balance: An equation linking opening stock, production, demand, and closing stock over time. Transportation Model: A network allocation model that sends supply to demand locations at minimum cost. 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 regular and overtime capacity?
Transportation Model: A network allocation model that sends supply to demand locations at minimum cost. Diet Model: A cost-minimisation model meeting lower bounds on nutrients or other requirements. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How should I practise assumptions and units?
Diet Model: A cost-minimisation model meeting lower bounds on nutrients or other requirements. Cash Flow: Money entering or leaving a period, with timing that must be represented explicitly. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Exam move
Build constraints by entity: one balance per source, destination, day, period, or month. Keep the time index visible because a repayment or inventory carry belongs to a specific later period. Rehearse the chapter method in this order: Define x_ij as units shipped from plant i to outlet j. Use one row balance for each plant and one column balance for each outlet. Let a=x_11.
The balances imply flows a, 4-a, 3-a, and 2+a, with zero at most a at most three. Substitute into cost: a+4(4-a)+3(3-a)+2(2+a)=29-4a. The cost falls as a rises, so a=3. The optimal flow matrix is [[3,1],[0,5]] and total cost is 17.
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