ETF2480 Chap.3 Business Applications of Linear Programs
Business Applications of Linear Programs
Many business linear programs share a conservation structure. Blending models total attributes contributed by ingredients and enforce composition bands. Multi-period production models reconcile opening inventory, production, demand and closing inventory for every period. Investment models carry cash between dates, while process models conserve material as it moves through transformations.
Recognising a pattern speeds formulation, but it does not justify copying coefficients or omitting context-specific rows. The model must still explain units, yield, timing, capacity and any final-state requirement. A solution should be reported at the same granularity as its variables: ingredient quantities for a blend, a period-by-period plan for inventory, or activity levels and material flows for a process.
What this chapter covers
- 01
Diet and blending requirements
- 02
Multi-period inventory balance
- 03
Cash and investment timing
- 04
Production-process conservation
- 05
Yield and conversion coefficients
- 06
Initial and terminal conditions
Worked example · free
Plan production across two periods
- +1Let p1,p2 be production and i1 be inventory after Period 1.
- +2Write p1 = 60 + i1 and i1 + p2 = 85, with 0 <= p1 <= 90 and 0 <= p2 <= 65.
- +2Period 2 needs 20 units carried in, so choose p1 = 80, i1 = 20 and p2 = 65.
- +1Cost is 80(10) + 20(1) + 65(12) = $1,600.
Key terms
- Blending model
- A model choosing ingredient quantities to meet composition limits at minimum cost or another objective.
- Inventory balance
- An equation reconciling opening stock and production with demand and closing stock.
- Yield coefficient
- The usable output produced by one unit of an input or process activity.
- Initial condition
- The known state, such as opening inventory or cash, at the start of the horizon.
- Terminal condition
- A required or valued state at the end of the planning horizon.
Business Applications of Linear Programs FAQ
How do blending percentages stay linear?
When total blend mass is fixed, attribute contribution is a weighted sum bounded by fixed requirements. If total mass varies, a percentage condition can often be written with the required proportion multiplied by total quantity on the other side, which remains linear after collecting terms.
Why is the inventory balance essential?
It is the conservation law linking consecutive periods. Without it, unused production can vanish or future demand can be met by stock that was never produced. The balance also makes holding cost and capacity trade-offs visible.
How should I model process losses?
Use a yield coefficient grounded in the process data. If one input unit produces 0.8 usable units, downstream use cannot exceed 0.8 times upstream activity unless another source exists. State whether unused output, scrap or inventory is permitted.
How can I audit a multi-period plan?
Build a table of opening stock, production, demand and closing stock for every period. Each row must balance, consecutive inventory values must link, and the objective must reproduce from the reported quantities.
Exam move
Create a pattern library organised by balance role rather than by company story. For a blend, write total mass and one row for each attribute band. For a multi-period plan, draw a timeline and write opening plus inflow equals use plus closing at every stage. For a process, draw arrows between activities and label each yield before writing equations.
Use different symbols for stock, production and demand so the period link remains visible. After solving, reconcile every period or processing node numerically. Then interpret why inventory, an ingredient or a process is used: capacity shortfall, cost difference, composition requirement or yield.
Finally, test a changed assumption, such as lower capacity or a terminal inventory requirement, and predict which part of the solution should move before re-solving. For blending, convert every percentage into an amount equation and check whether the denominator is fixed or variable. Construct paired minimum and maximum composition rows so their directions become meaningful rather than memorised.
For planning across time, write the initial condition, every transition and the terminal condition in separate lines; then sum all balances to confirm that internal inventories cancel. For processing networks, calculate the implied units at each stage and test whether any material is consumed twice. Practise adding realistic extensions one at a time: overtime, disposal, backlog, a safety-stock floor or a limited ingredient.
Each extension should have a named variable or row and an economic interpretation. Keep a small reconciliation table beside solver output showing production, carry-in, demand and carry-out by period. A solution is not ready to report until those columns balance and its cost can be reconstructed from the stated decisions.
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