MGMT20005 Chap.7 Business Applications of LP
Business Applications of LP
Week 7 reuses LP anatomy in three business settings. Operations models allocate production, workforce, inventory and capacity. Multi-period plans connect periods through a balance: opening stock plus production equals demand plus ending stock, and ending stock becomes the next period's opening quantity.
Finance applications allocate cash among investments with different returns, risk, liquidity and maturity dates; cash returning at a later node supports later decisions. Marketing applications allocate budget or placements across media while respecting availability, contract, minimum exposure and mix restrictions.
The equations change with units and timing, but the modelling sequence remains stable: define indexed variables, write the objective, preserve flow at every node, add policy constraints, solve, and interpret the result in management language.
What this chapter covers
- 01
Production scheduling and product-mix applications
- 02
Inventory balances across a planning horizon
- 03
Workforce and capacity allocation
- 04
Portfolio cash flows and maturity nodes
- 05
Media allocation, budgets and mix restrictions
- 06
Binding constraints and economic interpretation
Worked example · free
Allocate a campaign across two channels
- 2Formulate max R=3x+7y subject to x+2y at most 24, x at least y, and x,y non-negative.
- 2Transit has 3.5 reach per budget unit versus search at 3, so the model pushes toward transit until the mix rule x=y binds.
- 2Substitute x=y into the budget: y+2y=24, so y=8 and x=8.
- 2Reach is 3(8)+7(8)=24+56=80. The all-search corner gives 72, so the balanced plan is optimal.
Key terms
- Planning horizon
- The ordered set of periods over which linked production, inventory, cash or staffing decisions are modelled.
- Inventory balance
- A conservation equation linking opening stock and production to demand and ending stock in one period.
- Carryover inventory
- Ending stock from one period that becomes opening stock in the next period.
- Cash-flow node
- A point in time where available cash and maturing investments fund new investment decisions.
- Media allocation
- A model assigning budget or placements across channels to optimise reach or another campaign objective.
- Mix constraint
- A policy restriction controlling the relative quantities or shares of two or more decisions.
Business Applications of LP FAQ
What connects production decisions across periods?
The inventory balance. Opening inventory plus production must equal demand plus ending inventory, adjusted only for flows the model explicitly includes. Ending inventory then becomes the next period's opening stock. Omitting that link treats each period as independent and can create or destroy product inside the model.
How does a portfolio model represent maturity?
Each investment decision sends cash from its purchase node to the period when principal and return become available. The maturity amount enters that later cash-flow balance and can fund new decisions. Label variables by instrument and purchase period so the destination and return timing remain visible.
How should a percentage allocation rule be written?
Name the relevant total first. If channel A must receive at least 30% of a two-channel allocation, write A at least 0.30(A+B), then rearrange if useful. Writing A at least 0.30 imposes a small absolute floor, not a share, and changes the business meaning.
What should an application recommendation include?
State the indexed plan in operational units and timing, the objective achieved, and the constraint that shapes the result. Distinguish an exact optimisation output from an implementation recommendation, and identify which policy, budget or capacity assumption management should revisit if the proposed plan is impractical.
Exam move
Draw the flow before writing equations. For production, sketch periods with production inflow, demand outflow and inventory carryover. For finance, sketch purchase and maturity nodes. For media, list channels, costs, reach and any relative-share policy. Choose indices that match the business dimensions and write units beside each variable. Then formulate balances first, resource limits second and policy constraints last.
Check every balance dimensionally and read it back as a sentence. After solving, identify binding constraints and explain why they shape the plan. Do not quote a shadow price as a permanent resource value; its interpretation remains local to the range supplied by the sensitivity report. Practise a new scenario in each application family so the transferable conservation logic becomes more familiar than any one narrative.
Maintain a small library of balance patterns rather than memorising complete cases. Practise inventory carryover, cash maturity, shipment conservation and budget allocation with new symbols and units. For each pattern, identify the inflows, outflows and stock or balance that links periods or locations. Rebuilding the pattern from the business story is safer than forcing every application into one remembered equation.
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