ETF5248 Chap.2 Business Models with Linear Programming
Business Models with Linear Programming
Business linear programs often repeat a small number of structures. Multi-period plans use conservation to connect opening stock, production, demand and closing stock, while blending models route inputs into products under availability and quality restrictions.
Product-mix models combine resource rows with contribution or cost objectives.
The challenge is not recognising a keyword but assigning each variable one meaning and keeping physical flow separate from economic value.
This chapter shows how indexed variables scale a formulation, why initial and terminal conditions close a planning horizon, how recipe percentages become linear rows, and how a solved schedule should be checked period by period before it becomes advice.
Timing and composition create different forms of coupling.
In an inventory model, today's closing stock is tomorrow's opening stock, so a production decision has consequences beyond its own period. Initial inventory prevents the first balance from floating, while a terminal requirement stops the optimiser from treating the planning horizon as if operations end permanently. In a blend, each two-index variable identifies both an ingredient and a destination.
Availability constraints sum across destinations, and product rows sum the inputs reaching one blend. Quality limits then compare the delivered attribute with total output. Costs belong on the variables that create them: production cost on production, holding cost on closing inventory, and shortage penalties only when shortages are deliberately allowed.
After solution, a row-by-row schedule or input-output matrix should reproduce the claimed totals. This audit exposes missing destinations, double-counted costs, backward travel through time and recipe directions that were reversed during formulation.
What this chapter covers
- 01
Multi-period inventory balance equations
- 02
Initial and terminal boundary conditions
- 03
Production, holding and shortage costs
- 04
Blending variables with two indices
- 05
Availability, output and recipe rows
- 06
Product mix and whole-unit cautions
Worked example · free
Carry inventory through a three-period plan
- 1Period 1 closing stock is 12 + 18 - 20 = 10 units.
- 1Period 2 opens with 10 units and closes with 10 + 25 - 22 = 13 units.
- 1Period 3 opens with 13 units and closes with 13 + 20 - 26 = 7 units.
- 1All closing stocks are non-negative, so the schedule is feasible without shortages and finishes with seven units.
Key terms
- Balance equation
- An equality stating opening quantity plus inflow minus outflow equals closing quantity.
- Boundary condition
- A fixed initial or terminal value that closes a finite planning horizon.
- Holding cost
- The cost attached to inventory carried from one period into another.
- Recipe constraint
- A row enforcing the required composition or quality of a blended output.
- Indexed variable
- A decision quantity labelled by dimensions such as product, ingredient, location or period.
Business Models with Linear Programming FAQ
Why is one inventory constraint not enough?
Demand timing matters. A single total-production row can claim feasibility even when early demand occurs before enough production is available. One balance equation per period preserves chronology, and the closing value from one period becomes the opening value of the next.
When should demand be equality?
Equality is suitable when output must match demand exactly and there is no disposal, storage or lost-sales option. If overproduction, backlogging, disposal or unmet demand is possible, represent that choice explicitly with its own variable and cost rather than weakening the meaning of the demand row.
How do percentages remain linear in blending?
Multiply the required proportion by total blend weight and compare it with the total amount of the attribute supplied. When total output is a linear sum of ingredient variables, clearing a constant percentage keeps the row linear. Check the direction for minimum and maximum quality limits.
What should I report from a business LP?
Report the period schedule or product mix, total objective value, scarce resources and important assumptions. For a time model, include opening and closing stock. For a blend, include ingredient routing and quality verification. The explanation should allow a manager to reproduce the feasibility check.
Exam move
Create a reusable audit table for each recurring structure. For inventory, use columns for opening stock, production, demand and closing stock, and verify every carryover. For blending, create a matrix with inputs as rows and outputs as columns, then check each availability sum and each product recipe. Rewrite percentages as algebra only after deciding what the denominator represents.
Finally, attach every cost to the exact variable that creates it. These habits prevent double charging, missing destinations and off-by-one timing errors.
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