In short
Reducing a formula's cost while keeping its character is a constrained optimization: minimize cost subject to the formula summing to 100%, to limits you set, and to requirements that stand in for the smell. When everything is linear it is a linear program, its best answer lies at a corner of the feasible region, and each constraint has a shadow price — what that requirement costs. Its most important lesson is uncomfortable: the optimizer removes anything you did not tell it to keep.
What you will learn
- Formulate a cost reduction as an objective with equality and inequality constraints
- Solve a small linear program by checking the vertices of its feasible region
- Interpret a shadow price as the cost of a constraint
- Explain why an optimizer needs constraints that encode olfactory requirements
Before this lesson: A Formula as a Linear System, Reducing Cost Without Losing Character: Study Guide, Dose, Cost and Performance Trade-Offs
Before you start
This continues the formula as a linear system and assumes the cost reasoning of reducing cost without losing character and dose, cost and performance trade-offs.
Formulation
Let xi be the mass fraction of material i in an accord, and ci its cost per kg. Then:
- Objective: minimize Σ ci xi.
- Composition: Σ xi = 1, all xi ≥ 0.
- Limits you record: xi ≤ ui, for example a limit you have recorded from a current IFRA Standard for your product category, or a level above which a panel found the material intrusive. Do not take a limit from a lesson; take it from the current source.
- Character requirements: linear stand-ins for smell, such as "the contribution of the woody materials, weighted by their strength, is at least s".
Worked example
All figures are invented for the arithmetic, in currency units per kg. An accord uses an expensive natural (x1, 400), a cheap carrier material (x2, 40) and a booster (x3, 200). The character index requires x1 + 0.5x3 ≥ 0.4, and a panel found x2 intrusive above 0.5.
Eliminate x2 = 1 − x1 − x3. Cost becomes 40 + 360x1 + 160x3, subject to x1 + 0.5x3 ≥ 0.4, x1 + x3 ≥ 0.5 (from x2 ≤ 0.5), x1 + x3 ≤ 1 and non-negativity. Check the vertices:
| x1 | x2 | x3 | Cost |
|---|---|---|---|
| 0.3 | 0.5 | 0.2 | 180 |
| 0 | 0.2 | 0.8 | 168 |
| 0 | 0 | 1 | 200 |
The optimum removes the natural entirely, because per unit of the encoded character index the booster is cheaper. If the natural carried something the index does not measure — and in real accords it usually does — the model has just destroyed it. Add x1 ≥ 0.15, the least you judge necessary for its signature. The optimum moves to x1 = 0.15, x3 = 0.5, x2 = 0.35, at cost 174.
Shadow price. Each extra unit of x1 costs 360 but, through the character constraint, saves 2 units of x3 at 160: a net 40 per unit, or 6 for the 0.15 you required. That is the price of keeping the natural, and it is now an explicit, discussable number rather than a vague sense that naturals are expensive.
Where it stops
- Smell is not linear. Mixture suppression and power-law intensity make "character indices" rough local approximations. Re-smell every optimum.
- Integer and minimum-weight rules break linearity. "Either 0 or at least 0.5 g" is a mixed-integer constraint, which solvers handle but hand methods do not.
- Costs move. An optimum is only optimal at the prices you entered; record them with a date.
In professional practice
Constrained optimization is the formal version of what a perfumer does when asked to cut a formula's cost by a stated amount without a panel detecting the change. Its real value is making trade-offs explicit: a brief manager can see that a natural costs 40 per unit of character, and decide.
Open questions
Can panel data supply character constraints good enough to optimize against? Pairing a mixture design (the next lesson) with a fitted model, then optimizing that model under cost, is a research workflow used in food formulation; whether it transfers to fine fragrance, with its long drydowns, is an open and testable question.
Practical exercise: Optimize one accord, then smell the optimum
- Choose an accord of three or four materials you have, with their real costs per kg from your invoices, dated.
- Write the objective, the composition constraint, at least one recorded limit with its source, and at least one character requirement you can justify from your evaluations.
- Solve it: by vertices if it has three materials, or with a spreadsheet solver.
- Calculate the shadow price of one constraint by relaxing it slightly and re-solving.
- Make the original and the optimum and compare them in a coded triangle test with at least six people.
You have done this well if every cost and limit is dated and sourced, the optimum satisfies every constraint when you substitute it back, your shadow price is checked by re-solving, and you report whether the panel detected the change and what that sample size can and cannot show.
Common mistakes
- Trusting an optimum without smelling it. The model only knows its constraints.
- Using a limit from memory or a lesson. Use the current Standard for your category.
- Leaving out the constraint that keeps a signature material. The optimizer will remove it.
Key terms
- Linear program
- An optimization whose objective and constraints are all linear in the unknowns. Its optimum, if one exists, lies at a vertex of the feasible region.
- Feasible region
- The set of all compositions that satisfy every constraint. In a linear program it is a convex polygon or polytope.
- Shadow price
- How much the optimal objective would improve if one constraint were relaxed by one unit. It prices what a requirement costs.
Knowledge check
Written and reviewed by the RUŌOD Lab team. This is general education, not legal, regulatory, medical or safety advice. Example dilutions and exercises are for learning, are not validated products, and have not been safety-assessed. Completing a lesson confers no qualification. How we check these lessons.