Optimisation constraints: measuring their cost before declaring them untouchable.
Key distinctions
Three conditions before calculating.
Measurable constraint
A rule such as "stay visible" cannot be optimised. It must become a minimum coverage, a number of weeks, a share of voice or a demand threshold with a method of measurement.
Nature and authority
A legal obligation, a contract, a capacity and a management preference do not have the same status. The register states who can change the limit and within what delay.
Value of relaxation
The shadow price indicates the contribution gained for one additional unit of resource, or lost for a tightened requirement, as long as the change stays local.
Method
Turning management rules into revisable constraints
- 01Inventory the limits
Collect budget, minimums, maximums, contracts, dependencies, capacities, risks and delays; eliminate duplicates and purely narrative formulations.
- 02Specify each rule
Document variable, direction, threshold, unit, window, owner, justification and expiry date; identify the hard constraints and the penalisable preferences.
- 03Solve and diagnose
Compare the unconstrained solution, the feasible solution and the relaxation scenarios; compute active constraints, slack and shadow prices.
- 04Negotiate the relaxations
Compare the marginal value with the operational or political cost of change; revise first the active constraints with a high shadow price and a low cost of change.
The register of constraints and their shadow prices
Budget, brand floor, CRM ceiling and creative capacity are connected to a unit, a status, a marginal cost and an action.
Shadow price +0.42
Shadow price −0.25
Shadow price +0.68
Shadow price +€4.5k/asset
Shadow price 0
Relaxing CRM capacity by €20k creates €13.6k of contribution
The constrained solution produces €486k of contribution. The CRM ceiling is active and its shadow price is €0.68 per additional euro. Increasing capacity by €20k costs €9k. The brand floor has an opportunity cost of €0.25 per euro imposed.
Gross value of the CRM relaxation: 20,000 × 0.68 = €13,600. Net value after capacity: 13,600 − 9,000 = €4,600. Tightening the brand floor by €40k would destroy about 40,000 × 0.25 = €10,000.
The CRM extension is accepted within the local range. The brand floor is kept at its current level and its justification is reviewed before any increase.
Acceptance conditions
What must be true to act.
- 01
Every constraint has a unit, a period, an owner and a source.
- 02
Hard constraints are separated from negotiable preferences.
- 03
Slack and shadow price are examined for the active constraints.
- 04
Relaxation is decided on its net value and its range of validity.
Limits
What this analysis does not prove.
- The shadow price is local and can change after a large relaxation.
- Non-linear or discrete constraints require specific analyses.
- An economic value does not allow a legal obligation to be broken.
References
Works cited.
- Boyd, S. & Vandenberghe, L. (2004), Convex Optimization.
- Nocedal, J. & Wright, S. J. (2006), Numerical Optimization.
- Asadpour, A. et al. (2019), “Concise Bid Optimization Strategies with Multiple Budget Constraints”, Management Science, 65(12), 5785–5812.