Discrete Commitment Optimization¶
Choose among indivisible options or commitments when partial allocation is impossible.
The Diagnostic Story¶
Symptom: Each option looks attractive on its own, so the team picks the highest-ranked items one at a time — only to discover that the chosen set is over budget, incompatible, or leaves critical needs unmet. Rounding a fractional plan creates overruns. Selections that seemed independent turn out to share constraints, and the bundle that emerges fails even though every individual component passed review.
Pivot: Represent the decision in whole, indivisible units from the start; define the coupling constraints that link choices; and compare complete feasible bundles against an explicit objective rather than ranking options one at a time.
Resolution: The selected commitment bundle is globally feasible rather than locally appealing: budget, compatibility, and coverage constraints are satisfied together, and the tradeoffs among bundles are documented and visible. Fewer choices require emergency reversal after implementation begins.
Reach for this when you hear…¶
[capital budgeting] “We approved the three highest-NPV projects independently and then realized they all need the same engineering team in the same quarter — we have to re-run this as a bundle problem, not a ranking.”
[hospital staffing] “Shift scheduling looks fine on paper until you notice that three of the approved assignments require the same specialist to be in two places at once.”
[procurement] “The fractional relaxation said buy 2.7 of this equipment unit, which is not a thing, and when we rounded up we blew the budget and when we rounded down we missed minimum capacity.”
When This Archetype Applies¶
Partial catalog groundingSome structural conditions are represented by existing abstractions, but no sufficient condition set is fully represented.
Diagnostic problem
A decision contains indivisible or integer commitments coupled by constraints, but it is treated as smoothly divisible, fractionally allocable, or separable one option at a time.
What this problem means
The structural problem is false continuity. Decision-makers may reason as if options can be partially funded, smoothly adjusted, or independently ranked, while the real world demands whole commitments. A fractional or locally attractive answer then fails at implementation: projects cannot be half-launched, coverage requirements are missed, dependencies are broken, or a rounded answer violates capacity.
A second structural problem is bundle interaction. Each option may look good alone, but the selected set can be poor because choices share budgets, staff, space, deadlines, political legitimacy, or technical dependencies. Discrete commitment problems therefore require combination-level feasibility, not just option-level scoring.
Show the applicability expression
Applicability expression5 distinct conditions
groundedpartly groundedopen
5 conditions, all required.
5Required in every casenumbered 1–5
These hold no matter which pattern applies.
Indivisible options · grounded
Options are indivisible or all-or-nothing.
A decision involves indivisible commitments, assignments, selections, placements, or whole-number quantities, but decision-makers treat the problem as if options could be smoothly ranked, fractionally allocated, or considered one at a time. The narrower requirement in this condition set is: Options are indivisible or all-or-nothing.
Constraint-coupled choices · grounded · any one of 3
Choices interact through shared constraints.
The system must commit to whole choices while objectives and constraints span across those choices. The narrower requirement in this condition set is: Choices interact through shared constraints.
Integer commitment requirement · grounded
Whole-number quantities, slots, or commitments are required.
Use this archetype when choices are indivisible, when whole-number quantities matter, or when assignments and placements must be made as complete commitments. The narrower requirement in this condition set is: Whole-number quantities, slots, or commitments are required.
Binding allocation constraints · grounded · any one of 9
Fairness, eligibility, compatibility, or exclusion constraints bind.
This is a load-bearing situation condition in the diagnostic expression. The condition is: Fairness, eligibility, compatibility, or exclusion constraints bind. If it does not hold, this particular condition set is incomplete.
Continuous treatment mismatch · open
Decision-makers currently use a fractional, smoothly ranked, or one-option-at-a-time treatment that violates the discrete coupled structure.
This condition preserves a load-bearing part of the diagnostic problem that was not captured by a source-condition atom. It remains explicit because omitting it would weaken the sufficient condition set.
Other requirements and context (2)
Why these sit outside the expression
Supporting context — it may accompany or help interpret the situation, but it is not a load-bearing condition in a sufficient diagnostic set.
Supporting contextThe number of feasible combinations is too large for informal enumeration.
The archetype does not require a formal solver, but it does require explicit representation of the choice units, constraints, feasible combinations, and selection rule. In this archetype, the relevant contextual consideration is: The number of feasible combinations is too large for informal enumeration. It helps interpret the situation or strengthens the practical case for examining the archetype.
Supporting contextSelected commitments create durable implementation consequences.
Treating options independently creates local appeal but can produce globally infeasible, dominated, unfair, or strategically poor commitment bundles. In this archetype, the relevant contextual consideration is: Selected commitments create durable implementation consequences. It helps interpret the situation or strengthens the practical case for examining the archetype.
Coverage
4 of 5 conditions grounded · 1 open.
Mechanisms / Implementations¶
- Assignment Model: Represents pairings between agents and tasks as an eligibility grid, then commits each agent to exactly one compatible partner.
- Branch-and-Bound Procedure
- Constraint Satisfaction Search: Explores the space of discrete combinations to find any assignment that violates no constraint, driven by feasibility rather than an objective.
- Crew Scheduling Model: Builds legal duty bundles that cover every required shift under labor rules, minimum staffing, and equitable distribution of work.
- Facility Location Model: Chooses which whole sites to open so that demand is covered at acceptable cost and distance.
- Integer Programming Model: A formal declarative statement of a discrete decision — its binary and whole-number variables, objective, and constraints — written to be handed to a solver.
- Integer Programming Solver: Software that searches a formulated discrete model's feasible space and returns a proven or near-optimal commitment bundle.
- Project Selection Matrix: A structured table laying out candidate projects with their scores, costs, dependencies, and selection status for transparent human review.
- Selection Review Board: A standing human body that deliberates over a proposed commitment bundle, rules on exceptions and fairness, and takes accountable ownership of the decision.
- Solver Dashboard: A live interface that visualizes a running solver — its candidate solutions, objective values, remaining gap, and constraint violations.
Related Abstractions¶
Abstractions this archetype builds on — directly (a source ingredient) or as a related pattern. Links follow the typed catalog namespace.
Built directly on (3)
- Discreteness: Countable steps.
- Integer Linear Programming (ILP): Discrete optimization with integer variables.
- Optimization: Finds best solution under constraints.
Also references 6 related abstractions
- Cardinality: Size of sets.
- Complexity (Time/Space): Resource scaling with input size.
- Constraint: Limits possibilities to guide outcomes.
- Coupling: Interdependence among subsystems.
- Resource Management: Allocation of finite assets.
- Scheduling: Organizing tasks over time.
Variants¶
Narrower or domain-specific specializations that share this archetype's core structure. Recognized variants are established; candidate variants are provisional.
Binary Project Selection · subtype · recognized
Choose which indivisible projects, proposals, features, or initiatives to accept when each candidate is essentially yes-or-no.
Integer Quantity Commitment · subtype · recognized
Choose whole-number quantities of units, slots, staff, licenses, batches, or assets when fractional quantities are impossible or meaningless.
Assignment / Matching as Discrete Commitment · subtype · candidate
Assign agents, tasks, slots, resources, or roles to one another when each match is a discrete commitment constrained by capacity and compatibility.
Facility Location Commitment · domain variant · recognized
Choose which physical or logical sites to open, close, place, or reserve when each site is an indivisible commitment with network effects.
Editorial Notes¶
Problem Classification¶
Classification: Decision, Search & Optimization Failure → Allocation, Matching & Opportunity Cost
Problem kernel: indivisible commitments are treated as smoothly divisible
Rationale: Whole-number assignments and placements cannot be optimized by fractional intuition or independent ranking without violating feasibility.
Independent corroboration: The earliest necessary condition in the frozen evidence is: A decision involves indivisible commitments, assignments, selections, placements, or whole-number quantities, but decision-makers treat the problem as if options could be smoothly ranked, fractionally allocated, or considered one at a time. That is a allocation matching and opportunity cost problem because Discrete agents, tasks, resources, or commitments are assigned without counterpart fit, indivisibility, comparative advantage, or visibility into displaced alternatives.
Review outcome: Independent reviewer agreement; high confidence.