Plant sizing & Pareto fronts
The key idea
Plant sizing can produce several useful answers. One candidate costs less but leaves more demand unmet. Another uses less diesel but requires more capital. Before ranking them, distinguish a constraint from an objective.
A constraint decides whether a candidate is feasible: for example, unmet energy must be at most the stated percentage. An objective ranks the feasible candidates: for example, minimize lifetime cost and fuel use.
Remove choices that are worse on both axes
In the fixed example below, design C costs USD 60,000 and leaves 2% of energy unmet. Design D costs USD 67,000 and leaves 3% unmet. On cost and unmet energy, C is better in both respects. C dominates D.
Design B costs USD 52,000 but leaves 4% unmet. Neither B nor C improves both objectives relative to the other. They describe a trade-off.
All amounts and result values in this lesson are invented teaching summaries. They do not represent equipment quotations or Phasor optimization runs.
Change the question
Start with the 4% unmet-energy constraint and cost versus unmet energy. Select D and inspect its status. Then switch the second objective to fuel. Finally, reduce the unmet-energy limit to zero and see which candidate remains feasible.
Feasibility first, then the non-dominated front
Line: feasible non-dominated designs. Dashed outline: outside the unmet-energy limit. A line segment does not prove a design exists between candidates.
- Design C cost
- $60k
- Unmet energy
- 2%
- Fuel
- 10 kL/year
Design C: feasible and on this front. It remains a comparison option, not an automatic recommendation. Feasible front: B, C, E.
Data table: All candidate data and status
| Design | Cost USD k | Unmet % | Fuel kL/y | Status |
|---|---|---|---|---|
| A | 40 | 8 | 14 | Infeasible |
| B | 52 | 4 | 12 | Front |
| C | 60 | 2 | 10 | Front |
| D | 67 | 3 | 13 | Dominated |
| E | 78 | 0 | 8 | Front |
| F | 88 | 1 | 4 | Dominated |
Model note · Six invented candidate summaries, A–F. Cost is assumed lifetime cost in thousands of constant 2026 USD; fuel is kL/year. These are a fixed teaching table, not recorded optimization results. Both chosen objectives are minimized.
Design F uses less fuel than E but costs more and leaves more demand unmet. It can appear on the cost/fuel front while remaining dominated on the cost/unmet-energy front. A front only has meaning with its axes and feasibility rules attached.
A search only compares what it searched
A sizing result depends on the available component sizes, search bounds, dispatch method, weather and cost assumptions. A front from a finite candidate set does not establish a universal optimum. A line between two candidates is a visual guide; it is not evidence that an intermediate equipment configuration exists.
The final design also needs checks beyond the plotted objectives. Verify operating assumptions, the electrical network and project requirements before promoting a candidate.
The dominance testOptional — the page reads completely without it
With both objectives minimized, candidate A dominates candidate B when it is no worse on either axis and strictly better on at least one:
A dominates B
costA ≤ costB and fA ≤ fB · with at least one strict <
where f is the second objective, unmet energy or fuel. Equal candidates do not dominate each other.
step one: feasibility
feasible = [ candidates with unmet ≤ limit ]
step two: the front
front = [ feasible candidates not dominated by any other feasible candidate ]
Infeasible points stay in the diagram to explain rejected options, but they cannot become a feasible recommendation. The widget compares two objectives at a time; Phasor can use other objective and constraint combinations. This bounded comparison demonstrates the rule and does not recreate its optimization search.
See it in Phasor
Define the sizing bounds, objectives and feasibility limits. Review candidate results and their assumptions before choosing and saving the design you want to develop.