Distance distributions: each plot shows its own solution in black and the other side’s distribution in light gray. Both are frequency polygons joining the numbers of people in 20 equal-width distance bins. Both sides share bin boundaries and axis scales. The vertical lines mark average distance, maximum distance, and both selected objectives evaluated on that plot’s distances; coincident lines are merged.

Normalized markers: average distance is unchanged; top-k is the average of the k largest distances; the finite $L_p$ norm is divided by $n^{1/p}$; $L_\infty$ is the maximum. The convex combination is divided by $\lambda n+(1-\lambda)$. These are positive rescalings of the selected objectives: they preserve minimizers and equal d when every person travels distance d. Marker normalization does not change the precomputed solutions.

Closest facility: each person is connected to the nearest open site. Colors, metrics, and distributions use these same distances. Facility selections come from the solver; nearest assignments cannot worsen any of these objectives without financial constraints.

Distance colors: blue to red through a normalized sigmoid, with steepness 10 and midpoint at half the shared maximum distance. The same mapping applies to both plots and the legend. Only the colors are transformed; the metrics retain their original values.

Average distance / k-median: $L_1$, top-n, and $\lambda=1$ have the same minimizers. Total distance is n times the average; here n = 240.

k-means: $L_2$ minimizes the sum of squared Euclidean distances. Facilities here must be chosen from the 30 candidate sites, rather than placed freely.

k-center: $L_\infty$, top-1, and $\lambda=0$ minimize the maximum distance. The repository adds a small total-distance term to its $L_\infty$ objective. Equivalent maximum-distance endpoints share that precomputed solution.

Top-k: minimizes the sum of the k largest travel distances. This k counts people, not facilities. Dividing by k gives the same minimizers.

Convex combination: λ × total distance + (1 − λ) × maximum distance. Total distance is not normalized to an average, so λ is a coefficient rather than a percentage of practical influence.

Computed with the unmodified facloc FacilityLocationSolver and Gurobi. This synthetic region has 180 people in a dense center and 60 people dispersed across the wider region, overlapping the center’s edges. It was selected from eight screened layouts with continuous surrounding populations. The dense-center draws are bounded to the region, and the dispersed population is uniformly sampled across a broad area. Operating costs are zero, the subsidy allowance is zero, and all person–facility pairs are allowed. The facility limit stays at three. Sliders select precomputed solutions; objective values and locations are not interpolated. Optimality gaps refer to the objective passed to the solver.