Proximity Time (PT) in Facility Location

• Proximity Time (PT) is a framework that integrates equity by quantitatively modeling travel distances using the Kolm-Pollak Equally-Distributed Equivalent.
• The methodology employs a linearized proxy and MILP formulation to balance both mean efficiency and worst-off accessibility in facility location models.
• Computational experiments demonstrate that the approach maintains near-optimal average distances while significantly reducing the maximum travel distances in large-scale scenarios.

Proximity Time (PT) is not used as a term in the referenced literature; however, the supplied arXiv paper introduces a comprehensive framework for equitable facility location, where proximity—optimal assignment of users to amenities based on travel distance—is quantitatively modeled via the Kolm-Pollak Equally-Distributed Equivalent (EDE) and its linearized proxy. The Kolm-Pollak EDE is leveraged as a foundational metric for evaluating and optimizing facility locations in ways that robustly account for both average accessibility and equity among users. The approach achieves scalable, mixed-integer linear programming (MILP) formulations for large-scale, equity-focused location problems, delivering significant improvements in proximity for the worst-off populations while maintaining near-optimal mean experiences (Horton et al., 27 Jan 2024).

1. Kolm-Pollak Equally-Distributed Equivalent (EDE) in Facility Location

For a vector of undesirable outcomes $x=(x_1,\ldots,x_n)$, such as walking distances to a facility, the Kolm-Pollak EDE is defined using the exponential utility function:

• $u(z) = e^{-\kappa z}$, with $\kappa = \alpha \epsilon$ for $\epsilon<0$ (inequality aversion) and a data-driven normalization $\alpha$.
• The EDE for $N$ individuals is:

$$\mathcal{K}(z) = -\frac{1}{\kappa} \ln \left[ \frac{1}{N} \sum_{i=1}^N e^{-\kappa z_i} \right].$$

• In population-weighted settings, with blocks $r\in R$, block populations $p_r$, and total $T = \sum_r p_r$:

$$\mathcal{K}(z) = -\frac{1}{\kappa} \ln \left[ \frac{1}{T} \sum_{r\in R} p_r e^{-\kappa z_r} \right].$$

• The normalization $\alpha$ is computed as:

$\alpha = \frac{\sum_r p_r z_r}{\sum_r p_r z_r^2}.$

The Kolm-Pollak EDE emphasizes protecting the worst-off by making the EDE highly sensitive to higher distances as the inequality aversion $|\epsilon|$ increases.

2. Linearized Proxy Formulation and MILP Embedding

To render the EDE tractable for large-scale facility location, the nonlinear log-sum-exponential objective is linearized by exploiting assignment binary variables $y_{r,s}$ (block $r$ assigned to facility $s$):

• In standard p-median notation: $x_s\in\{0,1\}$ (facility open), $y_{r,s}\in\{0,1\}$ (block assignment). Each $z_r = \sum_s y_{r,s} d_{r,s}$.
• The nonlinear objective becomes:

$$\min\ \ \mathcal{K}(z) = -\frac{1}{\kappa} \ln \left[ \frac{1}{T} \sum_{r\in R} p_r e^{-\kappa \sum_s y_{r,s} d_{r,s}} \right].$$

Eliminating the logarithm, minimizing $\mathcal{K}(z)$ is equivalent to minimizing:

$$\widetilde{\mathcal{K}}(y) = \sum_{r \in R} p_r e^{-\kappa \sum_s y_{r,s} d_{r,s}}.$$

• The single-assignment structure yields $$e^{-\kappa \sum_s y_{r,s}d_{r,s}} = \sum_s y_{r,s} e^{-\kappa d_{r,s}}$$, so the entire objective is linear in $y_{r,s}$.

Final MILP formulation ("KPL"):

Variable	Domain	Interpretation
$x_s$	$\{0,1\}$	Facility $s$ open
$y_{r,s}$	$\{0,1\}$ or $[0,1]$	Block $r$ assigned to $s$ (split demand optional)
$q,\, v$	$\mathbb{R}$	Auxiliary for penalties (if needed)

The objective:

$$\min \sum_{r\in R}\sum_{s\in S} p_r y_{r,s} e^{-\kappa d_{r,s}} + T e^{-\kappa \bar{\mathcal{K}}} (v-1)$$

subject to standard assignment, capacity, and (optionally) penalty linearization constraints.

3. Model Extensions: Capacities, Split Assignment, and Location Penalties

The KPL framework accommodates key practical features:

• Facility capacities: $\sum_r p_r y_{r,s} \leq C_s x_s$ for all $s$
• Split demand assignment: $y_{r,s} \in [0,1]$
• Penalizing undesirable sites: Introduce penalty $c_s$ for $s \in U$, encode via $q = -\kappa \sum_{s \in U} c_s x_s$, auxiliary $v \ge e^q$, and add penalty term to the objective.
• Piecewise linearization for exponentials: For penalty linearization, tangent-based inequalities at points $\beta_i$ approximate $v \ge e^q$ with negligible error.

All extensions preserve MILP structure and ensure tractability even in massive real-world instances (Horton et al., 27 Jan 2024).

4. Computational Performance and Scalability

The linearized KPL MILP matches classical p-median models in combinatorial structure ($O(|R||S|)$ binaries):

• For instance, a nationwide supermarket placement problem (30,095 blocks, 7,618 sites, $\sim$248 million binaries) is solved at a 0.01% MIP gap with times comparable to p-median, and $5$–$10\times$ faster than the p-center.
• Experiments utilize Pyomo+Gurobi on large-memory HPC clusters, or SCIP for polling place problems.
• Penalty piecewise linearization introduces only $O(n)$ new continuous variables ($n =$ number of tangents), minimal overhead if penalties are sparse or uniform.
• All practical extensions, including capacities and split demand, retain full tractability at urban or national scale (Horton et al., 27 Jan 2024).

5. Equity-Performance Trade-offs: Empirical Findings

The Kolm-Pollak EDE-based approach yields marked improvements for equity:

• In synthetic examples, as the dispersion (inequality) of distances grows (with fixed mean), EDE rises above the mean and approaches the maximum as $|\epsilon|\to\infty$. For example, $z = [0,0,0,400]$ yields $\mathrm{EDE} \approx 389.0$ (mean 100, $\epsilon=-50$).
• In practical settings, e.g. across 500 U.S. cities, the KPL model sacrifices under 10 m mean walking distance to achieve 400–530 m reductions in the worst-case distance.
• For five real cities (polling location), the method attains lower EDE and maximum distance than the p-center, with means comparable to p-median, and consistently tighter (lower variance) distance distributions.
• Penalty application (Santa Rosa, CA): penalizing site openings with $c=4.86$ m and tangent spacing $w=0.001$ ensures selection of only 2 penalized sites (versus 8 unpenalized), with a total penalty error $<$0.04 m.

This demonstrates that the Kolm-Pollak EDE embedded in MILP can simultaneously balance efficiency and prioritize equity for the most disadvantaged, at nontrivial geographic scales (Horton et al., 27 Jan 2024).

6. Significance and Theoretical Implications

By bridging equity theory and large-scale operational optimization, the described framework provides a tractable, robust apparatus for equitable urban design. The Kolm-Pollak EDE, sensitive to outcome dispersion, directly mitigates the tendency of average-focused models to neglect severely underserved populations. Linearization and integrability enable deployment across domains—food desert remediation, polling site placement—without computational complexity exceeding that of classical facility models. A plausible implication is that similar techniques could extend to other resource allocation contexts emphasizing fairness, provided utility structures support comparable linearization. As $|\epsilon|$ increases, the method interpolates between mean-focused (p-median) and min-max (p-center) paradigms, offering a continuous spectrum of equity-aware solutions (Horton et al., 27 Jan 2024).