---
title: Proximity Time (PT) in Facility Location
url: https://www.emergentmind.com/topics/proximity-time-pt
type: topic
---

# Proximity Time (PT) in Facility Location

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 [2401.15452].

## 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 [2401.15452].

## 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 [2401.15452].

## 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 [2401.15452].

## 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 [2401.15452].

Source: https://www.emergentmind.com/topics/proximity-time-pt