Min-max optimal public service system design

This paper deals with designing a fair public service system. To achieve fairness, various schemes are be applied. The strongest criterion in the process is minimization of disutility of the worst situated users and then optimization of disutility of the better situated users under the condition that disutility of the worst situated users does not worsen, otherwise called lexicographical minimization. Focusing on the first step, this paper endeavours to find an effective solution to the weighted p-median problem based on radial formulation. Attempts at solving real instances when using a location-allocation model often fail due to enormous computational time or huge memory demands. Radial formulation can be implemented using commercial optimisation software. The main goal of this study is to show that the suitability solving of the min-max optimal public service system design can save computational time

Overview of Methods Implemented in MCA: Multiple Criteria Analysis of Discrete Alternatives with a Simple Preference Specification

Many methods have been developed for multiple criteria analysis and/or ranking of discrete alternatives. Most of them require complex specification of preferences. Therefore, they are not applicable for problems with numerous alternatives and/or criteria, where preference specification by the decisin makers can hardly be done in a way acceptable for small problems, e.g., for pair-wise comparisons. In this paper we describe several new methods implemented for a real-life application dealing with multi-criteria analysis of future energy technologies. This analysis involves large numbers of both alternatives and criteria. Moreover, the analysis was made by a large number of stakeholders without expeience in analytical methods. Therefore a simple method for interactive preference specification was condition for the analysis. The paper provides overview of several of new methods based on diverse concepts developed for multicriteria analysis, and summarizes a comparison of methods and experence of using them

A lexicographic minimax approach to the vehicle routing problem with route balancing

International audienceVehicle routing problems generally aim at designing routes that minimize transportation costs. However, in practical settings, many companies also pay attention at how the workload is distributed among its drivers. Accordingly, two main approaches for balancing the workload have been proposed in the literature. They are based on minimizing the duration of the longest route, or the difference between the longest and the shortest routes, respectively. Recently, it has been shown on several occasions that both approaches have some flaws. In order to model equity we investigate the lexicographic minimax approach, which is rooted in social choice theory. We define the leximax vehicle routing problem which considers the bi-objective optimization of transportation costs and of workload balancing. This problem is solved by a heuristic based on the multi-directional local search framework. It involves dedicated large neighborhood search operators. Several LNS operators are proposed and compared in experimentations

Convex Fairness Measures: Theory and Optimization

We propose a new parameterized class of fairness measures, convex fairness measures, suitable for optimization contexts. This class includes our new proposed order-based fairness measure and several popular measures (e.g., deviation-based measures, Gini deviation). We provide theoretical analyses and derive a dual representation of these measures. Importantly, this dual representation renders a unified mathematical expression and a geometric characterization for convex fairness measures through their dual sets. Moreover, we propose a generic framework for optimization problems with a convex fairness measure objective, including reformulations and solution methods. Finally, we provide a stability analysis on the choice of convex fairness measures in the objective of optimization models

Leximin Approximation: From Single-Objective to Multi-Objective

Leximin is a common approach to multi-objective optimization, frequently employed in fair division applications. In leximin optimization, one first aims to maximize the smallest objective value; subject to this, one maximizes the second-smallest objective; and so on. Often, even the single-objective problem of maximizing the smallest value cannot be solved accurately. What can we hope to accomplish for leximin optimization in this situation? Recently, Henzinger et al. (2022) defined a notion of \emph{approximate} leximin optimality. Their definition, however, considers only an additive approximation. In this work, we first define the notion of approximate leximin optimality, allowing both multiplicative and additive errors. We then show how to compute, in polynomial time, such an approximate leximin solution, using an oracle that finds an approximation to a single-objective problem. The approximation factors of the algorithms are closely related: an $(\alpha,\epsilon)$-approximation for the single-objective problem (where $\alpha \in (0,1]$ and $\epsilon \geq 0$ are the multiplicative and additive factors respectively) translates into an $\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\epsilon}{1-\alpha +\alpha^2}\right)$-approximation for the multi-objective leximin problem, regardless of the number of objectives. Finally, we apply our algorithm to obtain an approximate leximin solution for the problem of \emph{stochastic allocations of indivisible goods}. For this problem, assuming sub-modular objectives functions, the single-objective egalitarian welfare can be approximated, with only a multiplicative error, to an optimal $1-\frac{1}{e}\approx 0.632$ factor w.h.p. We show how to extend the approximation to leximin, over all the objective functions, to a multiplicative factor of $\frac{(e-1)^2}{e^2-e+1} \approx 0.52$ w.h.p or $\frac{1}{3}$ deterministically