Using ℓp-norms for fairness in combinatorial optimisation

https://doi.org/10.1016/j.cor.2020.104975Get rights and content

Highlights

  • We discuss the issue of fairness in combinatorial optimisation problems (COPs).

  • Convex mixed-integer nonlinear programs are presented for lp-norm problems.

  • Computational results are given for routing, location and network design problems.

  • An lp-norm approach tends to lead to fairer solutions for COPs.

Abstract

The issue of fairness has received attention from researchers in many fields, including combinatorial optimisation. One way to drive the solution toward fairness is to use a modified objective function that involves so-called ℓp-norms. If done in a naive way, this approach leads to large and symmetric mixed-integer nonlinear programs (MINLPs), that may be difficult to solve. We show that, for some problems, one can obtain alternative MINLP formulations that are much smaller, do not suffer from symmetry, and have a reasonably tight continuous relaxation. We give encouraging computational results for certain vehicle routing, facility location and network design problems.

Introduction

The issue of fairness has received considerable attention from researchers in many fields, including, for example, computer science (Jain et al., 1984), economics (Nash, Jr, F., 1950, Rabin, 1993, Weymark, 1981), marketing (Xia et al., 2004), operational research (Bertsimas, Farias, Trichakis, 2011, Kelly, 1997), philosophy (Rawls, 1971), psychology (Janssen, 2000) and recreational mathematics (Dubins and Spanier, 1961). As one might expect, it has also received attention from the combinatorial optimisation community (see, e.g., Ajtai, Aspnes, Naor, Rabani, Schulman, Waarts, 1998, Barbati, Piccolo, 2016, Bertsimas, Farias, Trichakis, 2011, Fernández, Pozo, Puerto, 2014, Filippi, Ogryczak, Speranza, 2017, Karsu, Azizoglu, 2019, Karsu, Morton, 2015, Kostreva, Ogryczak, Wierzbicki, 2004, Luss, 1999, Martin, Ouelhadj, Smet, Özcan, Berghe, 2013, Matl, Hartl, Vidal, 2018, Muklason, Parkes, Özcan, McCollum, McMullan, 2017, Nguyen, Weng, 2017, Nickel, Puerto, 2005, Ogryczak, 2009, Puerto, Rodrıguez-Chıa, Tamir, 2017, Zukerman, Mammadov, Tan, Ouveysi, Andrew, 2008).

As noted in Filippi et al. (2017), many combinatorial optimisation problems (COPs) of interest can be modelled as follows. We have a set of workers and a set of tasks. Each task must be assigned to one worker, and the total cost depends on the allocation of tasks to workers. The issue of fairness then arises immediately, since, if one worker has a significantly higher workload than another, the solution may be perceived to be unfair.

A variety of approaches have been proposed to avoid unfair solutions, without incurring a significant increase in cost. (We survey these in Section 2.2.) In this paper, we focus on an approach suggested in Kostreva et al. (2004), that uses a modified objective function involving so-called ℓp-norms. We make the following specific contributions:

  • We give an intuitive explanation for why the ℓp-norm approach tends to lead to fairer solutions.

  • We present a generic formulation for ℓp-norm problems as convex mixed-integer nonlinear programs (MINLPs).

  • For some specific COPs, we present alternative convex MINLP formulations that are much smaller, do not suffer from symmetry, and have a reasonably tight continuous relaxation.

  • We give some encouraging computational results for some simple vehicle routing, facility location and network design problems.

The paper has a simple structure. The literature is reviewed in Section 2. The intuitive explanation and generic formulation are presented in Section 3. The specialised formulations are described in Section 4. The computational results are given in Section 5, and some concluding remarks are made in Section 6.

Section snippets

Literature review

We now review the relevant literature. Section 2.1 covers fairness in general, and Section 2.2 covers fairness in combinatorial optimisation.

General remarks on the ℓp-Norm approach

In this section, we make some general remarks on the ℓp-norm approach. In Section 3.1, we try to explain why unfair solutions arise in the first place, and why the approach can help. In Sections 3.2 and 3.3, we present generic MINLP and 0–1 LP formulations for COPs with ℓp-norm objective.

From now on, we say that the workers are identical if cw(S)=cw(S) for all pairs of workers w, w′ and all sets ST. In this case, we write c(S) instead of cw(S).

Compact formulations for specific problems

It turns out that, for certain specific COPs, one can obtain alternative MINLP formulations that are both small and free from symmetry. In Sections 4.1 to 4.3, we show this for the multiple TSP, the capacitated minimum spanning tree problem, and the k-median problem. Then, in Section 4.4, we discuss how to strengthen the continuous relaxations of the formulations.

Computational results

In this section, we present computational results for the m-TSP, CMSTP and k-median problem, for varying values of p. All MILP and MINLP formulations were solved to optimality using the mixed-integer LP and SOCP solvers of CPLEX1 12.7, using default settings. The experiments were run on a MacBook Pro with a 2.3 GHz Intel Core i5 processor and with 8GB memory. A time limit of 3600 s was imposed on each instance.

We remark that preliminary

Discussion

Fairness has received a great deal of attention in many disciplines. To achieve fairness in combinatorial optimisation, we recommend using the ℓp-norm approach. We have shown that, for certain network optimisation problems, it is possible to derive compact and convex MINLP formulations of the ℓp variants, that do not suffer from symmetry. We have also shown that outer-approximating the MINLP, using a relatively small number of perspective cuts, often enables one to obtain results of acceptable

CRediT authorship contribution statement

Tolga Bektaş: Methodology, Formal analysis, Software, Validation, Writing - original draft, Writing - review & editing. Adam N. Letchford: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing.

Acknowledgements

We thank an anonymous reviewer for very useful comments, that have led to a significant improvement in the paper.

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