Computing leximin-optimal solutions in constraint networks

AbstractIn many real-world multiobjective optimization problems one needs to find solutions or alternatives that provide a fair compromise between different conflicting objective functions—which could be criteria in a multicriteria context, or agent utilities in a multiagent context—while being efficient (i.e. informally, ensuring the greatest possible overall agents' satisfaction). This is typically the case in problems implying human agents, where fairness and efficiency requirements must be met. Preference handling, resource allocation problems are another examples of the need for balanced compromises between several conflicting objectives. A way to characterize good solutions in such problems is to use the leximin preorder to compare the vectors of objective values, and to select the solutions which maximize this preorder. In this article, we describe five algorithms for finding leximin-optimal solutions using constraint programming. Three of these algorithms are original. Other ones are adapted, in constraint programming settings, from existing works. The algorithms are compared experimentally on three benchmark problems

Fairness in Communication and Computer Network Design

In communication networks, fair sharing of resources is an important issue for one main reason. The growth of network capacity is in general not matching the rapid growth of traffic. Consequently, the resources consumed by each user have to be limited. This implies that users cannot always be assigned the end-to-end bandwidth they ask for. Instead, the limited network resources should be distributed to users in a way that assures fair end-to-end bandwidth assignment among them. Obtaining fairness between network users and at the same time assuring efficient network utilization, is a source of non-trivial network optimization problems. Complicating factors are that each user has limited access to the (limited) network resources and that different users require and consume different amounts and types of resources. In this thesis different types of optimization problems associated with fair resource sharing in communication networks are studied. Initially, the notions of max-min fairness, proportional fairness, alpha-fairness etc., are put in a formal framework of fair rational preference relations. A clear, unified definition of fairness is presented. The theory is first applied to different types of allocation problems. Focus is put on convex and non-convex max-min fair traffic allocation problems, and a difference in difficulty between the two groups of problems is demonstrated. The studies are continued by an investigation of proportionally fair dimensioning. Two different cases are studied -- a simpler, when no resilience to failures is required, and a more complicated, assuming the possibility of link failures. In the context of fair sharing of the resources of a communication network, this thesis presents several original theoretical findings as well as solution algorithms for the studied problems. The results are accompanied by numerical results, illustrating algorithm efficiency for virtually all of the studied problems

Filtering Algorithms for the Multiset Ordering Constraint

Constraint programming (CP) has been used with great success to tackle a wide variety of constraint satisfaction problems which are computationally intractable in general. Global constraints are one of the important factors behind the success of CP. In this paper, we study a new global constraint, the multiset ordering constraint, which is shown to be useful in symmetry breaking and searching for leximin optimal solutions in CP. We propose efficient and effective filtering algorithms for propagating this global constraint. We show that the algorithms are sound and complete and we discuss possible extensions. We also consider alternative propagation methods based on existing constraints in CP toolkits. Our experimental results on a number of benchmark problems demonstrate that propagating the multiset ordering constraint via a dedicated algorithm can be very beneficial

Fair Resource Allocation in Macroscopic Evacuation Planning Using Mathematical Programming: Modeling and Optimization

Evacuation is essential in the case of natural and manmade disasters such as hurricanes, nuclear disasters, fire accidents, and terrorism epidemics. Random evacuation plans can increase risks and incur more losses. Hence, numerous simulation and mathematical programming models have been developed over the past few decades to help transportation planners make decisions to reduce costs and protect lives. However, the dynamic transportation process is inherently complex. Thus, modeling this process can be challenging and computationally demanding. The objective of this dissertation is to build a balanced model that reflects the realism of the dynamic transportation process and still be computationally tractable to be implemented in reality by the decision-makers. On the other hand, the users of the transportation network require reasonable travel time within the network to reach their destinations. This dissertation introduces a novel framework in the fields of fairness in network optimization and evacuation to provide better insight into the evacuation process and assist with decision making. The user of the transportation network is a critical element in this research. Thus, fairness and efficiency are the two primary objectives addressed in the work by considering the limited capacity of roads of the transportation network. Specifically, an approximation approach to the max-min fairness (MMF) problem is presented that provides lower computational time and high-quality output compared to the original algorithm. In addition, a new algorithm is developed to find the MMF resource allocation output in nonconvex structure problems. MMF is the fairness policy used in this research since it considers fairness and efficiency and gives priority to fairness. In addition, a new dynamic evacuation modeling approach is introduced that is capable of reporting more information about the evacuees compared to the conventional evacuation models such as their travel time, evacuation time, and departure time. Thus, the contribution of this dissertation is in the two areas of fairness and evacuation. The first part of the contribution of this dissertation is in the field of fairness. The objective in MMF is to allocate resources fairly among multiple demands given limited resources while utilizing the resources for higher efficiency. Fairness and efficiency are contradicting objectives, so they are translated into a bi-objective mathematical programming model and solved using the ϵ-constraint method, introduced by Vira and Haimes (1983). Although the solution is an approximation to the MMF, the model produces quality solutions, when ϵ is properly selected, in less computational time compared to the progressive-filling algorithm (PFA). In addition, a new algorithm is developed in this research called the θ progressive-filling algorithm that finds the MMF in resource allocation for general problems and works on problems with the nonconvex structure problems. The second part of the contribution is in evacuation modeling. The common dynamic evacuation models lack a piece of essential information for achieving fairness, which is the time each evacuee or group of evacuees spend in the network. Most evacuation models compute the total time for all evacuees to move from the endangered zone to the safe destination. Lack of information about the users of the transportation network is the motivation to develop a new optimization model that reports more information about the users of the network. The model finds the travel time, evacuation time, departure time, and the route selected for each group of evacuees. Given that the travel time function is a non-linear convex function of the traffic volume, the function is linearized through a piecewise linear approximation. The developed model is a mixed-integer linear programming (MILP) model with high complexity. Hence, the model is not capable of solving large scale problems. The complexity of the model was reduced by introducing a linear programming (LP) version of the full model. The complexity is significantly reduced while maintaining the exact output. In addition, the new θ-progressive-filling algorithm was implemented on the evacuation model to find a fair and efficient evacuation plan. The algorithm is also used to identify the optimal routes in the transportation network. Moreover, the robustness of the evacuation model was tested against demand uncertainty to observe the model behavior when the demand is uncertain. Finally, the robustness of the model is tested when the traffic flow is uncontrolled. In this case, the model's only decision is to distribute the evacuees on routes and has no control over the departure time

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

Sorted-pareto dominance and qualitative notions of optimality

Pareto dominance is often used in decision making to compare decisions that have multiple preference values – however it can produce an unmanageably large number of Pareto optimal decisions. When preference value scales can be made commensurate, then the Sorted-Pareto relation produces a smaller, more manageable set of decisions that are still Pareto optimal. Sorted-Pareto relies only on qualitative or ordinal preference information, which can be easier to obtain than quantitative information. This leads to a partial order on the decisions, and in such partially-ordered settings, there can be many different natural notions of optimality. In this paper, we look at these natural notions of optimality, applied to the Sorted-Pareto and min-sum of weights case; the Sorted-Pareto ordering has a semantics in decision making under uncertainty, being consistent with any possible order-preserving function that maps an ordinal scale to a numerical one. We show that these optimality classes and the relationships between them provide a meaningful way to categorise optimal decisions for presenting to a decision maker

Proportional and maxmin fairness for the sensor location problem with chance constraints

International audienceIn this paper we present a study on the Equitable Sensor Location Problem and we focus on the stochastic version of the problem where the surveying capacity of some sensors is measured as probability of intrusions detection. The Equitable Sensor Location Problem, which is an extension of the Equitable Facility Location Problem, considers installing surveying facilities as cameras/sensors in order to monitor and protect some important locations. Each location can be simultaneously protected by multiple facilities. Clearly this problem falls into the category of Maximal Coverage Location Problem and we focus on the equitable variant. The objective of the Equitable Sensor Location Problem is to provide equitable protection to all locations when the number of sensors that can be placed is limited. We study the resilient and ambiguous versions of this problem. The resilient sensor location problem considers the case when some sensors are assumed to fail partially or completely. The ambiguous version studies the case when the surveying probabilities are uncertain and represented by independent Bernouilli random variables with the corresponding ambiguity set containing the Bernouilli probability distributions. For each problem we consider two popular fairness measures which are the lexicographic optimal and proportionally fair solutions and provide an integer linear formulation together with the solution methodology. Numerical results for each studied problem are provided at the end of the paper

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

Robust competence assessment for job assignment

International audienceAllocating the right person to a task or job is a key issue for improving quality and performance of achievements, usually addressed using the concept of "competences". Nevertheless, providing an accurate assessment of the competences of an individual may be in practice a difficult task. We suggest in this paper to model the uncertainty on the competences possessed by a person using a possibility distribution, and the imprecision on the competences required for a task using a fuzzy constraint, taking into account the possible interactions between competences using a Choquet Integral. As a difference with comparable approaches, we then suggest to perform the allocation of persons to jobs using a Robust Optimisation approach, allowing to minimize the risk taken by the decision maker. We first apply this framework to the problem of selecting a candidate within n for a job, then extend the method to the problem of selecting c candidates for j jobs (c ≥ j) using the leximin criterion