A comparative performance analysis of evolutionary algorithms on $${\varvec{k}}$$ k -median and facility location problems

Although widely applied in optimisation, relatively little has been proven rigorously about the role and behaviour of populations in randomised search processes. This paper presents a new method to prove upper bounds on the expected optimisation time of population-based randomised search heuristics that use non-elitist selection mechanisms and unary variation operators. Our results follow from a detailed drift analysis of the population dynamics in these heuristics. This analysis shows that the optimisation time depends on the relationship between the strength of the selective pressure and the degree of variation introduced by the variation operator. Given limited variation, a surprisingly weak selective pressure suffices to optimise many functions in expected polynomial time. We derive upper bounds on the expected optimisation time of non-elitist evolutionary algorithms (EA) using various selection mechanisms, including fitness proportionate selection. We show that EAs using fitness proportionate selection can optimise standard benchmark functions in expected polynomial time given a sufficiently low mutation rate. As a second contribution, we consider an optimisation scenario with partial information, where fitness values of solutions are only partially available. We prove that non-elitist EAs under a set of specific conditions can optimise benchmark functions in expected polynomial time, even when vanishingly little information about the fitness values of individual solutions or populations is available. To our knowledge, this is the first runtime analysis of randomised search heuristics under partial information.

For problems involving uncertainties in design variables and parameters, a bi-objective evolutionary algorithm (EA) based approach to design optimization using evidence theory is proposed and implemented in this paper. In addition to a functional objective, a plausibility measure of failure of constraint satisfaction is minimized. Despite some interests in classical optimization literature, this is the first attempt to use evidence theory with an EA. Due to EA's flexibility in modifying its operators, nonrequirement of any gradient, its ability to handle multiple conflicting objectives, and ease of parallelization, evidence-based design optimization using an EA is promising. Results on a test problem and two engineering design problems show that the modified evolutionary multi-objective optimization algorithm is capable of finding a widely distributed trade-off frontier showing different optimal solutions corresponding to different levels of plausibility failure limits. Furthermore, a single-objective evidence-based EA is found to produce better optimal solutions than a previously reported classical optimization algorithm. Furthermore, the use of a graphical processing unit (GPU) based parallel computing platform demonstrates EA's performance enhancement around 160–700 times in implementing plausibility computations. Handling uncertainties of different types are getting increasingly popular in applied optimization studies and this EA based study is promising to be applied in real-world design optimization problems.

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a global simple multiobjective evolutionary algorithm (GSEMO) and apply it on a minimum weight general cover problem, which is general enough to subsume many important problems including the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is nonsubmodular. We show that GSEMO yields theoretically guaranteed approximation ratios matching those achievable by a greedy algorithm in expected polynomial time when the potential function g is polynomial in the input size and the minimum gap between different g-values is a constant. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by National Natural Science Foundation of China [11771013, U20A2068]; Zhejiang Provincial Natural Science Foundation of China [LD19A010001].

Facility location selection is a vital decision for companies that affects both cost and delivery time over the years. However, determination of the facility location is a NP-hard problem. A hybrid algorithm that combines revised weighted fuzzy c-means with Nelder Mead (RWFCM-NM) performs well when compared with well-known algorithms for the facility location problem (FLP) with deterministic customer demands and positions. The motivation of the study is both analyzing performance of the RWFCM-NM algorithm with probabilistic customer demands and positions and proposing a new approach for this problem. This paper develops two new algorithms for FLP when customer demands and positions are probabilistic. The proposed algorithms are a probabilistic fuzzy c-means algorithm and Nelder-Mead (Probabilistic FCM-NM), a probabilistic revised weighted fuzzy c-means algorithm and Nelder Mead (Probabilistic RWFCM-NM) for the un-capacitated planar multi-facility location problem when customer positions and customer demands are probabilistic with predetermined service level. Proposed algorithms performances were tested with 13 data sets and comparisons were made with four well known algorithms. According to the experimental results, probabilistic RWFCM-NM algorithm demonstrates superiority on compared algorithms in terms of total transportation costs.

The location of facilities in order to provide service for customers is a well-studied problem in the operations research literature. In the basic model, there is a predefined cost for opening a facility and also for connecting a customer to a facility, the goal being to minimize the total cost. Often, both in the case of public facilities (such as libraries, municipal swimming pools, fire stations,....) and private facilities (such as distribution centers, switching stations, ....), we may want to find a 'fair' allocation of the total cost to the customers-this is known as the cost allocation problem. A central question in cooperative game theory is whether the total cost can be allocated to the customers such that no coalition of customers has any incentive to build their own facility or to ask a competitor to service them. We establish strong connections between fair cost allocations and linear programming relaxations for several variants of the facility location problem. In particular, we show that a fair cost allocation exists if and only if there is no integrality gap for a corresponding linear programming relaxation; this was only known for the simplest unconstrained variant of the facility location problem. Moreover, we introduce a subtle variant of randomized rounding and derive new proofs for the existence of fair cost allocations for several classes of instances. We also show that it is in general NP-complete to decide whether a fair cost allocation exists and whether a given allocation is fair.

A globally convergent algorithm for facility location on a sphere

A self-supervised learning model for graph clustering optimization problems

Evolutionary Negative Selection Algorithms (ENSAs) could be regarded as hybrid algorithms of Evolutionary Algorithms (EAs) and Negative Selection Algorithms (NSAs). The average time complexity of ENSAs on combinational optimization problems has never been studied before. In this paper, the average time complexity of ENSAs on one combinational optimization problem is analyzed. The theoretical results demonstrate that, for the Two Max function, the ENSA with an appropriate matching threshold could perform better than the traditional (N+N) EA. Some simulation experiments on the combinational problem are also done, and the experimental results are consistent with theoretical results.

This paper presents a distributed O(1)-approximation algorithm in the $\mathcal{CONGEST}$ model for the metric facility location problem on a size-n clique network that has an expected running time of O(loglogn ·log*n) rounds. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. Since the facility location problem is specified by Ω(n2) bits of information, any fast solution in the $\mathcal{CONGEST}$ model must be truly distributed. Our paper makes three main technical contributions. First, we show a new lower bound for metric facility location. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an O(1)-ruling set of an appropriate spanning subgraph. Finally, we present a sub-logarithmic-round (in expectation) algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our algorithm accomplishes this by using a combination of randomized and deterministic sparsification.

Randomized decision making refers to the process of making decisions randomly according to the outcome of an independent randomization device, such as a dice roll or a coin flip. The concept is unconventional, and somehow counterintuitive, in the domain of mathematical programming, in which deterministic decisions are usually sought even when the problem parameters are uncertain. However, it has recently been shown that using a randomized, rather than a deterministic, strategy in nonconvex distributionally robust optimization (DRO) problems can lead to improvements in their objective values. It is still unknown, though, what is the magnitude of improvement that can be attained through randomization or how to numerically find the optimal randomized strategy. In this paper, we study the value of randomization in mixed-integer DRO problems and show that it is bounded by the improvement achievable through its continuous relaxation. Furthermore, we identify conditions under which the bound is tight. We then develop algorithmic procedures, based on column generation, for solving both single- and two-stage linear DRO problems with randomization that can be used with both moment-based and Wasserstein ambiguity sets. Finally, we apply the proposed algorithm to solve three classical discrete DRO problems: the assignment problem, the uncapacitated facility location problem, and the capacitated facility location problem and report numerical results that show the quality of our bounds, the computational efficiency of the proposed solution method, and the magnitude of performance improvement achieved by randomized decisions. Summary of Contribution: In this paper, we present both theoretical results and algorithmic tools to identify optimal randomized strategies for discrete distributionally robust optimization (DRO) problems and evaluate the performance improvements that can be achieved when using them rather than classical deterministic strategies. On the theory side, we provide improvement bounds based on continuous relaxation and identify the conditions under which these bound are tight. On the algorithmic side, we propose a finitely convergent, two-layer, column-generation algorithm that iterates between identifying feasible solutions and finding extreme realizations of the uncertain parameter. The proposed algorithm was implemented to solve distributionally robust stochastic versions of three classical optimization problems and extensive numerical results are reported. The paper extends a previous, purely theoretical work of the first author on the idea of randomized strategies in nonconvex DRO problems by providing useful bounds and algorithms to solve this kind of problems.

We discuss the 1-median and l-antimedJan problems with probabilistic demand. It is assumed that demand weights of users generated at nodes of the network are independent continuous random variables. In the 1-median problem with probabilistic demand the objective is to find a location of a desirable facility on a network that maximizes the probability that the weighted sum distance does not exceed some pre-detemiined value T. In the l-antimedian problem tbe objective is to locate an undesirable facility and tbus we maximize the probability tbat the total weighted distance is at least T. We discuss how to solve the problems under arbitrary distributions and for small and large networks.

The capacitated arc routing problem (CARP) is a classical NP-Hard combinatorial optimization problem. In this paper, we present a memetic algorithm based on Extension step and Statistical filtering for Large-Scale CARP (ESMAENS) which has improvements in both the global search strategy and the local search strategy. Firstly, ESMAENS adopts the Route Distance Grouping decomposition scheme (RDG) to decompose the large-scale CARP into some independent sub-problems. Then, the initial population would evolve into a new group by using the Evolutionary Algorithm. Furthermore, ESMAENS introduces the extension step search strategy to search the potential solutions around the current solution. As a result, with the increase of the iteration number, it avoids getting to the premature convergence. Meanwhile, the diversity of the population can be increased. Finally, in the local search strategy, the statistical filter is used to filter out some solutions and make the algorithm get the lower bound at a faster convergence rate with a high stability. Compared with RDG-MAENS, experimental results on Beullen’C, D, E, F, egl and EGL-G test set show that ESMAENS has a better convergence rate, and the stability of solutions obtained is improved. Furthermore, ESMAENS achieves a global search for the solutions. Especially for the large-scale EGL-G test set, ESMAENS can converge to a lower bound at a faster convergence rate, and it is suitable for solving large-scale CARP.

We consider the dynamic facility location problem with generalized modular capacities, a multiperiod facility location problem in which the costs for capacity changes may differ for all pairs of capacity levels. The problem embeds a complex cost structure and generalizes several existing facility location problems, such as those that allow temporary facility closing or capacity expansion and reduction. As the model may become very large, general-purpose mixed-integer programming (MIP) solvers are limited to solving instances of small to medium size. In this paper, we extend the generalized model to the case of multiple commodities. We propose Lagrangian heuristics, based on subgradient and bundle methods, to find good quality solutions for large-scale instances with up to 250 facility locations and 1,000 customers. To improve the final solution quality, a restricted MIP model is solved based on the information collected through the solution of the Lagrangian dual. Computational results show that the Lagrangian-based heuristics provide highly reliable results for all problem variants considered. They produce good quality solutions in short computing times even for instances where state-of-the-art MIP solvers do not find feasible solutions. The strength of the formulation also allows the method to provide tight bounds on the optimal value. Data and the online appendix are available at https://doi.org/10.1287/ijoc.2016.0738 .

Cloud computing has emerged as a buzzword of IT-enabled business services. It makes business sense to choose an optimal location for setting up the cloud service facilities such as data centres. The distribution of services between data centre facilities and clients of a cloud can be devised mathematically as the well-known facility location problem (FLP). The decisions of location of data centres are a vital element in strategic planning for a wide range of cloud service providers. In this paper, we have applied facility location techniques in operations management for evaluating the available potential locations and zero down to the best optimal facility location, which can be used to minimise the cost associated with setting up of data centre facilities. A case study is also presented applying FLP in Indian cloud computing scenario. By using these techniques, one can ensure desired QoS and optimum use of data centre facilities.

An algorithm for the capacitated, multi-commodity multi-period facility location problem