191 research outputs found
A Decomposition Algorithm for Nested Resource Allocation Problems
We propose an exact polynomial algorithm for a resource allocation problem
with convex costs and constraints on partial sums of resource consumptions, in
the presence of either continuous or integer variables. No assumption of strict
convexity or differentiability is needed. The method solves a hierarchy of
resource allocation subproblems, whose solutions are used to convert
constraints on sums of resources into bounds for separate variables at higher
levels. The resulting time complexity for the integer problem is , and the complexity of obtaining an -approximate
solution for the continuous case is , being
the number of variables, the number of ascending constraints (such that ), a desired precision, and the total resource. This
algorithm attains the best-known complexity when , and improves it when
. Extensive experimental analyses are conducted with four
recent algorithms on various continuous problems issued from theory and
practice. The proposed method achieves a higher performance than previous
algorithms, addressing all problems with up to one million variables in less
than one minute on a modern computer.Comment: Working Paper -- MIT, 23 page
Separable Convex Optimization with Nested Lower and Upper Constraints
We study a convex resource allocation problem in which lower and upper bounds
are imposed on partial sums of allocations. This model is linked to a large
range of applications, including production planning, speed optimization,
stratified sampling, support vector machines, portfolio management, and
telecommunications. We propose an efficient gradient-free divide-and-conquer
algorithm, which uses monotonicity arguments to generate valid bounds from the
recursive calls, and eliminate linking constraints based on the information
from sub-problems. This algorithm does not need strict convexity or
differentiability. It produces an -approximate solution for the
continuous problem in time
and an integer solution in time, where is
the number of decision variables, is the number of constraints, and is
the resource bound. A complexity of is also achieved
for the linear and quadratic cases. These are the best complexities known to
date for this important problem class. Our experimental analyses confirm the
good performance of the method, which produces optimal solutions for problems
with up to 1,000,000 variables in a few seconds. Promising applications to the
support vector ordinal regression problem are also investigated
Hybrid Genetic Search for the CVRP: Open-Source Implementation and SWAP* Neighborhood
The vehicle routing problem is one of the most studied combinatorial
optimization topics, due to its practical importance and methodological
interest. Yet, despite extensive methodological progress, many recent studies
are hampered by the limited access to simple and efficient open-source solution
methods. Given the sophistication of current algorithms, reimplementation is
becoming a difficult and time-consuming exercise that requires extensive care
for details to be truly successful. Against this background, we use the
opportunity of this short paper to introduce a simple -- open-source --
implementation of the hybrid genetic search (HGS) specialized to the
capacitated vehicle routing problem (CVRP). This state-of-the-art algorithm
uses the same general methodology as Vidal et al. (2012) but also includes
additional methodological improvements and lessons learned over the past decade
of research. In particular, it includes an additional neighborhood called SWAP*
which consists in exchanging two customers between different routes without an
insertion in place. As highlighted in our study, an efficient exploration of
SWAP* moves significantly contributes to the performance of local searches.
Moreover, as observed in experimental comparisons with other recent approaches
on the classical instances of Uchoa et al. (2017), HGS still stands as a
leading metaheuristic regarding solution quality, convergence speed, and
conceptual simplicity
Workload Equity in Vehicle Routing Problems: A Survey and Analysis
Over the past two decades, equity aspects have been considered in a growing
number of models and methods for vehicle routing problems (VRPs). Equity
concerns most often relate to fairly allocating workloads and to balancing the
utilization of resources, and many practical applications have been reported in
the literature. However, there has been only limited discussion about how
workload equity should be modeled in VRPs, and various measures for optimizing
such objectives have been proposed and implemented without a critical
evaluation of their respective merits and consequences.
This article addresses this gap with an analysis of classical and alternative
equity functions for biobjective VRP models. In our survey, we review and
categorize the existing literature on equitable VRPs. In the analysis, we
identify a set of axiomatic properties that an ideal equity measure should
satisfy, collect six common measures, and point out important connections
between their properties and those of the resulting Pareto-optimal solutions.
To gauge the extent of these implications, we also conduct a numerical study on
small biobjective VRP instances solvable to optimality. Our study reveals two
undesirable consequences when optimizing equity with nonmonotonic functions:
Pareto-optimal solutions can consist of non-TSP-optimal tours, and even if all
tours are TSP optimal, Pareto-optimal solutions can be workload inconsistent,
i.e. composed of tours whose workloads are all equal to or longer than those of
other Pareto-optimal solutions. We show that the extent of these phenomena
should not be underestimated. The results of our biobjective analysis are valid
also for weighted sum, constraint-based, or single-objective models. Based on
this analysis, we conclude that monotonic equity functions are more appropriate
for certain types of VRP models, and suggest promising avenues for further
research.Comment: Accepted Manuscrip
Industrial and Tramp Ship Routing Problems: Closing the Gap for Real-Scale Instances
Recent studies in maritime logistics have introduced a general ship routing
problem and a benchmark suite based on real shipping segments, considering
pickups and deliveries, cargo selection, ship-dependent starting locations,
travel times and costs, time windows, and incompatibility constraints, among
other features. Together, these characteristics pose considerable challenges
for exact and heuristic methods, and some cases with as few as 18 cargoes
remain unsolved. To face this challenge, we propose an exact branch-and-price
(B&P) algorithm and a hybrid metaheuristic. Our exact method generates
elementary routes, but exploits decremental state-space relaxation to speed up
column generation, heuristic strong branching, as well as advanced
preprocessing and route enumeration techniques. Our metaheuristic is a
sophisticated extension of the unified hybrid genetic search. It exploits a
set-partitioning phase and uses problem-tailored variation operators to
efficiently handle all the problem characteristics. As shown in our
experimental analyses, the B&P optimally solves 239/240 existing instances
within one hour. Scalability experiments on even larger problems demonstrate
that it can optimally solve problems with around 60 ships and 200 cargoes
(i.e., 400 pickup and delivery services) and find optimality gaps below 1.04%
on the largest cases with up to 260 cargoes. The hybrid metaheuristic
outperforms all previous heuristics and produces near-optimal solutions within
minutes. These results are noteworthy, since these instances are comparable in
size with the largest problems routinely solved by shipping companies
Hybrid Metaheuristics for the Clustered Vehicle Routing Problem
The Clustered Vehicle Routing Problem (CluVRP) is a variant of the
Capacitated Vehicle Routing Problem in which customers are grouped into
clusters. Each cluster has to be visited once, and a vehicle entering a cluster
cannot leave it until all customers have been visited. This article presents
two alternative hybrid metaheuristic algorithms for the CluVRP. The first
algorithm is based on an Iterated Local Search algorithm, in which only
feasible solutions are explored and problem-specific local search moves are
utilized. The second algorithm is a Hybrid Genetic Search, for which the
shortest Hamiltonian path between each pair of vertices within each cluster
should be precomputed. Using this information, a sequence of clusters can be
used as a solution representation and large neighborhoods can be efficiently
explored by means of bi-directional dynamic programming, sequence
concatenations, by using appropriate data structures. Extensive computational
experiments are performed on benchmark instances from the literature, as well
as new large scale ones. Recommendations on promising algorithm choices are
provided relatively to average cluster size.Comment: Working Paper, MIT -- 22 page
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