Two-Stage Vehicle Routing Problems with Profits and Buffers: Analysis and Metaheuristic Optimization Algorithms

This thesis considers the Two-Stage Vehicle Routing Problem (VRP) with Profits and Buffers, which generalizes various optimization problems that are relevant for practical applications, such as the Two-Machine Flow Shop with Buffers and the Orienteering Problem. Two optimization problems are considered for the Two-Stage VRP with Profits and Buffers, namely the minimization of total time while respecting a profit constraint and the maximization of total profit under a budget constraint. The former generalizes the makespan minimization problem for the Two-Machine Flow Shop with Buffers, whereas the latter is comparable to the problem of maximizing score in the Orienteering Problem. For the three problems, a theoretical analysis is performed regarding computational complexity, existence of optimal permutation schedules (where all vehicles traverse the same nodes in the same order) and potential gaps in attainable solution quality between permutation schedules and non-permutation schedules. The obtained theoretical results are visualized in a table that gives an overview of various subproblems belonging to the Two-Stage VRP with Profits and Buffers, their theoretical properties and how they are connected. For the Two-Machine Flow Shop with Buffers and the Orienteering Problem, two metaheuristics 2BF-ILS and VNSOP are presented that obtain favorable results in computational experiments when compared to other state-of-the-art algorithms. For the Two-Stage VRP with Profits and Buffers, an algorithmic framework for Iterative Search Algorithms with Variable Neighborhoods (ISAVaN) is proposed that generalizes aspects from 2BF-ILS as well as VNSOP. Various algorithms derived from that framework are evaluated in an experimental study. The evaluation methodology used for all computational experiments in this thesis takes the performance during the run time into account and demonstrates that algorithms for structurally different problems, which are encompassed by the Two-Stage VRP with Profits and Buffers, can be evaluated with similar methods. The results show that the most suitable choice for the components in these algorithms is dependent on the properties of the problem and the considered evaluation criteria. However, a number of similarities to algorithms that perform well for the Two-Machine Flow Shop with Buffers and the Orienteering Problem can be identified. The framework unifies these characteristics, providing a spectrum of algorithms that can be adapted to the specifics of the considered Vehicle Routing Problem.:1 Introduction 2 Background 2.1 Problem Motivation 2.2 Formal Definition of the Two-Stage VRP with Profits and Buffers 2.3 Review of Literature on Related Vehicle Routing Problems 2.3.1 Two-Stage Vehicle Routing Problems 2.3.2 Vehicle Routing Problems with Profits 2.3.3 Vehicle Routing Problems with Capacity- or Resource-based Restrictions 2.4 Preliminary Remarks on Subsequent Chapters 3 The Two-Machine Flow Shop Problem with Buffers 3.1 Review of Literature on Flow Shop Problems with Buffers 3.1.1 Algorithms and Metaheuristics for Flow Shops with Buffers 3.1.2 Two-Machine Flow Shop Problems with Buffers 3.1.3 Blocking Flow Shops 3.1.4 Non-Permutation Schedules 3.1.5 Other Extensions and Variations of Flow Shop Problems 3.2 Theoretical Properties 3.2.1 Computational Complexity 3.2.2 The Existence of Optimal Permutation Schedules 3.2.3 The Gap Between Permutation Schedules an Non-Permutation 3.3 A Modification of the NEH Heuristic 3.4 An Iterated Local Search for the Two-Machine Flow Shop Problem with Buffers 3.5 Computational Evaluation 3.5.1 Algorithms for Comparison 3.5.2 Generation of Problem Instances 3.5.3 Parameter Values 3.5.4 Comparison of 2BF-ILS with other Metaheuristics 3.5.5 Comparison of 2BF-OPT with NEH 3.6 Summary 4 The Orienteering Problem 4.1 Review of Literature on Orienteering Problems 4.2 Theoretical Properties 4.3 A Variable Neighborhood Search for the Orienteering Problem 4.4 Computational Evaluation 4.4.1 Measurement of Algorithm Performance 4.4.2 Choice of Algorithms for Comparison 4.4.3 Problem Instances 4.4.4 Parameter Values 4.4.5 Experimental Setup 4.4.6 Comparison of VNSOP with other Metaheuristics 4.5 Summary 5 The Two-Stage Vehicle Routing Problem with Profits and Buffers 5.1 Theoretical Properties of the Two-Stage VRP with Profits and Buffers 5.1.1 Computational Complexity of the General Problem 5.1.2 Existence of Permutation Schedules in the Set of Optimal Solutions 5.1.3 The Gap Between Permutation Schedules an Non-Permutation Schedules 5.1.4 Remarks on Restricted Cases 5.1.5 Overview of Theoretical Results 5.2 A Metaheuristic Framework for the Two-Stage VRP with Profits and Buffers 5.3 Experimental Results 5.3.1 Problem Instances 5.3.2 Experimental Results for O_{max R, Cmax≤B} 5.3.3 Experimental Results for O_{min Cmax, R≥Q} 5.4 Summary Bibliography List of Figures List of Tables List of Algorithm

The Vehicle Routing Problem with Service Level Constraints

We consider a vehicle routing problem which seeks to minimize cost subject to service level constraints on several groups of deliveries. This problem captures some essential challenges faced by a logistics provider which operates transportation services for a limited number of partners and should respect contractual obligations on service levels. The problem also generalizes several important classes of vehicle routing problems with profits. To solve it, we propose a compact mathematical formulation, a branch-and-price algorithm, and a hybrid genetic algorithm with population management, which relies on problem-tailored solution representation, crossover and local search operators, as well as an adaptive penalization mechanism establishing a good balance between service levels and costs. Our computational experiments show that the proposed heuristic returns very high-quality solutions for this difficult problem, matches all optimal solutions found for small and medium-scale benchmark instances, and improves upon existing algorithms for two important special cases: the vehicle routing problem with private fleet and common carrier, and the capacitated profitable tour problem. The branch-and-price algorithm also produces new optimal solutions for all three problems

An Optimal Control Theory for the Traveling Salesman Problem and Its Variants

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.Comment: 24 pages, 8 figure

Stochastic Vehicle Routing with Recourse

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage optimization problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed -- but costs here become more expensive by a factor lambda. We present an O(log^2 n log(n lambda))-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of Theorem 1.

Algorithms and Adaptivity Gaps for Stochastic k-TSP

Given a metric $(V,d)$ and a $\textsf{root} \in V$, the classic \textsf{k-TSP} problem is to find a tour originating at the $\textsf{root}$ of minimum length that visits at least $k$ nodes in $V$. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of \textsf{k-TSP}. In Stoch-Reward $k$-TSP, originally defined by Ene-Nagarajan-Saket [ENS17], each vertex $v$ in the given metric $(V,d)$ contains a stochastic reward $R_v$. The goal is to adaptively find a tour of minimum expected length that collects at least reward $k$; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an $O(\log k)$-approximation adaptive algorithm for this problem, and left open if there is an $O(1)$-approximation algorithm. We totally resolve their open question and even give an $O(1)$-approximation \emph{non-adaptive} algorithm for this problem. We also introduce and obtain similar results for the Stoch-Cost $k$-TSP problem. In this problem each vertex $v$ has a stochastic cost $C_v$, and the goal is to visit and select at least $k$ vertices to minimize the expected \emph{sum} of tour length and cost of selected vertices. This problem generalizes the Price of Information framework [Singla18] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for our problems, if we truncate the random variables at an ideal threshold and repeat, then their expected values form a good surrogate. Unfortunately, this ideal threshold is adaptive as it depends on how far we are from achieving our target $k$, so we truncate at various different scales and identify a "critical" scale.Comment: ITCS 202

The Team Orienteering Problem: Formulations and Branch-Cut and Price

The Team Orienteering Problem is a routing problem on a graph with durations associated to the arcs and profits assigned to visiting the vertices. A fixed number of identical vehicles, with a limited total duration for their routes, is given. The total profit gathered by all routes is to be maximized. We devise an extended formulation where edges are indexed by the time they are placed in the route. A new class of inequalities, min cut, and the triangle clique cuts of Pessoa et. al., 2007 are added. The resulting formulation is solved by column generation. Branching is done following the work of Boussier et al. 2007, to which the branch-cut-and-price algorithm here proposed is compared. A few new upper bounds were obtained. Overall the presented approach has shown to be very competitive