A cutting plane algorithm for the General Routing Problem

The General Routing Problem (GRP) is the problem of finding a minimum cost route for a single vehicle, subject to the condition that the vehicle visits certain vertices and edges of a network. It contains the Rural Postman Problem, Chinese Postman Problem and Graphical Travelling Salesman Problem as special cases. We describe a cutting plane algorithm for the GRP based on facet-inducing inequalities and show that it is capable of providing very strong lower bounds and, in most cases, optimal solutions. Received: November 1998 / Accepted: September 2000?Published online March 22, 2001     

The Economic Lot Scheduling Problem without Capacity Constraints

This study presents a comprehensive analysis on the Economic Lot Scheduling Problem (ELSP) without capacity constraints. We explore the optimality structure of the ELSP without capacity constraints and discover that the curve for the optimal objective values is piecewise convex with repsect to B, i.e., the values of basic period. The theoretical properties of the junction points on the piecewise convex curve not only provides us the information on “which product i” to modify, but also on “where on the B-axis” to change the set of optimal multpliers in the search process. By making use of the junction points, we propose an effective search algorithm to secure a global optimal solution for the ELSP without capacity constraints. Also, we use random experiments to verify that the proposed algorithm is efficient. The results in this paper lay important foundation for deriving an efficient heuristic to solve the conventional ELSP with capacity constraints.     

Alternating control tree search for knapsack/covering problems

The Multidimensional Knapsack/Covering Problem (KCP) is a 0–1 Integer Programming Problem containing both knapsack and weighted covering constraints, subsuming the well-known Multidimensional Knapsack Problem (MKP) and the Generalized (weighted) Covering Problem. We propose an Alternating Control Tree Search (ACT) method for these problems that iteratively transfers control between the following three components: (1) ACT-1, a process that solves an LP relaxation of the current form of the KCP. (2) ACT-2, a method that partitions the variables according to 0, 1, and fractional values to create sub-problems that can be solved with relatively high efficiency. (3) ACT-3, an updating procedure that adjoins inequalities to produce successively more constrained versions of KCP, and in conjunction with the solution processes of ACT-1 and ACT-2, ensures finite convergence to optimality. The ACT method can also be used as a heuristic approach using early termination rules. Computational results show that the ACT-framework successfully enhances the performance of three widely different heuristics for the KCP. Our ACT-method involving scatter search performs better than any other known method on a large set of KCP-instances from the literature. The ACT-based methods are also found to be highly effective on the MKP.     

An information theory perspective on the balanced minimum evolution problem

We show that the Balanced Minimum Evolution Problem (BMEP) is a cross-entropy minimization problem. This new perspective both extends the previous interpretations of the BMEP length function described in the literature and enables the identification of an efficiently computable family of lower bounds on the value of the optimal solution to the problem.     

Using scenario trees and progressive hedging for stochastic inventory routing problems

The Stochastic Inventory Routing Problem is a challenging problem, combining inventory management and vehicle routing, as well as including stochastic customer demands. The problem can be described by a discounted, infinite horizon Markov Decision Problem, but it has been showed that this can be effectively approximated by solving a finite scenario tree based problem at each epoch. In this paper the use of the Progressive Hedging Algorithm for solving these scenario tree based problems is examined. The Progressive Hedging Algorithm can be suitable for large-scale problems, by giving an effective decomposition, but is not trivially implemented for non-convex problems. Attempting to improve the solution process, the standard algorithm is extended with locking mechanisms, dynamic multiple penalty parameters, and heuristic intermediate solutions. Extensive computational results are reported, giving further insights into the use of scenario trees as approximations of Markov Decision Problem formulations of the Stochastic Inventory Routing Problem.     

The Vertex Coloring Problem and its generalizations

This is a summary of the author’s PhD thesis supervised by Paolo Toth and defended on 29 May 2007 at the Università di Bologna. The thesis is written in English and is available from the author upon request. The first part of this work deals with the Vertex Coloring Problem and its generalizations, for which models, bounds and algorithms are proposed. The Second Part is dedicated to a different problem on graphs, namely a Routing Problem in telecommunication networks where not only the efficiency, but also the fairness of the solution is considered.      

How good are SPT schedules for fair optimality criteria

We consider the following scheduling setting: a set of n tasks have to be executed on a set of m identical machines. It is well known that shortest processing time (SPT) schedules are optimal for the problem of minimizing the total sum of completion times of the tasks. In this paper, we measure the quality of SPT schedules, from an approximation point of view, with respect to the following optimality criteria: sum of completion times per machine, global fairness, and individual fairness.     

Bayesian reliability demonstration with multiple independent tasks

^** Corresponding author. Email: frank.coolen{at}durham.ac.uk We consider optimal testing of a system in order to demonstratereliability with regard to its use in a process after testing,where the system has to function for different types of tasks,which we assume to be independent. We explicitly assume thattesting reveals zero failures. The optimal numbers of tasksto be tested are derived by optimisation of a cost criterion,taking into account the costs of testing and the costs of failuresin the process after testing, assuming that such failures arenot catastrophic to the system. Cost and time constraints ontesting are also included in the analysis. We focus on studyof the optimal numbers of tests for different types of tasks,depending on the arrival rate of tasks in the process and thecosts involved. We briefly compare the results of this studywith optimal test numbers in a similar setting, but with analternative optimality criterion which is more suitable in caseof catastrophic failures, as presented elsewhere. For thesetwo different optimality criteria, the optimal numbers to betested depend similarly on the costs of testing per type andon the arrival rates of tasks in the process after testing.     

Models and heuristic algorithms for a weighted vertex coloring problem

We consider a weighted version of the well-known Vertex Coloring Problem (VCP) in which each vertex i of a graph G has associated a positive weight w _i . Like in VCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used. While in VCP the cost of each color is equal to one, in the Weighted Vertex Coloring Problem (WVCP) the cost of each color depends on the weights of the vertices assigned to that color, and it equals the maximum of these weights. WVCP is known to be NP-hard and arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose three alternative Integer Linear Programming (ILP) formulations for WVCP: one is used to derive, dropping integrality requirement for the variables, a tight lower bound on the solution value, while a second one is used to derive a 2-phase heuristic algorithm, also embedding fast refinement procedures aimed at improving the quality of the solutions found. Computational results on a large set of instances from the literature are reported.     

Scheduling on same-speed processors with at most one downtime on each machine

We consider the problem of scheduling a set of independent tasks on multiple same-speed processors with planned shutdown times with the aim of minimizing the makespan. We give an LPT-based algorithm, LPTX, which yields a maximum completion time that is less than or equal to 3/2 the optimal maximum completion time or 3/2 the time that passes from the start of the schedule until the latest end of a downtime. For problems where the optimal schedule ends after the last downtime, and when the downtimes represent fixed jobs, the LPTX maximum completion time is within 3/2 of the optimal maximum completion time. In addition, we show that this result is asymptotically tight for the class of polynomial algorithms assuming that P≠NP. We also show that the bound obtained previously for a similar problem, when no more than half of the machines are shut down at the same time, for the LPT algorithm is asymptotically tight in the class of polynomial algorithms if P≠NP.     

Computing optimal preemptive schedules for parallel tasks: linear programming approaches

 We study the problem of scheduling a set of n independent parallel tasks on m processors, where in addition to the processing time there is a size associated with each task indicating that the task can be processed on any subset of processors of the given size. Based on a linear programming formulation, we propose an algorithm for computing a preemptive schedule with minimum makespan, and show that the running time of the algorithm depends polynomially on m and only linearly on n. Thus for any fixed m, an optimal preemptive schedule can be computed in O(n) time. We also present extensions of this approach to other (more general) scheduling problems with malleable tasks, due dates and maximum lateness minimization. Received: November 1999 / Accepted: November 2002 Publication online: December 19, 2002 RID="⋆" ID="⋆" This work was done while the authors were associated with the research institutes IDSIA Lugano and MPII Saarbrücken and were supported in part by the Swiss Office Fédéral de l'éducation et de la Science project n 97.0315 titled ``Platform' and by EU ESPRIT LTR Project No. 20244 (ALCOM-IT)     

MIP-based heuristic for non-standard 3D-packing problems

This paper is the continuation of a previous work (Fasano in 4OR 2: 161–174, 2004), dedicated to a MIP formulation for non-standard 3D-packing issues, with additional conditions. The Single Bin Packing problem (Basic Problem) is considered and its MIP formulation shortly surveyed, together with some possible extensions, including balancing, tetris-like items and non-standard domains. A MIP-based heuristic is proposed to solve efficiently the Basic Problem or any possible extension of it, susceptible to a MIP formulation. The heuristic is a recursive procedure based on a non-blind local search philosophy. The concept of abstract configuration, concerning the relative positions between items, is introduced: the relative positions of items, determined by any abstract configuration, give rise to a feasible solution in an unbounded domain. The heuristic generates a sequence of good abstract configurations and solves, step by step, a reduced MIP model by fixing the relative positions of items, corresponding to the current abstract configuration.      

New formulation for the high multiplicity asymmetric traveling salesman problem with application to the Chesapeake problem

This paper presents a new model for a special type of traveling salesman problem called the High Multiplicity Asymmetric Traveling Salesman Problem (HMATSP). The formulation adopts a flow-based subtour elimination structure and establishes its validity for this problem. Also, we present computational results to demonstrate the efficacy of our modeling approach. The model is then incorporated as a substructure in a formulation for the lot-sizing problem involving parallel machines and sequence-dependent setup costs, also known as the Chesapeake Problem, and related test problems are solved to optimality for the first time in the literature.     

Control Policies for Inventory with Service Time

Abstract

This article introduces an additional control policy—the N-policy–into (s, S) inventory system with positive service time. Under specified interarrival and service time distributions, which are independent of each other, we obtain the necessary and sufficient condition for the system to be stable. We also obtain the optimal values of the control variables s, S, and N; it is seen that the cost function attains the minimum value at s = 0. It is also shown that the cost function is separately convex in the variables S and N. Numerical illustrations are provided. Several measures of performance of the system are evaluated.     

Globally solving a nonlinear UAV task assignment problem by stochastic and deterministic optimization approaches

In this paper, we consider a task allocation model that consists of assigning a set of m unmanned aerial vehicles (UAVs) to a set of n tasks in an optimal way. The optimality is quantified by target scores. The mission is to maximize the target score while satisfying capacity constraints of both the UAVs and the tasks. This problem is known to be NP-hard. Existing algorithms are not suitable for the large scale setting. Scalability and robustness are recognized as two main issues. We deal with these issues by two optimization approaches. The first approach is the Cross-Entropy (CE) method, a generic and practical tool of stochastic optimization for solving NP-hard problem. The second one is Branch and Bound algorithm, an efficient classical tool of global deterministic optimization. The numerical results show the efficiency of our approaches, in particular the CE method for very large scale setting.     

Solution of the Illumination Problem for Bodies with md M = 2

We solve here the Gohberg—Markus—Hadwiger Covering Problem (or, what is the same, the illumination problem ) for compact, convex bodies M\subset R ⁿ with md M=2 . Moreover, we outline an idea for a complete solution, using md M . Received August 23, 2000, and in revised form January 22, 2001. Online publication August 9, 2001.     

A Simple Proof of the Restricted Isometry Property for Random Matrices

We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.      

High-multiplicity cyclic job shop scheduling

We consider the High-Multiplicity Cyclic Job Shop Scheduling Problem. There are two objectives of interest: the cycle time and the flow time. We give several approximation algorithms after showing that a very restricted case is APX-hard.     

An inexact algorithm for the sequential ordering problem

Given the directed G = (N, A) and the penalty matrix C, the Sequential Ordering Problem (hereafter, SOP) consists of finding the permutation of the nodes from the set N, such that it minimizes a C-based function and does not violate the precedence relationships given by the set A. Strong sufficient conditions for the infeasibility of a SOP's instance are embedded in a procedure for the SOP's pre-processing. Note that it is one of the key steps in any algorithm that attempts to solve SOP. By dropping the constraints related to the precedence relationships, SOP can be converted in the classical Asymmetric Traveling Salesman Problem (hereafter, ASTP). The algorithm obtains (hopefully) satisfactory solutions by modifying the optimal solution to the related Assignment Problem (hereafter, AP) if it is not a Feasible Sequential Ordering (hereafter, FSO). The new solution ‘patches’ the subtours (if any) giving preference to the patches with zero reduced cost in the linking arcs. The AP-based lower bound on the optimal solution to ATSP is tightened by using some of the procedures given in [1]. In any case, a local search for improving the initial FSO is performed; it uses 3- and 4-changed based procedures. Computational results on a broad set of cases are reported.     

Asymptotic analysis for multi-objective sequential stochastic assignment problems

ABSTRACT

We provide an asymptotic analysis of multi-objective sequential stochastic assignment problems (MOSSAP). In MOSSAP, a fixed number of tasks arrive sequentially, with an n-dimensional value vector revealed upon arrival. Each task is assigned to one of a group of known workers immediately upon arrival, with the reward given by an n-dimensional product-form vector. The objective is to maximize each component of the expected reward vector. We provide expressions for the asymptotic expected reward per task for each component of the reward vector and compare the convergence rates for three classes of Pareto optimal policies.