Eleni Hadjiconstantinou

An Exact Algorithm for the Capacitated Vehicle Routing Problem Based on a Two-Commodity Network Flow Formulation

The capacitated vehicle routing problem (CVRP) is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. In this paper, we describe a new integer programming formulation for the CVRP based on a two-commodity network flow approach. We present a lower bound derived from the linear programming (LP) relaxation of the new formulation which is improved by adding valid inequalities in a cutting-plane fashion. Moreover, we present a comparison between the new lower bound and lower bounds derived from the LP relaxations of different CVRP formulations proposed in the literature. A new branch-and-cut algorithm for the optimal solution of the CVRP is described. Computational results are reported for a set of test problems derived from the literature and for new randomly generated problems.

An exact algorithm for general, orthogonal, two-dimensional knapsack problems

Abstract We present a new exact tree-search procedure for solving two-dimensional knapsack problems in which a number of small rectangular pieces, each of a given size and value, are required to be cut from a large rectangular stock plate. The objective is to maximise the value of pieces cut or minimise the wastage. We consider the case where there is a maximum number of times that a piece may be used in a cutting pattern. The algorithm limits the size of the tree search by using a bound derived from a Langrangean relaxation of a 0–1 integer programming formulaton of the problem. Subgradient optimisation is used to optimise this bound. Reduction tests derived from both the original problem and the Lagrangean relaxation produce substantial computational gains. The computational performance of the algorithm indicates that it is an effective procedure capable of solving optimally practical two-dimensional cutting problems of medium size.

An exact algorithm for orthogonal 2-D cutting problems using guillotine cuts

Abstract We consider the two-dimensional cutting problem which requires cutting a number of small rectangular pieces, each of a given size and value, from a large reatangular stock plate. The objective is to maximise the value of pieces cut, or (for a slightly simpler problem) to minimise the wastage. We consider the case where there is a maximum number of times that a piece may be used in a cutting pattern. We present a tree-search algorithm for this problem in which the size of the tree search is limited by using a bound derived from a state space relaxation of a dynamic programming formulation of the problem. A state space ascent method is used to optimise this bound. The computational performance of the algorithm is investigated by tests performed on randomly generated problems with constraints of varying ‘tightness’. The results indicate that the algorithm is an effective procedure capable of solving optimally practical two-dimensional cutting problems of medium size.

An efficient implementation of an algorithm for finding K shortest simple paths

In this article, we present an efficient computational implementation of an algorithm for finding the K shortest simple paths connecting a pair of vertices in an undirected graph with n vertices, m arcs, and nonnegative arc lengths. A minimal number of intermediate paths is formed based on the method of Katoh, Ibaraki and Mine [Networks 12 (1982), 411–427], which has the lowest worst-case complexity of O(n2) among all other existing algorithms for this problem. A theoretical description of the algorithm is presented with detailed explanations and some examples of the more complicated steps. Efficient data structures for storing and retrieving a large number of paths are given. The results of wide-ranging experimentation with a large number of randomly generated graphs of varying size and density confirm the linear dependency of computing time on K, as proven in Katoh et al., and the quadratic dependency of computing time on graph size as suggested by the worst-case computational complexity.

A NEW EXACT ALGORITHM FOR THE VEHICLE ROUTING PROBLEM BASED ON Q-PATHS AND K-SHORTEST PATHS RELAXATIONS

We consider the basic Vehicle Routing Problem (VRP) in which a fleet ofM identical vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject only to vehicle capacity constraints. In this paper, we present an exact algorithm for solving the VRP that uses lower bounds obtained from a combination of two relaxations of the original problem which are based on the computation ofq-paths andk-shortest paths. A set of reduction tests derived from the computation of these bounds is applied to reduce the size of the problem and to improve the quality of the bounds. The resulting lower bounds are then embedded into a tree-search procedure to solve the problem optimally. Computational results are presented for a number of problems taken from the literature. The results demonstrate the effectiveness of the proposed method in solving problems involving up to about 50 customers and in providing tight lower bounds for problems up to about 150 customers.

A new method for solving capacitated location problems based on a set partitioning approach

Abstract We consider the capacitated p-median problem (CPMP) in which a set of n customers must be partitioned into p disjoint clusters so that the total dissimilarity within each cluster is minimized and constraints on maximum cluster capacities are met. The dissimilarity of a cluster is computed as the sum of the dissimilarities existing between each entity of the cluster and the median associated to such cluster. In this paper we present an exact algorithm for solving the CPMP based on a set partitioning formulation of the problem. A valid lower bound to the optimal solution cost is obtained by combining two different heuristic methods for solving the dual of the LP-relaxation of the exact formulation. Computational tests on problems proposed in the literature and on new sets of test problems show the effectiveness of the proposed algorithm. Scope and purpose A basic location problem consists of locating a number of facilities or depots to supply a set of customers. The objective is to minimize the cost of locating the facilities and assigning the customers to them. This problem has been extensively studied in the literature and is commonly referred to as the plant location problem, or facility location problem. When each potential facility has a constraint on the maximum demand that it can supply and the number of facilities to locate is specified, the problem is known as the Capacitated p-median problem (CPMP). The purpose of this paper is to present a new exact algorithm for the CPMP and evaluate its computational performance on a set of test problems taken from the literature and on a new set of test problems.

An exact algorithm for the Traveling Salesman Problem with Deliveries and Collections

In this paper, we describe a new integer programming formulation for the Traveling Salesman Problem with mixed Deliveries and Collections (TSPDC) based on a two-commodity network flow approach. We present new lower bounds that are derived from the linear relaxation of the new formulation by adding valid inequalities, in a cutting-plane fashion. The resulting lower bounds are embedded in a branch-and-cut algorithm for the optimal solution of the TSPDC. Computational results on different classes of test problems taken from the literature indicate the effectiveness of the proposed method.

A hybrid genetic algorithm for the two-dimensional single large object placement problem

In the two-dimensional single large object placement problem, we are given a rectangular master surface which has to be cut into a set of smaller rectangular items, with the aim of maximizing the total value of the pieces cut. We consider the special case in which the items cannot be rotated and must be cut with their edges always parallel to the edges of the surface. We present new greedy algorithms and a hybrid genetic approach with elitist theory, immigration rate, heuristics on-line and tailored crossover operators. Extensive computational results for a large number of small and large benchmark test problems are presented. The results show that our approach outperforms existing heuristic algorithms.

A decomposition-based stochastic programming approach for the project scheduling problem under time/cost trade-off settings and uncertain durations

Resource allocation in project networks allows for the control of the processing time of an activity under time-cost tradeoff settings. In practice, project decisions are made in advance when activity durations are still highly uncertain. Given an activity-on-node project network and a set of execution modes for each activity, we consider the problem of deciding how and when to execute each activity to minimize project completion time or total cost with respect to captured activity durations. The inherent stochasticity is represented using a set of discrete scenarios in which each scenario is associated with a probability of occurrence and a realization of activity durations. In this paper, we propose a path-based two-stage stochastic integer programming approach in which the execution modes are determined in the first stage while the second stage performs activity scheduling according to the realizations of activity durations, hence, providing flexibility in the scheduling process at subsequent stages. The solution methodology is based on a decomposition algorithm which has been implemented and widely tested using a large number of test instances of varying size and uncertainty. The reported computational results demonstrate that the proposed algorithm converges fast to the optimal solution and present the benefits of using the stochastic model as opposed to the traditional deterministic model that considers only expected values of activity durations.

Tools for reformulating logical forms into zero-one mixed integer programs

Abstract A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Programming (ILP) formulation or a Mixed Integer Programming (MIP) formulation is presented. A reformulation procedure which uses the extended reverse Polish notation to represent a compound logical form is then described. The syntax of an LP modelling language is extended to incorporate statements in propositional logic forms with linear algebraic forms whereby 0–1 MIP models can be automatically formulated. A prototype user interface by which logical forms can be reformulated and the corresponding MIP constructed and analysed within an existing Mathematical Programming modelling system is illustrated. Finally, the steps to formulate a discrete optimisation model in this way are illustrated by means of an example.