A hybrid scatter search for the discrete time/resource trade-off problem in project scheduling

We develop a heuristic procedure for solving the discrete time/resource trade-off problem in the field of project scheduling. In this problem, a project contains activities interrelated by finish-start-type precedence constraints with a time lag of zero, which require one or more constrained renewable resources. Each activity has a specified work content and can be performed in different modes, i.e. with different durations and resource requirements, as long as the required work content is met. The objective is to schedule each activity in one of its modes in order to minimize the project makespan. We use a scatter search algorithm to tackle this problem, using path relinking methodology as a solution combination method. Computational results on randomly generated problem sets are compared with the best available results indicating the efficiency of the proposed algorithm.     

A scatter search algorithm for project scheduling under partially renewable resources

In this paper we develop a heuristic algorithm, based on Scatter Search, for project scheduling problems under partially renewable resources. This new type of resource can be viewed as a generalization of renewable and non-renewable resources, and is very helpful in modelling conditions that do no fit into classical models, but which appear in real timetabling and labor scheduling problems. The Scatter Search algorithm is tested on existing test instances and obtains the best results known so far.     

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.     

Diversified local search strategy under scatter search framework for the probabilistic traveling salesman problem

This paper focuses on introducing a concept of diversified local search strategy under the scatter search framework for the probabilistic traveling salesman problem (PTSP). Different combinations of three commonly used local search methods in the PTSP, i.e., 1-shift, 2-opt, and 3-opt, were used to investigate its effects. A set of numerical experiments were conducted to test the validity of the proposed strategy based on randomly generated test instances. The numerical results and the permutation test showed that the diversified local search strategy, especially by combining 1-shift and 2-opt algorithms, can most effectively solve the homogeneous and heterogeneous PTSP in most of the tested instances in comparison with the single local search strategy under scatter search framework.     

A scatter search algorithm for the distributed permutation flowshop scheduling problem

The distributed permutation flowshop problem has been recently proposed as a generalization of the regular flowshop setting where more than one factory is available to process jobs. Distributed manufacturing is a common situation for large enterprises that compete in a globalized market. The problem has two dimensions: assigning jobs to factories and scheduling the jobs assigned to each factory. Despite being recently introduced, this interesting scheduling problem has attracted attention and several heuristic and metaheuristic methods have been proposed in the literature. In this paper we present a scatter search (SS) method for this problem to optimize makespan. SS has seldom been explored for flowshop settings. In the proposed algorithm we employ some advanced techniques like a reference set made up of complete and partial solutions along with other features like restarts and local search. A comprehensive computational campaign including 10 existing algorithms, together with statistical analyses, shows that the proposed scatter search algorithm produces better results than existing algorithms by a significant margin. Moreover all 720 known best solutions for this problem are improved.     

A hybrid single and dual population search procedure for the job shop scheduling problem

This paper presents a genetic algorithm and a scatter search procedure to solve the well-known job shop scheduling problem. In contrast to the single population search performed by the genetic algorithm, the scatter search algorithm splits the population of solutions in a diverse and high-quality set to exchange information between individuals in a controlled way. The extension from a single to a dual population, by taking problem specific characteristics into account, can be seen as a stimulator to add diversity in the search process. This has a positive influence on the important balance between intensification and diversification. Computational experiments verify the benefit of this diversity on the effectiveness of the meta-heuristic search process. Various algorithmic parameters from literature are embedded in both procedures and a detailed comparison is made. A set of standard instances is used to compare the different approaches and the best obtained results are benchmarked against heuristic solutions found in literature.     

Scatter search for the fleet size and mix vehicle routing problem with time windows

This work proposes a scatter search (SS) approach to solve the fleet size and mix vehicle routing problem with time windows (FSMVRPTW). In the FSMVRPTW the customers need to be serviced in their time windows at minimal costs by a heterogeneous fleet. Computational results on 168 benchmark problems are reported. Computational testing revealed that our algorithm presented better results compared to other methods published in the literature.     

Optimal procedures for the discrete time/cost trade-off problem in project networks

We describe two algorithms, based on dynamic programming logic, for optimally solving the discrete time/cost trade-off problem (DTCTP) in deterministic activity-on-arc networks of the CPM type, where the duration of each activity is a discrete, nonincreasing function of the amount of a single nonrenewable resource committed to it. The first algorithm is based on a procedure proposed by Bein, Kamburowski and Stallmann for finding the minimal number of reductions necessary to transform a general network to a series-parallel network. The second algorithm minimizes the estimated number of possibilities that need to be considered during the solution procedure. Both procedures have been programmed in C and tested on a large set of representative networks to give a good indication of their performance, and indicate the circumstances in which either algorithm performs best.     

An iterated local search algorithm for the time-dependent vehicle routing problem with time windows

We generalize the standard vehicle routing problem with time windows by allowing both traveling times and traveling costs to be time-dependent functions. In our algorithm, we use a local search to determine routes of the vehicles. When we evaluate a neighborhood solution, we must compute an optimal time schedule for each route. We show that this subproblem can be efficiently solved by dynamic programming, which is incorporated in the local search algorithm. The neighborhood of our local search consists of slight modifications of the standard neighborhoods called 2- opt^*, cross exchange and Or-opt. We propose an algorithm that evaluates solutions in these neighborhoods more efficiently than the ones computing the dynamic programming from scratch by utilizing the information from the past dynamic programming recursion used to evaluate the current solution. We further propose a filtering method that restricts the search space in the neighborhoods to avoid many solutions having no prospect of improvement. We then develop an iterated local search algorithm that incorporates all the above ingredients. Finally we report computational results of our iterated local search algorithm compared against existing methods, and confirm the effectiveness of the restriction of the neighborhoods and the benefits of the proposed generalization.     

Robust scheduling and robustness measures for the discrete time/cost trade-off problem

Projects are often subject to various sources of uncertainties that have a negative impact on activity durations and costs. Therefore, it is crucial to develop effective approaches to generate robust project schedules that are less vulnerable to disruptions caused by uncontrollable factors. In this paper, we investigate the robust discrete time/cost trade-off problem, which is a multi-mode project scheduling problem with important practical relevance. We introduce surrogate measures that aim at providing an accurate estimate of the schedule robustness. The pertinence of each proposed measure is assessed through computational experiments. Using the insights revealed by the computational study, we propose a two-stage robust scheduling algorithm. Finally, we provide evidence that the proposed approach can be extended to solve a complex robust problem with tardiness penalties and earliness revenues.     

A discrete time optimal control model with uncertainty for dynamic machine allocation problem and its application to manufacturing and construction industries

This paper proposed a discrete time optimal control model in which machine failure time is modeled assuming a Weibull distribution and machine productivity is regarded as a fuzzy variable for dealing with a dynamic machine allocation problem (DMAP) in manufacturing and construction industries. The aim is to maximize total production or construction throughput when uncertainties such as machine breakdowns are taken into account. A failure probability-work time equation is presented to describe the relationship between machine failure probability and mean time to work. To transform the uncertain optimal control model into a deterministic one, the expected value model (EVM) was introduced for forming an equivalent crisp model. The fuzzy variables in the model are also defuzzified by using an expected value operator with an optimistic–pessimistic index. Then a number of lemmas and theorems are presented and proved to formulate the theoretical algorithm so that the crisp model of the DMAP can be solved. Three actual construction and production projects are used as practical application examples. The theoretical algorithm results for the three project examples are compared with a particle swarm optimization approach and a genetic algorithm method, which demonstrates the practicality and efficiency of our optimization method.     

A local search using solution fragments for the 2-machine bicriteria scheduling problem

In this work, we introduce a local search strategy for combinatorial optimization problems which explores neighborhoods obtained using fragments of current solutions. We apply the approach to the well-known -hard 2-machine bicriteria flowshop scheduling problem. Computational experiments using benchmark data show the approach to be effective when compared to other algorithms available for the problem.     

An analytical solution for the discrete time single server system with semi-Markovian arrivals

In this paper we derive an analytical solution for the stationary distribution of the number of customers and the idle time in a single server system with semi-Markovian arrival processes in discrete time domain (SM/G/1). This kind of arrival process enables us to take autocorrelations into account, with various applications for the modeling of communication and manufacturing systems. It will be shown that the distribution of the customer number can be represented as a linear combination of geometric distributions. Thus a simple calculation of higher moments of the customer number is possible.     

A tabu search heuristic using genetic diversification for the clustered traveling salesman problem

The clustered traveling salesman problem is an extension of the classical traveling salesman problem where the set of vertices is partitioned into clusters. The objective is to find a least cost Hamiltonian cycle such that the vertices of each cluster are visited contiguously and the clusters are visited in a prespecified order. A tabu search heuristic is proposed to solve this problem. This algorithm periodically restarts its search by merging two elite solutions to form a new starting solution (in a manner reminiscent of genetic algorithms). Computational results are reported on sets of Euclidean problems with different characteristics.     

Lasso solution strategies for the vehicle routing problem with pickups and deliveries

This paper considers the vehicle routing problem with pickups and deliveries (VRPPD) where the same customer may require both a delivery and a pickup. This is the case, for instance, of breweries that deliver beer or mineral water bottles to a set of customers and collect empty bottles from the same customers. It is possible to relax the customary practice of performing a pickup when delivering at a customer, and postpone the pickup until the vehicle has sufficient free capacity. In the case of breweries, these solutions will often consist of routes in which bottles are first delivered until the vehicle is partly unloaded, then both pickup and delivery are performed at the remaining customers, and finally empty bottles are picked up from the first visited customers. These customers are revisited in reverse order, thus giving rise to lasso shaped solutions. Another possibility is to relax the traditional problem even more and allow customers to be visited twice either in two different routes or at different times on the same route, giving rise to a general solution. This article develops a tabu search algorithm capable of producing lasso solutions. A general solution can be reached by first duplicating each customer and generating a Hamiltonian solution on the extended set of customers. Test results show that while general solutions outperform other solution shapes in term of cost, their computation can be time consuming. The best lasso solution generated within a given time limit is generally better than the best general solution produced with the same computing effort.     

Algorithms for solving discrete optimal control problems with infinite time horizon and determining minimal mean cost cycles in a directed graph as decision support tool

Time-discrete systems with a finite set of states are considered. Discrete optimal control problems with infinite time horizon for such systems are formulated. We introduce a certain graph-theoretic structure to model the transitions of the dynamical system. Algorithms for finding the optimal stationary control parameters are presented. Furthermore, we determine the optimal mean cost cycles. This approach can be used as a decision support strategy within such a class of problems; especially so-called multilayered decision problems which occur within environmental emission trading procedures can be modelled by such an approach.     

A matheuristic for the truck and trailer routing problem

In the truck and trailer routing problems (TTRPs) a fleet of trucks and trailers serves a set of customers. Some customers with accessibility constraints must be served just by truck, while others can be served either by truck or by a complete vehicle (a truck pulling a trailer). We propose a simple, yet effective, two-phase matheuristic that uses the routes of the local optima of a hybrid GRASP × ILS as columns in a set-partitioning formulation of the TTRP. Using this matheuristic we solved both the classical TTRP with fixed fleet and the new variant with unlimited fleet. This matheuristic outperforms state-of-the-art methods both in terms of solution quality and computing time. While the best variant of the matheuristic found new best-known solutions for several test instances from the literature, the fastest variant of the matheuristic achieved results of comparable quality to those of all previous method from the literature with an average speed-up of at least 2.5.     

A hybrid heuristic ordering and variable neighbourhood search for the nurse rostering problem

This paper is concerned with the development of intelligent decision support methodologies for nurse rostering problems in large modern hospital environments. We present an approach which hybridises heuristic ordering with variable neighbourhood search. We show that the search can be extended and the solution quality can be significantly improved by the careful combination and repeated use of heuristic ordering, variable neighbourhood search and back-tracking. The amount of computational time that is allowed plays a significant role and we analyse and discuss this. The algorithms are evaluated against a commercial Genetic Algorithm on commercial data. We demonstrate that this methodology can significantly outperform the commercial algorithm. This paper is one of the few in the scientific nurse rostering literature which deal with commercial data and which compare against a commercially implemented algorithm.     

Tabu search for the single row facility layout problem using exhaustive 2-opt and insertion neighborhoods

The single row facility layout problem (SRFLP) is the problem of arranging facilities with given lengths on a line, while minimizing the weighted sum of the distances between all pairs of facilities. The problem is NP-hard. In this paper, we present two tabu search implementations, one involving an exhaustive search of the 2-opt neighborhood and the other involving an exhaustive search of the insertion neighborhood. We also present techniques to significantly speed up the search of the two neighborhoods. Our computational experiments show that the speed up techniques are effective, and our tabu search implementations are competitive. Our tabu search implementations improved previously known best solutions for 23 out of the 43 large sized SRFLP benchmark instances.     

A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm

The Vehicle Routing Problem (VRP) is one of the most well studied problems in operations research, both in real life problems and for scientific research purposes. During the last 50 years a number of different formulations have been proposed, together with an even greater number of algorithms for the solution of the problem. In this paper, the VRP is formulated as a problem of two decision levels. In the first level, the decision maker assigns customers to the vehicles checking the feasibility of the constructed routes (vehicle capacity constraints) and without taking into account the sequence by which the vehicles will visit the customers. In the second level, the decision maker finds the optimal routes of these assignments. The decision maker of the first level, once the cost of each routing has been calculated in the second level, estimates which assignment is the better one to choose. Based on this formulation, a bilevel genetic algorithm is proposed. In the first level of the proposed algorithm, a genetic algorithm is used for calculating the population of the most promising assignments of customers to vehicles. In the second level of the proposed algorithm, a Traveling Salesman Problem (TSP) is solved, independently for each member of the population and for each assignment to vehicles. The algorithm was tested on two sets of benchmark instances and gave very satisfactory results. In both sets of instances the average quality is less than 1%. More specifically in the set with the 14 classic instances proposed by Christofides, the quality is 0.479% and in the second set with the 20 large scale vehicle routing problems, the quality is 0.826%. The algorithm is ranked in the tenth place among the 36 most known and effective algorithms in the literature for the first set of instances and in the sixth place among the 16 algorithms for the second set of instances. The computational time of the algorithm is decreased significantly compared to other heuristic and metaheuristic algorithms due to the fact that the Expanding Neighborhood Search Strategy is used.