Surrogate constraint normalization for the set covering problem

The set covering problem (SCP) is central in a wide variety of practical applications for which finding good feasible solutions quickly (often in real-time) is crucial. Surrogate constraint normalization is a classical technique used to derive appropriate weights for surrogate constraint relaxations in mathematical programming. This framework remains the core of the most effective constructive heuristics for the solution of the SCP chiefly represented by the widely-used Chvátal method. This paper introduces a number of normalization rules and demonstrates their superiority to the classical Chvátal rule, especially when solving large scale and real-world instances. Directions for new advances on the creation of more elaborate normalization rules for surrogate heuristics are also provided.     

Using Lagrangian dual information to generate degree constrained spanning trees

In this paper, a Lagrangian-based heuristic is proposed for the degree constrained minimum spanning tree problem. The heuristic uses Lagrangian relaxation information to guide the construction of feasible solutions to the problem. The scheme operates, within a Lagrangian relaxation framework, with calls to a greedy construction heuristic, followed by a heuristic improvement procedure. A look ahead infeasibility prevention mechanism, introduced into the greedy heuristic, allowed us to solve instances of the problem where some of the vertices are restricted to having degrees 1 or 2. Furthermore, in order to cut down on CPU time, a restricted version of the original problem is formulated and used to generate feasible solutions. Extensive computational experiments were conducted and indicate that the proposed heuristic is competitive with the best heuristics and metaheuristics in the literature.     

A comparison of algorithm RS with algorithm OPTSOL70

In a very recent paper (Almiñana and Pastor (1997)) we proposed a new lagrangian surrogate heuristic, called RS, for solving the location (or unicost) set covering problem. In that paper we show that RS is more accurate than the pair of greedy type heuristics FMC/CMA and that RS outperforms the surrogate heuristic SH. Here we are going to compare algorithms RS with the best designed hybrid algorithm for the location set covering problem, known as OPTSOL70.     

Designing greedy algorithms for the flow-shop problem by means of Empirically Adjusted Greedy Heuristics (EAGH)

This paper introduces Empirically Adjusted Greedy Heuristics (EAGH), a procedure for designing greedy algorithms for a given combinatorial optimization problem and illustrates the way in which EAGH works with an application to minimize the makespan in the permutation flow-shop problem. The basic idea behind EAGH is that a greedy heuristic can be seen as a member of an infinite set of heuristics, this set being defined by a function that depends on several parameters. Each set of values of the parameters corresponds to a specific greedy heuristic. Then, the best element of the set, for a training set of instances of the problem, is found by applying a non-linear optimization algorithm to a function that measures the quality of the obtained solutions to the instances of the training set, and which depends on the parameters that characterize each specific algorithm. EAGH allows improving known heuristics or finding good new ones.     

Worst case analysis of Max-Regret,Greedy and other heuristics for Multidimensional Assignment and Traveling Salesman Problems

Optimization heuristics are often compared with each other to determine which one performs best by means of worst-case performance ratio reflecting the quality of returned solution in the worst case. The domination number is a complement parameter indicating the quality of the heuristic in hand by determining how many feasible solutions are dominated by the heuristic solution. We prove that the Max-Regret heuristic introduced by Balas and Saltzman (Oper. Res. 39:150–161, 1991) finds the unique worst possible solution for some instances of the s-dimensional (s≥3) assignment and asymmetric traveling salesman problems of each possible size. We show that the Triple Interchange heuristic (for s=3) also introduced by Balas and Saltzman and two new heuristics (Part and Recursive Opt Matching) have factorial domination numbers for the s-dimensional (s≥3) assignment problem.     

Solving large set covering problems for crew scheduling

This paper presents a computationally effective heuristic method which produces good-quality solutions for large-scale set covering problems with thousands of constraints and about one million variables. The need to solve such large-scale problems arises from a crew scheduling problem of mass transit agencies where the number of work shifts required has to be minimized. This problem may be formulated as a large-scale non-unicost set covering problem whose rows are trips to be performed while columns stand for round trips. The proposed method is mainly based on lagragian relaxation and sub-gradient optimization. After the reduction of the number of rows and columns by the logical tests, “greedy” heuristic algorithms provide upper and lower bounds which are continuously improved to produce goodquality solutions. Computational results, regarding randomly generated problems and real life problems concerning crew scheduling at Italian Railways Company, show that good-quality solutions can be obtained at an acceptable computational cost. This work was supported by the project “Progetto Finalizzato Transporti 2” of National Research Council of Italy (C.N.R.) contract No. 94.01436PF74 and by “Ferrovie dello Stato S.p.A.”     

A probabilistic greedy search algorithm for combinatorial optimisation with application to the set covering problem

We present a probabilistic greedy search method for combinatorial optimisation problems. This approach is implemented and evaluated for the Set Covering Problem (SCP) and shown to yield a simple, robust, and quite fast heuristic. Tests performed on a large set of benchmark instances with up to 1000 rows and 10?000 columns show that the algorithm consistently yields near-optimal solutions.     

A hybrid adaptive large neighborhood search heuristic for lot-sizing with setup times

This paper presents a hybrid of a general heuristic framework and a general purpose mixed-integer programming (MIP) solver. The framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched. A mixed integer programming solver and its built-in feasibility heuristics is used to search a neighborhood for improving solutions. The general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods. The hybrid heuristic framework is applied to the multi-item capacitated lot sizing problem with setup times, where experiments have been conducted on a series of instances from the literature and a newly generated extension of these. On average the presented heuristic outperforms the best heuristics from the literature, and the upper bounds found by the commercial MIP solver ILOG CPLEX using state-of-the-art MIP formulations. Furthermore, we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature.     

Maximizing the Net Present Value of a Project Under Resource Constraints Using a Lagrangian Relaxation Based Heuristic with Tight Upper Bounds

Resource-constrained project scheduling under a net present value objective attracts growing interest. Because this is an NP-hard problem, it is unlikely that optimum solutions can be computed for large instances within reasonable computation time. Thus, heuristics have become a popular research field. Up to now, however, upper bounds are not well researched. Therefore, most researchers evaluate their heuristics on the basis of a best known lower bound, but it is unclear how good the performance really is. With this contribution we close this gap and derive tight upper bounds on the basis of a Lagrangian relaxation of the resource constraints. We also use this approach as a basis for a heuristic and show that our heuristic as well as the cash flow weight heuristic proposed by Baroum and Patterson yield solutions very close to the optimum result. Furthermore, we discuss the proper choice of a test-bed and emphasize that discount rates must be carefully chosen to give realistic instances.     

Minimising the number of gap-zeros in binary matrices

We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is referred to as the Consecutive Ones Matrix Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, which is linear-time in the number of rows if the number of columns is constant, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows.     

Hybrid genetic algorithm for permutation flowshop scheduling problems with total flowtime minimization

In this paper, a HGA (hybrid genetic algorithm) is proposed for permutation flowshop scheduling problems (PFSP) with total flowtime minimization, which are known to be NP-hard. One of the chromosomes in the initial population is constructed by a suitable heuristic and the others are yielded randomly. An artificial chromosome is generated by a weighted simple mining gene structure, with which a new crossover operator is presented. Additionally, two effective heuristics are adopted as local search to improve all generated chromosomes in each generation. The HGA is compared with one of the most effective heuristics and a recent meta-heuristic on 120 benchmark instances. Experimental results show that the HGA outperforms the other two algorithms for all cases. Furthermore, HGA obtains 115 best solutions for the benchmark instances, 92 of which are newly discovered.     

Model-hierarchical column generation and heuristic for the routing and wavelength assignment problem

The routing and wavelength assignment (RWA) problem typically occurs in wavelength division multiplexing optical networks. Given a number of available wavelengths, we consider here the problem of maximising the number of accepted connections with respect to the clash and continuity constraints. We first propose a new strategy which combines two existing models. This leads to an improved column generation scheme. We also present two heuristics to compute feasible solutions: a hybrid heuristic and the integer solution at the root node of the column generation. Our approaches are compared with the best existing results on a set of classic RWA instances.     

Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem

The unconstrained binary quadratic programming problem (BQP) is known to be NP-hard and has many practical applications. This paper presents a simulated annealing (SA)-based heuristic for the BQP. The new SA heuristic for the BQP is based on a simple (1-opt) local search heuristic and designed with a simple cooling schedule, but the multiple annealing processes are adopted. To show practical performances of the SA, we test on publicly available benchmark instances of large size ranging from 500 to 2500 variables and compare them with other heuristics such as multi-start local search, the previous SA, tabu search, and genetic algorithm incorporating the 1-opt local search. Computational results indicate that our SA leads to high-quality solutions with short times and is more effective than the competitors particularly for the largest benchmark set. Furthermore, the values of new best-known solutions found by the SA for several large instances are also reported.     

On the Significance of the Initial Solution in Travelling Salesman Heuristics

The quality requirements set by edge exchange heuristics on their initial solutions are evaluated in connection with the travelling salesman problem. The performance of the heuristics is measured using the expected value of the best solution achievable in a certain computing time. The computational results show that the use of initial solutions generated by applying a construction heuristic, instead of random initial solutions, typically improves the performance of edge exchange heuristics. The improvement, however, is dependent on the edge exchange heuristic to be used, the properties of the problem, and the computing time available.     

Single-commodity robust network design problem: Complexity,instances and heuristic solutions

We study a single-commodity Robust Network Design problem (RND) in which an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. In each scenario, a subset of the nodes is exchanging flow. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. Previously conducted computational investigations on the problem motivated the study of the complexity of some special cases and we present complexity results on them, including hypercubes. In turn, these results lead to the definition of new instances (random graphs with {−1, 0, 1} balances) that are computationally hard for the natural flow formulation. These instances are then solved by means of a new heuristic algorithm for RND, which consists of three phases. In the first phase the graph representing the network is reduced by heuristically deleting a subset of the arcs, and a feasible solution is built. The second phase consists of a neighborhood search on the reduced graph based on a Mixed-Integer (Linear) Programming (MIP) flow model. Finally, the third phase applies a proximity search approach to further improve the solution, taking into account the original graph. The heuristic is tested on the new instances, and the comparison with the solutions obtained by Cplex on a natural flow formulation shows the effectiveness of the proposed method.     

Scheduling of nonresumable jobs and flexible maintenance activities on a single machine to minimize makespan

This study addresses a single machine scheduling problem with periodic maintenance, where the machine is assumed to be stopped periodically for maintenance for a constant time w during the scheduling period. Meanwhile, the maintenance period [u, v] is assumed to have been previously arranged and the time w is assumed not to exceed the available maintenance period [u, v] (i.e. w ? v − u). The time u(v) is the earliest (latest) time at which the machine starts (stops) its maintenance. The objective is to minimize the makespan. Two mixed binary integer programming (BIP) models are provided for deriving the optimal solution. Additionally, an efficient heuristic is proposed for finding the near-optimal solution for large-sized problems. Finally, computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics. The mixed BIP model can optimally solve up to 100-job instances, while the average percentage error of the heuristic is below 1%.     

Experimental analysis of heuristics for the bottleneck traveling salesman problem

In this paper we develop efficient heuristic algorithms to solve the bottleneck traveling salesman problem (BTSP). Results of extensive computational experiments are reported. Our heuristics produced optimal solutions for all the test problems considered from TSPLIB, JM-instances, National TSP instances, and VLSI TSP instances in very reasonable running time. We also conducted experiments with specially constructed ‘hard’ instances of the BTSP that produced optimal solutions for all but seven problems. Some fast construction heuristics are also discussed. Our algorithms could easily be modified to solve related problems such as the maximum scatter TSP and testing hamiltonicity of a graph.     

Local search intensified: Very large-scale variable neighborhood search for the multi-resource generalized assignment problem

We introduce a heuristic for the Multi-Resource Generalized Assignment Problem (MRGAP) based on the concepts of Very Large-Scale Neighborhood Search and Variable Neighborhood Search. The heuristic is a simplified version of the Very Large-Scale Variable Neighborhood Search for the Generalized Assignment Problem. Our algorithm can be viewed as a k-exchange heuristic; but unlike traditional k-exchange algorithms, we choose larger values of k resulting in neighborhoods of very large size with high probability. Searching this large neighborhood (approximately) amounts to solving a sequence of smaller MRGAPs either by exact algorithms or by heuristics. Computational results on benchmark test problems are presented. We obtained improved solutions for many instances compared to some of the best known heuristics for the MRGAP within reasonable running time. The central idea of our heuristic can be used to develop efficient heuristics for other hard combinatorial optimization problems as well.     

Application of the out-of-kilter algorithm to the asymmetric traveling salesman problem

This paper presents a heuristic method that finds optimum or near-optimum solutions to the asymmetric traveling salesman problem. The method uses the out-of-kilter algorithm to search for a neighbourhood. When subtours are produced by a flow-augmenting path of the out-of-kilter algorithm, it patches them into a Hamiltonian cycle. It extends the neighbourhood space by exchanging an even number of arcs, and it also exchanges arcs by a non-sequential primary change. Instances from real applications were used to test the algorithm, along with randomly generated problems. The new heuristic algorithm produced optimum solutions for 16 out of 28 real-world instances from TSPLIB and other sources. Also, compared with four efficient heuristics, it produced the best solutions for all except six instances. It also produced relatively good solutions in reasonable times for 216 randomly generated instances from nine instance generators.     

Heuristics and memetic algorithm for the two-dimensional loading capacitated vehicle routing problem with time windows

This work deals with a new combinatorial optimization problem, the two-dimensional loading capacitated vehicle routing problem with time windows which is a realistic extension of the well known vehicle routing problem. The studied problem consists in determining vehicle trips to deliver rectangular objects to a set of customers with known time windows, using a homogeneous fleet of vehicles, while ensuring a feasible loading of each vehicle used. Since it includes NP-hard routing and packing sub-problems, six heuristics are firstly designed to quickly compute good solutions for realistic instances. They are obtained by combining algorithms for the vehicle routing problem with time windows with heuristics for packing rectangles. Then, a Memetic algorithm is developed to improve the heuristic solutions. The quality and the efficiency of the proposed heuristics and metaheuristic are evaluated by adding time windows to a set of 144 instances with 15–255 customers and 15–786 items, designed by Iori et al. (Transport Sci 41:253–264, 2007) for the case without time windows.