Difficult Combinatorial Optimization Problems Research Articles

Routing problems with loading constraints

We consider difficult combinatorial optimization problems arising in transportation logistics when one is interested in optimizing both the routing of vehicles and the loading of goods into them. The separate problems (routing and loading) are already $\mathcal{NP}$ -hard, and very difficult to solve in practice. A fortiori their combination is extremely challenging and stimulating. Although the specific literature is still quite limited, a first attempt to a systematic view of this field can be useful both to academic researchers and to practitioners. We review vehicle routing problems with two- and three-dimensional loading constraints. Other combinations of routing and special loading constraints arising from industrial applications are also considered.

Round robin tournaments and three index assignments

Scheduling a sports league can be seen as a difficult combinatorial optimization problem. We study some variants of round robin tournaments and analyze the relationship with the planar three-index assignment problem. The complexity of scheduling a minimum cost round robin tournament is established by a reduction from the planar three-index assignment problem. Furthermore, we introduce integer programming models. We pick up a popular idea and decompose the overall problem in order to obtain two subproblems which can be solved sequentially. We show that the latter subproblem can be casted as a planar three-index assignment problem. This makes existing solution techniques for the planar three-index assignment problem amenable to sports league scheduling.

A hybrid particle swarm optimization algorithm for the vehicle routing problem

This paper introduces a new hybrid algorithmic nature inspired approach based on particle swarm optimization, for successfully solving one of the most popular supply chain management problems, the vehicle routing problem. The vehicle routing problem is considered one of the most well studied problems in operations research. The proposed algorithm for the solution of the vehicle routing problem, the hybrid particle swarm optimization (HybPSO), combines a particle swarm optimization (PSO) algorithm, the multiple phase neighborhood search–greedy randomized adaptive search procedure (MPNS–GRASP) algorithm, the expanding neighborhood search (ENS) strategy and a path relinking (PR) strategy. The algorithm is suitable for solving very large-scale vehicle routing problems as well as other, more difficult combinatorial optimization problems, within short computational time. It is tested on a set of benchmark instances and produced very satisfactory results. The algorithm is ranked in the fifth place among the 39 most known and effective algorithms in the literature and in the first place among all nature inspired methods that have ever been used for this set of instances.

Incremental demand rerouting: optimization models and algorithms

This investigation addresses an important and difficult combinatorial optimization problem in the design and management of telecommunication systems known as the path assignment problem. The path assignment problem is specified by a set of demands on a network with given link capacities. Each demand must be assigned to exactly one routing path in such a way that the total amount of traffic routed over any link does not exceed that link's capacity. Given a network and a path assignment, the problem of interest is to maximize the available bandwidth (throughput) for a new demand. Network managers are concerned with the quality of service provided by their networks, and so they are reluctant to make major changes to an operating network that may inadvertently result in service disruptions for numerous clients simultaneously. Therefore, the objective of this investigation is to develop and test optimization models and algorithms that produce a series of throughput-improving modifications to the original path assignment. This allows network managers to implement the improved routing in stages with minimal changes to the overall routing plan between any two consecutive stages. Although the procedure may only reroute a few demands at each stage, the routing after the last stage should be as close as possible to an optimal routing. That is, it must maximize the throughput for the new demand. We present integer programming models and associated solution algorithms to maximize the throughput with a sequence of incremental path modifications. The algorithms were implemented with the AMPL modeling language and the CPLEX ILP solver, and tested on a family of five different data sets based on a European network widely studied in the literature. Empirical results show that the incremental approach finds near-optimal results within reasonable limits on CPU time.

An Effective Clustering Algorithm With Ant Colony

This paper proposes a new clustering algorithm based on ant colony to solve the unsupervised clustering problem. Ant colony optimization (ACO) is a population-based meta-heuristic that can be used to find approximate solutions to difficult combinatorial optimization problems. Clustering Analysis, which is an important method in data mining, classifies a set of observations into two or more mutually exclusive unknown groups. This paper presents an effective clustering algorithm with ant colony which is based on stochastic best solution kept--ESacc. The algorithm is based on Sacc algorithm that was proposed by P.S.Shelokar. It’s mainly virtue that best values iteratively are kept stochastically. Moreover, the new algorithm using Jaccard index to identify the optimal cluster number. The results of several times experiments in three datasets show that the new algorithm-ESacc is less in running time, is better in clustering effect and more stable than Sacc. Experimental results validate the novel algorithm’s efficiency. In addition, Three indices of clustering validity analysis are selected and used to evaluate the clustering solutions of ESacc and Sacc.

Cache-aware optimization of BAN applications

Body-area sensor network or BAN-based health monitoring is increasingly becoming a popular alternative to traditional wired bio-monitoring techniques. However, most biomonitoring applications need continuous processing of large volumes of data, as a result of which both power consumption and computation bandwidth turn out to be serious constraints for sensor network platforms. This has resulted in a lot of recent interest in design methods, modeling and software analysis techniques specifically targeted towards BANs and applications running on them. In this paper we show that appropriate optimization of the application running on the communication gateway of a wireless BAN and accurate modeling of the microarchitectural details of the gateway processor can lead to significantly better resource usage and power savings. In particular, we propose a method for deriving the optimal order in which the different sensors feeding the gateway processor should be sampled, to maximize cache reuse. In addition, we also investigate the effects on cache reuse of different memory layouts of the code processing the different sensor data. The joint optimization of code layout and the order in which the different sensors should be sampled--in order to maximize code cache reuse--turns out to be a difficult combinatorial optimization problem. But our experiments show that optimizing the sampling order of the sensors has a much larger influence on cache reuse, compared to the effects that different code layouts have. Based on this, we also propose a heuristic that obtains near-optimal solutions in jointly optimizing both code layout as well the sensor sampling order. Our case study using a faint fall detection application--from the geriatric care domain--which is fed by a number of smart sensors to detect physiological and physical gait signals of a patient show very attractive power consumption in the underlying processor. Alternatively, our method can be used to improve the sampling frequency of the sensors, leading to higher reliability and better response time of the application.

A filter-and-fan approach to the job shop scheduling problem

The job shop scheduling problem (JSSP) is a notoriously difficult problem in combinatorial optimization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new heuristic algorithm for the JSSP that effectively combines the classical shifting bottleneck procedure (SBP) with a dynamic and adaptive neighborhood search procedure. Our new search method, based on a filter-and-fan (F&F) procedure, uses the SBP as a subroutine to generate a starting solution and to enhance the best schedules produced. The F&F approach is a local search procedure that generates compound moves by a strategically abbreviated form of tree search. Computational results carried out on a standard set of 43 benchmark problems show that our F&F algorithm performs more robustly and effectively than a number of leading metaheuristic algorithms and rivals the best of these algorithms.

On the sensitivity of the control structure selection problem in large-scale multivariable systems

In complex manufacturing processes, control structure selection poses a serious challenge for the control engineer. Such systems are typically interacting, large scale and may comprise many loops. Choosing a correct control structure is not often a straightforward process, especially, if as it often is the case, one is faced with the challenge of having to integrate additional proposed control systems into an already existing plant-wide control system. Current approaches to the control structure selection problem rely either on the control engineer's experience and intuition or on solutions of difficult and non-convex combinatorial optimisation problems. A set of new computationally inexpensive graphical analysis tools based on basis pursuit regularisation methods are presented that provide significant insight into the control structure selection problem. The tools presented build a centralised controller by incrementally adding one loop at a time to the structure of a baseline decentralised controller. There are no constraints as to where new loops can be added which results in a rich class of sparse controllers. With the aid of an example, it is shown that how these tools are used directly to assess the local structural sensitivity of the problem and to identify key and important connections within the controller.

Certainty closure

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimization problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterizing the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modeling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.

Relaxed maximum a posteriori fault identification

We consider the problem of estimating a pattern of faults, represented as a binary vector, from a set of measurements. The measurements can be noise corrupted real values, or quantized versions of noise corrupted signals, including even 1-bit (sign) measurements. Maximum a posteriori probability (MAP) estimation of the fault pattern leads to a difficult combinatorial optimization problem, so we propose a variation in which an approximate maximum a posteriori probability estimate is found instead, by solving a convex relaxation of the original problem, followed by rounding and simple local optimization. Our method is extremely efficient, and scales to very large problems, involving thousands (or more) of possible faults and measurements. Using synthetic examples, we show that the method performs extremely well, both in identifying the true fault pattern, and in identifying an ambiguity group, i.e., a set of alternate fault patterns that explain the observed measurements almost as well as our estimate.

Routing optimization heuristics algorithms for urban solid waste transportation management

During the last decade, metaheuristics have become increasingly popular for effectively confronting difficult combinatorial optimization problems. In the present paper, two individual meatheuristic algorithmic solutions, the ArcGIS Network Analyst and the Ant Colony System (ACS) algorithm, are introduced, implemented and discussed for the identification of optimal routes in the case of Municipal Solid Waste (MSW) collection. Both proposed applications are based on a geo-referenced spatial database supported by a Geographic Information System (GIS). GIS are increasingly becoming a central element for coordinating, planning and managing transportation systems, and so in collaboration with combinatorial optimization techniques they can be used to improve aspects of transit planning in urban regions. Here, the GIS takes into account all the required parameters for the MSW collection (i.e. positions of waste bins, road network and the related traffic, truck capacities, etc) and its desktop users are able to model realistic network conditions and scenarios. In this case, the simulation consists of scenarios of visiting varied waste collection spots in the Municipality of Athens (MoA). The user, in both applications, is able to define or modify all the required dynamic factors for the creation of an initial scenario, and by modifying these particular parameters, alternative scenarios can be generated. Finally, the optimal solution is estimated by each routing optimization algorithm, followed by a comparison between these two algorithmic approaches on the newly designed collection routes. Furthermore, the proposed interactive design of both approaches has potential application in many other environmental planning and management problems.

A hybrid approach combining an improved genetic algorithm and optimization strategies for the asymmetric traveling salesman problem

The asymmetric traveling salesman problem (ATSP) appears in various applications. Although there are several heuristic approaches to its solution, the problem is still a difficult combinatorial optimization problem. This work proposes a novel hybrid approach specialized for the ATSP. The proposed method incorporates an improved genetic algorithm (IGA) and some optimization strategies that contribute to its effectiveness. In the IGA, both the crossover operation and the mutation operation are improved by selecting the optimum from a set of solutions. Three strategies: immigration, local optimization and global optimization are established based on several empirical optimization strategies to improve the evolution of the IGA. Computational experiments are conducted on 16 ATSP instances available in the TSPLIB (traveling salesman problem library). The comparative study shows that our proposed approach outperforms several other published algorithms.

PedMine—a simulated annealing algorithm to identify maximally unrelated individuals in population isolates

In family-based genetic studies, it is often useful to identify a subset of unrelated individuals. When such studies are conducted in population isolates, however, most if not all individuals are often detectably related to each other. To identify a set of maximally unrelated (or equivalently, minimally related) individuals, we have implemented simulated annealing, a general-purpose algorithm for solving difficult combinatorial optimization problems. We illustrate our method on data from a genetic study in the Old Order Amish of Lancaster County, Pennsylvania, a population isolate derived from a modest number of founders. Given one or more pedigrees, our program automatically and rapidly extracts a fixed number of maximally unrelated individuals. http://www.hg.med.umich.edu/labs/douglaslab/software.html (version 1.0.0).

Soft computing and cooperative strategies for optimization

In the last years, metaheuristics have emerged as powerful algorithmic approaches which have been applied with great success to difficult combinatorial optimization problems. However, this does not mean that metaheuristics can be applied blindly to any new problem. In this contribution we showed how the most basic ingredients of Soft Computing, namely fuzzy sets and fuzzy rules, are used in the context of a simple metaheuristic and a cooperative strategy based on it, to obtain successful results for the p-median problem.

Darwinian approach for dynamic spectrum allocation in next generation systems

The authors present the use of a genetic algorithm (GA) model as a solution approach to the dynamic spectrum allocation (DSA) problem considered as a difficult combinatorial optimisation problem. The proposed multi-objective GA model enhances overall spectral efficiency of the network, while optimising its own spectrum utilisation to generate accessible spectrum opportunities for other radio technologies. A novel two-dimensional encoding technique is defined to represent solutions in the problem domain and the technique enables significantly shorter convergence times. A simulation tool has been developed to model the GA-based DSA and to compare the new scheme with the conventional fixed spectrum allocation (FSA) scheme under both uniform and non-uniform traffic distributions. The proposed scheme significantly outperformed the FSA scheme both in terms of spectral efficiency gain and spectral utilisation.

Combination of Multiple Ant Colony System and Simulated Annealing for the Multidepot Vehicle-Routing Problem with Time Windows

The vehicle-routing problem (VRP) is an important management problem in the field of physical distribution and logistics. In practice, the logistics system usually includes more than one depot, and the start of the service at each customer must be within a given time window. Hence, the multidepot vehicle-routing problem with time windows (MDVRPTW) is an important variant of the VRP. The MDVRPTW is a difficult combinatorial optimization problem due to the many complex constraints involved. The research presented in this paper proposes a multiple ant colony system (MACS) to solve the problem. In addition, two hybrid algorithms, which combine the strengths of MACS and simulated annealing, are developed to improve solution quality. The performance of the proposed algorithms is tested on several benchmark instances and compared with that of other algorithms in the literature. The results indicate that the proposed algorithms are effective in solving the MDVRPTW, and six new best solutions are found.

Solving the resource-constrained project scheduling problem using filter-and-fan approach

The resource-constrained project scheduling problem is a notoriously difficult problem in combinatorial optimization for which extensive investigation has been devoted to the development of efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new heuristic algorithm for this problem based on filter-and-fan method incorporated with a local search, exploring in the defined neighborhood space. In the algorithm, the local search is used to generate a starting solution as well as to re-optimize the best schedules produced by the filter-and-fan method. The filter-and-fan is itself a local search procedure that generates compound moves in a tree search fashion. Computational results applied on a standard set of 2040 benchmark problems from the literature demonstrate the effectiveness of the approach.

A Hybrid Genetic Algorithm with Boltzmann Convergence Properties

Stochastic global search algorithms such as genetic algorithms are used to attack difficult combinatorial optimization problems. However, genetic algorithms suffer from the lack of a convergence proof. This means that it is difficult to establish reliable algorithm braking criteria without extensive a priori knowledge of the solution space. The hybrid genetic algorithm presented here combines a genetic algorithm with simulated annealing in order to overcome the algorithm convergence problem. The genetic algorithm runs inside the simulated annealing algorithm and provides convergence via a Boltzmann cooling process. The hybrid algorithm was used successfully to solve a classical 30-city traveling salesman problem; it consistently outperformed both a conventional genetic algorithm and a conventional simulated annealing algorithm.

Network Reliability Optimization via the Cross-Entropy Method

Consider a network of unreliable links, each of which comes with a certain price, and reliability. Given a fixed budget, which links should be purchased in order to maximize the system's reliability? We introduce a new approach, based on the cross-entropy method, which can deal effectively with the constraints, and noise introduced when estimating the reliabilities via simulation, in this difficult combinatorial optimization problem. Numerical results demonstrate the effectiveness of the proposed technique

Dynamic vehicle routing using genetic algorithms

Many difficult combinatorial optimization problems have been modeled as static problems. However, in practice, many problems are dynamic and changing, while some decisions have to be made before all the design data are known. For example, in the Dynamic Vehicle Routing Problem (DVRP), new customer orders appear over time, and new routes must be reconfigured while executing the current solution. Montemanni et al. [1] considered a DVRP as an extension to the standard vehicle routing problem (VRP) by decomposing a DVRP as a sequence of static VRPs, and then solving them with an ant colony system (ACS) algorithm. This paper presents a genetic algorithm (GA) methodology for providing solutions for the DVRP model employed in [1]. The effectiveness of the proposed GA is evaluated using a set of benchmarks found in the literature. Compared with a tabu search approach implemented herein and the aforementioned ACS, the proposed GA methodology performs better in minimizing travel costs.