Solving multiobjective,multiconstraint knapsack problems using mathematical programming and evolutionary algorithms

In this paper, we solve instances of the multiobjective multiconstraint (or multidimensional) knapsack problem (MOMCKP) from the literature, with three objective functions and three constraints. We use exact as well as approximate algorithms. The exact algorithm is a properly modified version of the multicriteria branch and bound (MCBB) algorithm, which is further customized by suitable heuristics. Three branching heuristics and a more general purpose composite branching and construction heuristic are devised. Comparison is made to the published results from another exact algorithm, the adaptive ε-constraint method [Laumanns, M., Thiele, L., Zitzler, E., 2006. An efficient, adaptive parameter variation scheme for Metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169, 932–942], using the same data sets. Furthermore, the same problems are solved using standard multiobjective evolutionary algorithms (MOEA), namely, the SPEA2 and the NSGAII. The results from the exact case show that the branching heuristics greatly improve the performance of the MCBB algorithm, which becomes faster than the adaptive ε -constraint. Regarding the performance of the MOEA algorithms in the specific problems, SPEA2 outperforms NSGAII in the degree of approximation of the Pareto front, as measured by the coverage metric (especially for the largest instance).     

Multicriteria scheduling optimization using an elitist multiobjective population heuristic: the h-NSDE algorithm

In today’s manufacturing industry more than one performance criteria are considered for optimization to various degrees simultaneously. To deal with such hard competitive environments it is essential to develop appropriate multicriteria scheduling approaches. In this paper consideration is given to the problem of scheduling n independent jobs on a single machine with due dates and objective to simultaneously minimize three performance criteria namely, total weighted tardiness (TWT), maximum tardiness and maximum earliness. In the single machine scheduling literature no previous studies have been performed on test problems examining these criteria simultaneously. After positioning the problem within the relevant research field, we present a new heuristic algorithm for its solution. The developed algorithm termed the hybrid non-dominated sorting differential evolution (h-NSDE) is an extension of the author’s previous algorithm for the single-machine mono-criterion TWT problem. h-NSDE is devoted to the search for Pareto-optimal solutions. To enable the decision maker for evaluating a greater number of alternative non-dominated solutions, three multiobjective optimization approaches have been implemented and tested within the context of h-NSDE: including a weighted-sum based approach, a fuzzy-measures based approach which takes into account the interaction among the criteria as well as a Pareto-based approach. Experiments conducted on existing data set benchmarks problems show the effect of these approaches on the performance of the h-NSDE algorithm. Moreover, comparative results between h-NSDE and some of the most popular multiobjective metaheuristics including SPEA2 and NSGA-II show clear superiority for h-NSDE in terms of both solution quality and solution diversity.     

Approximation, integration and differentiation of time functions using a set of orthogonal hybrid functions (HF) and their application to solution of first order differential equations

Differential equations of different types and orders are of utmost importance for mathematical modeling of control system problems. State variable method uses the concept of expressing n number of first order differential equations in vector matrix form to model and analyze/synthesize control systems.The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of sample-and-hold functions (SHF) and triangular functions (TF). This HF set is used to approximate a time function in a piecewise linear manner with the mean integral square error (MISE) much less than block pulse function based approximation which always provides staircase solutions.The operational matrices for integration and differentiation in HF domain are also derived and employed for solving non-homogeneous and homogeneous differential equations of the first order as well as state equations. The results are compared with exact solutions, the 4th order Runge-Kutta method and its further improved versions proposed by Simos [6]. The presented HF domain theory is well supported by a few illustrations.     

A unified exact method for solving different classes of vehicle routing problems

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.      

Minimizing shifts for personnel task scheduling problems: A three-phase algorithm

The personnel task scheduling problem is a subject of commercial interest which has been investigated since the 1950s. This paper proposes an effective and efficient three-phase algorithm for solving the shift minimization personnel task scheduling problem (SMPTSP). To illustrate the increased efficacy of the proposed algorithm over an existing algorithm, computational experiments are performed on a test problem set with characteristics motivated by employee scheduling applications. Experimental results show that the proposed algorithm outperforms the existing algorithm in terms of providing optimal solutions, improving upon most of the best-known solutions and revealing high-quality feasible solutions for those unsolved test instances in the literature.     

Integrating heuristic information into exact methods: The case of the vertex p-centre problem

We solve the vertex p-centre problem optimally using an exact method that considers both upper and lower bounds as part of its search engine. Tight upper bounds are generated quickly via an efficient three-level heuristic, which are then used to derive potential ‘lower bounds’ accordingly. These two pieces of information when used together make our chosen exact method more efficient at obtaining optimal solutions relatively quickly. The proposed implementation produced excellent results when tested on the OR Library data set. This integrated approach can be adopted for those exact methods that consider both upper and lower bounds within their search engine and hence provide a wider spectrum of applicability in other hard combinatorial problems.     

A new method for mean-variance portfolio optimization with cardinality constraints

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM) model, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model based on a reformulation as a Standard Quadratic Program, on a new lower bound that we establish, and on other recent theoretical and computational results for such problem. These results lead to an exact algorithm for solving the LAM model for small size problems. For larger problems, such algorithm can be relaxed to an efficient and accurate heuristic procedure that is able to find the optimal or the best-known solutions for problems based on some standard financial data sets that are used by several other authors. We also test our method on five new data sets involving real-world capital market indices from major stock markets. We compare our results with those of CPLEX and with those obtained with very recent heuristic approaches in order to illustrate the effectiveness of our method in terms of solution quality and of computation time. All our data sets and results are publicly available for use by other researchers.     

A genetic algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The algorithm is adapted from a genetic algorithm which has been successfully used on set covering problems, but instead of genetically improving a set of feasible solutions it tries to genetically restore feasibility to a set of near-optimal ones. Thus it may be regarded as operating in a dual sense to the more familiar genetic approach. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach.     

Approximated algorithms for the minimum dilation triangulation problem

The complexity status of the minimum dilation triangulation (MDT) problem for a general point set is unknown. Therefore, we focus on the development of approximated algorithms to find high quality triangulations of minimum dilation. For an initial approach, we design a greedy strategy able to obtain approximate solutions to the optimal ones in a simple way. We also propose an operator to generate the neighborhood which is used in different algorithms: Local Search, Iterated Local Search, and Simulated Annealing. Besides, we present an algorithm called Random Local Search where good and bad solutions are accepted using the previous mentioned operator. For the experimental study we have created a set of problem instances since no reference to benchmarks for these problems were found in the literature. We use the sequential parameter optimization toolbox for tuning the parameters of the SA algorithm. We compare our results with those obtained by the OV-MDT algorithm that uses the obstacle value to sort the edges in the constructive process. This is the only available algorithm found in the literature. Through the experimental evaluation and statistical analysis, we assess the performance of the proposed algorithms using this operator.     

A deterministic tabu search algorithm for the fleet size and mix vehicle routing problem

The fleet size and mix vehicle routing problem consists of defining the type, the number of vehicles of each type, as well as the order in which to serve the customers with each vehicle when a company has to distribute goods to a set of customers geographically spread, with the objective of minimizing the total costs. In this paper, a heuristic algorithm based on tabu search is proposed and tested on several benchmark instances. The computational results show that the proposed algorithm produces high quality results within a reasonable computing time. Some new best solutions are reported for a set of test problems used in the literature.     

A mathematical formulation and complexity considerations for the blocks relocation problem

The blocks relocation problem (BRP) may be defined as follows: given a set of homogeneous blocks stored in a two-dimensional stock, which relocations are necessary to retrieve the blocks from the stock in a predefined order while minimizing the number of those relocations? In this paper, we first prove NP-hardness of the BRP as well as a special case, closing open research questions. Moreover, we propose different solution approaches. First, a mathematical model is presented that provides optimal solutions to the general BRP in cases where instances are small. To overcome such limitation, some realistic assumption taken from the literature is introduced, leading to the definition of a binary linear programming model. In terms of computational time, this approach is reasonably fast to be used to solve medium-sized instances. In addition, we propose a simple heuristic based upon a set of relocation rules. This heuristic is used to generate “good” quality solutions for larger instances in very short computational time, and, consequently, is proposed for tackling problem instances where solutions are required (almost) immediately. Solution quality of the heuristic is measured against optimal solutions obtained using a state-of-the-art commercial solver and both of them are compared with reference results from literature.     

On spanning tree problems with multiple objectives

We investigate two versions of multiple objective minimum spanning tree problems defined on a network with vectorial weights. First, we want to minimize the maximum ofQ linear objective functions taken over the set of all spanning trees (max-linear spanning tree problem, ML-ST). Secondly, we look for efficient spanning trees (multi-criteria spanning tree problem, MC-ST).Problem ML-ST is shown to be NP-complete. An exact algorithm which is based on ranking is presented. The procedure can also be used as an approximation scheme. For solving the bicriterion MC-ST, which in the worst case may have an exponential number of efficient trees, a two-phase procedure is presented. Based on the computation of extremal efficient spanning trees we use neighbourhood search to determine a sequence of solutions with the property that the distance between two consecutive solutions is less than a given accuracy.Partially supported by Deutsche Forschungsgemeinschaft and HCº Contract no. ERBCHRXCT 930087.Partially supported by Alexander von Humboldt-Stiftung.     

Verified computation of solutions for obstacle problems with guaranteed L∞ error bound

In this paper, we consider a numerical enclosure method with guaranteed L^∞ error bound for the solutions of obstacle problems. Using the finite-element approximations and the explicit a priori error estimates for obstacle problems, we present an effective verification procedure that automatically generates on a computer a set which includes the exact solution. A particular emphasis is that our method needs no assumption of the existence of the solution of the original obstacle problems, but it follows as the result of computation itself. A numerical example for an obstacle problem is presented.     

An evolutionary heuristic for quadratic 0–1 programming

In this paper we present a heuristic algorithm for the well-known Unconstrained Quadratic 0–1 Programming Problem. The approach is based on combining solutions in a genetic paradigm and incorporates intensification algorithms used to improve solutions and speed up the method. Extensive computational experiments on instances with up to 500 variables are presented and we compare our approach both with powerful heuristic and exact algorithms from the literature establishing the effectiveness of the method in terms of solutions quality and computing time.     

A self-adaptive global best harmony search algorithm for continuous optimization problems

This paper presents a self-adaptive global best harmony search (SGHS) algorithm for solving continuous optimization problems. In the proposed SGHS algorithm, a new improvisation scheme is developed so that the good information captured in the current global best solution can be well utilized to generate new harmonies. The harmony memory consideration rate (HMCR) and pitch adjustment rate (PAR) are dynamically adapted by the learning mechanisms proposed. The distance bandwidth (BW) is dynamically adjusted to favor exploration in the early stages and exploitation during the final stages of the search process. Extensive computational simulations and comparisons are carried out by employing a set of 16 benchmark problems from literature. The computational results show that the proposed SGHS algorithm is more effective in finding better solutions than the state-of-the-art harmony search (HS) variants.     

Multiple criteria districting problems

Districting problems are of high importance in many different fields. Multiple criteria models seem a more adequate representation of districting problems in real-world situations. Real-life decision situations are by their very nature multidimensional. This paper deals with the problem of partitioning a territory into “homogeneous” zones. Each zone is composed of a set of elementary territorial units. A district map is formed by partitioning the set of elementary units into connected zones without inclusions. When multiple criteria are considered, the problem of enumerating all the efficient solutions for such a model is known as being NP-hard, which is why we decided to avoid using exact methods to solve large-size instances. In this paper, we propose a new method to approximate the Pareto front based on an evolutionary algorithm with local search. The algorithm presents a new solution representation and the crossover/mutation operators. Its main features are the following: it deals with multiple criteria; it allows to solve large-size instances in a reasonable CPU time and generates high quality solutions. The algorithm was applied to a real-world problem, that of the Paris region public transportation. Results will be used for a discussion about the reform of its current pricing system.     

The seriation problem and the travelling salesman problem

The relationship between two combinatorial problems is explained in this paper. The first problem is the well known travelling salesman problem. The other problem, the seriation problem, is frequently encountered in archeology and has applications in genetics and in graph theory. Recent efforts to solve the seriation problem have produced numerous statistical and heuristic solutions. However algorithms that produce global optima remain very unsatisfactory. A new exact algorithm based on the interdependence between the travelling salesman problem and the seriation problem is presented here. It can handle larger problems than any of the existing exact algorithms.     

A reactive tabu search for the vehicle routing problem

The classical vehicle routing problem (VRP) involves determining a fleet of homogeneous size vehicles and designing an associated set of routes that minimizes the total cost. Our tabu search (TS) algorithm to solve the VRP is based on reactive tabu search (RTS) with a new escape mechanism, which manipulates different neighbourhood schemes in a very sophisticated way in order to get a balanced intensification and diversification continuously during the search process. We compare our algorithm with the best methods in the literature using different data sets and report results including new best known solutions for several well-known benchmark problems.     

A clique algorithm for standard quadratic programming

A standard Quadratic Programming problem (StQP) consists in minimizing a (nonconvex) quadratic form over the standard simplex. For solving a StQP we present an exact and a heuristic algorithm, that are based on new theoretical results for quadratic and convex optimization problems. With these results a StQP is reduced to a constrained nonlinear minimum weight clique problem in an associated graph. Such a clique problem, which does not seem to have been studied before, is then solved with an exact and a heuristic algorithm. Some computational experience shows that our algorithms are able to solve StQP problems of at least one order of magnitude larger than those reported in the literature.     

2+1维广义浅水波方程的类孤子解与周期解

该文基于一个Riccati方程组，提出了一个新的广义投影Ric cati展开法，该方法直接简单并能构造非线性微分方程更多的新的解析解。利用该算法研究了(2+1)维广义浅水波方程，并求得了许多新的精确解，包括类孤子解和周期解。该算法也能应用到其它非线性微分方程中。