Genetic algorithms for solving the discrete ordered median problem

In this paper we present two new heuristic approaches to solve the Discrete Ordered Median Problem (DOMP). Described heuristic methods, named HGA1 and HGA2 are based on a hybrid of genetic algorithms (GA) and a generalization of the well-known Fast Interchange heuristic (GFI). In order to investigate the effect of encoding on GA performance, two different encoding schemes are implemented: binary encoding in HGA1, and integer representation in HGA2. If binary encoding is used (HGA1), new genetic operators that keep the feasibility of individuals are proposed. Integer representation keeps the individuals feasible by default, so HGA2 uses slightly modified standard genetic operators. In both methods, caching GA technique was integrated with the GFI heuristic to improve computational performance. The algorithms are tested on standard ORLIB p-median instances with up to 900 nodes. The obtained results are also compared with the results of existing methods for solving DOMP in order to assess their merits.     

Constraint Handling in Genetic Algorithms: The Set Partitioning Problem

In this paper we present a genetic algorithm-based heuristic for solving the set partitioning problem (SPP). The SPP is an important combinatorial optimisation problem used by many airlines as a mathematical model for flight crew scheduling.A key feature of the SPP is that it is a highly constrained problem, all constraints being equalities. New genetic algorithm (GA) components: separate fitness and unfitness scores, adaptive mutation, matching selection and ranking replacement, are introduced to enable a GA to effectively handle such constraints. These components are generalisable to any GA for constrained problems.We present a steady-state GA in conjunction with a specialised heuristic improvement operator for solving the SPP. The performance of our algorithm is evaluated on a large set of real-world problems. Computational results show that the genetic algorithm-based heuristic is capable of producing high-quality solutions.     

An empirical study of hybrid genetic algorithms for the set covering problem

The purpose of this paper is to explore the computational performance of several hybrid algorithms that are extensions of a basic genetic algorithm (GA) approach for solving the set covering problem (SCP). We start by making several enhancements to a GA for the SCP that was proposed by Beasley and Chu. Next, several hybrid solution approaches are introduced that combine the GA with various local neighbourhood search approaches, with a form of the greedy randomized adaptive search procedure, and with an estimation of distribution algorithms approach. Using Beasley's library of SCPs extensive computational results are generated for the hybrid solution approaches defined in this paper. Statistical analyses of the results are performed.     

Genetic algorithms for a supply management problem: MIP-recombination vs greedy decoder

Two variants of genetic algorithm (GA) for solving the Supply Management Problem with Lower-Bounded Demands (SMPLD) are proposed and experimentally tested. The SMPLD problem consists in planning the shipments from a set of suppliers to a set of customers minimizing the total cost, given lower and upper bounds on shipment sizes, lower-bounded consumption and linear costs for opened deliveries. The first variant of GA uses the standard binary representation of solutions and a new recombination operator based on the mixed integer programming (MIP) techniques. The second GA is based on the permutation representation and a greedy decoder. Our experiments indicate that the GA with MIP-recombination compares favorably to the other GA and to the MIP-solver CPLEX 9.0 in terms of cost of obtained solutions. The GA based on greedy decoder is shown to be the most robust in finding feasible solutions.     

The equilibrium generalized assignment problem and genetic algorithm

The well-known generalized assignment problem (GAP) is to minimize the costs of assigning n jobs to m capacity constrained agents (or machines) such that each job is assigned to exactly one agent. This problem is known to be NP-hard and it is hard from a computational point of view as well. In this paper, follows from practical point of view in real systems, the GAP is extended to the equilibrium generalized assignment problem (EGAP) and the equilibrium constrained generalized assignment problem (ECGAP). A heuristic equilibrium strategy based genetic algorithm (GA) is designed for solving the proposed EGAP. Finally, to verify the computational efficiency of the designed GA, some numerical experiments are performed on some known benchmarks. The test results show that the designed GA is very valid for solving EGAP.     

A new crossover approach for solving the multiple travelling salesmen problem using genetic algorithms

This paper proposes a new crossover operator called two-part chromosome crossover (TCX) for solving the multiple travelling salesmen problem (MTSP) using a genetic algorithm (GA) for near-optimal solutions. We adopt the two-part chromosome representation technique which has been proven to minimise the size of the problem search space. Nevertheless, the existing crossover method for the two-part chromosome representation has two limitations. Firstly, it has extremely limited diversity in the second part of the chromosome, which greatly restricts the search ability of the GA. Secondly, the existing crossover approach tends to break useful building blocks in the first part of the chromosome, which reduces the GA’s effectiveness and solution quality. Therefore, in order to improve the GA search performance with the two-part chromosome representation, we propose TCX to overcome these two limitations and improve solution quality. Moreover, we evaluate and compare the proposed TCX with three different crossover methods for two MTSP objective functions, namely, minimising total travel distance and minimising longest tour. The experimental results show that TCX can improve the solution quality of the GA compared to three existing crossover approaches.     

Single row facility layout problem using a permutation-based genetic algorithm

In this paper, a permutation-based genetic algorithm (GA) is applied to the NP-hard problem of arranging a number of facilities on a line with minimum cost, known as the single row facility layout problem (SRFLP). The GA individuals are obtained by using some rule-based as well as random permutations of the facilities, which are then improved towards the optimum by means of specially designed crossover and mutation operators. Such schemes led the GA to handle the SRFLP as an unconstrained optimization problem. In the computational experiments carried out with large-size instances of sizes from 60 to 80, available in the literature, the proposed GA improved several previously known best solutions.     

Extensive Testing of a Hybrid Genetic Algorithm for Solving Quadratic Assignment Problems

A robust search algorithm should ideally exhibit reasonable performance on a diverse and varied set of problems. In an earlier paper Lim et al. (Computational Optimization and Applications, vol. 15, no. 3, 2000), we outlined a class of hybrid genetic algorithms based on the k-gene exchange local search for solving the quadratic assignment problem (QAP). We follow up on our development of the algorithms by reporting in this paper the results of comprehensive testing of the hybrid genetic algorithms (GA) in solving QAP. Over a hundred instances of QAP benchmarks were tested using a standard set of parameters setting and the results are presented along with the results obtained using simple GA for comparisons. Results of our testing on all the benchmarks show that the hybrid GA can obtain good quality solutions of within 2.5% above the best-known solution for 98% of the instances of QAP benchmarks tested. The computation time is also reasonable. For all the instances tested, all except for one require computation time not exceeding one hour. The results will serve as a useful baseline for performance comparison against other algorithms using the QAP benchmarks as a basis for testing.     

The use of genetic algorithms to solve the economic lot size scheduling problem

The purpose of this paper is to investigate the use genetic algorithms (GAs) for solving the Economic Lot Size Scheduling Problem (ELSP). The ELSP is formulated using the Basic Period (BP) approach which results in a problem having one continuous decision variable and a number of integer decision variables equal to the number of products being produced. This formulation is ideally suited for using GAs. The GA is tested on Bomberger's classical problem. The resulting solutions were better than those obtained using an iterative dynamic programming (DP) approach. The total cost of GA solutions to the problem with utilization up to 65% were within 3.4% of the lower bound. The GA also performed well for higher utilization yielding solutions within 13.87% of the lower bound for utilization up to 86%. The GA was tested on a 30-item problem and good solutions were obtained. The results of the GA under different binary representations, crossover methods, and initialization methods are compared to identify the best settings. The results indicate that for this particular problem, binary representation works better than Gray coding, 2-point crossover is best, and an infeasible starting population is better than feasible.     

Solving large-scale maximum expected covering location problems by genetic algorithms: A comparative study

This paper compares the performance of genetic algorithms (GAs) on large-scale maximum expected coverage problems to other heuristic approaches. We focus our attention on a particular formulation with a nonlinear objective function to be optimized over a convex set. The solutions obtained by the best genetic algorithm are compared to Daskin's heuristic and the optimal or best solutions obtained by solving the corresponding integer linear programming (ILP) problems. We show that at least one of the GAs yields optimal or near-optimal solutions in a reasonable amount of time.     

Performance of a genetic algorithm for the graph partitioning problem

The performance of the genetic algorithm (GA) for the graph partitioning problem (GPP) is investigated by comparison with standard heuristics on well-known benchmark graphs. In general, there is a case where a practical performance of a conventional genetic approach, which performs only simple operations without a local search strategy, is not sufficient. However, it is known that a combination of GA and local search can produce better solutions. From this practice, we incorporate a simple local search algorithm into the GA. In particular, the search ability of the GA is compared with standard heuristics such as multistart local search and simulated annealing, which use the same neighborhood structure of the simple local search, for solving the GPP. Experimental results show that the GA performs better than its competitors.     

Generating ε-efficient solutions in multiobjective programming

Scalarization approaches to purposely generating ε-efficient solutions of multiobjective programs are investigated and a generic procedure for computing these solutions is proposed and illustrated with an example. Real-life decision making situations in which the solutions are of significance are described.     

Solving the Dial-a-Ride problem using genetic algorithms

In the Dial-a-Ride problem (DARP), customers request transportation from an operator. A request consists of a specified pickup location and destination location along with a desired departure or arrival time and capacity demand. The aim of DARP is to minimize transportation cost while satisfying customer service level constraints (Quality of Service). In this paper, we present a genetic algorithm (GA) for solving the DARP. The algorithm is based on the classical cluster-first, route-second approach, where it alternates between assigning customers to vehicles using a GA and solving independent routing problems for the vehicles using a routing heuristic. The algorithm is implemented in Java and tested on publicly available data sets. The new solution method has achieved solutions comparable with the current state-of-the-art methods.     

A memetic algorithm approach for solving the multidimensional multi-way number partitioning problem

In this paper, we describe a generalization of the multidimensional two-way number partitioning problem (MDTWNPP) where a set of vectors has to be partitioned into p sets (parts) such that the sums per every coordinate should be exactly or approximately equal. We will call this generalization the multidimensional multi-way number partitioning problem (MDMWNPP). Also, an efficient memetic algorithm (MA) heuristic is developed to solve the multidimensional multi-way number partitioning problem obtained by combining a genetic algorithm (GA) with a powerful local search (LS) procedure. The performances of our memetic algorithm have been compared with the existing numerical results obtained by CPLEX based on an integer linear programming formulation of the problem. The solution reveals that our proposed methodology performs very well in terms of both quality of the solutions obtained and the computational time compared with the previous method of solving the multidimensional two-way number partitioning problem.     

A genetic algorithm approach to the product line design problem using the seller's return criterion: An extensive comparative computational study

In this paper we deal with the product line design problem employing the seller's marginal return criterion. Because this problem is NP-Hard, many researchers proposed heuristic methods. We present a genetic algorithm (GA) based heuristic for solving the above problem. In the implementation, the GA is initialized in two different ways. In the first way, the GA is initialized with a random population. We call this algorithm GA1. In the second way, the solution of the beam search (BS) method is included in the first population of the GA. We call this algorithm GA2. We compare GA1, a recently developed BS method and GA2 on randomly generated problems. GA1 seems to be substantially better than the BS method in terms of CPU time. Also, the solutions found by GA1 are substantially better than those found by the BS method in comparable times. In many cases, GA2 improves the solution found by the BS method. Consequently, it is a good second step of the BS method.     

Nature inspired genetic algorithms for hard packing problems

This paper presents two novel genetic algorithms (GAs) for hard industrially relevant packing problems. The design of both algorithms is inspired by aspects of molecular genetics, in particular, the modular exon-intron structure of eukaryotic genes. Two representative packing problems are used to test the utility of the proposed approach: the bin packing problem (BPP) and the multiple knapsack problem (MKP). The algorithm for the BPP, the exon shuffling GA (ESGA), is a steady-state GA with a sophisticated crossover operator that makes maximum use of the principle of natural selection to evolve feasible solutions with no explicit verification of constraint violations. The second algorithm, the Exonic GA (ExGA), implements an RNA inspired adaptive repair function necessary for the highly constrained MKP. Three different variants of this algorithm are presented and compared, which evolve a partial ordering of items using a segmented encoding that is utilised in the repair of infeasible solutions. All algorithms are tested on a range of benchmark problems, and the results indicate a very high degree of accuracy and reliability compared to other approaches in the literature.     

A hybrid genetic algorithm with dominance properties for single machine scheduling with dependent penalties

In this paper, a hybrid genetic algorithm is developed to solve the single machine scheduling problem with the objective to minimize the weighted sum of earliness and tardiness costs. First, dominance properties of (the conditions on) the optimal schedule are developed based on the switching of two adjacent jobs i and j. These dominance properties are only necessary conditions and not sufficient conditions for any given schedule to be optimal. Therefore, these dominance properties are further embedded in the genetic algorithm and we call it genetic algorithm with dominance properties (GADP). This GADP is a hybrid genetic algorithm. The initial populations of schedules in the genetic algorithm are generated using these dominance properties. GA can further improve the performance of these initial solutions after the evolving procedures. The performances of hybrid genetic algorithm (GADP) have been compared with simple genetic algorithm (SGA) using benchmark instances. It is shown that this hybrid genetic algorithm (GADP) performs very well when compared with DP or SGA alone.     

The efficiency of hybrid mutation genetic algorithm for the travelling salesman problem

In this paper, we present an efficient genetic algorithm (GA) for solving the travelling salesman problem (TSP) as a combinatorial optimization problem. In our computational model, we propose a complete subtour exchange crossover that does not break as some good subtours as possible, because the good subtours are worth preserving for descendants. Generally speaking, global search GA is considered to be better approaches than local searches. However, it is necessary to strengthen the ability of local search as well as global ones in order to increase a GA total efficiency. In this study, our GA applies a stochastic hill climbing procedure in the mutation process of the GA. Experimental results showed that the GA leads good convergence as high as 99 percent even for 500 cities TSP.     

Multiagent visual area coverage using a new genetic algorithm selection scheme

Using genetic algorithms (GA) for solving NP-hard problems is becoming more and more frequent. This paper presents a use of GA with a new selection approach called the queen GA. The main idea is not to select both parents from the entire population, but to create a subgroup of better solutions (the queen cohort), and to use at least one of its members in each performed crossover. We demonstrate the use of the queen GA for the problem of repositioning observers across a polygonal area with obstacles in order to maximize the visual area coverage for a given time horizon. The queen GA gives superior results over a GA with different selection methods (i.e. proportion, ranking and tournament) at the 0.01 significance level. These comparative results were duplicated when elitism was included.     

Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities

Q-conditional symmetries (nonclassical symmetries) for the general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction-diffusion systems admitting such symmetry is derived. The results obtained for the reaction-diffusion systems are compared with those for the scalar reaction-diffusion equations. The symmetries found for reducing reaction-diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction-diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.