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.     

Using genetic algorithms for the coordinated scheduling problem of a batching machine and two-stage transportation

This paper considers a coordinated scheduling problem. For the first-stage transportation there is a crane available to transport the product from the warehouse to a batching machine. For the second-stage transportation there is a vehicle available to deliver the completed jobs from the machine shop floor to the customer. The coordinated scheduling problem of production and transportation deals with sequencing the transportation of the jobs and combining them into batches to be processed. The problem of minimizing the sum of the makespan and the total setup cost was proven by Tang and Gong [1] to be strongly NP-hard. This paper proposes two genetic algorithm (GA) approaches for this scheduling problem, with different result representations. The experimental results demonstrate that a regular GA and a modified GA (MGA) can find near-optimal solutions within an acceptable amount of computational time. Among the two proposed metaheuristic approaches, the MGA is superior to the GA both in terms of computing time and the quality of the solution.     

A genetic algorithm for service level based vehicle scheduling

In many practical applications, vehicle scheduling problems involve more complex evaluation criteria than simple distance or travel time minimization. Scheduling to minimize delays between the accumulation and delivery of correspondence represents a class of vehicle scheduling problems, where: the evaluation of candidate solutions is costly, there are no efficient schemes for evaluation of partial solutions or perturbations to existing solutions, and dimensionality is limiting even for problems with relatively few locations. Several features of genetic algorithms (GA's) suggest that they may have advantages relative to alternative heuristic solution algorithms for such problems. These include ease of implementation through efficient coding of solution alternatives, simultaneous emphasis on global as well as local search, and the use of randomization in the search process. In addition, a GA may realize advantages usually associated with interactive methods by replicating the positive attributes of existing solutions in the search process, without explicitly defining or measuring these attributes. This study investigates these potential advantages through application of a GA to a service level based vehicle scheduling problem. The procedure is demonstrated for a vehicle scheduling problem with 15 locations where the objective is to minimize the time between the accumulation of correspondence at each location and delivery to destination locations. The results suggest that genetic algorithms can be effective for finding good quality scheduling solutions with only limited search of the solution space.     

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.     

Combining Metaheuristics and Exact Methods for Solving Exactly Multi-objective Problems on the Grid

This paper presents a parallel hybrid exact multi-objective approach which combines two metaheuristics – a genetic algorithm (GA) and a memetic algorithm (MA), with an exact method – a branch and bound (B&B) algorithm. Such approach profits from both the exploration power of the GA, the intensification capability of the MA and the ability of the B&B to provide optimal solutions with proof of optimality. To fully exploit the resources of a computational grid, the hybrid method is parallelized according to three well-known parallel models – the island model for the GA, the multi-start model for the MA and the parallel tree exploration model for the B&B. The obtained method has been experimented and validated on a bi-objective flow-shop scheduling problem. The approach allowed to solve exactly for the first time an instance of the problem – 50 jobs on 5 machines. More than 400 processors belonging to 4 different administrative domains have contributed to the resolution process during more than 6 days.      

A multi-objective hot-rolling scheduling problem in the compact strip production

This paper addresses a hot-rolling scheduling problem from compact strip production processes. At first, a mathematical model that consists of two coupled sub-problems is presented. The first sub-problem is the sheet-strip assignment problem that is about how to assign sheet-strips to rolling-turns with the objective of minimizing virtual sheet-strips. The second is the sheet-strip sequencing problem that is about how to sort the sheet-strips in each rolling-turn with the objective of minimizing the maximal changes in thickness between adjacent sheet-strips and the change times of the thickness so as to ensure high quality sheet-strips to be produced. And then, an improved hot-rolling scheduling heuristic is proposed to solve the sheet-strip assignment problem. A multi-objective evolutionary algorithm is developed to find the Pareto optimal or near-optimal solutions for the sheet-strip sequencing problem. Besides, the problem-specific knowledge is explored. The key operators including crossover operator, mutation operator and repair operator are designed for the multi-objective evolutionary algorithm. At last, extensive experiments based on real-world instances from a compact strip production process are carried out. The results demonstrate the effectiveness of the proposed algorithms for solving the hot-rolling scheduling problem under consideration.     

Soft computing optimization methods applied to logistic processes

This paper discusses the methodologies that can be used to optimize a logistic process of a supply chain described as a scheduling problem. First, a model of the system based on a real-world example is presented. Then, a new objective function called Global Expected Lateness is proposed, in order to describe multiple optimization criteria. Finally, three different optimization methodologies are proposed: a classical dispatching rule, and two soft computing techniques, Genetic Algorithms (GA) and Ant Colony Optimization (ACO). These methodologies are compared to the dispatching policy in the real-world example. The results show that dispatching heuristics are outperformed by the GA and ACO meta-heuristics. Further, it is shown that GA and ACO provide statistically identical scheduling solutions and from the optimization performance point of view, it is equivalent to use any of the meta-heuristics.     

Four Payment Models for the Multi-Mode Resource Constrained Project Scheduling Problem with Discounted Cash Flows

In this paper, the multi-mode resource constrained project scheduling problem with discounted cash flows is considered. The objective is the maximization of the net present value of all cash flows. Time value of money is taken into consideration, and cash in- and out-flows are associated with activities and/or events. The resources can be of renewable, nonrenewable, and doubly constrained resource types. Four payment models are considered: lump sum payment at the terminal event, payments at prespecified event nodes, payments at prespecified time points and progress payments. For finding solutions to problems proposed, a genetic algorithm (GA) approach is employed, which uses a special crossover operator that can exploit the multi-component nature of the problem. The models are investigated at the hand of an example problem. Sensitivity analyses are performed over the mark up and the discount rate. A set of 93 problems from literature are solved under the four different payment models and resource type combinations with the GA approach employed resulting in satisfactory computation times. The GA approach is compared with a domain specific heuristic for the lump sum payment case with renewable resources and is shown to outperform it.     

Multiprocessor task scheduling in multistage hybrid flow-shops: a genetic algorithm approach

This paper considers multiprocessor task scheduling in a multistage hybrid flow-shop environment. The objective is to minimize the make-span, that is, the completion time of all the tasks in the last stage. This problem is of practical interest in the textile and process industries. A genetic algorithm (GA) is developed to solve the problem. The GA is tested against a lower bound from the literature as well as against heuristic rules on a test bed comprising 400 problems with up to 100 jobs, 10 stages, and with up to five processors on each stage. For small problems, solutions found by the GA are compared to optimal solutions, which are obtained by total enumeration. For larger problems, optimum solutions are estimated by a statistical prediction technique. Computational results show that the GA is both effective and efficient for the current problem. Test problems are provided in a web site at www.benchmark.ibu.edu.tr/mpt-hfsp.     

Computing the metric dimension of graphs by genetic algorithms

In this paper we consider the NP-hard problem of determining the metric dimension of graphs. We propose a genetic algorithm (GA) that uses the binary encoding and the standard genetic operators adapted to the problem. The feasibility is enforced by repairing the individuals. The overall performance of the GA implementation is improved by a caching technique. Since the metric dimension problem up to now has been considered only theoretically, standard test instances for this problem do not exist. For that reason, we present the results of the computational experience on several sets of test instances for other NP-hard problems: pseudo boolean, crew scheduling and graph coloring. Testing on instances with up to 1534 nodes shows that GA relatively quickly obtains approximate solutions. For smaller instances, GA solutions are compared with CPLEX results. We have also modified our implementation to handle theoretically challenging large-scale classes of hypercubes and Hamming graphs. In this case the presented approach reaches optimal or best known solutions for hypercubes up to 131072 nodes and Hamming graphs up to 4913 nodes.     

A hybrid metaheuristic for the resource-constrained project scheduling problem

This study presents a hybrid metaheuristic ANGEL for the resource-constrained project scheduling problem (RCPSP). ANGEL combines ant colony optimization (ACO), genetic algorithm (GA) and local search strategy. The procedures of ANGEL are as follows. First, ACO searches the solution space and generates activity lists to provide the initial population for GA. Next, GA is executed and the pheromone set in ACO is updated when GA obtains a better solution. When GA terminates, ACO searches again by using a new pheromone set. ACO and GA search alternately and cooperatively in the solution space. This study also proposes an efficient local search procedure which is applied to yield a better solution when ACO or GA obtains a solution. A final search is applied upon the termination of ACO and GA. The experimental results of ANGEL on the standard sets of the project instances show that ANGEL is an effective method for solving the RCPSP.     

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.     

Job scheduling in virtual manufacturing cells with lot-streaming strategy: a new mathematical model formulation and a genetic algorithm approach

This paper discusses the job scheduling problem in virtual manufacturing cells (VMCs) with the objective of makespan minimization. In the VMC scheduling problem, each job undergoes different processing routes and there is a set of machines to process any operation. Jobs are produced in lot and lot-streaming is permitted. In addition, machines are distributed through the facility, which raises the travelling time issue. For this reason, the decisions are machine assignments, starting times and sub-lot sizes of the operations. We develop a new Mixed Integer Linear Programming (MILP) formulation that considers all aspects of the problem. Owing to the intractability matter, it is unlikely that the MILP could provide solutions for big-sized instances within a reasonable amount of time. We therefore present a Genetic Algorithm (GA) with a new chromosome structure for the VMC environment. Based on a wide range of examinations, comparative results show that GA is quite favourable and that it obtains the optimum solution for any of the instances in the case where sub-lot number equals 1.     

A tabu-search heuristic for the flexible-resource flow shop scheduling problem

This paper presents a tabu-search heuristic for the flexible-resource flow shop scheduling (FRFS) problem [7]. The FRFS problem generalizes the classic flow shop scheduling problem by allowing job-operation processing times to depend on the amount of resource assigned to an operation; the objective is to determine simultaneously the job sequence, resource-allocation policy, and operation start times that optimize system performance. The tabu-search heuristic (TSH) employs a nested-search strategy based on a decomposition of the FRFS problem into these three components. We discuss computational experience with the THS procedure on more than 1600 test problems. The results show that TSH is effective in obtaining near-optimal solutions to the FRFS test problems. In particular, TSH generates optimal solutions for more than 70% of the test problems for which optimal solutions are known; overall, these solutions are within 0.3% of optimality on the average, and within 2.5% of optimality in the worst case.This research was supported in part by National Science Foundation grant SES90-22940.     

An Improved Genetic Algorithm for the Distributed and Flexible Job-shop Scheduling problem

The Distributed and Flexible Job-shop Scheduling problem (DFJS) considers the scheduling of distributed manufacturing environments, where jobs are processed by a system of several Flexible Manufacturing Units (FMUs). Distributed scheduling problems deal with the assignment of jobs to FMUs and with determining the scheduling of each FMU, in terms of assignment of each job operation to one of the machines able to work it (job-routing flexibility) and sequence of operations on each machine. The objective is to minimize the global makespan over all the FMUs. This paper proposes an Improved Genetic Algorithm to solve the Distributed and Flexible Job-shop Scheduling problem. With respect to the solution representation for non-distributed job-shop scheduling, gene encoding is extended to include information on job-to-FMU assignment, and a greedy decoding procedure exploits flexibility and determines the job routings. Besides traditional crossover and mutation operators, a new local search based operator is used to improve available solutions by refining the most promising individuals of each generation. The proposed approach has been compared with other algorithms for distributed scheduling and evaluated with satisfactory results on a large set of distributed-and-flexible scheduling problems derived from classical job-shop scheduling benchmarks.     

Solving resource-constrained project scheduling problem with evolutionary programming

This paper presents an evolutionary programming (EP)-based approach to solving the resource-constrained project scheduling problem (RCPSP), a well-known NP-hard problem in scheduling, with minimization of project duration as the objective subject to precedence and resource constraints. The individual representation of EP for the problem is based on random keys. The serial generation scheme is used in the decoding scheme to generate the project plan. Experimental analyses are presented to investigate the performance of the proposed EP-based methodology, including comparison of the four variants of EP, namely, CEP, FEP, MCEP and IMCEP, with each other and GA to find the best variant of EP for the RCPSP, and comparison of this best variant of EP (MCEP) with other approaches using the J30 standard instances set in PSPLIB. The computational results validate the effectiveness of the proposed algorithm.     

Metaheuristics for scheduling jobs with incompatible families on parallel batching machines

In this paper, we discuss the scheduling of jobs with incompatible families on parallel batching machines. The performance measure is total weighted tardiness. This research is motivated by a scheduling problem found in the diffusion and oxidation areas of semiconductor wafer fabrication where the machines can be modelled as parallel batch processors. Given that this scheduling problem is NP-hard, we suggest an ant colony optimization (ACO) and a variable neighbourhood search (VNS) approach. Both metaheuristics are hybridized with a decomposition heuristic and a local search scheme. We compare the performance of the two algorithms with that of a genetic algorithm (GA) based on extensive computational experiments. The VNS approach outperforms the ACO and GA approach with respect to time and solution quality.     

A Simulated Annealing Approach to the Solution of Flexible Labour Scheduling Problems

This paper presents a new solution approach to the discontinuous labour tour scheduling problem where the objective is to minimize the number of full-time employees required to satisfy forecast demand. Previous heuristic approaches have often limited the number of allowable tours by restricting labour scheduling flexibility in terms of shift length, shift start-times, days-off, meal-break placement, and other factors. These restrictions were essential to the tractability of the heuristic approaches but often resulted in solutions that contained a substantial amount of excess labour. In this study, we relaxed many of the restrictions on scheduling flexibility assumed in previous studies. The resulting problem environment contained more than two billion allowable tours, precluding the use of previous heuristic methods. Consequently, we developed a simulated annealing heuristic for solving the problem. An important facet of this new approach is an ‘intelligent’ improvement routine which eliminates the need for long run-times typically associated with simulated annealing algorithms. The simulated annealing framework does not rely on a special problem structure and our implementation rapidly converged to near-optimal solutions for all problems in the test environment.     

Multiple-objective heuristics for scheduling unrelated parallel machines

This research proposes two heuristics and a Genetic Algorithm (GA) to find non-dominated solutions to multiple-objective unrelated parallel machine scheduling problems. Three criteria are of interest, namely: makespan, total weighted completion time, and total weighted tardiness. Each heuristic seeks to simultaneously minimize a pair of these criteria; the GA seeks to simultaneously minimize all three. The computational results show that the proposed heuristics are computationally efficient and provide solutions of reasonable quality. The proposed GA outperforms other algorithms in terms of the number of non-dominated solutions and the quality of its solutions.     

Helper-objectives: Using multi-objective evolutionary algorithms for single-objective optimisation

This paper investigates the use of multi-objective methods to guide the search when solving single-objective optimisation problems with genetic algorithms. Using the job shop scheduling and travelling salesman problems as examples, experiments demonstrate that the use of helper-objectives (additional objectives guiding the search) significantly improves the average performance of a standard GA. The helper-objectives guide the search towards solutions containing good building blocks and help the algorithm escape local optima. The experiments reveal that the approach works if the number of simultaneously used helper-objectives is low. However, a high number of helper-objectives can be used in the same run by changing the helper-objectives dynamically. The experiments reveal that for the majority of problem instances studied, the proposed approach significantly outperforms a traditional GA.The experiments also demonstrate that controlling the proportion of non-dominated solutions in the population is very important when using helper-objectives, since the presence of too many non-dominated solutions removes the selection pressure in the algorithm.