A multi-objective production scheduling case study solved by simulated annealing

During several decades, research in production scheduling mainly concerns a single criterion to optimize. However, the analysis of the performance of a schedule often involves more than one aspect and therefore requires multi-objective analysis. Such situation appears in the real case study considered here.     

On the characterization and generation of nurse scheduling problem instances

Due to its complexity and relevance in practice, many different procedures have been proposed in the operations research literature to solve the well-known nurse scheduling problem (NSP). The NSP assigns nurses to shifts per day maximizing the overall quality of the roster while taking various constraints into account. The often highly case-specific workplace conditions in hospital departments have resulted in the development of dedicated (meta-)heuristics to find a workable schedule in an acceptable time limit. However, in spite of research community posing a growing need for benchmarking, these procedures lack any base for comparison.     

An electromagnetic meta-heuristic for the nurse scheduling problem

In this paper, we present a novel meta-heuristic technique for the nurse scheduling problem (NSP). This well-known scheduling problem assigns nurses to shifts per day maximizing the overall quality of the roster while taking various constraints into account. The problem is known to be NP-hard. Due to its complexity and relevance, many algorithms have been developed to solve practical and often case-specific models of the NSP. The huge variety of constraints and the several objective function possibilities have led to exact and meta-heuristic procedures in various guises, and hence comparison and state-of-the-art reporting of standard results seem to be a utopian idea. We present a meta-heuristic procedure for the NSP based on the framework proposed by Birbil and Fang (J. Glob. Opt. 25, 263–282, 2003). The Electromagnetic (EM) approach is based on the theory of physics, and simulates attraction and repulsion of sample points in order to move towards a promising solution. Moreover, we present computational experiments on a standard benchmark dataset, and solve problem instances under different assumptions. We show that the proposed procedure performs consistently well under many different circumstances, and hence, can be considered as robust against case-specific constraints.     

Comparison and hybridization of crossover operators for the nurse scheduling problem

In this paper, we present a hybrid genetic algorithm for the well-known nurse scheduling problem (NSP). The NSP involves the construction of roster schedules for nursing staff in order to maximize the quality of the roster schedule subject to various hard constraints. In the literature, several genetic algorithms have been proposed to solve the NSP under various assumptions. The contribution of this paper is twofold. First, we extensively compare the various crossover operators and test them on a standard dataset in a solitary approach. Second, we propose several options to hybridize the various crossover operators.     

Design of cooperative algorithms for multi-objective optimization: application to the flow-shop scheduling problem

This is a summary of the main results presented in the author’s PhD thesis. This thesis was supervised by El-Ghazali Talbi, and defended on 21 June 2005 at the University of Lille (France). It is written in French and is available at http://www.lifl.fr/~basseur/These.pdf. This work deals with the conception of cooperative methods in order to solve multi-objective combinatorial optimization problems. Many cooperation schemes between exact and/or heuristic methods have been proposed in the literature. We propose a classification of such schemes. We propose a new heuristic called adaptive genetic algorithm (AGA), that is designed for an efficient exploration of the search space. We consider several cooperation schemes between AGA and other methods (exact or heuristic). The performance of these schemes are tested on a bi-objective permutation flow-shop scheduling problem, in order to evaluate the interest of each type of cooperation.     

Solving a bi-objective nurse rerostering problem by using a utopic Pareto genetic heuristic

Nurse rerostering arises when at least one nurse announces that she will be unable to undertake the tasks previously assigned to her. The problem amounts to building a new roster that satisfies the hard constraints already met by the current one and, as much as possible, fulfils two groups of soft constraints which define the two objectives to be attained. A bi-objective genetic heuristic was designed on the basis of a population of individuals characterised by pairs of chromosomes, whose fitness complies with the Pareto ranking of the respective decoded solution. It includes an elitist policy, as well as a new utopic strategy, introduced for purposes of diversification. The computational experiments produced promising results for the practical application of this approach to real life instances arising from a public hospital in Lisbon.     

Solving a manpower scheduling problem for airline catering using metaheuristics

We study a manpower scheduling problem with job time windows and job-skills compatibility constraints. This problem is motivated by airline catering operations, whereby airline meals and other supplies are delivered to aircrafts on the tarmac just before the flights take-off. Jobs (flights) must be serviced within a given time-window by a team consisting of a driver and loader. Each driver/loader has the skills to service some, but not all, of the airline/aircraft/configuration of the jobs. Given the jobs to be serviced and the roster of workers for each shift, the problem is to form teams and assign teams and start-times for the jobs, so as to service as many flights as possible. Only teams with the appropriate skills can be assigned to a flight. Workload balance among the teams is also a consideration. We present model formulations and investigate a tabu search heuristic and a simulated annealing heuristic approach to solve the problem. Computational experiments show that the tabu search approach outperforms the simulated annealing approach, and is capable of finding good solutions.     

Solving the flight perturbation problem with meta heuristics

When there is a perturbation in a carefully constructed aircraft schedule, e.g. an aircraft breakdown, it is important to minimize the negative consequences of this disturbance. Here, a tabu search and a simulated annealing approach to the flight perturbation problem are presented. The heuristics use a tree-search algorithm to find new schedules for the aircraft, and utilize a path relinking strategy to explore paths between structurally different solutions. The computational results indicate that the solution strategies, especially the tabu search, can be successfully used to solve the flight perturbation problem.     

Solving the discrete-continuous project scheduling problem via its discretization

In this paper a discrete-continuous project scheduling problem is considered. In this problem activities simultaneously require discrete and continuous resources. The processing rate of each activity depends on the amount of the continuous resource allotted to this activity at a time. All the resources are renewable ones. The activities are nonpreemtable and the objective is to minimize the makespan. Discretization of this problem leading to a classical (i.e. discrete) project scheduling problem in the multi-mode version is presented. A simulated annealing (SA) approach to solving this problem is described and tested computationally in two versions: with and without finding an optimal continuous resource allocation for the final schedule. In the former case a nonlinear solver is used for solving a corresponding convex programming problem. The results are compared with the results obtained using SA for the discrete-continuous project scheduling problem where the nonlinear solver is used for exact solving the continuous part in each iteration. The results of a computational experiment are analyzed and some conclusions are included.     

Solving a school bus scheduling problem with integer programming

In many rural areas in Germany pupils on the way to school are a large if not the largest group of customers in public transport. If all schools start more or less at the same time then the bus companies need a high number of vehicles to serve the customer peak in the morning rush hours. In this article, we present an integer programming model for the integrated coordination of the school starting times and the public bus services. We discuss preprocessing techniques, model reformulations, and cutting planes that can be incorporated into a branch-and-cut algorithm. Computational results show that in our test counties a much lower number of buses would be sufficient if the schools start at different times.     

Solving multi-objective parallel machine scheduling problem by a modified NSGA-II

In this paper, we modify a Multi-Objective Evolutionary Algorithm, known as Nondominated sorting Genetic Algorithm-II (NSGA-II) for a parallel machine scheduling problem with three objectives. The objectives are – (1) minimization of total cost due tardiness, (2) minimization of the deterioration cost and (3) minimization of makespan. The formulated problem has been solved by three Multi-Objective Evolutionary Algorithms which are: (1) the original NSGA-II (Non-dominated Sorting Genetic Algorithm–II), (2) SPEA2 (Strength Pareto Evolutionary Algorithm-2) and (3) a modified version of NSGA-II as proposed in this paper. A new mutation algorithm has also been proposed depending on the type of problem and embedded in the modified NSGA-II. The results of the three algorithms have been compared and conclusions have been drawn. The modified NSGA-II is observed to perform better than the original NSGA-II. Besides, the proposed mutation algorithm also works effectively, as evident from the experimental results.     

Solving a dynamic cell formation problem using metaheuristics

In this paper, solving a cell formation (CF) problem in dynamic condition is going to be discussed by using some traditional metaheuristic methods such as genetic algorithm (GA), simulated annealing (SA) and tabu search (TS). Most of previous researches were done under the static condition. Due to the fact that CF is a NP-hard problem, then solving the model using classical optimization methods needs a long computational time. In this research, a nonlinear integer model of CF is first given and then solved by GA, SA and TS. Then, the results are compared with the optimal solution and the efficiency of the proposed algorithms is discussed.     

An estimation of distribution algorithm for nurse scheduling

Schedules can be built in a similar way to a human scheduler by using a set of rules that involve domain knowledge. This paper presents an Estimation of Distribution Algorithm (EDA) for the nurse scheduling problem, which involves choosing a suitable scheduling rule from a set for the assignment of each nurse. Unlike previous work that used Genetic Algorithms (GAs) to implement implicit learning, the learning in the proposed algorithm is explicit, i.e. we identify and mix building blocks directly. The EDA is applied to implement such explicit learning by building a Bayesian network of the joint distribution of solutions. The conditional probability of each variable in the network is computed according to an initial set of promising solutions. Subsequently, each new instance for each variable is generated by using the corresponding conditional probabilities, until all variables have been generated, i.e. in our case, a new rule string has been obtained. Another set of rule strings will be generated in this way, some of which will replace previous strings based on fitness selection. If stopping conditions are not met, the conditional probabilities for all nodes in the Bayesian network are updated again using the current set of promising rule strings. Computational results from 52 real data instances demonstrate the success of this approach. It is also suggested that the learning mechanism in the proposed approach might be suitable for other scheduling problems.     

Approximate solution of a resource-constrained scheduling problem

This paper is devoted to the approximate solution of a strongly NP-hard resource-constrained scheduling problem which arises in relation to the operability of certain high availability real time distributed systems. We present an algorithm based on the simulated annealing metaheuristic and, building on previous research on exact solution methods, extensive computational results demonstrating its practical ability to produce acceptable solutions, in a precisely defined sense. Additionally, our experiments are in remarkable agreement with certain theoretical properties of our simulated annealing scheme. The paper concludes with a short discussion on further research. This research was supported in part by Association Nationale de la Recherche Technique grant CIFRE-121/2004.     

Solving a multi-objective multi-skilled manpower scheduling model by a fuzzy goal programming approach

Manpower scheduling is an intricate problem in production and service environments with the purpose of generating fair schedules that consider employers’ objectives and employees’ preferences as much as possible. However, sometimes, vagueness of information related to employers’ objectives and employees’ preferences leads to the fuzzy nature of the problem. This paper presents a multi-objective manpower scheduling model regarding the lack of clarity on the target values of employers’ objectives and employees’ preferences. Hence, a fuzzy goal programming model is developed for the presented model. Afterwards, two fuzzy solution approaches are used to convert the fuzzy goal programming model to two single-objective models. Finally, the results obtained by both single-objective models are compared with each other to select the solution that has the greatest degree of the satisfaction level of employers’ objectives and employees’ preferences.     

Solving a nurse scheduling problem with knapsacks,networks and tabu search

This paper illustrates how a modern heuristic and two classical integer programming models have been combined to provide a solution to a nurse rostering problem at a major UK hospital. Neither a heuristic nor an exact approach based on a standard IP package was able to meet all the practical requirements. This was overcome by using a variant of tabu search as the core method, but applying knapsack and network flow models in pre- and post-processing phases. The result is a successful software tool that frees senior nursing staff from a time consuming administrative task.     

Solving a chemical batch scheduling problem by local search

In this paper the following chemical batch scheduling problem is considered: a set of orders has to be processed on a set of facilities. For each order a given amount of a product must be produced by means of chemical reactions before a given deadline. The production consists of a sequence of processes whereby each process has to be performed by one facility out of a given subset of facilities allowed for this process. The processing times depend on the choice of the facility and the processing is done in batch mode with given minimum and maximum sizes. The problem is to assign the processes to the facilities, splitting them into batches, and scheduling these batches in order to produce the demands within the given deadlines. For the scheduling part of the problem we present an approach based on the following steps. First, a procedure to calculate the minimum number of batches needed to satisfy the demands is presented. Based on this, the given problem is modeled in two different ways: as a general shop scheduling problem with set-up times or as scheduling problem with positive time-lags. Finally, a two-phase tabu search method is presented which is based on the two different formulations of the problem. The method is tested on some real world data. This revised version was published online in June 2006 with corrections to the Cover Date.     

Modeling a stochastic multi-objective supplier quota allocation problem with price-dependent ordering

One of the interesting subjects in supply chain management is supply management, which generally relates to the activities regarding suppliers such as empowerment, evaluation, partnerships and so on. A major objective of supplier evaluation involves buyers determining the optimal quota allocated to each supplier when placing an order. In this paper, we propose a multi-objective model in which purchasing cost, rejected units, and late delivered units are minimized, while the obtained total score from the supplier evaluation process is maximized. We assume that the buyer obtains multiple products from a number of predetermined suppliers. The buyer faces a stochastic demand with a probability distribution of Poisson regarding each product type. A major assumption is that the supplier prices are linearly dependent on the order size of each product. Since demand is stochastic, the buyer may incur holding and stockout costs in addition to the regular purchasing cost. We use the well-known L-1 metric method to solve the supplier evaluation problem by utilizing two meta-heuristic algorithms to solve the corresponding mathematical problems.     

Minimizing total weighted late work in the resource-constrained project scheduling problem

This paper deals with resource-constrained project scheduling problem under the weighted late work criterion. Late work objective functions estimate the quality of a schedule based on durations of late parts of activities, not taking into account the amount of delay for fully late activities. It is assume that a project contains activities interrelated by finish-to-start type precedence relations with time lag of zero, which require one or more constrained renewable resources. The objective is to schedule each activity such that the total weighted late work is minimized. The problem has been formulated using a linear integer programming model and solved by the CPLEX. Also, a set of priority rules have been designed to quickly generate a set of initial solutions. In order to solve the problem optimally, a depth-first branch-and-bound algorithm is applied based on idea of minimal delaying alternatives. The branching order of nodes that belong to the same level of the search tree is determined on the basis of the developed priority rules. This results in generation six different versions of the branch-and-bound algorithm. Computational results on randomly generated problem sets are provided to analyze the efficiency of the priority rules and the branch-and-bound algorithm.     

Solving the short-term electrical generation scheduling problem by an adaptive evolutionary approach

In this paper, we introduce an adaptive evolutionary approach to solve the short-term electrical generation scheduling problem (STEGS). The STEGS is a hard constraint satisfaction optimization problem. The algorithm includes various strategies proposed in the literature to tackle hard problems with constraints such as: the representation used a non-binary coding scheme that drastically reduces the search space compared with the traditional evolutionary approaches. Specialized operators are especially designed for this problem and for this kind of representation, which also includes a local search procedure. Furthermore, the algorithm is guided by an adaptive parameter control strategy. We used some very well known benchmarks for STEGS to evaluate our approach. The results are very encouraging and we have obtained new better values for all the systems tested. Our aim here is to show that evolutionary approaches can be considered as good techniques to be used to solve real-world highly constrained problems.