'Explicit Learning: an Effort towards Human Scheduling Algorithms'

Abstract Scheduling problems are generally NP-hard combinatorial problems, and a lot of research has been done to solve these problems heuristically. However, most of the previous approaches are problem-specific and research into the development of a general scheduling algorithm is still in its infancy. Mimicking the natural evolutionary process of the survival of the fittest, Genetic Algorithms (GAs) have attracted much attention in solving difficult scheduling problems in recent years. Some obstacles exist when using GAs: there is no canonical mechanism to deal with constraints, which are commonly met in most real-world scheduling problems, and small changes to a solution are difficult. To overcome both difficulties, indirect approaches have been presented (in [1] and [2]) for nurse scheduling and driver scheduling, where GAs are used by mapping the solution space, and separate decoding routines then build solutions to the original problem. In our previous indirect GAs, learning is implicit and is restricted to the efficient adjustment of weights for a set of rules that are used to construct schedules. The major limitation of those approaches is that they learn in a non-human way: like most existing construction algorithms, once the best weight combination is found, the rules used in the construction process are fixed at each iteration. However, normally a long sequence of moves is needed to construct a schedule and using fixed rules at each move is thus unreasonable and not coherent with human learning processes. When a human scheduler is working, he normally builds a schedule step by step following a set of rules. After much practice, the scheduler gradually masters the knowledge of which solution parts go well with others. He can identify good parts and is aware of the solution quality even if the scheduling process is not completed yet, thus having the ability to finish a schedule by using flexible, rather than fixed, rules. In this research we intend to design more human-like scheduling algorithms, by using ideas derived from Bayesian Optimization Algorithms (BOA) and Learning Classifier Systems (LCS) to implement explicit learning from past solutions. BOA can be applied to learn to identify good partial solutions and to complete them by building a Bayesian network of the joint distribution of solutions [3]. A Bayesian network is a directed acyclic graph with each node corresponding to one variable, and each variable corresponding to individual rule by which a schedule will be constructed step by step. The conditional probabilities are computed according to an initial set of promising solutions. Subsequently, each new instance for each node is generated by using the corresponding conditional probabilities, until values for all nodes have been generated. 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 Bayesian network is updated again using the current set of good rule strings. The algorithm thereby tries to explicitly identify and mix promising building blocks. It should be noted that for most scheduling problems the structure of the network model is known and all the variables are fully observed. In this case, the goal of learning is to find the rule values that maximize the likelihood of the training data. Thus learning can amount to 'counting' in the case of multinomial distributions. In the LCS approach, each rule has its strength showing its current usefulness in the system, and this strength is constantly assessed [4]. To implement sophisticated learning based on previous solutions, an improved LCS-based algorithm is designed, which consists of the following three steps. The initialization step is to assign each rule at each stage a constant initial strength. Then rules are selected by using the Roulette Wheel strategy. The next step is to reinforce the strengths of the rules used in the previous solution, keeping the strength of unused rules unchanged. The selection step is to select fitter rules for the next generation. It is envisaged that the LCS part of the algorithm will be used as a hill climber to the BOA algorithm. This is exciting and ambitious research, which might provide the stepping-stone for a new class of scheduling algorithms. Data sets from nurse scheduling and mall problems will be used as test-beds. It is envisaged that once the concept has been proven successful, it will be implemented into general scheduling algorithms. It is also hoped that this research will give some preliminary answers about how to include human-like learning into scheduling algorithms and may therefore be of interest to researchers and practitioners in areas of scheduling and evolutionary computation. References 1. Aickelin, U. and Dowsland, K. (2003) 'Indirect Genetic Algorithm for a Nurse Scheduling Problem', Computer & Operational Research (in print). 2. Li, J. and Kwan, R.S.K. (2003), 'Fuzzy Genetic Algorithm for Driver Scheduling', European Journal of Operational Research 147(2): 334-344. 3. Pelikan, M., Goldberg, D. and Cantu-Paz, E. (1999) 'BOA: The Bayesian Optimization Algorithm', IlliGAL Report No 99003, University of Illinois. 4. Wilson, S. (1994) 'ZCS: A Zeroth-level Classifier System', Evolutionary Computation 2(1), pp 1-18.

Approximation Algorithms for Network Design and Orienteering

This thesis presents approximation algorithms for some NP-Hard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widely-believed complexity-theoretic assumption that P is not equal to NP, there are no efficient (i.e., polynomial-time) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an NP-Hard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing fault-tolerant networks. Two vertices u,v in a network are said to be k-edge-connected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are k-vertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning that even if a small number of edges and nodes fail, the remaining nodes will still be able to communicate. A brief description of some of our results is given below. We study the problem of building 2-vertex-connected networks that are large and have low cost. Given an n-node graph with costs on its edges and any integer k, we give an O(log n log k) approximation for the problem of finding a minimum-cost 2-vertex-connected subgraph containing at least k nodes. We also give an algorithm of similar approximation ratio for maximizing the number of nodes in a 2-vertex-connected subgraph subject to a budget constraint on the total cost of its edges. Our algorithms are based on a pruning process that, given a 2-vertex-connected graph, finds a 2-vertex-connected subgraph of any desired size and of density comparable to the input graph, where the density of a graph is the ratio of its cost to the number of vertices it contains. This pruning algorithm is simple and efficient, and is likely to find additional applications. Recent breakthroughs on vertex-connectivity have made use of algorithms for element-connectivity problems. We develop an algorithm that, given a graph with some vertices marked as terminals, significantly simplifies the graph while preserving the pairwise element-connectivity of all terminals; in fact, the resulting graph is bipartite. We believe that our simplification/reduction algorithm will be a useful tool in many settings. We illustrate its applicability by giving algorithms to find many trees that each span a given terminal set, while being disjoint on edges and non-terminal vertices; such problems have applications in VLSI design and other areas. We also use this reduction algorithm to analyze simple algorithms for single-sink network design problems with high vertex-connectivity requirements; we give an O(k log n)-approximation for the problem of k-connecting a given set of terminals to a common sink. We study similar problems in which different types of links, of varying capacities and costs, can be used to connect nodes; assuming there are economies of scale, we give algorithms to construct low-cost networks with sufficient capacity or bandwidth to simultaneously support flow from each terminal to the common sink along many vertex-disjoint paths. We further investigate capacitated network design, where edges may have arbitrary costs and capacities. Given a connectivity requirement R_uv for each pair of vertices u,v, the goal is to find a low-cost network which, for each uv, can support a flow of R_uv units of traffic between u and v. We study several special cases of this problem, giving both algorithmic and hardness results. In addition to Network Design, we consider certain Traveling Salesperson-like problems, where the goal is to find short walks that visit many distinct vertices. We give a (2 + epsilon)-approximation for Orienteering in undirected graphs, achieving the best known approximation ratio, and the first approximation algorithm for Orienteering in directed graphs. We also give improved algorithms for Orienteering with time windows, in which vertices must be visited between specified release times and deadlines, and other related problems. These problems are motivated by applications in the fields of vehicle routing, delivery and transportation of goods, and robot path planning.

'Explicit Learning: an Effort towards Human Scheduling Algorithms'

Abstract Scheduling problems are generally NP-hard combinatorial problems, and a lot of research has been done to solve these problems heuristically. However, most of the previous approaches are problem-specific and research into the development of a general scheduling algorithm is still in its infancy. Mimicking the natural evolutionary process of the survival of the fittest, Genetic Algorithms (GAs) have attracted much attention in solving difficult scheduling problems in recent years. Some obstacles exist when using GAs: there is no canonical mechanism to deal with constraints, which are commonly met in most real-world scheduling problems, and small changes to a solution are difficult. To overcome both difficulties, indirect approaches have been presented (in [1] and [2]) for nurse scheduling and driver scheduling, where GAs are used by mapping the solution space, and separate decoding routines then build solutions to the original problem. In our previous indirect GAs, learning is implicit and is restricted to the efficient adjustment of weights for a set of rules that are used to construct schedules. The major limitation of those approaches is that they learn in a non-human way: like most existing construction algorithms, once the best weight combination is found, the rules used in the construction process are fixed at each iteration. However, normally a long sequence of moves is needed to construct a schedule and using fixed rules at each move is thus unreasonable and not coherent with human learning processes. When a human scheduler is working, he normally builds a schedule step by step following a set of rules. After much practice, the scheduler gradually masters the knowledge of which solution parts go well with others. He can identify good parts and is aware of the solution quality even if the scheduling process is not completed yet, thus having the ability to finish a schedule by using flexible, rather than fixed, rules. In this research we intend to design more human-like scheduling algorithms, by using ideas derived from Bayesian Optimization Algorithms (BOA) and Learning Classifier Systems (LCS) to implement explicit learning from past solutions. BOA can be applied to learn to identify good partial solutions and to complete them by building a Bayesian network of the joint distribution of solutions [3]. A Bayesian network is a directed acyclic graph with each node corresponding to one variable, and each variable corresponding to individual rule by which a schedule will be constructed step by step. The conditional probabilities are computed according to an initial set of promising solutions. Subsequently, each new instance for each node is generated by using the corresponding conditional probabilities, until values for all nodes have been generated. 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 Bayesian network is updated again using the current set of good rule strings. The algorithm thereby tries to explicitly identify and mix promising building blocks. It should be noted that for most scheduling problems the structure of the network model is known and all the variables are fully observed. In this case, the goal of learning is to find the rule values that maximize the likelihood of the training data. Thus learning can amount to 'counting' in the case of multinomial distributions. In the LCS approach, each rule has its strength showing its current usefulness in the system, and this strength is constantly assessed [4]. To implement sophisticated learning based on previous solutions, an improved LCS-based algorithm is designed, which consists of the following three steps. The initialization step is to assign each rule at each stage a constant initial strength. Then rules are selected by using the Roulette Wheel strategy. The next step is to reinforce the strengths of the rules used in the previous solution, keeping the strength of unused rules unchanged. The selection step is to select fitter rules for the next generation. It is envisaged that the LCS part of the algorithm will be used as a hill climber to the BOA algorithm. This is exciting and ambitious research, which might provide the stepping-stone for a new class of scheduling algorithms. Data sets from nurse scheduling and mall problems will be used as test-beds. It is envisaged that once the concept has been proven successful, it will be implemented into general scheduling algorithms. It is also hoped that this research will give some preliminary answers about how to include human-like learning into scheduling algorithms and may therefore be of interest to researchers and practitioners in areas of scheduling and evolutionary computation. References 1. Aickelin, U. and Dowsland, K. (2003) 'Indirect Genetic Algorithm for a Nurse Scheduling Problem', Computer & Operational Research (in print). 2. Li, J. and Kwan, R.S.K. (2003), 'Fuzzy Genetic Algorithm for Driver Scheduling', European Journal of Operational Research 147(2): 334-344. 3. Pelikan, M., Goldberg, D. and Cantu-Paz, E. (1999) 'BOA: The Bayesian Optimization Algorithm', IlliGAL Report No 99003, University of Illinois. 4. Wilson, S. (1994) 'ZCS: A Zeroth-level Classifier System', Evolutionary Computation 2(1), pp 1-18.

An experimental study of adaptive control for evolutionary algorithms

International audience

Interrelated product design activities sequencing with efficient tabu search algorithms

This paper proposes and investigates a metaheuristic tabu search algorithm (TSA) that generates optimal or near optimal solutions sequences for the feedback length minimization problem (FLMP) associated to a design structure matrix (DSM). The FLMP is a non-linear combinatorial optimization problem, belonging to the NP-hard class, and therefore finding an exact optimal solution is very hard and time consuming, especially on medium and large problem instances. First, we introduce the subject and provide a review of the related literature and problem definitions. Using the tabu search method (TSM) paradigm, this paper presents a new tabu search algorithm that generates optimal or sub-optimal solutions for the feedback length minimization problem, using two different neighborhoods based on swaps of two activities and shifting an activity to a different position. Furthermore, this paper includes numerical results for analyzing the performance of the proposed TSA and for fixing the proper values of its parameters. Then we compare our results on benchmarked problems with those already published in the literature. We conclude that the proposed tabu search algorithm is very promising because it outperforms the existing methods, and because no other tabu search method for the FLMP is reported in the literature. The proposed tabu search algorithm applied to the process layer of the multidimensional design structure matrices proves to be a key optimization method for an optimal product development.

Interrelated product design activities sequencing with efficient tabu search algorithms

This paper proposes and investigates a metaheuristic tabu search algorithm (TSA) that generates optimal or near optimal solutions sequences for the feedback length minimization problem (FLMP) associated to a design structure matrix (DSM). The FLMP is a non-linear combinatorial optimization problem, belonging to the NP-hard class, and therefore finding an exact optimal solution is very hard and time consuming, especially on medium and large problem instances. First, we introduce the subject and provide a review of the related literature and problem definitions. Using the tabu search method (TSM) paradigm, this paper presents a new tabu search algorithm that generates optimal or sub-optimal solutions for the feedback length minimization problem, using two different neighborhoods based on swaps of two activities and shifting an activity to a different position. Furthermore, this paper includes numerical results for analyzing the performance of the proposed TSA and for fixing the proper values of its parameters. Then we compare our results on benchmarked problems with those already published in the literature. We conclude that the proposed tabu search algorithm is very promising because it outperforms the existing methods, and because no other tabu search method for the FLMP is reported in the literature. The proposed tabu search algorithm applied to the process layer of the multidimensional design structure matrices proves to be a key optimization method for an optimal product development.

A hybrid genetic algorithm for the optimal constrained web service selection problem in web service composition

Web service composition is an important problem in web service based systems. It is about how to build a new value-added web service using existing web services. A web service may have many implementations, all of which have the same functionality, but may have different QoS values. Thus, a significant research problem in web service composition is how to select a web service implementation for each of the web services such that the composite web service gives the best overall performance. This is so-called optimal web service selection problem. There may be mutual constraints between some web service implementations. Sometimes when an implementation is selected for one web service, a particular implementation for another web service must be selected. This is so called dependency constraint. Sometimes when an implementation for one web service is selected, a set of implementations for another web service must be excluded in the web service composition. This is so called conflict constraint. Thus, the optimal web service selection is a typical constrained ombinatorial optimization problem from the computational point of view. This paper proposes a new hybrid genetic algorithm for the optimal web service selection problem. The hybrid genetic algorithm has been implemented and evaluated. The evaluation results have shown that the hybrid genetic algorithm outperforms other two existing genetic algorithms when the number of web services and the number of constraints are large.

Cross-entropy optimization for scaling factors of a fuzzy controller : a see-and-avoid approach for unmanned aerial systems

The Cross-Entropy (CE) is an efficient method for the estimation of rare-event probabilities and combinatorial optimization. This work presents a novel approach of the CE for optimization of a Soft-Computing controller. A Fuzzy controller was designed to command an unmanned aerial system (UAS) for avoiding collision task. The only sensor used to accomplish this task was a forward camera. The CE is used to reach a near-optimal controller by modifying the scaling factors of the controller inputs. The optimization was realized using the ROS-Gazebo simulation system. In order to evaluate the optimization a big amount of tests were carried out with a real quadcopter.

Balancing exploration and exploitation in particle swarm optimization on search tasking

In this study we present a combinatorial optimization method based on particle swarm optimization and local search algorithm on the multi-robot search system. Under this method, in order to create a balance between exploration and exploitation and guarantee the global convergence, at each iteration step if the distance between target and the robot become less than specific measure then a local search algorithm is performed. The local search encourages the particle to explore the local region beyond to reach the target in lesser search time. Experimental results obtained in a simulated environment show that biological and sociological inspiration could be useful to meet the challenges of robotic applications that can be described as optimization problems.

Ant colony optimization for reporting cell planning in mobile computing using selective paging strategy

Location management problem that arise in mobile computing networks is addressed. One method used in location management is to designate sonic of the cells in the network as "reporting cells". The other cells in the network are "non-reporting cells". Finding an optimal set of reporting cells (or reporting cell configuration) for a given network. is a difficult combinatorial optimization problem. In fact this is shown to be an NP-complete problem. in an earlier study. In this paper, we use the selective paging strategy and use an ant colony optimization method to obtain the best/optimal set of reporting cells for a given a network.

Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization

In this work, we explore simultaneous geometry design and material selection for statically determinate trusses by posing it as a continuous optimization problem. The underlying principles of our approach are structural optimization and Ashby’s procedure for material selection from a database. For simplicity and ease of initial implementation, only static loads are considered in this work with the intent of maximum stiffness, minimum weight/cost, and safety against failure. Safety of tensile and compression members in the truss is treated differently to prevent yield and buckling failures, respectively. Geometry variables such as lengths and orientations of members are taken to be the design variables in an assumed layout. Areas of cross-section of the members are determined to satisfy the failure constraints in each member. Along the lines of Ashby’s material indices, a new design index is derived for trusses. The design index helps in choosing the most suitable material for any geometry of the truss. Using the design index, both the design space and the material database are searched simultaneously using gradient-based optimization algorithms. The important feature of our approach is that the formulated optimization problem is continuous, although the material selection from a database is an inherently discrete problem. A few illustrative examples are included. It is observed that the method is capable of determining the optimal topology in addition to optimal geometry when the assumed layout contains more links than are necessary for optimality.

A real coded genetic algorithm for data partitioning and scheduling in networks with arbitrary processor release time

The problem of scheduling divisible loads in distributed computing systems, in presence of processor release time is considered. The objective is to find the optimal sequence of load distribution and the optimal load fractions assigned to each processor in the system such that the processing time of the entire processing load is a minimum. This is a difficult combinatorial optimization problem and hence genetic algorithms approach is presented for its solution.

Pareto-optimal solutions for multi-objective optimization of fed-batch bioreactors using nondominated sorting genetic algorithm

Many optimal control problems are characterized by their multiple performance measures that are often noncommensurable and competing with each other. The presence of multiple objectives in a problem usually give rise to a set of optimal solutions, largely known as Pareto-optimal solutions. Evolutionary algorithms have been recognized to be well suited for multi-objective optimization because of their capability to evolve a set of nondominated solutions distributed along the Pareto front. This has led to the development of many evolutionary multi-objective optimization algorithms among which Nondominated Sorting Genetic Algorithm (NSGA and its enhanced version NSGA-II) has been found effective in solving a wide variety of problems. Recently, we reported a genetic algorithm based technique for solving dynamic single-objective optimization problems, with single as well as multiple control variables, that appear in fed-batch bioreactor applications. The purpose of this study is to extend this methodology for solution of multi-objective optimal control problems under the framework of NSGA-II. The applicability of the technique is illustrated by solving two optimal control problems, taken from literature, which have usually been solved by several methods as single-objective dynamic optimization problems. (C) 2004 Elsevier Ltd. All rights reserved.

Genetic algorithm for satellite customer assignment

The problem of assigning customers to satellite channels is considered. Finding an optimal allocation of customers to satellite channels is a difficult combinatorial optimization problem and is shown to be NP-complete in an earlier study. We propose a genetic algorithm (GA) approach to search for the best/optimal assignment of customers to satellite channels. Various issues related to genetic algorithms such as solution representation, selection methods, genetic operators and repair of invalid solutions are presented. A comparison of this approach with the standard optimization method is presented to show the advantages of this approach in terms of computation time

Multiple UAV Coalitions for a Search and Prosecute Mission

Unmanned aerial vehicles (UAVs) have the potential to carry resources in support of search and prosecute operations. Often to completely prosecute a target, UAVs may have to simultaneously attack the target with various resources with different capacities. However, the UAVs are capable of carrying only limited resources in small quantities, hence, a group of UAVs (coalition) needs to be assigned that satisfies the target resource requirement. The assigned coalition must be such that it minimizes the target prosecution delay and the size of the coalition. The problem of forming coalitions is computationally intensive due to the combinatorial nature of the problem, but for real-time applications computationally cheap solutions are required. In this paper, we propose decentralized sub-optimal (polynomial time) and decentralized optimal coalition formation algorithms that generate coalitions for a single target with low computational complexity. We compare the performance of the proposed algorithms to that of a global optimal solution for which we need to solve a centralized combinatorial optimization problem. This problem is computationally intensive because the solution has to (a) provide a coalition for each target, (b) design a sequence in which targets need to be prosecuted, and (c) take into account reduction of UAV resources with usage. To solve this problem we use the Particle Swarm Optimization (PSO) technique. Through simulations, we study the performance of the proposed algorithms in terms of mission performance, complexity of the algorithms and the time taken to form the coalition. The simulation results show that the solution provided by the proposed algorithms is close to the global optimal solution and requires far less computational resources.