Evolutionary algorithms (EAs) are often well-suited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EAs are compared quantitatively where an extended 0/1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the strength Pareto EA (SPEA), that combines several features of previous multiobjective EAs in a unique manner. It is characterized by (a) storing nondominated solutions externally in a second, continuously updated population, (b) evaluating an individual's fitness dependent on the number of external nondominated points that dominate it, (c) preserving population diversity using the Pareto dominance relationship, and (d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EAs on the 0/1 knapsack problem.

Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach

The paradigmatic shift from a Web of manual interactions to a Web of programmatic interactions driven by Web services is creating unprecedented opportunities for the formation of online business-to-business (B2B) collaborations. In particular, the creation of value-added services by composition of existing ones is gaining a significant momentum. Since many available Web services provide overlapping or identical functionality, albeit with different quality of service (QoS), a choice needs to be made to determine which services are to participate in a given composite service. This paper presents a middleware platform which addresses the issue of selecting Web services for the purpose of their composition in a way that maximizes user satisfaction expressed as utility functions over QoS attributes, while satisfying the constraints set by the user and by the structure of the composite service. Two selection approaches are described and compared: one based on local (task-level) selection of services and the other based on global allocation of tasks to services using integer programming.

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QoS-aware middleware for Web services composition

Since 1985 various evolutionary approaches to multiobjective optimization have been developed, capable of searching for multiple solutions concurrently in a single run. But the few comparative studies of different methods available to date are mostly qualitative and restricted to two approaches. In this paper an extensive, quantitative comparison is presented, applying four multiobjective evolutionary algorithms to an extended 0/1 knapsack problem.

Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study

Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.

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Greedy Randomized Adaptive Search Procedures

The clonal selection principle is used to explain the basic features of an adaptive immune response to an antigenic stimulus. It establishes the idea that only those cells that recognize the antigens (Ag's) are selected to proliferate. The selected cells are subject to an affinity maturation process, which improves their affinity to the selective Ag's. This paper proposes a computational implementation of the clonal selection principle that explicitly takes into account the affinity maturation of the immune response. The general algorithm, named CLONALG, is derived primarily to perform machine learning and pattern recognition tasks, and then it is adapted to solve optimization problems, emphasizing multimodal and combinatorial optimization. Two versions of the algorithm are derived, their computational cost per iteration is presented, and a sensitivity analysis in relation to the user-defined parameters is given. CLONALG is also contrasted with evolutionary algorithms. Several benchmark problems are considered to evaluate the performance of CLONALG and it is also compared to a niching method for multimodal function optimization.

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Learning and optimization using the clonal selection principle

We consider the generalization of the classical P∥Cmax problem arising when a given limit k is imposed on the number of jobs that can be assigned to any machine. This generalization has practical interest in the optimization of assembly lines for printed circuit boards (PCB). The problem is strongly NP-hard for general k, it is solvable in O(n log n) time for fixed k=2, while it remains strongly NP-hard for any fixed k⩾3. We consider immediate adaptations of simple upper and lower bounds for P∥Cmax, and analyse their worst-case behaviour. We show that the cardinality constraint does not strengthen the LP relaxation of the problem, and that the worst-case performance of the bounds for P∥Cmax generally worsen when they are adapted to the new problem. New specifically tailored lower bounds are introduced, and their average tightness is evaluated through extensive computational experiments on randomly generated test instances. Copyright © 2001 John Wiley & Sons, Ltd.

Bounds for the cardinality constrained P∥Cmax problem

We consider generalized assignment problems with different objective functions: min-sum, max-sum, min-max, max-min. We review transformations, bounds, approximation algorithms and exact algorithms. The results of extensive computational experiments are given.

Generalized Assignment Problems

: Transportation problems involving routing and loading at the same time are currently a hot topic in combinatorial optimization. The interest of researchers and practitioners is motivated by the intrinsic difficulty of this research area, which combines two computationally hard problems, and by its practical relevance in important real world applications. This annotated bibliography aims at collecting, in a systematic way, the most relevant results obtained in the area of vehicle routing with loading constraints, with the objective of stimulating further research in this promising area. Keywords: Vehicle routing, Loading, Two-dimensional packing, Three-dimensional packing, Traveling salesman, Pickup and delivery. MSC: 90B06, 90C10, 90C27, 90C59

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An annotated bibliography of combined routing and loading problems

The FORTRAN implementation of an efficient algorithm which solves the 0–1 single knapsack problem is given. Computational results are presented, showing the proposed method to be generally superior to the best known algorithms.

Algorithm 37 Algorithm for the solution of the 0–1 single knapsack problem

An important variant of two-dimensional bin packing problem (2BP), which is also used in some approximation algorithms for its solution, is the strip packing problem (2SP), in which the items have to be packed in a strip of width W and infinite height, so as to minimize the height at which the strip is used. This chapter starts by reviewing classical mathematical models which have implications relevant to the topic of the present survey, and discuss more recent results. It devotes to upper bounds; including metaheuristics and approximation algorithms, lower bound, and exact algorithms. Most of the off-line heuristic algorithms from the literature are of the greedy type, and can be classified in two families: one-phase algorithms and two-phase algorithms. Good lower bounds on the optimal solution value are important both in the implementation of exact enumerative approaches and in the empirical evaluation of approximate solutions.

Silvano Martello

Papers

Bounds for the cardinality constrained P∥Cmax problem

Generalized Assignment Problems

An annotated bibliography of combined routing and loading problems

Algorithm 37 Algorithm for the solution of the 0–1 single knapsack problem

Two‐Dimensional Bin Packing Problems