Random-key genetic algorithms were introduced by Bean (ORSA J. Comput. 6:154---160, 1994) for solving sequencing problems in combinatorial optimization. Since then, they have been extended to handle a wide class of combinatorial optimization problems. This paper presents a tutorial on the implementation and use of biased random-key genetic algorithms for solving combinatorial optimization problems. Biased random-key genetic algorithms are a variant of random-key genetic algorithms, where one of the parents used for mating is biased to be of higher fitness than the other parent. After introducing the basics of biased random-key genetic algorithms, the paper discusses in some detail implementation issues, illustrating the ease in which sequential and parallel heuristics based on biased random-key genetic algorithms can be developed. A survey of applications that have recently appeared in the literature is also given.

Biased random-key genetic algorithms for combinatorial optimization

Constraints in container loading – A state-of-the-art review

We consider a special case of the symmetric capacitated vehicle routing problem, in which a fleet of K identical vehicles must serve n customers, each with a given demand consisting in a set of rectangular two-dimensional weighted items. The vehicles have a two-dimensional loading surface and a maximum weight capacity. The aim is to find a partition of the customers into routes of minimum total cost such that, for each vehicle, the weight capacity is taken into account and a feasible two-dimensional allocation of the items into the loading surface exists.

The problem has several practical applications in freight transportation, and it is NP-hard in the strong sense. We propose an exact approach, based on a branch-and-cut algorithm, for the minimization of the routing cost that iteratively calls a branch-and-bound algorithm for checking the feasibility of the loadings. Heuristics are also used to improve the overall performance of the algorithm. The effectiveness of the approach is shown by means of computational results.

An Exact Approach for the Vehicle Routing Problem with Two-Dimensional Loading Constraints

https://www.inf.utfsm.cl/~mcriff/SP/articulo1.pdf

A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces

We consider difficult combinatorial optimization problems arising in transportation logistics when one is interested in optimizing both the routing of vehicles and the loading of goods into them. The separate problems (routing and loading) are already \(\mathcal{NP}\)-hard, and very difficult to solve in practice. A fortiori their combination is extremely challenging and stimulating. Although the specific literature is still quite limited, a first attempt to a systematic view of this field can be useful both to academic researchers and to practitioners. We review vehicle routing problems with two- and three-dimensional loading constraints. Other combinations of routing and special loading constraints arising from industrial applications are also considered.

Routing problems with loading constraints

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Combining the use of our data structure for characterizing feasible packings with our new classes of lower bounds, and other heuristics, we develop a two-level tree search algorithm for solving higher-dimensional packing problems to optimality. Computational results are reported, including optimal solutions for all two-dimensional test problems from recent literature.

This is the third in a series of articles describing new approaches to higher-dimensional packing.

/pdf/an-exact-algorithm-for-higher-dimensional-orthogonal-packing-3zcnp6urot.pdf

An Exact Algorithm for Higher-Dimensional Orthogonal Packing

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for instance in two- or higher-dimensional space. We present a new approach for modeling packings, using a graph-theoretical characterization of feasible packings. Our characterization allows it to deal with classes of packings that share a certain combinatorial structure, instead of having to consider one packing at a time. In addition, we can make use of elegant algorithmic properties of certain classes of graphs. This allows our characterization to be the basis for a successful branch-and-bound framework.This is the first in a series of papers describing new approaches to higher-dimensional packing.

A Combinatorial Characterization of Higher-Dimensional Orthogonal Packing

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for instances in two- or higher-dimensional space. We present a new approach for modeling packings, using a graph-theoretical characterization of feasible packings. Our characterization allows it to deal with classes of packings that share a certain combinatorial structure, instead of having to consider one packing at a time. In addition, we can make use of elegant algorithmic properties of certain classes of graphs. This allows our characterization to be the basis for a successful branch-and-bound framework. 
This is the first in a series of papers describing new approaches to higher-dimensional packing.

A combinatorial characterization of higher-dimensional orthogonal packing

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Combining the use of our data structure for characterizing feasible packings with our new classes of lower bounds, and other heuristics, we develop a two-level tree search algorithm for solving higher-dimensional packing problems to optimality. Computational results are reported, including optimal solutions for all two--dimensional test problems from recent literature. 
This is the third in a series of articles describing new approaches to higher-dimensional packing; see cs.DS/0310032 and cs.DS/0402044.

pdf/an-exact-algorithm-for-higher-dimensional-orthogonal-packing-3gt70cgekn.pdf

An exact algorithm for higher-dimensional orthogonal packing

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An Exact Algorithm for Higher-Dimensional Orthogonal Packing

A Combinatorial Characterization of Higher-Dimensional Orthogonal Packing

A combinatorial characterization of higher-dimensional orthogonal packing

An exact algorithm for higher-dimensional orthogonal packing