Self-adaptive metaheuristics for solving a multi-objective 2-dimensional vector packing problem

In this paper, a multi-objective 2-dimensional vector packing problem is presented. It consists in packing a set of items, each having two sizes in two independent dimensions, say, a weight and a length into a finite number of bins, while concurrently optimizing three cost functions. The first objective is the minimization of the number of used bins. The second one is the minimization of the maximum length of a bin. The third objective consists in balancing the load overall the bins by minimizing the difference between the maximum length and the minimum length of a bin. Two population-based metaheuristics are performed to tackle this problem. These metaheuristics use different indirect encoding approaches in order to find good permutations of items which are then packed by a separate decoder routine whose parameters are embedded in the solution encoding. It leads to a self-adaptive metaheuristic where the parameters are adjusted during the search process. The performance of these strategies is assessed and compared against benchmarks inspired from the literature.     

Exact algorithms for the two-dimensional guillotine knapsack

The two-dimensional knapsack problem requires to pack a maximum profit subset of “small” rectangular items into a unique “large” rectangular sheet. Packing must be orthogonal without rotation, i.e., all the rectangle heights must be parallel in the packing, and parallel to the height of the sheet. In addition, we require that each item can be unloaded from the sheet in stages, i.e., by unloading simultaneously all items packed at the same either y or x coordinate. This corresponds to use guillotine cuts in the associated cutting problem.In this paper we present a recursive exact procedure that, given a set of items and a unique sheet, constructs the set of associated guillotine packings. Such a procedure is then embedded into two exact algorithms for solving the guillotine two-dimensional knapsack problem. The algorithms are computationally evaluated on well-known benchmark instances from the literature.The C++ source code of the recursive procedure is available upon request from the authors.     

Facility layout using weighted association rule-based data mining algorithms: Evaluation with simulation

Facility layout has considerable effects on the operational productivity and efficiency of a facility because of its direct effect on material handling costs. The objective of this study is to propose new weighted association rule-based data mining approaches for facility layout problem. Classic association rule-based approaches assume that each item has the same level of significance. On the other hand, in weighted association rule-based approaches, each item is assigned a weight according to its significance with respect to some user defined criteria. In this study, different weighted association rule-based data mining approaches, namely MINWAL(O), MINWAL(W), WARM and BWARM, are applied to facility layout problem. To confirm the viability of the proposed approaches, two case studies are presented. The approaches are compared in terms of general performance criteria for the facility layout problems using simulation. This is the first study that applies weighted association rule-based data mining approaches to facility layout problem. To address the needs in practice, “demand”, “part handling factor” and “efficiency of material handling equipment” are used as the weighting criteria. Then, this study differs from the previous works in that it considers the three key location factors together.     

A constraint programming approach to the multiple-venue,sport-scheduling problem

In this paper, we consider the problem of scheduling sports competitions over several venues which are not associated with any of the competitors. A two-phase, constraint programming approach is developed, first identifying a solution that designates the participants and schedules each of the competitions, then assigning each competitor as the “home” or the “away” team. Computational experiments are conducted and the results are compared with an integer goal programming approach. The constraint programming approach achieves optimal solutions for problems with up to sixteen teams, and near-optimal solutions for problems with up to thirty teams.     

Packing items from a triangular distribution

We consider the problem of packing n items, which are drawn according to a probability distribution whose density function is triangular in shape, into bins of unit capacity. For triangles which represent density functions whose expectation is 1/p for p = 3, 4, 5,…, we give a packing strategy for which the ratio of the number of bins used in the packing to the expected total size of the items asymptotically approaches 1.     

Solving the variable size bin packing problem with discretized formulations

In this paper we study the use of a discretized formulation for solving the variable size bin packing problem (VSBPP). The VSBPP is a generalization of the bin packing problem where bins of different capacities (and different costs) are available for packing a set of items. The objective is to pack all the items minimizing the total cost associated with the bins. We start by presenting a straightforward integer programming formulation to the problem and later on, propose a less straightforward formulation obtained by using a so-called discretized model reformulation technique proposed for other problems (see [Gouveia L. A 2n constraint formulation for the capacitated minimal spanning tree problem. Operations Research 1995; 43:130–141; Gouveia L, Saldanha-da-Gama F. On the capacitated concentrator location problem: a reformulation by discretization. Computers and Operations Research 2006; 33:1242–1258]). New valid inequalities suggested by the variables of the discretized model are also proposed to strengthen the original linear relaxation bounds. Computational results (see Section 4) with up to 1000 items show that these valid inequalities not only enhance the linear programming relaxation bound but may also be extremely helpful when using a commercial package for solving optimally VSBPP.     

Degeneracy of the EM algorithm for the MLE of multivariate Gaussian mixtures and dynamic constraints

EM algorithms for multivariate normal mixture decomposition have been recently proposed in order to maximize the likelihood function in a constrained parameter space having no singularities and a reduced number of spurious local maxima. However, such approaches require some a priori information about the eigenvalues of the covariance matrices. The behavior of the EM algorithm near a degenerated solution is investigated. The obtained theoretical results would suggest a new kind of constraint based on the dissimilarity between two consecutive updates of the eigenvalues of each covariance matrix. The performances of such a “dynamic” constraint are evaluated on the grounds of some numerical experiments.     

The min-conflict packing problem

In the classical bin-packing problem with conflicts (BPC), the goal is to minimize the number of bins used to pack a set of items subject to disjunction constraints. In this paper, we study a new version of BPC: the min-conflict packing problem (MCBP), in which we minimize the number of violated conflicts when the number of bins is fixed. In order to find a tradeoff between the number of bins used and the violation of the conflict constraints, we also consider a bi-objective version of this problem. We show that the special structure of its Pareto front allows to reformulate the problem as a small set of MCBP. We solved these two problems through heuristics, column-generation methods, and a tabu search. Computational experiments are reported to assess the quality of our methods.     

Tree-decomposition based heuristics for the two-dimensional bin packing problem with conflicts

This paper deals with the two-dimensional bin packing problem with conflicts (BPC-2D). Given a finite set of rectangular items, an unlimited number of rectangular bins and a conflict graph, the goal is to find a conflict-free packing of the items minimizing the number of bins used. In this paper, we propose a new framework based on a tree-decomposition for solving this problem. It proceeds by decomposing a BPC-2D instance into subproblems to be solved independently. Applying this decomposition method is not straightforward, since merging partial solutions is hard. Several heuristic strategies are proposed to make an effective use of the decomposition. Computational experiments show the practical effectiveness of our approach.     

Exactly solving packing problems with fragmentation

In packing problems with fragmentation a set of items of known weight is given, together with a set of bins of limited capacity; the task is to find an assignment of items to bins such that the sum of items assigned to the same bin does not exceed its capacity. As a distinctive feature, items can be split at a price, and fractionally assigned to different bins. Arising in diverse application fields, packing with fragmentation has been investigated in the literature from both theoretical, modeling, approximation and exact optimization points of view.We improve the theoretical understanding of the problem and we introduce new models by exploiting only its combinatorial nature. We design new exact solution algorithms and heuristics based on these models. We consider also variants from the literature with different objective functions and the option of handling weight overhead after splitting. We present experimental results on both datasets from the literature and new, more challenging, ones. These show that our algorithms are both flexible and effective, outperforming by orders of magnitude previous approaches from the literature for all the variants considered. By using our algorithms we could also assess the impact of explicitly handling split overhead, in terms of both solutions quality and computing effort.     

Solving a real-time allocation problem with constraint programming

In this paper, we present an original approach (CPRTA for “Constraint Programming for solving Real-Time Allocation”) based on constraint programming to solve a static allocation problem of hard real-time tasks. This problem consists in assigning periodic tasks to distributed processors in the context of fixed priority preemptive scheduling. CPRTA is built on dynamic constraint programming together with a learning method to find a feasible processor allocation under constraints. Two efficient new approaches are proposed and validated with experimental results. Moreover, CPRTA exhibits very interesting properties. It is complete (if a problem has no solution, the algorithm is able to prove it); it is non-parametric (it does not require specific tuning) thus allowing a large diversity of models to be easily considered. Finally, thanks to its capacity to explain failures, it offers attractive perspectives for guiding the architectural design process.     

An investigation of mating and population maintenance strategies in hybrid genetic heuristics for product line designs

This research builds on prior work on developing near optimal solutions to the product line design problems within the conjoint analysis framework. In this research, we investigate and compare different genetic algorithm operators; in particular, we examine systematically the impact of employing alternative population maintenance strategies and mutation techniques within our problem context. Two alternative population maintenance methods, that we term “Emigration” and “Malthusian” strategies, are deployed to govern how individual product lines in one generation are carried over to the next generation. We also allow for two different types of reproduction methods termed “Equal Opportunity” in which the parents to be paired for mating are selected with equal opportunity and a second based on always choosing the best string in the current generation as one of the parents which is referred to as the “Queen bee”, while the other parent is randomly selected from the set of parent strings. We also look at the impact of integrating the artificial intelligence approach with a traditional optimization approach by seeding the GA with solutions obtained from a Dynamic Programming heuristic proposed by others. A detailed statistical analysis is also carried out to determine the impact of various problem and technique aspects on multiple measures of performance through means of a Monte Carlo simulation study. Our results indicate that such proposed procedures are able to provide multiple “good” solutions. This provides more flexibility for the decision makers as they now have the opportunity to select from a number of very good product lines. The results obtained using our approaches are encouraging, with statistically significant improvements averaging 5% or more, when compared to the traditional benchmark of the heuristic dynamic programming technique.     

Packing first,routing second—a heuristic for the vehicle routing and loading problem

The Vehicle Routing and Loading Problem (VRLP) results by combining vehicle routing, possibly with time windows, and three-dimensional loading. Some packing constraints of high practical relevance, among them an unloading sequence constraint and a support constraint, are also part of the VRLP. Different formulations of the VRLP are considered and the issue is discussed under which circumstances routing and packing should be tackled as a combined task. A two-stage heuristic is presented following a “packing first, routing second” approach, i.e. the packing of goods and the routing of vehicles is done in two strictly separated stages. High quality results are achieved in short computation times for the 46 VRLP instances recently introduced by Moura and Oliveira. Moreover 120 new large benchmark instances including up to 1000 customers and 50,000 boxes are introduced and results for these instances are also reported.     

Chunky and equal-spaced polynomial multiplication

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many “easy” cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing “sparse” and “dense” methods. We prove that these approaches provide significant improvements in many cases but in the worst case are still comparable to the fastest existing algorithms.     

Multidimensional Cube Packing

We consider the d-dimensional cube packing problem (d-CPP): given a list L of d-dimensional cubes and (an unlimited quantity of) d-dimensional unit-capacity cubes, called bins, find a packing of L into the minimum number of bins. We present two approximation algorithms for d-CPP, for fixed d. The first algorithm has an asymptotic performance bound that can be made arbitrarily close to 2 – (1/2)^d . The second algorithm is an improvement of the first and has an asymptotic performance bound that can be made arbitrarily close to 2 – (2/3)^d . To our knowledge, these results improve the bounds known so far for d = 2 and d = 3, and are the first results with bounds that are not exponential in the dimension.     

互联网信息组织和规划中的带拒绝装箱问题

讨论如下定义的带拒绝装箱问题：设有许多等长的一维箱子，给定一个物品集，每个物品有两个参数：长度和罚值．物品可以放入箱子也可被拒绝放入箱子．如果将物品放入箱子，则使该箱剩余长度减少．一旦需将某一物品放入某一箱中，而该箱的剩余长度不够时，则需启用新箱子．如果物品被拒绝放入任何箱中，则产生惩罚．问怎样安排物品使所用箱子数与未装箱的物品总罚值之和最小．该问题是一个新的组合优化问题，来源于内部互联网的信息组织和规划．该文首先给出一个最优解值的下界估计，它可用于分枝定界法求最优解．由于该问题是强NP-难的，该文进一步研究它的离线和在线近似算法的设计与分析．文中给出一个离线算法，其绝对性能比为2；同时给出一个在线算法，其绝对性能比不超过3，渐近性能比为2，还对算法性能比的下界进行了讨论．     

Mixed integer programming approaches to exact minimization of total treatment time in cancer radiotherapy using multileaf collimators

The effectiveness of radiation therapy for cancer depends on the patient remaining still during treatment. It is thus important to minimize the total treatment time (TTT). When such treatment is delivered using multileaf collimators in “step-and-shoot” mode, it consists of a sequence of collimator configurations, or patterns; for each, the patient is exposed to radiation for a specified time, or beam-on time. The TTT can thus be divided into the total beam-on time and the time spent reconfiguring the collimators. The latter can reasonably be approximated by the number of patterns, multiplied by a constant overhead time per pattern. Previous approaches to this problem have all been heuristic; in particular none of them actually use the pattern overhead time to ascertain the best trade-off between beam-on time and number of patterns. In this paper, we develop exact solution approaches, based on mixed integer programming (MIP) formulations, which minimize the TTT. We consider direct solution of MIP formulations, and then exploit the bicriteria structure of the objective to derive an algorithm that “steps up” through the number of patterns used, leading to substantial computational savings.     

Robotic vocabulary building using extension inference and implicit contrast

TWIG (“Transportable Word Intension Generator”) is a system that allows a robot to learn compositional meanings for new words that are grounded in its sensory capabilities. The system is novel in its use of logical semantics to infer which entities in the environment are the referents (extensions) of unfamiliar words; its ability to learn the meanings of deictic (“I,” “this”) pronouns in a real sensory environment; its use of decision trees to implicitly contrast new word definitions with existing ones, thereby creating more complex definitions than if each word were treated as a separate learning problem; and its ability to use words learned in an unsupervised manner in complete grammatical sentences for production, comprehension, or referent inference. In an experiment with a physically embodied robot, TWIG learns grounded meanings for the words “I” and “you,” learns that “this” and “that” refer to objects of varying proximity, that “he” is someone talked about in the third person, and that “above” and “below” refer to height differences between objects. Follow-up experiments demonstrate the system's ability to learn different conjugations of “to be”; show that removing either the extension inference or implicit contrast components of the system results in worse definitions; and demonstrate how decision trees can be used to model shifts in meaning based on context in the case of color words.     

Minimum-cost single-source 2-splittable flow

In the single-source unsplittable flow problem, commodities must be routed simultaneously from a common source vertex to certain sinks in a given directed graph with edge capacities and costs. The demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. Moreover the cost of the solution should not exceed a given budget. An important open question is whether a simultaneous (2,1)-approximation can be achieved for minimizing congestion and cost, i.e., the budget constraint should not be violated. In this note we show that this is possible for the case of 2-splittable flows, i.e., flows where the demand of each commodity is routed along at most two paths. Our result holds under the “no-bottleneck” assumption, i.e., the maximum demand does not exceed the minimum capacity.     

Finding optimal satisficing strategies for and-or trees

Many tasks require evaluating a specified Boolean expression φ over a set of probabilistic tests whose costs and success probabilities are each known. A strategy specifies when to perform which test, towards determining the overall outcome of φ. We are interested in finding the strategy with the minimum expected cost.As this task is typically NP-hard—for example, when tests can occur many times within φ, or when there are probabilistic correlations between the test outcomes—we consider those cases in which the tests are probabilistically independent and each appears only once in φ. In such cases, φ can be written as an and-or tree, where each internal node corresponds to either the “and” or “or” of its children, and each leaf node is a probabilistic test. In this paper we investigate “probabilistic and-or tree resolution” (PAOTR), namely the problem of finding optimal strategies for and-or trees.We first consider a depth-first approach: evaluate each penultimate rooted subtree in isolation, replace each such subtree with a single “mega-test”, and recurse on the resulting reduced tree. We show that the strategies produced by this approach are optimal for and-or trees with depth at most two but can be arbitrarily sub-optimal for deeper trees.Each depth-first strategy can be described by giving the linear relative order in which tests are to be executed, with the understanding that any test whose outcome becomes irrelevant is skipped. The class of linear strategies is strictly larger than depth-first strategies. We show that even the best linear strategy can also be arbitrarily sub-optimal.We next prove that an optimal strategy honors a natural partial order among tests with a common parent node (“leaf-sibling tests”), and use this to produce a dynamic programming algorithm that finds the optimal strategy in time O(d²d(r+1)), where r is the maximum number of leaf-siblings and d is the number of leaf-parents; hence, for trees with a bounded number of internal nodes, this run-time is polynomial in the tree size. We also present another special class of and-or trees for which this task takes polynomial time.We close by presenting a number of other plausible approaches to PAOTR, together with counterexamples to show their limitations.