A Methodology for the Range Ordering of Alternatives using a Bayesian Decision Model with Applications to the Space Program

The primary objective of this paper is to provide a reasonably general and essentially unified approach to those problems involving value judgments and subjective decision making, without regard to excessive rigor. The principle areas and methods of attack developed are: (1) The selection of a value measure which emphasizes the fact that the criterion of optimum performance is quite arbitrary, its merits reflecting only the constraints on the problem and the objectives sought, (2) The utilization of statistical decision theory as a basis for the solution and subsequent ev aluation of a class of problems in which a priori value judgments must be assigned by an individual or committee under uncertainty, and (3) The application of the methodology to those areas in which the relative uncertainty level of a decision need be assessed in terms of a cost or penalty incurred in reaching the conclusion. A parti cularly important application is the selection of alternatives (ioe., projects by corporate executives) and the subsequent sensitivity analysis of the decision

Exploring search space trees using an adapted version of Monte Carlo tree search for combinatorial optimization problems

In this article, a novel approach to solve combinatorial optimization problems is proposed. This approach makes use of a heuristic algorithm to explore the search space tree of a problem instance. The algorithm is based on Monte Carlo tree search, a popular algorithm in game playing that is used to explore game trees. By leveraging the combinatorial structure of a problem, several enhancements to the algorithm are proposed. These enhancements aim to efficiently explore the search space tree by pruning subtrees, using a heuristic simulation policy, reducing the domains of variables by eliminating dominated value assignments and using a beam width. They are demonstrated for two specific combinatorial optimization problems: the quay crane scheduling problem with non-crossing constraints and the 0-1 knapsack problem. Computational results show that the algorithm achieves promising results for both problems and eight new best solutions for a benchmark set of instances are found for the former problem. These results indicate that the algorithm is competitive with the state-of-the-art. Apart from this, the results also show evidence that the algorithm is able to learn to correct the incorrect choices made by constructive heuristics

Reinforcement Learning Enhanced Heuristic Search for CombinatorialOptimizationReinforcement Learning Enhanced Heuristic Search for CombinatorialOptimization (Reinforcement Learning gebaseerde heuristieken voorcombinatorische optimalisatie)

The present thesis describes the use of reinforcement learning to enhance heuristic search for solving complex (real-world) optimization problems such as (project) scheduling, routing and assignment. Heuristic search methods are known to deliver good results in a reasonable amount of calculation time, without any guarantee of optimality. Often they require careful parameter tuning to obtain good results.  Reinforcement learning methods on the other hand, learn to act in a possible unknown random environment on a trial-and-error basis. The goal of the hybridizationof heuristic search and reinforcement learning is to generate intelligent search methods which are adaptive and generally applicable while keeping eventual extra overhead to a minimum.Three levels of inclusion of reinforcement learning into heuristic search methods are defined: the direct, the metaheuristic and the hyperheuristic level. At the direct level, the reinforcement learning method searches directly for good quality solutions, while at the metaheuristic and hyperheuristic level, the reinforcement learning component is added for learning good starting solutions, good parameter values, good objective functions, good heuristics, etc. For each level, one or more learning enhanced methods are demonstrated on benchmark and/or real-world problems. A general methodology for learning permutations without any domain knowledge is described. Additionally, a method for learning to select heuristics during search is described and tested on several hard combinatorial optimization problems such as the traveling tournament problem, the patient admission scheduling problem, and the machine reassignment problem. It is shown that this learning selection method performs significantly better than selecting the heuristics at random.From an application point of view, this thesis is mainly, though not exclusively, devoted to scheduling problems. We tackled the multi-mode resource-constrained project scheduling problem, the decentralized resource-constrained multi-project scheduling problem, the (dynamic) flexible job shop scheduling problem and a real-world production scheduling problem from the food industry. Real-world problems often hold a rich structure allowing for learning valuable information. We show that a multi-agent reinforcement learning architecture, morespecifically a network of learning automata with a common reward signal, is very well suited to design new hybrid well performing methods. Manynew best results for project scheduling benchmarks are generated using the proposed GT-MAS approach.status: publishe

10 years of Eternity II – from $2 million puzzle to challenging optimization problem

The Eternity II (EII) puzzle is a commercial edge matching puzzle in which 256 square tiles with four coloured edges must be arranged on a 16 by 16 grid such that all tile edges are matched. In addition, a complete solution requires that the `grey' patterns, which appear only on a subset of the tiles, should be matched to the outer edges of the grid. The puzzle belongs to the more general class of Edge Matching Puzzles, which have been shown to be NP-complete. In July 2007, toy distributor Tomy UK Ltd. released this challenging edge matching puzzle with a $2 million prize. However, to the best of our knowledge, no complete solution has ever been found. Meanwhile, the final scrutiny date for the cash price, 31 December 2010, has passed, leaving the large money prize unclaimed. In its 10 years of existence many people tried to solve EII and some are still trying. Many approaches to Edge Matching Puzzles are reported in the literature. Among these approaches are constraint programming and backtracking, metaheuristics, and evolutionary methods. Other approaches translate the problem into SAT, MILP or max-clique and then solve it with appropriate state of the art solvers. Some approaches have also been implemented on parallel computing or dedicated hardware.status: publishe

A comparison of mathematical formulations for the superpermutation problem

A superpermutation on n symbols is a string that contains each permutation of n symbols as a substring. Due to its numerous applications, it is often desirable to find the smallest superpermutation for a given number of symbols. The smallest superpermutaion and optimal length have already been established for up to five symbols utilizing a clever exhaustive enumeration. However, for more than five symbols, establishing the length of minimal superpermutation and finding a superpermutation for a given value of n continues to be an open problem. So far, the asymmetric traveling salesman formulation of the problems has been able to produce high quality solutions for six symbols. But being a hard optimization problem, this exact technique fails to solve the problem to optimality. Interestingly, multiple formulations are possible for this problem and the present work aims at studying these formulations and identifying formulations which are capable of producing optimal or near optimal superpermutations on six or more than six symbols. Formulations are compared based on the solution quality and are utilized to generate meaningful bounds on the optimal length of the superpermutation. To be able to cope with the increasing complexity of the problem as value of n increases, multiple-phase techniques are proposed which utilize different formulations in each phase. Another contribution of this work is a constructive matheuristic (CMH) strategy for the superpermutation problem. A CMH strategy is a decomposition based method which utilizes optimal solutions of the subproblems to construct a solution for the full problem. Being a heuristic, CMH does not guarantee optimal solutions, but is capable of generating quality solutions for problems with large number of symbols where exact techniques fail to be efficient. CMHs which use different formulations to solve the subproblems are compared and their performances are contrasted. Finally, a comprehensive evaluation of all the proposed formulations based on solution quality, algorithm runtime and generated bounds on the length of the superpermutation is presented. Best solutions found and newly generated bounds on the length of minimal superpermutation are also presented.status: Published onlin

A parallel branch and bound approach for the 3D Cutting and Packing Problem

status: accepte

A parallel tree based approach for the 3D Cutting and Packing Problem

status: accepte