Neuroevolution for solving multiobjective knapsack problems

The multiobjective knapsack problem (MOKP) is an important combinatorial problem that arises in various applications, including resource allocation, computer science and finance. When tackling this problem by evolutionary multiobjective optimization algorithms (EMOAs), it has been demonstrated that traditional recombination operators acting on binary solution representations are susceptible to a loss of diversity and poor scalability. To address those issues, we propose to use artificial neural networks for generating solutions by performing a binary classification of items using the information about their profits and weights. As gradient-based learning cannot be used when target values are unknown, neuroevolution is adapted to adjust the neural network parameters. The main contribution of this study resides in developing a solution encoding and genotype-phenotype mapping for EMOAs to solve MOKPs. The proposal is implemented within a state-of-the-art EMOA and benchmarked against traditional variation operators based on binary crossovers. The obtained experimental results indicate a superior performance of the proposed approach. Furthermore, it is advantageous in terms of scalability and can be readily incorporated into different EMOAs.Portuguese “Fundação para a Ciência e Tecnologia” under grant PEst-C/CTM/LA0025/2013 (Projecto Estratégico - LA 25 - 2013-2014 - Strategic Project - LA 25 - 2013-2014

An Evolutionary Approach for Solving the Rubik’s Cube Incorporating Exact Methods

Abstract. Solutions calculated by Evolutionary Algorithms have come to surpass exact methods for solving various problems. The Rubik’s Cube multiobjective optimization problem is one such area. In this work we present an evolutionary approach to solve the Rubik’s Cube with a low number of moves by building upon the classic Thistlethwaite’s approach. We provide a group theoretic analysis of the subproblem complexity in-duced by Thistlethwaite’s group transitions and design an Evolutionary Algorithm from the ground up including detailed derivation of our cus-tom fitness functions. The implementation resulting from these observa-tions is thoroughly tested for integrity and random scrambles, revealing performance that is competitive with exact methods without the need for pre-calculated lookup-tables.

Hybrid tabu search – strawberry algorithm for multidimensional knapsack problem

Multidimensional Knapsack Problem (MKP) has been widely used to model real-life combinatorial problems. It is also used extensively in experiments to test the performances of metaheuristic algorithms and their hybrids. For example, Tabu Search (TS) has been successfully hybridized with other techniques, including particle swarm optimization (PSO) algorithm and the two-stage TS algorithm to solve MKP. In 2011, a new metaheuristic known as Strawberry algorithm (SBA) was initiated. Since then, it has been vastly applied to solve engineering problems. However, SBA has never been deployed to solve MKP. Therefore, a new hybrid of TS-SBA is proposed in this study to solve MKP with the objective of maximizing the total profit. The Greedy heuristics by ratio was employed to construct an initial solution. Next, the solution was enhanced by using the hybrid TS-SBA. The parameters setting to run the hybrid TS-SBA was determined by using a combination of Factorial Design of Experiments and Decision Tree Data Mining methods. Finally, the hybrid TS-SBA was evaluated using an MKP benchmark problem. It consisted of 270 test problems with different sizes of constraints and decision variables. The findings revealed that on average the hybrid TS-SBA was able to increase 1.97% profit of the initial solution. However, the best-known solution from past studies seemed to outperform the hybrid TS-SBA with an average difference of 3.69%. Notably, the novel hybrid TS-SBA proposed in this study may facilitate decisionmakers to solve real applications of MKP. It may also be applied to solve other variants of knapsack problems (KPs) with minor modifications

A Product Recommendation Algorithm Based on Knapsack Optimization

Personalized Recommender System becomes an important research field in Electronic Commerce, and the main goal of current recommendation models is provide Best-Service to users. But, from enterprise’s viewpoint, the Max-Earning strategy is necessary to improve the benefit of enterprise. To solving this problem, knapsack model is applied to describe the commonly used Top-N recommend mechanism firstly. Then, the enterprise’s earnings are described as a constraint in knapsack model, a product recommended algorithm is proposed at the basis of optimization of knapsack problem. Experimental results show the proposed algorithm has similar performance with CF model when earning requirement and amount of recommended products is lower. So, both user’s value and enterprise’s value are improved through the proposed algorithm

A Random Forest Assisted Evolutionary Algorithm for Data-Driven Constrained Multi-Objective Combinatorial Optimization of Trauma Systems for publication

Many real-world optimization problems can be solved by using the data-driven approach only, simply because no analytic objective functions are available for evaluating candidate solutions. In this work, we address a class of expensive datadriven constrained multi-objective combinatorial optimization problems, where the objectives and constraints can be calculated only on the basis of large amount of data. To solve this class of problems, we propose to use random forests and radial basis function networks as surrogates to approximate both objective and constraint functions. In addition, logistic regression models are introduced to rectify the surrogate-assisted fitness evaluations and a stochastic ranking selection is adopted to further reduce the influences of the approximated constraint functions. Three variants of the proposed algorithm are empirically evaluated on multi-objective knapsack benchmark problems and two realworld trauma system design problems. Experimental results demonstrate that the variant using random forest models as the surrogates are effective and efficient in solving data-driven constrained multi-objective combinatorial optimization problems

A Binary differential search algorithm for the 0-1 multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is known to be NP-hard in operations research and it has a wide range of applications in engineering and management. In this study, we propose a binary differential search method to solve 0-1 MKPs where the stochastic search is guided by a Brownian motion-like random walk. Our proposed method comprises two main operations: discrete solution generation and feasible solution production. Discrete solutions are generated by integrating Brownian motion-like random search with an integer-rounding operation. However, the rounded discrete variables may violate the constraints. Thus, a feasible solution production strategy is used to maintain the feasibility of the rounded discrete variables. To demonstrate the efficiency of our proposed algorithm, we solved various 0-1 MKPs using our proposed algorithm as well as some existing meta-heuristic methods. The numerical results obtained demonstrated that our algorithm performs better than existing meta-heuristic methods. Furthermore, our algorithm has the capacity to solve large-scale 0-1 MKPs

Incorporating Memory and Learning Mechanisms Into Meta-RaPS

Due to the rapid increase of dimensions and complexity of real life problems, it has become more difficult to find optimal solutions using only exact mathematical methods. The need to find near-optimal solutions in an acceptable amount of time is a challenge when developing more sophisticated approaches. A proper answer to this challenge can be through the implementation of metaheuristic approaches. However, a more powerful answer might be reached by incorporating intelligence into metaheuristics. Meta-RaPS (Metaheuristic for Randomized Priority Search) is a metaheuristic that creates high quality solutions for discrete optimization problems. It is proposed that incorporating memory and learning mechanisms into Meta-RaPS, which is currently classified as a memoryless metaheuristic, can help the algorithm produce higher quality results. The proposed Meta-RaPS versions were created by taking different perspectives of learning. The first approach taken is Estimation of Distribution Algorithms (EDA), a stochastic learning technique that creates a probability distribution for each decision variable to generate new solutions. The second Meta-RaPS version was developed by utilizing a machine learning algorithm, Q Learning, which has been successfully applied to optimization problems whose output is a sequence of actions. In the third Meta-RaPS version, Path Relinking (PR) was implemented as a post-optimization method in which the new algorithm learns the good attributes by memorizing best solutions, and follows them to reach better solutions. The fourth proposed version of Meta-RaPS presented another form of learning with its ability to adaptively tune parameters. The efficiency of these approaches motivated us to redesign Meta-RaPS by removing the improvement phase and adding a more sophisticated Path Relinking method. The new Meta-RaPS could solve even the largest problems in much less time while keeping up the quality of its solutions. To evaluate their performance, all introduced versions were tested using the 0-1 Multidimensional Knapsack Problem (MKP). After comparing the proposed algorithms, Meta-RaPS PR and Meta-RaPS Q Learning appeared to be the algorithms with the best and worst performance, respectively. On the other hand, they could all show superior performance than other approaches to the 0-1 MKP in the literature

Approximate and Exact Merging of Knapsack Constraints with Cover Inequalities

This paper presents both approximate and exact merged knapsack cover inequalities, a class of cutting planes for knapsack and multiple knapsack integer programs. These inequalities combine the information from knapsack constraints and cover inequalities. Approximate merged knapsack cover inequalities can be generated through a O(n log n) algorithm, where n is the number of variables. This class of inequalities can be strengthened to an exact version with a pseudo-polynomial time algorithm. Computational experiments demonstrate an average improvement of approximately 8% in solution time and 5% in the number of ticks from CPLEX when approximate merged knapsack cover inequalities are implemented as preprocessing cuts to solve some benchmark multiple knapsack problems. Furthermore, exact merged knapsack cover inequalities improve the solution time and number of ticks of some random multiple knapsack instances by 15% and 5%, respectively

A hybrid multi-objective approach to capacitated facility location with flexible store allocation for green logistics modeling

We propose an efficient evolutionary multi-objective optimization approach to the capacitated facility location–allocation problem (CFLP) for solving large instances that considers flexibility at the allocation level, where financial costs and CO2 emissions are considered simultaneously. Our approach utilizes suitably adapted Lagrangian Relaxation models for dealing with costs and CO2 emissions at the allocation level, within a multi-objective evolutionary framework at the location level. Thus our method assesses the robustness of each location solution with respect to our two objectives for customer allocation. We extend our exploration of selected solutions by considering a range of trade-offs for customer allocation