A New Constructive Heuristic Driven by Machine Learning for the Traveling Salesman Problem

Recent systems applying Machine Learning (ML) to solve the Traveling Salesman Problem (TSP) exhibit issues when they try to scale up to real case scenarios with several hundred vertices. The use of Candidate Lists (CLs) has been brought up to cope with the issues. A CL is defined as a subset of all the edges linked to a given vertex such that it contains mainly edges that are believed to be found in the optimal tour. The initialization procedure that identifies a CL for each vertex in the TSP aids the solver by restricting the search space during solution creation. It results in a reduction of the computational burden as well, which is highly recommended when solving large TSPs. So far, ML was engaged to create CLs and values on the elements of these CLs by expressing ML preferences at solution insertion. Although promising, these systems do not restrict what the ML learns and does to create solutions, bringing with them some generalization issues. Therefore, motivated by exploratory and statistical studies of the CL behavior in multiple TSP solutions, in this work, we rethink the usage of ML by purposely employing this system just on a task that avoids well-known ML weaknesses, such as training in presence of frequent outliers and the detection of under-represented events. The task is to confirm inclusion in a solution just for edges that are most likely optimal. The CLs of the edge considered for inclusion are employed as an input of the neural network, and the ML is in charge of distinguishing when such edge is in the optimal solution from when it is not. The proposed approach enables a reasonable generalization and unveils an efficient balance between ML and optimization techniques. Our ML-Constructive heuristic is trained on small instances. Then, it is able to produce solutions—without losing quality—for large problems as well. We compare our method against classic constructive heuristics, showing that the new approach performs well for TSPLIB instances up to 1748 cities. Although ML-Constructive exhibits an expensive constant computation time due to training, we proved that the computational complexity in the worst-case scenario—for the solution construction after training—is O(n2logn2), n being the number of vertices in the TSP instance.

GA-BASED IMPROVEMENT HEURISTIC FOR MINIMIZING MAKESPAN IN A GENERAL FLOW-SHOP SCHEDULING PROBLEM

The study proposes an evolutionary algorithm-based improvement heuristic for the permutation flow-shop problem. The method uses a constructive heuristic to arrive at a good first solution. The GA-based improvement heuristic is used in conjunction with CDS, Gupta's algorithm, and Palmer's Slope Index, which are all well-known constructive heuristics. The method is put to the test on a series of ten issues that vary from 4 to 25 tasks and 4 to 30 machines. The outcomes are also compared to some of the most well-known lower-bound options

A simple and effective algorithm for the maximum happy vertices problem

In a recent paper, a solution approach to the Maximum Happy Vertices Problem has been proposed. The approach is based on a constructive heuristic improved by a matheuristic local search phase. We propose a new procedure able to outperform the previous solution algorithm both in terms of solution quality and computational time. Our approach is based on simple ingredients implying as starting solution generator an approximation algorithm and as an improving phase a new matheuristic local search. The procedure is then extended to a multi-start configuration, able to further improve the solution quality at the cost of an acceptable increase in computational time.

Solution procedures for block selection and sequencing in flat-bedded potash underground mines

Phosphates, and especially potash, play an essential role in the increase in crop yields. Potash is mined in Germany in underground mines using a conventional drill-and-blast technique. The most commercially valuable mineral contained in potash is the potassium chloride that is separated from the potash in aboveground processing plants. The processing plants perform economically best if the amount of potassium contained in the output is equal to a specific value, the so-called optimal operating point. Therefore, quality-oriented extraction plays a decisive role in reducing processing costs. In this paper, we mathematically formulate a block selection and sequencing problem with a quality-oriented objective function that aims at an even extraction of potash regarding the potassium content. We, thereby, have to observe some precedence relations, maximum and minimum limits of the output, and a quality tolerance range within a given planning horizon. We model the problem as a mixed-integer nonlinear program which is then linearized. We show that our problem is $${\mathcal {NP}}$$ NP -hard in the strong sense with the result that a MILP-solver cannot find feasible solutions for the most challenging problem instances at hand. Accordingly, we develop a problem-specific constructive heuristic that finds feasible solutions for each of our test instances. A comprehensive experimental performance analysis shows that a sophisticated combination of the proposed heuristic with the mathematical program improves the feasible solutions achieved by the heuristic, on average, by $$92.5\%$$ 92.5 % .

A Constructive Heuristic approach for solving Mixed Integer Nonlinear cascade optimization considering an hydrological uncertainty model

<p>The quest for a lower operational cost of daily scheduling cascade hydropower generation is subject to imperfections in the representation of reality due to it multiple uncertainties. One way to reduce uncertainties is to improve the representation of the hydrological processes and the operation of the electrical components of the system. A complementary strategy to deal with the difficulty of representing hydrological processes is to move from a deterministic approach to a probabilistic approach considering scenarios. These two proposals usually lead to high-dimensional problems that require Mixed Integer Nonlinear Programming for solving. This work proposes the use of simple Constructive Heuristics to solve this kind of problem guarding the non-linear formulations. The GAMS software was used to represent a hydropower cascade with individualized turbine with multiple inflow scenarios as presented in the formulation bellow. For each scenario s:</p><p><img src="https://contentmanager.copernicus.org/fileStorageProxy.php?f=gepj.d08e229d6cff52468730161/sdaolpUECMynit/12UGE&app=m&a=0&c=b47773da2e45f98cd009ccae1c68e910&ct=x&pn=gepj.elif&d=1" alt=""></p><p>Z is the objective function. a is a coefficient considered in the proposed Constructive Heuristic. Q is the outflow of the turbine. H is the set of hydropower plants. U is the set of turbines. t is the set of hours considered in the problem. s is the set of considered scenarios. α is a conversion term. τ is the time delay of the hydropower plants. β is the set of hydropower plants located upstream of a hydropower plant. I is the incremental flow. V is the volume of the reservoir. D is the power demand. Pg is power generated per turbine. Pmin and Pmax are the thresholds of power of the turbine. Pst is the net power generated by the turbine. Pgg are the electrical losses of the generator. Pmt are the mechanical losses of the generator. f1, f2, f3, f4, f5, f6, f7 and f8 are functions. HH is the hydraulic net head. UP is the upstream level. Down is the downstream level. Lo are the hydraulic and mechanical losses. ρ is the turbine efficiency curve. The CONOPT solver was selected for it resolution. The  Constructive Heuristic scheme considers a continuous variable “a” to represent if the turbine is opened or closed. After the first resolution of the problem with Nonlinear Programming, the integer part number of turbines for each power plant is fixed. The fractional part is tested for all the possible configurations and the best outcome is chosen. The results of the Constructive Heuristic were compared with other similar works with perfect correspondence. The system was tested for 51 ECMWF precipitation forecast scenarios as input of the hydrologic model MGB (Large Basin Model) with coherent results. The proposal is a feasible approach to reach unique optimal solutions for high-dimensional non-linear problems considering integer variables.</p>

The capacitated maximal covering location problem with heterogeneous facilities and vehicles and different setup costs: An effective heuristic approach

In this research, a maximal covering location problem (MCLP) with real-world constraints such as multiple types of facilities and vehicles with different setup costs is taken into account. An original mixed integer linear programming (MILP) model is constructed in order to find the optimal solution. Since the problem at hand is shown to be NP-hard, a constructive heuristic method and a meta-heuristic approach based on genetic algorithm (GA) are developed to solve the problem. To find the most effective solution technique, a set of problems of different sizes is randomly generated and solved by the proposed solution methods. Computational results demonstrate that the heuristic method is capable of producing optimal or near-optimal solutions in a rational execution time.