Block-centric GRAPH processing systems have received significant attention in recent years. To produce the required partitions, most of these systems use general-purpose PARTITIONING methods. As a result, the performance of them has been limited. To face this PROBLEM, special PARTITIONING algorithms have been proposed by researchers. However, these methods focused on traditional PARTITIONING measures like the number of cutting-edges and the load-balance. In return, the power of block-centric GRAPH processing systems is due to unique characteristics that are focused on the design of them. According to basic and important characteristics of these systems, in this paper two new measures are proposed as PARTITIONING goals. To the best of our knowledge, the proposed method is the first work that considers the diameter and size of the high-level GRAPH as optimization factors for PARTITIONING purposes. The evaluation of the proposed method over real GRAPHs showed that we could significantly reduce the diameter of the high-level GRAPH. Moreover, the number of cutting-edges of the proposed method are very close to Metis, one of most popular centralized PARTITIONING methods. Since the number of required supersteps in block-centric GRAPH processing systems mainly depends on the diameter of the high-level GRAPH, the proposed method can significantly improve the performance of these systems.

‎GRAPH coloring is the assignment of one color to each vertex of a GRAPH so that two adjacent vertices are not of the same color‎. ‎The GRAPH coloring PROBLEM (GCP) is a matter of combinatorial optimization‎, ‎and the goal of GCP is determining the chromatic number $\chi(G)$‎. ‎Since GCP is an NP-hard PROBLEM‎, ‎then in this paper‎, ‎we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a GRAPH adjacency matrix to colorize or separate a GRAPH‎. ‎To prove the correctness of the proposed algorithm‎, ‎we implement it in MATLAB software‎, ‎and for analysis in terms of solution and execution time‎, ‎we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software‎, ‎and we present the results in tables of various GRAPHs‎. ‎Several available algorithms used the largest degree selection strategy‎, ‎while our proposed algorithm uses the GRAPH adjacency matrix to select the vertex that has the smallest degree for coloring‎. ‎We provide some examples to compare the performance of our algorithm to other available methods‎. ‎We make use of the Dolan-Mor\'e performance profiles to assess the performance of the numerical algorithms‎, ‎and demonstrate the efficiency of our proposed approach in comparison with some existing methods‎.

The dramatic decline in renewable water resources, leakage and pollution in water distribution systems has led to a significant increase in the focus on leakage management and control approaches in most parts of the world. For this purpose, water distribution networks can be subdivided into manageable subdivisions with connecting pipes of these subdivisions equipped with flow meters to control leakage and better manage the water distribution network. In the present study, based on GRAPH theory, the concept of District Meter Area (DMA) is expressed. In order to rank the optimal design of the water distribution network, AHP has been used to minimize the balance in the subdivisions, the number of boundary pipes, the number of pipes equipped with flow meters and for maximization of both flexibility and minimum pressure indices. In this paper, by evaluating different algorithms for creating DMAs of water distribution networks, the best method is suggested. Indexes were ranked by studying the experts' opinion and forming the matrix of paired comparisons, so the first rank for maximizing the resilience index IR was 0. 401 and the last one was for minimizing the number of flow meters with a score of 0. 063. Based on the weight and criteria ranking, the water distribution network algorithms were scored. In terms of weighted GRAPH, the highest score belonged to EBC algorithm and the lowest score to FGC algorithm. In terms of the unweighted GRAPH spectral clustering algorithms rank first and FGC and MA algorithms rank last. In the unweighted GRAPH, some algorithms have equal scores, so more indices are needed to compare them. Due to the simplification of the PROBLEM and the pairwise comparison of the criteria with each other, according to the experts, this method offers an optimal and desirable result for selecting the appropriate method for converting the water distribution network into DMAs. In this paper, the EBC algorithm with a score of 0. 182 for the weighted GRAPH, the spectral clustering algorithms with a score of 0. 145 for the weighted GRAPH were ranked first.

GRAPH coloring is one of the issues that has been most noticed among combinatorial optimization issues. Many useful utility issues can be modeled as GRAPH coloring issues. The general form of this application is to form a GRAPH with nodes representing our favorite parts. The main PROBLEM of coloring the GRAPH is the grouping of vertex GRAPHs in small groups, so that no two heterogeneous vertices are in the same group. An important part of the application of GRAPH coloring PROBLEM in management science is. The concept of traffic lights includes controlling the system of a traffic light so that a safe level of safety can be obtained. Modeling the PROBLEM of traffic lights has been proposed as a PROBLEM of assignment in combinatorial theory. This PROBLEM is also modeled as a GRAPH coloring PROBLEM. In this paper, we have tried to model these PROBLEMs in practical examples as the PROBLEM of staining the fuzzy GRAPH and compare them with the proposed methods.

This paper addresses a bi-objective mixed integer optimization model under uncertainty for population PARTITIONING PROBLEM. The objective functions are to minimize the number of communications between partitions and to balance their population. The main constraints are defined for creating contiguous and compact partitions as well as assigning uniquely each basic unit to one partition. To deal with the uncertainty of parameters, a robust programming method is proposed that causes the uncertainty parameters lie between the interval of bestcase (the deterministic mode) and worst-case (the highest uncertainty level for all parameters). As the suggested method is NP-Hard, three meta-heuristic algorithms NSGAII, PESA, and SPEA are developed and, to evaluate the efficiency of the algorithms, 10 small-size examples, 10 medium-size examples and, 10 large-size examples are generated and solved. According to computational results, the SPEA has the best performance. The method is examined for a real-world application, as a case study in Iran.

This paper begins with a review of the GRAPH Colouring PROBLEM and its literature. Next developed methods for solving the PROBLEM are examined. Intelligent methods have demonstrated superior efficiency, since the PROBLEM is a NP-hard benchmark. A co-evolutionary algorithm for solving this PROBLEM is presented in this paper. The algorithm consists of two populations that simultaneously evolve using genetic operations. Eventually, the complete solution is found through merging individual members of the two populations. The effectiveness of the procedure has been compared to that of a simple genetic algorithm. The obtained results show significant advantage over the traditional routines.

Automated Guided Vehicle System (AGVS)provides the flexibility and automation demanded byFlexible Manufacturing System (FMS). However, with thegrowing concern on responsible management of resourceuse, it is crucial to manage these vehicles in an efficientway in order reduces travel time and controls conflicts andcongestions. This paper presents the development processof a new Memetic Algorithm (MA) for optimizing PARTITIONINGPROBLEM of tandem AGVS. MAs employ a GeneticAlgorithm (GA), as a global search, and apply a localsearch to bring the solutions to a local optimum point. Anew Tabu Search (TS) has been developed and combinedwith a GA to refine the newly generated individuals by GA. The aim of the proposed algorithm is to minimize themaximum workload of the system. After all, the performanceof the proposed algorithm is evaluated using Matlab. This study also compared the objective function of theproposed MA with GA. The results showed that the TS, asa local search, significantly improves the objective functionof the GA for different system sizes with large and smallnumbers of zone by 1. 26 in average.