This paper investigates the nonidentical parallel production line scheduling problem derived from an axle housing machining workshop of an axle manufacturer. The characteristics of axle housing machining lines are analyzed, and a nonidentical parallel line scheduling model with a jumping process operation (NPPLS-JP), which considers mixed model production, machine eligibility constraints, and fuzzy due dates, is established so as to minimize the makespan and earliness/tardiness penalty cost. While the physical structures of the parallel lines in the NPPLS-JP model are symmetric, the production capacities and process capabilities are asymmetric for different models. Different from the general parallel line scheduling problem, NPPLS-JP allows for a job to transfer to another production line to complete the subsequent operations (i.e., jumping process operations), and the transfer is unidirectional. The significance of the NPPLS-JP model is that it meets the demands of multivariety mixed model production and makes full use of the capacities of parallel production lines. Aiming to solve the NPPLS-JP problem, we propose a hybrid algorithm named the multi-objective grey wolf optimizer based on decomposition (MOGWO/D). This new algorithm combines the GWO with the multi-objective evolutionary algorithm based on decomposition (MOEA/D) to balance the exploration and exploitation abilities of the original MOEA/D. Furthermore, coding and decoding rules are developed according to the features of the NPPLS-JP problem. To evaluate the effectiveness of the proposed MOGWO/D algorithm, a set of instances with different job scales, job types, and production scenarios is designed, and the results are compared with those of three other famous multi-objective optimization algorithms. The experimental results show that the proposed MOGWO/D algorithm exhibits superiority in most instances. Full article

This paper discusses the power loss minimization problem in asymmetric distribution systems (ADS) based on phase swapping. This problem is presented using a mixed-integer nonlinear programming model, which is resolved by applying a master–slave methodology. The master stage consists of an improved version of the crow search algorithm. This stage is based on the generation of candidate solutions using a normal Gaussian probability distribution. The master stage is responsible for providing the connection settings for the system loads using integer coding. The slave stage uses a power flow for ADSs based on the three-phase version of the iterative sweep method, which is used to determine the network power losses for each load connection supplied by the master stage. Numerical results on the 8-, 25-, and 37-node test systems show the efficiency of the proposed approach when compared to the classical version of the crow search algorithm, the Chu and Beasley genetic algorithm, and the vortex search algorithm. All simulations were obtained using MATLAB and validated in the DigSILENT power system analysis software. Full article

Let G be a graph with no isolated vertex and let $N\left(v\right)$ be the open neighbourhood of $v\in V\left(G\right)$. Let $f:V\left(G\right)\to \{0,1,2\}$ be a function and $V_i=\{v\in V\left(G\right):f\left(v\right)=i\}$ for every $i\in \{0,1,2\}$. We say that f is a strongly total Roman dominating function on G if the subgraph induced by $V_1\cup V_2$ has no isolated vertex and $N\left(v\right)\cap V_2\ne \varnothing$ for every $v\in V\left(G\right)\backslash V_2$. The strongly total Roman domination number of G, denoted by ${\gamma }_{tR}^s\left(G\right)$, is defined as the minimum weight $\omega \left(f\right)={\sum }_{x\in V\left(G\right)}f\left(x\right)$ among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing ${\gamma }_{tR}^s\left(G\right)$ is NP-hard. Full article

The problem of the optimal load redistribution in electrical three-phase medium-voltage grids is addressed in this research from the point of view of mixed-integer convex optimization. The mathematical formulation of the load redistribution problem is developed in terminals of the distribution node by accumulating all active and reactive power loads per phase. These loads are used to propose an objective function in terms of minimization of the average unbalanced (asymmetry) grade of the network with respect to the ideal mean consumption per-phase. The objective function is defined as the $l_1$-norm which is a convex function. As the constraints consider the binary nature of the decision variable, each node is conformed by a $3\times 3$ matrix where each row and column have to sum 1, and two equations associated with the load redistribution at each phase for each of the network nodes. Numerical results demonstrate the efficiency of the proposed mixed-integer convex model to equilibrate the power consumption per phase in regards with the ideal value in three different test feeders, which are composed of 4, 15, and 37 buses, respectively. Full article

In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical semigroups of each given genus. We analyze the number of nodes at each depth in the forest and propose new conjectures. Some properties of the quasi-ordinarization transform are presented, as well as some relations between the ordinarization and quasi-ordinarization transforms. Full article

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex $x\in V\left(G\right)$ of a graph G, the neighbourhood of x is denoted by $N\left(x\right)$. The neighbourhood of a set $X\subseteq V\left(G\right)$ is defined to be $N\left(X\right)={\bigcup }_{x\in X}N\left(x\right)$, while the external neighbourhood of X is defined to be $N_e\left(X\right)=N\left(X\right)\setminus X$. Now, for every set $X\subseteq V\left(G\right)$ and every vertex $x\in X$, the external private neighbourhood of x with respect to X is defined as the set $P_e(x,X)=\{y\in V\left(G\right)\setminus X:N\left(y\right)\cap X=\left\{x\right\}\}.$ Let $X_w=\{x\in X:P_e(x,X)\ne ⌀\}$. The strong differential of X is defined to be ${\partial }_s\left(X\right)=|N_e\left(X\right)|-|X_w|$, while the quasi-total strong differential of G is defined to be ${\partial }_{s^{*}}\left(G\right)=max\{{\partial }_s\left(X\right):X\subseteq V\left(G\right)\mathrm{and}{X}_w\subseteq N\left(X\right)\}.$ We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard. Full article

The collection of personal data is exponentially growing and, as a result, individual privacy is endangered accordingly. With the aim to lessen privacy risks whilst maintaining high degrees of data utility, a variety of techniques have been proposed, being microaggregation a very popular one. Microaggregation is a family of perturbation methods, in which its principle is to aggregate personal data records (i.e., microdata) in groups so as to preserve privacy through k-anonymity. The multivariate microaggregation problem is known to be NP-Hard; however, its univariate version could be optimally solved in polynomial time using the Hansen-Mukherjee (HM) algorithm. In this article, we propose a heuristic solution to the multivariate microaggregation problem inspired by the Traveling Salesman Problem (TSP) and the optimal univariate microaggregation solution. Given a multivariate dataset, first, we apply a TSP-tour construction heuristic to generate a Hamiltonian path through all dataset records. Next, we use the order provided by this Hamiltonian path (i.e., a given permutation of the records) as input to the Hansen-Mukherjee algorithm, virtually transforming it into a multivariate microaggregation solver we call Multivariate Hansen-Mukherjee (MHM). Our intuition is that good solutions to the TSP would yield Hamiltonian paths allowing the Hansen-Mukherjee algorithm to find good solutions to the multivariate microaggregation problem. We have tested our method with well-known benchmark datasets. Moreover, with the aim to show the usefulness of our approach to protecting location privacy, we have tested our solution with real-life trajectories datasets, too. We have compared the results of our algorithm with those of the best performing solutions, and we show that our proposal reduces the information loss resulting from the microaggregation. Overall, results suggest that transforming the multivariate microaggregation problem into its univariate counterpart by ordering microdata records with a proper Hamiltonian path and applying an optimal univariate solution leads to a reduction of the perturbation error whilst keeping the same privacy guarantees. Full article

In recent years, the use of Genetic Algorithms (GAs) in symmetric cryptography, in particular in the cryptanalysis of block ciphers, has increased. In this work, the study of certain parameters that intervene in GAs was carried out, such as the time it takes to execute a certain number of iterations, so that a number of generations to be carried out in an available time can be estimated. Accordingly, the size of the set of individuals that constitute admissible solutions for GAs can be chosen. On the other hand, several fitness functions were introduced, and which ones led to better results was analyzed. The experiments were performed with the block ciphers AES(t), for $t\in \{3,4,7\}$. Full article

Fermat’s Factoring Algorithm (FFA) is an integer factorisation methods factoring the modulus N using exhaustive search. The appearance of the Estimated Prime Factor (EPF) method reduces the cost of FFA’s loop count. However, the EPF does not work for balanced primes. This paper proposed the modified Fermat’s Factoring Algorithm 1-Estimated Prime Factor (mFFA1-EPF) that improves the EPF method. The algorithm works for factoring a modulus with unbalanced and balanced primes, respectively. The main results of mFFA1-EPF focused on three criteria: (i) the approach to select good candidates from a list of convergent continued fraction, (ii) the establishment of new potential initial values based on EPF, and (iii) the application of the above modification upon FFA. The resulting study shows the significant improvement that reduces the loop count of FFA1 via (improved) EPF compared to existing methods. The proposed algorithm can be executed without failure and caters for both the modulus N with unbalanced and balanced primes factor. The algorithm works for factoring a modulus with unbalanced and balanced primes. Full article

Topology optimization is a modern method for optimizing the material distribution in a given space, automatically searching for the ideal design of the product. The method aims to maximize the design performance of the system regarding given conditions. In engineering practice, a given space is first described using the finite element method and, subsequently, density-based method with solid isotropic material with penalty. Then, the final shape is found using a gradient-based method, such as the optimality criteria algorithm. However, obtaining the ideal shape is highly dependent on the correct setting of numerical parameters. This paper focuses on the sensitivity analysis of key formulations of topology optimization using the implementation of mathematical programming techniques in MATLAB software. For the purposes of the study, sensitivity analysis of a simple spatial task—cantilever bending—is performed. This paper aims to present the formulations of the optimization problem—in this case, minimization of compliance. It should be noted that this paper does not present any new mathematical formulas but rather provides an introduction into the mathematical theory (including filtering methods and calculating large-size problems using the symmetry of matrices) as well as a step-by step guideline for the minimization of compliance within the density-based topology optimization and search for an optimal shape. The results can be used for complex commercial applications produced by traditional manufacturing processes or by additive manufacturing methods. Full article

In this paper, we analyze the security of a group key establishment scheme proposed by López-Ramos et al. This proposal aims at allowing a group of users to agree on a common key. We present several attacks against the security of the proposed protocol. In particular, an active attack is presented, and it is also proved that the protocol does not provide forward secrecy. Full article

This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where $w=(w_0,w_1,\dots ,w_l)$ is a vector of nonnegative integers such that $w_0\ge 1$. The secure w-domination number is defined as follows. Let G be a graph and $N\left(v\right)$ the open neighborhood of $v\in V\left(G\right)$. We say that a function $f:V\left(G\right)⟶\{0,1,\dots ,l\}$ is a w-dominating function if $f\left(N\left(v\right)\right)={\sum }_{u\in N\left(v\right)}f\left(u\right)\ge w_i$ for every vertex v with $f\left(v\right)=i$. The weight of f is defined to be $\omega \left(f\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$. Given a w-dominating function f and any pair of adjacent vertices $v,u\in V\left(G\right)$ with $f\left(v\right)=0$ and $f\left(u\right)>0$, the function $f_{u\to v}$ is defined by $f_{u\to v}\left(v\right)=1$, $f_{u\to v}\left(u\right)=f\left(u\right)-1$ and $f_{u\to v}\left(x\right)=f\left(x\right)$ for every $x\in V\left(G\right)\backslash \{u,v\}$. We say that a w-dominating function f is a secure w-dominating function if for every v with $f\left(v\right)=0$, there exists $u\in N\left(v\right)$ such that $f\left(u\right)>0$ and $f_{u\to v}$ is a w-dominating function as well. The secure w-domination number of G, denoted by ${\gamma }_w^s\left(G\right)$, is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on ${\gamma }_w^s\left(G\right)$ and raises the challenge of conducting a detailed study of the topic. Full article

During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions. Full article

Let G be a graph with no isolated vertex and $f:V\left(G\right)\to \{0,1,2\}$ a function. Let $V_i=\{v\in V\left(G\right):f\left(v\right)=i\}$ for every $i\in \{0,1,2\}$. The function f is an outer-independent Roman dominating function on G if $V_0$ is an independent set and every vertex in $V_0$ is adjacent to at least one vertex in $V_2$. The minimum weight $\omega \left(f\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ among all outer-independent Roman dominating functions f on G is the outer-independent Roman domination number of G. This paper is devoted to the study of the outer-independent Roman domination number of a graph, and it is a contribution to the special issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we obtain new tight bounds for this parameter, and some of them improve some well-known results. We also provide closed formulas for the outer-independent Roman domination number of rooted product graphs. Full article

The following paper presents deductive theories of n-Pythagorean fuzzy sets (n-PFS). N-PFS objects are a generalization of the intuitionistic fuzzy sets (IFSs) and the Yager Pythagorean fuzzy sets (PFSs). Until now, the values of membership and non-membership functions have been described on a one-to-one scale and a quadratic function scale. There is a symmetry between the values of this membership and non-membership functions. The scales of any power functions are used here in order to increase the scope of the decision-making problems. The theory of n-PFS introduces a conceptual apparatus analogous to the classic theory of Zadeh fuzzy sets, consistently striving to correctly define the n-PFS algebra. Full article