The facility layout problem (FLP) is a very important class of NP-hard problems in operations research that deals with the optimal assignment of facilities to minimize transportation costs. The quadratic assignment problem (QAP) can model the FLP effectively. One of the FLPs is the hospital facility layout problem that aims to place comprehensive clinics, laboratories, and radiology units within predefined boundaries in a way that minimizes the cost of movement of patients and healthcare personnel. We are going to develop a hybrid method based on discrete differential evolution (DDE) algorithm for solving the QAP. In the existing DDE algorithms, certain issues such as premature convergence, stagnation, and exploitation mechanism have not been properly addressed. In this study, we first aim to discover the issues that make the current problem worse and to identify the best solution to the problem, and then we propose to develop a hybrid algorithm (HDDETS) by combining the DDE and tabu search (TS) algorithms to enhance the exploitation mechanism in the DDE algorithm. Then, the performance of the proposed HDDETS algorithm is evaluated by implementing on the benchmark instances from the QAPLIB website and by comparing with DDE and TS algorithms on the benchmark instances. It is found that the HDDETS algorithm has better performance than both the DDE and TS algorithms where the HDDETS has obtained 42 optimal and best-known solutions from 56 instances, while the DDE and TS algorithms have obtained 15 and 18 optimal and best-known solutions out of 56 instances, respectively. Finally, we propose to apply the proposed algorithm to find the optimal distributions of the advisory clinics inside the Azadi Hospital in Iraq that minimizes the total travel distance for patients when they move among these clinics. Our application shows that the proposed algorithm could find the best distribution of the hospital’s rooms, which are modeled as a QAP, with reduced total distance traveled by the patients.

In this paper, we establish the existence and multiplicity results of solutions for parametric quasi-linear systems of the gradient-type on the Sierpiński gasket is proved. Our technical approach is based on variational methods and critical points theory and on certain analytic and geometrical properties of the Sierpiński fractal. Indeed, using a consequence of the local minimum theorem due to Bonanno, the Palais-Smale condition cut off upper at r, and the Palais-Smale condition for the Euler functional we investigate the existence of one and two solutions for our problem under algebraic conditions on the nonlinear part. Moreover by applying a different three critical point theorem due to Bonanno and Marano we guarantee the existence of third solution for our problem.

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The problem of communication design has been defined as one of the problems that belong to the category of NP-hard problem, and the aim of the topological communication network design is to identify component placement locations and connectivity aspects. On the other hand, the Reliable Communication Network Design (RCND) is a popular optimization problem used for maximizing network reliability. In addition, finding an accurate calculation of RCND explains the problem of NP-hard problem. To this end, literature studies suggested various metaheuristic algorithms that have been used as approximation methods to find the best solution to this problem. Some of these algorithms belong to the Evolutionary Algorithms (EAs) category, such as Genetic Algorithms (GAs), and some belong to the Swarm Intelligence Algorithms (SIAs) category, such as Ant Colony Optimization (ACO). However, to the best of our knowledge, the Ant Colony System (ACS) algorithm, which is considered an updated version of ACO, has not yet been used to design reliability-constrained communication network topologies. Therefore, this study aims to apply the updated version of the ACS algorithm for solving RCND in small, medium, and large networks. The proposed algorithm was benchmarked against present state-of-the-art techniques that address this challenge. The research findings show that the proposed algorithm is an optimal solution for a fully connected small network size (n=6, 7, 8, and 9) and it has been achieved as an optimal solution for all not fully connected sets (n=14, 16, and 20). In each case, the results for medium-sized networks were better than the benchmark results