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23 pages, 7313 KiB

Open AccessArticle

Mathematical Model of the Evolution of a Simple Dynamic System with Dry Friction

by Stelian Alaci, Florina-Carmen Ciornei, Costica Lupascu and Ionut-Cristian Romanu

Axioms 2024, 13(6), 372; https://doi.org/10.3390/axioms13060372 - 31 May 2024

Viewed by 683

A simple dynamic system with dry friction is studied theoretically and numerically. Models of systems including dry friction are not easily obtained, as defining the relationship between the friction force and the relative velocity presents a significant challenge. It is known that friction [...] Read more.

A simple dynamic system with dry friction is studied theoretically and numerically. Models of systems including dry friction are not easily obtained, as defining the relationship between the friction force and the relative velocity presents a significant challenge. It is known that friction forces exhibit notable discontinuities when there is a change in the direction of motion. Additionally, when the relative motion ceases, the friction force can assume any value within a certain range. In the literature, numerous models of dry friction are presented, and most of them assume a biunivocal dependency of the friction force with respect to relative velocity. The dynamic system considered here is a tilted rod with spherical ends, initially at rest. Dry friction forces are evident at the contact point with the horizontal plane. The ball–plane contact highlights the rolling friction or/and sliding friction. The problem is theoretically solved after adopting one of the two cases of friction: rolling friction or sliding friction. The nonlinear differential equations of motion have been derived, along with expressions for the magnitude of the normal reaction and the friction force. The results of the model are displayed graphically for three different sets of values for the coefficient of friction. It is revealed that there is a critical value of the coefficient of friction that determines the transition from rolling to sliding regimes. To validate the theoretical model, dynamic simulation software was utilised. The excellent match between the theoretical predictions and the results from the numerical simulation confirms the accuracy of the proposed analytical solution. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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19 pages, 3174 KiB

Open AccessArticle

Optimization of Location-Routing for Multi-Vehicle Combinations with Capacity Constraints Based on Binary Equilibrium Optimizers

by Rui Xi, Danju Lv, Yueyun Yu, Xin Huang, Ziqian Wang, Lianglian Gu, Zhicheng Zhu and Yan Zhang

Axioms 2024, 13(1), 31; https://doi.org/10.3390/axioms13010031 - 31 Dec 2023

Cited by 1 | Viewed by 1784

Abstract

The Location-Routing Problem (LRP) becomes a more intricate subject when the limits of capacities of vehicles and warehouses are considered, which is an NP-hard problem. Moreover, as the number of vehicles increases, the solution to LRP is exacerbated because of the complexity of [...] Read more.

The Location-Routing Problem (LRP) becomes a more intricate subject when the limits of capacities of vehicles and warehouses are considered, which is an NP-hard problem. Moreover, as the number of vehicles increases, the solution to LRP is exacerbated because of the complexity of transportation and the combination of routes. To solve the problem, this paper proposed a Discrete Assembly Combination-Delivery (DACA) strategy based on, the Binary Equilibrium Optimizer (BiEO) algorithm, in addition, this paper also proposes a mixed-integer linear programming model for the problem of this paper. Our primary objective is to address both the route optimization problem and the assembly group sum problem concurrently. Our BiEO algorithm was designed as discrete in decision space to meet the requirements of the LRP represented by the DACA strategy catering to the multi-vehicle LRP scenario. The efficacy of the BiEO algorithm with the DACA strategy is demonstrated. through empirical analysis utilizing authentic data from Changchun City, China, Remarkably, the experiments reveal that the BiEO algorithm outperforms conventional methods, specifically GA, PSO, and DE algorithms, resulting in reduced costs. Notably, the results show the DACA strategy enables the simultaneous optimization of the LRP and the vehicle routing problem (VRP), ultimately leading to cost reduction. This innovative algorithm proficiently tackles both the assembly group sum and route optimization problems intrinsic to multi-level LRP instances. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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24 pages, 4518 KiB

Open AccessArticle

A New Geometrical Design to Overcome the Asymmetric Pressure Problem and the Resulting Response of Rotor-Bearing System to Unbalance Excitation

by Hazim U. Jamali, H. S. S. Aljibori, Adnan Naji Jameel Al-Tamimi, Oday I. Abdullah, Adolfo Senatore and M. N. Mohammed

Axioms 2023, 12(9), 812; https://doi.org/10.3390/axioms12090812 - 24 Aug 2023

Cited by 3 | Viewed by 1449

Abstract

Improving the bearing design helps in reducing the negative consequences related to errors in installation, manufacturing, deflections under severe loading conditions, progressive wear of machine elements, and many other aspects. One of the methods of such a design improvement effort is changing the [...] Read more.

Improving the bearing design helps in reducing the negative consequences related to errors in installation, manufacturing, deflections under severe loading conditions, progressive wear of machine elements, and many other aspects. One of the methods of such a design improvement effort is changing the bearing profile along the bearing width to compensate for the reduction in the geometrical gap between the shaft and the bearing inner surface due to the aforementioned causes. Since in all rotating machinery, unbalance usually exists at some level, this paper deals with the response of this modified bearing to unbalanced excitation to evaluate the effectiveness of such geometrical design on the dynamic characteristics of the rotor-bearing system. The numerical solution is performed using the finite difference method by assuming Reynolds boundary conditions to determine the cavitation limits, and the 4th-order Range-Kutta method is used to determine the time responses resulting from the unbalance excitation. The time responses to this type of excitation show that the rotor-bearing with the improved geometrical design is more stable, particularly at high speeds. In addition, this modification leads to an improvement in the lubricant layer thickness and the reduction in the levels of the generated pressure between the surfaces despite the presence of large deviations from the perfectly aligned bearing system. Furthermore, the suggested geometrical design overcomes the problem of asymmetricity in the pressure field resulting from the shaft deviation to a large extent. The results of this work (the enhancement in the level of the film thickness and the improvement in the dynamic response of the system as well as the reduction of the maximum pressure value) extend the range of misalignment in which the rotor bearing systems can operate safely which represents a significant step in designing the rotor-bearing system. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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19 pages, 10268 KiB

Open AccessArticle

Solving Location Assignment and Order Picker-Routing Problems in Warehouse Management

by Johanna Bolaños-Zuñiga, M. Angélica Salazar-Aguilar and Jania Astrid Saucedo-Martínez

Axioms 2023, 12(7), 711; https://doi.org/10.3390/axioms12070711 - 22 Jul 2023

Cited by 2 | Viewed by 2946

Abstract

One of the critical warehousing processes is the order-picking process. This activity consists of retrieving items from their storage locations to fulfill the demand specified in the pick lists. Therefore, the storage location assignment affects the picking time and, consequently, reduces the operating [...] Read more.

One of the critical warehousing processes is the order-picking process. This activity consists of retrieving items from their storage locations to fulfill the demand specified in the pick lists. Therefore, the storage location assignment affects the picking time and, consequently, reduces the operating costs of the warehouse. This work presents two alternative mixed-integer linear models and an adaptive multi-start heuristic (AMH) for solving the integrated storage location and picker-routing problem. The problem considers a warehouse with a general layout and precedence constraints for picking according to the products weight. Experimental work confirms the efficiency of the proposed reformulations since we found out a total of 334 tested instances and optimal solutions for 51 new cases and 62 new feasible solutions. The proposed AMH improved more than 29% of the best-known solutions and required an average execution time of 117 s. Consequently, our proposed algorithm is an attractive decision-making tool to achieve efficiency when solving practical situations in a warehouse. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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23 pages, 997 KiB

Open AccessArticle

Optimizing Supplier Selection and Order Lot-Sizing Decisions in a Two-Stage Supply Chain

by José A. Ventura and Qingyuan Lu

Axioms 2023, 12(7), 615; https://doi.org/10.3390/axioms12070615 - 21 Jun 2023

Cited by 2 | Viewed by 2357

Abstract

This paper analyzes different lot-sizing policies for the supplier selection and order allocation problem in a two-stage supply chain. The supply chain consists of multiple candidate suppliers and a single buyer. In this system, selected suppliers produce a product in batches at finite [...] Read more.

This paper analyzes different lot-sizing policies for the supplier selection and order allocation problem in a two-stage supply chain. The supply chain consists of multiple candidate suppliers and a single buyer. In this system, selected suppliers produce a product in batches at finite production rates, ship it to the buyer, and the buyer sells it to the market at a constant demand rate. Our goal is to evaluate two lot-sizing policies and select the one that optimizes the supply chain by minimizing the total cost and maximizing supplier efficiency. A bi-objective mixed-integer nonlinear programming (BOMINLP) model is proposed. The first objective consists of the development of a coordination mechanism for supplier selection and order allocation that minimizes the entire supply chain cost, and the second objective comprises a data envelopment analysis (DEA) approach to evaluate the overall performance of suppliers to optimize supplier efficiency. Then, the lot-for-lot and order frequency policies are applied to the BOMINLP model separately to determine the set of selected suppliers as well as the corresponding order quantities and number of orders allocated to each selected supplier per replenishment cycle. Numerical examples that illustrate the solution approach and compare the two lot-sizing policies are provided. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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15 pages, 877 KiB

Open AccessArticle

A Mathematical Model for an Inventory Management and Order Quantity Allocation Problem with Nonlinear Quantity Discounts and Nonlinear Price-Dependent Demand

by Avelina Alejo-Reyes, Abraham Mendoza, Erik Cuevas and Miguel Alcaraz-Rivera

Axioms 2023, 12(6), 547; https://doi.org/10.3390/axioms12060547 - 1 Jun 2023

Cited by 2 | Viewed by 4969

Abstract

This article focuses on solving the order quantity allocation problem for retailers. It considers factors such as quality constraints, nonlinear quantity discounts, and price-dependent demand. By formulating it as a nonlinear maximization problem, the article aims to find the best combination of suppliers [...] Read more.

This article focuses on solving the order quantity allocation problem for retailers. It considers factors such as quality constraints, nonlinear quantity discounts, and price-dependent demand. By formulating it as a nonlinear maximization problem, the article aims to find the best combination of suppliers and order quantity out of infinite solutions to maximize the retailer’s profit. The main contribution of this research is a new mathematical model that can solve the problem of quality constraint and demand in a single step. This problem is complex due to the number of equations, their nonlinear nature, and the various trade-offs given by the market. Additionally, this research considers demand as output and includes price-dependent demand, which is more realistic for retailers. The proposed model was tested using an example from the recent literature and showed better results than the previously published best solution regarding profit maximization. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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17 pages, 2743 KiB

Open AccessArticle

A Modified Simulated Annealing (MSA) Algorithm to Solve the Supplier Selection and Order Quantity Allocation Problem with Non-Linear Freight Rates

by Paulina Gonzalez-Ayala, Avelina Alejo-Reyes, Erik Cuevas and Abraham Mendoza

Axioms 2023, 12(5), 459; https://doi.org/10.3390/axioms12050459 - 9 May 2023

Cited by 8 | Viewed by 2950

Abstract

Economic Order Quantity (EOQ) is an important optimization problem for inventory management with an impact on various industries; however, their mathematical models may be complex with non-convex, non-linear, and non-differentiable objective functions. Metaheuristic algorithms have emerged as powerful tools for solving complex optimization [...] Read more.

Economic Order Quantity (EOQ) is an important optimization problem for inventory management with an impact on various industries; however, their mathematical models may be complex with non-convex, non-linear, and non-differentiable objective functions. Metaheuristic algorithms have emerged as powerful tools for solving complex optimization problems (including EOQ). They are iterative search techniques that can efficiently explore large solution spaces and obtain near-optimal solutions. Simulated Annealing (SA) is a widely used metaheuristic method able to avoid local suboptimal solutions. The traditional SA algorithm is based on a single agent, which may result in a low convergence rate for complex problems. This article proposes a modified multiple-agent (population-based) adaptive SA algorithm; the adaptive algorithm imposes a slight attraction of all agents to the current best solution. As a proof of concept, the proposed algorithm was tested on a particular EOQ problem (recently studied in the literature and interesting by itself) in which the objective function is non-linear, non-convex, and non-differentiable. With these new mechanisms, the algorithm allows for the exploration of different regions of the solution space and determines the global optimum in a faster manner. The analysis showed that the proposed algorithm performed well in finding good solutions in a reasonably short amount of time. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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25 pages, 539 KiB

Open AccessArticle

Efficiency of Orthogonal Matching Pursuit for Group Sparse Recovery

by Chunfang Shao, Xiujie Wei, Peixin Ye and Shuo Xing

Axioms 2023, 12(4), 389; https://doi.org/10.3390/axioms12040389 - 17 Apr 2023

Cited by 2 | Viewed by 1692

Abstract

We propose the Group Orthogonal Matching Pursuit (GOMP) algorithm to recover group sparse signals from noisy measurements. Under the group restricted isometry property (GRIP), we prove the instance optimality of the GOMP algorithm for any decomposable approximation norm. Meanwhile, we show the robustness [...] Read more.

We propose the Group Orthogonal Matching Pursuit (GOMP) algorithm to recover group sparse signals from noisy measurements. Under the group restricted isometry property (GRIP), we prove the instance optimality of the GOMP algorithm for any decomposable approximation norm. Meanwhile, we show the robustness of the GOMP under the measurement error. Compared with the P-norm minimization approach, the GOMP is easier to implement, and the assumption of

γ

-decomposability is not required. The simulation results show that the GOMP is very efficient for group sparse signal recovery and significantly outperforms Basis Pursuit in both scalability and solution quality. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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21 pages, 5698 KiB

Open AccessArticle

Mathematical Model to Calculate Heat Transfer in Cylindrical Vessels with Temperature-Dependent Materials

by Martina Fernández-Gracía, Juan Francisco Sánchez-Pérez, Francisco del Cerro and Manuel Conesa

Axioms 2023, 12(4), 335; https://doi.org/10.3390/axioms12040335 - 29 Mar 2023

Cited by 4 | Viewed by 3252

Abstract

In this article, a mathematical model capable of simulating the heat transfer of cylindrical vessels whose properties are dependent on temperature is proposed. As a case study, it compares, from an approach of their heat transfer and chemical migration characteristics as a function [...] Read more.

In this article, a mathematical model capable of simulating the heat transfer of cylindrical vessels whose properties are dependent on temperature is proposed. As a case study, it compares, from an approach of their heat transfer and chemical migration characteristics as a function of the temperature reached, different materials commonly used for the manufacture of water bottles. More specifically, the materials studied were aluminium, polyethylene terephthalate, and polypropylene. The validation of the model consists of an experiment carried out in the laboratory with three water bottles of each of the materials under study, as well as simulations using the Network Simulation Method to recreate the heat transfer that occurs through the walls of the bottles. On the other hand, the nondimensionalization technique is also applied, which allows us to obtain the weight of each of the variables on the problem, as well as the existing relationship between them. Finally, an outside temperature of 30 °C to 50 °C is simulated, which is a common temperature range in southern Europe during the summer season, and an initial temperature of 20 °C for the water contained in the bottle to know the behaviour of the materials and what the final temperature of the water would be after one hour. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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27 pages, 593 KiB

Open AccessArticle

Incentive Contracts for a Queueing System with a Strategic Server: A Principal-Agent Perspective

by Jun Tu, Xiaoying Hu and Min Huang

Axioms 2023, 12(3), 272; https://doi.org/10.3390/axioms12030272 - 6 Mar 2023

Viewed by 2022

Abstract

Queueing systems with strategic servers are common in the service industry. The self-interested service rate decision of the strategic server will be detrimental to the queueing system. To improve the service rates, designing incentive contracts for the server from the queueing system owner’s [...] Read more.

Queueing systems with strategic servers are common in the service industry. The self-interested service rate decision of the strategic server will be detrimental to the queueing system. To improve the service rates, designing incentive contracts for the server from the queueing system owner’s perspective is critical. This study investigates the incentive contracts of queueing systems under exogenous and endogenous price scenarios. The unit-price and cost-sharing contracts are introduced to coordinate the queueing system. The effects of pricing mechanisms and contract types on the queueing system are investigated theoretically and experimentally. The results reveal that regardless of whether the price scenario is exogenous or endogenous, the cost-sharing contract is more effective than the unit-price contract in incentivizing the server to make a service effort. The cost-sharing contract with endogenous price can reduce the service price. The cost-sharing contract can boost profits for both the owner and server, albeit with conditions. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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20 pages, 14432 KiB

Open AccessArticle

Robot Time-Optimal Trajectory Planning Based on Quintic Polynomial Interpolation and Improved Harris Hawks Algorithm

by Jing Xu, Chaofan Ren and Xiaonan Chang

Axioms 2023, 12(3), 245; https://doi.org/10.3390/axioms12030245 - 27 Feb 2023

Cited by 8 | Viewed by 4060

Abstract

Time-optimal trajectory planning is one of the most important ways to improve work efficiency and reduce cost and plays an important role in practical application scenarios of robots. Therefore, it is necessary to optimize the running time of the trajectory. In this paper, [...] Read more.

Time-optimal trajectory planning is one of the most important ways to improve work efficiency and reduce cost and plays an important role in practical application scenarios of robots. Therefore, it is necessary to optimize the running time of the trajectory. In this paper, a robot time-optimal trajectory planning method based on quintic polynomial interpolation and an improved Harris hawks algorithm is proposed. Interpolation with a quintic polynomial has a smooth angular velocity and no acceleration jumps. It has widespread application in the realm of robot trajectory planning. However, the interpolation time is usually obtained by testing experience, and there is no unified criterion to determine it, so it is difficult to obtain the optimal trajectory running time. Because the Harris hawks algorithm adopts a multi-population search strategy, compared with other swarm intelligent optimization algorithms such as the particle swarm optimization algorithm and the fruit fly optimization algorithm, it can avoid problems such as single population diversity, low mutation probability, and easily falling into the local optimum. Therefore, the Harris hawks algorithm is introduced to overcome this problem. However, because some key parameters in HHO are simply set to constant or linear attenuation, efficient optimization cannot be achieved. Therefore, the nonlinear energy decrement strategy is introduced in the basic Harris hawks algorithm to improve the convergence speed and accuracy. The results show that the optimal time of the proposed algorithm is reduced by 1.1062 s, 0.5705 s, and 0.3133 s, respectively, and improved by 33.39%, 19.66%, and 12.24% compared with those based on particle swarm optimization, fruit fly algorithm, and Harris hawks algorithms, respectively. In multiple groups of repeated experiments, compared with particle swarm optimization, the fruit fly algorithm, and the Harris hawks algorithm, the computational efficiency was reduced by 4.7019 s, 1.2016 s, and 0.2875 s, respectively, and increased by 52.40%, 21.96%, and 6.30%. Under the optimal time, the maximum angular displacement, angular velocity, and angular acceleration of each joint trajectory meet the constraint conditions, and their average values are only 75.51%, 38.41%, and 28.73% of the maximum constraint. Finally, the robot end-effector trajectory passes through the pose points steadily and continuously under the cartesian space optimal time. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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