Modeling Of Real-world Problems Research Articles

Optimization modeling and verification from problem specifications using a multi-agent multi-stage LLM framework

This paper explores the use of Large Language Models (LLMs) in modeling real-world optimization problems. We concretely define the task of translating natural language descriptions into optimization models (NL2OPT) and provide criteria for classifying optimization problems for the NL2OPT task. Our novel multi-agent modeling framework leverages relations identifier agents and a multi-agent verification mechanism, eliminating the need for solver execution. Additionally, we introduce a straightforward and practical evaluation framework, offering a more effective assessment method compared to traditional execution-based evaluations. We have created a unique dataset tailored for optimization modeling, featuring Problem Specifications as a structured representation of optimization problems. Through comprehensive experiments, our study compares our modeling framework with existing LLM reasoning strategies, highlighting their relative effectiveness in optimization modeling tasks. We also perform ablation studies to explore the effect of different components of our modeling framework. Experimental results demonstrate that our multi-agent framework outperforms many common LLM prompting strategies.

Bi-objective optimization of flexible flow shop scheduling problem with multi-skilled human resources

In this article, a Multi-skilled Resource-Constrained Flexible Flow Shop Scheduling Problem (MSRC-FFSSP) is introduced. To process each job at every machine within this problem category, it is imperative to possess at least one set of multi-skilled human resources. Each human resource can operate a minimum of 1 and a maximum of N jobs depending on their skills and each job requires a minimum set of different skills. For this problem, the model is formulated as a MILP model with two objectives: i) minimizing the total completion time and ii) minimizing the total idle time for human resources. Moreover, a case study in a preventive maintenance environment as a flow shop is performed to establish the applicability of the model in real-world problems. Given that the problem is NP-Hard, a Genetic Algorithm (GA) and a Simulated Annealing (SA) algorithm in which the related parameters are tuned by the Taguchi method are presented. Finally, numerical results have been presented to verify the model, in addition to the analysis of the idle time of human resources. Additionally, to solve this problem, it was shown that the GA has better efficiency and performance compared to the SA algorithm. However, the SA algorithm outperforms the GA in terms of running time.

A Study and Application Analysis Exploring Pythagorean Fuzzy Set Distance Metrics in Decision Making

The ever-increasing demand for high-quality solutions drives research toward more sophisticated decision-making solutions. In the field of decision making, the ability to solve complex real-world problems is of paramount importance. To this end, fuzzy sets are used, which offer the possibility of incorporating uncertainty into the values describing decision options. This study focuses on Pythagorean fuzzy sets, an extension of classical fuzzy sets, providing even more tools for modeling real-world problems by presenting a distance measure for these specific sets. A verification of the characteristics of the proposed distance measure has been carried out, proving its validity. The proposed measure is characterized by a more straightforward formula and thus simplifies the calculations. Furthermore, to confirm its usability, a multi-criteria decision-making methodology is presented, the results of which are compared with two multi-criteria decision-making methods, namely, PF-TOPSIS and PF-VIKOR, and another distance measure previously presented in the literature. The comparative analysis highlights lower variability in terms of preference values calculated using the proposed distance measure, which confirms the stability and reliability of the newly proposed distance measure while maintaining low computational complexity. Moreover, a high correlation with rankings calculated using PF-TOPSIS ensures its utility in terms of decision making.

Neutrosophic Laplace Distribution with Properties and Applications in Decision Making

This paper introduces the concept of the neutrosophic Laplace distribution ( ), a probability distribution derived from the Laplace distribution. The offers a versatile framework for describing various real-world problems. We highlight the neutrosophic extension of the Laplace distribution and explore its applications in different areas. Extensive investigations into the mathematical properties of the distribution are presented, including the derivation of its probability density function, mean, variance, raw moment, skewness, and kurtosis. To estimate the parameters of the , we employ the method of maximum likelihood (ML) estimation within a neutrosophic environment. Furthermore, we conduct a simulation study to assess the effectiveness of the maximum likelihood approach in estimating the parameters of this new distribution. The findings demonstrate the potential of the in modeling and analyzing real-world phenomena. Eventually, some illustrative examples related to system reliability are provided to clarify further the implementation of the neutrosophic probabilistic model in real-world problems.

A distributed optimization approach via symmetric group actions under time-varying communication networks

The widespread adoption of edge computing has revolutionized data processing by decentralizing computational power, resulting in faster response times, decreased network latency, and enhanced privacy. This paradigm shift is crucial due to the increasing demand for real-time analytics and the growth of Internet of Things devices. Optimizing resource allocation and decision-making in on-board computing presents challenges, especially in dynamic and heterogeneous environments. Distributed optimization techniques address these challenges by allowing multiple agents to collaborate and make decisions based on local knowledge. This paper focuses on distributed optimization in multi-agent systems with time-varying communication networks and explores the advantages of considering symmetric group action to reach global optimization. In particular, we focus our study into a class of Integer Linear Programming (ILP) for modeling real-world optimization problems in decentralized environments. The paper introduces a novel approach that leverages group actions and probabilistic selection of initial states to enhance convergence to desirable solutions. The method guarantees feasibility while minimizing computational effort for the agents. Additionally, numerical simulations have been conducted to validate the proposed algorithm, demonstrating its efficacy in optimizing resource allocation for a class of scheduling problems.

COMPUTATIONAL FLUID–STRUCTURE INTERACTION SIMULATION OF HEMODYNAMICS OF ARTERIOVENOUS FISTULA

The objective of this study is to show how important a compliant wall technique is in simulating non-Newtonian and pulsatile blood flows in arteriovenous fistula (AVF). The three-dimensional idealized geometry is used to investigate the local hemodynamics in the end-to-side, radio-cephalic AVF using computational fluid-stricture interaction (FSI) simulation. The third-order Yeoh law is used to model the behavior of the hyperelatic vessel walls. Hemodynamic parameters such as velocity, wall shear stress (WSS), oscillatory shear index (OSI), vorticity, and venous outflow rate are calculated. The results extracted for WSS on comparison of rigid and compliance wall, rigid wall WSS are 45–48.5% larger values than the compliance wall. The difference between compliance wall and rigid wall OSI is 11.5% and the rigid wall is a 10.86% decrease in compliance wall. The difference between the rigid wall AVF vorticity and the compliance wall vorticity is 18.34%, and the vorticity in the compliance wall is a 20.2% increase over the rigid wall. Maximum principle stress occurs at anastomosis (1335[Formula: see text]Pa) and 507[Formula: see text]Pa, 93[Formula: see text]Pa on vein and artery, respectively. Finally, we conclude that the artery bed and heel of AVF are prone areas of Intimal Hyperplasia. The structural result shows the tendency of the inflammation pattern of the vein required for AVF maturation. ANSYS is a highly effective commercial tool for modeling real-world problems and performing multi-physics numerical simulations.

Approximability and efficient algorithms for constrained fixed-horizon POMDPs with durative actions

Partially Observable Markov Decision Process (POMDP) is a fundamental model for probabilistic planning in stochastic domains. More recently, constrained POMDP and chance-constrained POMDP extend the model allowing constraints to be specified on some aspects of the policy in addition to the objective function. Despite their expressive power, these models assume all actions take a fixed duration, which poses a limitation in modeling real-world planning problems. In this work, we propose a unified model for durative POMDP and its constrained extensions. First, we convert these extensions into an Integer Linear Programming (ILP) formulation, which can be solved using existing solvers in the ILP literature. Second, a heuristic search approach is provided that can efficiently prune the search space, guided by solving successive partial ILP programs. Third, we give a theoretical analysis of the problem: unlike short-horizon POMDPs, with policies of a constant depth, which can be solved in polynomial time, the constrained extensions are NP-Hard even with a planning horizon of two and non-negative rewards. To alleviate that, we propose a Fully Polynomial Time Approximation Scheme (FPTAS) that computes (near) optimal deterministic policies in polynomial time. The FPTAS is among the best achievable in theory in terms of approximation ratio. Finally, evaluation results show that our approach is empirically superior to the state-of-the-art fixed-horizon chance-constrained POMDP solver.

Reinforcement Learning Framework for Server Placement and Workload Allocation in Multiaccess Edge Computing

Cloud computing is a reliable solution to provide distributed computation power. However, real-time response is still challenging regarding the enormous amount of data generated by the IoT devices in 5G and 6G networks. Thus, multi-access edge computing (MEC), which consists of distributing the edge servers in the proximity of end-users to have low latency besides the higher processing power, is increasingly becoming a vital factor for the success of modern applications. This paper addresses the problem of minimizing both, the network delay, which is the main objective of MEC, and the number of edge servers to provide a MEC design with minimum cost. This MEC design consists of edge servers placement and base stations allocation, which makes it a joint combinatorial optimization problem (COP). Recently, reinforcement learning (RL) has shown promising results for COPs. However, modeling real-world problems using RL when the state and action spaces are large still needs investigation. We propose a novel RL framework with an efficient representation and modeling of the state space, action space and the penalty function in the design of the underlying Markov Decision Process (MDP) for solving our problem.

Optimal sampling-based neural networks for uncertainty quantification and stochastic optimization

In recent times, neural networks (NN) have been successfully utilized to model real-world engineering problems. However, complexities in constructing an optimal NN architecture and limited training data for expensive function evaluations still limit the wide application of NNs. In addition, the distribution of the training samples affects the network's ability to capture the behavior of true responses. Therefore, this study proposes a novel optimal-sampling-based, multi-response, NN framework with adaptive modification of the network's hyperparameters to alleviate these difficulties. The D-optimal design, which maximizes the determinant of the information matrix, is proposed as a sampling technique. The information matrix is built by using the samples and activation values of the neurons in the hidden layers. The optimal sampling is further integrated with an uniform max-min sampling to reduce the recurrence of sampling points and improve the space-filling property. Moreover, a network updating strategy is presented where the model hyperparameters are optimized using a Gaussian process-based Bayesian optimization. An additional feature of the framework is that the same optimal sampling location can be utilized for evaluation of different responses to improve the overall multi-response NN. The proposed framework was applied for uncertainty quantification and stochastic optimization, and the high accuracy and computational efficiency based on the results demonstrated the significance of this research.

A New Modified Logistic Distribution: Properties and Applications in Uncertainty Data Modeling

The logistic distribution is widely used to model various types of applied data. The modified logistic distribution under neutrosophic statistics is introduced in this work. The neutrosophic logistic distribution (NLD) and its engineering applications are mainly emphasized. An appealing characteristic of the suggested NLD is that it is useful to many widely utilized survival assessment metrics, including the reliability function, hazard function, and survival function. Applications of some mathematical and statistical properties of the suggested model are discussed. Numerical investigations on simulated data are used to validate the theoretical findings experimentally. From an application point of view, it is inferred that the proposed distribution fits data with imprecise, hazy, and fuzzy information better than the usual model. In addition, the maximum likelihood (ML) technique for the proposed model is discussed under the neutrosophic inference framework. Eventually, some illustrative examples related to system reliability are provided to clarify further the implementation of the neutrosophic probabilistic model in real-world problems.

Analysis of fractional global differential equations with power law

<abstract><p>We have considered a special class of ordinary differential equations in which the differential operators are those of the Caputo fractional global derivative. These equations are generalizations of the well-known differential equations with the Caputo fractional derivative. Due to the various possible applications of these equations to model real-world problems we have first introduced some new inequalities that will be used in all fields of science, technology and engineering where these equations could be applied. We used Nagumo's principles to establish the existence and uniqueness of the solution for this class of equations with additional conditions. We have applied the midpoint principle to obtain a numerical scheme that will be used to solve these equations numerically. Some illustrative examples are presented with excellent results.</p></abstract>

Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation

Hyperbolic partial differential equations are frequently referenced in modeling real-world problems in mathematics and engineering. Therefore, in this study, an initial-boundary value issue is proposed for the pseudo-hyperbolic telegraph equation. By operator method, converting the PDE to an ODE provides an exact answer to this problem. After that, the finite difference method is applied to construct first-order finite difference schemes to calculate approximate numerical solutions. The stability estimations of finite difference schemes are shown, as well as some numerical tests to check the correctness in comparison to the precise solution. The numerical solution is subjected to error analysis. As a result of the error analysis, the maximum norm errors tend to decrease as we increase the grid points. It can be drawn that the established scheme is accurate and effective

PIECEWISE MODELING OF THE TRANSMISSION DYNAMICS OF CONTAGIOUS BOVINE PLEUROPNEUMONIA DEPENDING ON VACCINATION AND ANTIBIOTIC TREATMENT

Atangana and Araz have introduced a new term called piecewise differentiation and integration which has been successfully applied to epidemiology in these days. This concept paves the way of seeing different patterns of modeling real-world problems. In this study, a mathematical model for the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) is considered and modified based on vaccination and antibiotic therapy by using these operators. This model is classified into six classes namely susceptible, exposed, vaccinated, infectious, continuously infected, and treated classes. The model is solved by a numerical scheme based on Newton interpolation polynomial. The existence and uniqueness of the model are investigated. Various patterns of piecewise concepts have been presented through the numerical figures.

Method for encoding and decoding variables in engineering problems

Dimensional analysis can be used in those cases, where the system of equations describing the problem to solve is unknown. The setup of the trial function F(Q1,…,QM)=0 relies on the expertise of the researcher. The researcher is confronted with the questions: which Qm is the dependent variable; what is the value of M ; are the chosen Qm effective. The new encoding-decoding method disclosed has the goal to answer these three questions and belongs to dimensional exploration techniques that can help in discovering the governing equations. This new method is based on low complexity, high performing, and well-established computer algorithms of number theoretic functions. The encoding-decoding method is exemplified on a real-world problem by searching for the positively homogeneous measurement function that models wave phenomena, electromagnetic phenomena, electromechanical phenomena, and thermodynamic phenomena of the future power grids. The temporal variation of the power density is considered in its form of the kind of quantity called second order partial derivative of the energy density with respect to time denoted ∂2W(r,t)∂t2. The new method generates a trial function F(∂2W(r,t)∂t2,Q2,…,Q19)=0 and a positively homogeneous measurement function u(π1,…,π9)=0 for the design of experiments. The validation of this new method is performed through its application on two cases: the simple pendulum and the kind of quantity energy. The efficiency, effectiveness, and completeness of the encoding-decoding method are compared with classical and modern dimensional analysis. The new method has the advantage over those state-of-the-art methods in requiring less dimensionless quantities πk as arguments of the measurement function u(π1,…,πK)=0 when modeling real-world problems. The encoding-decoding method is based on lattice theory while classical and modern dimensional analysis are based on linear algebra.

Incomplete Multiple View Fuzzy Inference System With Missing View Imputation and Cooperative Learning

Advancement of technology has made available data of different modalities that can be integrated effectively through multiple view learning for modeling real-world problems. Although multiple view learning has achieved great success in many applications, it still faces several challenges. One of them is how to reduce the negative impact of the missing views in incomplete multiple view datasets by fully exploiting the information available. Another challenge is how to enhance the interpretability of the multiple view model for scenarios with high transparency requirement. To address these challenges, this article proposes a novel modeling method for incomplete multiple view fuzzy system. Based on fuzzy interpretable rules, the method integrates missing view imputation and hidden view learning as one single process to yield a model of high interpretability, where cooperative learning is used to mine the complementary information between the visible views and the hidden view. The proposed method has four advantages when compared with existing approaches: 1) the method is more interpretable, attributed to the fuzzy interpretable rules that it is based on, 2) missing view imputation is integrated into the modeling to make it more efficient than the existing two-step strategy, 3) the method not only imputes missing views, but also mines the hidden view shared by the multiple visible views, and 4) cooperative learning is used to mine the complementary information, which significantly reduces the negative impact of missing views. Experiments on real datasets demonstrate the advantages of the proposed method.

The capacity constraint physarum solver

Physarum polycephalum inspired algorithm (PPA) has attracted immense attention recently. By modeling real-world problems into a graph and adopting proper adaption equation, PPA could solve system optimization or user equilibrium problems. However, few PPA-based algorithms can tackle problems with capacity constraints, lacking the ability to control link flows in the PPA. To tackle this problem, a novel framework, the capacitated PPA (CPPA), is proposed to allow capacity constraints toward link flow in the PPA. To demonstrate the validity of the CPPA, we have developed three applications of the CPPA, i.e., the CPPA for the maximum-flow problem (CPPA-MF), the CPPA for the minimum-cost–maximum-flow problem (CPPA-MCMF), and the CPPA for the link-capacitated traffic assignment problem (CPPA-CTAP). In the experiments, the CPPA-MF shows competitive performance among four baseline algorithms; the CPPA-MCMF is approximately 50% faster than the baseline algorithms; the CPPA-CTAP reduce the oversaturated degree from approximately 40% to 1.5% compared to the original algorithm. The experimental results prove the validation of the CPPA framework. The CPPA is also robust since it have been successfully applied in three different scenarios. The proposed CPPA extends the PPA’s application to capacity-constrain problems and might shed some lights on its future applications in complex real-world problems.

Software implementation of intuitionistic fuzzy sets and some operators

In this paper, we present a software implementation of the framework of Intuitionistic Fuzzy Sets (IFSs). The presented implementation allows the user to interactively shape an IFS, to compute, plot and visualize various of operators for IFS and allows for the modeling of real world problems.

An Interval Type-2 Fuzzy Model of Computing with Words via Interval Type-2 Fuzzy Finite Rough Automata with Application in COVID-19 Deduction

Classical automata, fuzzy automata, and rough automata with input alphabets as numbers or symbols are formal computing models with values. Fuzzy automata and rough automata are computation models with uncertain or imprecise information about the next state and can only process the string of input symbols or numbers. To process words and propositions involved in natural languages, we need a computation model to model real-world problems by capturing the uncertainties involved in a word. In this paper, we have shown that computing with word methodology deals with perceptions rather than measurements and allows the use of words in place of numbers and symbols while describing the real-world problems together with interval type-2 (IT2) fuzzy sets which have the capacity to capture uncertainties involved in word using its footprint of uncertainty. The rough set theory, which has potential of modeling vagueness in the imprecise and ill-defined environment, introduces a computation model, namely, IT2 fuzzy rough finite automata, which is efficient to process uncertainties involved in words. Further, we have shown the application of introduced IT2 fuzzy finite rough automaton in the medical diagnosis of COVID-19 patients.

Deterministic-Stochastic modeling: A new direction in modeling real world problems with crossover effect.

Many real world problems depict processes following crossover behaviours. Modelling processes following crossover behaviors have been a great challenge to mankind. Indeed real world problems following crossover from Markovian to randomness processes have been observed in many scenarios, for example in epidemiology with spread of infectious diseases and even some chaos. Deterministic and stochastic methods have been developed independently to develop the future state of the system and randomness respectively. Very recently, Atangana and Seda introduced a new concept called piecewise differentiation and integration, this approach helps to capture processes with crossover effects. In this paper, an example of piecewise modelling is presented with illustration to chaos problems. Some important analysis including a piecewise existence and uniqueness and piecewise numerical scheme are presented. Numerical simulations are performed for different cases.

A new approach based on inventory control using interval differential equation with application to manufacturing system

<p style='text-indent:20px;'>Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In this regard, the aim of this study is two folded. Firstly, the concept of system of interval differential equations and its solution procedure in the parametric approach have been proposed. To serve this purpose, using parametric representation of interval and its arithmetic, system of linear interval differential equations is converted to the system of differential equations in parametric form. Then, a mixing problem with three liquids is considered and the mixing process is governed by system of interval differential equations. Thereafter, the mixing liquid is used in the production process of a manufacturing firm. Secondly, using this concept, a production inventory model for single item has been developed by employing mixture of liquids and the proposed production system is formulated mathematically by using system of interval differential equations.The corresponding interval valued average profit of the proposed model has been obtained in parametric form and it is maximized by centre-radius optimization technique. Then to validate the proposed model, two numerical examples have been solved using MATHEMATICA software. In addition, we have shown the concavity of the objective function graphically using the code of 3D plot in MATHEMATICA. Finally, the post optimality analyses are carried out with respect to different system parameters.</p>