Equivalent Deterministic Model Research Articles

Stochastic resource leveling in projects with flexible structures

In project management, efficient utilization of resources plays a key role in project success, and resource leveling is an effective technique to optimize resource usage. During project execution, there are often uncertainties that complicate resource leveling. Furthermore, existing research on resource leveling typically assumes a fixed project structure. However, this is not always the case in practice, because there may be a variety of optional technical solutions for some activities, leading to a flexible project structure. Therefore, considering both stochastic activity durations and flexible project structures, we propose and study the stochastic resource leveling problem with flexible project structures (SRLP-PS). The solution of the SRLP-PS is in the form of a scheduling policy. We design two algorithms for solving the NP-hard SRLP-PS: (a) an exact algorithm based on stochastic programming, in which we formulate a scenario-based non-linear stochastic programming model and linearize it into an equivalent deterministic mixed-integer linear programming model that can be directly solved by CPLEX; and (b) an improved differential evolution algorithm, which is equipped with several problem-specific components, such as two mutation operators balancing exploration and exploitation, initialization, and local improvement search. Extensive computational experiments on a large number of benchmark instances are performed to validate our algorithms, which are also compared with state-of-the-art meta-heuristics. The computational results reveal the effectiveness and competitiveness of our algorithms. We also analyze the value of stochastic information based on the exact algorithm and the meta-heuristics, respectively.

Research on Optimization of International Multimodal Transportation Routes Based on Uncertainty Theory

Delays in international multimodal marine transport can result in subsequent connection delays, leading to prolonged transportation time and increased uncertainty. To identify more reliable and cost-effective robust transportation routes, this study establishes a constrained planning model for multimodal transportation routes. Employing uncertainty theory, the model is transformed into a deterministic equivalent class model and ultimately incorporates an adaptive differential evolution (ADE) algorithm. Results of a case study indicate that: (i) the proposed model, compared with deterministic models, exhibits greater robustness and better aligns with practical transportation scenarios, resulting in substantial reductions in actual transportation time and cost; (ii) the solution efficiency of the ADE algorithm surpasses that of the genetic algorithm and Dijkstra algorithm; (iii) the start time of transportation and the confidence level of uncertainty also play crucial roles in influencing route selection. Therefore, decision-makers should consider a multifaceted approach when formulating transportation routes.

Dynamics for a Nonlinear Stochastic Cholera Epidemic Model under Lévy Noise

In this study, we develop a comprehensive mathematical model to analyze the dynamics of epidemic cholera, characterized by acute diarrhea due to pathogen overabundance in the human body. The model is first developed from a deterministic point of view, and then it is modified to include the randomness by stochastic differential equations. The study selected Lévy noise above other well-known types of noise, emphasizing its importance in epidemic modeling. Besides presenting a biological justification for the stochastic system, we demonstrate that the equivalent deterministic model exhibits possible equilibria. The introduction is followed by theoretical analysis of the model. Through rigorous analysis, we establish that the stochastic model ensures a unique global solution. Lyapunov function theory is applied to construct necessary conditions, which on average, guarantee the model’s stability for R0s>1. Our findings suggest the likelihood of eradicating the disease when Rs is below one, a significant insight supported by graphical simulations of the model. Graphical illustrations were generated from simulating the model in order to increase the analytical results’ robustness. This work provides a strong theoretical framework for a thorough comprehension of a range of such diseases. This research not only provides a deeper understanding of cholera dynamics but also offers a robust theoretical framework applicable to a range of similar diseases, alongside a novel approach for constructing Lyapunov functions for nonlinear models with random disturbances.

Beyond green borders: an innovative model for sustainable transportation in supply chains

Modern requirements necessitate the establishment of sustainable transportation systems, considering the substantial growth in transportation activities over recent years, which is expected to continue. Companies are facing the challenge of modeling their system transport to align with green principles. Sustainable transport relied on involving diverse stakeholders, particularly scientific research, in the development of this field. In light of this, maintaining sustainable transport quality involves conducting thorough investigations into an innovative study focusing on an uncertain interval programming model for a multi-stage, multi-objective, multi-product transportation challenge within budget constraints and safety measures in a green supply chain. Human languages often contain imperfect or unknown information, inherently lacking certainty; achieving precision in describing existing states or future outcomes is frequently unattainable. In probability theory, sufficient historical information is crucial for estimating probability distributions; while in fuzzy theory, determining a reliable membership function proves challenging; hence, there is often a hesitant estimation of the degree of belief in the occurrence of each condition. Addressing such uncertainties, the theory of uncertain intervals proves highly valuable. Given these considerations, the elements of the specified problem are recognized as uncertain intervals. To manage this lack of assurance, a fusion of interval theory and methods from uncertain programming is used to formulate two distinct models: an expected value model and a chance-constrained model. The equivalent deterministic models are then formulated and solved utilizing Weighted Sum Method, fuzzy programming, and goal programming. Following this, a numerical example is utilized to assess the model’s performance, and the results obtained are compared. Finally, the document concludes with a sensitivity analysis and outlines future directions.

Fourth Party Logistics Routing Problem Under Uncertain Time and Random Demand

With the rapid development of the economy, the fourth party logistics (4PL) has become an indispensable part of economic development, and the path optimization is an important topic in the research of 4PL routing problem. This paper studies the fourth party logistics routing problem with random demand and uncertain transportation time and transfer time by considering the randomness and uncertainty of the environment. In order to solve the path optimization strategy that minimizes the total transportation cost, this paper establishes an uncertain random programming model under the constraints of the total transportation time, the carrying capacity of the logistics provider, the transfer capacity of the transfer station, and the demand of the distributor. Subsequently, this paper transforms the uncertain random programming model into an equivalent deterministic programming model based on the distribution functions of random demand and uncertain time, and verifies the effectiveness of the model through two examples. The results show that the uncertain random programming model can give a reasonable path optimization strategy.

A Type-2 fuzzy hybrid preference optimization methodology for electric vehicle charging station location

This work focuses on a hybrid preference-based electric vehicle charging station location problem, which considers multiple optimization preferences of distribution network operators, charge station owners, and electric vehicle users. The problem is formulated by an uncertain mixed-integer programming model. Due to the multi-fold uncertainty of the charging process, the uncertain model parameters are expressed as Type-2 fuzzy variables (T2-FVs). The critical value-based type reduction method is adopted to handle the high computational complexity. The proposed uncertain model is converted to its equivalent deterministic chance-constrained programming model. The deterministic counterpart is solved by General Algebraic Modeling System (GAMS). At last, numerical simulations are performed to demonstrate the proposed location strategy as well as some sensitivity analyses. The results indicate that for any given parameters, the equivalent deterministic model follows the general form of mixed-integer programming one that can be easily solved by GAMS. The proposed methodology can effectively handle the multi-fold uncertainty of the charging process. Compared with crisp models, the proposed location strategy can provide more robust location decisions for electric vehicle charging stations (EVCSs). In addition, we also found that different interest groups have conflicting preferences for the locations of EVCSs, so considering multiple hybrid optimization preferences is necessary.

Optimizing Chance Constraint Multiple-Objective Fractional Mathematical Programming Problem Involving Dependent Random Variable

This manuscript suggests a methodology to solve chance constraint multiple-objective linear fractional mathematical programming problem in which the parameters are dependent random variables to each other. The proposed problem is formulated by taking few of the parameters as continuous dependent random variables. The proposed model cannot be solved directly by using existing methodology. Thus in order to solve the proposed model, an equivalent deterministic model is derived. The procedure to solve the proposed model is accomplished in two main steps. Initially, the proposed multiple-objective chance constraint linear fractional mathematical problem is transformed to deterministic equivalent multiple-objective linear fractional mathematical programming by the help of chance constrained method. In the second step, multiple-objective functions, which consist of fractional functions is solved by using lexicographic programming approach. Finally, an example is mentioned to illustrate the methodology.

A study on uncertain fixed charge bi-objective 4-dimensional transportation problems under budget constraints

<span lang="EN-US">Generally, in traditional transportation problems, single items are transported from a set of origins to a set of destinations in such a way that the transportation cost is minimized. To face the challenges that arise in modelling transportation problem various parameters are to be considered. In this work to provide an optimal transportation policy to the decision maker, we have studied a fixed-charge bi-objective multi-product <br /> 4-dimensional transportation problem with vehicle speed under budget constraints under uncertain environment. The objective of the proposed model is to have the maximum profit with the minimum time taken of transporting the goods. An equivalent deterministic model of the proposed model is obtained to deal the uncertain variable using expected value-chance constraint properties on uncertainty theory and its compromise solution is obtained by using the goal programming technique. A numerical example is discussed to provide more clarity on the proposed model.</span>

A dynamic ripple-spreading algorithm for solving mean–variance of shortest path model in uncertain random networks

Combinatorial optimization involves more and more evaluation indicators, and some parameters cannot be accurately described. This paper considers a shortest path problem where arc costs include both uncertainty and randomness, and the decision-maker wishes to minimize both the expected cost and the variance of this cost. Firstly, a mean–variance model for the shortest path problem with uncertain arc cost and random arc cost is proposed, and the equivalent deterministic model of the model is deduced. Secondly, we develop a dynamic ripple spreading algorithm (DRSA) to solve the model, based on the ripple spreading patterns on the natural water surface. Then, the ripple spreading speed of the algorithm is simulated and predicted by hybrid prediction algorithm (HPA) on the basis of obtaining real urban traffic data, and it is verified by theoretical proof that DRSA can find the Pareto optimal path from the source node to the destination node within a single run. Finally, the proposed mean–variance shortest path problem model and DRSA are verified by numerical experiments.

A Joint Location–Allocation–Inventory Spare Part Optimization Model for Base-Level Support System with Uncertain Demands

This paper copes with a joint Location-Allocation-Inventory problem in a three-echelon base-level spare part support system with epistemic uncertainty in uncertain demands of bases. The aim of the paper is to propose an optimization model under the uncertainty theory to minimize the total cost, which integrates crucial characterizations of the inventory control decisions and the location-allocation scheme arrangement under a periodic review order-up-to-S (T, S) policy. Uncertainty theory is introduced in this paper to characterize epistemic uncertainty, where demands are treated as uncertain variables and stockout loss is represented by value-at-risk in uncertain measurement. To solve the original uncertain optimization model, an equivalent deterministic model is derived and addressed by an improved bilevel genetic algorithm. Moreover, the proposed models and algorithm are encoded into numerical examples for supply chain programming. The results highlight the applicability of the model and the algorithm’s effectiveness in approaching the optimal solution compared with traditional genetic algorithm. Sensitivity analyses are further made for the impacts of review time and inventory capacity on different cost components.

A target-based optimization model for bike-sharing systems: From the perspective of service efficiency and equity

The emergence of bike-sharing systems has considerably improved last- and first-mile transportation systems. To ensure attractiveness to end users, operators aim to design effective service-oriented operational strategies to meet the desired service targets for users. Most existing studies focus on the service efficiency of bike-sharing systems, while service equity is overlooked. In this study, we propose a target-based stochastic distributionally robust optimization (TSDRO) model that addresses both the efficiency and equity of the service level in docked bike-sharing systems under demand uncertainty. We first employ a dissatisfaction risk measure to jointly quantify the probability and magnitude of user dissatisfaction in a zone. Then, we apply a lexicographic-order approach to define the objective function to achieve equity of service among different zones. This lexicographic approach optimizes the worst-off individual and the second-worst zone in an iterative manner. To address demand ambiguity, we use a data-driven method to explore the relationship between the demand distribution and several exogenous factors, including weather and weekends, and then construct a scenario-based distributionally robust optimization model. Based on duality theory and linear decision approximation, this model can be reformulated as a tractable equivalent deterministic model, which can be solved via a bisection-search approach to optimality. Numerical experiments based on real operational data show that compared with the benchmark models, the TSDRO model achieves (i) better out-of-sample performance in terms of service efficiency and (ii) higher service equity among users in different zones. Moreover, setting a lower target level may generate a better solution.

Uncertain Public R&D Project Portfolio Selection Considering Sectoral Balancing and Project Failure

In order to promote scientific and technological innovation and sustainable development, public funding agencies select and fund a large number of R&D projects every year. To guarantee the performance of the resulting project portfolio and the government’s investment benefits, the decision maker needs to select appropriate projects and determine a reasonable funding amount for each selected project. In the process of project selection, it is necessary to consider the balance of funding allocated to different scientific sectors as well as the failure probability of the projects in future execution, so that the expected performance of the project portfolio is maximized as much as possible. In view of this, we propose and study the uncertain public R&D project portfolio selection problem considering sectoral balancing and project failure. We formulate a stochastic programming model for the problem to support the portfolio decisions of the funding agencies. We also transform the model into an equivalent deterministic second-order cone programming model that can be directly solved by exact solvers. We generate datasets reflecting different scenarios through simulation and perform computational experiments to validate our model. The impacts of various factors (i.e., the number of project proposals, project failure probability, the upper limit of the budget allocated to each project, and the decision maker’s tolerance for project failure) on the project portfolio performance are analyzed.

Application of Uncertain Programming in Hardware/Software Partitioning: Model and Algorithm

Hardware/software partitioning is a typical multi-stage decision optimization problem; most existing hardware/software partitioning methods ignore a fact that real-life decisions are usually made in an uncertain state. We should model the hardware/software partitioning problem in uncertain environments and deal with uncertainty. The state-of-the-art work proposed an uncertainty conversion method for hardware/software partitioning, but this method does not include the equivalent deterministic model and is not suitable for dealing with different types of uncertainties. In order to cope with different situations with various uncertainties, we should apply uncertain programming to build a model in uncertain environments and give different equivalent deterministic models to convert different uncertainties theoretically. In this paper, we present the process of applying uncertain programming to solve the hardware/software partitioning problem, including the model and algorithm. We convert the uncertain programming model into its equivalent deterministic models, including the expected value model and the chance-constrained programming model; we give details for the conversion methods of these two models. We present the custom genetic algorithm to solve the converted model, by incorporating a greedy idea in two steps of the genetic algorithm. Experimental results show that the custom genetic algorithm can find a high-quality approximate solution while running much faster for large input scales, compared with the exact algorithm.

Types of Maintenance Based on Uncertain Data Envelope Analysis

Nowadays, one of the main challenges of marine maintenance is how to select an optimum maintenance strategy for each component of the complex ship machinery system. The uncertainty of the parameters is one of the main difficulties encountered. For example, engineers and experts are always questioning the credibility and integrity of the data collected, and historical maintenance records may also be lost during maintenance. The above data asymmetry will also lead to parameter uncertainty, which directly affects the accuracy of the prediction results. This paper proposes a method for determining maintenance types of ship equipment based on uncertainty theory and the Data Envelopment Analysis (DEA) model. Firstly, this paper constructs an uncertain maintenance optimization model (UMOM) based on the classic Data Envelopment Analysis model. Then, we converted the UMOM into an equivalent deterministic model for easy calculation by the uncertainty theory. Finally, a case study is given to verify this model. The results will conclude that the UMOM can meet the need for a reasonable classification of maintenance types of mechanical equipment in a marine system and provide valuable information for systemic management and storage.

Investigating alternative power supply solutions under long term uncertainty for offgrid-offshore fish farm: The case of Hinnøya island, Norway

The aim of this paper is to explore potential least-cost decarbonisation solutions for an off-grid and offshore fish farm power supply system under long term uncertainty. The Hinnøya island in Norway is used as a use case. Three distinctive off-grid and grid-based alternative power supply solutions were proposed and studied as a replacement to the existing diesel power solution in a three-stage stochastic model and under two critical long-term uncertainties: (1) access to strong grid and (2) storage battery cost. The TIMES modelling framework is applied. The stochastic model results reveal that grid integrated storage is an optimal near-term investment for a storage battery cost of 295 €/kWh and less by 2025. In scenarios with no access to strong grid, grid integrated storage continues to be a least-cost solution in the long-term as well, whereas in those scenarios with strong grid access, new investment in storage is not required after 2030. Contrary to the stochastic model runs, the equivalent deterministic model runs showed that a hybrid wind and diesel solution is an optimal near-term investment. From 2030, however, a similar technology pattern in both stochastic and deterministic model runs are observed. Nevertheless, the results are very sensitive to the assumed storage battery costs. Higher storage costs (as high as 704 €/kWh by 2025) would make the hybrid wind and diesel solution an optimal solution instead of the grid integrated storage solution in near-term investment in both stochastic and deterministic model runs.

Robust simulation-optimization framework for synthesis and design of natural gas downstream Incorporating renewable hydrogen network under uncertainty

Many intrinsic uncertainties are part of natural gas downstream synthesis, design, and operation. This work extends the earlier developed deterministic model to account for new decision variables, unit operations, and uncertainty in the network. The uncertainty includes the feedstock composition and flowrate, utility requirements, by-products, and final product market price. The proposed two-stage stochastic model includes new major processing units (helium recovery units), hydrogen production via renewable technologies (biomass gasification, water electrolysis), and operating modes. The stochastic formulation simultaneously considers uncertainty in 12 different model parameters. The model's first stage aims to find the optimal network's design variables, such as processing units, technologies, and operating modes common to all scenarios. The second stage is concerned with deriving operational variables, for example, the network's overall inlet feedstock flowrates, individual units' inlet flowrates, and units' capacity. Outcomes obtained for a range of uncertain material/energy feedstock and products/by-products market prices result in higher expected operating profits than the equivalent deterministic model. The analysis's key takeaways show that the most prevailing network consists of two syngas technologies: (1) auto-thermal reforming to feed a low-temperature Fischer Tropsch synthesis unit and (2) steam methane reforming for hydrogen production. Additionally, renewable hydrogen is produced via electrolysis and biomass gasification. There is an improvement of 2.22% in the objective function value using the stochastic approach instead of the deterministic.

A new uncertain random portfolio optimization model for complex systems with downside risks and diversification

There are different types of securities yields in the financial market. The yields of these securities can be described as uncertain variables or random variables. This paper considers an uncertain random portfolio selection problem, in which uncertain and random return rates exist simultaneously. First, by considering downside risks and diversification constraints, an uncertain random bi-objective mean-variance-VaR-entropy model for portfolio selection problems is proposed. Here, investment return and risk are, respectively, quantified by uncertain random expected value and variance. Then the formulated uncertain random model is transformed into two equivalent deterministic models. Furthermore, we use the NSGA-II algorithm to solve the equivalent bi-objective model, and propose a new optimal solution criterion to find a single optimal solution in the Pareto optimal solution set. Finally, a numerical simulation is performed to verify the validity and the practicality of the proposed model and the NSGA-II algorithm.

Spare Parts Transportation Optimization Considering Supportability Based on Uncertainty Theory

Ensuring a consistent, continuous, and efficient spare parts supply is a critical issue that must be addressed in the equipment support system. In order to effectively improve the coverage level and handle the common asymmetry information present in practical applications, the spare parts transport vehicle routing and scheduling model was further optimized. We integrated supportability requirements and uncertainty theory into the model to better describe the actual uncertain demand of each site. We selected three critical supportability indicators as constraints, redefined them with uncertain variables, and then completed the chance-constrained model on this basis. Once the confidence level is specified, the uncertain constraints can be transformed into deterministic constraints, and finally, the equivalent deterministic model can be solved easily. In addition, a feasible solution can be found through a genetic algorithm, and a numerical example is provided to validate the model’s rationality. The proposed method successfully seeks the balance between the total cost and supportability.

A robust approach to integrated wireless charging infrastructure design and bus fleet size optimization

Due to the advantage of low emissions, electric buses are an appealing option for public bus systems. However, the driving range restriction and long charging time requirement of electric bus systems limit their development in practice. Wireless power transfer (WPT) is able to charge buses en route and makes electric bus systems more flexible. We consider a multiline transit system in which each bus line is associated with a given number of trips. We propose a new WPT location problem incorporating operational-level information, such as the number of trips for each line. The WPT location is restricted to bus stops that have a less negative impact on normal traffic and are good candidates for WPT installation. The problem is formulated as mixed integer programming (MIP). It simultaneously determines the WPT location, the length of the charging lane, the number and line assignment of buses and battery size. In addition, due to widespread uncertainty in energy consumption and charging, we propose a robust model, which can be reformulated into a tractable deterministic equivalent MIP model so that reliable bus operation can be obtained. Numerical experiments are conducted with two real bus systems. The solutions of the robust model indicate high robustness against uncertainty.

Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

The fractional Brownian motion (FBM) is a common model for long and short-range dependent phenomena that appears in different fields, including physics, biology, and finance. In the current work, a new spectral technique named the fractional Wiener Hermite Expansion (FWHE) is developed to analyze stochastic models with FBM. The technique has a theoretical background in the literature and proof of convergence. A new complete orthogonal Hermite basis set is developed. Calculus derivations and statistical analysis are performed to handle the mixed multi-dimensional fractional and/or integer-order integrals that appear in the analysis. Formulas for the mean and variance are deduced and are found to be based on fractional integrals. Using the developed expansion with the statistical properties of the basis functionals will help to reduce the stochastic model to equivalent deterministic fractional models that can be analyzed numerically or analytically with the well-known techniques. A numerical algorithm is developed to be used in case there is no available analytical solution. The numerical algorithm is compared with the fractional Euler-Maruyama (EM) technique to verify the results. In comparison to sampling based techniques, FWHE provides an efficient analytical or numerical alternative. The applicability of FWHE is demonstrated by solving different examples with additive and multiplicative FBM.