Optimal Equilibrium Research Articles

A Project Scheduling Game Equilibrium Problem Based on Dynamic Resource Supply

In a resource-constrained project scheduling problem, most studies ignore that resource supply is a separate optimization problem, which is not in line with the actual situation. In this study, the project scheduling problem and the resource supply problem are regarded as a dynamic game system, with interactive influences and constraints. This study proposes a Stackelberg dynamic game model based on the engineering supply chain perspective. In this model, the inherent conflicts and complex interactions between the Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP) and the Multi-Period Supply Chain Problem (MPSCP) are studied to determine the optimal equilibrium strategy. A two-level multi-objective programming method is used to solve the problem. The MRCPSP is the upper-level planning used to optimize project scheduling and activity mode selection to minimize project cost and duration; MPSCP is a lower-level planning method that seeks to make resource transportation decisions at a lower cost. A two-layer hybrid algorithm, consisting of Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), is proposed to determine the optimal equilibrium strategy. Finally, the applicability and effectiveness of the proposed optimization method are evaluated through a case study of a large hydropower construction project, and management suggestions for related departments are provided.

Post-combustion capture of CO2 using novel aqueous Triethylenetetramine and 2-Dimethylaminoethanol amine blend: Equilibrium CO2 loading-empirical model and optimization, CO2 desorption, absorption heat, and 13C NMR analysis

The equilibrium CO2 loading was estimated by absorption of CO2 gas in the bubble column reactor for novel aqueous amine blend of triethylenetetramine (TETA) and 2-dimethylaminoethanol (DMAE). The absorption study was performed at a temperature of 298.15 to 333.15 K, mole fraction of TETA from 0.05 to 0.2, solution concentration varied from 1 to 3 mol/L, and CO2 partial pressure ranging from 10.13 to 25.33 kPa. The maximum experimental equilibrium CO2 loading was 0.92 mol CO2/mol amine at 315.65 K temperature, 17.73 kPa CO2 partial pressure, 0.13 mol fraction of TETA, and 1 mol/L solution concentration. The absorption results were validated by an empirical model with an average absolute relative deviation of 5.631 %. The desorption study was performed at a constant temperature and pressure of 393.15 K and 17.73 kPa, respectively. The cyclic loading capacity of this blend at 3 mol/L concentration showed 55.03 % higher results than that of 30 wt% (5 mol/L) monoethanolamine (MEA). The Gibbs-Helmholtz equation calculated the heat of CO2 absorption, and for this amine blend, it was found to be −67.135 KJ/mol. 13C nuclear magnetic resonance (NMR) spectroscopy was adopted to identify the chemical species in CO2-loaded and unloaded amine blends. Response surface methodology software optimized the final response. It predicted optimal equilibrium CO2 loading of 0.926207 mol CO2/mol amine at 304.36 K temperature, 0.14 mol fraction of TETA, 17.24 kPa partial pressure of CO2, and 1 mol/L solution concentration.

A dual industrial chain model and optimal equilibrium condition analysis based on information computation

A dual industrial chain model and optimal equilibrium condition analysis based on information computation

Linear Tracking MPC for Nonlinear Systems—Part I: The Model-Based Case

We develop a tracking model predictive control (MPC) scheme for nonlinear systems using the linearized dynamics at the current state as a prediction model. Under reasonable assumptions on the linearized dynamics, we prove that the proposed MPC scheme exponentially stabilizes the optimal reachable equilibrium w.r.t. a desired target setpoint. Our theoretical results rely on the fact that, close to the steady-state manifold, the prediction error of the linearization is small and hence, we can slide along the steady-state manifold towards the optimal reachable equilibrium. The closed-loop stability properties mainly depend on a cost matrix which allows us to trade off performance, robustness, and the size of the region of attraction. In an application to a nonlinear continuous stirred tank reactor, we show that the scheme, which only requires solving a convex quadratic program online, has comparable performance to a nonlinear MPC scheme while being computationally significantly more efficient. Further, our results provide the basis for controlling nonlinear systems based on data-dependent linear prediction models, which we explore in our companion paper.

Modeling and Optimization of Triple Diode Model of Dye-Sensitized Solar Panel Using Heterogeneous Marine Predators Algorithm

The reliable mathematical model construction of dye-sensitized solar cells (DSSCs) using the triple-diode model (TDM) is proposed but it is a challenge due to its complexity. This work implements a novel method incorporating a recent meta-heuristic optimizer called the heterogeneous marine predators algorithm (H-MPA) to identify the nine parameters of the triple-diode equivalent circuit of DSSCs. In the optimization procedure, the nine unknown parameters of TDM are employed as decision variables, but the objective function to be minimized is the root mean square error (RMSE) between the experimental data and the estimated data. To prove the superiority of the H-MPA, the obtained results are compared with the slime mold algorithm (SMA), Transient search optimizer (TSO), Manta-Ray Foraging Optimization algorithm (MRFO), Forensic-Based Investigation (FBI), Equilibrium optimizer (EO), and Artificial ecosystem-based optimizer. The primary findings demonstrated the superiority of the proposed strategy in building a consistent model of the triple-diode model of DSSCs.

Linear Tracking MPC for Nonlinear Systems—Part II: The Data-Driven Case

In this article, we present a novel data-driven model predictive control (MPC) approach to control unknown nonlinear systems using only measured input–output data with closed-loop stability guarantees. Our scheme relies on the data-driven system parameterization provided by the fundamental lemma of Willems <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> We use new input–output measurements online to update the data, exploiting local linear approximations of the underlying system. We prove that our MPC scheme, which only requires solving strictly convex quadratic programs online, ensures that the closed loop (practically) converges to the (unknown) optimal reachable equilibrium that tracks a desired output reference while satisfying polytopic input constraints. As intermediate results of independent interest, we extend the fundamental lemma to affine systems and we derive novel robustness bounds w.r.t. noisy data for the open-loop optimal control problem, which are directly transferable to other data-driven MPC schemes in the literature. The applicability of our approach is illustrated with a numerical application to a continuous stirred tank reactor.

Payments for environmental services strategy for transboundary air pollution: A stochastic differential game perspective

Air pollution has become a global threat to societal development. The main challenges of transboundary air pollution control include two perspectives: uneven socioeconomic development of regions and the diffusion of air pollution. This paper proposes an PES strategy to alleviate transboundary air pollution by coordinating regional economic interests and environmental preferences within the joint prevention and control of air pollution region. To make the model design more realistic, we introduce the stochastic differential game model to characterize the diffusion and uncertainty of air pollution. The optimal feedback Nash equilibrium is derived in three PES scenarios (no PES, dynamic PES, and fixed-fee PES) by using the Hamilton-Jacobi-Bellman equation. Numerical simulations and sensitivity analysis are implemented to compare the optimal strategies under the three PES scenarios. The dynamic PES strategy is shown to outperform the no PES strategy and the fixed-fee PES strategy by encouraging the backward region to cut more emissions. Besides, the confidence interval theory is used to estimate the variation range of air pollution stocks, which provides a powerful diagnostic tool for policy-makers.

Approach to an Equivalent Freight-Based Sustainable Joint-Quotation Strategy for Shipping Blockchain Alliance

To improve the sustainability of the shipping industry, a practice of establishing a new type of shipping alliance based on blockchain has been implemented. In this practice, the following question emerges: How will shipping lines achieve sustainable profit improvement? This paper focuses on the freight decision-making problem by constructing a multi-round joint-quotation strategy. This paper also demonstrates the potential impact of a joint quotation strategy on the blockchain-based open freight market from a theoretical perspective. The numerical experiment results show that, compared with the initial state, the joint quotation strategy can help to stabilize shipping demands and reduce the fluctuation in overall demands. In this strategy, the freight level needs to be high to maximize profits. However, part of the demands will be sacrificed as a result. Moreover, the optimal equilibrium solution under the joint quotation strategy is relatively vulnerable to changes in the competitive relationship among the members of an alliance. In addition, the joint quotation may also be resolutely resisted by the shipper due to monopoly risk, resulting in a major risk of a sharp reduction in demand. The findings in this paper offer a decision-making reference for the sustainable development of the shipping industry.

Multivariate systemic optimal risk transfer equilibrium

A Systemic Optimal Risk Transfer Equilibrium (SORTE) was introduced in: “Systemic optimal risk transfer equilibrium”, Mathematics and Financial Economics (2021), for the analysis of the equilibrium among financial institutions or in insurance-reinsurance markets. A SORTE conjugates the classical Bühlmann’s notion of a risk exchange equilibrium with a capital allocation principle based on systemic expected utility optimization. In this paper we extend such a notion to the case when the value function to be optimized is multivariate in a general sense, and it is not simply given by the sum of univariate utility functions. This takes into account the fact that preferences of single agents might depend on the actions of other participants in the game. Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general framework allows us to introduce and study a Nash Equilibrium property of the optimizer. We prove existence, uniqueness, and the Nash Equilibrium property of the newly defined Multivariate Systemic Optimal Risk Transfer Equilibrium.

Multi-agent based optimal equilibrium selection with resilience constraints for traffic flow

Traffic guidance and traffic control are effective means to alleviate traffic problems. Formulating effective traffic guidance measures can improve the utilization rate of road resources and optimize the performance of the entire traffic network. Assuming that the traffic coordinator can capture traffic flow information in real-time utilizing sensors installed on each road, we consider the strong resilience constraints to construct an optimal selection problem of balanced flow in the traffic network. Based on multi-agent modeling, each agent has access to the corresponding traffic information of each link. We design a distributed optimization algorithm to tackle this optimization problem. In addition to the inherent advantages of distributed multi-agent algorithms, the communication topology among the sensors is allowed to be time-varying, which is more consistent with reality. To prove the effectiveness of the proposed algorithm, we also give a numerical simulation in the multi-agent environment. It should be reiterated that the optimization problem studied in this paper mainly focuses on traffic managers’ perspectives. The goal of the studied optimization problem is to minimize the overall cost of the traffic network by adjusting the optimal equilibrium traffic flow. This study provides a reference for solving congestion optimization in today’s cities.

Characteristics Analysis and Equilibrium Optimization of Mixed Traffic Flow considering Connected Automated and Human-Driven Vehicles

Considering the impact of informatization condition, vehicles on the road network are divided into connected automated vehicles (CAVs) and human-driven vehicles (HDVs), which follow the principle of system optimization and stochastic user equilibrium, respectively. Taking the road network reserve capacity maximization model under the condition of road capacity constraint as the upper-level programming and the traffic assignment model under heterogeneous flow environment as the lower level programming, then a bilevel programming model is constructed. Among them, the nonuniform demand growth multiplier is adopted for each OD pair to reflect the inconsistency of traffic demand structure growth, and the calculation of link capacity is related to the market penetration of CAVs. The incremental method, method of successive averages, and simulated annealing algorithm are used to solve the model, and the effects of different market penetration on road network capacity, travel time, and saturation are analyzed through a numerical example. The relevant data under different weights are normalized and the optimal deployment scheme of CAVs and HDVs in different periods is obtained by comprehensive evaluation. Meanwhile, the mixed equilibrium flow state is explored under the premise of given market penetration to verify the feasibility of the model and algorithm.

Phase equilibrium of the CaO–SiO2–Al2O3–Ce2O3 system: A basis for recovering REEs from RE-Containing slag

With the massive consumption of rare earth resources, the recovery of rare earth elements (REEs) from secondary resources has become a focus of attention. It is economically necessary to recover REEs from Bayan Obo RE-containing blast furnace slag (RE-BFS), which contains considerable REEs. However, due to the lack of relevant thermodynamic data, it is difficult to effectively recover REEs from RE-BFS. Therefore, the isothermal phase diagram of CaO–SiO2–Al2O3–Ce2O3 at 1100 °C was constructed in this work, according to the principle of phase diagram construction combined with the method of equilibrium experiments. The optimal rare earth equilibrium phase in this system was defined as Ce4.667-xCax(SiO4)3O1-0.5x. According to the actual composition of the RE-BFS, a dominant separation region of L + Ce4.667-xCax(SiO4)3O1-0.5x was obtained in the phase diagram. Subsequently, the extraction of REEs in the RE-BFS could be completed by solid–liquid separation, which allowed economic benefits without affecting the environment. The data reported in this work provide a theoretical basis for solving the problem of recovery from RE-containing secondary resources and developing high value-added materials.

A Tri-dimensional Equilibrium-based stochastic optimal dispatching model for a novel virtual power plant incorporating carbon Capture, Power-to-Gas and electric vehicle aggregator

This study proposes a novel structure of carbon-to-power-based virtual power plant (C2P-VPP) considering the flexible demand response and electric vehicle-to-grid aggregators (EVA). C2P is integrated by gas-power plant carbon capture (GPPCC), carbon storage devices (CS), and power-to-gas (P2G). Then, to balance the multiple objectives of the dispatching cost, the carbon emission and the output fluctuation, a tri-dimensional coordinated optimal dispatching model is construct from the perspective of the energy impossibility triangle problem. And the robust optimization theory is applied to characterize the uncertainty of wind power plant (WPP) and photovoltaic power generation (PV). Thirdly, to solve the above multi-objective model, an improved fuzzy equilibrium coordination-based-model solution algorithm is proposed. The algorithm integrates the fuzzy satisfaction theory, the multi-objective input income table, the entropy weight method into the complete information static game model. Finally, the CIGRE medium voltage distribution system is chosen for case study, the results show: (1) GPPCC could capture and store CO2 in carbon storage devices, and P2G could convert the CO2 into CH4 for conventional gas turbines (CGT) providing flexible power output. Compared with the scenario only with CS or P2G, when they are both introduced, the carbon emissions reduced by 7.75 % and 3.82 %, the dispatching cost reduced by 2.90 % and 7.77 %, and the output fluctuation reduced by 10.03 % and 2.80 %. (2) When robust coefficient Г is from 0.85 to 0.95, the decision-maker is the risk preference type and willing to bear risk to pursue excess benefit. (3) Sensitivity analysis shows when the rated capacity ratio of CGT, GPPCC and P2G is from 8:4:1 to 2:2:1, the dispatching scheme are optimal. When the capacity ratio of WPP, PV and EVA is from 20:1 to 10:1, the objective value changes significantly, and the optimal equilibrium scheme could be established when the capacity ratio is higher than 10:1. Therefore, the proposed optimal dispatching model could provide an effective tools for decision makers to achieve the optimal dispatching schemes.

Recycling light-emitting-diode waste quartz sand and stone sludge for the synthesis of amine grafted mesoporous humidity control material

This article proposes a cost-effective development method to recycle two types of waste material (i.e., light-emitting-diode (LED) waste quartz and stone sludge) to produce mesoporous materials. The proposed method is simple yet effective. Extraction of the precursor liquid from the silicon and aluminum sources respectively contained in the waste quartz and stone sludge is based on the alkali fusion method, and the mesoporous material is synthesized after the hydrothermal reaction. The novelty of this work lies in our functionalization of Mobil Composition of Matter No. 41 (MCM-41) through amine group grafting (AGG-MCM-41). The results show that the structure of the mesoporous MCM-41 grafted by the amine functional groups remains orderly. The 29Si nuclear magnetic resonance spectrum confirmed that the supported surface of the MCM-41 features a large number of silanol groups, which verifies that modification of the amine functional group improved the effect. The results show that when the addition amount of amine functional group is 5 vol% (when relative humidity = 95%), the highest equilibrium moisture content of the material is 26.5 kg/kg. The wet performance of the material was studied, and it was proved that 5.0-AGG-MCM-41 complies with the standard of Japanese Industrial Standard (JIS A 1475) (equilibrium moisture content>5 m3/m3). The research results also show that the optimal equilibrium moisture content is more than 44 times that of metal-organic frameworks humidity pump. Therefore, the synthesized AGG-MCM-41 offers excellent humidity control performance in the practical application of building materials as it narrows the indoor humidity fluctuation spontaneously without any energy consumption. This creates new opportunities for smart indoor humidity control. The amine-group-grafted MCM-41 effectively reduces the shortcomings of high energy consumption, high costs, and insufficient space caused by mechanical dehumidifiers. In addition, the production of AGG-MCM-41 from LED waste and stone sludge achieves 100% recycling. Transformation of two problematic industrial byproducts into humidity-conditioning materials with beneficial technical and environmental properties represents a significant contribution towards the goals of sustainable development.

New robust hybrid Jarratt-Butterfly optimization algorithm for nonlinear models

The nonlinear system of equations (NSE) is the basis of many engineering and scientific models. However, these models must have accurate solutions to be successful. Several optimization algorithms, including Butterfly Optimization Algorithm (BOA), can be applied to solving NSE problems. However, optimization algorithms lack accurate solutions due to their limitations, including trapping at local optima and divergence problems. In this work, a novel hybrid approach that is called Jarratt Butterfly Optimization Algorithm (JBOA) is presented to solve NSE. The proposed JBOA is developed by combining Jarratt’s iterative method and the BOA. Combining the two methods has enhanced BOA’s search mechanism and convergence speed. It also overcomes Jarratt’s limitations, including selecting initial values, trapping at local optima, and divergence problems. This results in solving NSE in more accurate solutions and fewer iterations. JBOA’s efficiency was evaluated based on eight benchmark systems, two of which represent real-world applications. Further, JBOA was compared to several optimization algorithms, including the original BOA algorithm, Particle Swarm Optimization (PSO), Harris Hawk Optimization (HHO), Equilibrium Optimization (EO), and Ant Lion Optimizer (ALO). Furthermore, the performance of JBOA and Jarratt’s method has been compared. Finally, the experimental results support the superiority of JBOA as it significantly outperformed all other compared optimization algorithms in all benchmark systems in terms of best solutions, average fitness values, stability, and convergence speed.

Routing of autonomous vehicles for system optimal flows and average travel time equilibrium over time

We envisage a transportation management center (TMC) for future autonomous vehicles that distributes the traffic over a cycle comprising many days in such a manner that the average travel time (AVT) of vehicles of the same origin–destination (OD) over the cycle is identical at equilibrium while the entire network achieves system optimal (SO) flows every day. Vehicles are assigned to a set of paths, with different proportions of time within the cycle on different paths, but with the same AVT over the cycle at equilibrium. Since AVT is the same for each user, it is inherently fair to every-one. Thus, not only do users gain by subscribing to this system but the overall system also, as SO flows are followed each day. We then model the case of mixed equilibrium (ME), wherein drivers are given the option to subscribe to the TMC path combinations and hence assist in achieving SO, while non-subscribers only minimize their own private costs and follow their user equilibrium (UE) paths, which typically have shorter times than the corresponding AVT. To counter this, a tolling scheme is imposed on non-subscribers, i.e., UE users, based on their value-of-time (VOT) distribution and the travel time difference between subscribers' AVT and non-subscribers' UE time at ME. In the end, the total cost of travel of non-subscribers, toll plus UE time, is equal to the corresponding AVT of subscribers. The efficiency of this system is shown through numerical studies. Finally, the impact of market penetration and different VOT distributions on AVT and toll are investigated through sensitivity analysis.

Optimal voting rules for international organizations, with an application to the United Nations

AbstractThis paper studies a self‐enforcing mechanism for an international organization that interacts repeatedly over time. A random shock determines which countries would be in favor of or against taking a collective action. If the organization wants to take the action, incentives must be provided. The optimal stationary equilibrium is equivalent to a mechanism characterized by voting weights. When the discount factor is not too high, the voting weights are random. Moreover, within a class of parameters, the optimal mechanism mimics the way voting power is distributed among United Nations members via Security Council seats. That is, the model rationalizes the existence of a council with permanent members who have veto power, nonpermanent members, and the exact number of permanent and nonpermanent members observed at the Security Council.

Optimal equilibrium contracts in the infinite horizon with no commitment across periods

Optimal equilibrium contracts in the infinite horizon with no commitment across periods

Resource allocation for D2D-assisted haptic communications

Haptic communications is recognized as a promising enabler of extensive services by enabling real-time haptic control and feedback in remote environments, e.g., teleoperation and autonomous driving. Considering the strict transmission requirements on reliability and latency, Device-to-Device (D2D) communications is introduced to assist haptic communications. In particular, the teleoperators with poor channel quality are assisted by auxiliaries, and each auxiliary and its corresponding teleoperator constitute a D2D pair. However, the haptic interaction and the scarcity of radio resources pose severe challenges to the resource allocation, especially facing the sporadic packet arrivals. First, the contention-based access scheme is applied to achieve low-latency transmission, where the resource scheduling latency is omitted and users can directly access available resources. In this context, we derive the reliability index of D2D pairs under the contention-based access scheme, i.e., closed-loop packet error probability. Then, the reliability performance is guaranteed by bidirectional power control, which aims to minimize the sum packet error probability of all D2D pairs. Potential game theory is introduced to solve the problem with low complexity. Accordingly, a distributed power control algorithm based on synchronous log-linear learning is proposed to converge to the optimal Nash Equilibrium. Experimental results demonstrate the superiority of the proposed learning algorithm.

A Mathematical Modeling Approach of Target Damage Strategy on the Intersection Confrontation

The efficiency of target damage is to measure the effect of projectile attacks. Because there is antagonism at the intersection of projectiles and targets, it is very difficult to establish scientific damage assessment strategy model and a numerical calculation method. To scientifically evaluate the target damage effect when the projectile attacks the aircraft target, this paper introduces a game confrontation mechanism and proposes a numerical calculation method of aircraft target damage game strategy under the intersection confrontation. According to the theory of two-person non-cooperative games, the projectile is regarded as one of the players in the confrontation damage game, and the aircraft target is regarded as the other player. The damage gain model and gain function of the intersection confrontation of both sides are established. The effective expectation function and the minimum and maximum theorem are used to obtain the optimal Nash equilibrium solution of the game matrix strategy, and the Nash equilibrium point of the strategy space set is derived. Through calculation and comparative analysis, the results show that the optimal gain degree composed of the damage strategy selected by both the projectile and aircraft target under the Nash equilibrium solution brings the greatest gain. The proposed hybrid game strategy of damage assessment can be an effective reference for air combat decision making in the state of intersection confrontation.