Equilibrium Solution Research Articles

Pricing decisions and fairness concerns: supply chain power structures

PurposeThis paper divides the dyadic supply chain into three power structures according to the relative channel power of the supply chain members and consequently examines the optimal supply chain pricing decisions when both suppliers and retailers are concerned with fairness issues.Design/methodology/approachThree models are constructed, namely the Stackelberg game model with the supplier as the leader, the Nash game model with the balance of power and another Stackelberg game model with the retailer as the leader. The equilibrium solutions are solved, and their results are analyzed.FindingsThe retail price of a product increases with an increase in the fairness concerns of the leader in a supply chain in which the supplier or retailer is the leader, while the fairness concerns of the member with less channel power have no effect on the retail price. In a power-balanced supply chain, both suppliers and retailers increase their retail prices as their fairness concerns increase. The relative size of the members’ fairness concerns affects member profits and total supply chain profits.Originality/valueThe main contributions are as follows: First, this paper proposes a new approach to studying supply chain pricing strategy, considering fairness concerns and power structure. Secondly, three game models are constructed. The Nash equilibrium solution is introduced to study the fairness of supply chain participants in pricing decisions and overall supply chain profitability. Finally, the supply chain management theory is expanded by this study on pricing decisions and supply chain performance.

Cooperation Dynamics in Multiagent Systems: Modeling Vehicular Cooperation through Game Theory

<div>Cooperation lies at the core of multiagent systems (MAS) and multiagent reinforcement learning (MARL), where agents must navigate between individual interests and collective benefits. Advanced driver assistance systems (ADAS), like collision avoidance systems and adaptive cruise control, exemplify agents striving to optimize personal and collective outcomes in multiagent environments. The study focuses on strategies aimed at fostering cooperation with the aid of game-theoretic scenarios, particularly the iterated prisoner’s dilemma, where agents aim to optimize personal and group outcomes. Existing cooperative strategies, such as tit-for-tat and win-stay lose-shift, while effective in certain contexts, often struggle with scalability and adaptability in dynamic, large-scale environments. The research investigates these limitations and proposes modifications to align individual gains with collective rewards, addressing real-world dilemmas in distributed systems. By analyzing existing cooperative strategies, the research investigates their effectiveness in encouraging group-oriented behavior in repeated games. It suggests modifications to align individual gains with collective rewards, addressing real-world dilemmas in distributed systems. Furthermore, it extends to scenarios with exponentially growing agent populations (<i>N</i> → +∞), addressing computational challenges using mean-field game theory to establish equilibrium solutions and reward structures tailored for infinitely large agent sets. Practical insights are provided by adapting simulation algorithms to create scenarios conducive to cooperation for group rewards. Additionally, the research advocates for incorporating vehicular behavior as a metric to assess the induction of cooperation, bridging theoretical constructs with real-world applications.</div>

Complexity Analysis and Control of Output Competition in a Closed-Loop Supply Chain of Cross-Border E-Commerce Under Different Logistics Modes Considering Chain-to-Chain Information Asymmetry

To investigate the dynamic complexity of chain-to-chain output decisions in a closed-loop supply chain system of cross-border e-commerce (CBEC), this study decomposes the system into four product–market (PM) chains, based on the e-commerce platform’s information-sharing strategy and the manufacturer’s selected logistics mode (direct mail or bonded warehouse). By combining game theory with complex systems theory, discrete dynamic models for output competition among PM chains under four scenarios are constructed. The Nash equilibrium solution and stability conditions of the models are derived according to the principles of nonlinear dynamics. The stability of the model under the four scenarios, as well as the impacts of the initial output level and comprehensive tax rates on the stability and stability control of the system, are analyzed using numerical simulation methods. Our findings suggest that maintaining system stability requires controlling the initial output levels, the output adjustment speeds, and tariff rates to remain within specific thresholds. When these thresholds are exceeded, the entropy value of the model increases, and the system outputs decisions to enter a chaotic or uncontrollable state via period-doubling bifurcations. When the output adjustment speed of the four PM chains is high, the direct-mail logistics mode exhibits greater stability. Furthermore, under increased tariff rates for CBEC, the bonded warehouse mode has a stronger ability to maintain stability in system output decisions. Conversely, when the general import tax rate increases, the direct-mail mode demonstrates better stability. Regardless of the logistics mode, the information-sharing strategy can enhance the stability of system output decisions, while increased e-commerce platform commission rates tend to reduce stability. Interestingly, the use of a non-information-sharing strategy and the direct-mail logistics mode may be more conducive to increasing the profit levels of overseas manufacturers. Finally, the delayed feedback control method can effectively reduce the entropy value, suppress chaotic phenomena in the system, and restore stability to output decisions from a fluctuating state.

Dynamic Decisions of Quality and Goodwill in a Two-Echelon Supply Chain with Delay Effect

Improving product quality is an effective way for a company to gain goodwill as well as to increase sales. In this paper, the focus is on a two-echelon supply chain consisting of a manufacturer and a retailer. Considering the delayed effect of quality improvement on product quality, the differential equations on the levels of quality and goodwill as the state variables are established to explore the impact of delay time on members’ strategy inputs, supply chain profits and performance under decentralized and centralized decision models. Additionally, in this paper, how different profit maximization scenarios affect supply chain revenue under decentralized models is examined. The equilibrium solutions are obtained by numerical simulation. The results show that both quality improvement efforts and marketing efforts are decreasing functions of delay time, and the delay effect affects the motivation of supply chain members’ strategic inputs. Importantly, supply chain profit under centralized decision is not always higher than that under decentralized decision when the manufacturer’s concern degree and delay time are taken into account. The Rubinstein bargaining model is then used to optimize the profit distribution between the manufacturer and the retailer under centralized decision. In this paper, it is also identified that the longer the delay time, the lower the levels of quality and goodwill. The dual effect of delay effect and concern degree will exacerbate the negative impact on the levels of quality and goodwill.

Carbon emission reduction dynamic decision-making in the electricity supply chain with different carbon emission allowance principles

To address the impacts of typical principles for allocating initial carbon emission allowance, namely, the grandfathering and benchmarking principles on long-term emission reduction strategies, this paper presents four differential game models for an electricity supply chain consisting of a single electricity supplier and a single electricity retailer. The equilibrium solutions for different models are obtained and comparatively analyzed. The results show that the power supplier exhibits greater emission reduction efforts and, therefore, that the retailer has higher profits under the benchmarking principle than under the grandfathering principle. Moreover, under the two principles, the level of the power supplier’s emission reduction efforts is related to the decision-making modes of the electricity supply chain. Under specific conditions, the power supplier’s profit is also higher under the benchmarking principle than under the grandfathering principle. In addition, this study extends the models to investigate the constrained differential game models and obtains the game equilibrium solutions, which can be proved as saddle point equilibria.

A dynamic analysis of a green closed-loop supply chain with different on-line platform smart recycling and selling models

Global warming and waste product recycling have become increasingly serious. Green innovation of a closed-loop supply chain (GCLSC) and selection of on-line platform recycling and sales models need to be solved as a matter of urgency. To achieve this purpose, dynamic influences of green innovation, platform smart marketing, and recycling investments on goodwill are considered. Dynamic game models are used to solve the equilibrium solutions and optimal trajectories under different models, and influences of commission rates on the profits and key variables are compared via numerical simulations. The results show that the evolution paths of goodwill under the four decision models all descend at first, then stabilize; the optimal strategy can be selected based on the intervals of sales and recycling commission rates. Under the dual-agency model (agency selling and platform agency recycling (Model AP)), the higher the sales commission rate (SCR) is, the more likely the manufacturer chooses the reselling and platform agency recycling (Model RP), which, however, may lead to insufficient green innovation investment. The higher the RCR, the greater the platform profit when selecting agency selling. The system profit first increases, then decreases with growing sales commission, and Model AP represents the optimal strategy. The research offers reference for the selection of operation models and investment decision, is conducive to improving the enterprise goodwill, and promotes the sustainable development of the GCLSC.

Thermal solutions of strongly magnetized disks and the hysteresis in X-ray binaries

Context. X-ray binaries (XRBs) exhibit a spectral hysteresis for luminosities in the range 10−2 ≲ L/LEdd ≲ 0.3, with a hard X-ray spectral state that persists from quiescent luminosities up to ≳0.3LEdd, transitioning to a soft spectral state that survives with decreasing luminosities down to ∼10−2LEdd. Aims. We present a possible approach to explain this behavior based on the thermal properties of a magnetically arrested disk simulation. Methods. By post-processing the simulation to include radiative effects, we solved for all the thermal equilibrium solutions as the accretion rate, Ṁ, varies during the XRB outburst. Results. For an assumed scaling of the disk scale height and accretion speed with temperature, we find that two solutions exist in the range of 10−3 ≲ Ṁ/ṀEddington ≲ 0.1 at r = 8 rg (4 × 10−2 ≲ Ṁ/ṀEddington ≲ 0.5 at r = 3 rg): a cold, optically thick solution, and a hot, optically thin one. This opens the possibility of a natural thermal hysteresis in the right range of luminosities for XRBs. We stress that our scenario for the hysteresis does not require us to invoke the strong advection-dominated accretion flow principle, nor does it require the magnetization of the disk to change during the XRB outburst. In fact, our scenario requires a highly magnetized disk in the cold soft state to reproduce the transition from soft to hard state at the right luminosities. Our scenario therefore predicts a jet, although possibly very weakly dissipative, in the soft state of XRBs. We also predict that if active galactic nuclei have similar hysteresis cycles and are strongly magnetized, they undergo a transition from soft to hard state at much lower L/LEdd than XRBs.

On the Equations of Compressible Fluid Dynamics With Cattaneo‐Type Extensions for the Heat Flux: Symmetrizability and Relaxation Structure

ABSTRACTThe aim of this work is twofold. From a mathematical point of view, we show the existence of a hyperbolic system of equations that is not symmetrizable in the sense of Friedrichs. Such system appears in the theory of compressible fluid dynamics with Cattaneo‐type extensions for the heat flux. In contrast, the linearizations of such system around constant equilibrium solutions have Friedrichs symmetrizers. Then, from a physical perspective, we aim to understand the relaxation term appearing in this system. By noticing the violation of the Kawashima–Shizuta condition, locally and smoothly, with respect to the Fourier frequencies, we construct persistent waves, that is, solutions preserving the norm for all times that are not dissipated by the relaxation terms.

Investment Decision and Coordination of Fresh Supply Chain Blockchain Technology Considering Consumer Preference

In this paper, we study the decision-making and coordination problem of a two-tier fresh food supply chain consisting of a supplier and a retailer. Considering the influencing factors of consumers’ information preference, freshness, and misrepresentation, we construct a centralized decision-making model and a decentralized decision-making Stackelberg game model. We also analyze the changes in the equilibrium solution of the supply chain before and after the input of blockchain technology, identify the conditions for the investment in blockchain technology, and design a “cost-sharing + benefit-sharing” combination contract for the coordination of the blockchain. The results are as follows: Firstly, under decentralized decision-making, if the fresh supplier misreports the freshness of the product, it will mislead the retailer to increase the order quantity, and its own profit will rise. Therefore, the fresh supplier has the motivation to misreport freshness. However, the backlog of fresh products will eventually damage the retailer‘s profit, and the overall profit of the supply chain will also be damaged. Therefore, the increase in the profit of the fresh supplier is at the expense of the overall interests and stability of the supply chain. Second, when the investment cost of blockchain technology is within a certain threshold, it is feasible to invest in blockchain technology. Consumers’ preference for traceable fresh products will encourage the fresh supply chain to improve the level of information traceability and increase investment in blockchain technology. Finally, there are double marginal effects in the fresh supply chain under decentralized decision-making. The combined contract of “cost-sharing + revenue-sharing” can coordinate the overall revenue of the supply chain to the level of centralized decision-making. When the contract parameters meet certain conditions, Pareto improvement in revenue can be achieved for all parties involved in the fresh supply chain. The willingness of retailers to invest in blockchain technology will change with the change in contract parameters. When the proportion of retailers’ costs and the proportion of shared income are higher, the level of retailers’ investment in blockchain technology will decrease. Therefore, the interests of supply chain members need to be balanced in the process of contract coordination.

Hybrid game model for electricity trading and pricing among multiple microgrids and consumers based on demand-side complex networks

Local energy markets (LEMs) are pivotal for enhancing renewable energy consumption and facilitating green electricity by linking multiple microgrids (MMGs) and consumers. Existing studies often overlook the evolution of consumer purchasing strategies influenced by social spatial structures and social learning dynamics. This paper addresses this gap by introducing a hybrid game model for LEMs electricity trading and dynamic pricing, integrating demand-side complex network structures. This model encompasses dynamic interactions between MMGs and electricity and carbon markets, employing non-cooperative games among MMGs, complex network evolutionary game among consumers, and Stackelberg games between MMGs and consumers. It promotes efficient transactions and adaptive pricing through distributed algorithms optimizing equilibrium solutions. Key findings include: (1) Compared to Power-to-Grid and Peer-to-Peer models, our model increased MMGs' revenues by 44.10 % and 10.60 %, reduced consumers’ costs by 8.76 % and 1.15 %, and cut carbon emissions by 93.32 % and 77.30 %, significantly boosting economic and environmental benefits. (2) Complex network analysis underscores the importance of social learning in adjusting power consumption strategies and pricing mechanisms. (3) Multidimensional consumer decision-making enhances renewable energy valuation, fosters price competition among MMGs, and accelerates low-carbon transitions in LEMs, providing crucial theoretical support for LEM structure design and policy to enhance renewable energy use.

Blockchain’s role in operation strategy of power battery closed-loop supply chain

With the rapid development of the electric vehicle industry, the issue of recycling retired power batteries is becoming increasingly prominent. Considering the adoption of blockchain technology to enhance information traceability for retired power batteries, we construct three closed-loop supply chain decision-making models: a supply chain that does not adopt blockchain technology, a manufacturing enterprise that independently bears the input cost of blockchain technology, and a scenario which both manufacturing and selling enterprises jointly bear the blockchain technology input costs. Through the comparative analysis of the equilibrium solutions of these models, including the profits of each supply chain member, consumer surplus, environmental impact, and social welfare, it is concluded that the level of information traceability increases with the increase of the blockchain technology input cost-sharing ratio λ by the selling enterprise. However, there is a threshold for λ and λ cannot be infinitely increased. The profits of both manufacturing and selling enterprises increase with the adoption of blockchain technology, yet the ladder-utilizing enterprise achieves the highest profits when not adopting blockchain technology. Furthermore, the adoption of blockchain technology can increase consumer surplus and social welfare while reducing the impact on the environment. These studies provide theoretical guidance on both economic and social environment for the operation decision of each member in the closed-loop supply chain of power batteries.

Optimization Strategy for Integrated Energy Microgrids Based on Shared Energy Storage and Stackelberg Game Theory

The implementation of community power generation technology not only increases the flexibility of electricity use but also improves the power system’s load distribution, increases the overall system efficiency, and optimizes energy allocation. This article first outlines the operational context of the system and analyzes the roles and missions of the various participants. Subsequently, optimization models are developed for microgrid operators, community power storage facility service providers and load aggregators. Next, the paper explores the game relationship between microgrid operators and load aggregators, proposing a model based on the Stackelberg game theory and proving the presence and singularity of the Stackelberg equilibrium solution. Finally, simulations are conducted using Yalmip tools and the CPLEX solution on the MATLAB R2023a software platform. A combination of heuristic algorithms and solver methods is employed to optimize the strategies of microgrid operators and load aggregators. The research findings show that the proposed framework is not only able to achieve an effective balance of interests between microgrid operators and load aggregators but also creates a win–win situation between load aggregators and shared energy storage operators. Additionally, the solution algorithms used ensure the protection of data privacy.

Differential Game Theory in Electric Vehicle Charging Optimization

Differential Game Theory is a strong way to solve the hard optimization problems that come up when charging electric vehicles (EVs) in smart grids. This method takes into account the strategic exchanges between many parties, such as EV users, charging station operators, and grid operators, each of whom has their own goals. In this study, we look at how differential game theory can be used to find the best charging plans for electric vehicles. Our goal is to lower total energy costs, make the grid less crowded, and get more energy from green sources. We take into account how power prices change, the different types of green energy that are available, and how EVs move around by modeling the charging process as a difference game. The game-theoretic method lets you come up with plans where each player maximizes their own reward while taking into account what other players do. To find the best price methods for different situations, such as peak and off-peak hours, different amounts of green energy usage, and different pricing plans, we use Nash equilibrium solutions. Our findings show that differential game theory can successfully balance the different needs of parties, which can lead to EV charging systems that work better and last longer. Furthermore, the suggested way not only makes smart grids work better, but it also helps lower carbon emissions by supporting the use of clean energy. In addition, this method helps us understand how to create reward systems that match individuals' goals with world improvement goals. This makes it possible for EV charging strategies to work together better. This study shows that differential game theory could be a very useful tool in the development of smart and environmentally friendly transportation systems.

The Navier-Stokes equation and a fully developed turbulence

In fairly general conditions we give explicit (smooth) solutions for the potential flow. We show that, rigorously speaking, the equations of the fluid mechanics have not rotational solutions. However, within the usual approximations of an incompressible fluid and an isentropic flow, the Navier-Stokes equation has approximate vorticial (rotational) solutions, generated by viscosity. However, in general, these vortices are unstable, and a discrete distribution of vorticial solutions is not in me chanical equilibrium; it forms an unstable vorticial liquid. On the other hand, these solutions may exhibit turbulent, fluctuating instabilities for large variations of the velocity over short distances. We represent a fully developed turbulence as a homogeneous, isotropic and highly-fluctuating distribution of singular centres of turbulence. A regular mean flow can be included. In these circumstances the Navier-Stokes equation exhibits three time scales. The equations of the mean flow can be disentangled from the equations of the fluctuating part, which is reduced to a vanishing inertial term. This latter equation is not satisfied after averaging out the temporal fluctuations. For a homogeneous and isotropic distribution of non-singular turbulence centres the equation for the inertial term is satisfied trivially, i.e. both the average fluctuating velocity and the average fluctuating inertial term are zero. If the velocity is singular at the turbulence centres, we are left with a quasi-ideal classical gas of singularities, or a solution of singularities (solute) in quasi thermal equilibrium in the background fluid (solvent). This is an example of an emergent dynamics. We give three examples of vorticial liquids.

Static analysis of masonry polygonal domes with membrane forces

Masonry polygonal domes are parts of several historic monuments and masonry buildings. The analysis of polygonal domes with a static approach is here presented, considering the structure made of vault cells and ribs. The vaults are portions of a cylindrical surface with single curvature in the meridian direction, and translation of the generatrix curve in the direction of the cylinder axis. The system is divided into rigid elements with interfaces. The first part of the work regards the geometry of the rigid elements, in particular the angles that characterize the shape of these. Then the equilibrium equations applied to every element are presented, with a procedure to calculate the membrane response for the self-weight in a non-redundant system, under symmetry conditions. The sum of the solution for the vaults is combined with effects of the weight of ribs and radial walls. The use of the equilibrium solution for pointed domes, domes with a lantern and oculus or additional load to the dome self-weight is shown. A limit analysis static approach is formulated calculating the maximum load relative to the equilibrium of the rigid elements considering the interfaces tensile strength and limited cohesion and friction for the shear resistance.

Signaling the Capability of New Counterterrorism Technologies in the Face of a Strategic Threat

ABSTRACTWhen implementing new security technologies (e.g., cargo scanners at ports), security and defense agencies face important decisions regarding the disclosure of related information to the public. Public disclosure of these technologies can grant adversarial organizations access to critical security‐related information, enabling them to potentially develop or adjust their attack strategies to circumvent defense systems. Alternatively, withholding information about newly deployed technologies can create uncertainties for adversaries regarding the effectiveness of defense systems. In this work, we study defensive information disclosure related to the deployment of new security technology, focusing on the level of technical information released by the defender and how this affects the adversary's (i) belief about the technology's capability (in detecting or otherwise thwarting an attack) and (ii) attack plan. Specifically, we introduce a novel signaling model in which the defender has private information about the (level of) capability of her technology, and decides on the level of related technical detail to be disclosed. The attacker, upon receiving the defender's signal, updates his belief about the capability of the technology and subsequently decides whether to launch an attack. We obtain and prove closed‐form equilibrium solutions to the model and conduct numerical sensitivity analyses to understand how the equilibrium strategies change as the model's data varies. Based on the equilibrium conditions, we also characterize and analyze four unique scenarios that partition the solution space based on relations between the defender's signaling costs.

Optimization of vitrification methods for equine oocytes

An important method for preserving equine germplasm is the cryopreservation of equine oocytes. Due to its ease, rapidity and affordability, vitrification freezing has taken over as the primary method of horse oocyte cryopreservation. The vitrification cryoprotectants utilized in this investigation were Ethylene glycol (E), Dimethyl sulfoxide (D), Sucrose (S), and Ficoll (F). According to the oocyte volume alteration, the treatment time was 39 s in equilibrium solution ED10 (10 % EG + 10 % DMSO), 32 s in equilibrium solution ED15 (15 % EG + 15 % DMSO), while 20 s in equilibrium solution ED20 (20 % EG + 20 % DMSO). We prepared three kinds of cryosolutions EDFS30 (E15 %+D15 %+70 %FS), EDFS35 (E17.5 % + D17.5 % + 65 %FS), EDFS40 (E20 % + D20 % + 60 %FS) according to the proportion of protectant components. Among 27 freezing protocols, we selected protocol ED10 (39 s) + EDFS30 + 80 s which has the highest in vitro culture maturation rate of 19.3 % while protocol ED20 (20 s) + EDFS40 + 120 s is the worst. Apoptosis gene analysis revealed that BAX, BID, BOK, and TP53 expression was substantially higher in oocytes from the ED20 (20 s) + EDFS40 + 120 s group than in oocytes from the ED10 (39 s) + EDFS30 + 80 s and control groups (p<0.01). This study investigated several vitrification schemes for equine oocytes.

Chance-constrained bi-level optimal scheduling model for distribution network with thermal controllable load aggregators

An increasing amount of distributed renewable energy is being integrated into distribution networks to achieve decarbonization. It is essential to exploit demand-side flexible resources further to enhance system flexibility in response to the intermittency and unpredictability of renewable energy sources. This paper introduces a polytope-based aggregation method for thermostatically controlled loads (TCLs), aggregating numerous individual TCLs into a unified virtual battery via the aggregator (AGG). This approach avoids the dimensionality curse faced by the distribution system operator when directly controlling each TCL, while efficiently utilizing TCL flexibility. Subsequently, a bi-level optimization model is established, where AGGs are treated as independent stakeholders participating in the distribution network scheduling optimization through the local energy market. This model incorporates chance constraints to address the uncertainty of renewable energy sources. Finally, the distributionally robust chance constraint (DRCC) method is used to convert chance constraints into a linear form, and strong duality theory and Karush–Kuhn–Tucker conditions are applied to transform the bi-level model into a single-level model with equilibrium solutions. Case studies on the IEEE 33-bus network demonstrate that the proposed polytope-based aggregation method substantially improves computational efficiency with minimal optimality loss. Additionally, the DRCC method offers superior economic performance compared to robust and deterministic optimization approaches, while maintaining robustness.

Automatic construction and verification algorithm for smart contracts based on formal verification

As an emerging technology, blockchain demonstrates strong potential for applications in digital finance. As a core component of blockchain, the security and reliability of smart contracts is crucial. To ensure the high reliability of smart contracts, this study employs formal construction and verification techniques based on game theory. Initially, the profit function is defined using distortion techniques, and a game model for supply chain participation is designed. However, the equilibrium solution of the two-party game does not represent the optimal solution for the supply chain system. Therefore, the study introduces third-party participation to optimize the equilibrium solution. Finally, a probability model detection method is used to verify the constructed smart contract model. The results show that the supply chain model, analyzed through formal methods, has attributes consistent with theoretical analysis. Consequently, the research on automatic construction and verification algorithms for smart contracts based on formal verification proves to be effective and feasible in practical applications.

ADP-Based Decentralized Load Frequency Control Schemes to Multiarea Asynchronous Markov Jumping Power Systems With Experience Replay.

This article investigates the problem of robust decentralized load frequency control (LFC) in multiarea interconnection power systems, in the presence of the external disturbances and stochastic abrupt variations, such as component failures and different load demands. To capture the different operating conditions of the load in the multiarea power systems, a Markov superposition technique is skillfully employed to model the system's component matrices. To solve the Nash equilibrium solution of the zero-sum differential game problem, an improved online adaptive dynamic programming (ADP) algorithm based on the experience replay technique (ERT) is developed, which addresses the nonlinear coupling difficulties encountered in solving the game algebraic Riccati equations (Game AREs). The proposed algorithm weakens the condition that traditional policy iteration online solutions of zero-sum games require an initial stabilizing control policy pair, and considers the fact that only the transition probability matrix is known, while the prior knowledge of the dynamics is completely unknown. The algorithm is used to obtain both a minimum LFC and maximum disturbance policy for a given disturbance rejection level. Additionally, a safer LFC controller of the power systems with the Markov jumping parameters is designed. The stability and convergence of the proposed algorithm are demonstrated. Finally, the effectiveness and good performance of the proposed design method are validated using a two-area four-mode simulation example.