Hierarchical Decision Problem Research Articles

Optimal Control and Filtering for Hierarchical Decision Problems With Constraint Based on Stackelberg Strategy

Optimal Control and Filtering for Hierarchical Decision Problems With Constraint Based on Stackelberg Strategy

Integral reinforcement learning-based dynamic event-triggered safety control for multiplayer Stackelberg–Nash games with time-varying state constraints

In this article, an integral reinforcement learning-based dynamic event-triggered safety control scheme is proposed to tackle the multiplayer Stackelberg–Nash games (MSNG) problem in continuous-time nonlinear systems with time-varying state constraints. Initially, a new barrier function (BF) is introduced by integrating traditional barrier functions with a novel smooth function to address the challenge of time-varying state constraints. Then, the constrained MSNG system is transformed into an unconstrained system through the application of the state transformation technique, which helps to characterize hierarchical decision problems as MSNG problems with the leader and followers. Meanwhile, the integral reinforcement learning (IRL) technique is also applied to ease the demand for precise system dynamics. Moreover, a new dynamic event-triggered control (DETC) mechanism is designed, resulting in coupled dynamic event-triggered Hamilton–Jacobi (HJ) equations. A single critic neural network (NN) is constructed to learn the optimal control laws for the leader and followers. By employing Lyapunov theory, the integral reinforcement learning-based dynamic event-triggered safety control method ensures that the system state and the critic NN weight errors are uniformly ultimately bounded (UUB). Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed algorithm.

Adaptive dynamic programming-based hierarchical decision-making of non-affine systems

In this paper, the problem of multiplayer hierarchical decision-making problem for non-affine systems is solved by adaptive dynamic programming. Firstly, the control dynamics are obtained according to the theory of dynamic feedback and combined with the original system dynamics to construct the affine augmented system. Thus, the non-affine multiplayer system is transformed into a general affine form. Then, the hierarchical decision problem is modeled as a Stackelberg game. In the Stackelberg game, the leader makes a decision based on the information of all followers, whereas the followers do not know each other’s information and only obtain their optimal control strategy based on the leader’s decision. Then, the augmented system is reconstructed by a neural network (NN) using input–output data. Moreover, a single critic NN is used to approximate the value function to obtain the optimal control strategy for each player. An extra term added to the weight update law makes the initial admissible control law no longer needed. According to the Lyapunov theory, the state of the system and the error of the weights of the NN are both uniformly ultimately bounded. Finally, the feasibility and validity of the algorithm are confirmed by simulation.

Extended Kalman Filter Based Resilient Formation Tracking Control of Multiple Unmanned Vehicles via Game-Theoretical Reinforcement Learning

In this paper, we discuss the resilient formation tracking control problem of multiple unmanned vehicles (MUV). A dynamic leader-follower distributed control structure is utilized to optimize the performance of the formation tracking. For the follower of the MUV, the leader is a cooperative unmanned vehicle, and the target of formation tracking is a non-cooperative unmanned vehicle with a nonlinear trajectory. Therefore, an extended Kalman filter (EKF) observer is designed to estimate the state of the target. Then the leader of the MUV is adjusted dynamically according to the state of the target. In order to describe the interactions between the follower and dynamic leader, a Stackelberg game model is constructed to handle the hierarchical decision problems. At the lower layer, each follower responds by observing the leader's strategy, and the potential game is used to prove a Nash equilibrium among all followers. At the upper layer, the dynamic leader makes decisions depending on the response of all followers to reaching the Stackelberg equilibrium. Moreover, the Stackelberg-Nash equilibrium of the designed game theoretical model is proven. A novel reinforcement learning-based algorithm is designed to achieve the Stackelberg-Nash equilibrium of the game. Finally, the effectiveness of the method is verified by a variety of formation tracking simulation experiments.

Interval-parameter bi-level programming for energy system management under uncertainty: Towards a deep-decarbonized and sustainable future in China

To reach the goal of carbon-neutral society, synergistic approaches based on mathematical models that are capable of planning energy system associated with complex uncertainty and hierarchical structure are desired. In this study, an interval-parameter bi-level programming (IBP) method that is effective for solving hierarchical decision-making problems with conflict objectives and uncertainty is developed. Then, an IBP based China's energy system model (abbreviated as IBP_CES) is formulated for mainland China during 2021–2050, where the higher-level objective of minimizing carbon dioxide (CO2) emission is given priority, following by the lower-level objective of minimizing system cost. Results from IBP_CES show that, although China's energy supply would still rely heavily on fossil fuels, the share of coal would decrease to 33.9% by 2050. Results also disclose that the national CO2 emission would reach the peak value during 2026-2030 and then decline rapidly due to development of renewable energy and promotion of carbon-treatment technology which is beneficial for government to achieve carbon neutrality goals. In addition, compared to the traditional single-level model, CO2 emission from IBP_CES can decrease by 5.9%, demonstrating that IBP_CES is more superior in energy system decarbonization. The findings can help decision makers to well understand the interactions and tradeoffs between climate mitigation and economic development, and provide effective information on clean, sustainable and deep-decarbonized development strategies.

Managing Product Transitions: A Bilevel Programming Approach

We model the hierarchical and decentralized nature of product transitions using a mixed-integer bilevel program with two followers, a manufacturing unit and an engineering unit. The leader, corporate management, seeks to maximize revenue over a finite planning horizon. The manufacturing unit uses factory capacity to satisfy the demand for current products. The demand for new products, however, cannot be fulfilled until the engineering unit completes their development, which, in turn, requires factory capacity for prototype fabrication. We model this interdependency between the engineering and manufacturing units as a generalized Nash equilibrium game at the lower level of the proposed bilevel model. We present a reformulation where the interdependency between the followers is resolved through the leader’s coordination, and we derive a solution method based on constraint and column generation. Our computational experiments show that the proposed approach can solve realistic instances to optimality in a reasonable time. We provide managerial insights into how the allocation of decision authority between corporate leadership and functional units affects the objective function performance. This paper presents the first exact solution algorithm to mixed-integer bilevel programs with interdependent followers, providing a flexible framework to study decentralized, hierarchical decision-making problems.

A reformulation strategy for mixed-integer linear bi-level programming problems

Bi-level programming has been used widely to model interactions between hierarchical decision-making problems, and their solution is challenging, especially when the lower-level problem contains discrete decisions. The solution of such mixed-integer linear bi-level problems typically need decomposition, approximation or heuristic-based strategies which either require high computational effort or cannot guarantee a global optimal solution. To overcome these issues, this paper proposes a two-step reformulation strategy in which the first part consists of reformulating the inner mixed-integer problem into a nonlinear one, while in the second step the well-known Karush-Kuhn-Tucker conditions for the nonlinear problem are formulated. This results in a mixed-integer nonlinear problem that can be solved with a global optimiser. The computational and numerical benefits of the proposed reformulation strategy are demonstrated by solving five examples from the literature.

Decomposition Algorithms for Some Deterministic and Two-Stage Stochastic Single-Leader Multi-Follower Games

We consider a certain class of hierarchical decision problems that can be viewed as single-leader multi-follower games, and be represented by a virtual market coordinator trying to set a price system for traded goods, according to some criterion that balances supply and demand. The objective function of the market coordinator involves the decisions of many agents, which are taken independently by solving convex optimization problems that depend on the price configuration and on realizations of future states of the economy. One traditional way of solving this problem is via a mixed complementarity formulation. However, this approach can become impractical when the numbers of agents and/or scenarios become large. This work concerns agent-wise and scenario-wise decomposition algorithms to solve the equilibrium problems in question, assuming that the solutions of the agents’ problems are unique, which is natural in many applications (when solutions are not unique, the approximating problems are still well-defined, but the convergence properties of the algorithm are not established). The algorithm is based on a previous work of the authors, where a suitable regularization of solution mappings of fully parameterized convex problems is developed. Here, we show one specific strategy to manage the regularization parameter, extend some theoretical results to the current setting, and prove that the smooth approximations of the market coordinator’s problem converge epigraphically to the original problem. Numerical experiments and some comparisons with the complementarity solver PATH are shown for the two-stage stochastic Walrasian equilibrium problem.

A Hierarchical Decision-Making Strategy for the Energy Management of Smart Cities

This paper presents a hierarchical decision-making strategy for the energy management of a smart city. The proposed decision process supports the city energy manager and local policy makers in taking energy retrofit decisions on different urban sectors by an integrated, structured, and transparent management. To this aim, in the proposed decision strategy, a bilevel programming model integrates several local decision-making units, each focusing on the energy retrofit optimization of a specific urban subsystem, and a central decision unit. We solve the hierarchical decision problem by a game theoretic distributed algorithm. We apply the developed decision model to the case study of the city of Bari (Italy), where a smart city program has recently been launched.

RDM interval arithmetic based decision making on port selection

In this article, we consider a hierarchical multiattribute decision problem of an optimal port choice under interval information. The solution of the problem is based on the use of RDM interval arithmetic. The obtained results show validity of the suggested approach.

Decision Making on an Optimal Port Choice under Z-information

The decision of shipping lines as to which port to use is a strategic one, and it is one of the most crucial factors to influence the operational and business performance of the organizations. The existing decision theories are not sufficiently adequate to account for imprecision and partial reliability of decision-relevant information in real-world problems. Prof. Zadeh suggested the concept of a Z-number which is able to formalize imprecision and partial reliability of information. In this article, we consider a hierarchical multiattribute decision problem of an optimal port choice under Z-number-based information. The solution of the problem is based on the use of Z-number-valued weighted average aggregation operator. The obtained results show validity of the suggested approach.

Hierarchical Decision-Making Problems – Modeling and Solutions

Abstract Article illustrates the state of the work conducted at the Faculty of Management Warsaw University of Technology on the issue of modeling hierarchical decision-making problems in the context of administrative and infrastructural conditions of the various forms of public safety. The aim is to develop a universal methodology of conduct for the management needs of the public administration, whose powers are focused on maintaining the continuity of the critical infrastructure of the State. The key issues covered by the article are: modeling of hierarchical issues and decision-making processes in the multi-layered organizational structures; harmonization of scales significance of decisionmaking areas with significance weights of elementary decisions in these decision areas; and a priori contradictions of elementary decisions from different decision areas and value assessments of taken problem decisions.

Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods

Pythagorean fuzzy set initially developed by Yager (2014) is a new tool to model imprecise and ambiguous information in multicriteria decision making problems. In this paper, we propose a novel closeness index for Pythagorean fuzzy number (PFN) and also introduce a closeness index-based ranking method for PFNs. Next, we extend the Pythagorean fuzzy set to present the concept of interval-valued Pythagorean fuzzy set (IVPFS) which is parallel to interval-valued intuitionistic fuzzy set. The elements in IVPFS are called interval-valued Pythagorean fuzzy numbers (IVPFNs). We further introduce the basic operations of IVPFNs and investigate their desirable properties. Meanwhile, we also explore the ranking method and the distance measure for IVPFNs. Afterwards, we develop a closeness index-based Pythagorean fuzzy QUALIFLEX method to address hierarchical multicriteria decision making problems within Pythagorean fuzzy environment based on PFNs and IVPFNs. This hierarchical decision problem includes the main-criteria layer and the sub-criteria layer in which the relationships among main-criteria are interdependent, the relationships among sub-criteria are independent and the weights of sub-criteria take the form of IVPFNs. Therefore, in the developed method we first define the concept of concordance/discordance index based on the closeness index-based ranking methods and compute the sub-weighted concordance/discordance indices by employing the weighted averaging aggregation operator based on the closeness indices of IVPFNs. In order to take main-criteria interactions into account, we further employ Choquet integral to calculate the main-weighted concordance/discordance indices. By investigating all possible permutations of alternatives with the level of concordance and discordance of the complete preference order, we finally obtain the optimal rankings of alternatives. The proposed method is implemented in a risk evaluation problem in order to demonstrate its applicability and superiority. The salient features of the proposed method, compared to the state-of-the-art QUALIFLEX-based methods, are: (1) it can take the interactive phenomena among criteria into account; (2) it can manage simultaneously the PFN and IVPFN decision data; (3) it can deal effectively with the hierarchal structure of criteria. The proposed method provides us with a useful way for hierarchical multicriteria decision making problems within Pythagorean fuzzy contexts. In addition, we also extend the proposed method to manage heterogeneous information which includes five different types of information such as real numbers, interval numbers, fuzzy numbers, PFNs and hesitant fuzzy elements.

Machine learning-based coding unit depth decisions for flexible complexity allocation in high efficiency video coding.

In this paper, we propose a machine learning-based fast coding unit (CU) depth decision method for High Efficiency Video Coding (HEVC), which optimizes the complexity allocation at CU level with given rate-distortion (RD) cost constraints. First, we analyze quad-tree CU depth decision process in HEVC and model it as a three-level of hierarchical binary decision problem. Second, a flexible CU depth decision structure is presented, which allows the performances of each CU depth decision be smoothly transferred between the coding complexity and RD performance. Then, a three-output joint classifier consists of multiple binary classifiers with different parameters is designed to control the risk of false prediction. Finally, a sophisticated RD-complexity model is derived to determine the optimal parameters for the joint classifier, which is capable of minimizing the complexity in each CU depth at given RD degradation constraints. Comparative experiments over various sequences show that the proposed CU depth decision algorithm can reduce the computational complexity from 28.82% to 70.93%, and 51.45% on average when compared with the original HEVC test model. The Bjøntegaard delta peak signal-to-noise ratio and Bjøntegaard delta bit rate are -0.061 dB and 1.98% on average, which is negligible. The overall performance of the proposed algorithm outperforms those of the state-of-the-art schemes.

Autonomous Energy Harvesting Base Stations With Minimum Storage Requirements

In this work, optimal energy and resource allocation for the downlink of an autonomous energy-harvesting base station is investigated. In particular, the joint maximization of the users' utilities and the base station's revenue is considered, while these hierarchical decision problems are matched to the framework of a generalized Stackelberg game. Also, an efficient iterative method is proposed that enables all the players to reach the variational equilibrium, i.e., the optimal solution of the game. Simulation results validate the effectiveness of the proposed method.

Cost-Benefit Analysis for Optimization of Risk Protection Under Budget Constraints.

Cost-benefit analysis (CBA) is commonly applied as a tool for deciding on risk protection. With CBA, one can identify risk mitigation strategies that lead to an optimal tradeoff between the costs of the mitigation measures and the achieved risk reduction. In practical applications of CBA, the strategies are typically evaluated through efficiency indicators such as the benefit-cost ratio (BCR) and the marginal cost (MC) criterion. In many of these applications, the BCR is not consistently defined, which, as we demonstrate in this article, can lead to the identification of suboptimal solutions. This is of particular relevance when the overall budget for risk reduction measures is limited and an optimal allocation of resources among different subsystems is necessary. We show that this problem can be formulated as a hierarchical decision problem, where the general rules and decisions on the available budget are made at a central level (e.g., central government agency, top management), whereas the decisions on the specific measures are made at the subsystem level (e.g., local communities, company division). It is shown that the MC criterion provides optimal solutions in such hierarchical optimization. Since most practical applications only include a discrete set of possible risk protection measures, the MC criterion is extended to this situation. The findings are illustrated through a hypothetical numerical example. This study was prepared as part of our work on the optimal management of natural hazard risks, but its conclusions also apply to other fields of risk management.

Application of AHP and ELECTRE III/IV Methods to Multiple Level, Multiple Criteria Evaluation of Urban Transportation Projects

In this paper the authors carry out a multiple level, multiple criteria evaluation of 18 urban transportation projects. At each level of the hierarchical decision problem different multiple criteria ranking sub-problems have been structured and solved with the application of AHP method, ELECTRE III/IV method and their combination (AHP/ELECTRE III/IV). Thus, the computational phase allowed for testing the above mentioned multiple criteria ranking methods, i. e.: AHP and ELECTRE III/IV and analyzing their suitability for performing a multiple level, multiple criteria evaluation of transportation projects. Due to axiomatic differences between methods alternative aggregation formulas of the generated rankings by ELECTRE III/IV and AHP methods have been proposed. A multi - aspect discussion and comparison of generated results and applied methods have been presented.

Multi-Criteria Evaluation Approach of Mobile Text Entry Methods

Current evaluations on mobile text entry methods focus only on speed and accuracy performance but fail to consider subjective preferences of users. Such evaluation is ill-defined and does not constitute a systems approach since it requires a complex multi-criteria decision process that must encompass both objective and subjective criteria. Thus, this paper proposes the use of Analytic Network Process (ANP) in the evaluation of five mobile text entry methods in Android smartphones. A hierarchical network decision problem is developed from a focus group discussion of keen smartphone users. Using ANP, new insights in research and design of text entry methods are proposed. Results show that QWERTY method is the most preferred text entry method and is influenced by criteria on perceived text entry speed and number of keys. A survey is conducted to validate the results of the evaluation process using ANP. The main contribution of this work is in introducing a multi-criteria evaluation approach for mobile text entry research that holistically addresses speed-accuracy performance along with relevant subjective criteria that may potentially influence decision-making.

Combining Information for Quantitative Decision-making in Drug Development

Taking decisions during the development of a new drug requires combining many and varying pieces of information. The interconnections between these different pieces of information are often only partially known and in some cases merely conjectured. Despite this uncertainty, decisions must be taken – often with regard to the balance of risk and benefit – in order to make progress. A clear, consistent and efficient methodology for describing the structure of a project and the comprehensive state of knowledge at any point in time is required in order to add transparency to the decision process. A viable methodology must allow for a natural characterization of the uncertainty inherent in the drug development process and a relatively easy implementation. We present a possible solution that satisfies these requirements. The foundation of the proposed approach is based on a beta-binomial update mechanism well known in Bayesian statistics. This work provides a consistent framework for solving hierarchical multicriteria decision problems. Copyright © 2014 John Wiley & Sons, Ltd.

A two-level hierarchical Markov decision model with considering interaction between levels

Decision in reality often have the characteristic of hierarchy because of the hierarchy of an organization’s structure. In this paper, we propose a two-level hierarchic Markov decision model that considers the interactions of agents in different levels and different time scales of levels. A backward induction algorithm is given for the model to solve the optimal policy of finite stage hierarchic decision problem. The proposed model and its algorithm are illustrated with an example about two-level hierarchical decision problem of infrastructure maintenance. The optimal policy of the example is solved and the impacts of interactions between levels on decision making are analyzed.