Nonlinear Bilevel Programming Research Articles

A novel approach to optimize an integrated network design and pricing of a healthcare supply chain

Adopting various pricing policies has been highly regarded in recent years for setting prices and increasing firms' profits. One of the most common steps in pricing is to identify costs. Since a significant part of costs is related to the corresponding supply chain, many researchers in different fields have used decision-making for simultaneous pricing and network design. However, there is no such approach in the field of healthcare. This paper tries to fill this gap by formulating a mixed-integer nonlinear bi-level programming model examining the interaction of hospitals and their medicines suppliers. At the upper level, there is a competitive market where a new firm (entrant) intends to enter the market and faces the challenge of pricing medicines and making network design decisions. At the lower level, there is a private hospital competing with a public hospital, and it also struggles with healthcare services pricing and supplier selection. A comprehensive utility function that considers healthcare services prices, quality, waiting time, health insurance, readmission rate, and referral rate is extended at this level. Three novel meta-heuristic algorithms are recommended, including bi-level, improved fruit fly, jellyfish optimization, and forensic-based investigation optimization algorithms to solve the presented complex mathematical problem.

A nonmonton active interior point trust region algorithm based on CHKS smoothing function for solving nonlinear bilevel programming problems

<abstract><p>In this paper, an approach is suggested to solve nonlinear bilevel programming (NBLP) problems. In the suggested method, we convert the NBLP problem into a standard nonlinear programming problem with complementary constraints by applying the Karush-Kuhn-Tucker condition to the lower-level problem. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the nonlinear programming problem is successively smoothed. A nonmonton active interior-point trust-region algorithm is introduced to solve the smoothed nonlinear programming problem to obtain an approximately optimal solution to the NBLP problem. Results from simulations on several benchmark problems and a real-world case about a watershed trading decision-making problem show how the effectiveness of the suggested approach in NBLP solution development.</p></abstract>

An Integrated Pricing, QoS-Aware Sensor Location Model for Security Protection in Society 5.0

Society 5.0 refers to an advanced society based on big data, artificial intelligence, sensors, and robots to improve many aspects of life in a smart city. The role of sensors in Society 5.0 is critical. Sensors and the Internet of Things can be considered to work as a service system. Specifically, sensors can track millions of objects to support city security. This article considers the competitive sensor networks in terms of Quality of Service, which can be quantified by service price. The sensors are modeled as an <formula><tex>$M/M/1/n$</tex></formula> queueing service system with a location on a secured grid area. A competitive admission fee is considered for tracking orders from society, which makes the arriving tracking demands price-sensitive. Moreover, a bi-level nonlinear program is developed wherein the first level maximizes the sensors&#x2019; revenue and social benefit revenue while also minimizing the wait time of tracking orders from society; in the second level, the expected damage cost is minimized from a disruptive scenario toward valuable infrastructure. Moreover, the second-level model is linearized, and it solves the problem by a branch and bound and enumeration algorithm. Finally, an illustrative example of the proposed model is presented.

An Optimized Decision Model for Electric Vehicle Aggregator Participation in the Electricity Market Based on the Stackelberg Game

With the growing popularity of charging pile infrastructure and the development of smart electronic devices and 5G communication technologies, the electric vehicle aggregator (EVA) as a bidding entity can aggregate numerous electric vehicle (EV) resources to participate in the electricity market. Moreover, as the number of grid-connected EVs increases, EVA will have an impact on the nodal marginal prices of electricity market clearing. Aiming at the bidding and offering problem of EVA participation in the day-ahead and intra-day electricity markets, based on the Stackelberg game theory, this paper establishes a bilevel optimization model for EVA participation in the two-stage electricity market as a price-maker. In the proposed bilevel model, the upper-level and lower-level models are constructed as an operational problem for EVA and a market-clearing problem for independent system operator (ISO), respectively. In the day-ahead stage, EVA is optimized to maximize its own expected benefits, and ISO aims to improve the social benefits. In the intra-day stage, EVA is optimized to maximize its self-interest, and the ISO aims to make it possible to minimize the cost of expenditures to maintain the system’s supply–demand balance. Karush–Kuhn–Tucker (KKT) conditions and dual theory are used to transform the nonlinear bilevel programming model into a mixed-integer single-level linear programming model. In order to verify the validity of the proposed bilevel model as well as to comparatively analyze the impact of EVA’s participation in the electricity market on the market clearing results. Two scenarios are set up where EVA is seen as the price-taker in Scenario 1 and EVA is seen as the price-maker in Scenario 2. ISO’s revenue under Scenario 2 increased by USD 2262.66 compared to Scenario 1. In addition, the EVA acts as an energy consumer in Scenario 1 with a charging cost of USD 26,432.95, whereas in Scenario 2, the EVA can profit by participating in the electricity market with a revenue of USD 26,432.95, at which point the EVA acts like a virtual power plant. The simulation examples verify that the proposed bilevel optimization model can improve the benefits of ISO and EVA at the same time, achieving mutual benefits for both parties. In addition, the simulation analyzes the impact of abandonment penalty price on ISO and EVA intra-day revenues. Comparing the scenarios where the abandonment penalty price is 0 with USD 10/MW, the ISO’s revenue in the intra-day market decreases by USD 197.5. Correspondingly, EVA’s reserve capacity is dispatched to consume wind power in the intra-day market, and its revenue increases by USD 197.5. The proposed two-stage bilevel optimization model can provide a reference for EVA to develop scheduling strategies in the day-ahead and intra-day electricity markets.

Hierarchical coordinated energy management strategy for electricity-hydrogen integrated charging stations based on IGDT and hybrid game

The popularity of environment-friendly vehicles promotes the rapid development of electric and hydrogen integrated charging stations (EHI-CSs), which makes the power system face unprecedented challenges. Optimized and effective management strategies are needed to coordinate the operation among EHI-CSs and the distribution system operator (DSO). To this end, a bi-level energy management framework is proposed for DSO and EHI-CSs. The framework takes DSO as the leader of the upper game and provides dynamic differential pricing for EHI-CSs, while EHI-CSs work as the followers, react to the leader's decision in the manner of Nash-Harsanyi bargaining game. Moreover, aiming at the uncertainty of photovoltaic, a risk aversion robustness model for EHI-CSs is designed with information gap decision theory. A two-stage solution is proposed to solve the nonlinear bi-level programming problem with hierarchical energy interaction. Firstly, the lower-level energy cooperation is equivalent to two sub-problems of coalition's profit maximization and payment bargaining, the bisection approach is employed to distributed solve the energy pricing problem in the first stage with tie-line power and prices being the only exchange information. Then the second stage determines the payment according to the scheduling results. Numerical simulations verify the effectiveness of the proposed model and algorithm.

Distributed Dispatch of Integrated Electricity-Heat Systems With Variable Mass Flow

The constant mass flow assumption has dominated distributed dispatch of integrated electricity-heat systems (IEHSs), which ensures the simplicity of decomposition while incurring opportunity costs. In contrast, a heat operation strategy with variable flow and variable temperature (VF-VT) enhances flexibility and optimality. However, VF-VT renders the IEHS dispatch problem into a mixed-integer nonlinear bilevel nested program, which leaves a critical yet unresolved challenge for distributed autonomous dispatch. Therefore, this paper proposes a two-stage alternating procedure embedded with sequential equivalent techniques. A feasible initial point is obtained in the first stage, and the total costs are minimized thereafter. In each iteration, the heat sector optimizes both hydraulic and thermal states based on a surrogate model, and submits the heat equivalent to the electricity sector; the electricity sector solves the reduced IEHS dispatch problem and then updates the boundary. The feasibility is proved theoretically, while numerical tests validate the effectiveness.

An interval-based nested optimization framework for deriving flexibility from smart buildings and electric vehicle fleets in the TSO-DSO coordination

Emerging renewable-based transmission and distribution systems, despite many environmental and economic benefits, due to the intermittent nature of their production resources, compared to traditional systems, need more flexibility capacities, which necessitates the need for more suppliers of flexibility. To deal with these challenges, a nested framework is presented to derive the required flexibility of the transmission system operator (TSO) from distributed energy resources (DERs) and active end-users such as smart buildings (SBs) and electric vehicle (EV) fleets at the distribution level. To this end, a novel mechanism to design the demand response program (DRP) is introduced in which tariffs with time-varying rewards are built based on flexibility requirements. The coordination between TSO and distribution system operator (DSO) is initially modeled as a bi-level non-linear programming (NLP) problem, in which the upper-level is day-ahead (DA) operational planning of DS considering the schedules received from SBs, while the lower-level is DA operational planning of the TS. The bi-level NIL problem is transformed into a single-level linear programming (LP) problem by Krush Kuhn Tucker (KKT) conditions, Big-M method and Strong Duality Theory (SDT), which makes it computationally tractable. Finally, a two-stage interval-based algorithm solves the obtained single-level problem to secure the planning against uncertainties where battery energy storage systems (BESSs) are responsible for dealing with extreme conditions. The simulation results testify that the proposed interval-based nested framework has improved the economic, technical and security aspects of the TSO-DSO coordination since it has reduced the daily costs of the energy and flexibility markets, relieved lines congestion and improved voltage characteristics.

Capacitated Preventive Health Infrastructure Planning with Accessibility-Based Service Equity

Hard-to-access health infrastructure is likely to lead to increased morbidity and mortality. The optimal layout of health facilities is significant for disease control and prevention. This study aims to propose a method to provide equitable access to capacitated preventive health facilities, which captures the key features of the effect of congestion in a competitive choice environment. The problem is formulated as a bilevel nonlinear integer programming model. The upper level is a biobjective programming model subject to investment budget (B) constraint, and the lower level is a user equilibrium analogous model resulting from the users’ choice of facility location. An efficient and operable heuristic algorithm was designed according to the bilevel decision structure where a genetic algorithm (GA) with elite strategy is developed to solve the upper-level problem and the method of successive averages (MSA) is adopted to solve the lower-level problem. A case study is employed to validate the performance of the proposed method. The results show that the method is robust and could reach an equal service quality in a reasonable computation time. However, the sensitivity analysis indicated that the marginal benefit of the investment decreased. There is an optimal B beyond which further increments in investment would not offset the benefits. In addition, the proposed method could be beneficial for other congested public service facilities that users are free to choose from.

Improved unscented Kalman filter based interval dynamic state estimation of active distribution network considering uncertainty of photovoltaic and load

State estimation of active distribution network (ADN) plays an important role in distribution energy management system. The increase penetration of distributed generations, especially the distributed photovoltaic (PV), in ADN leads to high uncertainty of ADN’s operation and the state of the ADN varies with the variation of the PV output power. For the uncertainty of PV power output, an interval dynamic state estimation (IDSE) method, which estimates the interval of ADN state variables is proposed in this paper. Firstly, considering the slow computation speed of the Unscented Kalman Filter (UKF), the square root UKF is used to predict the real-time operating level of the state variables. Secondly, since the power output of PV has features of the variation randomly, the neural network-based prediction intervals is employed to predict the power output interval of PV. Finally, the normal fluctuation range of ADN state is modelled as a bilevel non-linear programming problem to perform IDSE, which in turn monitors the operating state of ADN. The proposed method is evaluated on the IEEE 33-node and IEEE 123-node systems, respectively. The test results demonstrate that the dynamic status of the ADN can be tracked accurately using the proposed method.

Location and capacity planning for preventive healthcare facilities with congestion effects

&lt;p style="text-indent:20px;"&gt;A painful lesson got from pandemic COVID-19 is that preventive healthcare service is of utmost importance to governments since it can make massive savings on healthcare expenditure and promote the welfare of the society. Recognizing the importance of preventive healthcare, this research aims to present a methodology for designing a network of preventive healthcare facilities in order to prevent diseases early. The problem is formulated as a bilevel non-linear integer programming model. The upper level is a facility location and capacity planning problem under a limited budget, while the lower level is a user choice problem that determines the allocation of clients to facilities. A genetic algorithm (GA) is developed to solve the upper level problem and a method of successive averages (MSA) is adopted to solve the lower level problem. The model and algorithm is applied to analyze an illustrative case in the Sioux Falls transport network and a number of interesting results and managerial insights are provided. It shows that solutions to medium-scale instances can be obtained in a reasonable time and the marginal benefit of investment is decreasing.&lt;/p&gt;

Multi-Oligopoly Sequential Pricing Mechanisms and Their Game Analysis in Raw Material Supply Chains

The sequential pricing game model is an approach that can be effectively used to solve the problem with multi-oligopoly pricing mechanisms in raw material supply chains. However, the existing sequential pricing mechanism does not fully consider constraints such as the purchase volume of downstream firms and the change information of each parameter, which leads to the pricing mechanism being detached from the real market. According to the concept of the sequential pricing game model being used among multi-oligopolies under constraints, we constructed the constrained sequential pricing game model by incorporating the parameters related to the product demand function, marginal production cost, dominant coefficient, following coefficient, and agreed minimum purchase volume as constraints, and the model was converted into a nonlinear bilevel programming model to facilitate model solving. Furthermore, we provided the analytical solution formulas for six special cases, thus making the model more similar to the real market. In addition, the effects of the agreed minimum purchase volume and the dominant and following coefficients on the equilibrium quoted prices and profits of the firms were analyzed. The results of the numerical simulation show that the constrained sequential pricing game model is more effective than the unconstrained sequential pricing game model in solving the problem with the multi-oligopoly pricing mechanism, which means that it can be used to establish a better pricing mechanism and provide a more reasonable and scientific basis for market operation and policymakers in solving practical problems.

Value chain design through synergistic optimisation of configuration and operation of value activities: an application case of cold chain logistics

The ‘no breakage chain’ characteristics of cold chain logistics make it important to study the synergistic optimisation between its activities. This study analyses the value chain of cold/frozen goods logistics businesses and proposes a value chain-based synergistic framework: intra-enterprise synergy, and synergy of internal and external enterprises. Based on this framework, a cost optimisation model for intra-enterprise synergy of value activities is established; then with consideration of the market-oriented value activity configuration, a synergistic optimisation model for internal and external enterprise between customer utility and enterprise cost through value activity configuration and operations is established. The synergistic optimisation model is a 0–1 nonlinear bilevel programming (BLP) problem, which is solved by using a bilevel nested genetic algorithm (BNGA). Finally, the established synergistic optimisation model is applied to case analysis of a cold chain logistics company as a computational example to demonstrate the effectiveness of the model.

An evolutionary algorithm based on approximation method and related techniques for solving bilevel programming problems.

In the engineering and economic management fields, optimisation models frequently involve different decision-making levels. These are known as multi-level optimisation problems. Because the decision-making process of such problems are hierarchical, they are also called a hierarchical optimisation problems. When the problem involves only two-level decision-making, the corresponding optimisation model is referred to as a bilevel programming problem(BLPP). To address the complex nonlinear bilevel programming problem, in this study, we design an evolutionary algorithm embedded with a surrogate model-that it is a approximation method and correlation coefficients. First, the isodata method is used to group the initial population, and the correlation coefficients of the individuals in each group are determined based on the rank of the leader and follower objective functions. Second, for the offspring individuals produced by the evolutionary operator, the surrogate model is used to approximate the solution of the follower's programming problem, during which the points in the population are screened by combining the correlation coefficients. Finally, a new crossover operator is designed by the spherical search method, which diversifies the generated offspring. The simulation experimental results demonstrate that the proposed algorithm can effectively obtain an optimal solution.

An Active-Set Fischer–Burmeister Trust-Region Algorithm to Solve a Nonlinear Bilevel Optimization Problem

In this paper, the Fischer–Burmeister active-set trust-region (FBACTR) algorithm is introduced to solve the nonlinear bilevel programming problems. In FBACTR algorithm, a Karush–Kuhn–Tucker (KKT) condition is used with the Fischer–Burmeister function to transform a nonlinear bilevel programming (NBLP) problem into an equivalent smooth single objective nonlinear programming problem. To ensure global convergence for the FBACTR algorithm, an active-set strategy is used with a trust-region globalization strategy. The theory of global convergence for the FBACTR algorithm is presented. To clarify the effectiveness of the proposed FBACTR algorithm, applications of mathematical programs with equilibrium constraints are tested.

A Mathematical Negotiation Mechanism for Distributed Procurement Problems and a Hybrid Algorithm for its Solution

In this paper, a mathematical negotiation mechanism is designed to minimize the negotiators’ costs in a distributed procurement problem at two echelons of an automotive supply chain. The buyer’s costs are procurement cost and shortage penalty in a one-period contract. On the other hand, the suppliers intend to solve a multi-period, multi-product production planning to minimize their costs. Such a mechanism provides an alignment among suppliers’ production planning and order allocation, also supports the partnership with the valued suppliers by taking suppliers’ capacities into account. Such a circumstance has been modeled via bi-level programming, in which the buyer acts as a leader, and the suppliers individually appear as followers in the lower level. To solve this nonlinear bi-level programming model, a hybrid algorithm by combining the particle swarm optimization (PSO) algorithm with a heuristic algorithm based on A* search is proposed. The heuristic A* algorithm is embedded to solve the mixed-integer nonlinear programming (MINLP) sub-problems for each supplier according to the received variable values determined by PSO system particles (buyer’s request for quotations (RFQs)). The computational analyses have shown that the proposed hybrid algorithm called PSO-A* outperforms PSO-SA and PSO-Greedy algorithms.

A flexible rolling regression framework for the identification of time-varying SIRD models

The present paper introduces a rolling regression approach to fitting the time-varying parameters of a susceptible-infected-recovered-deceased (SIRD) model. Our approach provides a domain-aware optimization environment applicable to any epidemiology compartmental model that is linear with respect to those parameters. Our framework is presented as a mixed integer bilevel nonlinear programming problem that is tackled with a real-valued genetic algorithm where we optimize the window size used for regression and estimate the hidden population of infected, recovered, and dead individuals that is not captured by the data available. We test our optimization framework with 2020 COVID-19 data from the state of Minnesota and compare our results both quantitatively and qualitatively, concluding that they are consistent with previously published research.Finally, we compare our proposed method against one of our benchmarks and show that the flexibility of our approach offers better system identification, thus highlighting the importance of domain-aware models that can infer the size of hidden populations from the recorded data and with the aid of equations that describe efficiently the dynamics of an epidemic.

Nonlinear bilevel programming approach for decentralized supply chain using a hybrid state transition algorithm

This paper investigates a decentralized supply chain that is composed of one manufacturer and multiple distributors. The manufacturer produces goods and wholesales them to multiple distributors and then the distributors sell products to various markets. The entire production period of the manufacturer is divided into several intervals. The decision-making problem of the entire decentralized supply chain is presented as a two-echelon coordination game network, in which each decision-maker can influence decision-making of other levels. A Stackelberg game framework is proposed to coordinate the decision-making process. And then two nonlinear bi-level programming (BLP) models are developed to find the optimal equilibrium decision scheme by switching the leader and follower roles between the manufacturer and the distributors. The models consider the manufacturer’s budget constraints in each interval and the market demands are affected by distributors’ selling price and advertising strategies. According to the hierarchy and complexity of bi-level programming problem (BLPP), a nested bi-level method based on hybrid state transition algorithm is proposed to address the BLP models, and mapping approximation strategy is utilized to improve computational efficiency. Finally, the numerical experiments are performed to demonstrate the superiority of the proposed method in terms of accuracy and computational efficiency.

An interior-point trust-region algorithm to solve a nonlinear bilevel programming problem

&lt;abstract&gt;&lt;p&gt;In this paper, a nonlinear bilevel programming (NBLP) problem is transformed into an equivalent smooth single objective nonlinear programming (SONP) problem utilized slack variable with a Karush-Kuhn-Tucker (KKT) condition. To solve the equivalent smooth SONP problem effectively, an interior-point Newton's method with Das scaling matrix is used. This method is locally method and to guarantee convergence from any starting point, a trust-region strategy is used. The proposed algorithm is proved to be stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem.&lt;/p&gt; &lt;p&gt;A global convergence theory of the proposed algorithm is introduced and applications to mathematical programs with equilibrium constraints are given to clarify the effectiveness of the proposed approach.&lt;/p&gt;&lt;/abstract&gt;

An active-set with barrier method and trust-region mechanism to solve a nonlinear Bilevel programming problem

&lt;abstract&gt;&lt;p&gt;Nonlinear Bilevel programming (NBLP) problem is a hard problem and very difficult to be resolved by using the classical method. In this paper, Karush-Kuhn-Tucker (KKT) condition is used with Fischer-Burmeister function to convert NBLP problem to an equivalent smooth single objective nonlinear programming (SONP) problem. An active-set strategy is used with Barrier method and trust-region technique to solve the smooth SONP problem effectively and guarantee a convergence to optimal solution from any starting point. A global convergence theory for the active-set barrier trust-region (ACBTR) algorithm is studied under five standard assumptions. An applications to mathematical programs are introduced to clarify the effectiveness of ACBTR algorithm. The results show that ACBTR algorithm is stable and capable of generating approximal optimal solution to the NBLP problem.&lt;/p&gt;&lt;/abstract&gt;

The hub location and pricing problem

This paper introduces the joint problem of locating hubs on a network and determining transportation prices between the hubs. Two levels of decision makers are present in the problem acting non-cooperatively: hub transportation provider and customers. The objective of the hub transportation provider is to locate hubs and to set the prices (per unit of commodity) of crossing the hub arcs maximizing its profit, whereas the customers aim is to send their commodities, in the cheapest way, having the possibility of using the hub arcs at the price set by the hub transportation provider or using the existing network at a predefined tariff. The problem is modeled as a nonlinear bilevel programming formulation, which is in turn linearized, and strengthened through variable reductions as well as valid inequalities. The case in which the price of each hub arc is determined by applying a common discount factor to the predefined tariff in the existing network is also studied. Computational results of mixed integer programming models and a metaheuristic on instances adapted from the literature are presented.