Bi-level Mathematical Model Research Articles

Economy-equity equilibrium based bi-level provincial renewable portfolio standard target allocation: Perspective from China

Renewable portfolio standards (RPS) are policy tools used in many countries to promote the use of renewable energy. However, balancing the equity and costs of RPS target allocations is very difficult. Therefore, this study proposed a bi-level framework to characterize this dilemma, in which the upper-level government, as the RPS target allocator, focuses on policy equity and maximizing global renewable energy utilization, and the lower-level government, as the policy implementer, considers the costs to achieve the RPS target. Only when the two types of decision-makers fully interact can they establish economical and equitable goals for a successful implementation. The bi-level mathematical model uses responsibility, capacity, and equality principles to measure equity and optimize the RPS target allocations. The proposed approach was applied to a Chinese case, from which it was found that the inclusion of hydropower in the RPS targets made equitable allocations more challenging, and equitable RPS target allocations have higher costs. Besides, central governments can determine equity principle weights to encourage greater renewable energy generation.

An efficient recurrent neural network for defensive Stackelberg game

This study presents a robust and efficient recurrent neural network (RNN) to solve the continuous defensive location problem (CDLP). CDLP is a bilevel mathematical model, where a planner locates various types of defense facilities for stopping a offender from reaching a critical vertex called core in the network. Due to the NP-hardness of the CDLP, RNN is implemented to find satisfactory solutions on four small-sized, two medium-sized and two larger-sized examples of problem. The numerical experiments show that RNN is able to produce promising results with good execution time and precision when compared with the best previous methods including tabu search (TS), imperialist competitive algorithm (ICA) and hybridization of TS with Levenberg–Marquardt algorithm (HTS) and hybridization of ICA with BFGS algorithm (HICA). The objective values of RNN show that this method stops the attacker at a further distance from the core in comparison with other methods. As well, to validate the efficiency of the proposed RNN, statistical criteria are considered and results are analyzed. For small-sized examples, the computational experiments show that the RNN can obtain solutions as the same as exact approaches but in a shorter runtime.

Analyzing the Vulnerabilities in a Transshipment Network: A Bilevel Programming Approach

As one of the most common types of networks, transshipment networks face certain risks in their daily operations. This risk either stems from a deliberate intervention in the process or from an unpredictable natural effect. The owner of the network normally aims to minimize the total cost of the transportation, while a rival or an opponent might attempt to intervene in the process and cause an increase in the costs. Unpredictable natural setbacks, on the other hand, can also have a negative impact on the network necessitating decision-makers to be more vigilant, especially for the worst-case consequences of network disruptions. The problem can be designed as a bilevel mathematical model. In this study, we develop a bilevel max-min model considering that the arcs in a transshipment network are liable to a disruption imposed by an intervenor. We present an original model with a novel solution approach to identify vulnerabilities in the arcs of a given transshipment network. The results reveal the most vulnerable arcs in the network, which have the highest negative impact on transportation costs when a disruption occurs. The decision-makers can use such results to analyze the risk in distribution networks such as electrical grids and develop associated measures to decrease the cost and increase the flexibility of the distribution systems. The model also enables decision makers with a methodical understanding of worst cases if the distribution network is interrupted for some reason.

Government intervention in municipal waste collection with a sustainable approach: a robust bi-level problem.

Conservation of the environment has taken a prime position among areas of concern for managers and practitioners worldwide. This study aims to provide a bi-level mathematical model for municipal waste collection considering the sustainability approach. The mathematical model with conflicting objects was proposed at the upper level of the model of maximizing government revenue from waste recycling and at the lower level of minimizing waste collection and recycling costs, which had stochastic parameters and was scenario based. A case study was conducted in the Saveh processing site (Iran). Due to the complexity of the bi-level model, the KKT approach was adopted to unify the model. Finally, the relevant calculations were performed based on actual information. The results of the problem in the case study showed the efficiency of the proposed method. Several computational analyses randomly generated different waste recycling rates and obtained significant management results.

A bi-level mathematical model to protect gateways in underwater wireless sensor networks

A bi-level mathematical model to protect gateways in underwater wireless sensor networks

A bi-level mathematical model to protect gateways in underwater wireless sensor networks

A bi-level mathematical model to protect gateways in underwater wireless sensor networks

A unified activity-based framework for one-way car-sharing services in multi-modal transportation networks

We propose in this paper a novel activity-based bi-level mathematical model to study the interaction between one-way car-sharing services and travel demand as well as their impacts on a multi-modal network performance. At the upper level, car-sharing operators decide the price of service and the plan of relocating shared vehicles among the depots. At the lower level, driven by their own activities, the decision of travelers is made bases on the equilibrium choice of routes and associated transport modes that include private cars, public transport and shared cars. Furthermore, we develop a scalable link-based method that is capable of solving large-scale network with capacitated link flow. We show via numerical examples the impact of the one-way car-sharing services on both route choices and network performance, and demonstrate the significant improvements in the obtained solution both in terms of accuracy and computational complexity.

A novel bi-level multi-objective genetic algorithm for integrated assembly line balancing and part feeding problem

The manufacturing industry has been pursuing an efficient and economical assembly system. By considering assembly line balancing (ALB) and part feeding (PF) as an integrated problem and programming them simultaneously opens additional opportunities to improve the performance of the entire assembly system. However, the integrated ALB and PF problem is a non-deterministic polynomial (NP) hard problem. This implies that exact solutions cannot be obtained in a reasonable computation time and its near-optimal solutions can only be realised by meta-heuristics. In this study, we propose a novel bi-level multi-objective genetic algorithm (NBMGA) to solve the integrated ALB and PF problem. First, a bi-level mathematical model is established to simultaneously minimise the number of stations and workload smoothness of ALB in the upper level as well as the number of supermarkets of PF in the lower level. Second, the NBMGA with two modified strategies, including extending fitness evaluation and adaptive termination condition, is designed for problem solving. Finally, a series of computational experiments are conducted to demonstrate the efficacy of the proposed algorithm. The computational results indicate that the proposed algorithm outperforms the bi-level nondominated sorting genetic algorithm (NSGA) II in terms of the approximation to the true frontier without sacrificing computational efficiency.

Selective catalytic reduction system

We study the interdiction of smuggling network that arranging the activities of the police in order to successfully interdict criminals in smuggling goods. This work contributes to the literature of maximum flow network interdiction problems by addressing asymmetric information, uncertain conditions, multi commodity, and with multiple sources (origins) and sinks (destinations). Information Asymmetry realistically occurs due to incomplete information of interdictor (police) and operator (smuggler) about each other's performance, which is adapted from the real-world condition. We propose two mixed-integer programming models by reformulating a Min–Max bi-level mathematical model. In the first model, the type of interdiction is discrete (zero and one), while in the second model, the interdiction is assumed continuous, meaning that the partial interdiction is possible. The asymmetry type of the smuggler's information towards the police have formulated through a linear function while the asymmetry of the police information to the smuggler is formulated using an uncertain parameter through a two-stage stochastic programming framework. To solve the first model, an innovative exact hybrid method is proposed combining of a Decomposition Method and Progressive Hedging Algorithm (DM-PHA). An augmented Karush-Kuhn-Tucker (KKT) method is also used to solve the second model. Several sensitivity analyses are then conducted, and the results demonstrate the applicability and effectiveness of the proposed models as well as the solving approach. It is also shown that the proposed models can be used as a suitable approach in uncertain environment and under asymmetric information to determine the optimal interdiction decisions of police to prevent further smuggling.

A maximum-flow network interdiction problem in an uncertain environment under information asymmetry condition: Application to smuggling goods

We study the interdiction of smuggling network that arranging the activities of the police in order to successfully interdict criminals in smuggling goods. This work contributes to the literature of maximum flow network interdiction problems by addressing asymmetric information, uncertain conditions, multi commodity, and with multiple sources (origins) and sinks (destinations). Information Asymmetry realistically occurs due to incomplete information of interdictor (police) and operator (smuggler) about each other's performance, which is adapted from the real-world condition. We propose two mixed-integer programming models by reformulating a Min–Max bi-level mathematical model. In the first model, the type of interdiction is discrete (zero and one), while in the second model, the interdiction is assumed continuous, meaning that the partial interdiction is possible. The asymmetry type of the smuggler's information towards the police have formulated through a linear function while the asymmetry of the police information to the smuggler is formulated using an uncertain parameter through a two-stage stochastic programming framework. To solve the first model, an innovative exact hybrid method is proposed combining of a Decomposition Method and Progressive Hedging Algorithm (DM-PHA). An augmented Karush-Kuhn-Tucker (KKT) method is also used to solve the second model. Several sensitivity analyses are then conducted, and the results demonstrate the applicability and effectiveness of the proposed models as well as the solving approach. It is also shown that the proposed models can be used as a suitable approach in uncertain environment and under asymmetric information to determine the optimal interdiction decisions of police to prevent further smuggling.

A competitive stochastic bi-level inventory location problem

ABSTRACT This paper aims to address an inventory competitive facility location problem with the order quantity in a stochastic environment. In this problem, the facility location, customer design, and inventory control decisions have been integrated for two decision makers, i.e. the leader and the follower. To this end, a bi-level nonlinear nonconvex mixed integer mathematical model was proposed. Next, an alternative bi-level mixed-integer linear model was obtained. The enumeration branch-and-cut method in the field of bi-level linear programming was developed to come up with optimal or near-optimal solutions. To determine the branching choice and specialized cuts, a preference function was used. Computational results of some data tests were reported, and the results were compared with those of previous famous methods to show the efficiency of the algorithm. According to the computational results, the proposed algorithm could find effective solutions for solving such problems, which could compete with other well-known published algorithms.

Game theory–based bilevel model for multiplayer pavement maintenance management

Pavement maintenance is widely thought to be critical for promoting their sustainability. It plays a pivotal role in sustainable and resilient transportation infrastructure for economic development and social inclusion improvement. Existing research mainly focuses on optimizing the maintenance strategies based on one player but ignores the inherent conflicts and complex interactions among multiplayer decision-makers. To address these issues, this research examined the problem of pavement maintenance involving the highway agency and maintenance service providers under the uncertainty of pavement conditions with the aim of optimizing maintenance strategies. A novel bilevel mathematical model was proposed to handle the inherent conflicts and complex interactions among decision-makers based on the Stackelberg game and Nash game. The goal is to obtain the Stackelberg–Nash equilibrium solution, where the highway agency, as the upper-level leader, determines the maintenance tasks, whereas the service providers, as the lower-level followers, provide the corresponding maintenance activities. To formulate this complex model, a bilevel particle swarm optimization (BLPSO) algorithm was developed for pavement maintenance. Finally, a practical case was used to demonstrate the practicality and efficiency of the proposed model. The results show that the mathematical model can provide feasible and effective pavement maintenance strategies for multiplayer decision-makers in real-world applications.

Evaluating Connectable Capacity of Distributed Wind Generation in Distribution Networks Through a Bayesian Integrated Optimization Method

The effective evaluation of the connectable capacity of renewable energy plays a vital role in the development of a sustainable distribution network. Quantifying the capacity while considering network security and the local renewable accommodation policy is a challenge. This article proposes a scenario-based bilevel mathematical model using a Bayesian integrated optimization method to evaluate and quantify the connectable capacity of distributed wind generation in distribution networks, which effectively integrates the characteristics of wind power and the local accommodation policy. The constraint on the generation curtailment ratio (CR) is innovatively designed to represent the renewable accommodation policy and integrated with network security constraints to coordinatively quantify the capacity. The model is solved by the Bayesian integrated optimization method. The regression-based algorithm greatly reduces the complexity of alternating iteration and improves the calculation efficiency. Practical cases are used to verify the effectiveness of the proposed method. Results indicate that the method is more efficient than traditional optimization algorithms, and CR integration ensures that the connectable capacity fits local renewable energy development policies well.

On Conflicts between Pharmaceutical Patent Protection and the Right to Life and Health Based on a Stackelberg Game.

To solve the conflict between pharmaceutical patent protection and the right to life, health and increased patient satisfaction, drug prices can be regulated by levying an excess profit tax. An optimal tax strategy was formulated that not only could lower drug prices and improve public health and welfare, but also considers companies’ earnings. The strategy was based on the Stackelberg game theory as a bi-level mathematical model. In the model, the government is the leader, with patient satisfaction as the main goal, and pharmaceutical companies are the followers, with maximum drug revenue as the goal. The results show that under the premise of ensuring sufficient incentives for patent holders, the optimized tax on excess profit can effectively compensate for the shortcomings of pharmaceutical patent protection, alleviate the failure of market regulation of drug prices, improve patient satisfaction, and increase total social welfare.

A Bi-level Meta-heuristic Approach for a Hazardous Waste Management Problem

This study concentrates on designing a medical waste management system with a hierarchical structure, including a local government and a waste management planner. The upper-level seeks to design and control the waste management facilities by minimizing the environmental risks related to the disposal of medical waste. While, the lower-level model is to determine the waste collection plans by only minimizing its total operational costs. Therefore, this study develops a bi-level mathematical model, in which the benefits of the both stakeholders are taken into account. As this problem poses difficulty in searching for the optimal solution, a bi-level meta-heuristic approach based on the Genetic Algorithm (GA) is employed for solving the problem. Finally, a case study is conducted to show that the proposed model and solution approach are practical and efficient.

Planning and Optimal Scheduling Method of Regional Integrated Energy System Based on Gray Wolf Optimizer Algorithm

For the regional integrated energy system (RIES) with multi-energy complementary, it is key to plan and optimize scheduling, which can improve energy efficiency. This paper focuses on the planning and scheduling of the RIES with self-provided waste incineration power plant. A bi-level mathematical model of the RIES with electricity, heat, cooling and gas interconnection is built. The upper level takes the minimum annual comprehensive cost as the optimization objective to find the optimal configuration. Based on the configuration scheme provided by the upper level, the optimized scheduling scheme under the economic and environmental indicators is developed by using some constrains, such as the supply and demand balance and equipment output operation. At the same time, the policy subsidy of self-provided waste incineration power plant and the transaction cost of pollutant emission are considered in the optimal scheduling. Then the gray wolf optimizer algorithm is proposed to solve the bi-level optimization problem. The effectiveness and superiority of the planning and optimal sheduling model proposed are demonstrated by analyses in typical scene.

A New Bi-Level Mathematical Model and Algorithm for VONs Mapping Problem

Elastic optical networks (EONs) virtualization can allow the virtual optical networks (VONs) to utilize all the physical resources of EONs, and can attain a high performance of the networks. However, the optimal scheme for VONs mapping should be determined. To tackle these challenges, a bi-level mathematical model is established. leader's and follower's objectives are to minimize energy consumption and the maximum index of used frequency slots, respectively. The bi-level mathematical model can determine the optimal schemes of VONs mapping. To solve the mathematical model effectively, a uniform design method is applied to generate initial population for the lower level problem. In addition, To solve the whole model effectively, a tailor-made encoding, population initialization, genetic operators and local search operator are designed. An efficient genetic algorithm with local search operator is proposed for the bi-level mathematical model. To evaluate the mathematical model and the designed algorithm, a large number of experiments are performed on three kinds of the widely used networks, and the experimental results indicate that the effectiveness of the proposed bi-level mathematical model and designed algorithms.

A Cost-Sharing Approach for Decentralized Electricity–Heat Operation With Renewables

Pipeline energy storage in district heating systems (DHSs) has great potential to improve electricity-heat coordination and promote the accommodation of renewables in power systems. However, under a decentralized operation scheme, the DHS operator may not have the motivation to utilize pipeline energy storage because it will lead to a rise in temperature and thus increase the system's overall heat loss. To address this challenge, this paper proposes a decentralized electricity-heat coordination framework by sharing part of the benefits of renewables accommodation from a power system to a DHS. A bilevel mathematical model is proposed for a policy designer to evaluate the effects of the proposed framework and determine the best sharing ratio between the two systems. An end-to-end DHS model is developed and used in the bilevel programming to consider pipeline energy storage with an explicit relationship between heat generation and demand. Finally, a case study justifies the effectiveness of the decentralized framework and proves that its performance is almost identical to the centralized power and heat dispatch framework.

Optimal charging facility location and capacity for electric vehicles considering route choice and charging time equilibrium

In this study, the optimal design of location and capacity of charging facilities for electric vehicles (EVs) is investigated. A bi-level mathematical model is proposed to derive optimal design considering the equilibrium of route choice and waiting time for charging. The objective is to minimize the joint cost of facility constructions and EV drivers’ travel and waiting time over the network. The upper-level model allocates the facilities and their capacity, while the lower-level model characterizes equilibrium behavior of drivers’ route and charging facility choices. In particular, we model drivers at each charging facility as the M(t)/M/n queue and approximate the average queuing time and probability of waiting time as functions of facility capacity and demand arrival rate. The bi-level model is then converted into a single-level model, and the solution algorithm is proposed for iteratively solving the relaxed problems. Comprehensive experiments are conducted on three networks to evaluate algorithm performances, assess solution robustness and understand the scalability of the solution approach on large networks.

Modelling the effect of multi-stakeholder interactions on construction site layout planning using agent-based decentralized optimization

Abstract Construction site layout planning (CSLP) is important to ensure that cost, time, quality and environmental impacts are strictly controlled. Even though extensive research has been conducted on CSLP problems, the effect of the multi-stakeholder interactions on optimal site layout plans has been less studied. Therefore, this paper proposes an agent-based decentralized bi-level mathematical model that incorporates decisions from three stakeholders, to examine the CSLP multi-stakeholder interactions. To solve the model, a particle swarm optimization (PSO)-based solution approach is developed to generate the CSLP, construction material logistics planning (CMLP) and security planning (SP) solutions in an integrated CSLP-CMLP/SP system. Specifically, the stakeholders are modelled as individual agents with respective decisions and objective functions, and the inter-dependencies between the layout planner and the other two stakeholders (i.e. the logistics planner and the safety manager) are simultaneously incorporated and dealt with using the bi-level model. The multi-stakeholder interactions are realized by iteratively updating the individual decisions to determine an optimal integrated solution in which the potential stakeholder conflicts in the construction stage are minimized. A case study is given to validate the practicality of the designed model in addressing the CSLP-CMLP and CSLP-SP interactions. The contributions of this research are its applicability to model the effect of multi-stakeholder interactions in agent-based decentralized models, the design of a PSO-based solution approach to achieve an integrated CSLP-CMLP/SP solution, and the provision of academic and practical guidance for decision makers to further enrich global decentralized CSLP research.