Bi-level Mathematical Model Research Articles

Dynamic interdiction networks with applications in illicit supply chains

We study the interdiction of illegal product distribution in a network with multiple sources (origins) and sinks (destinations). This work contributes to the literature of dynamic maximum flow interdiction problems by addressing multiple commodities in a network of relationships. The related distribution network consists of (1) criminals, who are hierarchically connected, and seeking to maximize the total profit flow from origins to destinations, and (2) enforcement officers aiming to minimize criminals’ long-term success by monitoring and arresting them, using the limited resources at their disposal. Considering several real-world operational details, we first propose a mixed-integer programming model by reformulating a Min–Max bi-level mathematical model. We then propose a new formulation and discuss its efficiency compared with the traditional duality-based reformulation. This new formulation also has a higher compatibility with decomposition solution methods. Utilizing the new formulation, we design a solution method based on the Benders decomposition procedure and apply several accelerating strategies (e.g., Super Valid Inequalities) to solve larger instances for a better representation of reality. Lastly, we create a heuristic method based on real-world evidence, which is usually practiced by law enforcement officers. Our results show that the quality of the heuristic method declines quickly as the network size increases.

Network Reconfiguration for Increasing Transportation System Resilience Under Extreme Events.

Evacuating residents out of affected areas is an important strategy for mitigating the impact of natural disasters. However, the resulting abrupt increase in the travel demand during evacuation causes severe congestions across the transportation system, which thereby interrupts other commuters' regular activities. In this article, a bilevel mathematical optimization model is formulated to address this issue, and our research objective is to maximize the transportation system resilience and restore its performance through two network reconfiguration schemes: contraflow (also referred to as lane reversal) and crossing elimination at intersections. Mathematical models are developed to represent the two reconfiguration schemes and characterize the interactions between traffic operators and passengers. Specifically, traffic operators act as leaders to determine the optimal system reconfiguration to minimize the total travel time for all the users (both evacuees and regular commuters), while passengers act as followers by freely choosing the path with the minimum travel time, which eventually converges to a user equilibrium state. For each given network reconfiguration, the lower-level problem is formulated as a traffic assignment problem (TAP) where each user tries to minimize his/her own travel time. To tackle the lower-level optimization problem, a gradient projection method is leveraged to shift the flow from other nonshortest paths to the shortest path between each origin-destination pair, eventually converging to the user equilibrium traffic assignment. The upper-level problem is formulated as a constrained discrete optimization problem, and a probabilistic solution discovery algorithm is used to obtain the near-optimal solution. Two numerical examples are used to demonstrate the effectiveness of the proposed method in restoring the traffic system performance.

Modeling carbon regulation policies in inventory decisions of a multi-stage green supply chain: A game theory approach

Governments and policymakers around the world put their best efforts to control the pollutions and climate change. Thus, they set various regulations to reduce greenhouse gases and carbon footprints. It is expected that firms should follow these regulations while maintaining their profitability. In this regard, firms can manage their carbon emissions across their supply chain (SC) by inventory management, since operational adjustments may affect the amount of carbon emissions, such as changing the production quantity and the frequency of transportations. This study applies a Stackelberg game between the government and a multi-stage green supply chain (GSC), in which the government’s goal is to maximize social welfare and that of the GSC is to minimize its cost. First, we formulate the inventory cost and carbon emission of a multi-stage GSC under two decision-making structures: non-coordinated and coordinated GSCs. Second, we develop eight bi-level mathematical programming models considering the structure of the GSC and the regulations of the government including carbon cap, carbon tax, carbon trade, and carbon offset. Then two solution approaches will present for solving problems based on the type of the GSC structure. This study examines the effect of coordination and carbon regulations on inventory cost, carbon emission and the objective function of the government.

Limited-stop bus service: A strategy to reduce the unused capacity of a transit network

Abstract Flexible adaptation to the heterogeneous distribution of travel demand is a challenging issue in the transit network design and planning. Limited-stop service is a strategy used to deal with this problem. In this paper, a bi-level mathematical model is proposed in which the upper-level model minimizes the total fleet unused capacity of limited-stop services, while the objective function of the lower-level model is to minimize total expected travel time for all passengers. Since this model is operator-oriented, some constraints are embedded in it in order to prevent excessive fall of the passengers' service level. Actually, the upper level is a mixed integer linear programming model; but because of NP-hardness and the lower level model, a heuristic algorithm is developed in order to solve the proposed bi-level model. It's worth mentioning that this heuristic approach is based on iterated local search, simulated annealing, and tabu search metaheuristics. Finally, a vast variety of numerical experiments are carried out that are based on some virtual origin-destination demand matrices generated according to different characteristics. Computational results show that by replacing the local service with the set of limited-stop services, the fleet unused capacity can be reduced by up to 48%.

A bi-level bi-objective mathematical model for cellular manufacturing system applying evolutionary algorithms

The present study aimed to design a bi-objective bi-level mathematical model for multi-dimensional cellular manufacturing system. Minimizing the total number of voids and balancing the assigned workloads to cells are regarded as two objectives of the upper level of the model. However, the lower level attempts to maximize the workers' interest to work together in a special cell. To this aim, two nested bi-level metaheuristics including particle swarm optimization (NBL-PSO) and a population-based simulated annealing algorithm (NBL-PBSA) were implemented to solve the model. In addition, the goal programming approach was utilized in the upper level procedure of these algorithms. Further, nine numerical examples were applied to verify the suggested framework and the TOPSIS method was used to find the better algorithm. Furthermore, the best weights for upper level objectives were tuned by using a weight sensitivity analysis. Based on computational results, all three objectives were different from their ideal goals when decisions about inter and intra-cell layouts, and cell formation to balance the assigned workloads by considering voids and workers' interest were simultaneously madeby considering a wide assumption-made problem closer to the real world. Finally, NBL-PBSA could perform better than NBL-PSO, which confirmed the efficiency of the proposed framework.

Tiered transferable pollutant pricing for cooperative control of air quality to alleviate cross-regional air pollution in China

Cross-regional air pollution is a widespread concern worldwide. This problem is particularly acute in China, where it has caused a substantial public health crisis. We consider the problem of determining tiered transferable pollution price control for air quality based on a piecewise transfer cost. Under tiered pricing, the more pollution that is transferred, the higher the transfer cost per pollution unit that must be paid. The tiered transferable pollutant pricing (TTPP) control problem is first formulated as a bi-level mathematical program model in which the upper level consists of China's Ministry of Environmental Protection, which controls air pollution using a tiered transferable pollution price, and the lower level is represented by a group of administrative regions that aim to relieve pollution with respect to their generalized costs, including reduction and transfer costs. Based on optimal conditions, the TTPP control problem is theoretically reformulated as the minima of a solution set for a sequence of equations. Each equation sequence is structured such that all of its components are optimal reactions of the lower-level regions. Then, we apply a bisection-based algorithm to solve the sequence of equations. Finally, we use the pollutant SO2 control problem in the Beijing-Tianjin-Hebei delta as a numerical example to provide insight into the model and performance of the algorithm. The results reveal that all regions in that delta can benefit from the tiered transfer pricing scheme, which also implies that the scheme has potential for practical application.

A bi-level school bus routing problem with bus stops selection and possibility of demand outsourcing

This paper develops a bi-level mathematical model for the school bus routing problem aiming at designing an efficient transportation system considering the possibility of predicting the students’ response. In the real world, the demand for using private cars depends on how well public transportation systems are operating especially in metropolitan cities. An inefficient public transportation will lead to an increase in the demand for using private cars. This issue will result in problems such as increased traffics and urban pollutions. To address this issue, an efficient public transportation system is designed by developing a new bi-level mathematical model. In the proposed model, the designer of the public transportation system, as the upper-level decision-maker, will locate appropriate bus stops and identify bus navigation routes. Subsequently, the decision regarding the allocation of students to transportation systems or outsourcing them will be made at the lower level which is considered as an operational-level decision-making. To solve this problem, two hybrid metaheuristic approaches named GA-EX-TS and SA-EX-TS have been proposed based on location-allocation-routing (LAR) strategy. The performance of these proposed methods is compared with exact solutions achieved from an explicit enumeration approach followed in the small-scale instances. Finally, the proposed approaches are used to solve 50 random instance problems. Comparing the results of the two tuned hybrid algorithms and conducting the sensitivity analysis of the model provide evidence for the good performance of the proposed approach.

Assembly line design considering line balancing and part feeding

PurposeThis paper aims to optimize the configuration of assembly lines (ALs) considering the two long-term decision problems within the line balancing and part feeding (PF) contexts, when the supermarket concept is applied in PF.Design/methodology/approachTo this purpose, a bi-level mathematical model is proposed to deal with the assembly line balancing problem (ALBP) and supermarket location problem (SLP) during the strategic decision-making phase of ALs’ configuration. The proposed model is applied on the known test problems taken from the ALBP literature to verify its performance.FindingsThe computational results verify that when the proposed structure is applied, the resulting AL configurations are optimized from both ALBP and SLP considerations in terms of the number of stations and line efficiency as well as supermarket transportation and installation costs.Originality/valueNo study has yet dealt with the long-term decision problem of configuring ALs considering both ALBP and SLP. Also, this study validates the effect of the ALBP on the SLP solutions as two long-term interrelated decision problems.

A Non-linear Integer Bi-level Programming Model for Competitive Facility Location of Distribution Centers

The facility location problem is a strategic decision-making for a supply chain, which determines the profitability and sustainability of its components. This paper deals with a scenario where two supply chains, consisting of a producer, a number of distribution centers and several retailers provided with similar products, compete to maintain their market shares by opening new distribution centers because of increase in demand. The competition problem is formulated as a non-linear integer bilevel mathematical model, where the upper level represents the decisions of the leader producer and the lower level administrates the decisions of the follower producer. It has been shown that even small-scale, bilevel mathematical programming problems are strongly NP-hard, so an adapted bilevel ant colony algorithm with inter-level information sharing is developed to solve the problem. To evaluate the performance of the developed ant colony algorithm, the upper bound of the competitive facility location problem is determined by solving the upper-level problem as an integer linear programming model without considering the follower’s decision. Comparing the computational results of the developed ant colony algorithm with those of the determined upper bounds shows the satisfactory capability of the proposed approach of solving even medium- and large-scale problems.

Analytical Forecasting of Population Distribution over Years in a New Rail Transportation Corridor

AbstractFor fast-growing cities in developing countries, population distribution is significantly affected by a candidate rail transit line. To enjoy the travel convenience of rail service, households would like to move into residential locations in the candidate rail transportation corridor. Meanwhile, while more households are attracted to live in the candidate rail transportation corridor over years, the over-year performance of the candidate rail transit line would be better. That is, interaction exists between population distribution and performance of the candidate rail transit line over years. Unfortunately, the over-year interaction between population distribution and performance of the candidate rail transit line was seldom considered in previous studies. This paper tries to bridge this research gap and presents analytical forecasting of population distributions over years in a new candidate rail transportation corridor. A bilevel mathematical model is proposed, with a lower-level problem formula...

An Emissions-Based User Equilibrium Model and Algorithm for Left-turn Prohibition Planning

The left-turn of vehicles at intersections has significant impacts on urban traffic congestions and accidents, which have negative effect on vehicle emissions causing air pollution. Many urban traffic networks prohibit direct left-turns for transport planning to keep traffic moving efficiently on major roads. As such, this paper proposes a bi-level mathematical model for left-turn prohibition planning considering both travel times and traffic emissions. The lower-level and upper-level are respectively solved by using the Frank-Wolfe algorithm and an improved genetic algorithm. By numerical examples, this paper shows that the improved algorithm can effectively enhance the speed and accuracy of the calculation, and the traffic congestions and emissions can be alleviated by implementing the left-turn prohibition at some carefully selected intersections.

Quantum binary particle swarm optimization-based algorithm for solving a class of bi-level competitive facility location problems

This paper deals with a special class of competitive facility location problems, in which two non-cooperative firms compete to capture the most of a given market, in order to maximize their profit. This paper intends to present a simple and effective nested strategy based on the quantum binary particle swarm optimization (QBPSO) method for solving the bi-level mathematical model of the problem. In solution approach, an improvement procedure is embedded into QBPSO to increase the convergence speed and generate more accurate solutions. Taguchi's method is employed to systematically determine the optimal values of QBPSO parameters. Finally, computational results on large-scale instances with up to 300 locations and 350 clients (more than 100,000 variables and 300,000 constraints at each level) confirmed the method efficiency in terms of solution quality and time.

Two‐stage two‐dimensional guillotine cutting stock problems with usable leftover

AbstractIn this study, we solve the nonexact two‐stage two‐dimensional guillotine cutting problem considering usable leftovers, in which stock plates remainders of the cutting patterns (nonused material or trim loss) can be used in the future, if they are large enough to fulfill future demands for items (ordered smaller plates). This cutting problem can be characterized as a residual bin‐packing problem because of the possibility of putting back into stock residual pieces, as the trim loss of each cutting/packing pattern does not necessarily represent waste of material depending on its size. Two bilevel mathematical programming models to represent this nonexact two‐stage two‐dimensional residual bin‐packing problem are presented. The models basically consist of cutting/packing the ordered items using a set of plates of minimum cost and, among all possible solutions of minimum cost, choosing one that maximizes the value of the generated usable leftovers. Because of special characteristics of these bilevel models, they can be reformulated as one‐level mixed integer programming models. Results of some numerical experiments are presented to show that the models represent appropriately the problem and illustrate their performance.

Urban Traffic Congestion Pricing Model with the Consideration of Carbon Emissions Cost

As one of the most effective traffic demand management opinions, congestion pricing can reduce private car travel demand and the associated carbon dioxide emissions. First, we summarized the status quo of transport carbon dioxide emission charges and congestion pricing, and then, we analyzed the characteristics of urban transport carbon dioxide emissions. Then, we proposed a (pricing) framework in which carbon emission costs would be considered as part of the generalized cost of travel. Based on this framework, this paper developed a bi-level mathematical model to optimize consumer surplus, using congestion and carbon emission charges as the control variables. A dissect search algorithm was used to solve the bi-level program model, and a numerical example was given to illustrate the methodology. This paper incorporates the emission pricing into the congestion pricing model, while considering two modes, and puts forward suitable proposals for the implementation of an urban traffic congestion pricing policy in China.

An interval programming approach for the bilevel linear programming problem under fuzzy random environments

In this paper, we address a class of bilevel linear programming problems with fuzzy random variable coefficients in objective functions. To deal with such problems, we apply an interval programming approach based on the $$\alpha $$ ? -level set to construct a pair of bilevel mathematical programming models called the best and worst optimal models. Through expectation optimization model, the best and worst optimal problems are transformed into the deterministic problems. By means of the Kth best algorithm, we obtain the best and worst optimal solutions as well as the corresponding range of the objective function values. In this way, more information can be provided to the decision makers under fuzzy random circumstances. Finally, experiments on two examples are carried out, and the comparisons with two existing approaches are made. The results indicate the proposed approaches can get not only the best optimal solution (ideal solution) but also the worst optimal solution, and is more reasonable than the existing approaches which can only get a single solution (ideal solution).

Batch Splitting Activity in Scheduling of Virtual Cell with Capacity Constraints Based on Bi-Level Mathematical Model

Although there has been some researches about virtual cell manufacturing system, the existing literature lack of discussion about the scheduling model that considering with bottleneck machines in the virtual cells. In view of this deficiency and the new characteristics of the batch splitting problem, this paper considered the batch splitting (or lot splitting) problem in scheduling of virtual manufacturing cells with bottlenecks and multiple machine types, and each of which has several identical machines. In consideration of the hierarchical decision structure of the problem, we developed a bi-level multi-objective mathematical model. Scheduling results and batch splitting strategies of both bottleneck and non-bottleneck machines are given in separate decision levels and additional scheduling objectives are improved in the second model level, while maintaining the maximum use of the bottleneck machine ability. In order to demonstrate how this approach works, application example was shown in this paper.

Robust Design Optimization of Gear Parameters for a Wind Turbine Using Interval Analysis

A robust design optimization method is suggested to optimize the system parameters of a gear transmission for a wind turbine with minimum sensitivity of fatigue strength to variations in uncertain application factor, dynamic load coefficient, material property parameters, and other coefficients representing unknown working condition. Interval numbers are employed to model the uncertainties by which only the upper and lower bounds are needed. Based on interval mathematics, the original real-valued objective and constraint functions are replaced by the interval-valued functions, which directly represent the variation bounds of the new functions under uncertainty. The single-objective function is converted into two-objective functions for minimizing the mean value and the variation, and the constraint functions are reformulated with the acceptable robustness level, resulting in a bi-level mathematical model. The optimization results of gear parameter demonstrate the validity and feasibility of the presented method.

A novel non-probabilistic approach using interval analysis for robust design optimization

A technique for formulation of the objective and constraint functions with uncertainty plays a crucial role in robust design optimization. This paper presents the first application of interval methods for reformulating the robust optimization problem. Based on interval mathematics, the original real-valued objective and constraint functions are replaced with the interval-valued functions, which directly represent the upper and lower bounds of the new functions under uncertainty. The single objective function is converted into two objective functions for minimizing the mean value and the variation, and the constraint functions are reformulated with the acceptable robustness level, resulting in a bi-level mathematical model. Compared with other methods, this method is efficient and does not require presumed probability distribution of uncertain factors or gradient or continuous information of constraints. Two numerical examples are used to illustrate the validity and feasibility of the presented method.

Analysis of Electric Grid Security Under Terrorist Threat

We describe new analytical techniques to help mitigate the disruptions to electric power grids caused by terrorist attacks. New bilevel mathematical models and algorithms identify critical system components (e.g., transmission lines, generators, transformers) by creating maximally disruptive attack plans for terrorists assumed to have limited offensive resources. We report results for standard reliability test networks to show that the techniques identify critical components with modest computational effort.