Bi-level Multi-objective Programming Research Articles

Interval-valued trapezoidal neutrosophic bi-level multiobjective linear programming numbers: algorithm and its application to pharmaceutical production planning problem

Interval-valued trapezoidal neutrosophic bi-level multiobjective linear programming numbers: algorithm and its application to pharmaceutical production planning problem

Estimation of travel flux between urban blocks by combining spatio-temporal and purpose correlation

Understanding the travel flux between urban blocks is fundamental for traffic demand prediction, urban area planning and urban traffic management. However, the uncertainty of human mobility patterns and the complexity of urban transportation systems usually yield challenges in accurately estimating the travel flux within a city. Thus, we propose a novel travel flux estimation method that integrates traffic flow characteristics (traffic volume and travel time), spatio-temporal autocorrelation, and travel purpose correlation. First, the geographically weighted method was used to model and verify the spatio-temporal autocorrelation of origin–destination flows, whereas the purpose correlation of origin–destination flows was expressed through the function feature vectors of urban blocks. Then, a multi-objective bi-level programming model, according to the generalized least squares method, was constructed to estimate the travel flux between blocks. This was used to solve the problem of accurate estimation of travel flux by combining microscopic traffic flow characteristics with macroscopic spatio-temporal and purpose characteristics. Finally, an empirical analysis of the Hankou district, Wuhan City, demonstrated that in contrast to the existing method, the accuracy of the proposed method for predicting the human travel flux improved by approximately 20%. The estimated results were consistent with the spatial distribution pattern of human travel. Moreover, these results can provide targeted decision support for planning urban spaces, allocating urban resources, and guiding vehicular traffic.

Sustainable road network design considering hydrogen fuel cell vehicles

Environmental pollution and energy shortages have brought about an increased focus on new energy vehicles. Hydrogen fuel cell vehicles (HFCVs) have experienced rapid development due to the potential to alleviate energy pressures and reduce pollution emissions. Near zero-carbon emissions offer a promising avenue for promoting sustainable transportation development. To evaluate the impact of HFCVs on the transportation environment, this paper investigates the problem of sustainable transportation network design including HFCVs. Specifically, the problem is formulated as a bi-level multi-objective programming problem, with the upper level aimed at determining the optimal network design scheme considering multiple objectives, while the lower level addresses the mixed traffic flow that comprises both HFCVs and fuel vehicles. To solve the multi-objective sustainable network design problem, an integrated solution framework that combines the particle swarm optimization (PSO) algorithm with the Frank-Wolfe algorithm (FW) is developed. Specifically, the PSO algorithm is utilized to solve the upper-level model and identify the optimal network design schemes, while FW algorithm is adopted to handle mixed traffic flow assignments. Finally, the proposed model and algorithm are implemented in two numerical experiment to demonstrate their effectiveness and efficiency.

An improved bi-level programming model for water resources allocation under multiple uncertainties

Nowadays, with the scarcity of water resources, competition for water resources among different levels and water sectors is growing increasingly fierce. Furthermore, uncertainties are unavoidable in the water resources system. To address the aforementioned issues, a fuzzy max-min decision bi-level multi-objective interval programming model was proposed, which can not only focus on water conflicts at the same level or between different levels, but also pay attention to optimal allocation of water resources under uncertainty. The developed model was then applied to a case study in Wuwei City, Gansu Province, China, which selected fairness of water distribution and agricultural economic benefits as planning objectives. Based on the developed model, different water resources optimal allocation schemes under different representative hydrological years were provided. From the result, as representative hydrological years changed from wet (P = 25%) to dry (P = 75%), agricultural economic benefit and Gini coefficient of agriculture would vary from [35.19, 37.78] × 108 yuan to [31.12, 31.99] × 108 yuan and from [0.468, 0.429] to [0.505, 0.503], which indicates that as available water resources decrease, agricultural economic benefit would decrease and fairness of water distribution would also decrease. And the water distribution fairness of the upper bound water allocation scheme is higher than that of the lower bound water allocation scheme when in the same representative hydrological year. In addition, no matter what representative hydrological year, the results of the established bi-level programming model are always in the middle of the results of the upper and lower level individual objective, which means that the developed bi-level programming model has great advantage to deal with water competing conflict among different levels. Furthermore, based on the results of developed model, the reasonable water resources optimization schemes can be determined by the decision-makers when faced with multi-objective, bi-level and multiple uncertainties problems.

An acceptability-based multi-objective traffic flow adjustment method for environmental sustainability and equity

As a product of digital development, connected autonomous vehicles (CAVs) offer a unique prospective solution to alleviate the possible performance deterioration of road networks under the mixed environment of human-driven vehicles (HVs) and CAVs. In this paper, we propose a traffic flow adjustment method (TFAM) that treats CAVs as mobile regulators with the purpose to reshape traffic flow distribution on road networks by guiding rather than controlling CAVs. More specifically, we deploy subsidy nodes to briefly outline travel routes and achieve higher acceptability than traditional route-based control schemes. The TFAM is a multi-objective bi-level programming problem where the upper-level problem optimizes the network performance through regulating location and subsidy on the subsidy nodes. The lower-level problem is a dual dynamic traffic assignment (DDTA) model. Apart from the total travel time cost (TTTC), total emission cost (TEC) and network equity (NE) are also introduced as optimization objectives to highlight environmental sustainability and acceptability. To obtain the Pareto solution frontier, a meta-heuristic algorithm with an improved encoding process is proposed. Results of two numerical case studies demonstrate the effects of TFAM on traffic flow distribution and network performance, which yields valuable insights on the optimization of urban traffic systems.

Multiobjective bilevel programming model for multilayer perceptron neural networks

The architecture of multilayer perceptron (MLP) neural networks dictates the network's performance. However, aiming at the specific classification problems, suitable architectures of MLPs must be determined beforehand. More often, the design decisions rely on a trial and error learning and the experience knowledge. To automatically design an MLP's network architecture and optimize network parameters, a multiobjective bilevel programming model is built. In this model, a multiobjective optimization problem is constructed in the upper level for obtaining a set of Pareto optimal architectures of the MLPs, considering network complexity, training error rate, and validation error rate, while a single-objective optimization problem is established in the lower level to search for the optimum network parameters for a given network architecture. For dealing with this model efficiently, a novel multiobjective hierarchical learning algorithm (MOHLA) is proposed, in which an integer-coding NSGA-II is developed as the upper-level optimizer for a set of Pareto optimal network structures of the MLPs, while a non-iterative method is regarded as the lower-level solver for the MLPs' connection parameters. After a set of trained MLPs is obtained finally by using MOHLA, a selective ensemble strategy is adopted for improving identification accuracy. Three types of multiobjective bilevel programming models are investigated and compared in the experiments. Moreover, the proposed MOHLA is compared with several state-of-the-art learning approaches on various classification problems. The experimental results confirm that MOHLA performs well.

An inexact bi-level multi-objective modeling approach for managing agricultural water-food-carbon-land-ecology-nutrition nexus in an arid area

Multiple subsystems such as water, food, land, carbon, ecology, and nutrition are closely intertwined, and the resulting complexity and uncertainty pose challenges to water resource management. This study proposes an optimization framework from a novel water-food-carbon-land-ecology-nutrition (WONDER) nexus perspective for the integrated management of agricultural and ecological water resources. Firstly, the irrigation priority ranking of decision-making units (DMU) is obtained based on the remote sensing indices from the Google Earth Engine (GEE) platform. Secondly, the ecological satisfaction degree, which aggregates carbon sink satisfaction degree and ecological water demand satisfaction degree, is proposed to characterize the degree of achieving the pre-given ecological objectives. Then, the above two indicators (i.e., the irrigation priority ranking and the ecological satisfaction degree) are integrated with a fuzzy-interval credibility-constrained bi-level multi-objective programming model to realize: (1) reconciling conflicts in food nutrition, food security, economic development, and ecological conservation; (2) making trade-offs among the positions and preferences of different decision-makers; (3) dealing with the dual uncertainties of fuzzy-interval parameters; (4) obtaining optimal agricultural and ecological water allocation schemes. The applicability of the model is verified by taking Hongyashan Irrigation District of Minqin County as an example. The results show that: (1) the irrigation priority ranking for the DMUs is: DMU 10 > DMU 9 > DMU 4 > DMU 5 > DMU 1 > DMU 11 > DMU 6 > DMU 8 > DMU 7 > DMU 2 > DMU 3; (2) the ratio of agricultural to ecological water allocation in the irrigation district is about 3:1, which is consistent with the water use characteristics of this area; (3) the total water allocation during the whole growth period for different crops is mainly determined by planting area, and the ranking of the water allocation is: sunflower > maize > wheat > seed melon. Moreover, compared with the single-level models, the inexact bi-level multi-objective model can well coordinate the upper and lower decision-makers and the multiple decision objectives under the WONDER nexus. Therefore, the proposed framework can provide the scientific basis for balancing agricultural and ecological water management and conflicting objectives in similar irrigation districts through involving multiple subsystems and perspectives.

Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach.

A new procurement and supply contract for manufacturing technology standards’ diffusion from home to host countries

The rise of industry 4.0 has led to the implementation of new technology-based manufacturing standards (TMSs). This study proposes a new contract to support a manufacturer, Haier, in diffusing its home-country TMSs into host countries with technology gaps. The manufacturer conducts the diffusion by operating two-sided networks (i.e. consumer and supplier networks) and dual sourcing (i.e. placing orders from two suppliers in different countries). A procurement contract is designed to coordinate the two-sided networks; it defines a market-pricing and part-procurement problem to maximise the scale of customers and local suppliers, respectively. These suppliers decide whether to accept the contract after considering their production conditions and technology gaps. A part-supply contract is executed if the procurement contract is accepted. Surplus parts are procured from the manufacturer’s home country if the local supply is insufficient. The coupling of procurement and supply contract (PSC) design is inherent, while a hierarchical collaborative optimisation (HCO) model based on Stackelberg game theory is formulated. The HCO model deploys a multi-objective bi-level programming decision structure to examine the PSC coupling. A nested non-dominated sorting genetic algorithm II is designed to address this HCO model, while a case study of refrigerator TMS diffusion is analysed for verification.

Plantwide control design using latent variables: An integration between control allocation and a measurement combination approach

This paper presents a plantwide control design methodology based on a novel structure which consists of a decentralized strategy complemented by a control allocation (CA) module and a measurement combination (MC) block. Taking into account a principal components analysis (PCA) selection approach, the CA and MC modules perform a dimensional reduction of the original input–output variable space in order to obtain sets of latent variables (or principal components) as control actions and controlled variables. The use of principal components in the controller design provides several interesting features given that: (i) the conditioning of the subsystem to be controlled can be improved, (ii) when performing combinations of variables, the CA and MC modules act as steady-state decouplers and thus an apparently diagonal process is obtained, which favors the reduction of the variables interaction and the pairing problem is automatically solved, and (iii) they allow to naturally handle nonsquare systems. The proposed design procedure is implemented through a multiobjective bilevel mixed-integer nonlinear programming (BMINLP) optimization problem. The leader problem is based on the minimization of three functional costs: 1- the well-known sum of squared deviations (SSD) index, 2- the number of selected manipulated variables (actuators), and 3- the number of selected measurements (sensors). The inner optimization minimizes the relative gain array number (RGAN). This provides a good trade-off between the degree of conditioning/controllability and the complexity/cost of the resulting system. This problem is efficiently solved through genetic algorithms and allows to perform: (i) the selection of the manipulated variables (actuators) and the measurements (sensors) to be used, (ii) the computation of the matrices that characterize the CA and MC modules, and (iii) the stability analysis of the multivariable control structure. The overall design procedure only requires steady-state models of the process. The Tennessee Eastman case study is considered for the simulation and performance evaluation of the proposed solutions.

Multi-objective optimization of cordon sanitaire considering operation cost and queueing equity

ABSTRACT This paper aims topropose a method to determine the optimal deployment of cordon sanitaire for epidemic control. A bi-level multi-objective programming model is proposed where the lower level is transportation system equilibrium with queueing time to predict link traffic flow, and the upper level is queueing network design, which is a multi-objective integer nonlinear programming model. The primary objective is to minimize the total operation cost of servers with a predetermined waiting time constraint, and the secondary objective is to reach queueing equity. A heuristic algorithm is designed where the method of successive averages is adopted for the lower-level model, and a genetic algorithm is designed for the upper-level model. The computational results show that the methods can find a good heuristic optimal solution. These methods are useful for planners to determine the optimal deployment of cordon sanitaire for disease control and prevention purposes.

Optimizing water allocation in an inter-basin water diversion project with equity-efficiency tradeoff: A bi-level multiobjective programming model under uncertainty

Achieving optimal water allocation in an inter-basin water diversion project is a challenge that involves multilevel stakeholders, and the uncertainty and disequilibrium of water allocation further intensify competition among different stakeholders. This study proposes a bi-level multiobjective programming model for water allocation in inter-basin water diversions that considers equilibrium sustainability. The model integrates interval parameter programming, fuzzy credibility-constrained programming, bi-level multiobjective programming, and the Gini coefficient into a general framework. The model not only balances the stakeholders at different levels to ensure water allocation equity but also comprehensively considers the fuzzy uncertainty of parameters and constraints to obtain the interrelationship among water diversion, water allocation satisfaction, and efficiency objectives. The developed model was applied to allocate water for the Hanjiang-to-Weihe Water Diversion Project in Shaanxi Province, China. The results showed that 1) the model derived a feasible tradeoff between upper basin manager and lower regional managers under different water diversion scenarios; 2) water allocations and objectives at each level are sensitive to changes in the credibility of water diversion; 3) the new model improves the equity of water allocation and ensures the efficient allocation to various stakeholders at each level based on a comparison of the proposed model with a model that does not consider equilibrium. The proposed model can help decision makers choose the optimal scheme based on their acceptable risk level to achieve a more efficient and sustainable water allocation for inter-basin water diversions.

Optimization of facility location and size problem based on bi-level multi-objective programming

With the rapid urbanization, solving the facility location and size problem (FLSP) of general service infrastructure (GSI) has become an essential issue in spatial planning. Due to unreasonable location and regional scale, the satisfaction of residents has been seriously affected. This paper develops a bi-level multi-objective programming (BLMOP) to optimize both facility location and size. Three major problems have been addressed: (1) solving the contradiction between supply and demand; (2) keeping a balance of social, economic, and environmental benefits; and (3) designing a multi-objective particle swarm optimization (MOPSO) algorithm by modifying the parameters and learning strategies. To obtain feasible solutions, a combination of optimistic and pessimistic approaches is adopted. Taking the rural areas of Southwest China as an example, the results find that the proposed model enables to provide objective-oriented optimization schemes depending on the decision-maker’s (DM) preferences. Furthermore, the MOPSO algorithm can solve the BLMOP and provide Pareto-optimal solutions separately.

A Bi-level Multi-objective Programming on Data Envelopment Analysis by Penalty Approach

A Bi-level Multi-objective Programming on Data Envelopment Analysis by Penalty Approach

A bi-level multi-objective programming model for water resources management under compound uncertainties in Dongjiang River Basin, Greater Bay Area of China

To facilitate regional water resources allocation, an integrated bi-level multi-objective programming (IBMP) model with dual random fuzzy variables was developed in this research The proposed model was derived through incorporating dual random fuzzy variables, multi-objective programming, and interval parameter programming within a bi-level optimization framework. This approach improved upon the previous bi-level programming methods and had two advantages. Firstly, it was capable of reflecting tradeoffs among multiple conflict preferences for water related bi-level hierarchical decision-making processes. Secondly, random fuzzy variables were used to tackle the dual uncertainties in both sides of the constraints, which were characterize as probability density functions and discrete intervals. Then, a real-world water resources planning problem was employed for illustrating feasibility of the application of IBMP model in Dongjiang river watershed of south China. Results reflected the alternative decisions for water allocation schemes under a set of probability levels and fuzzy α - cut levels, which can support in-depth analysis of tradeoffs among multiple levels and objective values. Moreover, modeling comparison analysis was undertaken to illustrate the performances of the proposed model.

Solving Bi-Level Multi-Objective Quadratic Programming Problem with Neutrosophic Parameters in Objective Functions and Constraints Using Fuzzy Approach

Solving Bi-Level Multi-Objective Quadratic Programming Problem with Neutrosophic Parameters in Objective Functions and Constraints Using Fuzzy Approach

A hybrid multi-objective bi-level interactive fuzzy programming method for solving ECM-DWTA problem

Electronic countermeasure (ECM) has become one of the most significant factors in modern warfare, in the course of combat, the electronic jamming allocation tasks need to be flexibly adjusted with the change of combat stage, which puts forward higher requirements for the modeling and solution method of this kind of problems. To solve the ECM dynamic weapon target assignment (ECM-DWTA) problem, a hybrid multi-target bi-level programming model is established. The upper level takes the sum of the electronic jamming effects in the whole combat stage as an optimization objective, and locally optimizes the ECM weapon (ECM-WP) assignment scheme in each stage. The lower level takes the importance expectation value of the target subjected to interference and combat consumption as double optimization objectives to globally optimize the ECM-WP assignment scheme. Focus on solving this complex model, a hybrid multi-objective bi-level interactive fuzzy programming algorithm (HMOBIF) is proposed, in this method, exponential membership function is used to describe the satisfaction degree of each level. When solving the multi-objective optimization problem composed of membership functions in the upper and lower levels, we use the MOEA/D algorithm to obtain the Pareto Front (PF) solution set, and then each solution in PF is evaluated and selected by the TOPSIS multi-criteria evaluation method. This local and global interactive optimization process of bi-level model is actually the process of executing observation-orientation-decision-action loop in practical combat. According to the current example, we conduct numerical simulation on the parameters in the model and obtain the parameter values suitable for the model solution. The computational experiments on different scale ECM-DWTA problems show that HMOBIF method is superior to four bi-level programming algorithms in terms of performance index, and can better solve ECM-DWTA problems.

Optimal trade-off of integrated river basin water resources allocation considering water market: A bi-level multi-objective model with conditional value-at-risk constraints

The predictable water shortage crisis is intensifying due to frequent extreme weather caused by climate change and human activities. Motivated by urgent practical needs of more scientific and balanced river basin water resources allocation (RBWRA) schemes, a comprehensive bi-level multi-objective programming (BLMOP) model for RBWRA considering uncertain risks and water market is constructed in this paper, which pursues the objectives of environmental protection, equality, and economic development. River basin authority, as the leader of RBWRA, is committed to reducing pollutants while introducing Gini coefficient to maintain the equality of economic development between sub-areas. Meanwhile, the sub-areas (i.e., followers) are dedicated to improving the economic benefit. In addition, the water market is introduced as water redistribution mechanism and its effect is probed into in detail. Moreover, conditional value-at-risk (CVaR) constraints are integrated into the model to control the economic loss caused by the risks of uncertain water availability. To solve the BLMOP model, a novel bi-level interaction based adaptive global equilibrium optimization (BI-aGEO) approach is proposed, which can obtain the best equilibrium solution that satisfies all decision-making participants. Furthermore, the proposed model and approach are applied to the Hanjiang River Basin in central China to elaborate on their feasibility and effectiveness. Results show that the proposed model is effective in ensuring the sustainability, equality and efficiency of RBWRA. The introduction of water market is shown to be able to improve water efficiency and promote water conservation and CVaR constraints play a significant role in controlling economic risks. Finally, some managerial suggestions are provided for different RBWRA participants for more scientific and effective integrated water resources management (IWRM).

A novel multi-objective bi-level programming problem under intuitionistic fuzzy environment and its application in production planning problem

This paper presents an optimization method to solve a multi-objective model of a bi-level linear programming problem with intuitionistic fuzzy coefficients. The idea is based on TOPSIS (technique for order preference by similarity to ideal solution) method. TOPSIS method is a multiple criteria method that identifies a satisfactory solution from a given set of alternatives based on the minimization of distance from an ideal point and maximization of distance from the nadir point simultaneously. A new model of multi-objective bi-level programming problem in an intuitionistic fuzzy environment has been considered. The problem is first reduced to a conventional multi-objective bi-level linear programming problem using accuracy function. Then the modified TOPSIS method is proposed to solve the problem at both the leader and the follower level where various linear/non-linear membership functions are used to represent the flexibility in the approach of decision-makers (DMs). The problem is solved hierarchically, i.e., first the problem at the leader level is solved and then the feasible region is extended by relaxing the decision variables controlled by the leader. The feasible region is extended to obtain a satisfactory solution for the DMs at both levels. Finally, the application of the proposed approach in the production planning of a company has been presented. An illustrative numerical example is also given to explain the methodology defined in this paper.

Bi-level multi-objective programming approach for bioenergy production optimization towards co-digestion of kitchen waste and rice straw

The increase in kitchen waste (KW) generation and irresponsible rice straw (RS) disposal has accentuated problems regarding environmental pollution and resource wasting. Although anaerobic co-digestion has been shown to be a more efficient strategy for bioenergy production than single digestion, there is still significant resistance to its application and diffusion owing to RS collection and many related factors in practice. Therefore, this study proposes a cooperative scheme for the cross-regional problem involving authorities and disposal plants. Subsequently, a bi-level multi-objective programming is developed to express the complexity of conflicts relationship in a multi-hierarchical problem and to achieve economic development, environmental protection and societal satisfaction in the bioenergy optimization process. A practical case study is then investigated to demonstrate the applicability of methodology in co-digestion deployment and resource recovery improvement, whereby scenario analyses are conducted by adjusting associated parameters. The practical application observed that (1) RS purchase was positively correlated with carbon reduction and social satisfaction targets, while excessive pursuit of social satisfaction resulted in a rebounding of carbon emissions; (2) the most popular mix ratios in disposal plants were 4:1 and 3:1, but they favor the latter when RS is readily available in sufficient quantities. Finally, some targeted suggestions are proposed for all stakeholders, which are expected to be a reference for potential users in other areas.