Bi-level Multi-objective Programming Research Articles

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.

Location planning of electric vehicle charging station with users’ preferences and waiting time: multi-objective bi-level programming model and HNSGA-II algorithm

Interactive users’ preferences and waiting time together have great impact on charging station network design of electric vehicles (EVs), but only waiting time was considered in previous studies. To fill this research gap, this paper addresses a location planning problem for EV charging stations, which considers users’ preferences and waiting time simultaneously. The problem is formulated as a multi-objective bi-level programming model, the upper level model determines locations and capacity options of charging stations with the objectives of minimising total cost and minimising total service tardiness, and the lower level model determines the allocation of users to stations with the objective of minimising total travel time. A hybrid non-dominated sorting genetic algorithm II (HNSGA-II) with embedded level determination algorithm (LDA) and a partial enumeration algorithm (PEA) are proposed, respectively, to solve the model. Furthermore, managerial analysis is implemented to verify the advantages of considering users’ preferences in reducing charging service tardiness and saving cost compared with the mode of no considering users’ preferences. And sensitivity analysis is also performed to provide managerial insights for EV charging station location practice. Finally, a real-world case study is conducted to verify the applicability of the proposed approach in solving practical location planning problems.

Railway Alignment Optimization Based on Multiobjective Bi-Level Programming Considering Ecological Impact

Railway projects frequently pass through ecologically fragile regions and result in various ecological damages. The railway alignment determining the layout of structures plays an important role in ecological protection. This paper presents a railway alignment optimization model based on multiobjective bi-level programming (MBRAO). The upper level of MBRAO is a horizontal alignment optimization for the minimization of investment and ecological impacts from tunnel drainage and train noise. A multiobjective evolutionary algorithm based on decomposition (MOEA/D) solves the horizontal alignment optimization and naturally maintains population diversity. The lower level of MBRAO is a vertical alignment optimization for investment minimization. A self-adaptive differential evolution (DE) algorithm adjusting essential parameters based on optimization status solves the vertical alignment optimization efficiently. MBRAO is applied in multistage at both levels to find the suitable numbers of horizontal and vertical points of intersection. This paper introduces three kinds of vertical feature data generated corresponding to horizontal feature data to define different site modification costs on the bridge, subgrade, and tunnel structures. A real-world case study at Wolong Reserve is studied to verify the effectiveness of MBRAO based on a customized geographic information system (GIS).

A New Approach for Solving Bi-Level Multi-Objective NonLinear Programming Model under Neutrosophic Environment

A New Approach for Solving Bi-Level Multi-Objective NonLinear Programming Model under Neutrosophic Environment

Multi-Objective Optimization of Differentiated Urban Ring Road Bus Lines and Fares Based on Travelers’ Interactive Reinforcement Learning

This paper proposes a new multi-objective bi-level programming model for the ring road bus lines and fare design problems. The proposed model consists of two layers: the traffic management operator and travelers. In the upper level, we propose a multi-objective bus lines and fares optimization model in which the operator’s profit and travelers’ utility are set as objective functions. In the lower level, evolutionary multi agent model of travelers’ bounded rational reinforcement learning with social interaction is introduced. A solution algorithm for the multi-objective bi-level programming is developed on the basis of the equalization algorithm of OD matrix. A numerical example based on a real case was conducted to verify the proposed models and solution algorithm. The computational results indicated that travel choice models with different degrees of rationality significantly changed the optimization results of bus lines and the differentiated fares; furthermore, the multi-objective bi-level programming in this paper can generate the solution to reduce the maximum section flow, increase the profit, and reduce travelers’ generalized travel cost.

Optimality conditions for nonsmooth multiobjective bilevel optimization using tangential subdifferentials

In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex.

Decentralized Decision System for Closed-Loop Supply Chain: A Bi-Level Multi-Objective Risk-Based Robust Optimization Approach

This paper proposes a novel risk-based robust mixed-integer linear programming to design a decentralized closed-loop supply chain. The model is formulated as an uncertain bi-level multi-objective programming with multiple suppliers, manufacturers, and distributors, as the leader, and recovery, recycling, and disposal centers as the follower. A Scenario-based Conditional Value-at-Risk is employed to capture the demand uncertainty. The Karush-Kuhn-Tucker approach, ε−constraint, and LP-metric are leveraged to deal with the complexity of the bi-level coordination, and the multi-objectivity of the leader and follower. The performance of the model is compared with the performance of the deterministic decentralized model and the corresponding multi-objective model designed for the centralized system in both the robust and deterministic modes. Results indicate better performance of robust approaches compared to deterministic approaches. The decentralized approach provides better performance for the cost-sensitive decision-maker, especially the optimistic one, and those who are sensitive to social parameters prefer the centralized approach.

An optimal modeling approach for reducing carbon footprint in agricultural water-energy-food nexus system

Modern agriculture calls for efficient and environmental-friendly agricultural water and land management practice. Agricultural water and land utilization mode has a great impact on the amount of carbon emissions, and thus the aim of this work is to propose an optimal modeling approach for generating efficient agricultural water and land management alternatives and reducing carbon emission in agricultural water-energy-food nexus system. This study presents a novel approach consisting of carbon footprint lifecycle assessment method and bi-level multi-objective stochastic programming model. The proposed approach has contributions in following aspects: (1) the environmental impact of different resource allocation strategies can be measured in optimization (2) the spatial variability of spatial data (e.g., ET0 and precipitation); can be fully reflected via remote sensing information; (3) tradeoffs among two decision-making levels and their conflicting objectives under randomness of surface water availability can be addressed. The proposed approach was applied to the middle reaches of the Heihe River basin, northwest China. After solving the proposed model, the optimal water and land use alternatives under different hydrological years can be generated. Decision makers can plan agricultural production strategy according to optimization schemes, and reduce carbon emission through increasing vegetable cultivation and surface water utilization. Furthermore, the performance comparison of different models indicated that fierce conflicts exist between income fairness and economic benefits. The comprehensive evaluation value of bi-level multi-objective stochastic programming model is 0.7036, which is highest among single- and multi-objective models, showing that such model has obvious advantage in dealing with multiple decision-making levels and conflicting objectives. This approach can help decision makers of similar regions manage the agricultural production system in a more efficient and environment-friendly way.

A Bi-Level Model for Green Freight Transportation Pricing Strategy Considering Enterprise Profit and Carbon Emissions

Due to the impact of COVID-19, enterprises need effective pricing strategies to improve profits and viability. In order to fill research gaps in the literature relating to market competition among different freight modes and to adjust the freight transportation structure by optimizing transportation prices, we propose a multi-objective bi-level programming pricing model that considers market competition and the carbon emissions of the freight system in China. First, an objective function in upper-level planning was used to improve logistics enterprise profits and reduce the total carbon emissions of the freight system. Then, a generalized cost function for the freight transportation mode was designed to quantify the market competition among different transportation modes, and a user equilibrium assignment model was established to obtain the results of cargo flow assignment in lower-level planning. To solve the model, a sensitivity analysis algorithm was designed, and a logistics network example was used to verify the effectiveness of the model. The experimental results show that reasonable freight price adjustment can effectively increase enterprise profits and reduce the total carbon emissions of the freight system. In this paper, we provide a new method for freight pricing research, considering the market competition of multiple transportation modes, and provide a new idea for adjusting the transportation structure through freight price optimization, which will play a positive role in promoting the development of green freight.

K-means cluster interactive algorithm-based evolutionary approach for solving bilevel multi-objective programming problems

Solving bilevel multi-objective programming problems is one of the hardest tasks facing researchers in the optimization community. Bilevel multi-objective programming problems is an optimization problem consists of two interconnected hierarchical multi-objective programming problems: upper-level problem and lower-level problem. Difficulty in solving bilevel multi-objective programming problems is the need to solve lower-level multi-objective programming problem to know the feasible space of the upper-level problem. The proposed algorithm consists of two nested artificial multi-objective algorithms. One algorithm is for the upper-level problem and the other is for the lower-level problem. Also, the proposed algorithm is enriched with a k-means cluster scheme in two phases. The first phase is before starting two nested algorithms to help the algorithm to start with more appropriates solutions to the bi-level problem. The second phase is within the two nested algorithms to guide the algorithm to the most preferred solutions to the upper-level decision-maker. The performance of the proposed algorithm has been evaluated on different test problems including low dimension and high dimension test problems. The experimental results show that the proposed algorithm is a feasible and efficient method for solving the bilevel multi-objective programming problem.

A bi-level multi objective programming approach to solve grey problems: an application to closed loop supply chain network

PurposeThis paper aims to develop a grey decentralized bi-level multi-objective programming (MOP) model. A solution approach is also proposed for the given model. A production and transportation plan for a closed loop supply chain network under an uncertain environment and different scenarios is also developed.Design/methodology/approachIn this paper, we combined grey linear programming (GLP) and fuzzy set theory to present a solution approach for the problem. The proposed model first solves the given problem using GLP. Membership functions for the decision variables under the control of the leader and for the goals are created. These membership functions are then used to generate the final solutions.FindingsThis paper provides insight for fomenting the decision-making process while providing a more flexible approach in uncertain logistics problems. The deviations of the final solution from the individual best solutions of the two levels are very little. These deviations can further be reduced by adjusting the tolerances associated with the decision variables under the control of the leader.Practical implicationsThe proposed approach uses the concept of membership functions of linear form, and thus, requires less computational efforts while providing effective results. Most of the organizations exhibit decentralized decision-making under the presence of uncertainties. Therefore, the present study is helpful in dealing with such scenarios.Originality/valueThis is the first time, formulation of a decentralized bi-level multi-objective model under a grey environment is carried out as per the best knowledge of the authors. A solution approach is developed for bi-level MOP under grey uncertainty.

Optimal management of cultivated land coupling remote sensing-based expected irrigation water forecasting

This study proposed a novel cultivated land planning approach coupling expected irrigation water prediction and crop planting structure optimization. Under the guidance of the expected irrigation water prediction, the development of agricultural planting strategies was proposed through the cultivated land planning method to improve the efficiency of water use and ensure agricultural production. This approach was consisted of two models: a combination forecasting model was developed for estimating expected irrigation water with help of remote sensing data (RSCM); and a bi-level multiobjective programming model for planning crop planting structure (BMPSOM) was formulated for crop planting structure programming. Then, the proposed approach applied to a real-world case in Northwest China for verifying its validity. Results show that: 1) the forecast accuracy of the combination forecast model has been greatly improved compared to the single forecast model, the accuracy test results are RMSE = 0.25 × 108 m3, MAPE = 0.78%; 2) the use of remote sensing data can reflect the soil moisture with spatial variability in the forecast; 3) BMPSOM model can well reflect the leader-follower relationship and make tradeoff among multiple objectives of each decision maker in tackling practical problem. Application has proved that the agricultural planting strategy formulated by this approach can not only ensure economic benefits but also increase food production under limited water resources. Therefore, this method has good application prospects in areas where water resources are scarce.

A dual-randomness bi-level interval multi-objective programming model for regional water resources management

In this research, a dual-randomness bi-level interval multi-objective programming (DR-BIMP) model was developed for supporting water resources management among multiple water sectors under complexities and uncertainties. Techniques of bi-level multi-objective programming (BMOP), double-sided stochastic chance-constrained programming (DSCCP), and interval parameter programming (IPP) were incorporated into an integrated modeling framework to achieve comprehensive consideration of the complexities and uncertainties of water resources management systems. The DR-BIMP model can not only effectively deal with the interactive effects between multiple decision-makers in complex water management systems through the bi-level hierarchical strategies, but also can characterize the multiple uncertainties information expressed as interval format and probability density functions. It could thus improve upon the existing bi-level multi-objective programming through addressing discrete interval parameters and dual-randomness problems in optimization processes simultaneously. Then, the developed model was applied to a real-world case to optimally allocate water resources among three different water sectors in five sub-regions in the Dongjiang River basin, south China. The results of the model include determining values, interval values, and stochastic distribution information, which can assist bi-level decision-makers to plan future resources effectively to some extent. After comparing the variations of results, it is found that an increasing probability level can lead to higher system benefits, which is increased from [20,786.00, 26,425.92] × 108 CNY to [22,290.84, 27,492.57] × 108 CNY, while the Gini value is reduced from [0.365, 0.446] to [0.345, 0.405]. A set of increased probability levels gives rise to the lower-level objectives. Furthermore, the advantages of the DR-BIMP model were highlighted by comparing with the other models originated from the developed model. The comparison results indicated that the DR-BIMP model was a valuable tool for generating a range of decision alternatives and thus assists the bi-level decision-makers to identify the desired water resources allocation schemes under multiple scenarios.

Reliability-based equitable transit frequency design

Fairness is an important criterion for achieving sustainable urban development. While most existing studies focus on accessing or evaluating the fairness condition of a given transit network, this study explicitly incorporates fairness as an objective in the planning step. A multi-objective bilevel programming model is developed where the lower level problem is the reliability-based transit assignment problem and the upper level problem is to determine optimal frequency settings to simultaneously minimise the total effective travel cost and maximise the network fairness condition. A multi-objective Artificial Bee Colony algorithm is developed to solve the bilevel model. Numerical studies find that: (1) increasing the frequency may not improve the fairness condition; (2) there is a tradeoff between the two objectives (3) the effect of passengers’ risk aversion attitude on the fairness measurement depends on the frequency setting; it could either amplify the fairness measurements or have no impact.

Multiobjective Environmentally Sustainable Optimal Design of Dedicated Connected Autonomous Vehicle Lanes

The emergence of connected autonomous vehicles (CAVs) is not only improving the efficiency of transportation, but also providing new opportunities for the sustainable development of transportation. Taking advantage of the energy consumption of CAVs to promote the sustainable development of transportation has attracted extensive public attention in recent years. This paper develops a mathematical approach to investigating the problem of the optimal implementation of dedicated CAV lanes while simultaneously considering economic and environmental sustainability. Specifically, the problem is described as a multi-objective bi-level programming model, in which the upper level is to minimize the system-level costs including travel time costs, CAV lane construction cost, and emission cost, whereas the lower level characterizes the multi-class network equilibrium with a heterogeneous traffic stream consisting of both human-driven vehicle (HVs) and CAVs. To address the multi-objective dedicated CAV lane implement problem, we propose an integrated solution framework that integrates a non-dominated sorting genetic algorithm II (NSGA-II) algorithm, diagonalized algorithm, and Frank–Wolfe algorithm. The NSGA-II was adopted to solve the upper-level model, i.e., hunting for the optimal CAV lanes implementation schemes. The diagonalized Frank–Wolfe (DFW) algorithm is used to cope with multi-class network equilibrium. Finally, numerical experiments were conducted to demonstrate the effectiveness of the proposed model and solution method. The experimental results show that the total travel time cost, total emission cost, and total energy consumption were decreased by about 12.03%, 10.42%, and 9.4%, respectively, in the Nguyen–Dupuis network as a result of implementing the dedicated CAV lanes.

Active learning for multi-objective optimal road congestion pricing considering negative land use effect

The road congestion pricing policy is implemented to alleviate traffic congestion and improve the efficiency of the transportation system during peak hours. However, the negative land use effect caused by this policy could not be ignored. How to design the optimal congestion toll that can not only ensure its positive effect on the transportation system but also reduce its negative effect on land use is an urgent problem to be solved. Given this, this paper proposed a multi-objective bi-level programming road congestion pricing model based on the integrated land use and transportation model to optimize the regional average accessibility, regional average land use diversity, and regional total flow time. Since the proposed problem is NP-hard, this paper innovatively proposed an active learning optimization algorithm based on multi-objective Bayesian optimization, which improves the computation efficiency of the bi-level programming model by automatically finding the next sampling point (candidate solution) according to the probability information. An empirical analysis of Jiangyin City demonstrated the effectiveness of the proposed approach in coordinating the relationship between land use and transportation and alleviating the negative land use effect caused by road congestion pricing. Moreover, the algorithm proposed in this paper can also be used to solve other transportation-related black box problems with high computation complexity.

Optimality conditions and duality in terms of convexificators for multiobjective bilevel programming problem with equilibrium constraints

This paper is devoted to the investigation of a nonsmooth multiobjective bilevel programming problem with equilibrium constraints ((MBPP) for short) in terms of convexificators in finite-dimensional spaces. We present necessary optimality conditions for the local weak efficient solution to such problem. Under the Mangasarian–Fromovitz and generalized standard Abadie type constraint qualification in the sense of convexificators, we establish as an application the Wolfe and Mond-Weir type dual problem for the problem (MBPP). Besides, we provide strong and weak duality theorems for the original problem and its Wolfe and Mond–Weir type dual problem under suitable assumptions on the $$\partial ^*$$ -convexity and the upper semi-regularity of objective and constraint functions. Illustrative examples are also proposed to demonstrate the main results of the paper.

A Bi-Level Multi-Objective Optimization Model for Micro-Circulation Road Networks in the Open Block Area Considering Traffic Pollution and Intersection Delays

It is important for urban traffic micro-circulation to improve the density of urban branch road networks by opening roads inside blocks. To reasonably optimize the micro-circulation road network in the open block area, a bi-level multi-objective programming model that considers traffic pollution and intersection delays was developed. In this paper, the goals of minimizing traffic pollution and total travel cost are added to the upper-level programming model and the user equilibrium assignment model with the consideration of intersection delay was presented as the lower-level programming model. A modified genetic algorithm (GA) embedded with the Frank-Wolfe algorithm was designed to solve the established model. The traffic conditions of arterial roads and micro-circulation branch roads before and after optimizing the micro-circulation block road network were compared and analyzed by a numerical example. The results demonstrated that the bi-level programming model can effectively determine the traffic direction of branch roads and the forbidden situation of intersections in the micro-circulation network. Compared with the closed block, the average saturation of the main trunk road decreased from 0.97 to 0.83 with a decline ratio of 14.43% after optimizing the micro-circulation network in the open block area; the average saturation of the secondary trunk road decreased from 0.86 to 0.77, with a decline ratio of 10.47%. The travel time cost decreased by approximately 6.55%, and the traffic pollution decreased by approximately 3.40%, which verified the optimization effect of the model and the algorithm.

Scenario-Based Network Reconfiguration and Renewable Energy Resources Integration in Large-Scale Distribution Systems Considering Parameters Uncertainty

Renewable energy integration has been recently promoted by many countries as a cleaner alternative to fossil fuels. In many research works, the optimal allocation of distributed generations (DGs) has been modeled mathematically as a DG injecting power without considering its intermittent nature. In this work, a novel probabilistic bilevel multi-objective nonlinear programming optimization problem is formulated to maximize the penetration of renewable distributed generations via distribution network reconfiguration while ensuring the thermal line and voltage limits. Moreover, solar, wind, and load uncertainties are considered in this paper to provide a more realistic mathematical programming model for the optimization problem under study. Case studies are conducted on the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks, where the 59- and 83-node distribution networks are real distribution networks in Cairo and Taiwan, respectively. The obtained results validate the effectiveness of the proposed optimization approach in maximizing the hosting capacity of DGs and power loss reduction by greater than 17% and 74%, respectively, for the studied distribution networks.