Lower Level Decision Makers Research Articles

A bi-level multi-objective optimization model for inter-provincial carbon emissions transfer tax on electricity production

Low-carbon electricity development is crucial for achieving China's carbon peaking and carbon neutrality goals. The current carbon reduction territorial management model (CRTMM) can only bring short-term carbon reduction and is not enough to motivate provinces to further reduce carbon. However, few studies focus on the inter-provincial coordinated electricity carbon reduction. To overcome the shortcomings of the CRTMM and fill the research gap, this study creatively presents an original bi-level multi-objective inter-provincial carbon emissions transfer tax model (CETTM) on electricity generation in China, where the central government is the upper-level decision maker (leader) and provincial governments are the lower-level decision makers (followers). As the leader, the central government pursues overall maximum economic benefit and minimum pollutant emissions, and as the followers, provincial governments endeavor to maximize internal economic benefits. To ensure the scientificity and accuracy of the calculation results, an improved NSGA-II algorithm is designed to solve the model. Data from Anhui, Qinghai, Heilongjiang, and Jiangsu provinces are then used for empirical analysis to validate the model. The findings demonstrate that the CETTM is superior to the CRTMM in reducing carbon emissions, lowering pollutant emissions and improving economic benefits. The CETTM reduces CO2 emissions by 20.04 × 106 tons, lowers pollutant emissions by 10.65 × 104 tons, and increases benefits by 101.99 × 108 CNY in contrast to the CRTMM. The robustness of the model is verified using a sensitivity analysis. Finally, this study proposes policy recommendations for the CETTM implementation.

On complexity of finding strong-weak solutions in bilevel linear programming

We consider bilevel linear programs (BLPs) that model hierarchical decision-making settings with two independent decision-makers (DMs), referred to as a leader (an upper-level DM) and a follower (a lower-level DM). BLPs are strongly NP-hard. In general, the follower's rational reaction (i.e., a set that contains optimal solutions of the lower-level problem for a given leader's decision) is not a singleton. If we assume that, for a given leader's decision, the follower always selects a solution from the rational reaction set that is most (least) favorable to the leader, then we obtain the optimistic (pessimistic) model. It is known that the optimistic (pessimistic) model remains NP-hard even if an optimal pessimistic (optimistic) solution of the same BLP is known. One interesting generalization of these two approaches studied in the related literature is to consider the so-called α-strong-weak model, where parameter α∈[0,1] controls the leader's level of conservatism. That is, α provides the probability of the follower's “cooperation” in a sense that, the lower-level's rational decisions that are most and least favorable to the leader are picked with probabilities α and 1−α, respectively. In this note we show that for any fixed α∈(0,1) the problem of finding an optimal α-strong-weak solution remains strongly NP-hard, even when both optimal optimistic and pessimistic solutions for the same BLP are known.

Modelling a bi-level multi-objective post-disaster humanitarian relief logistics network design problem under uncertainty

Post-disaster humanitarian relief logistics focuses on managing the supply and shipment of relief items. Owing to the complexity arising from the hierarchical decision relationship and the uncertainties of demand and transportation, it is challenging to develop an effective humanitarian relief logistics network (HRLN). A globalized robust bi-level multi-objective HRLN model involving uncertainties in transportation time, transportation cost and demand is developed to address this problem, in which an upper-level decision-maker focuses on the allocation problem and a lower-level decision-maker considers the supply problem. For tractability, the goal programming approach is employed to trade off conflicting timeliness, satisfaction and operational cost objectives. The proposed globalized robust bi-level HRLN model is reformulated as its globalized robust counterpart formulations under two different perturbation structures. The primal–dual approach is adopted to transform the bi-level models into equivalent mixed-integer conic programs. A case study concerning an Iranian earthquake is conducted to prove the performance of the proposed model. Globalized robust models are more flexible and able to immunize against uncertainty by comparison with traditional robust optimization and deterministic solutions. The managerial insights of using the globalized robust bi-level optimization approach are reported for disaster response management. The design of an emergency aviation network to circumvent the effects of COVID-19 could be considered in future research.

Youth should decide: the principle of subsidiarity in paediatric transgender healthcare

Drawing on the principle of subsidiarity, this article develops a framework for allocating medical decision-making authority in the absence of capacity to consent and argues that decisional authority in paediatric...

Investigation of Climate Change Adaptation Impacts on Optimization of Water Allocation Using a Coupled SWAT-bi Level Programming Model

One way to deal with the future effects of climatic changes on the water resources and to cope with water shortages in basins is to have a clear understanding of the future climate change trends. To this end, this study proposes an integrated hydrological-water transfer and supply (HWTS) framework including a coupled SWAT-Bi level programming model to investigate future optimal water supply between different sectors with regard to transaction right. Indeed, Soil & Water Assessment Tool (SWAT) is applied to project the rate of streamflow under Representative Concentration Pathway scenarios of RCP2.6&RCP4.5 and future periods (2020–2040 & 204(Abbas et al. 2015)2060). In addition, a case study of the Hamoun wetland in southeastern of Iran is considered for calibration and validation of real historical data (2000–2016) and then simulation of future streamflow patterns (2020–2060). Next, simulated streamflow data extracted by SWAT is entered as the input of market based bi-level optimization model so that upper-level manager seeks optimization of the available water level in the reservoirs while the lower-level decision maker tries to minimize the economic loss due to water shortage between different sectors regarding transaction right. However, after solving the model with the Improved Particle Swarm Optimization (IBPSO) technique, the final results show that although not much economic profit will be made, but considering specific management strategies such as demand reduction schemes to conserve more water, the imbalance between supply and demand can be significantly improved.

Bilevel Multi-Objective Gray Wolf Algorithm Based on Packet Transport Network Optimization

Packet transport network (PTN), as an efficient transmission network technology in mobile communications in the big data era, is used by more and more communication operators. The existing PTN resource utilization rate is low, the network security is poor, so the existing PTN needs to be optimized in all aspects. For the optimization of the PTN, it is necessary to consider the decision of both the operator user and the service product supplier. Therefore, this paper proposes a bilevel multi-objective gray wolf algorithm based on PTN optimization problem. The operator user is the upper-level decision maker, and the objective function is to pay the product supplier the lowest cost. The product supplier is the lower-level decision maker, it mainly includes two major objective functions. The first objective function is to maximize the Label switching path overlap rate(LSPOR) evaluation score to solve the abnormal Label Switching Path (LSP) problem in the network, and the second is to maximize the committed bandwidth with utilizing rate(CBWUR) evaluation score to solve the problem of excessive Committed Information Rate(CIR) bandwidth usage in the network. According to the three scale network situation in Hubei, China, the improved multi-objective gray wolf algorithm is used to solve the PTN bilevel programming problem. The experimental results show that the model increases the utilization rate of network resources, and reduces the cost to be paid by the upper-level decision makers.

Iterative Parametric Approach for Quadratically Constrained Bi-Level Multiobjective Quadratic Fractional Programming

The paper proposed an iterative parametric approach procedure for solving Bi-level Multiobjective Quadratic Fractional Programming model. The Model is divided into two levels-upper and lower. In the first stage of the approach, a set of pareto optimal solutions of upper Level is obtained by converting the problem into equivalent single non-fractional parametric objective optimization problem by using parametric vector and ε-constraint method. Then for the second stage, the solution of upper level is followed by the lower level decision maker while finding solution with the proposed algorithm to obtain the best preferred solution. A numerical example is solved in the last to validate the feasibility of the approach.

Bilevel programming solution algorithms for optimal price-bidding of energy producers in multi-period day-ahead electricity markets with non-convexities

We consider the problem of devising optimal price-offers (bids) for an energy producer participating in a multi-period day-ahead electricity market which exhibits non-convexities due to the discrete nature of the generation units’ commitments and quantities. The problem definition assumes perfect knowledge of the market’s technical characteristics, as well as of the bidding offers of the remaining producers. The problem is formulated as a bilevel optimization model with integer decision variables and linear constraint sets at both levels. The producer acts as the upper-level decision maker, aiming to find the optimal bidding offers that will maximize his individual profit upon clearing of the market, while an independent system operator acts as the lower-level decision maker, aiming to ensure satisfaction of the demand for energy at the minimum total bid-cost. Utilizing the theoretical properties of this problem, we develop both a heuristic as well as an exact algorithmic solution methodology for tackling it. More effective between the two naturally turns out to be the heuristic approach, which works iteratively, optimizing a single price-offer at each iteration, given that the remaining ones are kept fixed at their current values. We present experimental results demonstrating that it provides high quality solutions, while exhibiting reasonable computational requirements. We also demonstrate how the underlying theory can be utilized for the generation of valid inequalities to a suitable relaxation of the original formulation, in which the so-called bilevel feasibility of the obtained solution is not guaranteed. These inequalities are exploited within a cutting-plane framework by the exact solution approach for identifying the global optimum of the problem.

Towards sustainable water management in an arid agricultural region: A multi-level multi-objective stochastic approach

Water-shortage crisis is threatening the food production and sustainable development around the world. Especially for arid agricultural regions, it is necessary to plan sustainable agricultural water management strategies for improving water use efficiency. But there are many complexities in it, such as multiple decision-making levels, objectives, water users and uncertainties. To effectively tackle these complexities, this study presents a novel optimization-modeling approach consisting of a multi-level multi-objective stochastic programming (MLMOSP) model and weighting quantification method for formulating sustainable water-allocation schemes in arid agricultural regions. The MLMOSP model incorporates multi-level programming, multi-objective programming, and stochastic expectation programming into a general framework. The proposed approach is capable of: 1) quantifying key factors affecting water-allocation systems through weighting quantification methods; 2) describing the main conflicting objectives of each decision-making level, including economic benefits, environment impacts, fairness, effectiveness, and crop yield; 3) considering tradeoffs among conflicting objectives, and 4) reflecting the leader-follower relationship under different scenarios of surface water availability at a regional scale and a monthly temporal resolution. The proposed approach is applied to a real-world case in a typical arid agricultural region of northwest China for verifying its validity. From this real-world case, it is found that: 1) optimization results corresponding to different flow-level scenarios of surface runoff can provide upper-, middle-, and lower-level decision makers with a set of decision alternatives to help identify the most appropriate management strategy; and 2) multiple model comparisons show that the MLMOSP approach can not only give more practical results guaranteeing the achievement of decision-making goals at different decision-making levels, but also help reduce groundwater extraction under different flow level scenarios of surface runoff.

Pessimistic optimisation for modelling microbial communities with uncertainty

Optimisation-based mathematical models provide ways to analyse and obtain predictions on microbial communities who play critical roles in the ecological system, human health and diseases. However, there are inherent model and data uncertainties from the existing knowledge and experiments so that the imposed models may not exactly reflect the reality in nature. Here, we aim to have a flexible framework to model microbial communities with uncertainty, and introduce P-OptCom, an extension of an existing method OptCom, based on pessimistic bilevel optimisation. This framework relies on the coordinated decision making between the single upper-level and multiple lower-level decision makers to better approximate community steady states even when the individual microorganisms' behavior deviate from the optimum in terms of their cellular fitness criteria. Our study demonstrates that without experimental knowledge in advance, we are able to analyse the trade-offs among the members of microbial communities and closely approximate the actual experimental measurements.

Pessimistic optimisation for modelling microbial communities with uncertainty

Optimisation-based mathematical models provide ways to analyse and obtain predictions on microbial communities who play critical roles in the ecological system, human health and diseases. However, there are inherent model and data uncertainties from the existing knowledge and experiments so that the imposed models may not exactly reflect the reality in nature. Here, we aim to have a flexible framework to model microbial communities with uncertainty, and introduce P-OptCom, an extension of an existing method OptCom, based on pessimistic bilevel optimisation. This framework relies on the coordinated decision making between the single upper-level and multiple lower-level decision makers to better approximate community steady states even when the individual microorganisms' behavior deviate from the optimum in terms of their cellular fitness criteria. Our study demonstrates that without experimental knowledge in advance, we are able to analyse the trade-offs among the members of microbial communities and closely approximate the actual experimental measurements.

A simulation-based bi-level multi-objective programming model for watershed water quality management under interval and stochastic uncertainties.

A simulation-based interval stochastic bi-level multi-objective programming (SISBLMOP) model was proposed in this research, through integrating the global nutrient export from watersheds model, interval parameter programming and stochastic chance-constrained programming into a general bi-level multi-objective programming framework. The SISBLMOP model can handle multiple uncertainties expressed as discrete intervals and probability density functions in both the simulation and optimization processes. System complexities, including the hierarchy structure of upper- and lower-level decision makers, can also be addressed in the model. The proposed model is applied to a real-world case study of the Xinfengjiang Reservoir Watershed in South China to identify the satisfactory implementation levels of multiple best management practices (BMPs). The model results show that multiple BMP schemes for water quality management can be obtained under different upper- and lower-level decision-making and risk-violation scenarios, reflecting the cooperation and gaming results of the two-level decision makers. Consequently, the corresponding BMP implementation costs are acceptable to both the upper- and lower-level decision makers. The model is widely applicable and can be effectively used for water quality management under multiple uncertainties and complexities.

Designing time-of-use tariffs in electricity retail markets using a bi-level model – Estimating bounds when the lower level problem cannot be exactly solved

Time-of-use tariffs are a pricing strategy for a product or service in which the supplier establishes time-differentiated prices. Dynamic (e.g., day-ahead) time-differentiated electricity prices can contribute to increase the retailer's profit, allow end-users to reduce the consumption costs and enhance grid efficiency.The electricity retailer and the consumer are hierarchically related. The interaction between them can be modeled by a bi-level (BL) optimization model – the retailer is the upper level decision maker and the consumer is the lower level decision maker. The retailer and the consumer have different and conflicting goals: the retailer establishes the pricing scheme to sell electricity to consumers to maximize his profit; the consumer reacts to these prices by determining the operation of the controllable loads in order to minimize the discomfort and the electricity bill.In this work, a BL optimization model incorporating shiftable, interruptible and thermostatic loads is proposed. The upper level problem is tackled by a particle swarm optimization algorithm while the lower level problem is solved by an exact mixed-integer programming solver. The inclusion of the thermostatic load in the lower level problem imposes a much higher computational burden. Therefore, it may not be possible to find the optimal lower level solution, and a sub-optimal lower level solution is infeasible to the BL problem. Considering a computational budget, this work proposes an approach to compute good quality estimates of bounds for the upper level objective function, providing the leader further information and allowing him to make sounder decisions in an adequate time frame.

On Various Approaches for Bi-level Optimization Problems

In this paper we review some different basic approaches for solving bi-level optimization problems (BLOP). Firstly, the formulation and some basic concepts of such BLOP are presented. Secondly, some conventional approaches for solving the BLOP such as; vertex enumeration method, branch and bound algorithm, Karush Kuhn-Tucker (KKT) transformation are exhibited. The vertex enumeration based approaches which use the important characteristic that at least one global optimal solution is attained at an extreme point of the constraints set. The KKT approaches in which a BLOP is transformed into a single level problem that solves the upper level decision maker (ULDM) problem while including the lower level decision maker (LLDM) optimality conditions as extra constraints. Fuzzy programming approach mainly based on the fuzzy set theory. Finally, formulation of the bi-level multi-objective decision making (BL-MODM) problem and recently developed approaches, such as; fuzzy goal programming (FGP) and technique for order preference by similarity to ideal solution (TOPSIS) approach, for solving such problem are presented. Numerical illustrations are presented for each technique to ensure the applicability and efficiency.

A bi-level programming model for sustainable supply chain network design that considers incentives for using cleaner technologies

Nowadays, the population growth leads environmental, legislative, and social concerns. As a consequence, government and companies started to be aware of the impact of supply chain operations in the environment and society. While most of research in Supply Chain Network Design (SCND) has focused on economic performance, recently some studies consider environmental and social aspects of the SCND. The majority of existing related papers deal with multi-objective optimization modelling approaches when considering sustainable aspects of the supply chain. In this study, the optimization of a hierarchized Sustainable Supply Chain Network Design (SSCND) is addressed. To model this situation, a bi-level programming approach is considered. In the proposed model, an Environmental Protection Agency from the government has the leader role and acts as the upper level decision maker. Also, the supply chain's manager assumes the follower role acting as the lower level decision maker. Leader's objective function considers environmental aspects, while follower's objective function considers economical aspects. In the modelled situation, the government tries to cause an impact in the SCND by offering financial incentives (subsidies) and by encouraging the supply chain's manager to use cleaner technologies. The resulting bi-level programming model is a mixed-integer linear program. Due to the structure of the lower level problem, the bi-level model cannot be reduced into an equivalent single-level one by using KKT conditions. Therefore, an algorithm based on the simulated annealing metaheuristic is proposed for solving the problem. The results show that an appropriate government's financial incentives strategy lead to significantly decrease the environmental impact of the SSCND. Moreover, a greater impact is achieved when the financial incentives do not depend on the total production obtained via the use of cleaner technologies (scenario 1) than assigning subsidies based on the production made with specific technologies (scenario 2). Furthermore, the proposed model could be considered as a new approach for SSCND procuring environmental and economic aspects under an assumed predefined hierarchy.

Feasibility reduction approach for hierarchical decision making with multiple objectives

Several decision making problems with multiple decision makers in a hierarchical decision making scenario can be best described using multilevel optimization problem. Different organizational structure adapt a hierarchical decision making system where each decision maker controls parts of the decision variables and is affected by the decision of other decision makers. Recently, inspired by natural adaptation, a metaheuristic based algorithm is proposed for these problems. The algorithm works by using an initial solution generated by solving the leader’s problem, where the leader is the decision maker in the upper level. That solution will go through each level by adjusting and evolving its components. Even though the approach is tested to be promising, the final solution can be very different from the initial one given by the leader. Furthermore, no cooperation mechanism between the decision makers is given. In addition, the decision makers may have multiple and conflicting objectives. In this paper a cooperation mechanism where fuzzy membership function is used to link the cooperation between the decision makers is used. That means once the solution received by the lower level decision makers, a cooperative feasible region will be determined which is a subset of the relaxed feasible region. To deal with the multiobjective optimization problem, preference free method called ideal point method will be used. Bi-level numerical examples are also given to demonstrate how the algorithm works.

Bi-level Linear Programming Problem with Neutrosophic Numbers

The paper presents a novel strategy for solving bi-level linear programming problem based on goal programming inneutrosophic numbers environment. Bi-level linear programming problem comprises of two levels namely upper or first leveland lower or second level with one objective at each level. The objective function of each level decision maker and the systemconstraints are considered as linear functions with neutrosophic numbers of the form [p + q I], where p, q are real numbers andI represents indeterminacy. In the decision making situation, we convert neutrosophic numbers into interval numbers and theoriginal problem transforms into bi-level interval linear programming problem. Using interval programming technique, the target interval of the objective function of each level is identified and the goal achieving function is developed. Since, the objectives of upper and lower level decision makers are generally conflicting in nature, a possible relaxation on the decision variables under the control of each level is taken into account for avoiding decision deadlock. Then, three novel goal programmingmodels are presented in neutrosophic numbers environment. Finally, a numerical problem is solved to demonstrate the feasibility, applicability and novelty of the proposed strategy.

A linear bi-level multi-objective program for optimal allocation of water resources.

This paper presents a simple bi-level multi-objective linear program (BLMOLP) with a hierarchical structure consisting of reservoir managers and several water use sectors under a multi-objective framework for the optimal allocation of limited water resources. Being the upper level decision makers (i.e., leader) in the hierarchy, the reservoir managers control the water allocation system and tend to create a balance among the competing water users thereby maximizing the total benefits to the society. On the other hand, the competing water use sectors, being the lower level decision makers (i.e., followers) in the hierarchy, aim only to maximize individual sectoral benefits. This multi-objective bi-level optimization problem can be solved using the simultaneous compromise constraint (SICCON) technique which creates a compromise between upper and lower level decision makers (DMs), and transforms the multi-objective function into a single decision-making problem. The bi-level model developed in this study has been applied to the Swat River basin in Pakistan for the optimal allocation of water resources among competing water demand sectors and different scenarios have been developed. The application of the model in this study shows that the SICCON is a simple, applicable and feasible approach to solve the BLMOLP problem. Finally, the comparisons of the model results show that the optimization model is practical and efficient when it is applied to different conditions with priorities assigned to various water users.

TOPSIS Approach for Solving Bi-Level Non-Linear Fractional MODM Problems

TOPSIS (technique for order preference similarity to ideal solution) is considered one of the known classical multiple criteria decision making (MCDM) methods to solve bi-level non-linear fractional multi-objective decision making (BL-NFMODM) problems, and in which the objective function at each level is considered nonlinear and maximization type fractional functions. The proposed approach presents the basic terminology of TOPSIS approach and the construction of membership function for the upper level decision variable vectors, the membership functions of the distance functions from the positive ideal solution (PIS) and of the distance functions from the negative ideal solution (NIS). Thereafter a fuzzy goal programming model is adopted to obtain compromise optimal solution of BL-NFMODM problems. The proposed approach avoids the decision deadlock situations in decision making process and possibility of rejecting the solution again and again by lower level decision makers. The presented TOPSIS technique for BL-NFMODM problems is a new fuzzy extension form of TOPSIS approach suggested by Baky and Abo-Sinna (2013) (Applied Mathematical Modelling, 37, 1004-1015, 2013) which dealt with bi -level multi-objective decision making (BL-MODM) problems. Also, an algorithm is presented of the new fuzzy TOPSIS approach for solving BL-NFMODM problems. Finally, an illustrative numerical example is given to demonstrate the approach.

An export coefficient based inexact fuzzy bi-level multi-objective programming model for the management of agricultural nonpoint source pollution under uncertainty

In this research, an export coefficient based inexact fuzzy bi-level multi-objective programming (EC-IFBLMOP) model was developed through integrating export coefficient model (ECM), interval parameter programming (IPP) and fuzzy parameter programming (FPP) within a bi-level multi-objective programming framework. The proposed EC-IFBLMOP model can effectively deal with the multiple uncertainties expressed as discrete intervals and fuzzy membership functions. Also, the complexities in agricultural systems, such as the cooperation and gaming relationship between the decision makers at different levels, can be fully considered in the model. The developed model was then applied to identify the optimal land use patterns and BMP implementing levels for agricultural nonpoint source (NPS) pollution management in a subcatchment in the upper stream watershed of the Miyun Reservoir in north China. The results of the model showed that the desired optimal land use patterns and implementing levels of best management of practices (BMPs) would be obtained. It is the gaming result between the upper- and lower-level decision makers, when the allowable discharge amounts of NPS pollutants were limited. Moreover, results corresponding to different decision scenarios could provide a set of decision alternatives for the upper- and lower-level decision makers to identify the most appropriate management strategy. The model has a good applicability and can be effectively utilized for agricultural NPS pollution management.