Upper-level Decision Maker Research Articles

P-facility capacitated location problem with customer-equilibrium decisions: a recreational case study in Mazandaran province

Purpose This paper aims to propose a bi-level mixed integer linear location-allocation problem. The upper-level objective function is dedicated to minimizing the total distances covered by customers to meet the p-selected facilities and the fixed cost values for establishing these facilities. While in the lower level, a customer preference function evaluates the priority of customers in selecting facilities. Design/methodology/approach The solution approach to the proposed model uses the Karush–Kuhn–Tucker (KKT) optimality conditions to the lower-level problem where a set of p-selected facilities are introduced as the selection of the upper-level decision maker. The bi-level model reduces to a single-level model with some added binary variables. Findings Sensitivity analysis of the proposed bi-level model concerning variations of such different parameters as customers’ preferences and the number of selected facilities have been provided, using some numerical examples. Also, locating a recreational facility in Mazandaran province, Iran, has been provided to evaluate the reliability of the proposed model and efficiency of the solution approach, as well. Originality/value To the best of the authors’ knowledge, this paper is original and its findings are not available elsewhere.

Determining an Optimal Government Subsidy Scheme for Shared Parking Management via a Bi-Level Programming Approach

A bi-level programming model is formulated to determine a government subsidy scheme for shared parking management. The government, as the upper-level decision-maker, seeks to maximize the social benefit by providing subsidy to the shared parking platform, incentivizing the platform to rent more shared parking slots, which can increase the utilization rate of idle parking slots and reduce the curbside parking cruising time of parking demanders. At the lower-level formulation, the shared parking platform, as a reseller, not only matches shared parking slot demanders but also determines which shared parking slots to rent, which is approached by a two-sided decision-making process. A “one-to-many” matching principle is adopted for the platform to maximize its revenue. A modified genetic algorithm is designed to solve the proposed model. Results indicate that the government subsidy has a positive impact on stimulating the shared parking market. Specifically, from the perspective of the government, the subsidy can increase the number of matched shared parking demanders, improve the supply of parking slots, and reduce curbside parking cruising time. From the perspective of the shared parking platform, the subsidy can lead to higher revenue.

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.

A simulation-based multi-objective two-level optimization decision-making approach for supporting balanced irrigation water management

In this study, a simulation-based multi-objective two-level optimization decision-making approach is developed for optimal irrigation water allocation, improving irrigation water productivity and controlling regional accumulated salts. Techniques of multi-objective programming, two-level programming and simulation model of water and salt physical movement process are incorporated into the modeling framework. The simulated processes are parameterized and calibrated with field experimental data while the optimization model is used to generate optimal solutions though predefined objectives and the associated constraints. This model is applied to a case study on irrigation water allocation in the Jiefangzha Irrigation Subarea in Hetao Irrigation District, Northwest China. Firstly, the study area is delineated into several homogeneous irrigation decision-making units (IDMUs) for better characterizing their spatial variability because it’s spatially heterogeneous. Afterwards, decomposition-coordination algorithm is introduced to solve such an integrated simulation-based multi-objective two-level optimization model. Finally, optimal solutions of irrigation water allocation for different crops during crop growth periods in different IDMUs can be obtained for supporting sustainable strategies of irrigation. The results can achieve balanced tradeoffs between different stakeholders (i.e., the upper-level decision-makers and the lower-level farmers) and between conflicting economic objectives and environmental objectives. Moreover, optimal solutions have a slight increase in economic returns over the baseline scenario (i.e., status quo), but the irrigation water productivity is increased by nearly 60% due to less irrigation water used. Regional accumulated salts can be controlled because the soil salinity is constrained within the predetermined salt accumulation constraint. Therefore, these findings can provide evidence for efficient use of irrigation water resources and further support of sustainable irrigated agriculture.

Neutrosophic programming approach to multilevel decision-making model for supplier selection problem in a fuzzy situation

In supply chain management, the selection of suppliers is vital and plays a valuable role in the performance of organizations. A hierarchical structure, with different levels in the selection of suppliers, is employed, wherein sequential decisions are made from the highest to the lowest level. Decision variables, called controlling factors, are divided into several categories. In the decision-making process, often because of the lack of confidence or uncertainty, It becomes challenging for decision-makers to give explicit/crisp values to any parameter, resulting in uncertainty in the problem. In this paper, we address a multi-level supplier selection problem with fuzzy supply and demand. To avoid decision conflicts, superior or upper-level decision-makers give tolerances that could be used as a possible relaxation. Thus, the problem is employed with fuzzy constraints. Based on a neutrosophic decision set, the neutrosophic compromise programming approach (NCPA) is used as a solution technique with the idea of an indeterminacy degree as well as different objectives for membership and non-membership degrees. Membership functions (Linear-type) are used to develop satisfactory solutions by fuzzily describing objective functions and controlling factors. A numerical illustration is provided to demonstrate the validity and appropriateness of NCPA.

A goal programming strategy for bi-level decentralised multi-objective linear programming problem with neutrosophic numbers

This paper develops a goal programming (GP) algorithm to evaluate bi-level decentralised multi-objective linear programming problem (BLDMOLPP) in neutrosophic number (NN) environment. In a BLDMOLPP, a single decision maker (DM) is present at the upper level and multiple decision makers at the lower level. Here the parameters of the problem are considered to be NNs in the form of [P+QI], where P and Q are real numbers and indeterminacy is represented through the symbol I. I is expressed in the form of a real interval as agreed upon by the DMs. The BLDMOLPP with NNs then gets converted into an interval BLDMOLPP. Using interval programming, the target intervals for the objective functions are identified and subsequently, the goal achievement functions are constructed. The upper level DM provides some possible relaxation on the decision variables under his/her control to cooperate with the lower level DMs to attain a compromise optimal solution. Thereafter, goal programming (GP) models are formulated by minimising the deviational variables and thereby obtaining the most satisfactory solution for all DMs. Finally, a numerical example demonstrates the feasibility and simplicity of the proposed strategy.

A 5G Hubs Location Hierarchized Problem that Balances the Connection of the Users

In this paper, a telecommunication network situation is studied. Particularly, the problem considers an upper-level decision maker that enables different types of hubs in the network and another one that connects the users to the enabled hubs. The objective of the upper-level decision maker is to balance the number of users connected to the enabled hubs. However, the connection of users is performed by another decision maker with lower hierarchy, the one associated to the lower-level problem. Hence, decisions at the upper level are given as parameters to the lower level. So that, the lower level is solved to obtain feasible bi-level solutions. In this case, the rational reaction of the decision-maker assigned to the lower level is obtained. Due to the complexity of this problem, an evolutionary algorithm is proposed to obtain a good approximation. Numerical experimentation shows the efficiency and robustness of the proposed solution scheme. Based on the results, it can be concluded that the proposed algorithm provides good alternatives for the bi-level decision-making process. Also, some managerial insights are given regarding the different types of enabled hubs and the connection of the users.

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 Bilevel Multiobjective Model for Optimal Allocation of Water Resources in the Punjab Province of Pakistan

In this study, a simple but efficient bilevel multiobjective model (BLMOM) has been formulated for the optimal allocation of available water (AW) among competing water users. Upper-level decision makers (DMs), being the leader in the hierarchy (i.e., river authorities), allocate AW to lower-level DMs (i.e., canal authorities) based on equity and stability, whereas lower-level DMs allocate AW among competing users based on two single- and one multiobjective functions. The first objective function (OF1) maximizes the satisfaction rate (SR) of various water users, whereas the second objective function (OF2) maximizes the net economic benefits (NEB). The multiobjective function (OF12) maximizes the combined effect of the first two single objectives. The multiobjective function has been solved by using the simultaneous compromise constraint (SICCON) technique which creates a compromise between single-objective functions. The model was applied at Taunsa Barrage, Pakistan, for the optimal allocation of AW. Various scenarios were analyzed by varying priorities assigned to different objective functions and water users to evaluate the model applicability under various conditions. When OF1 was considered, maximum SR of 61% was achieved. In case of priority given to OF2, maximum NEB of 77 million USD was attained. However, in the case of OF12, both SR and NEB were maximized, whereas wheat, cotton, sugarcane, rice, onion and sunflower water users attained NEB of 14.36, 5.03, 22.96, 27.85, 1.34 and 0.13 million USD, respectively, with overall NEB of 72 million USD against satisfaction rate of 52%.

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.

Fuzzy programming approach to Bi-level linear programming problems

In this study, we discussed a fuzzy programming approach to bi-level linear programming problems and their application. Bi-level linear programming is characterized as mathematical programming to solve decentralized problems with two decision-makers in the hierarchal organization. They become more important for the contemporary decentralized organization where each unit seeks to optimize its own objective. In addition to this, we have considered Bi-Level Linear Programming (BLPP) and applied the Fuzzy Mathematical Programming (FMP) approach to get the solution of the system. We have suggested the FMP method for the minimization of the objectives in terms of the linear membership functions. FMP is a supervised search procedure (supervised by the upper Decision Maker (DM)). The upper-level decision-maker provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).

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.

Modified Evolutionary Algorithm and Chaotic Search for Bilevel Programming Problems

Bi-level programming problem (BLPP) is an optimization problem consists of two interconnected hierarchical optimization problems. Solving BLPP is one of the hardest tasks facing the optimization community. This paper proposes a modified genetic algorithm and a chaotic search to solve BLPP. Firstly, the proposed algorithm solves the upper-level problem using a modified genetic algorithm. The genetic algorithm has modified with a new selection technique. The new selection technique helps the upper-level decision-maker to take an appropriate decision in anticipation of a lower level’s reaction. It distinguishes the proposed algorithm with a very small number of solving the lower-level problem, enhances the algorithm performance and fasts convergence to the solution. Secondly, a local search based on chaos theory has applied around the modified genetic algorithm solution. Chaotic local search enables the algorithm to escape from local solutions and increase convergence to the global solution. The proposed algorithm has evaluated on forty different test problems to show the proposed algorithm effectiveness. The results have analyzed to illustrate the new selection technique effect and the chaotic search effect on the algorithm performance. A comparison between the proposed algorithm results and other state-of-the-art algorithms results has introduced to show the proposed algorithm superiority.

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.

A robust decision support leader-follower framework for design of contamination warning system in water distribution network

In recent years, several models have been proposed to inoculate Water Distribution Systems (WDS) against impacts of accidental and/or intentional compromised water quality through optimal deployment of online monitoring sensors in the network, which is referred to as Contamination Warning Systems (CWS). Translating such modeling efforts to real-world practice is, however, a challenge as different involved parties may pursue conflicting goals and modeling-based recommendations may not justify all stakeholders' criteria. It is, hence, pivotal to develop conflict resolution methodologies to support engagement of different stakeholders in securing a safe water distribution. The decision making structure for CWS design is often of top-down nature, with the upper level decision maker concerned mainly about public safety and lower level stakeholders concerned about operational costs. In this study, a decision support framework based on Leader-Follower Game is proposed, given different power levels. Leader's objectives are focused on the CWS robustness, while followers have conflicting interests that are in turn resolved via Nash Bargaining method. Lamerd WDS (Fars, Iran) is selected to assess the proposed model's performance. The results show the proposed objective and parsimonious model provides a robust solution that complies with the leader's criteria and maximizes the followers' satisfaction. The proposed decision support system helps govern WDSs in a resilient and safe manner and warrants practical implementation of modeling-based security assurance policies to provide sustainable service to the society.

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.

A Branch and Bound-Based Algorithm for the Weak Linear Bilevel Programming Problems

Most real-world optimization problems are hierarchical involving non-cooperative objectives. Many of these problems can be formulated in terms of the first (upper level) objective function being minimized over the solution set mapping of the second (lower level) optimization problem. Often the upper level decision maker is risk-averse. The resulting class of problem is named weak bilevel programming problem. This paper presents a new algorithm which embeds a penalty function method into a branch and bound algorithm to deal with a weak linear bilevel programming problem. An example illustrates the feasibility of the proposed algorithm.