Bi-level Mixed Integer Linear Programming Research Articles

Core Pricing in Combinatorial Exchanges with Financially Constrained Buyers: Computational Hardness and Algorithmic Solutions

The computation of market equilibria is a fundamental and practically relevant problem. Although we know the computational complexity and the types of price functions necessary for combinatorial exchanges with quasilinear preferences, the respective literature does not consider financially constrained buyers. We show that computing market outcomes that respect budget constraints but are core stable is a problem in the second level of the polynomial hierarchy. Problems in this complexity class are rare, but ignoring budget constraints can lead to significant efficiency losses and instability. We introduce mixed integer bilevel linear programs (MIBLP) to compute core-stable market outcomes and provide effective column and constraint generation algorithms to solve these problems. Although full core stability quickly becomes intractable, we show that realistic problem sizes can actually be solved if the designer limits attention to deviations of small coalitions. This n-coalition stability is a practical approach to tame the computational complexity of the general problem and at the same time provides a reasonable level of stability for markets in the field where buyers have budget constraints.

A bilevel model formulation for solving a post-hurricane damaged timber management problem

Among other factors, the softwood industry is disrupted by a large number of hurricanes that landfall in the southern U.S. Lack of efficient tools to manage the wood market interactions in the post-hurricane situation increases timber salvage loss drastically. In this study, we propose a bi-level mixed-integer linear programming model that captures important features such as the hurricane’s degree, quality of damaged timbers, price-related issues, optimizes different critical decisions (e.g., purchasing, storage, and transportation decisions) of a post-hurricane damaged timber management problem. The overall goal is to provide an efficient decision-making tool for planning and recovering damaged timber to maximize its monetary value and mitigate its negative ecological impacts. Due to the complexity associated with solving the proposed model, we developed two exact solution methods, namely, the enhanced Benders decomposition and the Benders-based branch-and-cut algorithms, to efficiently solve the model in a reasonable timeframe. We use 15 coastal counties in southeast Mississippi to visualize and validate the algorithms’ performance. Key managerial insights are drawn on the sensitivity of a number of critical parameters, such as selling/purchasing prices offered by the landowners/mills, quality-level, and deterioration rate of the damaged timbers on their economic recovery following a natural catastrophe.

Best response-based individually look-ahead scheduling for natural gas and power systems

The natural gas system and the electric power system are becoming increasingly coupled in consideration of more interdependent components, e.g., natural gas-fired units, in the systems. Energy supplies for these two systems need to be dispatched economically from their own perspectives when these two systems belong to different owners. This paper investigates the best response-based individually look-ahead scheduling for the natural gas system and the electric power system. The optimal scheduling strategies for these two systems are the best responses of the opposite sides, and the best response of each side is achieved by an iterative best-response-search approach. Each best-response-search process is established as a bilevel mixed integer linear programming model, in which the upper level aims to minimize each side’s operational cost constrained by on–off states of generators/lines/gas sources, and the lower level aims to minimize the total energy consumption cost constrained by the operational constraints in consideration of the scheduling of generators/lines/gas sources of two systems in the upper level. To deal with solving difficulty caused by binary variables in the lower level, the original bilevel mixed integer linear programming model is converted into a constrained single-level formulation, which is solved by a decomposition algorithm based on an iterative column-and-constraint generation method. Two cases validate the model and the algorithm, and the costs with the best response-based model and the joint optimization-based model are compared.

The Role of Biowaste: A Multi-Objective Optimization Platform for Combined Heat, Power and Fuel

Biomass, bioenergy and negative emission technologies are inherent to the future design of energy systems. Urban clusters have a growing demand for fuel, heat and electricity, which is both a challenge and an opportunity for biomass-based technologies. Their deployment should meet demand, while minimizing environmental impact and staying cost-competitive. We develop a systematic approach for the design, evaluation and ranking of biomass-to-X production strategies under uncertain market conditions. We assemble state-of-the-art and innovative conversion technologies, based on feedstock, by-products and waste characteristics. Technical specifications, as well as economic and environmental aspects are estimated based on literature values and industry experts input. Embedded into a bi-level mixed-integer linear programming formulation, the framework identifies and assesses current and promising strategies, while establishing the most robust and resilient designs. The added value of this approach is the inclusion of sub-optimal routes which might outperform competing strategies under different market assumptions. The methodology is illustrated in the anaerobic digestion of food and green waste biomass used as a case study in the current Swiss market. By promoting a fair comparison between alternatives it highlights the benefits of energy integration and poly-generation in the energy transition, showing how biomass-based technologies can be deployed to achieve a more sustainable future.

Protection of large-scale smart grids against false data injection cyberattacks leading to blackouts

Power system is one of the most critical cyber–physical systems in a sustainable society, whose security and reliability can significantly impact other infrastructures. Therefore, security and resilience of smart grids are one of the most challenging research questions in protection of critical infrastructures. After cyberattacks on the U.S. power grid in 2018, the national security agency announced cyberattacks might cause a blackout in transmission systems. Therefore, it is critical to model cyberattack scenarios, propose a detection strategy to mitigate these threats in a realistic context, and study how the models can improve the defense of large-scale grids. In this paper, a bi-level mixed-integer linear programming (BMILP) model is developed to accurately model false data injections (FDIs) that are targeted to overflow multiple transmission lines and cause a blackout in large-scale grids. Compared to the existing research, the proposed model considers that attackers might have limited access to measurement buses and models attacks on targeted transmission lines without being detected by existing DC state-estimation. In addition, to protect the system against these attacks, it is proved that a detection framework based on recursive weighted least-square (WLS) state-estimation can detect the FDIs, unlike classical weighted least square (WLS) estimation that fails to identify stealthy FDIs. To validate the effectiveness of the proposed attack model and detection framework in practical grid infrastructures, the IEEE 118-bus benchmark and a 2000-bus synthetic grid replicating the electricity network of Texas, U.S. are used.

Improved [formula omitted]-space algorithm for min-max bilevel problems with an application to misinformation spread in social networks

In this work we propose an improvement of the x-space algorithm developed for solving a class of min–max bilevel optimization problems (Tang Y., Richard J.P.P., Smith J.C. (2016), A class of algorithms for mixed-integer bilevel min–max optimization. Journal of Global Optimization, 66(2), 225–262). In this setting, the leader of the upper level problem aims at restricting the follower’s decisions by minimizing an objective function, which the follower intends to maximize in the lower level problem by making decisions still available to her. The x-space algorithm solves upper and lower bound problems consecutively until convergence, and requires the dualization of an approximation of the follower’s problem in formulating the lower bound problem. We first reformulate the lower bound problem using the properties of an optimal solution to the original formulation, which makes the dualization step unnecessary. The reformulation makes possible the integration of a greedy covering heuristic into the solution scheme, which results in a considerable increase in the efficiency. The new algorithm referred to as the improved x-space algorithm is implemented and applied to a recent min–max bilevel optimization problem that arises in the context of reducing the misinformation spread in social networks. It is also assessed on the benchmark instances of two other bilevel problems: zero-one knapsack problem with interdiction and maximum clique problem with interdiction. Numerical results indicate that the performance of the new algorithm is superior to that of the original algorithm, and also compares favorably with a recent algorithm developed for mixed-integer bilevel linear programs.

Robustness improvement for cyber physical system based on an optimization model of interdependent constraints.

With the rapid development of information technology, traditional infrastructure networks have evolved into cyber physical systems (CPSs). However, this evolution has brought along with it cyber failures, in addition to physical failures, which can affect the safe and stable operation of the whole system. In light of this, in this paper, we propose an interdependence-constrained optimization model to improve the robustness of the cyber physical system. The proposed model includes not only the realistic physical law but also the interdependence between the physical network and the cyber network. However, this model is highly nonlinear and cannot be solved directly. Therefore, we transform the model into a bi-level mixed integer linear programming problem, which can be easily and effectively solved in polynomial time. We conduct the simulation based on standard Institute of Electrical and Electronics Engineers test cases and study the impact of the disaster level and coupling strength on the robustness of the whole system. The simulation results show that our proposed model can effectively improve the robustness of the cyber physical system. Moreover, we compare the performance of the power supply in different CPSs, which have different network structures of the cyber network. Our work can provide useful instructions for system operators to improve the robustness of CPSs after extreme events happen in them.

Bi-level mixed-integer linear programming algorithm for evaluating the impact of load-redistribution attacks on Volt-VAR optimization in high- and medium-voltage distribution systems

The increasing integration of distributed generators into distribution systems (DSs) poses significant challenges to the stable operation and cyber-physical security of integrated high- and medium-voltage distribution system (HMVDS), leaving Volt-VAR optimization (VVO) vulnerable to load-redistribution attacks. This study focused on VVO in an HMVDS subject to the load-redistribution attack. The VVO under load-redistribution attack was formulated as a single-leader-multi-follower bi-level mixed-integer linear programming (BMILP) model that captured the interaction between the attacker and DS operators. The attacker in the upper level aimed to maximize voltage deviations and load curtailment through load-redistribution attack, while the DS operators aimed to minimize the voltage deviations in the lower levels. A relaxation-based bi-level reformulation and decomposition algorithm was designed to solve the BMILP model through replacing each mixed-integer linear programming lower-level problem with multiple linear programming lower-level problems, and using nonnegative relaxation variables and penalty functions to ensure equivalence. Finally, case studies on a constructed HMVDS and a real-world HMVDS were conducted to illustrate the impact of load-redistribution attacks on VVO and verify the effectiveness of the proposed methods.

A Dynamic Robust Restoration Framework for Unbalanced Power Distribution Networks

The increasing penetration of photovoltaic (PV) generators has led to a reduction in the effectiveness of existing strategies for restoring the power distribution network. This article proposes a dynamic robust restoration (DRR) framework for the recovery of outage power considering uncertain PV outputs and demands. This framework is presented in two subsequent steps. In the first step, optimal decisions regarding the network configurations are generated. The second step then computes the modified dynamic Distflow equations and constraints under consideration of the worst operating conditions over the associated uncertainty sets with the aim of maximizing the recovery of outage power. The DRR model is formulated as a bilevel mixed-integer linear programming problem. A decomposition algorithm in a master–sub structure is used to solve the resulting system. The results of case studies show that the proposed DRR model yields obvious advantages over the existing deterministic dynamic restoration model in terms of robustness against system uncertainties.

Multiobjective economic and environmental optimization of global crude oil purchase and sale planning with noncooperative stakeholders

The petrochemical supply chain is a worldwide undertaking, where final products will often travel thousands of miles from oil well to gas station pump. Within the crude oil supply chain, various entities compete and attempt to maximize their profits by exploiting the demands and needs of the other companies within the supply chain. Each company or player in the crude oil industry has its own objective, and it will compete against other players trying to pursue their respective objectives. Due to the non-cooperative structure of the crude oil supply chain, the decisions that maximize profit often do not coincide with decisions that minimize environmental impact, as a reduction in environmental impact usually correlates with a reduction in profit. In this work, the crude oil supply chain from oil well to refinery is modelled as a mixed-integer bilevel linear program that accounts for conflicting objectives and interactions between different stakeholders. The composition, pricing, transportation distances, and environmental impacts of the different crude oils are taken into consideration in the model. In the bilevel problem, the crude oil producers aim to maximize their own profits from the sale of their crude oil, while the crude oil refiner has the dual objectives of both maximizing the profit made from the sale of distilled products to the market and minimizing the life cycle environmental impact of the refinery products, which is determined by the type of crude oil purchased by the refinery. The resulting model is then applied to two case studies, both based on a U.S. refinery purchasing oil from various crude oil-producing countries. Both case studies produce a set of pareto-optimal decisions for the refiner that display the inherent trade-offs between minimizing “cradle-to-gate” environmental impact and maximizing profit. At the trade-off point in the first case, a 4.4% decrease in profit leads to a 3.0% decrease in the kilograms of CO2 per megajoule of energy produced. Meanwhile, the trade-off point selected in the second case displays a 7.5% reduction in the total environmental impact while decreasing total profit by only 5.9%. Furthermore, the refiner’s profit at the trade-off point in the second case is $2.148 M, which is situated between the worst-case deterministic profit of $1.558 M and the best-case deterministic profit of $2.652 M.

A Multiobjective Mixed-Integer Bilevel Linear Programming Approach to Global Crude Oil Purchase and Sale with Noncooperative Stakeholders

The petrochemical industry is an integral component of the global today. The petrochemical supply chain itself is a worldwide undertaking, and the final products that are offered to consumers will often travel thousands of miles from oil well to gas station pump. In the crude oil supply chain, there are three main stages: crude oil extraction, oil separation and refinement, and final product delivery. Within this supply chain, various companies will compete and attempt to maximize their profits by exploiting the demands and needs of the other companies within the supply chain. Each company or player in the crude oil industry has its own objective, and it will compete against other companies trying to pursue their individual objectives. In this work, the crude oil supply chain from oil well to refinery is modelled as a mixed-integer bilevel linear program (MIBLP) which accounts for conflicting objectives and interactions between different stakeholders. The composition, pricing, transportation distances, and environmental impacts of the different oils are taken into consideration when constructing the model. The resulting model is then applied to a U.S.-based refinery that purchases oil from various countries surrounding the Persian Gulf. The case study produces a pareto-optimal curve that displays the trade-off between environmental impact and profit. One intermediate point on the curve is selected to show how a 4.4 % decrease in profit can lead to a 3.0 % decrease in total kg CO2-eq produced.

Trilateral Planning Model for Integrated Community Energy Systems and PV-Based Prosumers—A Bilevel Stochastic Programming Approach

This paper proposes a trilateral bilevel stochastic mixed integer bilevel linear programming (MIBLP) model to handle a joint master-slave operation-planning problem. This model considers the interactions among an integrated community energy system (ICES), prosumers, and the wholesale electricity market (WM). The ICES, in the upper level, makes a joint optimal photovoltaic (PV) and energy storage system plan taking into account the concurrent interactions with the WM and prosumers to maximize its profit. On the other hand, in the lower level, the prosumers aim at minimizing their bills via an optimal PV-package sizing while interacting with the ICES and the WM simultaneously. The strong duality theorem is used to recast this MIBLP model into an equivalent single-level model. Since the lower level is also an operation-planning problem, this recast results in a mixed integer nonlinear programming (MINLP) model that prevents finding a satisfactory solution. To remedy this issue, sensitivity analysis and a separation-based linearization technique are applied. Numerical results illustrate the effective performance of the proposed business model while providing valuable information for the ICES and consumers.

A robust bi-level optimization modelling approach for municipal solid waste management; a real case study of Iran

Abstract In recent years, reverse logistics have attracted many attentions due to many reasons, such as extension of new environmental laws, development of social responsibility, and economic interests. Reduction in natural resources and raw material reserves, increasing production costs, and the problems caused by an industrial waste landfill and consumer goods have also increased the focus on reverse logistics over the past decade. This paper proposes a municipal solid waste management network in order to minimize different costs. We develop a bi-level mixed integer linear programming model. The lower-level involves location and establishment costs of solid waste collection stations, and the upper-level includes the allocation of waste to the various centers. In order to deal with uncertainty in the amount of collected solid waste, we need to take a scenario-based robust optimization approach into account. The proposed model is evaluated using a case study in Babol, Mazandaran province, Iran. The results indicate that the collection stations are selected in regions that have less distance from their covered regions which leads to the optimal flow of the wastes/products. The proposed robust optimization approach ensures less sensitivity to model solutions for the provided scenarios. In addition, considering hierarchy in this model leads to a more appropriate analysis compared to when a multi-objective model is used.

A Multi-Parametric optimization approach for bilevel mixed-integer linear and quadratic programming problems

Optimization problems involving two decision makers at two different decision levels are referred to as bi-level programming problems. In this work, we present novel algorithms for the exact and global solution of two classes of bi-level programming problems, namely (i) bi-level mixed-integer linear programming problems (B-MILP) and (ii) bi-level mixed-integer convex quadratic programming problems (B-MIQP) containing both integer and bounded continuous variables at both optimization levels. Based on multi-parametric programming theory, the main idea is to recast the lower level problem as a multi-parametric programming problem, in which the optimization variables of the upper level problem are considered as bounded parameters for the lower level. The resulting exact multi-parametric mixed-integer linear or quadratic solutions are then substituted into the upper level problem, which can be solved as a set of single-level, independent, deterministic mixed-integer optimization problems. Extensions to problems including right-hand-side uncertainty on both lower and upper levels are also discussed. Finally, computational implementation and studies are presented through test problems.

Screening Hidden N-$k$ Line Contingencies in Smart Grids Using a Multi-Stage Model

This paper addresses an issue in the contingency screening that the probabilities of some high-order N- k contingencies involving multiple element failures may be significantly underestimated in the traditional screening processes. Considering the severe consequences of these hidden contingencies, this practice would impose a potential risk to the system operations. In this paper, we propose a multi-stage approach to screen out these N- k contingencies and reveal their hidden probabilities. The approach is formulated as a bi-level mixed integer linear programming problem and validated by simulations on the IEEE 14-bus and 118-bus systems.

Minimax-Regret Robust Defensive Strategy Against False Data Injection Attacks

This paper develops a multi-level game-theoretic framework for determining a cost-effective defensive strategy for protecting power systems from false data injection attacks like load redistribution attacks. First, a multi-level optimization problem considering interactions among defenders, attackers and operators is modeled based on the minimax-regret decision rule, which is then reformulated as an equivalent bi-level mixed-integer linear programming problem. Next, an implicit enumeration algorithm is developed to find a globally optimal solution to this complex bi-level problem. Several acceleration techniques are introduced to improve the computation efficiency of the proposed method for large-scale power system applications. Last, the proposed defensive strategy is validated by case studies based on a six-bus test system and a modified two-area RTS-96 system.

A projection-based reformulation and decomposition algorithm for global optimization of a class of mixed integer bilevel linear programs

We propose an extended variant of the reformulation and decomposition algorithm for solving a special class of mixed-integer bilevel linear programs (MIBLPs) where continuous and integer variables are involved in both upper- and lower-level problems. In particular, we consider MIBLPs with upper-level constraints that involve lower-level variables. We assume that the inducible region is nonempty and all variables are bounded. By using the reformulation and decomposition scheme, an MIBLP is first converted into its equivalent single-level formulation, then computed by a column-and-constraint generation based decomposition algorithm. The solution procedure is enhanced by a projection strategy that does not require the relatively complete response property. To ensure its performance, we prove that our new method converges to the global optimal solution in a finite number of iterations. A large-scale computational study on random instances and instances of hierarchical supply chain planning are presented to demonstrate the effectiveness of the algorithm.

A robust bi-level programming model to design a closed loop supply chain considering government collection’s policy

This study aims in providing a new approach regarding design of a closed loop supply chain network through emphasizing on the impact of the environmental government policies based on a bi-level mixed integer linear programming model. Government is considered as a leader in the first level and tends to set a collection rate policy which leads to collect more used products in order to ensure a minimum distribution ratio to satisfy a minimum demands. In the second level, private sector is considered as a follower and tries to maximize its profit by designing its own closed loop supply chain network according to the government used products collection policy. A heuristic algorithm and an adaptive genetic algorithm based on enumeration method are proposed and their performances are evaluated through computational experiences. The comparison among numerical examples reveals that there is an obvious conflict between the government and CLSC goals. Moreover, it shows that this conflict should be considered and elaborated in uncertain environment by applying Min-Max regret scenario based robust optimization approach. The results show the necessity of using robust bi-level programming in closed loop supply chain network design under the governmental legislative decisions as a leader-follower configuration.

Cyber-Attack on Overloading Multiple Lines: A Bilevel Mixed-Integer Linear Programming Model

In this letter, we propose a bilevel mixed-integer linear programming (BMILP) model for cyber-attack on multiple transmission lines in economic dispatch by injecting false load data. In contrast to the state of art research, the proposed model is able to overload a set of lines that are selected by the attacker in the framework of BMILP. Therefore, the proposed model is computationally efficient due to its bilevel linear programming property. The effectiveness of the proposed bilevel linear approach has been successfully validated on an IEEE 118-bus system.

Optimizing decentralized production\u2013distribution planning problem in a multi-period supply chain network under uncertainty

Decentralized supply chain management is found to be significantly relevant in today’s competitive markets. Production and distribution planning is posed as an important optimization problem in supply chain networks. Here, we propose a multi-period decentralized supply chain network model with uncertainty. The imprecision related to uncertain parameters like demand and price of the final product is appropriated with stochastic and fuzzy numbers. We provide mathematical formulation of the problem as a bi-level mixed integer linear programming model. Due to problem’s convolution, a structure to solve is developed that incorporates a novel heuristic algorithm based on Kth-best algorithm, fuzzy approach and chance constraint approach. Ultimately, a numerical example is constructed and worked through to demonstrate applicability of the optimization model. A sensitivity analysis is also made.