Bilevel Linear Programming Research Articles

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).

Data Envelopment Analysis with Lower Bound on Input to Measure Efficiency Performance of Department in Universitas Malikussaleh

DEA has become one of the most appropriate methods for comparing the various Decision-Making Units (DMU) associated with public services such as universities. There are two primary outputs that can be used to measure college performance, namely: the number of graduates and the number of publications. While the primary input of college efficiency measurement is the number of teaching staff and the number of students. The higher learning institution is Universitas Malikussaleh, located in Lhokseumawe, a city in the Aceh province of Indonesia. This paper develops a method to evaluate the efficiency of all departments in Universitas Malikussaleh using DEA with bounded input. The extreme dissimilarity between the weights often found in DEA applications. In this paper, we develop a new DEA model, which can be transformed into a particular case of the bi-level linear program to calculate the lower boundary of input from the pessimistic viewpoints based on the shortcomings of the existing approaches. In the case of having a single input, providing lower bounds for the input weights by imposing the conditions that it uses for the average input level of the DMU being assessed uses per unit of output. Accordingly, we present some essential differences inefficiency of those departments. Finally, we discuss the effort that should be made by these departments in order to become efficient

Estimating DLMP confidence intervals in distribution networks with AC power flow model and uncertain renewable generation

This study proposes a systematic approach for estimating intervals of distribution locational marginal price (DLMP) and corresponding confidence levels without requiring an exact probability distribution of renewable generation. The DLMP is evaluated based on a variant of the alternating current (AC) power flow model, namely, the branch flow model, in which network losses, bus voltage, and reactive power are taken into account. Based on the exactness of convex relaxation of optimal power flow, the authors developed a linear AC power flow model by applying second-order cone relaxation and global polyhedral approximation. Given a set of renewable power volatility, interval prediction of the DLMP is formulated as a bilevel linear program, which is solved via a mixed-integer linear program (MILP). Considering variances and unimodality of renewable power forecast error, a conservative estimation of the confidence level is expressed by a generalised Gauss inequality, which comes down to semidefinite programming. The proposed method is a natural extension of existing interval and probabilistic forecast methods, leveraging the proposed linear AC power flow model and inexact probability distributions of renewable power output, and could be promising in future distribution power markets. Case studies corroborate the effectiveness of the proposed method.

A two-level vertex-searching global algorithm framework for bilevel linear fractional programming problems

ABSTRACT We propose a new two-level vertex-searching algorithm framework that finds a global optimal solution to the continuous bilevel linear fractional programming problem over a compact polyhedron, in which both the upper and the lower objectives are linear fractional. Our solution method adopts the vertex-searching approach on the polyhedron, and the search space is determined by the set of candidates of optimal base to the lower level problem. In order to search base, a modified enumerative scheme, that is a new upper bound filter scheme inserted into the classical enumerative scheme, is proposed. The main solution procedure is designed on solving a sequence of upper and lower level mathematical programs; instead of a single-level problem reformulation approach, which is popularly and widely used in literature. An extension on general upper level objective functions such as quasiconvex/quasiconcave for the proposed vertex-searching approach is discussed. Numerical experiments show that our algorithm leads us to a global optimum. We conclude that our proposed algorithm framework has the simplest solution procedure and has potential efficiency advantages, which may reduce the complexity of enumerative schemes for medium or large-scale problems, while comparing with existing global algorithms such as the Kth-best algorithm (a two-level vertex-searching algorithm) and the single-level duality-based reformulation algorithm (a single-level vertex-searching algorithm).

Risk-Averse Models in Bilevel Stochastic Linear Programming

We consider a two-stage stochastic bilevel linear program where the leader contemplates the follower's reaction at the second stage optimistically. In this setting, the leader's objective function value can be modeled by a random variable, which we evaluate based on some law-invariant (quasi-)convex risk measure. After establishing Lipschitzian properties and existence results, we derive sufficient conditions for differentiability when the choice function is a Lipschitzian transformation of the expectation. This allows us to formulate first-order necessary optimality conditions for models involving certainty equivalents or expected disutilities. Moreover, a qualitative stability result under perturbation of the underlying probability distribution is presented. Finally, for finite discrete distributions, we reformulate the bilevel stochastic problems as standard bilevel problems and propose a regularization scheme for solving a deterministic bilevel programming problem.

ON THE PROBLEMS OF BILEVEL OPTIMIZATION UNDER RCPLD CONSTRAINT QUALIFICATIONS

Multilevel optimization problems often arise in various applications (in economics, ecology, power engineering and other areas) when modeling complex systems with a hierarchical structure associated with independent actions of subsystems. The difficulty of analyzing such complex systems requires first of all the study of bilevel models, the management of which would be an integral part of the analysis of more complex systems. In solving bilevel programming problems, an important role is played by the property of partial calmness, the presence of which allows us to reduce the bilevel problem to the classical nonlinear programming problem with a nonsmooth objective function. It is known that linear bilevel programming problems are partially stable. The proof of this property for more complex problems meets difficulties. In particular, our article shows the inaccuracy of some results in this area. The goal of the paper is to obtain some new results in the partial calmness of bilevel programming. In particular, new sufficient conditions for bilevel problems are proved. The results are obtained on the base of Lipschitz-like properties for multivalued mappings. In the paper we propose new sufficient conditions for partial calmness which are based on some modification of the known constraint qualification RCPLD which have been proposed by the researches Andreani, Haeser, Schuverdt and Silva.

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.

Bi-Level Multi-Objective Stochastic Linear Fractional Programming with General form of Distribution

This paper deals with the stochastic approach of bi-level multi-objective linear fractional programming problem.In this type of bi-level programming problem stochastic nature the right hand side resource vector is considered to follow a general form of distribution F (bi) = 1 − Bi^exp(Aih(bi))[13], which in itself includes many well known distributions such as Pareto distribution, Weibull distribution etc. After converting the problem into an equivalent deterministic form, each level of the problem is transformed into a single objective by using K-T conditions. Finally the problem is solved by Taylors series approach. A numerical example is also presented to illustrate how the proposed approach is utilized.

Bi-level programming problem in the supply chain and its solution algorithm

Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where the upper level of a hierarchy may have his objective function and decision space partly determined by other levels. In addition, each planner’s control instruments may allow him to influence the policies at other levels and thereby to improve his own objective function. As a tool, bi-level programming is applied for modeling decentralized decisions in which two decision makers make decisions successively. In this paper, we specifically address bi-level decision-making problems with budget constraint as an attractive feature in the context of enterprise-wide supply chain. We first describe the typical bi-level linear programming problem (BLLPP) and its optimal solution to the penalty function problem, and then, a cooperative decision-making problem in supply chain is modeled as BLLPP. A particle swarm optimization-based computational algorithm is designed to solve the problem, and the numerical example is presented to illustrate the proposed framework.

Multi‐objective bi‐level optimisation to design real‐time pricing for demand response programs in retail markets

Electric energy consumption is progressively increased requiring new investments in the network to meet. Demand response programs (DRPs) are of practical tools to defer and/or decrease these investments. In this study, a real-time DRP is proposed to promote consumer participation in the energy delivery. The proposed DRP is designed on behalf of the Load Serving Entity (LSE) and aims to maximise its profit. The LSE purchases energy from the up-stream wholesale market and offers to the consumers in the retail market. The consumers, with the objective of maximising the profit, have the choice to purchase energy from the retail market or the LSE. The problem is formulated as a mixed integer linear bi-level programming model wherein LSE and aggregators, on behalf of the consumers, act as the leader and followers of the problem. Also, to avoid creating a new peak profile in light load periods promoted by the designed LSE prices, the upper level problem is tailored as a multi-objective formulation wherein flattening load profile in addition to the maximising LSE profit is considered. The proposed model is implemented on a test case and the results show that the both LSE and consumers benefit is increased with a smoother load profile.

A multiple objective programming approach to linear bilevel multi-follower programming

In this paper, we investigate the relationship between a certain class of linear bilevel multifollower programming problems and multiple objective programming. We introduce two multiple objective linear programming problems with different objective functions and the same constraint region. We show that the extreme points of the set of efficient solutions for both problems are the same as those of the set of feasible solutions to the linear bilevel multi-follower programming problem. Based on this relationship, a new algorithm to find an optimal solution for the linear bilevel multifollower programming problem is developed. Some numerical examples are presented to show the feasibility of the proposed algorithm.

The rank pricing problem: Models and branch-and-cut algorithms

One of the main concerns in management and economic planning is to sell the right product to the right customer for the right price. Companies in retail and manufacturing employ pricing strategies to maximize their revenues. The Rank Pricing Problem considers a unit-demand model with unlimited supply and uniform budgets in which customers have a rank-buying behavior. Under these assumptions, the problem is first analyzed from the perspective of bilevel pricing models and formulated as a non linear bilevel program with multiple independent followers. We also present a direct non linear single level formulation bearing in mind the aim of the problem. Two different linearizations of the models are carried out and two families of valid inequalities are obtained which, embedded in the formulations by implementing a branch-and-cut algorithm, allow us to tighten the upper bound given by the linear relaxation of the models. We also study the polyhedral structure of the models, taking advantage of the fact that a subset of their constraints constitutes a special case of the Set Packing Problem, and characterize all the clique facets. Besides, we develop a preprocessing procedure to reduce the size of the instances. Finally, we show the efficiency of the formulations, the branch-and-cut algorithms and the preprocessing through extensive computational experiments.

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.

Severe Cyber Attack for Maximizing the Total Loadings of Large-Scale Attacked Branches

In this letter, a bilevel linear programming model is proposed to reveal the severe cyber-attack scheme in economic dispatch. By using this model, not only many branches including the targeted ones can be overloaded, but also the total loadings of all overloaded branches are maximized. This is a serious attack, since large power flows will be redistributed and it could further cause overloaded branches if those attacked branches are tripped due to overloading. Simulations on the IEEE 57-bus system show that the proposed model can successfully overload many branches including the targeted ones and maximize the total loadings of all overloaded branches.

The Backpropagation Artificial Neural Network Based on Elite Particle Swam Optimization Algorithm for Stochastic Linear Bilevel Programming Problem

For a class of stochastic linear bilevel programming problem, we firstly transform it into a deterministic linear bilevel covariance programming problem. Then, the deterministic bilevel covariance programming problem is solved by backpropagation artificial neural network based on elite particle swam optimization algorithm (BPANN-PSO). Finally, we perform the simulation experiments and the results show that the computational efficiency of the proposed algorithm has a potential upside compared with the classical algorithm.

Green Procurement Relationships Development under Carbon Emissions Regulations: A Bi-Level Programming Approach.

A multi-period Stackelberg game is adopted to study a green procurement relationship between manufacturers and suppliers in a supply chain. The manufacturers are considered as leaders, while the suppliers are modelled as followers in this Stackelberg game. Accordingly, a mixed binary linear bi-level programming model is developed to elaborate the game in consideration of carbon tax scheme. The upper level (the leader) aims at selecting a proper number of suitable suppliers to provide heterogeneous raw materials at the lowest operational cost. The objective of the lower level (the follower) is to find optimal purchasing quantities of raw materials. In addition, two lemmas are introduced to transform the mixed linear bi-level programming model into a single level linear programming model. The numerical example illustrates that: (1) the manufacturer prefers to adopt the multiple sourcing strategy due to the flexibility; (2) keeping stable supplies and large order volumes could effectively reduce carbon emissions for the suppliers and make the supply chain greener.

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 mixed-integer linear optimization model for a two-echelon agribusiness supply chain

We formulate and solve a mixed-integer linear program to model an agribusiness two-echelon supply chain between an agro-industrial producer and a wholesaler. The presented model is an equivalent reformulation of a bilevel linear program, using the well known KKT conditions and a disjunction approach on the complementary constraints. It includes a hierarchical decision structure that contemplates an agro-industrial producer in the upper level, who maximizes its profit by delineating the field into management zones and assigning to each one a specific crop rotation plan. In turn, in the lower level is the wholesaler who minimizes the penalization cost of lost sales that could experiment in order to satisfy the demand requirement according to the units provided by the producer. We present a case study to show the relevance of the proposed methodology.