Bilevel Linear Programming Research Articles

A robust bi-level programming model to enhance resilience in supply chain network considering government role, taxing, and raw sale: An iron ore case study

The mining industry plays a substantial role in any economy because of its enormous influence on other sections of a country. Despite the significant challenges in this industry, such as government regulations, taxing policies, raw material sales, and disruptions, there are rare studies that have considered mining supply chain network design. Thus, this paper proposes a mixed-integer linear bi-level programming model under operational and disruption risks to design a robust and resilient mining supply chain network. The government is considered as the leader that aims to decrease the sales of raw materials, boost the export of final products, and maximize its profit. Moreover, the closed-loop mining supply chain is introduced as the follower level that maximizes the total profit. In other words, this study presents a bi-level model that balances the export of raw materials and final products, for the first time. Additionally, a data envelopment analysis approach is applied to rank the candidate mine areas and reduce the model’s complexity. Finally, a real-world case study of the Iranian iron ore supply chain is applied to demonstrate the efficiency and applicability of the addressed model. The results indicate that the effect of raw material price on the upper level is 22% higher than the lower level under the same conditions. Moreover, the exchange rate has a great impact on the objective functions of both levels. Furthermore, Yazd province has great potential to be introduced as a steel pole in Iran.

Exploiting the polyhedral geometry of stochastic linear bilevel programming

Exploiting the polyhedral geometry of stochastic linear bilevel programming

How many clues to give? A bilevel formulation for the minimum Sudoku clue problem

It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.

A Problem based on Lagrange Relaxation Approach & Solving Optimization on Bi-level Model Programming Problem

Abstract: In this paper, a salutation method based on Lagrange relaxation for discrete-continuous bi-level problems, with binary variables in the leading problem, considering the optimistic approach in bi-level programming. Linearized taking advantage of the structure of the leading problem, an algorithm is solving two-dimensional bi-level linear programming problems, classification of constrains, algorithm removes all redundant constraints, cycling and solution of problem.

Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems

This article considers a bi-level linear programming with single valued trapezoidal fuzzy neutrosophic cost coefficient matrix and Pythagorean fuzzy parameters in the set of constraints both in the right and left sides. Based on the score functions of the neutrosophic numbers and Pythagorean fuzzy numbers, the model is changed to the corresponding crisp bi-level linear programming (BLP) problem. This problem is designated as a Pythagorean fuzzy bi-level linear programming (PFBLP) problem under neutrosophic environment. Kuhn-Tucker's conditions for optimality are necessary and sufficient for the existence of the optimal solution to a BLP problem. Using the suggested methodology, the problem is formulated as a single-objective non-linear programming problem with several variables and constraints. Two typical numerical examples are examined to illustrate the proposed approach.

War game problem considering the mobility of weapons and targets

War-Gaming is recognized as a valuable tool for commanders, leaders, and managers. Well-executed War-Games have delivered significant competitive advantages in numerous conflicts. The war-game confirmed the commanders’ knowledge of weapon systems and performance, as well as the time and space necessary to carry out battlefield maneuvers. One of the primary missions of each army on the battlefield is weapon target assignment. The weapon target assignment (WTA) is a critical problem to command to be solved in battlefield decisions. In a WTA problem, we should assign available weapons to determined targets appropriately to optimize the performance criteria. This study discusses a problem in relation to allocating and scheduling in WTA considering the mobility weapons and mobility targets. Bi-level linear programming problem is defined so that each level independently optimizes its own objective functions but is influenced by actions taken by another unit. To solve the under studied problem, three famous meta-heuristic algorithms including simulated annealing (SA), genetic algorithm (GA) and grey wolf optimizer (GWO) methods are proposed. Since the performance of meta-heuristic algorithms depends on setting the parameters, the Taguchi method has been used statistically for set parameters of the developed Algorithms. Performance evaluation of the presented algorithms is conducted through numerical experiments involving randomly generated test problems. To compare the results of proposed meta-heuristic algorithms, ANOVA and Tukey tests were used. The Computational results have shown that proposed GWO algorithm worked better than the SA and GA algorithms.

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.

Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications

We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds with a continuous wait-and-see decision after observing the leader’s action and revelation of uncertainty. We assume that only the information regarding the mean, covariance, and support is known. We formulate the problem as a distributionally robust (DR) two-stage problem. The pessimistic DR bilevel program is shown to be equivalent to a generic two-stage distributionally robust stochastic (nonlinear) program with both a random objective and random constraints under proper conditions of ambiguity sets. Under continuous distributions, using linear decision rule approaches, we construct upper bounds on the pessimistic DR bilevel program based on (1) a 0-1 semidefinite programming (SDP) approximation and (2) an exact 0-1 copositive programming reformulation. When the ambiguity set is restricted to discrete distributions, an exact 0-1 SDP reformulation is developed, and explicit construction of the worst-case distribution is derived. To further improve the computation of the proposed 0-1 SDPs, a cutting-plane framework is developed. Moreover, based on a mixed-integer linear programming approximation, another cutting-plane algorithm is proposed. Extensive numerical studies are conducted to demonstrate the effectiveness of the proposed approaches on a facility location problem. History: Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0168 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0168 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Investing in Wind Energy Using Bi-Level Linear Fractional Programming

Investing in wind energy is a tool to reduce greenhouse gas emissions without negatively impacting the environment to accelerate progress towards global net zero. The objective of this study is to present a methodology for efficiently solving the wind energy investment problem, which aims to identify an optimal wind farm placement and capacity based on fractional programming (FP). This study adopts a bi-level approach whereby a private price-taker investor seeks to maximize its profit at the upper level. Given the optimal placement and capacity of the wind farm, the lower level aims to optimize a fractional objective function defined as the ratio of total generation cost to total wind power output. To solve this problem, the Charnes-Cooper transformation is applied to reformulate the initial bi-level problem with a fractional objective function in the lower-level problem as a bi-level problem with a fractional objective function in the upper-level problem. Afterward, using the primal-dual formulation, a single-level linear FP model is created, which can be solved via a sequence of mixed-integer linear programming (MILP). The presented technique is implemented on the IEEE 118-bus power system, where the results show the model can achieve the best performance in terms of wind power output.

Setting closer targets based on non-dominated convex combinations of Pareto-efficient units: A bi-level linear programming approach in Data Envelopment Analysis

Data Envelopment Analysis (DEA) very often sets unrealistic targets, which require from the decision-making units (DMUs) a huge amount effort, perhaps non-assumable, for their achievement. For the identification of best practices in the benchmarking, this paper proposes considering as peers not only DEA efficient DMUs but also those that are Pareto efficient, and allowing for reference sets spanning convex combinations that are not dominated by observed DMUs. It is therefore an approach that somehow relaxes the convexity in DEA, and sets targets representing best practices in the sense that they define a course of action leading to results that are not worse than those of the real plans. A bi-level linear programming (BLP) DEA model is developed which finds the closest targets from a convex combination of the DMUs in a reference set satisfying such non-dominance requirement. This approach has proven to be successful in setting targets that require an effort significantly smaller than that needed to achieve the closest targets on the strong efficient frontier of both the DEA and the free disposal hull (FDH) technologies. In the empirical illustration, we have observed reductions in the order of 40 percentage points on average in the total effort required for their achievement, thus setting more realistically implementable directions for improving performance towards best practices than those provided by conventional DEA and FDH.

An analysis of distribution planning under a regulatory regime: An integrated framework

AbstractDistribution system planning is a multifaceted topic involving financial, regulatory, and system level analysis. The wide nature of the topic warrants a holistic study considering all aspects of analysis. The distribution utility is a natural monopoly that is subjected to utility regulation. The regulator can impact customer experience by strategically influencing the planning decisions of the utility. Hence, this paper reviews the existing utility regulation methods in the context of the distribution system and their efficacy in improving certain reliability and efficiency objectives. A two‐bus system is used to demonstrate the impact of classical models in alleviating reliability and efficiency issues through demand response. Further, a review is conducted on distribution system planning models without a regulatory regime, and suitable models for holistic analysis are identified. A two‐person complete information regulator and utility game with a comprehensive distribution system model at the lower level is proposed. A framework based on the Mixed Integer Bilevel Linear Program (MIBLP) is discussed to find the equilibrium point of the proposed game.

On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose a branch-and-cut algorithm for the problem class we study, and a cutting-plane method for the problem variant with only binary variables. We present an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our binary instances.

The follower optimality cuts for mixed integer linear bilevel programming problems

We study linear bilevel programming problems, where (some of) the leader and the follower variables are restricted to be integer. A discussion on the relationships between the optimistic and the pessimistic setting is presented, providing necessary and sufficient conditions for them to be equivalent. A new class of inequalities, the follower optimality cuts, is introduced. They are used to derive a single-level non-compact reformulation of a bilevel problem, both for the optimistic and for the pessimistic case. The same is done for a family of known inequalities, the no-good cuts, and a polyhedral comparison of the related formulations is carried out. Finally, for both the optimistic and the pessimistic approach, we present a branch-and-cut algorithm and discuss computational results.

Centroid for Solving Bi-Level Linear Fractional Programming Problems with Fuzzy Interval Coefficients by Utilizing MO-Technique

This paper presents an efficient and straightforward methodology with less computational complexities to title the bi-level objective linear fractional programming problem with fuzzy interval coefficients (BILOLFPP with FIC). To construct the methodology, the concept of mean technique is utilized to tackle the fuzzy numbers in addition to adding to α = [mean (ai) , mean (bi)] , i = 1, …, n, then. Accordingly, the fuzzy programming issue is converted into a single objective linear fractional programming problem (SOLFPP with FIC) by the utilize of weight function. The fuzzy technique has significant structural transform metamorphosis during the recent decades. Numerous to mention introduced have been undertaken to explanation fuzzy methodology for linear, non-linear programming issues. While, the previous finding that introduced have been conflicting, recent studies of competitive situations indicate that LFPP with fuzzy interval coefficients (LFPP with FIC) has an advantageous effect mostly on comparison situation. One of the suggestions which we found is interval approximations, closed interval approximation of sequential fuzzy number for resolving fuzzy number LFPP without changing it to a crisp issue. A new variant of modified simplex methodology is studied here just for resolving fuzzy number LFPP utilizing fuzzy arithmetic. Consequently, fuzzy representation of some important theories of fuzzy LFPP has been reproved. A fuzzy LFPP with FIC is worked out as numerical examples illustrate to the suggested methodology. On iterative processes, it decreases the overall processing time to explain, the modified simplex methodology for solving BILLFPP with FIC with out to crisp by taking numerical examples and compare with Nasseri, Verdegay and Mahmoudi methodology changing it to a crisp issue [9].

SOLVING THE BILEVEL LINEAR PROGRAMMING PROBLEM USING THE MONTE CARLO METHOD

In this paper we propose to use the "Monte Carlo" method for solving bilevel linear programming (the bilevel linear programming problem – BLP problem). In the BLP problem, each decision maker tries to optimize their own objective function without considering the objective of the other party, but the decision of each party affects the objective value of the other party as the decision space. The existing methods for solving the BLP problem can be grouped into four categories: a) methods based on vertices enumeration; b) methods based on Kuhn-Tuck conditions; c) the fuzzy approach; d) metaheuristics methods. Starting from Gnedenko's theorem, this paper uses the "Monte Carlo" method to determine an approximate solution for the BLP problem. The numerical example presents the performance of the proposed approach.

Integrated stochastic transmission network and wind farm investment considering maximum allowable capacity

• A stochastic model is presented for transmission and wind power investment. • The model considers demand and wind generation uncertainty. • The model considers maximum allowable capacity. • The model is formulated as a stochastic bilevel optimization problem. • Multi-parametric programming is used to solve the problem. This research presents a novel mathematical methodology for integrated transmission network and wind farm investment (ITWI) considering maximum allowable capacity (MAC). The joint problem of transmission and wind farm investment planning is carried out under a central planner perspective, where load and wind power uncertainties are managed using scenario-based stochastic programming. Distinct from the existing models in which the installation capacity of wind farms is restricted only by the available investment budget, the wind power capacity is limited by MAC considering strength measures. In this regard, the investment problem is addressed as a bi-level programming model where the upper level seeks to minimize the investment cost associated with transmission lines and wind farms plus the operation cost while the lower level determines the MAC of wind farms. The existence of integer decision variables in the lower level renders Karush-Kuhn-Tucker (KKT) conditions invalid. Thus, the multi-parametric programming (MPP) method is utilized to solve the mixed-integer bi-level linear programming (MIBLP). Numerical tests on two different power systems corroborate the efficiency of the proposed model. The comparable results demonstrate that ignoring the MAC of wind farms leads to inefficient solutions due to additional unnecessary investment in the wind energy sector.

Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach.

Solving bi-level linear fractional programming problems with fuzzy interval coefficients using game matrix

Solving bi-level linear fractional programming problems with fuzzy interval coefficients using game matrix

A method for solving interval type-2 triangular fuzzy bilevel linear programming problem

In this paper, we consider the bilevel linear programming problem (BLPP) where all the coefficients in the problem are interval type-2 triangular fuzzy numbers (IT2TFNs). First, we convert a BLPP with IT2TFN parameters to an interval BLPP. In the next step, we solve BLPPs and obtain optimal solution as an IT2TFN.

Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications

Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications