scholarly journals An Effective Branch-and-cut algorithm in Order to Solve the Mixed Integer Bi-level Programming

2017 ◽  
Vol 5 (1) ◽  
pp. 1 ◽  
Author(s):  
Arsalan Rahmani ◽  
Majid Yousefikhoshbakht
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Historical Perspective ◽  
Preference Function ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Branch And Bound Method ◽  
Computational Results ◽  
High Quality ◽  
Level Problem

<p>In this paper, a new branch-and-cut algorithm for mixed integer bi-level programming is proposed. For achieving this purpose, a historical perspective of the development of enumeration methods in the field of bi-level linear programming is considered. Then, we present some obstacles for using branch and bound method based on them, and an algorithm is developed to solve for mixed integer bi-level problem. Finally, we use a preference function to determine the choice of branching and specialized cuts in a branch and cut tree. Computational results are reported and compared favorably to those of previous methods and then implications discussed. The results show that not only the proposed algorithm can find high quality solutions for solving a number of the problems, but also it is competitive with other famous published algorithms.</p>

scholarly journals Closing the gap in linear bilevel optimization: a new valid primal-dual inequality

2020 ◽  
Author(s):  
Thomas Kleinert ◽  
Martine Labbé ◽  
Fränk Plein ◽  
Martin Schmidt
Keyword(s):  
Branch And Bound ◽  
Optimization Problems ◽  
Valid Inequalities ◽  
Bilevel Optimization ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Valid Inequality ◽  
Single Level ◽  
Primal Dual ◽  
Level Problem

Abstract Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.

scholarly journals Analysis of Mixed-Integer Linear Programming Formulations for a Fuel-Constrained Multiple Vehicle Routing Problem

2017 ◽  
Vol 05 (04) ◽  
pp. 197-207 ◽  
Author(s):  
Kaarthik Sundar ◽  
Saravanan Venkatachalam ◽  
Sivakumar Rathinam
Keyword(s):  
Linear Programming ◽  
Vehicle Routing ◽  
Integer Linear Programming ◽  
Vehicle Routing Problem ◽  
Mixed Integer Linear Programming ◽  
Optimal Solution ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Computational Results ◽  
Routing Problem

This paper addresses a fuel-constrained, multiple vehicle routing problem (FCMVRP) in the presence of multiple refueling stations. We are given a set of targets, a set of refueling stations, and a depot where [Formula: see text] vehicles are stationed. The vehicles are allowed to refuel at any refueling station, and the objective of the problem is to determine a route for each vehicle starting and terminating at the depot, such that each target is visited by at least one vehicle, the vehicles never run out of fuel while traversing their routes, and the total travel cost of all the routes is a minimum. We present four new mixed-integer linear programming (MILP) formulations for the problem. These formulations are compared both analytically and empirically, and a branch-and-cut algorithm is developed to compute an optimal solution. Extensive computational results on a large class of test instances that corroborate the effectiveness of the algorithm are also presented.

scholarly journals Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

2020 ◽  
pp. 233-250
Author(s):  
Ralph Bottesch ◽  
Max W. Haslbeck ◽  
Alban Reynaud ◽  
René Thiemann
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Decision Procedure ◽  
Simplex Algorithm ◽  
Upper Bounds ◽  
Mixed Integer ◽  
Small Solution ◽  
Code Generator ◽  
Integer Arithmetic ◽  
Termination Proofs

AbstractWe implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL. We further integrate this procedure into one application, namely into , a formally verified certifier to check untrusted termination proofs. This checking involves assertions of unsatisfiability of linear integer inequalities; previously, only a sufficient criterion for such checks was supported. To verify the soundness of the decision procedure, we first formalize the proof that every satisfiable set of linear integer inequalities also has a small solution, and give explicit upper bounds. To this end we mechanize several important theorems on linear programming, including statements on integrality and bounds. The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle’s code generator. It internally relies upon an adapted version of an existing verified incremental simplex algorithm.

A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

2022 ◽  
Author(s):  
Vahid Mahmoodian ◽  
Iman Dayarian ◽  
Payman Ghasemi Saghand ◽  
Yu Zhang ◽  
Hadi Charkhgard
Keyword(s):  
Objective Function ◽  
Branch And Bound ◽  
Capacity Allocation ◽  
Branch And Cut ◽  
Nash Bargaining ◽  
Mixed Integer ◽  
Bargaining Solution ◽  
Reliability Optimization ◽  
Criterion Space ◽  
Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

scholarly journals Algorithm and mathematical model of supply chain management for raw wood from the regions in Russia: Сomparison and analysis

2020 ◽  
Vol 15 (3) ◽  
pp. 385-404
Author(s):  
Rodion Sergeevich Rogulin ◽  
◽  
Lev Solomonovich Mazelis ◽  
Keyword(s):  
Genetic Algorithm ◽  
Supply Chain ◽  
Supply Chain Management ◽  
Branch And Bound ◽  
Raw Materials ◽  
Mixed Integer ◽  
Branch And Bound Method ◽  
Chain Management ◽  
Industrial Enterprises ◽  
Non Linear

Supply chain management is a burning issue for modern industrial enterprises. To handle this issue, non-linear stochastic models are successfully applied to find the reasonable and efficient solutions. A need to develop a unique method to find the solutions to supply chain management tasks defined as stochastic mixed-integer non-linear programming tasks is determined by the limitations imposed by the general models. The sum of the total raw procurement costs from the Commodity Exchange over the defined planning horizon is taken to be the target function of the unique model, while the binary variables which show whether a purchasing order is included into the procurement plan are used for optimization purposes. Some parameters of model’s limitations are stochastic and consider the uncertainty factor and risks in supplying the required raw materials to the manufacturing site. Branch-and-bound and genetic algorithms are applied at some steps in the developed heuristic algorithm. The algorithm and the model are tested at a major timber processing enterprise in Primorsky Area. Four types of processors over three planning horizons were applied to compare the efficiency of the proposed algorithm with partial application of the genetic algorithm or branch-and-bound method. The findings analysis shows that, unlike the genetic algorithm, the unique one is more stable in terms of uncertainty of the input parameters in comparison with the branch-and-bound method. It provides the solutions in the models with a great number of variables. The algorithm is shown to be universal enough for its further modification in solving more complicated problems of the same class, containing a significantly larger number of probabilistic parameters that describe other uncertainties in the supply of raw materials. Further research is seen to include the development of the proposed algorithm to increase the rate of convergence for its better efficiency.

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