# Benders' decomposition method

## Top PDF Benders' decomposition method: ### A Benders decomposition algorithm for multi-factory scheduling problem with batch delivery

This section contains the computational experiments for evaluation of the proposed Benders decomposition method for the mentioned problem. The algorithm is coded in commercial software GAMS and executed on a PC with Intel Core 2 Duo and 2 GB of RAM memory. To the best knowledge of the authors, this paper is a novel research in the scheduling eld; therefore, there is lack of benchmark or competent study on this problem for evaluating the method. Thus random datasets for dierent sizes of the problem are generated for assessing and checking eciency of the method against an adapted B&B presented in . Features of the generated test problems are described in the following; then, the computational experiments are presented. ### Benders’ decomposition algorithm to solve bi-level bi-objective scheduling of aircrafts and gate assignment under uncertainty

Management and scheduling of flights and assignment of gates to aircraft play a significant role in improving the procedure of the airport, due to the growing number of flights, decreasing the flight times. This research addresses assigning and scheduling of runways and gates in the main airport simultaneously. Moreover, this research considers the unavailability of runway’s constraint and the uncertain parameters relating to both areas of runway and gate assignment. The proposed model is formulated as a comprehensive bi-level bi-objective problem.The leader’s objective function minimizes the total waiting time for runways and gates for all aircrafts based on their importance coefficient. Meanwhile, the total distance traveled by all passengers in the airport terminal is minimized by a follower’s objective function. To solve the proposed model, the decomposition approach based on Bendersdecomposition method is applied. Empirical data are used to show the validation and application of our model. A comparison shows the effectiveness of the proposed model and its significant impact on cost decreasing. ### Reliable Network Design Problem under Node Failure with Benders Decomposition

The design of telecommunication network with capacity constraints of links, routers and ports of routers is con- sidered in this paper. Specially, we limit each demand flow traversed through a pre-specified maximal number of links (called hops) under node failure scenarios in IP layer network. Such a design must be the most cost-effec- tive and ensure that feasible flows continue to exist even when any relay node of the network fails. We propose a reliable mixed-integer programming (MIP) model with multi-scenario constraints to optimally design a mini- mum-cost survivable IP network that continues to support a good communication under any node failure scena- rio. Then we transform the MIP model into many single scenario models, that is, simplified MIPs, nonlinear programming (NLP) models and MIP models under Benders decomposition Then we transform the MIP model into many single scenario models, that is, simplified MIPs, nonlinear programming (NLP) models and MIP mod- els under Benders decomposition. Three heuristic methods are proposed to solve these models including branch-and-bound algorithm, global algorithm for NLP, and heuristic algorithm based on benders decomposi- tion. We mainly study the application of Benders decomposition method, where dual model and bounding pro- cedures are given for each MIP model under Benders decomposition at each scenario. The results of our compu- tational experiments validate the effectiveness of the proposed models and algorithms. ### An Improved Decomposition Technique for Solving Integer Linear Programming Problems

In this paper a new technique had presented for solving ILPPs. The idea of benders decomposition method had been used for developing this system. By using AMPL one computer code had been developed to solve ILPPs easily. Moreover, the graphical representations had been illustrated to show the Convergence of the master and the sub-problem values using MATHEMATICA. This improved technique will be extended to solve large scale Mixed-Integer Programming. Finally it is noted that the presented decomposition algorithm can be used as an effective tool for solving ILPPs to avoid the laborious calculations using row generation. ### A new mathematical model for intensity matrix decomposition using multileaf collimator

The details of the presented algorithm based on the developed model will be explained in this section. The matrix’s rows are first organized based on decreasing order ofTNMU(Engel, 2005) complexity C(A) and the most complex row, 𝑖𝑖 * (the first row in rank) will be selected. The minimum decomposition cardinality of each row will be calculated through the Bendersdecomposition method and so the lower bound of the cardinality of the total decomposition of the matrix ( 𝑘𝑘 ) will be obtained. Then through limiting the number of segment sets of the presented mathematical model to the lower bound (S:= 𝑘𝑘 ), the number of the rows to one ( 𝑀𝑀 = 1 ), and removing the objective function it would be possible to simultaneously examine the feasibility of the monitor units sequence 𝑏𝑏 𝑖𝑖 , 𝑠𝑠 ∈ {1, … , 𝑆𝑆}, which are generated through two related methods. In such manner that the sequences of monitor units 𝑏𝑏� 𝑖𝑖 are entered as parameter to a linear system of inequalities including (3-5) and the following constraint and its feasibility will be examined for all the rows one by one by tracking the predetermined manner: ### Kamal transform integrated with Adomian decomposition method for solving predator prey systems

In this paper, we use Kamal transform and Decomposition Method (KTADM), which is integrated of transform and Decomposition Method for solving the approximate numerical solutions of Prey systems. We can easily decompose the nonlinear terms by the help of special kind of Adomian polynomials. This technique provides a sequence of functions, which converges fast to the accurate solution of the problems. Finally, numerical examples prepared to illustrate and given to show the effectiveness and applicability of this method in solving these kind of systems. ### Vol 2014

Problems concerning system of linear equations are often occurred in engineering and sciences. Many researchers have developed several methods for solving the problems. Noor et al.  had used decom- position method for solving the system (1). Babolian et al. have used Adomian decomposition method to derive an iterative method which is similar to the Jacobi iteration. Allahviranloo  used Ado- mian decomposition method for fuzzy system. Keramati  and Liu  have used homotopy perturbation method to derive some iterative methods for solving the system. Improving Newton-Raphson method for non linear equations by modified Adomian decomposition method was discussed in . In this paper, we evoke a strong procedural con- dition for the decomposition method in  which is sufficient for the existence of system (1). ### Fingering Phenomenon Arising in Double Phase Flow Through Porous Media By Using Adomain Decomposition Method Patel Anilkumar J. 1, Khatri Ronak G.2 , Dr. Bhathawala P. H. 3

. Roessler J. and Husner H., (1997), "Numerical solution of 1+2 dimensional Fisher's equation by finite elements and Galerkin method", J. Math. Comput. Modeling, Vol. 25, No. 9, p. 57-67. . Wescot B.L. and Rizwan-uddin, (2001), "An ### Adomian Decomposition Study of Unidimensional Flow in Unsaturated Porous Media Khatri Ronak G. *1 , Patel Anilkumar. J. 1Dr. Bhathawala P. H.,2

. Roessler J. and Husner H., (1997), "Numerical solution of 1+2 dimensional Fisher's equation by finite elements and Galerkin method", J. Math. Comput. Modeling, Vol. 25, No. 9, p. 57-67. . Wescot B.L. and Rizwan-uddin, (2001), "An ### Parallel algorithms for two-stage stochastic optimization

This chapter presents our parallel algorithms for scalable stochastic integer optimization. Specifically, we are interested in problems with integer solutions, and hence, in BnB ap- proaches. Although BnB is a well-studied method, there has been little prior work in parallelizing or scaling two-stage, stochastic Integer Programs (IPs). Unlike typical, iter- ative scientific applications, we encounter some very interesting characteristics that make it challenging to realize a scalable design. The total amount of computation required to find optima is not constant across multiple runs. This challenges traditional thinking about scalability and parallel efficiency. It also implies that reducing idle time does not imply quicker runs. The sequential grains of computation are quite coarse. They display a wide variation and unpredictability in sizes. The structure of the branch-and-bound search tree is sensitive to several factors, any of which can cause significantly alter the search tree causing longer times to solution. We explore the causes for this fragility and evaluate the trade-offs between scalability and repeatability. ### Application of Modified Benders Decomposition to Single Stage Multi Commodity Multi Period Warehouse Location Problem: An Empirical Investigation

Sharma  solved the real life fertilizer distribution system problem using BendersDecomposition formu- lating it as a mixed 0 - 1 integer linear program (MILP). The concept of strong and weak formulations as given by Sharma & Verma  also gave the hybrid formulations for the same problem. The proposal of multi-commod- ity problem formulation for given by Elson . Capacitated plant location problem is also of interest to many researchers and many heuristic and exact approaches have been given for the same. Sharma & Agrawal  used vertical decomposition to solve the two staged capacitated warehouse location problem (TSCWLP) which was attempted by Keskin and Uster  using heuristic based scatter search algorithm. Some new strong formu- lations of the two staged location problem were shown by Sharma & Namdeo . Sharma et al.  have giv- en the strong formulations of SSSPMCWLP and shown the effectiveness of strong over weak in terms of num- ber of iterations using Branch and Bound solutions. They also developed the hybrid formulations for it to im- prove the computational time. ### A comparison of numerical methods for solving the bratu and bratu type problems

Keywords: Bratu problem, Bratu-Type problems, standard Adomian decomposition method, modified Adomian decomposition method, shooting method, finite difference method.... rvIasalah Jenis-[r] ### Numerical Solution of Sawada-Kotera equation by using Iterative Methods

In this paper, the Sawada-Kotera equation is solved by using the Adomian’s decomposition method, modiﬁed Adomian’s decomposition method, variational iteration method, mod- iﬁed variational iteration method, homotopy perturbation method, modiﬁed homotopy perturbation method and homotopy analysis method. The approximate solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods. ### A study on Degasperis - Procesi equation by iterative methods

The Degasperis-Procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa- Holm equation. In this paper, the Degasperis-Procesi equation is solved by using the Adomian’s decomposition method , modified Adomian’s decomposition method , variational iteration method , modified variational iteration method, homotopy perturbation method, modified homotopy perturbation method and homotopy analysis method. The existence and uniqueness of the solution and convergence of the proposed methods are proved in details. Finally an example shows the accuracy of these methods. ### Adomian decomposition method for two-dimensional nonlinear fredholm integral equation of the second kind

In  this method gives a better result when applied to the Volterra integral equation and it is shown in . In this dissertation, we try to solve the method to solving two-dimensional Fredholm integral equation (FIE) of the second kind in Xie.W and Lin.F , stated that ### The use of iterative methods for solving Black-Scholes equation

In this paper, the Black-Scholes equation is solved by using the Adomian’s decomposition method , modified Adomian’s decomposition method , variational iteration method , modified variational iteration method, ho- motopy perturbation method, modified homotopy perturbation method and homotopy analysis method. The existence and uniqueness of the solution and convergence of the proposed methods are proved in details. A numerical example is studied to demonstrate the accuracy of the presented methods. ### Convergence of iterative methods applied to Burgers-Huxley equation

In this paper, a Burgers-Huxley equation is solved by using variety of methods: the Ado- mian’s decomposition method , modiﬁed Adomian’s decomposition method , variational iteration method , modiﬁed variational iteration method, homotopy perturbation method, modiﬁed homotopy perturbation method and homotopy analysis method. The approxi- mate solution of this equation is calculated in the form of series whose components are computed by applying a recursive relation. Consequently, the existence and uniqueness of the solution and the convergence of the proposed methods are proved. Furthermore, a numerical example is studied to demonstrate the accuracy of the presented methods. ### Application of Iterative Methods for Solving General Riccati Equation

In this paper, the general Riccati diﬀerential equation is solved by using the Adomian’s decomposition method (ADM) , modiﬁed Adomian’s decomposition method (MADM), variational iteration method (VIM), modiﬁed variational iteration method (MVIM), ho- motopy perturbation method (HPM), modiﬁed homotopy perturbation method (MHPM) and homotopy analysis method (HAM). The existence and uniqueness of the solution and convergence of the proposed methods are proved in details. A numerical example is stud- ied to demonstrate the accuracy of the presented methods. ### FPGA Implementation OF Iterative Log Multiplier Using Operand Decomposition For Image Processing Application

Basically in a large number of DSP applications, speed is the significant criteria compared to accuracy. In these cases, the most suitable multiplier is the truncated and Logarithmic multipliers. In case of truncated multipliers, the less important partial products are left out and recompense is provided. So it partly compensates for left out terms. In the Logarithmic multipliers rounding of products can be done. It is alternatively used for fixed point number and floating point number. This translates multiplication into addition and division into subtraction. The main goal of Log Multiplier is to lessen errors with power utilization. One of the methods to calculate log and antilog is to use LUT method, but it requires more area to store the log and antilog values. The Mitchell log multiplier uses only some shift operations for log and antilog calculations .So it require fewer hardware overhead and results high speed with less power utilization. 