Branch and Bound Method to Solve The Sum of Two Objective Functions

In this paper, the problem of sequencing a set of n jobs on single machine was considered to minimize multiple objectives function (MOF). The objective  is to find the optimal  solution (scheduling) for n independent jobs to minimize the objective function consists of a sum  of  weighted number  of  early  jobs  and  total  weighted of completion time. This problem is strongly NP-hard and to resolve it we derived two lower bounds (LB1, LB2) and heuristic method to get an upper bound which are used in root node of branch and bound tree. Some special cases  and dominance rule were proposed and proved. Results of extensive computational tests show that the proposed (BAB) algorithm effective in solving problems with up to (30)  jobs in time less than or equal to (30) minutes

Новый подход к решению задачи «Минимизация суммарного взвешенного опоздания при выполнении независимых заданий с директивными сроками одним прибором»

На основании исследования свойств данной задачи предложен новый подход к ее решению и разработанный на его основе эффективный приближенный алгоритм. Сформулированы условия, при выполнении которых оптимальное решение достигается за полиномиальное время. При невыполнении этих условий предлагаются правила отсечений, позволяющие существенно сократить время решения задачи. Данный алгоритм позволил получить точные решения для ряда задач с числом переменных существенно большим 50-ти. Оптимальные значения функционала, полученные данным алгоритмом для тестовых примеров, совпали со значениями функционала, рассчитанными другими известными методами.На підставі дослідження властивостей даної задачі запропоновано новий підхід до її розв’язання і розроблено на його основі ефективний наближений алгоритм. Сформульовано умови, при виконанні яких оптимальне рішення досягається за поліноміальний час. При невиконанні цих умов пропонуються правила відсікань, що дозволяють суттєво скоротити час розв’язання задачі. Даний алгоритм дозволив одержати точні рішення для ряду задач з числом змінних суттєво більшим 50-ти. Оптимальні значення функціонала, отримані даним алгоритмом для тестових прикладів, збіглися зі значеннями функціонала, отриманими відомими методами.Based on the problem properties research, a new approach to its solution is offered and an effective approximate algorithm based upon it is developed. Conditions for optimal solution to be found in polynomial time are formulated. When these conditions are not satisfied, the truncation rules that allow to decrease the time for solving the task essentially are offered. This algorithm allowed to get exact solutions for a series of problems with a number of variables essentially more than 50. Optimal values of the criterion function got by this algorithm for test examples were congruent with criterion function values got by known methods

A survey of algorithms for the single machine total weighted tardiness scheduling problem

AbstractThis paper surveys algorithms for the problem of scheduling jobs on a single machine to minimize total weighted tardiness. Special attention is given to two dynamic programming and four branch and bound algorithms. The dynamic programming algorithms both use the same recursion defined on sets of jobs, but they generate the sets in lexicographic order and cardinality order respectively. Two of the branch and bound algorithms use the quickly computed but possibly rather weak lower bounds obtained from linear and exponential functions of completion times problems. These algorithms rely heavily on dominance rules to restrict the search. The other two branch and bound algorithms use lower bounds obtained from the Lagrangean relaxation of machine capacity constraints and from dynamic programming state-space relaxation. They invest a substantial amount of computation time at each node of the search tree in an attempt to generate tight lower bounds and thereby generate only small search trees. A computational comparison of all these algorithms on problems with up to 50 jobs is given

A new lower bounding scheme for the total weighted tardiness problem

Cataloged from PDF version of article.We propose a new dominance rule that provides a sufficient condition for local optimality for the 1\\Sigma w(i)T(i) problem. We prove that if any sequence violates the proposed dominance rule, then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged. Therefore, it can be used in reducing the number of alternatives for finding the optimal solution in any exact approach. We introduce an algorithm based on the dominance rule, which is compared to a number of competing approaches for a set of randomly generated problems. We also test the impact of the dominance rule on different lower bounding schemes. Our computational results over 30,000 problems indicate that the amount of improvement is statistically significant for both upper and lower bounding schemes. (C) 1998 Elsevier Science Ltd. All rights reserved

Constraint Programming for Scheduling

Our goal is to introduce the constraint programming (CP) approach within the context of scheduling. We start with an introduction to CP and its distinct technical vocabulary. We then present and illustrate a general algorithm for solving a CP problem with a simple scheduling example. Next, we review several published studies where CP has been used in scheduling problems so as to provide a feel for its applicability. We discuss the advantages of CP in modeling and solving certain types of scheduling problems. We then provide an illustration of the use of a commercial CP tool (OPL Studio) in modeling and designing a solution procedure for a classic problem in scheduling. We conclude with our speculations about the future of scheduling research using this approach

Evaluación de funciones de utilidad de GRASP en la programación de producción para minimizar la tardanza total ponderada en una máquina

This paper considers the total weighted tardiness minimization in a single machineenvironment (1|| wj Tj ) a scheduling problem which has been proved to be NP-Hard. The solution approach uses the Greedy Randomized Adaptive Search Procedure (GRASP) meta-heuristic known for the quality of the solutions it can generate and the selective ability of its utility function during the construction phase. This work proposes and analyses three different utility functions for the problem in question. A statistical study showed significant differences between the mean values obtained from the proposed utility functions. The computational experiments were carried out using problems instances found in the OR-LIBRARY, and the outcome of these experiments were competitive solutions compared to the best known values of the instances involved. This work also shows the ease of developing GRASP methods for solving scheduling problems in a simple spreadsheet software such as MS Excel.Este artículo aborda la minimización de la tardanza total ponderada en un entorno de producción (1|| wj Tj ) que es conocido en complejidad como de tipo NP-hard. El enfoque de solución propuesto utiliza la metaheurística Greedy Randomized Adaptive Search Procedure (GRASP), la cual es reconocida por la correlación existente entre la calidad de las soluciones y la capacidad discriminante de la función de utilidad empleada en su fase constructiva. Este trabajo propone y analiza tres diferentes funciones de utilidad para este problema en particular. El desempeño de estas funciones se evaluó mediante un estudio estadístico que evidenció diferencias significativas en los valores medios de tardanza total ponderada, explicadas por el factor función de utilidad. La fase experimental se desarrolló usando instancias de la librería OR-LIBRARY y permitió obtener soluciones competitivas en calidad con respecto a los mejores valores conocidos para las instancias de este problema. Este trabajo ilustra la potencialidad de uso de métodos GRASP implementados en una hoja de cálculo normal para hallar soluciones a problemas de programación de la producción

Time-dependent traveling salesman problem and application to the tardiness problem in one-machine scheduling

Shortest path approach -- Subgradient optimization -- Branch and bound structures -- Dominance test -- TDTSP formulation -- Branch and bound -- The weighted tardiness problem