Minimizing weighted total earliness, total tardiness and setup costs

The paper considers a (static) portfolio system that satisfies adding-up contraints and the gross substitution theorem. The paper shows the relationship of the two conditions to the weak dominant diagonal property of the matrix of interest rate elasticities. This enables to investigate the impact of simultaneous changes in interest rates on the asset demands.

Flow shop scheduling with earliness, tardiness and intermediate inventory holding costs

We consider the problem of scheduling customer orders in a flow shop with the objective of minimizing the sum of tardiness, earliness (finished goods inventory holding) and intermediate (work-in-process) inventory holding costs. We formulate this problem as an integer program, and based on approximate solutions to two di erent, but closely related, Dantzig-Wolfe reformulations, we develop heuristics to minimize the total cost. We exploit the duality between Dantzig-Wolfe reformulation and Lagrangian relaxation to enhance our heuristics. This combined approach enables us to develop two di erent lower bounds on the optimal integer solution, together with intuitive approaches for obtaining near-optimal feasible integer solutions. To the best of our knowledge, this is the first paper that applies column generation to a scheduling problem with di erent types of strongly NP-hard pricing problems which are solved heuristically. The computational study demonstrates that our algorithms have a significant speed advantage over alternate methods, yield good lower bounds, and generate near-optimal feasible integer solutions for problem instances with many machines and a realistically large number of jobs

The single machine earliness and tardiness scheduling problem: lower bounds and a branch-and-bound algorithm

This paper addresses the single machine scheduling problem with a common due date aiming to minimize earliness and tardiness penalties. Due to its complexity, most of the previous studies in the literature deal with this problem using heuristics and metaheuristics approaches. With the intention of contributing to the study of this problem, a branch-and-bound algorithm is proposed. Lower bounds and pruning rules that exploit properties of the problem are introduced. The proposed approach is examined through a computational comparative study with 280 problems involving different due date scenarios. In addition, the values of optimal solutions for small problems from a known benchmark are provided.Fundacao de Amparo a Pesquisa do Estado de Sao Paulo - FAPESP[06/03496-3]""Conselho Nacional de Desenvolvimento Cientifico e Tecnologico"" - CNPq[486124/2007-0]""Conselho Nacional de Desenvolvimento Cientifico e Tecnologico"" - CNPq[307399/2006-0

Exact and Heuristic Algorithms for the Job Shop Scheduling Problem with Earliness and Tardiness Over a Common Due Date

Scheduling has turned out to be a fundamental activity for both production and service organizations. As competitive markets emerge, Just-In-Time (JIT) production has obtained more importance as a way of rapidly responding to continuously changing market forces. Due to their realistic assumptions, job shop production environments have gained much research effort among scheduling researchers. This research develops exact and heuristic methods and algorithms to solve the job shop scheduling problem when the objective is to minimize both earliness and tardiness costs over a common due date. The objective function of minimizing earliness and tardiness costs captures the essence of the JIT approach in job shops. A dynamic programming procedure is developed to solve smaller instances of the problem, and a Multi-Agent Systems approach is developed and implemented to solve the problem for larger instances since this problem is known to be NP-Hard in a strong sense. A combinational auction-based approach using a Mixed-Integer Linear Programming (MILP) model to construct and evaluate the bids is proposed. The results showed that the proposed combinational auction-based algorithm is able to find optimal solutions for problems that are balanced in processing times across machines. A price discrimination process is successfully implemented to deal with unbalanced problems. The exact and heuristic procedures developed in this research are the first steps to create a structured approach to handle this problem and as a result, a set of benchmark problems will be available to the scheduling research community

Hybrid Genetic Bees Algorithm applied to Single Machine Scheduling with Earliness and Tardiness Penalties

This paper presents a hybrid Genetic-Bees Algorithm based optimised solution for the single machine scheduling problem. The enhancement of the Bees Algorithm (BA) is conducted using the Genetic Algorithm's (GA's) operators during the global search stage. The proposed enhancement aims to increase the global search capability of the BA gradually with new additions. Although the BA has very successful implementations on various type of optimisation problems, it has found that the algorithm suffers from weak global search ability which increases the computational complexities on NP-hard type optimisation problems e.g. combinatorial/permutational type optimisation problems. This weakness occurs due to using a simple global random search operation during the search process. To reinforce the global search process in the BA, the proposed enhancement is utilised to increase exploration capability by expanding the number of fittest solutions through the genetical variations of promising solutions. The hybridisation process is realised by including two strategies into the basic BA, named as â\u80\u9creinforced global searchâ\u80\u9d and â\u80\u9cjumping functionâ\u80\u9d strategies. The reinforced global search strategy is the first stage of the hybridisation process and contains the mutation operator of the GA. The second strategy, jumping function strategy, consists of four GA operators as single point crossover, multipoint crossover, mutation and randomisation. To demonstrate the strength of the proposed solution, several experiments were carried out on 280 well-known single machine benchmark instances, and the results are presented by comparing to other well-known heuristic algorithms. According to the experiments, the proposed enhancements provides better capability to basic BA to jump from local minima, and GBA performed better compared to BA in terms of convergence and the quality of results. The convergence time reduced about 60% with about 30% better results for highly constrained jobs

A new hybrid meta-heuristic algorithm for solving single machine scheduling problems

A dissertation submitted in partial ful lment of the degree of Master of Science in Engineering (Electrical) (50/50) in the Faculty of Engineering and the Built Environment Department of Electrical and Information Engineering May 2017Numerous applications in a wide variety of elds has resulted in a rich history of research into optimisation for scheduling. Although it is a fundamental form of the problem, the single machine scheduling problem with two or more objectives is known to be NP-hard. For this reason we consider the single machine problem a good test bed for solution algorithms. While there is a plethora of research into various aspects of scheduling problems, little has been done in evaluating the performance of the Simulated Annealing algorithm for the fundamental problem, or using it in combination with other techniques. Speci cally, this has not been done for minimising total weighted earliness and tardiness, which is the optimisation objective of this work. If we consider a mere ten jobs for scheduling, this results in over 3.6 million possible solution schedules. It is thus of de nite practical necessity to reduce the search space in order to nd an optimal or acceptable suboptimal solution in a shorter time, especially when scaling up the problem size. This is of particular importance in the application area of packet scheduling in wireless communications networks where the tolerance for computational delays is very low. The main contribution of this work is to investigate the hypothesis that inserting a step of pre-sampling by Markov Chain Monte Carlo methods before running the Simulated Annealing algorithm on the pruned search space can result in overall reduced running times. The search space is divided into a number of sections and Metropolis-Hastings Markov Chain Monte Carlo is performed over the sections in order to reduce the search space for Simulated Annealing by a factor of 20 to 100. Trade-o s are found between the run time and number of sections of the pre-sampling algorithm, and the run time of Simulated Annealing for minimising the percentage deviation of the nal result from the optimal solution cost. Algorithm performance is determined both by computational complexity and the quality of the solution (i.e. the percentage deviation from the optimal). We nd that the running time can be reduced by a factor of 4.5 to ensure a 2% deviation from the optimal, as compared to the basic Simulated Annealing algorithm on the full search space. More importantly, we are able to reduce the complexity of nding the optimal from O(n:n!) for a complete search to O(nNS) for Simulated Annealing to O(n(NMr +NS)+m) for the input variables n jobs, NS SA iterations, NM Metropolis- Hastings iterations, r inner samples and m sections.MT 201

Mixed integer formulations using natural variables for single machine scheduling around a common due date

34 pages, 10 figuresWhile almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness tardiness penalties around a common due-date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis are proposed to assess the practical relevance of this mixed integer programming based methods