Capacitated lot sizing problem with stochastic times

Bu makalede üretim ve kurulum süreleri stokastik olan kapasite kısıtlı çok ürünlü dinamik parti büyüklüğü belirleme problemi ele alınmıştır. Bu problemde tüm sürelerin stokastik olduğu durum göz önünde bulundurularak hem verimli hem de güvenilir üretim planları elde edilmektedir. Ele alınan problemin amacı klasik üretim maliyetleri ve ek mesai maliyetlerinden oluşan toplam maliyeti en küçüklemektir. Klasik maliyetler, üretim, kurulum ve envanter tutmaktan kaynaklanmaktadır. Ek mesai maliyetleri ise makinenin zaman kapasitesini aşacak şekilde kullanılmasından dolayı ortaya çıkmaktadır. Öncelikle, belirli bir üretim ve kurulum planı için beklenen ek mesai süresini kesin olarak hesaplayan bir prosedür önerilmiştir. Problemi etkin bir şekilde çözmek için tabu algoritmasına dayanan bir çözüm yaklaşımı geliştirilmiştir. Bu yaklaşım üç aşamadan oluşmaktadır: Başlangıç, iyileştirme ve planlama. Algoritmanın ilk aşamasında olurlu planlar üreten bir başlangıç metodu önerilmiştir. Bulunan planlar makalede önerilen tabu arama metoduyla iyileştirilmektedir. Planlama aşamasında, yerel arama metodunun bulduğu çözümleri iyileştirmek için bir doğrusal programlama modeli geliştirilmiştir. Çözüm yöntemimizin performansı literatürde yayınlanmış alt sınırlar kullanılarak onaylanmıştır. Ayrıca, sonuçlar tabu arama yöntemimizin makul sürelerde çok iyi çözümler elde ederek iyi performans sergilediğini göstermektedir.In this paper, we study a capacitated multi-item dynamic lot sizing problem with stochastic production and setup times. In this problem, we consider stochastic times to obtain production plans that are both efficient and reliable. The objective of the considered problem is to minimize the total cost including regular production costs and expected overtime costs. The regular costs result from production, setup and inventory holding. The expected overtime costs are incurred due to the excess usage of the machine capacity. First, a procedure that exactly computes the expected overtime for a given production and setup plan is developed. A solution procedure based on tabu search algorithm is proposed to effectively solve the problem. This procedure includes three main phases: initialization, improving, and scheduling. In the first phase of the algorithm, an initialization method is developed to construct feasible production plans. These plans are then improved by the proposed tabu search method. In the scheduling phase, a linear programming model is developed to further improve the solutions obtained by the local search method. The performance of our solution procedure is validated by the lower bounds reported in the literature. Moreover, results show that our tabu search method performs well by obtaining very good solutions in reasonable amount of times.TR - Dizi

A capacitated lot sizing problem with stochastic setup times and overtime

In this paper, we study a Capacitated Lot Sizing Problem with Stochastic Setup Times and Overtime (CLSP-SSTO). We describe a mathematical model that considers both regular costs (including production, setup and inventory holding costs) and expected overtime costs (related to the excess usage of capacity). The CLSP-SSTO is formulated as a two-stage stochastic programming problem. A procedure is proposed to exactly compute the expected overtime for a given setup and production plan when the setup times follow a Gamma distribution. A sample average approximation procedure is applied to obtain upper bounds and a statistical lower bound. This is then used to benchmark the performance of two additional heuristics. A first heuristic is based on changing the capacity in the deterministic counterpart, while the second heuristic artificially modifies the setup time. We conduct our computational experiments on well-known problem instances and provide comprehensive analyses to evaluate the performance of each heuristic

Multivariate chance-constrained method applied in multi-objective optimization problems of manufacturing processes

Agência 1In the multi-objective optimization problems of manufacturing processes, the responses of interest are often significantly correlated. In addition to the multivariate nature of the problems, product demands, productive capacities, cycle times, the costs of labor, machines, and tools are just some of the many random variables involved in the optimization model. In particular, when using Design of Experiments (DoE) techniques and regression methods, the estimated coefficients for the empirical models - such as response surface models - are also stochastic. However, it has been observed that most of the articles published in this research area are limited to represent the stochastic variables in a deterministic way. Within this context, the present study aimed to propose the use of stochastic programming techniques combined with multivariate statistical methods including some process capability indices widely used in the industry, such as the capacity index and the Parts Per Million () index. The use of the methods combined used resulted in the proposal of the Multivariate Chance-Constrained Programming (MCCP). To test the applicability of the MCCP method, a multi-objective optimization problem of the AISI 52100 hardened steel turning process was selected as a case study given its widespread use and relevance to the industry nowadays. As a starting point for this study, a set of experimental results obtained from a central composite design was used. The decision variables were the cutting speed (), the feed rate () and the depth of cut (). The responses of interest selected for this work were the total machining cost per part (), the material removal rate (), the tool life (), the average roughness () and the total roughness (). After analyzing the data and building the mathematical models for the responses of interest, three approaches were carried out. In the first approach, the index included the calculation of the variance of the response surface model of . In the second approach, the probability that is less than or equal to a predefined value was modelled as a stochastic objective function. Finally, the third approach described the application of the proposed MCCP method. In this approach, the index was calculated using a normal bivariate distribution for both and . The main results of this research were: a) the demonstration and validation of an equation used to calculate the variance of a continuous, derivable and dependent function of stochastic variables; b) the analysis of the impact of seven stochastic industrial variables (setup time, lot size, machine and labor costs, insert changing time, tool holder price, tool holder life and insert price) on the cost of the process; c) finding that maximizing tool life may reduce cost in some cases – for example when using Wiper tools – but the change of the cutting conditions alone does not necessarily reduce the cost of the process, as in what occurred in the case study analyzed

Modelo matemático para dimensionamento de lote e sequenciamento de produção com máquinas paralelas

A programação da produção é atividade primordial dentro das indústrias e passa a ser mais importante quanto mais complexa é a operação de cada empresa. À medida que o ambiente produtivo aumenta em recursos, produtos e restrições, torna-se mais difícil alcançar a programação ótima dada a quantidade de possibilidades e combinações distintas. A execução desta atividade com o devido cuidado pode gerar economia de recursos, maximização das horas de maquinário convertidas em produção, entre outros benefícios. O uso de modelos matemáticos e computadores para esta atividade possibilita testar uma grande quantidade de combinações das mais diferentes variáveis e, no limite, encontrar a proposta de solução ótima e que faz a melhor alocação dos recursos escassos. Dentro deste escopo, o presente trabalho apresenta um modelo matemático, acompanhado do código computacional em Python. Este é aplicável em indústrias que utilizam máquinas paralelas heterogêneas de capacidade finita, com setups dependentes (tanto em custo quanto em tempo) da sequência de programação, com produtos em estágio único de processamento, observando restrições de estoque de componentes, horas-homem disponíveis e capacidade de armazenagem de produtos acabados. Complementando as restrições mencionadas, o modelo tem por objetivo reduzir o custo de execução da programação. Para isto, considera os custos de setup, os custos de manutenção de estoque, a penalidade financeira por não atender o pedido no prazo especificado pelo cliente e o custo de produção do item vendido considerando que as máquinas são heterogêneas e, por consequência, apresentam custos diferentes para produzir o mesmo item. O modelo executado resolveu o problema para uma indústria com 18 máquinas, 350 produtos acabados, 4576 componentes, 2 macro períodos de programação, cada um contendo 5 micro períodos variáveis. A solução ótima foi encontrada em 10h de processamento utilizando um computador com processador Intel i5-2410M 2.30GHz quad-core com 6GB de memória RAM, solver Gurobi, tendo sido programado em linguagem Python com a biblioteca PuLP. Muito embora este tempo total de processamento possa ser considerado alto para uma aplicação prática, o modelo atingiu o gap de 1% do melhor resultado em aproximadamente 2h.Production scheduling is a primary activity within industries and becomes more important the more complex each company's operation is. As the productive environment increases in resources, products and constraints, it becomes harder to achieve the optimal lot-sizing and scheduling given the number of possibilities and different combinations. Carrying out this activity can save resources, maximize machine hours converted into production, among other benefits. The use of mathematical models and computers for this activity makes it possible to test a large number of combinations of different variables and, in the limit, to find the optimal solution that makes the best allocation of scarce resources. Within this scope, this dissertation presents a mathematical model, accompanied by computational code in Python, applicable in industries that use capacitated heterogeneous parallel machines, with dependent setups (both in cost and time) of the scheduling sequence, single-stage processing, observing component availability constraints, available man-power, and finished goods storage capacity. Complementing the mentioned restrictions, the model aims to reduce the cost of production plan execution. For this, it considers setup costs, inventory maintenance costs, back-ordering penalty and finished good production cost considering that the machines are heterogeneous and, consequently, have different costs to produce the same item. The model solved the problem for an industry with 19 machines, 480 products, 4576 components, 2 macro periods, each one containing 5 variable micro periods. The optimal solution was found in 10h of processing using a computer with an Intel i5-2410M 2.30GHz quad-core processor with 6GB of RAM, solved by Gurobi package, programmed using Python language and PuLP library. Although total processing time can be considered high for a practical application, the model reached the gap of 1% of the best result in approximately 2h

Essays on Shipment Consolidation Scheduling and Decision Making in the Context of Flexible Demand

This dissertation contains three essays related to shipment consolidation scheduling and decision making in the presence of flexible demand. The first essay is presented in Section 1. This essay introduces a new mathematical model for shipment consolidation scheduling for a two-echelon supply chain. The problem addresses shipment coordination and consolidation decisions that are made by a manufacturer who provides inventory replenishments to multiple downstream distribution centers. Unlike previous studies, the consolidation activities in this problem are not restricted to specific policies such as aggregation of shipments at regular times or consolidating when a predetermined quantity has accumulated. Rather, we consider the construction of a detailed shipment consolidation schedule over a planning horizon. We develop a mixed-integer quadratic optimization model to identify the shipment consolidation schedule that minimizes total cost. A genetic algorithm is developed to handle large problem instances. The other two essays explore the concept of flexible demand. In Section 2, we introduce a new variant of the vehicle routing problem (VRP): the vehicle routing problem with flexible repeat visits (VRP-FRV). This problem considers a set of customers at certain locations with certain maximum inter-visit time requirements. However, they are flexible in their visit times. The VRP-FRV has several real-world applications. One scenario is that of caretakers who provide service to elderly people at home. Each caretaker is assigned a number of elderly people to visit one or more times per day. Elderly people differ in their requirements and the minimum frequency at which they need to be visited every day. The VRP-FRV can also be imagined as a police patrol routing problem where the customers are various locations in the city that require frequent observations. Such locations could include known high-crime areas, high-profile residences, and/or safe houses. We develop a math model to minimize the total number of vehicles needed to cover the customer demands and determine the optimal customer visit schedules and vehicle routes. A heuristic method is developed to handle large problem instances. In the third study, presented in Section 3, we consider a single-item cyclic coordinated order fulfillment problem with batch supplies and flexible demands. The system in this study consists of multiple suppliers who each deliver a single item to a central node from which multiple demanders are then replenished. Importantly, demand is flexible and is a control action that the decision maker applies to optimize the system. The objective is to minimize total system cost subject to several operational constraints. The decisions include the timing and sizes of batches delivered by the suppliers to the central node and the timing and amounts by which demanders are replenished. We develop an integer programing model, provide several theoretical insights related to the model, and solve the math model for different problem sizes