Lot-sizing problem

Název práce: Lot-sizing problém Autor: Ondřej Kafka Katedra: Katedra pravděpodobnosti a matematické statistiky Vedoucí bakalářské práce: RNDr. Martin Branda, Ph.D. Abstrakt: V předložené práci se seznámíme se základními pojmy z oblasti lot-sizingu. Představíme si Wagner-Whitinův problém a odvodíme algoritmus dynamického programování, jak jej řešit. Dále se podíváme na případ problému PCLSP (Profit maximizing capacitated lot size problem) s pevný- mi cenami a zanedbatelnými přípravnými náklady a budeme jej řešit pomocí speciálního algoritmu lineárního programování. Vše se pokusíme vysvětlit na konkrétních příkladech. V závěru práce ověříme efektivitu uvedených algoritmů pomocí numerické studie na náhodných datech, porovnáme rychlost naprogramovaných algoritmů s profesionálním optimalizačním nástrojem Gurobi. Klíčová slova: Lot sizing, dynamické programování, lineární programováníTitle: Lot-sizing problem Author: Ondřej Kafka Department: Department of probability and mathematical statistics Supervisor: RNDr. Martin Branda, Ph.D. Abstract: In the present work, we define the basic concepts of lot-sizing. We introduce Wagner-Whitin's dynamic lot size problem and derive a dynamic programming algorithm for the solution. Next we look at the case of PCLSP (Profit maximizing capacitated lot size problem) problem with fixed prices and negligable setup costs and solve it using specialized linear programming algorithm. Everything we try to explain with concrete examples. In the end we verify the efficiency of those algorithms by numerical study on random data comparing the performance of programmed algorithms with the professional optimization solver Gurobi. Keywords: Lot-sizing, dynamic programming, linear programmingDepartment of Probability and Mathematical StatisticsKatedra pravděpodobnosti a matematické statistikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Loss of customer goodwill in the uncapacitated lot-sizing problem

Abstract Loss of customer goodwill in uncapacitated single level lot-sizing is studied with a mixed integer programming model extending the well-known Wagner-Whitin (WW) model. The objective is to maximize profit from production and sales of a single good over a finite planning horizon. Demand, costs, and prices vary with time. Unsatisfied demand cannot be backordered. It leads to the immediate loss of profit from sales. Previous models augment the total cost objective by this lost profit. The difference of the proposed model is that unsatisfied demand in a given period causes the demand in the next period to shrink due to the loss of customer goodwill. A neighborhood search and restoration heuristic is developed that tries to adjust the optimal lot sizes of the original no-goodwill-loss model to the situation with goodwill loss. Its performance is compared with the Wagner-Whitin solution, and with the commercial solver CPLEX 8.1 on 360 test problems of various period lengths

Integration of production, maintenance and quality : Modelling and solution approaches

Dans cette thèse, nous analysons le problème de l'intégration de la planification de production et de la maintenance préventive, ainsi que l'élaboration du système de contrôle de la qualité. Premièrement, on considère un système de production composé d'une machine et de plusieurs produits dans un contexte incertain, dont les prix et le coût changent d'une période à l'autre. La machine se détériore avec le temps et sa probabilité de défaillance, ainsi que le risque de passage à un état hors contrôle augmentent. Le taux de défaillance dans un état dégradé est plus élevé et donc, des coûts liés à la qualité s’imposent. Lorsque la machine tombe en panne, une maintenance corrective ou une réparation minimale seront initiées pour la remettre en marche sans influer ses conditions ou le processus de détérioration. L'augmentation du nombre de défaillances de la machine se traduit par un temps d'arrêt supérieur et un taux de disponibilité inférieur. D'autre part, la réalisation des plans de production est fortement influencée par la disponibilité et la fiabilité de la machine. Les interactions entre la planification de la maintenance et celle de la production sont incorporées dans notre modèle mathématique. Dans la première étape, l'effet de maintenance sur la qualité est pris en compte. La maintenance préventive est considérée comme imparfaite. La condition de la machine est définie par l’âge actuel, et la machine dispose de plusieurs niveaux de maintenance avec des caractéristiques différentes (coûts, délais d'exécution et impacts sur les conditions du système). La détermination des niveaux de maintenance préventive optimaux conduit à un problème d’optimisation difficile. Un modèle de maximisation du profit est développé, dans lequel la vente des produits conformes et non conformes, les coûts de la production, les stocks tenus, la rupture de stock, la configuration de la machine, la maintenance préventive et corrective, le remplacement de la machine et le coût de la qualité sont considérés dans la fonction de l’objectif. De plus, un système composé de plusieurs machines est étudié. Dans cette extension, les nombres optimaux d’inspections est également considéré. La fonction de l’objectif consiste à minimiser le coût total qui est la somme des coûts liés à la maintenance, la production et la qualité. Ensuite, en tenant compte de la complexité des modèles préposés, nous développons des méthodes de résolution efficaces qui sont fondées sur la combinaison d'algorithmes génétiques avec des méthodes de recherches locales. On présente un algorithme mimétique qui emploi l’algorithme Nelder-Mead, avec un logiciel d'optimisation pour déterminer les valeurs exactes de plusieurs variables de décisions à chaque évaluation. La méthode de résolution proposée est comparée, en termes de temps d’exécution et de qualités des solutions, avec plusieurs méthodes Métaheuristiques. Mots-clés : Planification de la production, Maintenance préventive imparfaite, Inspection, Qualité, Modèles intégrés, MétaheuristiquesIn this thesis, we study the integrated planning of production, maintenance, and quality in multi-product, multi-period imperfect systems. First, we consider a production system composed of one machine and several products in a time-varying context. The machine deteriorates with time and so, the probability of machine failure, or the risk of a shift to an out-of-control state, increases. The defective rate in the shifted state is higher and so, quality related costs will be imposed. When the machine fails, a corrective maintenance or a minimal repair will be initiated to bring the machine in operation without influencing on its conditions or on the deterioration process. Increasing the expected number of machine failures results in a higher downtime and a lower availability rate. On the other hand, realization of the production plans is significantly influenced by the machine availability and reliability. The interactions between maintenance scheduling and production planning are incorporated in the mathematical model. In the first step, the impact of maintenance on the expected quality level is addressed. The maintenance is also imperfect and the machine conditions after maintenance can be anywhere between as-good-as-new and as-bad-as-old situations. Machine conditions are stated by its effective age, and the machine has several maintenance levels with different costs, execution times, and impacts on the system conditions. High level maintenances on the one hand have greater influences on the improvement of the system state and on the other hand, they occupy more the available production time. The optimal determination of such preventive maintenance levels to be performed at each maintenance intrusion is a challenging problem. A profit maximization model is developed, where the sale of conforming and non-conforming products, costs of production, inventory holding, backorder, setup, preventive and corrective maintenance, machine replacement, and the quality cost are addressed in the objective function. Then, a system with multiple machines is taken into account. In this extension, the number of quality inspections is involved in the joint model. The objective function minimizes the total cost which is the sum of maintenance, production and quality costs. In order to reduce the gap between the theory and the application of joint models, and taking into account the complexity of the integrated problems, we have developed an efficient solution method that is based on the combination of genetic algorithms with local search and problem specific methods. The proposed memetic algorithm employs Nelder-Mead algorithm along with an optimization package for exact determination of the values of several decision variables in each chromosome evolution. The method extracts not only the positive knowledge in good solutions, but also the negative knowledge in poor individuals to determine the algorithm transitions. The method is compared in terms of the solution time and quality to several heuristic methods. Keywords : Multi-period production planning, Imperfect preventive maintenance, Inspection, Quality, Integrated model, Metaheuristic

A review of discrete-time optimization models for tactical production planning

This is an Accepted Manuscript of an article published in International Journal of Production Research on 27 Mar 2014, available online: http://doi.org/10.1080/00207543.2014.899721[EN] This study presents a review of optimization models for tactical production planning. The objective of this research is to identify streams and future research directions in this field based on the different classification criteria proposed. The major findings indicate that: (1) the most popular production-planning area is master production scheduling with a big-bucket time-type period; (2) most of the considered limited resources correspond to productive resources and, to a lesser extent, to inventory capacities; (3) the consideration of backlogs, set-up times, parallel machines, overtime capacities and network-type multisite configuration stand out in terms of extensions; (4) the most widely used modelling approach is linear/integer/mixed integer linear programming solved with exact algorithms, such as branch-and-bound, in commercial MIP solvers; (5) CPLEX, C and its variants and Lindo/Lingo are the most popular development tools among solvers, programming languages and modelling languages, respectively; (6) most works perform numerical experiments with random created instances, while a small number of works were validated by real-world data from industrial firms, of which the most popular are sawmills, wood and furniture, automobile and semiconductors and electronic devices.This study has been funded by the Universitat Politècnica de València projects: ‘Material Requirement Planning Fourth Generation (MRPIV)’ (Ref. PAID-05-12) and ‘Quantitative Models for the Design of Socially Responsible Supply Chains under Uncertainty Conditions. Application of Solution Strategies based on Hybrid Metaheuristics’ (PAID-06-12).Díaz-Madroñero Boluda, FM.; Mula, J.; Peidro Payá, D. (2014). A review of discrete-time optimization models for tactical production planning. International Journal of Production Research. 52(17):5171-5205. doi:10.1080/00207543.2014.899721S51715205521

Optimization in Logistics

Práce je zaměřena na optimalizaci modelů dopravního a přepravního problému s náhodností poptávky, přidáním hran a dynamickým oceňováním. V teoretické části práce jsou uvedeny matematické modely dopravy a popsán software GAMS, který je při řešení použit. Praktická část průběžně doplňuje teoretickou část a aplikuje popsané modely na reálná data.The thesis is focused on the optimization of models of transportation and transshipment problem with random demand, additional edges, and dynamic pricing. The theoretical part of the thesis introduces mathematical models of transportation. The software GAMS, which is used for the solution, is all so described. The practical part is a split among chapters and implements the described models by using real data.