A Hybrid Genetic Algorithm to Solve a Logt-Sizing and Scheduling Problem

Abstract: This paper reports a lot-sizing and scheduling problem, which minimizes inventory and backlog costs on m parallel machines with sequence-dependent set-up times over t periods. Problem solutions are represented as product subsets ordered and/or unordered for each machine m at each period t. The optimal lot sizes are determined applying a linear program. A genetic algorithm searches either over ordered or over unordered subsets (which are implicitly ordered using a fast ATSP-type heuristic) to identify an overall optimal solution. Initial computational results are presented, comparing the speed and solution quality of the ordered and unordered genetic algorithm approaches.

Lot production size problem and simulation of urban transport

OBJECTIVES AND STUDY METHOD: There are two subjects in this thesis: “Lot production size for a parallel machine scheduling problem with auxiliary equipment” and “Bus holding for a simulated traffic network”. Although these two themes seem unrelated, the main idea is the optimization of complex systems. The “Lot production size for a parallel machine scheduling problem with auxiliary equipment” deals with a manufacturing setting where sets of pieces form finished products. The aim is to maximize the profit of the finished products. Each piece may be processed in more than one mold. Molds must be mounted on machines with their corresponding installation setup times. The key point of our methodology is to solve the single period lot-sizing decisions for the finished products together with the piece-mold and the mold-machine assignments, relaxing the constraint that a single mold may not be used in two machines at the same time. For the “Bus holding for a simulated traffic network” we deal with One of the most annoying problems in urban bus operations is bus bunching, which happens when two or more buses arrive at a stop nose to tail. Bus bunching reflects an unreliable service that affects transit operations by increasing passenger-waiting times. This work proposes a linear mathematical programming model that establishes bus holding times at certain stops along a transit corridor to avoid bus bunching. Our approach needs real-time input, so we simulate a transit corridor and apply our mathematical model to the data generated. Thus, the inherent variability of a transit system is considered by the simulation, while the optimization model takes into account the key variables and constraints of the bus operation. CONTRIBUTIONS AND CONCLUSIONS: For the “Lot production size for a parallel machine scheduling problem with auxiliary equipment” the relaxation we propose able to find solutions more efficiently, moreover our experimental results show that most of the solutions verify that molds are non-overlapping even if they are installed on several machines. We propose an exact integer linear programming, a Relax&Fix heuristic, and a multistart greedy algorithm to solve this problem. Experimental results on instances based on real-world data show the efficiency of our approaches. The mathematical model and the algorithm for the lot production size problem, showed in this research, can be used for production planners to help in the scheduling of the manufacturing. For the “Bus holding for a simulated traffic network” most of the literature considers quadratic models that minimize passenger-waiting times, but they are harder to solve and therefore difficult to operate by real-time systems. On the other hand, our methodology reduces passenger-waiting times efficiently given our linear programming model, with the characteristic of applying control intervals just every 5 minutes.

装配车间的多品种变批量的生产调度优化模型

针对装配车间调度问题,提出了综合考虑生产中物料配送能力,车间存储物料能力以及装配工艺序列问题的多品种变批次的优化调度方法。主要是分两步解决:首先考虑不同类型产品的装配加工的顺序优化问题;然后,根据生产约束建立各类型产品的分批问题。最后,实现计划调度衔接,并应用于生产。该方法对于实际装配生产具有很强的可操作性。

Polyhedral study of mixed integer sets arising from inventory problems

“Branch-and-cut” algorithm is one of the most efficient exact approaches to solve mixed integer programs. This algorithm combines the advantages of a pure branch-and-bound approach and cutting planes scheme. Branch-and-cut algorithm computes the linear programming relaxation of the problem at each node of the search tree which is improved by the use of cuts, i.e. by the inclusion of valid inequalities. It should be taken into account that selection of strongest cuts is crucial for their effective use in branch-and-cut algorithm. In this thesis, we focus on the derivation and use of cutting planes to solve general mixed integer problems, and in particular inventory problems combined with other problems such as distribution, supplier selection, vehicle routing, etc. In order to achieve this goal, we first consider substructures (relaxations) of such problems which are obtained by the coherent loss of information. The polyhedral structure of those simpler mixed integer sets is studied to derive strong valid inequalities. Finally those strong inequalities are included in the cutting plane algorithms to solve the general mixed integer problems. We study three mixed integer sets in this dissertation. The first two mixed integer sets arise as a subproblem of the lot-sizing with supplier selection, the network design and the vendor-managed inventory routing problems. These sets are variants of the well-known single node fixed-charge network set where a binary or integer variable is associated with the node. The third set occurs as a subproblem of mixed integer sets where incompatibility between binary variables is considered. We generate families of valid inequalities for those sets, identify classes of facet-defining inequalities, and discuss the separation problems associated with the inequalities. Then cutting plane frameworks are implemented to solve some mixed integer programs. Preliminary computational experiments are presented in this direction.

Infeasibility handling in genetic algorithm using nested domains for production planning

In this paper we present a genetic algorithm with new components to tackle capacitated lot sizing and scheduling problems with sequence dependent setups that appear in a wide range of industries, from soft drink bottling to food manufacturing. Finding a feasible solution to highly constrained problems is often a very difficult task. Various strategies have been applied to deal with infeasible solutions throughout the search. We propose a new scheme of classifying individuals based on nested domains to determine the solutions according to the level of infeasibility, which in our case represents bands of additional production hours (overtime). Within each band, individuals are just differentiated by their fitness function. As iterations are conducted, the widths of the bands are dynamically adjusted to improve the convergence of the individuals into the feasible domain. The numerical experiments on highly capacitated instances show the effectiveness of this computational tractable approach to guide the search toward the feasible domain. Our approach outperforms other state-of-the-art approaches and commercial solvers. (C) 2009 Elsevier Ltd. All rights reserved.

A linear optimization approach to the combined production planning model

Two fundamental processes usually arise in the production planning of many industries. The first one consists of deciding how many final products of each type have to be produced in each period of a planning horizon, the well-known lot sizing problem. The other process consists of cutting raw materials in stock in order to produce smaller parts used in the assembly of final products, the well-studied cutting stock problem. In this paper the decision variables of these two problems are dependent of each other in order to obtain a global optimum solution. Setups that are typically present in lot sizing problems are relaxed together with integer frequencies of cutting patterns in the cutting problem. Therefore, a large scale linear optimizations problem arises, which is exactly solved by a column generated technique. It is worth noting that this new combined problem still takes the trade-off between storage costs (for final products and the parts) and trim losses (in the cutting process). We present some sets of computational tests, analyzed over three different scenarios. These results show that, by combining the problems and using an exact method, it is possible to obtain significant gains when compared to the usual industrial practice, which solve them in sequence. (C) 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

MRP-production planning in agile manufacturing

Over the years many researchers have been investigating the area of MRP production planning and still is an area of high interest today. This paper look at production planning where there is unpredictable demands due to the type of product and market a company produces for. Production planning becomes difficult when demand fluctuates unpredictably, and hence a historical sales forecast is used as the initial data for production planning. The sales from previous years, especially in a seasonal market, don't necessarily correlate well to the current and future sales for the next year. A planner working in such environment would have himself frustrating to create a feasible production plan that not only needs to meet customer's demands but also to built up the 'correct' amount of stock for that peak sell season. To overcome some of these problems, This work describes a production planning methodology that can be implemented robustly and quickly. This paper has studied two multi-item lot-sizing problems. We detailed the development of the planning problem mathematically and highlight some solutions to initial problems investigated.

Um modelo de otimização inteira mista e heurísticas relax and fix para a programação da produção de fábricas de refrigerantes de pequeno porte

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A linear optimization approach to the combined production planning model

Two fundamental processes usually arise in the production planning of many industries. The first one consists of deciding how many final products of each type have to be produced in each period of a planning horizon, the well-known lot sizing problem. The other process consists of cutting raw materials in stock in order to produce smaller parts used in the assembly of final products, the well-studied cutting stock problem. In this paper the decision variables of these two problems are dependent of each other in order to obtain a global optimum solution. Setups that are typically present in lot sizing problems are relaxed together with integer frequencies of cutting patterns in the cutting problem. Therefore, a large scale linear optimizations problem arises, which is exactly solved by a column generated technique. It is worth noting that this new combined problem still takes the trade-off between storage costs (for final products and the parts) and trim losses (in the cutting process). We present some sets of computational tests, analyzed over three different scenarios. These results show that, by combining the problems and using an exact method, it is possible to obtain significant gains when compared to the usual industrial practice, which solve them in sequence. (C) 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Meta-heuristic approaches for a soft drink industry problem

The present paper evaluates meta-heuristic approaches to solve a soft drink industry problem. This problem is motivated by a real situation found in soft drink companies, where the lot sizing and scheduling of raw materials in tanks and products in lines must be simultaneously determined. Tabu search, threshold accepting and genetic algorithms are used as procedures to solve the problem at hand. The methods are evaluated with a set of instance already available for this problem. This paper also proposes a new set of complex instances. The computational results comparing these approaches are reported. © 2008 IEEE.

Reformulação para um problema integrado de dimensionamento e sequenciamento de lotes

In this paper, some new constraints and an extended formulation are presented for a Lot Sizing and Scheduling Model proposed in the literature. In the production process considered a key material is prepared and is transformed into different final items. The sequencing decisions are related to the order in which the materials are processed and the lot sizing decisions are related to the final items production. The mathematical formulation considers sequence-dependent setup costs and times. Results of the computational tests executed using the software Cplex 10.0 showed that the performance of the branch-and-cut method can be improved by the proposed a priori reformulation.

Strategies to model scheduling decisions to plan the soft drink production process

In this paper we present a mixed integer model that integrates lot sizing and lot scheduling decisions for the production planning of a soft drink company. The main contribution of the paper is to present a model that differ from others in the literature for the constraints related to the scheduling decisions. The proposed strategy is compared to other strategies presented in the literature.

Genetic algorithm, MIP and improvement heuristic applied to the MLCLP with backlogging

The present paper solves the multi-level capacitated lot sizing problem with backlogging (MLCLSPB) combining a genetic algorithm with the solution of mixed-integer programming models and the improvement heuristic fix and optimize. This approach is evaluated over sets of benchmark instances and compared to methods from literature. Computational results indicate competitive results applying the proposed method when compared with other literature approaches. © 2013 IEEE.

Modelos matemáticos para problemas de dimensionamento de lotes com restrições de capacidade e custos de transporte

In this paper, capacitated lot sizing problems in which the classical lot sizing decisions are made considering the transportation costs of the manufactured products were studied. These costs are related to the necessary number of pallets or trucks to pack and/or transport the products from the factory to the warehouse. Three extensions of a mixed integer linear programming model from the literature are considered, representing general cases that are commonly found in companies. These models are tested and evaluated using an optimization package, and a Lagrangian heuristic was developed for one of the extensions proposed.

Modelos matemáticos para o problema integrado de dimensionamento de lotes e corte de estoque undimensional

Pós-graduação em Matemática - IBILCE