A knapsack problem as a tool to solve the production planning problem in small foundries

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Abstract

According to recent research carried out in the foundry sector, one of the most important concerns of the industries is to improve their production planning. A foundry production plan involves two dependent stages: (1) determining the alloys to be merged and (2) determining the lots that will be produced. The purpose of this study is to draw up plans of minimum production cost for the lot-sizing problem for small foundries. As suggested in the literature, the proposed heuristic addresses the problem stages in a hierarchical way. Firstly, the alloys are determined and, subsequently, the items that are produced from them. In this study, a knapsack problem as a tool to determine the items to be produced from furnace loading was proposed. Moreover, we proposed a genetic algorithm to explore some possible sets of alloys and to determine the production planning for a small foundry. Our method attempts to overcome the difficulties in finding good production planning presented by the method proposed in the literature. The computational experiments show that the proposed methods presented better results than the literature. Furthermore, the proposed methods do not need commercial software, which is favorable for small foundries.

Introduction

According to the Brazilian Foundry Association, Abifa [1], Brazil was the 7th major world producer of molten metals in 2005. At this time, the Brazilian foundry sector generated 60,000 jobs and 90% of them were in small and medium foundries (market-driven foundries). Therefore, to maintain the competitivity of this sector, it is necessary to improve the techniques to achieve high levels of efficiency and low operating costs. Production planning is one of the main factors that affects industrial productivity. In their research, Fernandes and Leite [2] discussed the importance of this planning for Brazilian foundries.

The aim of the production planning problem in market-driven foundries is to find a production plan with minimum costs of production, setup, inventory and backlogging, respecting limited resources. In the literature, there are few studies concerning the production planning problem in the foundry sector. Some of them focus on medium-sized foundries, Santos-Meza et al. [3], Araujo et al. [4], Duda [5] and Duda and Osyczka [6]. For small foundries, Silva and Morabito [7], Araujo et al. [8] and Tonaki and Toledo [9] can be cited. Good reviews for classical lot-sizing problems as Bahl et al. [10], Drexl and Kimms [11], Brahimi et al. [12] and Jans and Degraeve [13] can be cited.

In this study, we analyzed the production planning problem faced by a small foundry in the interior of Sao Paulo (Brazil), previously examined by Araujo et al. [8] and Tonaki and Toledo [9]. The lot size to be produced of each item during each period of the finite planning horizon had to be determined. This problem requires another fundamental decision, which is the choice of all the alloys that must be melted during each period. Silva and Morabito [7] present a greedy approach for the cutting and packing problem to solve this matter. Araujo et al. [8] proposed a model and a heuristic for the same problem. Based on Araujo's model, Tonaki and Toledo [9] suggested that this problem can be viewed as two sub-problems: the production planning problem of alloys and the production planning problem of items. These problems are solved in the hierarchical way. First, the alloys to be produced are determined and afterwards the items which required these alloys are defined. The authors propose Lagrangian heuristics for both of these problems. They indicate that the solution method to the problem that determines the alloy scheduling needs to be improved to create better production plans.

Our purpose is to consider the problem as suggested by Tonaki and Toledo [9], but we propose a method to evaluate different alloy production plans. We proposed a genetic algorithm as a solution method to determine the production plan of alloys to be melted. Once the alloys to be melted are determined, we have various lot-sizing as sub-problems. In order to solve the problem described previously, we use the knapsack problem as a tool to determine the items to be produced when the alloy is melted. This purpose focuses on presenting a solution method without commercial software and avoiding the deficiencies pointed out in the methods from Tonaki and Toledo [9]. Our solution method is considered simpler to be applied in industrial environments than the method proposed by Tonaki and Toledo [9] and computational experiments show that our purpose finds plans with lower costs.

This paper is organized as follows. In the next section, the problem is formally stated. In Section 3, we describe the solution method proposed. Test results are reported in Section 4 using instances based on real datasets. The conclusion and discussion of possible future work are presented in Section 5.

Section snippets

The production process of a small foundry

The main foundry processes melt alloys and mold metal in the required items. These processes must be scheduled at the same time. The transformation of ore and scrap metal into alloys with specified levels of carbon, silicon, zinc, etc. determines properties for items such as brittleness and corrosion resistance. The alloy in liquid state is poured into moulds to cool and produce the final items.

In this production planning problem, it is necessary to determine the lot size of the items and the

Solution methods proposed

To solve this production planning without using commercial software, Tonaki and Toledo [9] proposed heuristic algorithms called balance furnace loading heuristic (BFLH) based on Lagrangian heuristics. The heuristic called BFLH (15) solves the problem aiming to minimize the costs, while the heuristic called BFLH (19), besides minimizing the costs, penalizes the metal molten not used in molding items. The authors indicate that the solution method for the problem that determines the alloy

Computational experiments

In this section, we present the experiment design used for comparing the performance of the genetic algorithm presented above with the performance of the BFLH proposed by Tonaki and Toledo [9] which has shown to be effective in solving the two-step problem and has better results than those obtained in earlier studies—Araujo et al. [8]. All methods are written in C language and the experiments were conducted on a Pentium 4 computer with a 2.8 GHz processor and 1 GB RAM memory using Windows XP. The

Conclusion

In this paper, we studied the production planning problem in a small foundry. The problem has two fundamental decisions: to choose the alloys to be melted and to determine the quantity and which items are to be produced. In the literature, there is one solution method based on the Lagrangian heuristic to resolve this two-step problem, but it needs improvements to find better sequences of alloys. The genetic algorithm with the proposed knapsack problem attempts to overcome these difficulties. In

Acknowledgments

The authors would like to thank the Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) of Brazil for financial support. Furthermore, the authors would like to thank anonymous reviewers for their helpful comments that greatly improved the content of the paper.

References (19)

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