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Optimization of Mixed Casting Processes Considering Discrete Journal of Mechanical Science and Technology Journal of Mechanical Science and Technology 23 (2009) 1899~1910 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-009-0428-y Optimization of mixed casting processes considering discrete ingot sizes† Yong Kuk Park1,* and Jung-Min Yang2 1School of Mechanical and Automotive Engineering, Catholic University of Daegu, Hayang-Eup, Gyeongsan-si, Gyeongbuk, 712-702, Korea 2Department of Electrical Engineering, Catholic University of Daegu, Hayang-Eup, Gyeongsan-si, Gyeongbuk, 712-702, Korea (Manuscript Received September 18, 2008; Revised March 23, 2009; Accepted March 26, 2009) -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract Using linear programming (LP), this research devises a simple and comprehensive scheduling methodology for a complicated, yet typical, production situation in real foundries: a combination of expendable-mold casting, permanent- mold casting and automated casting for large-quantity castings. This scheduling technique to determine an optimal casting sequence is successfully applied to the most general case, in which various types of castings with different al- loys and masses are simultaneously produced by dissimilar casting processes within a predetermined period. The methodology proves to generate accurate scheduling results that maximize furnace or ingot efficiency. For multi- variable and multi-constraint optimization problems per se, it provides an extremely practical solution which is readily implemented in most real-world casting plants. In addition, incorporating ingot adjustment from the reality of discrete ingot size, this LP scheduling can assist the casting industry in strengthening its competence by heightening ingot utili- zation as well as satisfying due dates. Keywords: Expendable-mold and permanent-mold casting; Scheduling; Optimization; Linear programming (LP); Melting furnace; Discrete ingot size -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- wheels, various aircrafts and vessel components are 1. Introduction some of modern castings. Castings are among the Casting, or the process of a foundry, is a very effi- highest volume, mass-produced items manufactured cient and cost-effective manufacturing process which by the metalworking industry. Today, the foundry can transform raw materials into discrete products. It industry is a large and increasingly technical industry is also one of the earliest net shape manufacturing [1, 2]. methods because castings are usually close to final A foundryman must schedule casting sequences or shapes and thus require little post-processing. A use- establish an elaborate production plan by employing ful part is manufactured by simply introducing molten limited resources in a foundry. To improve productiv- metal into a mold with a cavity that has desired ge- ity and/or minimize production time and cost, optimi- ometry and letting the molten metal solidify. Casting zation of a production plan or optimal scheduling is a can produce intricate shapes in a single piece, ranging critical step. There have been extensive research ef- in various sizes. Cylinder blocks, cylinder heads, forts to develop a variety of optimization techniques transmission cases, pistons, turbine disks, automotive and apply them to numerous fields in mechanical †This paper was recommended for publication in revised form by Associate engineering including scheduling [3-9]. Also, in steel Editor Young-Seog Lee making, researchers have performed continual inves- * Corresponding author. Tel.: +82 53 850 2723, Fax.: +82 53 850 2710 tigations for scheduling or production planning of a E-mail address: [email protected] © KSME & Springer 2009 steel mill. Many of them adopted mathematical pro- 1900 Y. K. Park and J.-M. Yang / Journal of Mechanical Science and Technology 23 (2009) 1899~1910 gramming as a main tool to solve diverse scheduling complexity of GA itself [21]. As a deterministic op- problems associated with iron and steel manufactur- timization methodology, linear programming (LP) is ing [10-15]. Nevertheless, the vast majority of the regarded as far superior to a stochastic method like previous researches focused on continuous casting in GA if global optimization is guaranteed in the consid- a steel mill with a mass production environment. ered problem, because LP is simpler and easier to Very few studies were directed to scheduling of a apply than any other stochastic method [22]. Addi- foundry. On the other hand, some of them were ex- tionally, in the formulation of casting sequence opti- tended to scheduling of slab, billet or bloom making, mization, LP does not fail to include all major pa- and hot and cold rolling which follow the iron and rameters involved in casting as constraints and to steel manufacturing process in a large-scale steel describe precisely entire casting processes commonly plant [16-18]. observed in reality. This feature will be discussed Most ordinary foundries, medium to small-sized further in detail in the following sections. ones, practicing various types of casting processes This research aims to devise an LP-based optimiza- have a distinctive production environment, which is tion model for casting scheduling in most job-shop very different from that of a large-scale steel mill. For type foundries. The LP model generates an optimal instance, many foundries often manage job-shop type casting sequence resulting in the maximum use of production and start casting diverse kinds of castings molten alloy. Also, an augmented scheme considering upon receiving irregular orders from customers. discrete ingot sizes enhances the utilization percent- Therefore, short-term based scheduling is needed age further. Incorporating the amount of castings depending on each specific order from a customer. produced in each shift as the primary variable, we Frequently, yet irregularly, a foundryman faces com- prove that the objective function to maximize alloy plicated scheduling problems of manufacturing vari- utilization percentage and entire constraints reflecting ous kinds of products with current manufacturing real casting conditions can be perfectly represented in facilities in the foundry. Even worse, the scheduling linear forms. This paper presents appropriate argu- problems change every time according to the orders, ments and verification in the following order. De- and they have to be solved instantly for a feasible and scriptions of casting processes and scheduling in a optimal production plan to meet the order within a real foundry, and rigorous mathematical formulation due date. Hence a simple and practical scheduling (LP) for casting scheduling are in Sections 2 and 3, methodology is vital. Unfortunately, since the sophis- respectively. Detailed explanation on adjustment of ticated algorithms or approaches from the previous the quantity of ingots based on discrete ingot sizes researches mentioned above are designed for large- follows in Section 4. Section 5, a numerical experi- scale iron or steel-making in a mass production envi- ment, leads to the application of the proposed LP ronment, they cannot be practically applied to most model to scheduling of principal casting processes, job-shop type foundries in the real world. which are commonly practiced in a typical foundry. There are only a handful of studies available to Analysis of scheduling results and ingot adjustments solve the problem of casting scheduling in a foundry are also discussed in the case study. Lastly, brief [19, 20]. They use usage percentage of molten metal comments on the summary and contribution of the in casting to assess the efficiency of a scheduling research are addressed in Section 6. solution. In a case study of an oversimplified ficti- tious foundry, a foundryman receives an order of 60 2. Casting scheduling in a foundry castings made from a single alloy in 10 shifts [19]. He and his coworkers rigorously apply genetic algorithm The processes of a foundry, or casting processes as (GA) approaches for finding a solution to the maxi- they are often called, can be divided into expendable- mum metal utilization. However, their GA model mold casting process and permanent-mold casting cannot be applied to a foundry in the real world be- process. Expendable-mold casting can use a mold cause it is inappropriate to casting scheduling, exclud- only one time for each casting operation because the ing any time constraints and completely neglecting mold must be destroyed to acquire a casting after the production conditions of a real foundry such as solidification of molten alloy. In permanent-mold presence of discrete ingots, multiple types of alloys, casting, on the other hand, a mold usually made of furnaces and machines, as well as the computational metal or graphite is repeatedly used-thus ‘permanent’- Y. K. Park and J.-M. Yang / Journal of Mechanical Science and Technology 23 (2009) 1899~1910 1901 to cast numerous identical castings successively. To Suppose a foundry receives a mixed order with a remove (or ‘eject’) the casting, the permanent-mold due date of producing rn for each type n casting (n = should be dismantled. To facilitate this dismantling
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