INNOVATIVE ACCESS TO SOLUTION "TWO-DIMENSIONAL" CUTTING PROBLEM IN METALLURGICAL OPERATIONS. Martin LAMPA, Kamila JANOVSKÁ, Petr BESTA

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1 INNOVATIVE ACCESS TO SOLUTION "TWO-DIMENSIONAL" CUTTING PROBLEM IN METALLURGICAL OPERATIONS Martin LAMPA, Kamila JANOVSKÁ, Petr BESTA VŠB Technical University of Ostrava, 17. listopadu 15, Ostrava - Poruba, Czech republic, EU, martin.lampa@vsb.cz, kamila.janovska@vsb.cz, petr.besta@vsb.cz Abstract Post has documented the assumption that a relatively simple methodology applicable modeling of linear programming can still bring appropriate element optimization in managerial decision making (the exact basis). These are mainly used in the operational and tactical management of industrial enterprises, where the in the "real" time can provide relevant information for the choice of optimal, or approaching the optimum variants of solving problems. Is cited as an example the problem of optimal cutting steel strips in coils and its solution as a problem of linear programming in MS Excel. Keywords: linear programming, optimization, cutting problem 1. INTRODUCTION This type of management decision making is now possible without the the connection with modern computer equipment and managing equipment. Computer technology is used for modeling, analysis, experimentation and the search for optimal or approaching optimum structure, functioning of real systems and their parts. Nowdays the technical specifications of computers allow their use not only as an off-line problem solving tool, but as a tool for on-line direct proceeding of the projected control systems, in relation on the sources for data collection, recording and transmission of information.one of the problem, where computer off-line support can be successfully used in decision-making processes, is the issue of cutting problem [1]. Further, there is as an example introduced a solving problem of optimal cutting steel strips as a linear programming problem - the task of optimal cutting material by Simplex method algorithm in MS Excel [3]. 2. DEFINITION OF PROBLEMS In a metallurgical company are made cold steel sheets in coils of required thickness of 0.3 mm and a width of 1200 mm and a length of 2000 m, which should be divided into three required dimensions: 512 x with a total of 42% of the parts, 730 x 1152 mm with the total amount of components 20% and 450 x with a total of 38% of parts. We assume that there is limited capacity of the separator machine. The total annual number of coils is pcs. The amount (quantity) of each size sheets in the required percentage amount, subject to minimum input material, to be cutted. 3. VARIANTS OF POSSIBLE DIVISION OF COILS We will divide the steel roll of the length 2000 m to three required length so that as far as possible we will respect the ratio of 42: 20: 38 and at the same time achieved low loss of the belt area. Variations of division represent the file lengths, which allows to respect the different amounts of appropriate ratio between them for one coil [4]. Analogy then cut all 88,500 pieces of coils.

2 Variantion 1: 1152 mm 20 mm 450 mm Waste: 9 x 0,512 x 0,02 = 0,09216 m 2 Variantion 2: 20 mm 450 mm Waste: 0,512 x 0,02 = 0,01024 m 2

3 Variantion 3: 176 mm Waste: 0,730 x 0,176 = 0,12848 m 2 Variantion 4: 48 mm 1200 mm Waste: 0,730 x 0,048 = 0,03504 m 2

4 4. THE MATHEMATICAL MODEL DEFINED BY CUTTING PROBLEM Mathematical displaying we express the requirement of finding non-negative values x i, to respect individual constraints conditions and at the same time the target function as a criterion of optimality decision to acquire the smallest possible value [5]. x i - the frequency of the i-th division variants 2000 m coil (strip), i = 1, 2, 3, Number of parts of the 2000 m strip: 42 % 512 x 730 x 2 + 2x 3 20 % 730 x x 1 + x 4 38 % 450 x 512 9x 1 + x 2 Total quantity of parts from 2000 m band: 13x 1 + 2x 2 + 2x 3 + x 4 42 % parts of the dimention 512 x 730: x 2 + 2x 3 /13x 1 + 2x 2 + 2x 3 + x 4 = 42/ x x 3 42(13x 1 + 2x 2 + 2x 3 + x 4 ) = 0-546x x x 3 42x 4 = 0 /:2-273x 1 + 8x 2 + 8x 3 21x 4 = 0 20 % parts of the dimention 730 x 1152: 4x 1 + x 4 /13x 1 + 2x 2 + 2x 3 + x 4 = 20/ x x 4 20(13x 1 + 2x 2 + 2x 3 + x 4 ) = 0 7x 1-2x 2-2x 3 + 4x 4 = 0 Binding conditions 4,608x 1 + 0,512x 2 + 0,730x 3 + 0,730x 4 = x 1 + 8x 2 + 8x 3 21x 4 = 0 7x 1-2x 2-2x 3 + 4x 4 = 0 Effectiveness function 0,09216x 1 + 0,01024x 2 + 0,12848x 3 + 0,03504x 4

5 5. SOLUTION The solution was found by using of te algorithm of linear programming in the spreadsheet processor MS Excel [2]. By rounding into the integral number we obtained the following solution: x 1 = 0 x 2 = 2139 x 3 = 112 For every from the pcs of coils x 4 = 1126 Z min = 75,85336 m 2 Total quantity of cut parts: 13x 1 + 2x 2 + 2x 3 + x 4 = 5628 ks Dimention 512 x 730: x 2 + 2x 3 = = 2363 ks 2363/5628 x 100 = 41,99 42 % Dimention 730 x 1152: 4x 1 + x 4 = = 1126 ks 1126/5628 x 100 = 20,00 20 % Dimention 450 x 512: 9x 1 + x 2 = = 2139 ks 2139/5628 x 100 = 38,00 38 % Waste (material losts): 75,85336/(2000 x 1,2) x 100 = 3,16 % In the final solution was cut pcs of three dimentions of which 2363 pcs of dimention 512 x, 1126 pcs of dimention 730 x 1152 mm and pcs of dimention 450 x. The total is 75,85 m 2, what is 3,16 % of the total lenght of the strip. 6. CONCLUSION The example illustrates on the specific problems of practical metallurgical plant the meaningfulness of applications of linear programming to obtain informations - recommendations for decision making. It represents the formulation and the solution of cutting problem by linear programming algorithms. This example of the use of linear programming algorithm proves, among other things, that it is not always necessary to grab to complicated algorithms to obtain solutions of the decision problem.

6 ACKNOWLEDGEMENT The work was supported by the specific university research of Ministry of Education, Youth and Sports of the Czech Republic No. SP2013/19. REFERENCES [1] GROS I.: Kvantitativní metody v manažerském rozhodování, Praha: GRADA Publishing: s. [2] LENORT, R., SAMOLEJOVÁ, A.: Analysis and identification of floating capacity bottlenecks in metallurgical production, METALURGIJA, 2007, vol. 46, no. 1, pp [3] LAMPA, M.; SAMOLEJOVÁ, A.; KRAUSOVÁ, E. Optimal cutting of input material of metallurgical operations as a linear programming problems. In METAL 2012: 21st Anniversary International Conference on Metallurgy and Materials. Ostrava: TANGER, 2012, pp ISBN [4] LAMPA, M.; SAMOLEJOVÁ, A.; ROŽNOVSKÝ, L. Optimizing of Pallet Metallurgical Products Layout on Truck Loading Space. In METAL 2011: 20th Anniversary International Conference on Metallurgy and Materials. Ostrava: TANGER, 2011, pp ISBN [5] JANOVSKÁ, K.; VILAMOVÁ, Š.; SAMOLEJOVÁ, A.; VOZŇÁKOVÁ, I. Determination of complex energy consumption of metallurgical production on the base of mathematical modelling of interdisciplinary linkages, METALURGIJA, 2012, vol. 51, no. 4, pp