Automatic Determination of Optimal Length of Casting Steel Blocks in the Context of an Imprecise Manufacturing

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1 st International Conference on Control Systems and Computer Science Automatic Determination of Length of Steel Blocks in the Context of an Imprecise Manufacturing Iulia Clitan, Vlad Muresan, Mihail Abrudean and Sas Diana Monica Automation Department, Technical University of Cluj Napoca, Cluj Napoca, Romania Abstract This paper describes an algorithm usable for automatic determination of of casting steel, of different shapes and sizes, in the context of an imprecise manufacturing due to an unpredictable end-user demand, meaning a supply order that is constantly changing. The proposed algorithm is a general solution that takes into account the technological limitation of a steelmaking plant and of a rolling process, based on the steel type and the cross section of a steel block. The solution is the automatic computing of casting s for steel and the selection of the casting s combination in order to comply with the maximum number of casting that can be managed by the steelmaking plant. For obtaining an overall minimum waste of material, integer linear programming is applied for the optimization problem. The algorithm developed by the authors is implemented using MATLAB. Testing and validation is done via computer simulation and different scenarios. The overall steel waste is computed, all performed tests are compared based on this value and the procedure that generated the best result is described throughout the paper. Keywords casting ; steel block; casting process; imprecise manufacturing; integer linear programming; minimum steel waste. I. INTRODUCTION The manufacturing processes are constantly evolving. This article focuses on the rolling process, a manufacturing process that intends to generate steel products of different shapes and sizes from solid steel block. The rolling process is primarily based on three technological stages [1],[2]: Steelmaking, a stage where iron and scrap metal are melted, generating liquid steel, that is further refined. The main aggregate used in steelmaking is the electric arc furnace [3],[4] Fig. 1. Solid steel of different shapes and sizes Continuous casting, used for obtaining solid steel of different shapes and sizes, as seen in Fig. 1 [5], [6]. This stage is based on a ladle furnace and a distributor that continuously casts the refined liquid steel in a water-cooled copper form, to achieve the solidification [7]. These first two stages are usually carried out in a steel making plant. Processing of final steel product, from which the rolling process represents an important manufacturing method of converting a steel piece with a larger cross section into a smaller section steel product, suitable with the end user demand, carried out in a steel rolling mill [8]. The steel obtained by continuous casting have certain s and cross sections, which are determined by the technological constraints and physical limitation of the casting process. The required casting varies from one order of supply to another, since in order to comply with the variety of end-user specification, different input s are required for the technological process of rolling [9]. The input of the technological process of rolling differs from the casting, due to the fact that there is a limited casting range set by the casting equipment from the steel making plant. That being said, first the input rolling is determined and from this the proper casting is derived. The end-user demand for a final product is converted into a number that are needed at a certain input, as to cover that specific demand. The input is generated such as during the technological process of rolling minimum material loss occurs, taking into account the tolerances required for each rolling step. Thus, for each received order from an end-user, the number needed at a certain input, required for the production process, is known. If this falls into the restricted casting range and if the maximum number of s that can be overall by the casting equipment is not excedeed, then there is no problem. However, in most cases the problem of selecting the casting s occurs, since at least one of the two technological limitation are not fulfilled in the context of a supply order that is constantly changing [10] /17 $ IEEE DOI /CSCS

2 This paper is topical for the steel manufacturing process, and the aim of it is to provide a solution for obtaining, in an automated manner, the casting s, in the context of an imprecise and umpredictable order, taking into consideration the restrictions of the casting process (the technological limitation on the s as a limited casting range; the number of casting s that are managable at the steel making plant). This article tackles the case when the two technological limitation are not fulfilled, and the proposed solution for achieving an overall minimum loss of material, in linear terms, consists in sorting the casting s of each specific order in an ascending manner, in terms of the waste material amount for each order. From the ascending sorted set of casting orders the first 6 distinct s are chosen (considering that the maximum number of s that can be at the steelmaking plant equals 6), and the remaining casting s are optimized using linear programming according to the previously selected s. The result is going to be in terms of a minimum loss of linear material, a minimum steel waste that remains unprocessed by the rolling process (raw steel). II. DETERMINIG THE REQUIRED CASTING LENGTH FOR A STEEL BLOCK Nowadays, orders from end-users received at the rolling mill are very diverse and constantly changing, which makes the rolling supply demand to be very imprecise. The supply demand consists in the number needed and the corresponding input required for the production process, as presented in the previous section. The input can be obtained by cutting it from steel of a greater. The casting process is often technologically limited [4]. The technological limitations from the steel making plant are known; mainly the limitation of the casting equipment, such as the maximum number of s that can be equals 6. The range of casting s is also limited, and is given as a minimum at which a block can be poured at, due to the limitation of the continuous casting equipment, and a maximum casting due to physical limitation imposed by shipping, meaning the maximum of a steel block that can be transported from the steel making plant to the rolling mill, regardless of the type of transport used. In order to comply with the casting range the input is converted into a casting that belongs to the restricted casting range. The casting is determined as a multiple of the input, taking into consideration the number of steel needed at the input, such that the remaining amount of raw steel is minimum. The following algorithm is proposed by the authors such that the casting for each supply demand is generated automatically. The input variables are the input and the number needed for each order. Step 1. Iteratively compute all multiples of the input, of a certain order, that fit between the minimum and the maximum casting s. The algorithm starts from the minimum. The is then increased by 1 millimeter until it reaches the maximum possible of casting. At each stage it is verified if the selected represents a multiple value of the input, and the value is kept if so. Step 2. Calculate based on the number required for each order, the multiple that must be chosen as the required casting so the waste of material, computed using (1), is minimal, respectively how many pieces must be at the required casting to cover a specific order. waste _ l = mult _ l no _ mult input _ l no _ block (1) Where: waste_l represents the waste of material, mult_l represents the multiple s, no_mult represents the number of needed for a specific multiple to cover a specific order, input_l represents the input generated for that specific order and no_ represents the number of input steel needed for that specific order. The proposed algorithm returns as output variables a required casting and the number of that must be, for each order. III. DETERMINING THE OPTIMAL CASTING LENGTHS Since the end-user demands vary, also the type, meaning the steel chemistry, needed for each steel block can vary. Another technological limitation of the casting process is the fact that it can only cast one cross section and one type of steel at a time. In order to determine the casting that takes into consideration all presented limitation the following steps must be applied. First The supply order set must be grouped into sections depending on the type needed; Second Each type sections are further grouped into subsections depending on the cross section of the block s. Third The casting s are computed for each subsection from the supply order set. In the following, a set of orders represent a previously grouped subsection. A. Determine the Maximum Lengths Further Used in the Optimization Problem The solution to determine the casting s as to obtain an overall minim steel residue is to ensure a minimum residue for the majority of orders, in terms of waste of steel. The steel waste is given as the difference between the amount of casting steel needed to cover one order and the amount of material needed for the rolling process, in terms of. The steel waste represents the of a steel block that remains unused in the process of rolling. For each order the amount needed for the casting process is computed: the casting of the steel block multiplied by the number of that must be ; and the amount necessary for the rolling process is computed: the input of the steel block multiplied by the 100

3 number of. The difference between the two generates the steel waste for each order. Further, the casting set of orders is sorted in an ascended manner, in terms waste for each order at a time. In the case of more orders that have the same value waste, those orders are rearranged by sorting them in a descending manner is respect to another criterion, the steel residue (the remaining excess from the casting amount and the amount necessary for the rolling process, in respect to the amount of steel required for casting). In order to ensure overall minimum residue, from the previously ordered set, the first 6 distinct casting s are chosen as the final casting s, which have the lowest values obtained for the steel waste. For the remaining orders, linear optimization is applied, with the required s as the 6 previously selected. The result is a combination of s for each remaining order, and minimum residue in terms of unused of a steel block. This will guaranty an overall value waste that will be of minimum value. B. Integer Linear Programming for Determining the Combination of Lengths for the Remainig Orders A linear programming problem can be defined as the problem of maximizing or minimizing an objective criterion subject to linear constraints [11], which represents an indicator that measures the result of the problem [12]. In this paper integer linear programming is used in order to minimize the steel waste for certain orders, the that are and remain unused by the rolling process, the canonical form for such a system is described in (2). The linear programming model presented is applied for every remaining casting order. min( W1 x1 + W2x W6 x6) a1x1 + a2x a6x6 b no _ block xi 0, i = 1,..., xi Z i 6 The objective function is represented by the waste function (3). For the combination of casting s from the previously selected casting s array, in order to achieve minimum material loss for each casting order, the variables x i represent the number of needed to be at each corresponding casting and must be positive integers. The variables a i, i=1..6 represent the integer part of the ratio between the i th casting and the input of a specific order (number at a full ideal that can be used in the rolling process from a casting ). Minimize Waste = min( W x + W x W ) (3) _ x6 Where W j represents the waste of material of each final casting, computed using (4) as the remainder, amount left over after division between cast_l j (that represents the j th (2) casting from the final casting array) and input_l (that represents the input of a specific order). cast _ l j W =, = 1,...,6 _ j remainder j input l The first technological constraint for this problem is presented in the second row of the model presented in (2), where b represents the number that need to be at the ideal for a specific order to be fulfilled. The second constraint is represented by the limitation of each x i value to an integer one, having the lower boundary set at zero and the upper boundary set to equal the number needed at the input. The result of the integer linear programming algorithm is going to generate a combination of number of casting steel for each final casting so as to obtain overall minimum loss of material, for each casting order. IV. TESTING THE PROPOSED SOLUTION The algorithm developed by the authors is implemented in MATLAB computer software, in order to generate the required data automatically, both the final casting s and the combination of casting s so as to obtain an overall minimum steel waste and not to exceed the maximum number of s that can be at the steelmaking plant. In order to test the proposed algorithm and to better highlight the results of the presented solution, taking into consideration the technological constraints of the continuous casting process, the authors implemented some slight variations of the proposed algorithm, meaning tests are performed on the same supply order set, arranged in different manners. The authors use a fictitious set of supply orders, given in Table 1, selected in such a way that the resulted number of casting s exceeds the maximum number of 6 that can be at the steelmaking plant, to test and validate the optimization part of the algorithm. Also the authors choose to have two orders with the same input, but a different number needed, in order to test and validate the casting computing part of the algorithm. TABLE I. THE SUPPLY ORDER SET steel (4) 101

4 It is considered that the supply order set belongs to the same steel type and have the same cross section, thus the orders proposed for testing are already grouped into a corresponding subsection. The supply order in this case is considered to consist of an input and the number of steel that are needed for a certain order of an end-user demand to be accomplished. A. Test the Length Computing The proposed algorithm from section 2 for computing the casting is applied for the order set given in Table 1, considering the limited casting range between a minimum equal to 8000 mm and a maximum equal to mm. Taking into consideration that the aim of the algorithm is to provide a result with minimum steel waste, for each order at a time, the resulting casting s and the number of needed for each casting, are given in Table 2. The steel waste is computed as the difference between the casting multiplied by number of and the input multiplied by number. It can be noted that for the input of 2753 we obtain two different casting s since there are two orders with this input, each having a unique number attached, which demonstrates proper implementation of the algorithm. B. Test for a Set of Orders Arranged in a Descending Manner in Terms of Amount of Casted Steel Required The casting orders from Table 2 are further arranged in a descending manner, in terms of amount of steel required (the casting multiplied by the number of ). The arranged set of orders is given in Table 3, and from this ordered set it results that the six final casting s are the first six distinct s, such as: mm, 8259 mm, mm, 8931 mm, mm and mm. TABLE II. THE RESULTS OF THE CASTING LENGTH COMPUTING ALGORITHM Length Results For the remaining orders, integer linear programming is applied in order to determine the combination of casting, from the casting array, for each remaining order. The MATLAB function intlinprog() is used [13]. Also the MATLAB function floor() is used for computing the integer part of the ratio required for the first technological constraint. In Table 4 the results obtained from the optimization problem are shown. After applying the algorithm described on a set of descending sorted orders a total waste block equal to mm results. C. Test for a Set of Orders Arranged in an Ascending Manner in Terms of Amount of Casted Steel Required The authors test the linear programming optimization problem on the same supply order set arranged in an ascending manner. The new sorted order set is given in Table 5. The same MATLAB functions are used to implement de optimization problem for the remaining orders, however for this new sorted order set, the six final casting s differ, and the casting array consists of: 8360 mm, 9099 mm, mm, mm, mm and 8931 mm. In Table 6 the results obtained from the optimization problem for the remaining orders are shown. After applying the algorithm described on a set of ascending sorted orders a total residue of mm results. This means that an ascended sorted order set generates a worsen result by 95% from the descended sorted order set. TABLE III. THE SUPPLY ORDER SET ARRANGED IN A DESCENDING MANNER Length Results TABLE IV. THE OPTIMAL CASTING LENGTH FOR TEST B

5 TABLE V. THE SUPPLY ORDER SET ARRANGED IN AN ASCENDING MANNER Length Results The authors also test the case when for orders with equal steel waste, the sorting is done in an ascending manner is respect to steel residue. This means that the six final casting s are 9099 mm, mm, mm, 8360 mm, mm and 8259 mm. Applying the linear programming problem on this new array of casting, for the remaining orders, will generate the results presented in Table 9, and a overall steel waste of 8744 mm. The overall waste value is higher than the overall waste obtained for the previous test, proving that the previously performed test generated the best result. TABLE VII. THE SUPPLY ORDER SET ARRANGED IN AN ASCENDING MANNER IN TERMS OF STEEL WASTE Length Results TABLE VI. THE OPTIMAL CASTING LENGTH FOR TEST C combination , 3 98, 2 149, D. Test for a Set of Orders Arranged in an Ascending Manner in Terms of The authors test the linear programming optimization problem on the same supply order set but this time arranged in an ascending manner in respect to the waste block, as presented in section 3. The general idea is to obtain minimum residue overall, thus the approach presented in this paper is also tested and compared against the previously performed tests. If for some cases the steel waste in mm is equal, then the orders are sorted in a descending manner in respect to steel residue (the remaining excess from the casting amount and the amount necessary for the rolling process, in respect to the amount of steel required for casting). The new sorted order set is given in Table 7. For the optimization problem the same MATLAB function intlingprog() is used to implement de optimization problem for the remaining orders and the six final casting s are the first six distinct s from Table 7 (9099 mm, mm, mm, 8360 mm, mm, mm). The results of the optimization are presented in Table 8. In order to make a comparison between the results obtain for all tested versions of the proposed algorithm the overall steel waste is computed, and the value equals 8059 mm. This denotes an improvement from the descended sorted order set with 125.5%. TABLE VIII. THE OPTIMAL CASTING LENGTH FOR TEST D, CASE FOR ORDERS WITH EQUAL WASTE SORTED DESCENDING IN RESPECT TO STEEL RESIDUE combination 11012, 9099, , 74 2, 2 3, TABLE IX. THE OPTIMAL CASTING LENGTH FOR TEST D, CASE FOR ORDERS WITH EQUAL WASTE SORTED ASCENDING IN RESPECT TO STEEL RESIDUE combination , , 149 2, 2 3,

6 V. RELATED WORK The presented results were elaborated based on a prior research [14], [15]. First the integer linear programming was tested for fixed casting s, evenly distributed among the possible casting range [14] and the results were correlated to the number of casting batches [15]. From previous research the general solution presented in this paper was elaborated. This means that the authors contribution consists of analyzing and converting the real linear problem into a mathematical one, developing, implementing and testing different solutions for obtaining the combination of necessary casting s, in order to prove which is the most suitable solution (in terms of minimum consumption of material). All testing were performed on the same input data, consisting in a set of orders, grouped as the necessary for rolling and the number of required billets, cut at the required, in order to satisfy a final client s demand, of final rolled products. Another author s contribution is the developing of the algorithm that determines, in an automatic manner, the casting most appropriate for a specific order, such that there is a minimal waste of billets. A billet waste is considered a piece of a billet that remains unused in the rolling process, and it occurs whenever the amount of steel billets is higher than the amount billets that is necessary to be rolled in order for a final specific demand to be covered. Since from the performed tests, for the set of orders arranged in an ascending manner in terms waste, and in a descending manner in terms residue, the overall value waste is minimum proves the applicability of the solution presented in this paper, and validates the algorithm proposed by the authors. VI. CONCLUSIONS The authors present by means of this paper an intelligent system in order to generate in an automated manner the solution in terms of minimal loss of material, minimal steel waste, for the casting at which the need to be in order to satisfy a variety of supply orders needed for the rolling process, and to comply with the technological limitation of either the steelmaking plant or the rolling mill. The general solution that generated the best result, meaning minimum loss of material that remains unused in the rolling process, is presented and the method describe is validated by means of tests performed using computer simulation. Different approaches of sorting the supply order set were carried out, and it was demonstrated that the worst case scenario is to sort the order set into an ascending manner, in terms of amount of steel required, and to select the first casting s as the final casting array, then to apply the optimization problem on the remaining orders. The best case scenario was to arrange the order set in an ascending manner in respect to the waste block s, if the steel waste is equal, then the orders are further sorted in a descending manner in respect to the remaining excess from the casting amount and the amount necessary for the rolling process, in respect to the amount required for casting. Such a method generates an improvement of % from the case of a descending sorted order set, in terms of amount of steel required, and an improvement of over 300 % from the case of an ascending sorted order set in terms of amount of steel required. Thus the obtained results validated the algorithm elaborated by the authors, meaning that the solution complies with the maximum number of s that can be at the steelmaking plant and minimum steel waste in terms of is obtained overall. The presented problem is a complex one with multiple variables that influence the outcome. Such variables can be for instance the number of cuts and the cutting procedure required for each steel block, in order to obtain the input for the rolling process. The cutting procedure depends on the chemistry and for each type of procedure different tolerances need to be added to the input. In order to present a general solution the cutting tolerance was neglected, the required input rolling was considered throughout this paper since the cutting tolerance can be added at the final casting, depending on the cutting procedure used. The final tolerance that needs to be added equals the cutting tolerance multiplied by the number of cuts minus one. ACKNOWLEDGMENT The research activity is sponsored by the research project no / , financed by the Technical University of Cluj-Napoca. REFERENCES [1] G.E. Totten, Steel Heat Treatment Handbook. Metallurgy and technologies, CRC Taylor & Francis Group, [2] V. Geant, Rafinarea oelului (Steel refinery). Bucureti, Romania: Ed. Printech, [3] M. 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