Minimization of Billet Remnant Using Zero-One Integer Programming

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1 Minimization of Billet Remnant Using Zero-One Integer Programming Julsiri Jaroenpuntaruk*, Chartchai Matrakul** Department of Industrial Engineering Faculty of Engineering Thammasat University, Rangsit Campus Pathum-thani 12121, Thailand Abstract The traditional production of billet continuous casting process results in production loss due to operator s decision made on the length of cuts of the last billets in each strand. This paper presents an application of linear programming for minimizing remnant in billet casting process at Siam Iron and Steel Company. The mathematical model using zero-one variables is formulated to minimize the remnant in cutting the last billets in each strand and obtain the maximum number of billets at the specified length. The model has been programmed using LINGO software. The computer communication system has made it possible to automatically use the decision variables obtained from the computation to achieve the maximum number of good billets with minimum remnant. The result of the implementation shows that the maximum number of good billets can be achieved with zero remnant. Compared to the conventional production system, this increases in yield as much as tons/month/line. And finally the computer program can work successfully in real time. Keywords Minimization; Continuous casting process; Zero-one variable. *Corresponding author; jjulsiri@engr.tu.ac.th **Former graduate student 1. Introduction Linear programming (LP) is the most widely used in mathematical programming real-world applications. Realworld examples solved by linear programming techniques include steelmaking-continuous casting production problem in which production loss is to be minimized. Recent applications of the steel foundries and continuous casting production solved by linear programming include Voorhis et al. [1] and Tang et al. [2]. This paper presents an application of linear programming concerning the project of Siam Iron and Steel Company (SISCO), one of the largest steel producers in Thailand, whose goal is to minimize remnant and in the mean time to achieve the maximum number of good billets in its continuous casting production. Since the conventional production of the continuous casting process results in production loss due to operator s decision made on the length of cuts of the last four billets in each strand, the production system improvement using integer linear programming has been studied and implemented. To accomplish this task, the mathematical model using zero one variables is formulated to minimize the remnant in cutting the last four billets in each strand and obtain the maximum number of billets at the specified length. The computer program, LINGO, was used to solve this problem and yielded the optimal solution. Computer communication system has been built in order to provide interfacing of computers and machines at various control levels. Accordingly, the decision variables achieved by the computer program can be automatically transferred and real-time utilized in the billet continuous casting production system. 2. The Continuous Casting Process

2 The Siam Iron and Steel Company (SISCO) was established in 1967 at Saraburi Province, Thailand. It becomes one of the major reinforcing bar mills in Thailand. At present, SISCO produces billet from which the company s products such as round bar, deformed bar, angle, channel and wire rod are made. The billet sizes produced by the company are 100 x 100 mm 2, 120 x 120 mm 2 and 300 x 170 mm Manufacturing Process The billet continuous casting process begins with feeding scrap into an Electric Arc Furnace. Then, Ferro alloys are added to the refining molten steel obtained from the Electric Arc Furnace and degassing is made by injecting argon or nitrogen at Ladle Furnace. Afterwards, the ladle containing casting liquid steel is moved to tundish at Continuous Casting Machine (CCM). The liquid steel flows through the ladle gate to the tundish where the liquid steel begins to transform into billets. The continuous casting machine can be described by the flow chart depicted in Figure 1. Ladle Tundish Mold S ( Slag Weight ) T ( Steel Weight ) D Withdrawal Cutting Torch Gi Figure 1. Continuous billet casting process. H 2.2 Problem Naturally, operators make their decisions upon their skills and experiences on the length of cuts of the last four billets in each strand. These manual operations cause production loss of undesired remnant in each strand of Continuous Casting Machine. For this reason, the company project goal is set to improve the production system for minimum remnant. 3. A Billet Continuous Casting Model Algorithms based on simplex method for solving linear programming problems have gained popularity as a result of optimality guarantee and many techniques which reduce complexity and computation time. Recent techniques are Eiselt et al. [3] and Kharab [4]. Different form of linear programming problems can be solved by many different approaches. And different approaches have their own appeal in certain situations. Ozan [5] states that most integer programming problems can be modeled effectively by using a special kind of variable that assumes only two values, 0 or 1. Such variables are called 0-1 (zero-one) variables. Also stated by Ignizio and Cavalier [6], the zero-one values of the binary variables are used to mathematically characterize the yes-no decisions in the model. In multiplant, multiproduct

3 scheduling application, for instance, Jaroenpuntaruk and Kroll [7] demonstrates how to perform disaggregate planning by the application of the integer programming model (0-1 variables). In the billet continuous casting model, our objective is to minimize remnant and in the meantime satisfy desired conditions and assumptions described below. 3.1 Notation and Assumptions The model assumptions are as follows: - the manufacture of high carbon billet of 130x170 mm 2 is selected to develop; - remnants are any billets go below minimum allowable weight; - first priority is given to the billet at maximum allowable weight; - computation begins when the last ladle gate is closed. The notation is introduced as follows: Z the measure which is proportional to total amount by which all billets might fall below the value of maximum allowable weight i index for the strands 1, 2,..., m j index for the cuts 1, 2,..., n T i casting liquid steel withdrawn from remaining casting liquid steel in tundish to strand i (kg) X ij billet of cut j in strand i (kg) R ij amount by which billet of cut j in strand i might fall below the value of maximum allowable weight (kg) T remaining casting liquid steel (kg) in tundish when the last ladle gate is closed Gi billet-in-process (kg) already passed through cutting torch, in strand i, when the last ladle gate is closed S slag (kg) to be removed from liquid steel in tundish L minimum allowable weight of billet (kg) H maximum allowable weight of billet (kg) D billet-in-process between mold and cutting torch in each strand (kg) E billet end to be cut off in each strand (kg). 3.2 Mathematical Model Subject to the following constraints: Minimize Z = Σ Σ ( mn-m-k) R ij (1) Constraint of net weight of steel after slag removal m Σ T i = T S (2) i = 1 Constraints of billets in each strand n Σ X ij = T i + D + Gi E (3) j = 1 Constraints of billets on the first cut in each strand Constraints of billets on second cut in each strand mn m 1 n m j = 2 i=1 k = i+3j-7 X i1 = H (4) X i2 L (5) X i2 + R i2 = H (6)

4 Constraints of billets on each cut from the third to the last cut in each strand Constraint to ensure that no cut to be made after the last no cut Constraint to ensure that no billets go beyond maximum allowable weight and finally and X ij L.b ij (7) X ij + R ij = H.b ij (8) B ij b ij (9) X ij H for all i,j (10) X ij, R ij 0 for all i,j b ij are 0 1 variables. 4. Process Controller The block diagram of the CCM controller at SISCO is illustrated in Figure 2. The controller which comprises of Man machine interface (MMI), Central Programmable Logic Controller (PLC) and three Strand Programmable Logic Controller is connected to Industrial PC. Man-Machine Interface (MMI) Central PLC Industrial PC Strand # 1 PLC Strand # 2 PLC Strand # 3 PLC Figure 2. Block diagram of continuous billet casting process control As soon as the last ladle gate is closed, the Industrial PC receives necessary process data and execute the computer program for optimal solution. To this point, the optimal lengths of cuts are sent to enable the controller to manufacture the optimal last billets at each strand. This can be accomplished by computer communication system

5 developed for the controller. In a similar manner, Turbo Pascal Version 6 program makes the effective communication possible between the Controller and Industrial PC. A thorough discussion of interfacing techniques and machine and system control can be found by Asfahl [8] and Rehg [9]. 5. Conclusions This paper presents the integrated approach of optimization and automation to solve the problem of undesired remnant in the manufacture of billet in continuous casting production system. The goal is to improve the production system to minimize billet remnant. Hence, an optimization model was built to minimize remnant in cutting the last billets in each strand and obtain the maximum number of billets at the specified length or weight. The model has been formulated and implemented on the continuous casting machine at Siam Iron and Steel Company. The machine has three strands (m=3) and the number of the last cuts to be taken into account is 4 (n=4). In addition to the computer communication system, software was developed using LINGO package and Turbo Pascal V.6 programming language to automate the casting process and ensure real-time utilization of optimal solution in the production. The successful implementation reveals that the developed production system is capable of handling realistically sized casting problem in real-time environment. The result shows an increase in yield as much as 5,344 kg /month. In conclusion, this system could be applied to many other steel or similar plants. However, precaution must be taken on the problem size which may exceed the capability of regular mixed integer programming code due to many integer variables in the model. If this is the case, the heuristics and heuristic programming such as Simulated Annealing, Tabu Search and Genetic Algorithms may be more appropriate approach to acceptable solution instead. References 1. Voorhis, T.V., Peters, F. and Johnson, D., 2001, Developing Software for Generating Pouring Schedules for Steel Foundries, Computers & Industrial Engineering, 39(3-4), Tang, L., Liu, J., Rong, A. and Yang, Z., 2000, A Mathematical Programming Model for Scheduling Steelmaking-Continuous Casting Production, European Journal of Operational Research, 120(2), Eiselt, H.A. and Sandblom, C.L., 1999, Price Probing in the Simplex Method, Computers & Industrial Engineering, 37(1-2), Kharab, A., 2000, An Advanced Macro Spreadsheet Program for the Simplex Method, Computers & Operations Research, 27(3), Ozan, T.M., 1986, Applied Mathematical Programming for Production and Engineering Management, Prentice Hall, New Jersey. 6. Ignizio, J.P. and Cavalier, T.M., 1994, Linear Programming, Prentice Hall, New Jersey. 7. Jaroenpuntaruk, J. and Kroll, D.E., 1991, A Disaggregation Problem for a Multiplant, Multiproduct Scheduling Application, Production Planning & Control, 2(4), Asfahl, C.R., 1992, Robot and Manufacturing Automation, Second Ed., John Wiley & sons, New York. 9. Rehg, J.A., 1994, Computer-Integrated Manufacturing, Prentice Hall, New Jersey.