OPTIMIZATION OF SPECIFIC FACTORS TO PRODUCE SPECIAL ALLOYS

Transcription

1 5 th International Conference Coputational Mechanics and Virtual Engineering COMEC October 2013, Braşov, Roania OPTIMIZATION OF SPECIFIC FACTORS TO PRODUCE SPECIAL ALLOYS I. Milosan 1 1 Transilvania University of Brasov, ROMANIA, ilosan@unitbv.ro Abstract: Finding the best solution fro all the industrial solution is an optiization proble. The paper presents an experiantal study regarding the casting of 5 special alloys parts using. For this process it was calculate the optiization of the specific hourly productivity and the cost of each line in hand, aiing to achieve axiu benefit, using the Phases I proble of the Siplex algorith. The nuber of unknowns coponents of calculation using the classical ethod is large, difficult, requiring a large volue of work and are insufficiently precise, all work was done whit the optiization by the linear prograing, using a personal C-Soft. Keywords: optiizatio linear prograing, obective functio C-Soft, special alloys 1. NTRODUCTION In optiization probles, we have to find solutions which are optial or near-optial with respect to soe goals. Usually, we are not able to solve probles in one step, but we follow soe process which guides us through proble solving. Ofte the solution process is separated into different steps which are executed one after the other. Coonly used steps are recognizing and defining probles, constructing and solving odels, and evaluating and ipleenting solutions [1]. In an optiization proble, the types of atheatical relationships between the obective and constraints and the decision variables deterine how hard it is to solve, the solution ethods or algoriths that can be used for optiizatio and the confidence you can have that the solution is truly optial. The optiization of production processes outside aterials ust be ade in relation to an econoic criterio so the function should be an obective indicator of econoic efficiency of the process analyzed [1]. The ain optiization criteria are econoic, technical and econoic nature. Planning processes to solve planning or optiization probles have been of aor interest in operations research [1-5]. Planning is viewed as a systeatic, rational, and theory-guided process to analyze and solve planning and optiization probles. The planning process consists of several steps: 1. Recognizing the proble, 2. Defining the proble, 3. Constructing a odel for the proble, 4. Solving the odel, 5. Validating the obtained solutions, and 6. Ipleenting one solution. 2. STANDARD FORM A first stage of optiization is to deterine the atheatical odel and the second step is finding the optiu coordinates in the ultifactorial space. This eans deterining the extree values (axiu or iniu) optiized paraeters and factors, which receives the optiized paraeter values, this step is even calling optiization. Optiizing a process in ters of technological restriction require the use of a atheatical odel which contains the optiized and restrictions of type equality and inequality [1]. 154

2 If the optiized function and the restrictions are linear, then linear prograing is used and when the optiized function and constraints are used nonlinear prograing [3, 4]. The ost used ethod to optiize the restriction is the Siplex algorith ethod [4, 5]. Siplex algorith applies when the nuber of equations () and nuber of variables (n) is large and full description of the ethod is cubersoe. To optiize an industrial process using linear prograing, the atheatical odel consists of three parts [1]: obective functio liitations and non negative restrictions proble: a) The obective function: F = n 1 c x ; = 1, 2,...,, +1,.., n (1) where: x (x 1, x 2,...x n ) = syste variables-process s paraeters; c - connection coefficients< b) The inequality constrants proble (functional conditions of the process): where: = n; i =. ( n) 1 i 1 1 i 1 1 i 1 a x i a x i a x i 155 b i (2) b i (3) = b i (4) where: a i are called coefficients technology and can be each positive, negative or null. c) the nonnegative ters: x 0 (5) Relations (1) - (5) for a Canonical Linear progra (CLP) Optiization of such probles it ade through the following steps [1]: 1. Establishent of Canonical Linear Progra (CLP); 2. Perfor linear Canonical Linear Progra (CLP) in the Standard Linear Progra (SLP) by adding or subtracting (depending on the shape of each ineq uality:,, =) of variable spacing (x ie ) or artificial variables (x ia ) variables are added in order to easily get value syste (for finding the solution to start). 3. Next step in the resolving this optiization proble is to finding initial starting solutio by equating to 0 the next equation: (n-) n = 0 (6) where: n = nuber of the unknown of the proble and = nuber of the equations Fro the relation presented, it was deterine the base variables (BV) and the values of base variables (VBV). 4. Siplex table is built, starting by iteration 0 (iteration= changing of base) presented in table 1 VB VVB Table 1: Siplex Tableau, iteration 0 c 1 c 2... c r... c c c k... c n c i x 1 x 2... x r... x x x k... x n c 1 x 1 x a 1, a 1k... a 1n c 2 x 2 x a 2, a 2k... a 2n c r x r x r a r, a rk... a rn c x x a, a k... a n z z 0 z 1 z 2 z r... z z z k... z n z - c z 1 -c 1 z 2 -c 2... z r -c r... z -c z +1 -c z k -c k... z n -c n where: BV - the base variables

3 VBV - the values of base variables x and x i = syste variables-process s paraeters; x (x 1, x 2, x,...x n ) and x i (x 1, x 2, x ); = n; i =. ( n) After the iteratio considering differences z - c, analysis is accoplished according to the shape of the progra (axiu or iniu) in resolving the case [1]. This results on this study optiization by linear prograing, it was verified using software in C-SOFT for rapid optiization of operating plants with liited representative saple utilization.to solve, the one-phase approach is applied [1]. 3. EXPERIMENTHAL RESEARCHES A first stage of optiization is to deterine the atheatical odel and the second step is finding the optiu coordinates in the ultifactorial space. This eans deterining the extree values (axiu or iniu) optiized paraeters and factors, which receives the optiized paraeter values, this step is even calling optiization. In this experient it was intended to study the obtaining of 5 cast iron landarks (R1-R5) using olybdenu, nichel and copper as a alloying eleents of cast iron. For this process were used in the following specifications: specific consuption for each ilestone achieved daily, quantity available of alloying cast iron eleents, aiing to achieve axiu benefit, using the Siplex algorith - the Phases I proble [1, 2]. The optiization of the obtaining of 5 cast iron parts is ade in relation to an econoic criterio so the function is an obective indicator of econoic efficiency of the process analyzed [6, 7]. The presentation of the input data is presented in table 2. Alloys Table 2: The presentation of the input data Specific consuption for each ilestone achieved daily, [kg] R1 R2 R3 R4 R5 156 Quantity available, [kg] Mo Cast iron Ni Cast iron Cu Cast iron Benefit [Euro] Analyzing Table 2 entions the following: - specific daily consuption to achieve a specific type pieces R1, consuing 10 kg of Mo Cast iron, 20 kg of Ni Cast iron and 20 kg of Cu Cast iro brings a benefit of 10 Euro; - specific daily consuption to achieve a specific type pieces R2, consuing 20 kg of Mo Cast iro brings a benefit of 20 Euro; - specific daily consuption to achieve a specific type pieces R3, consuing 40 kg of Mo Cast iro 20 kg of Ni Cast iron and 20 kg of Cu Cast iro brings a benefit of 20 Euro; - specific daily consuption to achieve a specific type pieces R4, consuing 10 kg of Ni Cast iron and 20 kg of Cu Cast iro brings a benefit of 40 Euro; - specific daily consuption to achieve a specific type pieces R5, consuing 20 kg of Mo Cast iro 20 kg of Ni Cast iron and 10 kg of Cu Cast iro brings a benefit of 10 Euro; 1) Establishent of linear canonical progra (CLP) a) The obective function (function to be optiized is the beneficial): F= 10x x x x 4 +10x 5 = ax (7) b) The restrictions proble is: 10x x x 3 +20x (8) 20x x x 4 +20x (9) 20x x x 4 +10x (10) c) The non negativity conditions: x 1 0 ; x 2 0; x 3 0; x 4 0; x 5 0 (11) 2) Perfor linear canonical transforation progra (CLP) in the standard linear progra (SLP) by adding or subtracting (depending on the shape of each inequality:,, =) of variable spacing (x ie ) variables are added in this care in order to easily get value syste (for finding the solution to start). a) The obective function

4 F= F= 10x x x x 4 +10x 5 = ax (12) b) The restrictions proble is: 10x x x 3 +20x 5 + x 1e = (13) 20x x x 4 +20x 5 + x 2e = 5000 (14) 20x x x 4 +10x 5 + x 3e = (15) c) The non negativity conditions: x 1 0 ; x 2 0; x 3 0; x 4 0; x 5 0; x 1e 0 ; x 2e 0; x 3e 0; (16) 3) Fro the relation presented, it was deterine the base variables (BV) and the values of base variables (VBV), presented in table 3. Table 3. Base variables (BV) and values of base variables (VBV) x 1e x 2e 5000 x 3e The values fro VBV are considered to be an adissible basic solution [1]; 4) Siplex table is built, starting by iteration 0 (iteration= changing of base), presented in table 4. z Table 4. Siplex Tableau, Phase I, iteration 0 0 x 1e x 2e x 3e z - c Because is an axiu progra optiizatio it was analyzed all differents z - c - establish a procedure that allows oving fro one base to another; - basic changes are ade by decreasing values (proble solved is axiu) optiization function; - stops the iteration process (oving fro one base to another), when it is not possible to increase the value of optiization function. - enter the base, the x 4 is the entering variable in the base and x 2e is the leaving variable of the base; - value of 20 fro the colun of x 4 are called the pivot operation. The siplex algorith proceeds by perforing successive pivot operations which each give an iproved basic feasible solution; the choice of pivot eleent at each step is largely deterined by the requireent that this pivot iprove the solution. Stops the iteration process (oving fro one base to another) when it is not possibl e to decrease the value of optiization functio so to reach the optial solutio F=ax, with solution x i optiu and all differents z - c 0, so it reached the optial solutio results presented in table 5. z Table 5. Siplex Tableau, iteration 1 0 x 1e x 2e x / z - c enter the base, the x 2 is the entering variable in the base and x 1e is the leaving variable of the base; - value of 20 fro the colun of x 2 are called the pivot operation. In table 6 are presented Siplex table, iteration 2. Table 6. Siplex Tableau, iteration 2 20 x /

5 z 0 x 2e x / z - c Because all differents z - c 0, so it reached the optial (axiu) solution The atheatical results of the optial solutins are: z 0 = F ax = (benefit in Euro), with solutions: x 2 optiu = 1000 (the daily specific consuption of landark R 2, in kg); x 4 optiu = 50 (the daily specific consuption of landark R 4, in kg); 5. CONCLUSIONS Analyzing all data taken into account, there can say the following: - For this process it was calculate the optiization of the specific hourly productivity and the cost of each line in hand, aiing to achieve axiu benefit, using the Siplex algorith the Phases I Method. - In this case, because the nuber of coponents is large, the above ethods are cubersoe, requiring a large volue of work, using the classical calculation. - By using this software in place, reduce the coputing tie to several hours using traditional ethod to 2-3 inutes, getting an accurate result, respecting both the econoic and technical coponent, without affecting the sooth running of the etallurgical process. - The atheatical results of the optial solutins of this application are: the benefit is Euro, producing landak 2 and 4, consuing the quantities of 1000 and 50 kg respectively. REFERENCES [1]. Taloi, D.: Optiization of etallurgical processes. Application in etalurgy, Didactic and Pedagogical Publishing House, Bucharest, (1987). [2]. Babu, B., V., Angira, R.: Optiization of Industrial Proceses Using Iproved and Modified Differential Ev olutio Springer-Verlag, Berli (2008). [3]. Anderso C., G.: Applied etallurgical process testing and plant optiization with design of experientation software. Minerals, Metals & Materials Society, part I, 1-26, (2006). [4]. Klees, J., Friedler, F., Bulatov, I., Varbanov, P.: Sustainability in the Process Industry: Integration and Optiizatio Green Manufacturing & Siste Engineering, Manchester, (2011). [5]. Liptak, B., G.: Optiization of Industrial Unit Processes-Second Editio CRC Press, Boca Rato (1998). [6]. Gui, W-I., Yang, C.-H., Che X.-F.,Wang. Y.-L.: Modeling and optiization probles and challenges arising in Nonferrous Metallurgical Processes. Acta Autoatica Sinica,, 39(3), , (2013). [7]. Parhi, P.K.; Sarangi, K.; Mohanty, S.: Process optiization and extraction of nickel by hollow fiber ebrane using response surface ethodology, Minerals & Metallurgical Processing, Vol. 29, No. 4, pp , (2012). 158