Process Capability Proposal. with Polynomial Profile

Transcription

1 Contemporary Engineering Sciences, Vo. 11, 2018, no. 85, HIKARI Ltd, Process Capabiity Proposa with Poynomia Profie Roberto José Herrera Gestión de a Caidad Research Group Universidad de Atántico, Coombia Mendoza Mendoza Ade Afonso 3i+d Research Group Universidad de Atántico, Coombia Cabarcas Reyes Juan Caros 3i+d Research Group Universidad de Atántico, Coombia Copyright 2018 Roberto José Herrera et a. This artice is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Abstract This proposa for a capabiity indicator aims to evauate the processes that have a poynomia profie, evauated in terms of quaity characteristics with a significant degree of association. Among the capacity proposas with a inear profie is that of Shahriari and Sarafian, where they demonstrate that the use of capabiity indices with simpe profies eads to better accuracy in the evauation of the indicator. These study shows, in contrast to the simpe inear profie indicator, that some processes are not suitabe for simpe inear association, the quadratic profie or the poynomia profie in genera is more appropriate. The research shows a specific case study in a food company; the capabiity assessment with poynomia profie presents a better fit, in contrast to the capabiity indices with simpe inear profie. Keywords: Capabiity indices, Linear profies, Quaity statistica contro 1 Introduction Statistica process contro (SPC) is a too used as an anaysis and monitoring too

2 4228 Roberto José Herrera et a. for variabes that may cause inconsistencies within the process. This behavior can be corrected or prevented by using statistica quaity measures such as capabiity indices; expressing with this indicator a possibe effect on performance, generay caused by factors other than natura conditions. These factors prevent compiance with the minimum quaity requirements, incuding design specifications, estabished for each of the variabes or quaity characteristics of the product, considered by an important number of the mutivariate capabiity indicators, which are not correated. This research presents a modification of the capabiity indices of simpe inear profies impemented by Nemati Keshteia, R. Baradaran Kazemzadeha and A. Amirib and R. Noorossan [9], appying indicators with poynomia profies, in this way it is possibe to make an unbiased estimate of the performance when the variabes present an important association with a different profie. Linear profies are commony represented as parametric modes, such as simpe inear regression, mutipe inear regression, poynomia regression, ogistic regression and the new modes. There are other investigations such as the one suggested by Wooda [11], which aso presents an evauation of the capabiity of the process using indicators with inear profies. However, there are few documents on these types of indicators. This artice aims to make a sma contribution on this topic, with the purpose of opening the discussion of the appicabiity or advantage in its impementation. 2 Theory framework 2.1 Profie monitoring In situations where the quaity of a process or product is characterized by a functiona reationship between a response variabe and one or more independent variabes, it is caed profie monitoring. Most research on profie monitoring refers to the simpe inear profie, i.e. a singe independent variabe. Guevara et a. [7] The mode of the singe inear profie is defined by the foowing expression, y ij = A 0 + A 1 X i + ε ij, i = 1,2,, n, j = 1,2,, k (1) where ε ij are the residuas defined as independent random variabes normay distributed with mean zero and variance σ 2 NID(0, σ 2 ). The sope and intersection are caed regression coefficients or profie, it is assumed that the x vaues are fixed and take the same set of vaues for each sampe, Guevara et a. [6] The capabiities indices are used to evauate the performance of the process, initiay presented by Kane [10], symboizing it as C p in charge of measuring the potentia without incuding the process mean, C p = USL LSL UNTL LNTL = USL LSL (μ+3σ) (μ 3σ) (2)

3 Process capabiity proposa with poynomia profie 4229 where σ is the process standard deviation, USL and LSL are the ower and upper specification imits respectivey. UNTL is the upper natura toerance imit and LNTL is the ower natura toerance imit. The other side, when considering evauating the process dispacement or position with respect to the centraization measure, the appropriate indicator is the C pk defined by the foowing equation [3] C pk = min {C pu = USL μ 3σ, C p = μ LSL 3σ } (3) Shahriari and Sarafian [3] present a method for cacuating the process capabiity indices when monitoring a simpe inear profie, considering the response variabe as a characteristic with known distribution and specification imits. This C pk of the response variabe is cacuated for each eve of the expanatory variabe, then the C pk is entered as the process capabiity indice. Ebadi and Shahriari [4] [5] repaced the response variabe with a predicted variabe at each eve of the expanatory variabe, then used a mutipe process capabiity indices to measure the process capabiity, resuting in underestimated C pk vaues at the expanatory variabe eves in the study, and suggested a Bothe [1] method, which uses the proportion of nonconforming eements for its measurement, P = min{p 1, P 2, P p }, the capabiity indice is defined, C p = 1 3 φ 1 {P} (4) where φ 1 being the inverse standard norma distribution. The process capabiity indice C p that was defined in equation (2) is a comparison between the natura toerance imits and the specification imits of a process. In a singe inear profie y = A 0 + A 1 X is the process reference ine, a 0 + a 1 x is the conditiona mean of y and x, then µ is cacuated as foows: μ = a 0 + a 1 x (5) where y is a norma random variabe with mean a 0 + a 1 x and variance of σ 2, a 0 k and a 1 are estimates of A 0 and A 1 and are cacuated as a 0 = ( j a 0j )/k k a 1 = ( j a 1j )/k respectivey, a 0j and a 1j the intercepts and sopes estimated in the j-th sampe profie. The variance of the σ 2 process is estimated using MSE and cacuated as k MSE = ( j MSE j )/k, where MSE j is the estimated variance in the umpteenth sampe profie. Therefore, you can define the toerance imits UNTL and LNTL of the response variabe as,

4 4230 Roberto José Herrera et a. UNTL y (x) = μ + 3σ = a 0 + a 1 x + 3σ (6) LNTL(x) = μ + 3σ = a 0 + a 1 x 3σ (7) UNTL and LNTL are two parae ines where the distance between them is equa to the capabiity 6σ. The specification imits of y are found as a function of the variabe x as, USL y (x) = a 0u + a 1u x (8) LSL y (x) = a 0u + a 1u x (9) The C p of a singe inear profie has a functiona form, as presented in foowing equation, C p = USL y(x) LSL y (x) UNTL y (x) LNTL(x) x [x 1, x 2 ] (10) By choosing C p (x) as the process capabiity indices, it is possibe to evauate the capabiity at each eve of the expanatory variabe x, obtaining detaied process information. However, it is necessary in quaity contro to have a unique capabiity indices vaue to give an overa judgment of the process. It is therefore recommended to use the imited area between USL y and LSL y to cacuate USL y (x) and LSL y (x) and aso the imited area between UNTL y and LNTL y to cacuate UNTL y (x) and LNTL(x). To determine a singe vaue for the C p of a singe inear profie is, C p (profie) = [USL y(x) LSL y (x)]dx x [UNTL y (x) LNTL(x)]dx x x [x, x u ] (11) UNTL y (x) and LNTL(x) are two parae ines, simiary USL y (x) and LSL y (x) are defined in equations (8) and (9). The distance of these parae ines can be considered as their difference, so the capabiity indice of the inear profie C p (profie) is cacuated as foows, C p (profie) = a 0u a 0 6σ (12) and C pk is described as: C pk (x) = min { USL y(x) μ y (x) UNTL y (x) μ y (x), μ y(x) LSL y (x) μ y (x) LNTL(x) } x [x, x u ] (13) where μ y (x) is the function of the reference ine. C pk (x) gives the vaue of C pk of

5 Process capabiity proposa with poynomia profie 4231 a singe process at each eve of x, the C pk of a singe inear profie is cacuated: C pk (profie) = min { x [LSL y (x) μ y (x)]dx, x [μy(x) LSLy(x)]dx [UNTL y (x) μ y (x)]dx x [μ y (x) LNTL(x)]dx x } (14) The capabiity indice of process C pk when ony the upper or ower functiona specification imits are avaiabe can be cacuated using the foowing equations: x [USL y (x) μ y (x)]dx C pk(profie)= x [UNTL y (x) μ y (x)]dx (15) x [μ y (x) LSL y (x)]dx C p(profie)= x [μ y (x) LNTL(x)]dx (16) when the vaue of LSL y (x) is greater than μ y (x) in [x, x m ] and ess than μ y (x) in [x m, x u ], then the minimum capabiity indice is cacuated as foows: C pk (profie) = min { xm x [USL y (x) μ y (x)]dx [μ y (x) USL y (x)]dx xm, x [μy(x) LSLy(x)]dx [UNTL y (x) μ y (x)]dx x [μ y (x) LNTL(x)]dx x } (17) 2.2 Mutivariate capabiity indice assuming independent variabes The authors Chen et a. [2] have deveoped capabiity indices that evauate the quaity characteristics that intervene within a productive process, showing in their proposa the degree or proportion with which a product can compy integray with each of the specifications and requirements of the market. Based on the compiance ratio of each of the process quaity characteristics, the capabiity indice for mutipe characteristics are obtained as foows, C T pk = 1 3 φ 1 [ [π p j=1 (2φ(3Cpk ) 1)+1] 2 ] (18) where C pk denotes the vaue of the j-nth characteristic for j = 1,2,..., N, p is the number of characteristics and φ is the standard norma distribution. 2.3 Proposed poynomia capabiity indice The new proposa presents a modification of equations (6), (7), (8) and (9) of R. Nemati et a. [9], the specification imits USL y and LSL y are obtained from the confidence intervas, for the poynomia case, defined as foows:

6 4232 Roberto José Herrera et a. USL y (x) = a 0u + a 1u x + a 1u x 2 (19) LSL y (x) = a 0 + a 1 x + a 1 x 2 (20) UNTL y (x) = μ + 3σ = a 0 + a 1 x + 3σ (21) LNTL(x) = μ + 3σ = a 0 + a 1 x 3σ (22) 3 Methodoogy The evauation of the mutivariate capabiity indices is verified by a case study in a food company, where two variabes choride and Brix are seected. To determine the capabiity indice with inear profies, proceed as foows: 1) Tabuate the data taken in the manufacturing process of the food product. 2) Cacuate the traditiona univariate capabiity indice C pk for both variabes. 3) Evauate the capacity indices taking the proposas of Chen et a. [2], considering the independent variabes, Nemati et a. [9], with a simpe inear profie and the proposa of this study that considers a poynomia profie. The information comes from a food company, the variabes studied were Brix and Choride shown in Tabe 1, whose specifications are Brix [ ] and Choride [ ]. The physicochemica characteristics of the fina product and the data were coected over a period of 9 months; see Tabe 1, R. Herrera et a. [8]. Tabe 1. Coded partia measurements of the Historica Brix and Choride variabes. By the Authors No. Brix Choride No. Brix Choride 1 29,8 2, ,8 2, ,2 2, ,0 2, ,3 2, ,5 2, ,2 2, ,0 2, ,9 2, ,4 2, ,9 2, ,3 2, , ,2 2, , , ,7 2, ,3 2, ,9 2, , ,2 2, ,9 2, ,5 2, ,5 2, ,2 2, ,5 2, ,6 2, ,5 2, ,8 2, ,4 2,61

7 Process capabiity proposa with poynomia profie Resuts Initiay the traditiona univariate capabiity indices of equation (1) were determined for each variabe independenty, as shown in Figures 1 and 2, obtaining, evauated in equation (1), C p variabe Brix: 0.59 and a C p of the Choride variabe: 0.77 Figure 1. Índice de capabiity for variabe Brix C p = Figure 2. Índice de capabiity for variabe Choride C p = 0.77 Both [1] indicators show a ower resut than the unit, in this case the process doesn't meet the specifications in the two (2) variabes.

8 4234 Roberto José Herrera et a. Tabe 2. Comparison of the vaues obtained in the capabiity indices of the Brix and Choride variabes. By the authors Capabiity indice with singe inear profie R. Namiti Mutivariate Mutivariate capabiity indice Chen et a. Proposed poynomia capabiity indice Capabiity indice proportion of nonconforming eements Bothe Univariate Univariate capabiity indice T Cp(profie) Cpk(profie) C pk CpP CpkP P = min{0.80, 0.56 } Brix Choride In contrast to the poynomia profie, the variabes have a strong reationship with an R-squared of which guarantees a high correation between them. The equation of the poynomia mode adjusted for the case study is, Brix = 26, ,66607 Choride 0, Choride^2 The mutivariate capabiity indice is cacuated by modifying the proposa of R. Nemati et a.[9] of the process, by a poynomia mode that is denoted as CpP, for this purpose the upper and ower specification imits of the process (USL and LSL) and the upper and ower process imits (UNTL and LNTL) are determined in the foowing way, the USL and LSL imits are obtained with the confidence intervas of the process which are: Tabe 3. Confidence intervas for poynomia mode coefficients. By the Authors Parametric Estimates Error Standard Limit ow Limit up Constant 26,948 1, , ,113 Choride 1, , , ,6482 Choride ^2-0, , , ,1611 The specification imits based on equations (19) and (20) USL = 29, ,6482 x 0,1611 x 2 and LSL = 24, , x 0, x 2, where μ is the equation of the variabe Y, σ is the root of the mean square of the error which is 0,9251. The equations of the process imits equations (21) and (22) evauated in the information are respectivey for the upper and ower imit UNTL = 29, ,66607x 0,218416x 2 and LNTL = 24, ,66607x 0,218416x 2. In this way, the process capabiity índice is cacuated taking into account the specifications of the company's Choride variabe which are [ ] and based on equation (14), (15) and (16) modifying a inear poynomia mode, equations (19), (20), (21) and (22).

9 Process capabiity proposa with poynomia profie 4235 Performing the cacuations corresponding to the above ratio gives a capabiity indice with a poynomia profie of CpP = 1.816, indicating that the process satisfactoriy meets the estabished specifications. CpkP is cacuated, equations (19), (20), (21), (22) and (23), in order to identify where the process is dispaced. The vaues of the centering indicators CpkP = min{1.816, 1.814}, indicate a centered process. 5 Discussion The univariate capabiity index for each of the variabes indicates that the design specifications are not met. The mutivariate capabiity indice for independent events gives the same resut as the univariate capabiity. However, appying the simpe inear profie, as we as the poynomia regression profie, the case study resuts indicate that the process is capabe of meeting the design requirements. The capabiity indice of the poynomia profie presents satisfactory resuts according to the experience of those responsibe for evauating the quaity of the product, which indicates that this technique is more accurate for the cacuation of this indicator. References [1] D. Bothe. A Capabiity study for an entire product, Rev. ASQC Quaity Contro Transactions, 46 (1991), [2] H. F. Chen, A mutivariate process capabiity index over a rectanguar soid toerance zone, Statistica Sinica, 4 (1994), [3] H. Shahriari and M. Sarafian, Evauación de proceso - Evauación de a capacidad en perfies íneaes, Conferencia Internaciona de Ingeniería Industria, Teherán, Irán (2009). [4] M. Ebadi and A. Amiri, Evauation of process capabiity in mutivariate simpe inear profies, Scientia Iranica, 19 (2012), [5] M. Ebadi and H. Shahriari, A process capabiity index for simpe inear profie, Internationa Journa of Advanced Manufacturing Technoogy, 64 (2013), [6] R. D. Guevara and J. A. Vargas, Evauation of process capabiity in mutivariate noninear profies, Internationa Journa of Advanced Manufacturing Technoogy. Submitted for pubication (2014).

10 4236 Roberto José Herrera et a. [7] R. D. Guevara and J. A. Vargas, Process capabiity anaysis for noninear profies using depth functions, Quaity and Reiabiity Engineering Internationa, 31 (2013), [8] R. Herrera, S. Ruiz, L. Sacedo, Desarroo de índices de capacidad de proceso para un perfi poinómico en una empresa de eaboración de sasas, Investigación e Innovación en Ingenierías, 6 (2018), no. 2, [9] R. Nemati Keshtei, R. Baradaran Kazemzadeh, A. Amiri and R. Noorossana, Deveoping functiona process capabiity índices for simpe inear profie. Scientia Iranica, Sharif University of Technoogy, [10] V.E. Kane, Process capabiity indices, Journa of Quaity Technoogy, 18 (1986), [11] W. H. Wooda, D. J. Spitzner, D. C. Montgomery, and S. Gupta, Using contro charts to monitor process and product quaity profies, Journa of Quaity Technoogy, 36 (2004), Received: September 7, 2018; Pubished: September 26, 2018