ELABORATION OF A MATHEMATICAL MODEL FOR DESIGNING ATTRIBUTE ACCEPTANCE PLANS

Transcription

1 D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić Razrada matematičkog modela za izradu atributivnih planova prijema ELABORATION OF A MATHEMATICAL MODEL FOR DESIGNING ATTRIBUTE ACCEPTANCE PLANS Danijel Kapusta, Goran Kozina, Damir Vusić, Živko Kondić, Ante Čikić ISSN UDC/UDK : Professional paper In this artile, attribute aeptane plans are explained and elaborated, together with their design by using standards and nomograms. The elaboration of a mathematial model for designing speifi aeptane plans is also presented as well as the implementation of these models in a software appliation. Their possible appliation in speifi ases is shown on several examples. The results obtained by using a software appliation are ompared with the standards and are presented with adequate urves. Keywords: AQL, attribute aeptane plans, HRN ISO :1989, Larson nomogram, operating harateristi urve of the aeptane plan, sampling, statistial quality ontrol Razrada matematičkog modela za izradu atributivnih planova prijema Stručni članak U ovome radu obrađeni su i pojašnjeni planovi prijema za atributne karakteristike proizvoda pomoću normi i nomograma, te razrađen matematički model za izradu speifičnih planova prijema koji je prilagođen računalnoj aplikaiji. Na nekoliko primjera prikazani su mogući načini primjene u konkretnim slučajevima. Rezultati dobiveni pomoću računalne aplikaije uspoređeni su s normama i prikazani s pripadajućim krivuljama. Ključne riječi: AQL, HRN ISO :1989, Larsonov nomogram, operativna krivulja plana prijema, planovi prijema za atributivne karakteristike, statistička kontrola kvalitete, uzorkovanje 1 Introdution Uvod Quality is a notion that has always been present in nature and an be simply desribed as the survival of the fittest. In today's sense, quality ontrol has existed sine the day men started making things. It evolved with time in order to satisfy larger prodution of goods, to ut expenses, and to satisfy market demand for higher quality. One of the definitions of quality is: "Quality is the level at whih a set of distintive harateristis satisfies requirements." [1] In order to satisfy the above mentioned definition, it is neessary to determine the mode and ontrol parameters. Two basi types of ontrol offer themselves: 100 % ontrol Statistial ontrol. 100 % ontrol is a type of inspetion applied to ertain harateristis of all produts or materials in a group to determine whether a produt meets imposed requirements. The main disadvantage of the 100 % method is that eah element needs to be inspeted separately, whih results in high osts of this inspetion type. Moreover, it is impossible to apply ontrol during whih the objet of ontrol gets destroyed. Statistial ontrol presents a set of methods and proedures for gathering, proessing, analysis, and interpretation of data to ensure the quality of a produt, proess and servie. It an be divided into ontrol during a proess (ontrol harts) and ontrol after the finished proess (aeptane plans). Aeptane plans an be further divided depending on the ontrolled harateristis into variable aeptane plans and attribute aeptane plans. The advantage of statistial ontrol presents signifiantly smaller ontrol volume in the form of element numbers whih need to be ontrolled. Tests are simpler and a large number of objets an be ontrolled in a relatively short time. Aeptane plans present an only possibility in ase the ontrol is destroying an objet. 2 Attribute aeptane plans Planovi prijema za atributivne karakteristike An aeptane plan (sampling plan) presents a set of regulations whih determine the size of a sample (or samples, when it is a matter of double or multiple systems), whih needs to selet aeptane or rejetion numbers from the basi lot. [2, 8] Within the system of sample seletion, one or more samples are seleted from the delivery by using any proedure whih ensures randomness or adequate representative quality of the whole delivery. If the subjet are smaller parts or parts pakaged in a loose ondition (e.g. srews, nuts, and similar), it is enough to randomly selet from the ontainer as many piees as neessary aording to the reeived size of the sample. In ase the subjet are parts whih are separately pakaged or plaed in separate ontainers, after labeling eah piee in any numerial order, random numbers are used for hoosing representative piees [7]. Aeptane plans an be further divided into single (Fig. 1), double (Fig. 2), and multiple (Fig. 3). The deision on the hoie of an aeptane plan is usually based on eonomi and administrative fators. Double and multiple aeptane plans enable the use of smaller sample size than the size used with single aeptane plans. However, administrative diffiulties and the prie per unit in a sample are usually smaller with single aeptane plans. With multiple aeptane plans, the proedure is repeated until the right ondition for aeptane or rejetion of the shipment is ahieved. The main advantage here is that the lot is sorted out during the inspetion proess. Aeptane number ( A or ) is the maximum number of defetive units whih an be tolerated in a sample to Tehnial Gazette 18, 2(2011),

2 Elaboration of a mathematial model for designing attribute aeptane plans D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić onsider the basi lot satisfying. 2.1 Operating harateristi urve of an aeptane plan Operativna krivulja plana prijema Operating harateristi urve represents the probability of aepting the sample Pa depending on the fration of defets in sample p (Fig. 4). Figure 1 Single aeptane plan Slika 1. Jednostruki plan prijema Figure 4 Operating harateristi urve Slika 4. Operativna krivulja Figure 2 Double aeptane plan Slika 2. Dvostruki plan prijema When it omes to 100 % ontrol, operating harateristi urve represents an ideal ase (Fig. 4, left). This means that eah lot that has a lower quality level (the quality level represents the perentage of defetiveness, what means lower is better) from the aeptable level ( p< pa) will be aepted with the probability of 100 %, while every shipment with a quality level higher than the aeptable ( p> pa) will be rejeted. The borderline ase ( p= pa) requires an agreed upon proedure. When it omes to sampling (Fig. 4, right), the situation hanges sine there is no more 100 % ertainty for aeptane or rejetion of the shipment, depending on the fration of defets in the sample. Operating harateristi urve in that ase presents the probability that a lot of ertain quality level is aepted or rejeted. Depending on the aeptane plan parameters there is a ertain probability for a good lot to be rejeted or for a bad lot to be aepted. These values are defined by using α and β risks for the predetermined perentage of defetive parts [9, 10]. Figure 5 Operating harateristi urve in the sampling ase Slika 5. Operativna krivulja u slučaju uzorkovanja Figure 3 Multiple aeptane plan Slika 3. Višestruki plan prijema pa=aql presents the aeptable quality level for whih there is so alled α produer's risk that a delivery of good quality is rejeted (as a rule 5 %). AQL presents the highest tolerated perent defetive to onsider the shipment satisfying. pt=rql presents the tolerated quality level for whih there is so alled β onsumer's risk that a delivery of a signifiantly worse quality is aepted (as a rule 10 %). 288 Tehnič ki vjesnik 18, 2(2011),

3 D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić Razrada matematičkog modela za izradu atributivnih planova prijema pn is neutral quality or % of a lot disard for whih there is a 50 % probability that a good delivery is rejeted or a bad delivery aepted. 2.2 Average outgoing quality urve Krivulja prosječne izlazne kvalitete Average Outgoing Quality (AOQ) is a perent defetive at the entrane warehouse. (After applying the sampling, the deliveries are taken to the entrane warehouse.) AOQ an be taken into onsideration when the prodution is stable and the number of deliveries is large enough. The value of average outgoing quality an be obtained by using the equation (1), and when the basi lot size N is muh bigger than sample n it an be simplified and alulated by using the equation (2), by inserting in the equation the data with units as shown in Fig. 6, where pi equals the perent defetive and Pai the probability of aeptane at a speifi point. Pai pi ( N n) AOQ, (1) N AOQ p P F ( 2) i i ai i. The average outgoing quality urve presents the graph of dependene of average outgoing quality in relation to the perent defetive. The value AOQL is the maximum value of average quality of an outgoing produt (AOQ) for every possible quality tolerated aording to the given aeptane plan. Fig. 6 presents the operating harateristi urve (up) and the urve of average outgoing quality (down). It an be seen in the graphs that for the presented aeptane plan and alpha risk of 5 % AQL equals 1 %, while the maximum amount of average outgoing quality aording to the given aeptane plan amounts approximately to 1,25 %. 2.3 HRN ISO :1989 HRN ISO :1989 Standard HRN ISO :1989 speifies sampling plans and proedures for the inspetion of disrete elements aording to the attributes. The sampling plans in this standard are indexed aording to the values of aeptable quality levelaql. [3] The purpose of this standard is to affet the supplier by putting eonomi and psyhologial pressure on him with not aepting the delivery in order to maintain the proess average at least on the level of speified quality of AQL, at the same time giving the onsumer upper border for the risk of oasionally aepting a defetive lot. Appliation: Final produts Components and raw materials Proedures Materials in proess Warehouse stoks Maintenane proedures Data or reords Administrative proedures. Fig. 7 shows the proess of aeptane plan design by using the standard. Figure 7 Choosing an aeptane plan by using the standard Slika 7. Odabir plana prijema pomoću norme 3 Statistial bases Statističke osnove Figure 6. Average outgoing quality urve Slika 6. Krivulja prosječne izlazne kvalitete Standard HRN ISO :1989 is foused on the protetion of a produer and the aeptane of shipments whose quality level is the same as AQL, while the buyer's need to identify a bad shipment is put in a position of seondary importane (alpha risk is always higher than beta risk). In order to develop a mathematial model whih enables arbitrary hoie of sampling parameters, it is neessary to study data distribution. For the elaboration of attribute aeptane plans disontinuous distributions are used (Hypergeometri, Binomial and Poisson), while for the making of variable aeptane plans ontinuous distributions are used (Normal, Student's- t, χ 2, Fisher)[11]. Tehnial Gazette 18, 2(2011),

4 Elaboration of a mathematial model for designing attribute aeptane plans D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić 3.1 Hypergeometri distribution Hipergeometrijska razdioba In the sample of N elements there are M elements with attribute A ( ) and NM - elements with attribute Ā (o), as shown in Fig. 8. If a random sample is seleted from n piees, there will be d elements in it with attribute A( ) and nd - elements with attribute Ā (o) [7, 8, 11]. The trials are independent, whih means the outome of one trial does not affet the outome of the other. In ase all the abovementioned points are satisfied, the probability of suess is given in equation (5) whih presents the distribution funtion of binomial distribution. n i ni Pr ( d ) p q. i0 i (5) Cumulative distribution funtion of binomial distribution is given in equation (6) and it presents the probability or less suess realized in n trials, when in the sample there is a fration of defetive parts given in p. Figure 8 Lots of hypergeometri distribution Slika 8. Skupovi hipergeometrijske razdiobe Hypergeometri distribution presents the probability of appearane of ertain elements with attribute A in a random sample size n seleted from the basi lot size N. The distribution funtion is given in equation (3) and it presents the probability of appearane of exatly d elements with attribute A in a representative sample of size n whih is randomly seleted from lot N. M N M (,,, ) d n d H d N M n. (3) N n Cumulative distribution funtion is given in equation (4) and it presents the probability of seleting or less elements with attributea in a representative sample of size n whih is randomly seleted from lot N. M N M ( ) i n i P r d. ( 4) i0 N n Hypergeometri distribution is applied for isolated shipments when the basi lot is small, and by seleting samples proportions of good and bad parts in the basi lot are hanged. 3.2 Binomial distribution Binomna razdioba For easier understanding of binomial distribution it is neessary to explain the binomial experiment whih satisfies these onditions: The experiment onsists of n trials Eah trial an result in only two possible outomes whih an be alled suess or failure The probability of suess p is the same in eah trial (the probability of failure q=1 p) n i ni Pr ( d ) p q. i0 i ( 6) In ase the basi lot Nis big in relation to the sample n, so that seleting the sample from the basi lot does not signifiantly hange the rest of the basi lot, independently on how many defets there are in the basi lot, the defet distribution an be onsidered binomial with parameters n and p [7, 8] Larson nomogram Larsonov nomogram Nomograms are onsidered to be an out-of-date tehnology and are mostly replaed with standards and software, but there are ases where they present a fast and elegant solution. Fig. 9 shows the Larson nomogram whih presents a graphi illustration of umulative distribution funtion of binomial distribution. On the basis of the following example, the way of designing an aeptane plan by using the Larson nomogram is explained: It is neessary to reate suh a sampling plan whih aepts 95 % of samples ontaining 2 % of defetive lots and aepts only 5 % of samples ontaining 9 % of defetive lots. 1. Determine the allowed fration of defetive lots pα to onsider the shipment satisfying: pα = 2 % (0,02) 2. Determine the probability needed for the aeptane of a good shipment: Pα= 1 α = 95 % (0,95) 3. Determine the size of fration pβ of defetive lots to onsider the shipment unsatisfying: pβ = 9 % (0,09) 4. Determine the probability for aepting a bad shipment with Pβ whih an be tolerated: Pβ = 5 % (0,05) 5. Connet the points pα and Pα=1 α with a straight line 6. Connet the points pβ and Pβ= β with a straight line 7. Read the values n and in the nomogram. It an be seen in Fig. 10 that the solution of this problem presents aeptane plan given with parameters n = 100 and = Tehnič ki vjesnik 18, 2(2011),

5 D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić Razrada matematičkog modela za izradu atributivnih planova prijema Q P( d; ) e x!. (7) Cumulative distribution funtion of the Poisson distribution is given in equation (8). P r ( d ; ) e. ( 8) i! i0 i The Poisson distribution is used for approximating the binomial with parameters n and p when n is big and p is small, e.g. n 20 and p 5 and exellent overlap for n 100 and n p The elaboration of mathematial model Razrada matematičkog modela 3.3 Poisson distribution Poissonova razdioba Figure 9 Larson nomogram [4] Slika 9. Larsonov nomogram [4] Figure 10 The solution by using the Larson nomogram Slika 10. Rješenje pomoću Larsonovog nomograma Poisson distribution an be seen as a borderline ase of binomial with parameters n and p, in ase the number of trials approahes infinity ( n ) while p approahes zero ( p 0) where λ = n p =onst. A disrete random variable has the Poisson distribution with parameter λ if its distribution funtion is given in equation (7). P ( d ; ) r i0 i e i!, For an easier understanding of mathematial formulation for the sampling plans design, it is neessary to remember operating harateristi of an aeptane plan. The graph of operating harateristi urve presents the probability of aepting a delivery with a ertain perentage of defetive parts for the mentioned sampling plan. Given that the size of delivery N is big in relation to the sample size n, and that seleting a sample n from the delivery does not signifiantly hange the average struture of the rest of the delivery (independently on how many defetive lots there are in the sample), the distribution of defets d in a random sample n is binomial with parameters n and p, where p presents the fration of defets per delivery. The probability of observing exatly d effets aording to the binomial formula is given in equation (5). The probability of aeptane is the probability at whih the number of defets found in a sample d is smaller or equal to the aeptane number and for the binomial distribution is given in equation (6). In order to design a sampling plan with assoiated operating harateristi urve two points are needed. The first point presents the probability of aeptane Pα=1 α for deliveries with the fration of defets p1= AQL. The other point an be alled Pβ, where the fration of defets is p2= RQL. [5] Given that the binomial distribution is adequate, parameters n and an be obtained by solving nonlinear system of equations obtained in (6): n d nd B( d, ; n, p1) p q, d 0 d (9) 1 1 n d nd B( d, ; n, p2) p2 q. d0 d ( 10) Equations (9) (binomial aeptane probability of a good shipment) and (10) (binomial aeptane probability of a shipment of signifiantly worse quality) present nonlinear system of equations, whih means a diret solution does not exist. There are several iterative methods for solving this system, thus for the design of the aeptane plan a omputer is essential. Equivalently to the equation systems (9) and (10) it is possible, with minor hanges, to use the systems obtained by using equation (4) both for hypergeometri or by using equation (8) for the Poisson Tehnial Gazette 18, 2(2011),

6 Elaboration of a mathematial model for designing attribute aeptane plans D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić distribution. 4.1 Integration of the mathematial model in software appliation Integraija matematičkog modela u računalnu aplikaiju In 1969, W. C. Guenthier found appropriate algorithm for searhing the solution for the system of equations (9) and (10). One of the forms of this algorithm appliable in programming in software appliations is given in the flow hart shown in Fig. 11 [6]. The example of integration of the mathematial model for binomial distribution by using m- file in MATLAB is shown in Fig. 12. On the basis of the ode in Fig. 12, a user interfae is reated for easier input and printout of results and urves, whih is shown in Fig. 13. Figure 11 Chart flow of the algorithm for iterative solution finding Slika 11. Dijagram toka algoritma za iterativno traženje rješenja Figure 13 The example of user interfae designed in Matlab Slika 13. Primjer korisničkog sučelja izrađenog u Matlabu 5 Appliation Primjena Figure 12 The example of algorithm adaptation for implementation in Matlab Slika 12. Primjer prilagodbe algoritma za izvođenje u Matlabu In this hapter, on the basis of examples, lassi examples of usage and omparison with the Larson nomogram and standard are presented. Sine the opying of Croatian standards or their parts is prohibited, tables from the anelled Amerian military standard MIL STD-105E will be used for omparison, together with the name of analogue table from the HRN ISO :1989 standard (standard HRN ISO :1989 derived from the Amerian standard MIL STD-105, with osmeti hanges so the tables and results orrespond). 292 Tehnič ki vjesnik 18, 2(2011),

7 D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić Razrada matematičkog modela za izradu atributivnih planova prijema 5.1 Comparison with the Larson Nomogram Usporedba s Larsonovim nomogramom It is neessary to reate suh a sampling plan whih aepts 95 % of samples ontaining 2 % of defetive lots and aepts only 5 % of samples ontaining 9 % of defetive lots. Sine the task is analogue to the former example solved with the Larson nomogram, following onlusions an be drawn, based on Fig. 13: Operating harateristi urve derived from the data in the Larson nomogram (urve marked with irles in Fig. 13) is plaed below the urve derived by using the mathematial model for the input data (urve marked with squares in Fig. 13), whih means that it will result in striter riteria for aeptane with the same perentage of defetive lots The average outgoing quality urve for the data from the Larson nomogram an also be found below the outgoing quality urve for the input data, whih means that after a number of shipments big enough at the entrane warehouse would be less defetive lots for the aeptane plan derived from the Larson nomogram. number whih desribes the size of the basi lot for the alulation neessary for the printout of theaoq urve. Obtained sampling plan has parameters n = 20 and = 2 whih orrespond to the data in the standard (Fig. 15). 5.2 Comparison with the standard ISO HRN :1989 Usporedba s normom ISO HRN :1989 In order to make the omparison with standards, it is neessary to find input data for the software appliation first. Sine in the standard only AQL and the level of review are defined, it is neessary to inspet arefully the tables and operating harateristi urves for the appropriate ode letter and take out the needed data. As an example, a ode letter F is hosen: HRN ISO :1989, graph F, Table X-F-1, MIL STD-105E, graph F, table X-F-1, Sample size , AQL = 4, n=20, =2, shown in Fig. 14. The following data an be read from the diagram and the table in Fig. 14: AQL =4%; α 5% RQL =24,5%for β =10%. Figure 15 Printout of the results for the ode letter F Slika 15. Ispis rezultata za kodno slovo F Figure 14 Table and operating harateristi urve with ode letter F Slika 14. Tabela i operativna krivulja kodnog slova F When this data is inserted into the appliation, obtained results are shown in Fig. 15. The sample size N when using binomial distribution does not play any part during the design of sampling plans, but it is neessary to insert the Figure 16 Comparison of the overlap of operating harateristi urves Slika 16. Usporedba poklapanja operativnih krivulja Tehnial Gazette 18, 2(2011),

8 Elaboration of a mathematial model for designing attribute aeptane plans D. Kapusta, G. Kozina, D. Vusić, Ž. Kondić, A. Čikić When the operating harateristi urve from Fig. 15 is opied onto the graph in Fig. 14, it is visible that the overlap of the urves is more than satisfying (Fig. 16). Based on the proedure presented in the previous example, it is neessary to design sampling plans for the data given in Tab. 1. Table 1 Input data for omparison Tablia 1. Ulazni podai za usporedbu Ord. Code numb. letter Sample size AQL α RQL β 1 D G ,5 5 15, J , L , ,94 10 Table 2 Compared solutions by using the standard and software appliation Tablia 2. Usporedna rješenja pomoću norme i računalne aplikaije Ord. numb. 1 Standard n=8, =2 Appliation B: n=8, =2 P: n=12, =3 2 n=32, =2 H: n=7, =2 B: n=32, =2 P: n=43, =3 3 n=80, =2 H: n=32; =2 B: n=80, =2 P: n=103, =3 4 n=200, =1 H: n=79, =2 B: n=200, =1 P: n=201, =1 H: n=197, =1 B Binomial distribution, P Poisson distribution, H Hypergeometri distribution It is visible from Tab. 2 that the appliation gives results whih orrespond to the standard. It is also visible that the Poisson distribution approximates well the binomial for large samples and smaller probabilities. 6 Conlusion Zaključak Standard HRN ISO :1989 is reated with the aim to enable a simple and reliable system for the design of sampling plans. The main problem presents the fat that it is formed around the aeptable level of quality AQL and is stern when it omes to the hoie of other parameters for the design of aeptane plans. Larson nomogram presents a fast and simple way for the design of aeptane plans, whih is with parameters adapted to the speifi ase. However, beause of the network distribution, some disrepany of parameters is possible, whih results in insuffiiently ritial or overly ritial aeptane plan. As opposed to the mentioned standard, statistial approah enables the design of mathematial models whih orrespond to speifi situations, but it presents a problem sine there is extensive and slow alulation of umulative distribution funtions in relation to data distribution. This paper presents an overview of one of the possible ways of development and implementation of mathematial model in a software appliation whih is not limited with sampling parameters or with the volume of mathematial alulation and it presents a simple and fast way for the design of aeptane plans adapted for the use in onrete situations. The emphasis is put on design of optimal aeptane plans whih ensure that the obtained parameters and operating harateristi urves faithfully follow requirements ( AQL, RQL, alpha and beta risks, and sample sizes) and by that favor both the produer and the onsumer. 7 Referenes Literatura [1] HRN EN ISO 9000:2008. Sustavi upravljanja kvalitetom - Temeljna načela i terminološki rječnik (ISO 9000:2005; EN ISO 9000:2005) [2] Dusman, F.; Stanče, R. Odabrana poglavlja iz kontrole kvalitete, Sveučilišna naklada Liber, [3] HRN ISO :1989. Postupi uzorkovanja za pregled prema atributima - 1. dio: Planovi uzorkovanja indeksirani prema prihvatljivoj razini kakvoće (AQL) za pregled partija po partija (ISO :1989) [4] Mathews, P.; Mathews, M.; Mathews, B. Design of Single Sampling Plans for Defetives, Malnar and Bailey, In., Fairport Harbor, [5] Natrella, M. NIST/SEMATECH e-handbook of Statistial Methods, U. S. Commere Department, Washington, [6] Joahim, H. M.; Rinne, H. Statistihe methoden der Qualitätssiherung, Carl Hanser Verlag, Wien, [7] Kondić, Ž. ; Čikić, A. Upravljanje kvalitetom u mehatronii, Visoka tehnička škola Bjelovar, Bjelovar, [8] Kondić, Ž. Kontrola i metode poboljšavanja, Zrinski, Čakove, [9] Crossley, M. L. Statistial Quality Methods, ASQ, Wisonsin, [10] Grant, E. L.; Leavenworth, R. S. Statistial Quality Control, MGraw Hill, New York, [11] Dumičić, K. Značaj ispravnog korištenja teorije i metode uzoraka u praktičnom istraživanju. // Ekonomski analitičar, 5(1991), Autors' addresses Adrese autora Danijel Kapusta, ing. stroj. Mr. s. Goran Kozina, dipl. ing. Damir Vusić, dipl. ing. Do. dr. s. Živko Kondić Veleučilište u Varaždinu Križanićeva 33, Varaždin ; zkondi@velv.hr Do. dr. s. Ante Čikić Visoka tehnička škola u Bjelovaru, Trg. E. Kvaternika 4, Bjelovar aiki@vtsbj.hr 294 Tehnič ki vjesnik 18, 2(2011),