Kalite kontrolunda parametre analizi ve markov zincirlerinin kalite kontrolu problemine bir uygulaması
Parameter analysis in quality control and an application of markov chains to a quality control problem
- Tez No: 22019
- Danışmanlar: DOÇ. DR. CEVDET CERİT
- Tez Türü: Doktora
- Konular: İşletme, Business Administration
- Anahtar Kelimeler: Kalite kontrol, Markov zinciri, Parametre analizi, Quality control, Markov chain, Parameter analysis
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Günümüzde mamul yapısının ve imalât yöntemlerinin arz ettiği karmaşık yapı ve üretim miktarlarının büyüklüğü, kalite problemi, sonuç itibariyle de kalite kontrol olayı üzerinde dikkatlerin odaklanmasına neden olmaktadır. Rekabetin giderek artması, kalite unsurunun bu sahada ürüne güç katması, kalitenin itici bir güç olarak ortaya çıkmasında kalite kontrolünün önemli bir rol oynaması; sonuçta kalite kontrol faaliyetlerinin öneminin hızla artma sına neden olmuştur. Kalite kontrolü, doğal olarak beraberinde kalite maliyeti problemini getirmektedir. Bu çalışmada kalite kontrol maliyetlerinden faydalanarak, kalite kontrolü problemine bazı yeni bakış açılarının kazandırılması amaçlanmaktadır. Bölüm. 1 de konuyla ilgili temel kavramlar açıklanmış, kalite kontrol maliyetlerinin toplam kalite maliyeti için deki yeri ve önemi vurgulanmıştır. Bölüm. 2 de, toplam kalite maliyetini minimum kılacak bir q0(t) kalite seviyesinin bulunabileceği gerçeği ifade edilmiş, firmanın net geliri kalite kontrol maliyetleri cinsinden formüle edilmiştir, t anında firmanın net gelirini gösteren n(t) fonksiyonunun maksimize edilebilmesi için izlenmesi gereken kalite yolu araştırılmıştır. Bölüm. 3 de, n(t) net geliri bir birim ürünün kontrol maliyetinin ve piyasaya intikal eden bozuk malların maliye tinin de dikkate alınmasını sağlayacak tarzda yeniden formüle edilmiş, üretilen malın tamamının satıldığı varsayılıp, talebe olumlu yönde katkıda bulunan faktörler kazanç (varış), olumsuz yönde katkıda bulunan faktörler kayıp (servis) olarak vurgulanıp, probleme bir kuyruk problemi olarak bakılabilmesi sağlanmıştır. p kusurlu oranını ve örnekteki kabul edilebilir r adet bozuk mal sayısını dikkate alarak kazancın beklenen değeri ve parametrelerin ala cağı muhtelif değerlere göre E[n(t)] nin izleyeceği yol incelenmiştir. Bölüm. 4 de, Kaleporselen Elektroteknik Sanayi A.Ş. de üretilen DIII sigorta buşonlarının 1976-1986 yılları arasında gerçekleştirilen üretim miktarları veri olarak kullanılıp, Bölüm. 3 de geliştirilen modelin parametrelerinin mü tekabillerinin analizi yapılmış, kümülatif fonksiyonları oluşturulmuştur. Üretim miktarının muhtelif aralıklar üzerindeki hareketi, bu aralıklara ait noktaların gerçekleşme si olasılıkları vasıtasıyla incelenmiştir. xii
Özet (Çeviri)
PARAMETER ANALYSIS IN QUALITY CONTROL AND AN APPLICATION OF MARKOV CHAINS TO A QUALITY CONTROL PROBLEM SUMMARY Statistical methods and several other techniques im proved in relation with these methods are used widely in industry and in industrial quality control. The complexity of the structure of the product and the production methods of today and the hugeness of the amount of the production have caused us to focus our attention on the problem of quality control and quality concepts. The increase in competition and the importance of the support of the quality of the product in this area, the role of the quality control as for a supportive element in quality have increased the importance of quality control activities. With the problem of quality control the prob lem of quality control costs have arisen naturally. Our aim in this work is to add some new perspectives to the analysis of the quality control costs problems. The prob lem of the optimization of the income of the firm by using the quality costs as inputs and surveys realized on the optimum quality path were investigated by formulating the problem as a problem of calculus of variations and solving it with the aid of some techniques involving the concept of quality control. The problems of an increase or a decrease in the amount of production in a sufficiently small inter val At and the changes observed in the amount of production on several intervals are formulated as birth and death chains and investigated by means of the techniques Markov chains provide. In Chapter One, basic concepts related to the topic were explained, the role and the importance of the quality control costs in total quality cost was determined. In Chapter Two, optimum quality costs and the zero de fects concept were explained. The investigations on the xiiianalysis of zero defects concept were briefly summarized. In the investigations concerning zero defects concept, to tal quality cost has been used as a starting point and this approach was called as the traditional approach. Total quality cost (C12[q(t)] ) which depends on the variable qual ity level of the product [q(t)] was expressed as a function of quality costs. It is assumed that the accumulation of knowledge concerning quality (bx[z(t)] and b2[z(t)] ) and the technological advancements in this field (a^t) and a2(t)) will decrease the internal and external failure costs (Cx£q(t)] ) and prevention and appraisal costs (C2[q(t)] ) per unit of the product at quality level q(t), (O^q(t)^l). The costs in question are expressed as q (qr, t) =a1 ( t) bx [z ( t) ] Cx [q{ t) ] C, (qr, t) =a2 ( t) 2>2 [z ( fc) ] C, [g( t) ] The net income ix(t) of the firm at time t was formu lated as n ( t) ={P[q( t) ] -ax ( t) bx [z(t)] Cx [g( t) ] - -a2 (t)b2[z(t)]C2 [q( t) ] -C3 ( t) )D[q{ t) ] and the best of the paths which are available for q(t) was explored by maximizing n(t). P[q(t)] is the price per unit of the product at q(t) quality level. C3(t) is the produc tion cost per unit of the product except the costs concern ing quality. D[q(t)] is the amount of the demand at q(t) quality level. A functional given below was defined in order to use in maximization of it(t), <p(D =/Vrt7t(c)dt:. XXVq>(T) is the present value of the total income which is sup posed to accumulate at the end of a planning period of T, ix(t) is the net income at time t and r is the discount fac tor. It is necessary to optimize <p(T) in order to optimize n(t). If the necessary condition which is formulated under the name of Euler equation for a functional to have an ex- tremum is applied to cp(T), we get the following result: [q*(t)]' is non-negative if and only if (d/dt)(C1D)^0. This means that the net profit n(t) of the firm will keep on increasing as the quality level q(t) increases as long as (d/dtMCiDJâO. The problem was also investigated when the demand is constant and it is found that as long as ux+u2 is smaller than vx+v2 the best strategy is to increase the level of quality. Here ui=a'1bi(z)C1 and vi=haib'i(z)* *(Ci-qC1)/ (i-1,2). In Chapter Three, it is assumed that the most important of the factors which determine the demand is the product ' s quality which has a varying character in time and the a- mount of production is determined only by the amount of de mand. It is also assumed that the production is equal to the demand, that is D[q( t)]=X(t). The other factors in the determination of the demand are listed below: - consumers who have used faulty products and decide not to repeat purchase, - consumers who have switched to other products, - consumers who have been affected by advertising, seller reputation, warranties, servicing reputation and switched from competitors to this firm, - consumers who have been affected by those customers who have used products free from defects, - consumers who have decided not to repeat purchase due to negative word-of -mouth effect. The effects of these factors to the problem are reflected by the symbols a, 6, 6, a and p respectively. The rate of defective items p is considered as a random variable and it is assumed that Ax of the product, whose amount is x, has been the subject of the final control; A is the rate of product which has been the subject of the final control in the total product and 0<A£1. In this chapter, the net profit of the firm is reformulated by xvconsidering the cost of control of one unit of product (Ci[q(t)]) and the loss caused by one unit of product (Ca[q(t)] ) which has not been the subject of the final con trol. If the number of defective items in Ax items which were inspected is less than a predetermined number r, it is assumed that the level of a desirable quality is attained and the whole lot except those found defective in the sam ple is sold. If the number of defective items exceeds r, the whole lot is inspected, the defective ones are discard ed and the non-defective ones are sold. If C(x,p) is re garded as the inspection cost of x units, when the rate of defective is p, the inspection cost can be formulated as (AxC±+ {x-Ax) pCs, yzr xCt, y>r Let ix1(t) be the income obtained from the sale of a product, then the income obtained from the sale of a prod uct can be formulated as TCj.it) =P[q(t)] -Ck[g(t)] -Cjq(t) ] -C3(t) Ck[q(t)] is the internal failure costs per unit of the product arising due to the quality level and Ck[q(t)]=C1[q(t)]-Cs[q(t)] Ce[q(t)] is the prevention and appraisal costs except inspection cost per unit of the product at q(t) quality level and Ce[q(t)]=C2[q(t)]-Ci[q(t)] The net profit of the firm at time t will be formulated as % ( t) =1^ (x-Axp) - [AxC±+ (x-Ax) pCs], y<,r -Ttxx{l-p) -xC±, y>z xviand its expected value as E [it ( t) \p] =E[x(t) ] {{n1 (1-Ap) - {AC±+ (l-A)pCs] )P(y*r) + [n1(l-p)+Ci]P(y>r)}. The firm in question may lose or gain a customer in unit time. The losses may be due to the dissatisfactions caused by faulty products or to the tendency which exists in the nature of human beings of switching to different products serving to the same purpose. Let a and 6 be the parameters representing the affects of these two actions. In the field determined by these assumptions the rate of loss which will occur for one product may be formulated as u=a (l-A)pP(yzr) +6 (1-p) The gain which is a combination of the actions symbol ized by the parameters 9, o and p will form the base for formulating the rate of gain for one product as X=Q+a (1-p) -p (l-A)pP(yir) If we consider the rate of gain as rate of arrival and the rate of loss as rate of service, then we can formulate the problem as a linear queuing problem and make use of the results already obtained in this field. The expected value of the amount of the product manufactured by this firm at time t, in other words the share of the firm in the market as far as this product is concerned will be obtained in term of the parameters a, 6, 0, a and (3 as £[x(t)]=i*exp{[8+(a-6) (1-p) - -(a+p) (l-A)pP(yzr)] Û. xviiIn the case for p having a degenerate probability density function the expected value of the profit is formulated as E[% ( t) ] »i [ (1-A) (njt+Ci-pC.) V+itj. (1-p) -C±] * *exp{[6+(o-6) (l-p)-(a+P) (1-A)pv]t). Here v=P(y£r). The optimum value of r which is the upper boundary for the faulty products in the sample is investigated for the cases where v=0 or 0<v<l or v=l in order to maximize the expected value of the profit. If a-6<0 then the optimum value of p is C,- *i 1 (0-8) t For p*, E[it(t)] will be maximum. In Chapter Four birth and death probabilities of a lin ear queuing chain were related to the birth and death rates of Chapter Three as px=QAt+a{x-xp) At-p (x-n)pP(y^r) At. qx=a {x-n) pP (yzx) A fc+6 (x-xp) A t. and the probability of having no occurrence as rx=1-Px-Gx xviiiwhen the share of the firm in the market is x, the proba bility of decreasing to a share a before increasing to a share b is formulated as £-1 u(x) =PATa<Tb) =^ -, a±x<b 1 y ya Here u(a)=l, u(b)=0, rv_x=l and _ Q.if' yfy'+i " ^-y p -.v ^i _^y # V-Ky<d. y Pv--Pv+i - P. y For an application of the theoretical knowledge ex- presssed in Chapter Three and Chapter Four the data about Dili fuse links produced by Kaleporselen Elektroteknik Sa nayi A.Ş. during the period 1976-1986 were used. By means of these data the probability density functions for the pa rameters a, 6, 9, a and p were obtained. The values of these parameters are determined by means of simulation and used in u(x). XIX
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