Parameter analysis in quality control and an application of markov chains to a quality control problem

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), (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 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 0Keyword: Kalite kontrol = Quality control ; Markov zinciri = Markov chain ; Parametre analizi = Parameter analysis