Designing Control Charts for Minimum Total Quality Costs

The basic control chart consists of sampling from a process over time and charting process measurements. For as long as the measurements fall inside control limits (L), the process is considered in-control. In feed production, it is customary to set the control limits at two standard deviations. This practice does not consider the probability of both Type I and Type II errors in addition to all the costs incurred. The objective of this research was to derive a general methodology to determine optimum sample size (n), sampling period (h), and location of L for an X chart used to monitor production processes. A quality cycle is defined as the time between the start of successive in-control periods. Let A1 be the average production length while in-control and A2 when the process has shifted to an out-of-control state. The total cost of quality is given by C = {C0/ + C1 (-t + nE + h(A2) + 1T1 + 2T2) + sY/A1 + W} {1/ + (1 1)sT0/A1 t + nE + h(A2) + T1 + T2} + {[a + bn)/h] x [1/ t + nE + h(A2) + 1T1 + 2T2]} {1/ + (1 1)sT0/A1 t + nE + h(A2) + T1 + T2}, with s = exp(h) / (1 exp(h)), t = [1 1 + h)exp(h)] / [ (1 exp(h)], A1 = 1/α, A2 = 1/(1 ), α = Pr(X CL>L /√n) + Pr(X CL <-L /√n < X CL < L /√n), where t = expected time of occurrence of the assignable cause; s = expected number of samples taken while in-control; = 1/mean time process is in-control; E = time to sample and chart one item; T0 = expected search time when false alarm; T1 = expected time to discover the assignable cause; T2 = expected time to repair the process; 1 = 1 if production continues during searches, 0 otherwise; 2 = 1 if production continues during repair, 0 otherwise; C0 = quality cost/ hour while producing in-control; C1 = quality cost/hour while producing out-ofcontrol; Y = cost per false alarm; W = cost to locate and repair the assignable cause; a = fixed cost per sample; b = cost per unit sampled. The values n*, h* and L* that give the minimum C provides the optimum control design. The procedure is easily implemented on a modern spreadsheet. 1 For more information, contact at: The Ohio State University, 221A Animal Science Building, 2029 Fyffe Road, Columbus, OH 43210; 614-292-6507; e-mail: [email protected] Back | Forward | Table of