Abstract—In some real applications of Statistical Process Control it is necessary to design a control chart to not detect small process shifts, but keeping a good performance to detect moderate and large shifts in the quality. In this work we develop a new quality control chart, the synthetic T 2 control chart, that can be designed to cope with this objective. A multi-objective optimization is carried out employing Genetic Algorithms, finding the Pareto-optimal front of non-dominated solutions for this optimization problem. Keywords—Multi-objective optimization, Quality Control, SPC, Synthetic T 2 control chart. I. INTRODUCTION OWADAYS it begins to be common to face problems or applications where the mathematical modeling produces an optimization problem with several objectives. The multi-objective optimization consists of optimizing simultaneously several objective functions. In many cases, some of the objective functions represent conflicting criteria. Obviously, in these cases no unique solution can be found because the entire objective functions cannot be optimized (maximized or minimized) without considering the effect of the experimental changes in the other response functions. In general terms, the optimization problem can be formulated as follows, being n the number of decision variables, x j , m restrictions and p objectives: Find x (x 1 , x 2 , …, x n ) that Maximize / minimize Z = ( z 1 (x), z 2 (x), …, z n (x)) Subject to x ∈ F With F ⊂ R n , F feasible region of solutions space R n and Z = z(F) ⊂ R p , Z feasible region of objectives space R p . Many times the set F can be written as F={ x ∈ R n : g i (x) , 0 ≤ x j , 0 ≤ j i, ∀ } when g i functions are the restrictions. In some cases, variables z k are called objective functions or objectives. In some real applications of Statistical Process Control it is necessary to design a control chart to not detect small shifts in Manuscript received September 9, 2007.This work was supported by the Spanish “Ministerio de Educación y Ciencia” and European FEDER funding, under Grant DPI 2006-06124. Francisco Aparisi is with the Departamento de Estadística e Investigación Operativa Aplicadas y Calidad. Universidad Politécnica de Valencia, 46022 Valencia. Spain (+34-963877490; e-mail: faparisi@eio.upv.es). Marco A. de Luna is with the Departamento de Ingeniería Industrial y Mecánica. ITESM. 45140 Guadalajara, Mexico (email: mdeluna@itesm.mx). the process mean, but keeping a good performance to detect moderate and large shifts. This design was first posed by Woodall (1985) and it is known as the design for In-control and Out-of-control regions. Although the idea is not new, it is quite difficult to design a quality control chart that can solve this problem. The typical control charts can not be adapted and the optimization problem it is rather difficult. In this work we develop a new chart, the synthetic T 2 control chart, an improvement of the standard Hotelling’s T 2 chart. This new chart can be designed to solve this multi-objective problem. Therefore, the objective of this work is to apply Evolutionary Multi-Criterion Optimization to the design of the synthetic T 2 control chart to solve the problem of In-control and Out-of-control regions and to find the Pareto-optimal front. II. THE SYNTHETIC T 2 CONTROL CHART A. Defining the Synthetic T 2 Control Chart The main objective of quality control charts is to detect shifts in the production process that are due to assignable causes. Samples are taken from the process and the calculated statistic is plotted in a chart. Is the point is plotted outside the control limit(s) we have to assume that there is an assignable cause of variation; we assume that there is a shift in the variable(s) we are monitoring, see [1]. A measure of performance of a control chart is the ARL (Average Run Length). It is the average number of points (or samples) that we have to plot until the chart signals. If there is really a shift in the process the ARL has to be as minimum as possible. However, when there is no a shift, the ARL must be as maximum as possible. In order to control shifts in the process mean, the Shewhart control chart is the most widely used control chart. However, its performance to detect small shifts is not good (large ARL values). The univariate synthetic chart (only one variable is monitored) was introduced in [2] as an alternative to improve the performance of the Shewhart control chart to detect small process shifts. It is the result of combining a Shewhart chart and a CRL chart (a chart originally designed to detect increments in the percentage of defective units). The synthetic- X chart shows better ARL values to detect process shifts, for any shift magnitude, than the X control chart. In some cases, especially for moderate and large shifts, the synthetic- chart has better performance than the EWMA control chart [2]. The Synthetic T 2 Quality Control Chart and its Multi-Objective Optimization Francisco Aparisi, and Marco A. de Luna N World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:1, No:11, 2007 3447 International Scholarly and Scientific Research & Innovation 1(11) 2007 scholar.waset.org/1307-6892/4917 International Science Index, Computer and Information Engineering Vol:1, No:11, 2007 waset.org/Publication/4917