OPTIMAL RESPONSE SURFACE DESIGNS IN THE PRESENCE OF DISPERSION EFFECTS 61 it is not necessary for the optimal design to require the variances at the low and middle levels of the vari- runs at all nine design locations. In addition to requiring three levels of each vari- g the varia structure type for each variable,are linear-linear inear-quadra at its low level,a value of o when the variable is at andatqusdhai vae ofI when the variable nverse quadratic sidered for eacl complete variance structure.Half of the cases involve eao大区hogon n which a me for each variable ance ratio but are。co parameters Th our cases conside d include ca s in which th Variance Structures as 1:1.1:4 and 1:2.1:8.Fortunately.the general con heariance ons depend on the size of the variance ratios,no rogeneous variance patter u in th of onof th hetero ussion,only six variance ratio cases are presented for alled"variance structure types",and the degrees of each variance structure simplest involves homo heterogeneity,callec ratios follows is a th based on ariable Small to moderate degre of dispersio models specified in (1).(2).and (3) 1:5.15 (0 by varianc ratios rsion cat ed,and th sented which fall within this moderate disp egory include 1:4,1:4:1:2.13:and 1:1,1:4).Severe by stating the appropriate variance structure type rati for each variable. 6.8ad22 The individ of the l riances are d but not presented include 1:6,1:6:1:8,1:8:and 1:2. to the highest variane ×2.denoted 1:8 1:V.The values of V and their interpretations are The variances that result from the specification presented shortly nce ratio t One variance structure type is a"linear"varianc and different variance ratios.This standardization (V)/2.and the high level of the variable hasa efactorandceat that are all positive and which sum to nine ariance of V Us- ng a three-lev variance struc MM type is a"quadratic variance structure,which has in the design region ble being os the low and high leve VLH VMH VHH VLM VMM VHM at the VLL VML -VHL ing at the middle level of the variable.Finally cubic"variance structures are considered,in which As an example,consider the case with variance Vol.29.No.1.January 1997 Joura!of Quality Technology Reproduced with permission of the copyright owner.Further reproduction prohibited without permission.Reproduced with permission of the copyright owner. Further reproduction prohibited without permission