Scheduling methods for the statistical setup adjustment problem (IJPR, 41(7), pp. 1467-1481): a correction and clarification

In the first equation on page 1471 of our recent paper (Pan and Del Castillo, 2003) the right hand side should say (d− d̂0)/σε and not d− d̂0. In addition, we point out that the cost function (4) is not the same as that used by Trietsch (2000). As explained in the paper, equation (4) is obtained by assuming the setup error or offset d is an unknown, non-random constant. Trietsch (2000) assumes d to be a random variable with known mean and variance. If d is a constant, our derivation gives Var(Yt)/σ ε ( = 1 + t−1 (σ2 ε/P0+t−1) ) since Xi is known at time t(> i). Trietsch (2000), in contrast, assumed d random and arrives at σ 2 ε/P0+t σ2 ε/P0+t−1 which is really Var(Yt|Yt−1, Yt−2, ..., Y1). Our expression for Var(Yt) can be much smaller than Var(Yt|Yt−1, ..., Y1) when σ2 ε/P0 is small, particularly for the crucial first adjustment. This makes sense because in Trietsch (2000) P0 is Var(d) (the setup variance) whereas in our paper P0 is an a priori measure of confidence in the initial guess d̂0 (thus, under constant d, Var(Y1) always equals σ2 ε since the setup error is not random and all variability comes from the part to part error ε1). The comparisons in section 4 of our paper assume that d is an unknown constant. It is noteworthy that Trietsch’s method performs quite well despite the different assumption made on d. We believe it is more appropriate to model d as a constant when interest is on a single lot of product, thus there is only one instance for observing the effects of the error. A random d is probably more appropriate when the same part is produced over several consecutive lots, with a setup before each lot. Finally, we point out that Trietsch (2000) also discussed an optimal network model approach to the setup adjustment scheduling problem (under d random), not considered in our paper, although the differences between such method and his approximate method considered in our paper should be minimal. The analogy with an inventory control problem, which is exploited in our paper, is ours.