Sequential bifurcation

Bettonvil (1995) discussed a particular form of the multiple group screening idea called sequential bifurcation, in which... 

Chapter 13 reviews sequential bifurcation in more detail. 

Table 2. Example of experimental runs added at each stage and used in each analysis sequential bifurcation ...

Bettonvil, B. (1995). Factor screening by sequential bifurcation. Communications in Statistics—Simulation and Computation, 24, 165-185. 

Screening for the Important Factors in Large Discrete-Event Simulation Models Sequential Bifurcation and Its Applications... 

In this chapter, we explain the technique of sequential bifurcation and add some new results for random (as opposed to deterministic) simulations. In a detailed case study, we apply the resulting method to a simulation model developed for Ericsson in Sweden. In Sections 1.1 to 1.3, we give our definition of screening, discuss our view of simulation versus real-world experiments, and give a brief indication of various screening procedures. 

Our case study is introduced in Section 1.4. The assumptions behind, and the steps involved in, sequential bifurcation are described in Sections 2.2 and 2.3 and the steps are illustrated using a first-order polynomial model with random noise. In Section 2.4, a more realistic model involving interactions is used for screening the important factors in the case study. Issues of programming are addressed in Section 3. 

Originally, sequential bifurcation was developed in the doctoral dissertation of Bettonvil (1990) which was later summarized by Bettonvil and Kleijnen (1996) and updated by Campolongo et al. (2000) to include a discussion of applications. Other authors have also studied sequential bifurcation (see, for example, Cheng,... 

Cheng and Holland, 1999). Sequential bifurcation is related to binary search (Wan et al., 2004) which searches a sorted array by repeatedly dividing the search interval in half, beginning with an interval covering the whole array. First, we give an outline of the sequential bifurcation procedure (Section 2.1). Second, we present the assumptions and notation of sequential bifurcation (Section 2.2), and, third, we illustrate the procedure through our case study (Section 2.3). 

The efficiency of sequential bifurcation, as measured by the number of observations (that is, simulation runs and hence simulation time), increases if the individual factors are renumbered to be in increasing order of importance (see Bettonvil 1990, page 44), so that... 

We introduce the following additional sequential bifurcation notation adapted for replicated random responses. We use y( ) r to represent the observed (simulation) output in replicate r when factors 1 to j are set at their high levels and the remaining factors are set at their low levels (r = 1,..., m). 

The sequential bifurcation procedure starts by observing (simulating) the two most extreme scenarios. In scenario 1, all factors are at their low levels and, in scenario 2, all factors are at their high levels. From the metamodel (1), we obtain the expected values of the response variables as... 

Figure 3. The steps of sequential bifurcation applied to the Old supply chain configuration, assuming a first-order polynomial metamodel. Estimates of the Pf-j defined in equation (6) are indicated at each step.

Table 1. Observations for the first two scenarios simulated in sequential bifurcation for the Old supply chain...

In hindsight, we might have used fewer replications in the early steps of the procedure, as these steps have higher signal/noise ratio due to the fact that the signal decreases as a result of less aggregation of main effects as the sequential bifurcation progresses. 

The aggregated or group effect is an upper limit U for the value of any individual main effect. The goal of sequential bifurcation is to And the most important factors, that is, the factors that have significant main effects. If, however, we terminate our screening prematurely (for example, because the computer breaks down or our clients get impatient), then sequential bifurcation still allows identification of the factors with the largest main effects. 

For this case study, sequential bifurcation stops after 21 steps. The upper limit, denoted by 1/(21), for the main effect of any remaining individual factor is then... 

In the next section we consider a more realistic metamodel that includes interactions and we illustrate the design and analysis of experiments for sequential bifurcation under such models. 

Table 3. The important factors found from sequential bifurcation under two meta-models for the Old supply chain simulation. Details of the factors are given in Table 2...

Two-Factor Interactions and Foldover Designs in Sequential Bifurcation... 

Sequential bifurcation may give misleading results if, say, two factors have unimportant main effects but the interaction between them is important (see, also, Lewis and Dean, 2001). However, we only consider situations in which the following strong heredity assumption of Wu and Hamada (2000) holds. 

Figure 4. Sequential bifurcation assuming a first-order polynomial plus two-factor interactions metamodel, applied to the Old supply chain simulation includes upper limits for parameter values.

Analogous to Figure 3, Figure 4 shows the sequential bifurcation steps when we do allow for interactions. A comparison of these two figures and the two lists of important factors in Table 3 shows that, in our case study, we find the same shortlist. The individual values, however, do differ interactions are important. 

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