Location models

Heat transfer models are a powerful tool for developing autoclave process cycles. They are especially useful in aiding tool designers in choosing tooling materials, thicknesses, and thermocouple locations. Models can also be used to determine if a tooling concept would be detrimental in a specific position in the autoclave and the types of tools that should be processed together to optimize the cure cycle. 

FIGURE 11 Segment of simulation model illustrating New Delhi campus network. This is where the business process outsourcing team and related servers in our distributed system are located. Modeling tool is OPNET. 

Figure 16 contains the most important criteria to classify facility location problems. Regarding solution space, as Francis et al. (1983, pp. 221, 240) explain, discrete location models are the most realistic (especially be-... 

Kouvelis et al. (2004) present a relatively simple multi-period MILP plant location model for global production network design with investment decisions only allowed in the first period. The production system consists of component-dedicated manufacturing sites and final assembly sites. It is limited to two production levels and one final product. The objective function maximizes the NPV of the production network. The main purpose of the model is to analyze the effects financing subsidies, tax regimes, tariff structures and local content requirements have on optimal network design. The analysis is based on theoretical considerations and a numerical example. More complex aspects of international trade such as duty drawbacks are not considered. 

Aikens CH (1985) Facility location models for distribution planning. European Journal of Operational Research 22 263-279... 

Antunes A, Peeters D (2001) On solving complex multi-period location models using simulated annealing. European Journal of Operational Research 130 190-201... 

Eiselt HA (1992) Location modeling in practice. American Journal of Mathematical and Management Sciences 12 3-18 Eiselt HA, Laporte G (1995) Objectives in Location Problems. In Drezner Z (ed) Facility Location. Springer, Berlin et al., pp 151-180 Eiteman DK, Stonehill AI, Moffett MH (2006) Multinational Business Finance, 11th edn. Pearson Education, Boston et al. 

Hamacher HW, Nickel S (1998) Classification of location models. Location Science 6 229-242... 

Figure 16 Chiral recognition based on a four-location model in the case of 3-methyl-2-butanol. (a) Two enantiomeric alcohol molecules in the channel of NDCA (b) an electron density map of the alcohol on a plane composed of Cl, C2 and C3 carbons.

Locate model cSscontimities 1 Rewlialize after variable step-change... 

Figure 13.24 Structure of food webs and organisms in the Gulf of Bothnia at three locations. Model structure and arrows correspond to the outer component in figure 13.23. Total dissolved organic matter = TDOM. (Modified from Sandberg et al., 2004.)...

A plot of the residuals versus the fitted values from the location model with A and F only is shown in Figure 4. The spread of the residuals is seen to increase with the fitted value, a feature that might be explained by dispersion effects of the experimental factors, in particular those in the location model. We examine this conjecture with the various methods that have been suggested for screening dispersion effects. 

Figure 4. Plot of residuals versus fitted values from the 2s experiment on hue. The location model includes the main effects of factors A and F only.

Box and Meyer (1986) considered the identification of dispersion effects in unreplicated 2k p designs. The first step in their approach is to identify and estimate the active location effects. Let r,(i = 1,..., ) denote the residuals from the fitted location model. To examine whether factor j has a dispersion effect (or, equivalently, whether there is a dispersion effect associated with the jth main effect contrast), compute the sums of squared residuals at the two levels of this factor ... 

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

Figure 5. Normal probability plot of the Box-Meyer dispersion statistics for the 2s experiment on hue, with A and F in the location model.

Bergman and Hynen (1997) developed a method similar to that of Box and Meyer (1986), but with a simple and exact distribution theory for inference from the test statistic. The important observation of Bergman and Hynen was that the residuals from the fitted location model could complicate inference for the Box-Meyer statistic in two ways. First, the residuals in the two sums of squares could be correlated. Second, the residuals at the high (low) level of factor j typically depend on the actual variances at both levels of the factor, not just the level at which the run was made. 

Table 1. Expanded location models for the Bergman-Hynen method for several potential dispersion effects when the location model includes the main effects of factors A and B. The location model always includes main effects for A and B and also includes the main effect of the dispersion candidate and its interactions with A and with B...

On the dyestuffs example, the Bergman-Hynen method also signals factor F as being related to dispersion. With the main effects of A and F in the location model, F has a Bergman-Hynen statistic of 3.27 (p-value = 0.001). The next strongest effects, as with the Box-Meyer method, are the ADEFinteraction, with a statistic of 2.24 (p-value = 0.017) and the CEF interaction, with a statistic of 0.45 (p-value = 0.035). 

Brenneman (2000) found that Harvey s method could underestimate the dispersion effect of factor j if that factor was left out of the location model. This result led Brenneman and Nair (2001) to propose a modified version of Harvey s method for two-level factorial experiments that is based on the results of Bergman and Hynen (1997). In the modified version, the dispersion statistic for factor j is computed from residuals from an expanded location model that includes the effect of factor j and all its interactions with other effects in the location model. For two-level designs, the modified Harvey s statistic for factor j is then... 

The dyestuffs example illustrates the potential for problems with Harvey s method. With only A and F in the location model, the ABDEF interaction clearly stands out with the largest value of the statistic D ". However, subsequent... 

An unbiased estimator of j by... 

For the dyestuffs experiment, we applied the McGrath-Lin method to all 15 pairs of main effects with factors A and F in the location model. Factor F consistently had a strong dispersion effect, with p-values of 0.0001 to 0.004, depending on the second factor in the pair. Factor A was also found to have a potential dispersion effect, with values of 0.028 to 0.16. The />value for factor A was less than 0.05 except when paired with factor C. So the analyst is left with a practical problem of deciding whether factor A has a dispersion effect. The analysis provides conflicting evidence about factor A and it is not clear which pairing(s) should dictate the decision. 

Brenneman and Nair (2001) proposed a strategy that combines their modified version of Harvey s method with joint location and dispersion modeling for a log-linear dispersion model. After fitting a location model with ordinary least squares regression, they recommended an initial check to see if there are sufficient degrees of freedom even to consider looking for dispersion effects. The condition they... 

Liao (2000) derived a test statistic for single dispersion effects in 2" k designs. He applied the generalized likelihood ratio test for a normal model to the residuals after fitting a location model, which results in Bartlett s (1937) classical test for comparing variances in one-way layouts. The test is then applied, in turn, to compare the variances at the two levels of each of the k experimental factors. We caution that the test statistic (equation (3) in Liao) is written incorrectly. 

Correct identification of the location model is a serious problem that affects all the methods for identifying and estimating dispersion effects. Pan (1999) showed that small to moderate location effects that are undetected can seriously impair... 

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