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45 direction of greatest ease is perpendicular to that of greatest difficulty. Figure 10.1 shows that for d, with p 2, these directions are at 45° angles to one another. Indeed, the bestfit parameters characterize road networks. Consider ufj. If and p = 1, the network is basically a rectangular grid. The ratio of travel distance to straightline distance increases from 1 when travel is parallel to an axis, to fl when travel is at a 45° angle to an axis. As increases while p == 1, the form and degree of directional bias remain the same, but there is an increasing nonlinearity in the roadway. The degree of rectangular bias reduces as p increases to 2. If p = 2, the road network is highly developed and relatively free of directional bias. If p > 2 the directional bias is inverted because travel is most efficient at 45° angles to the axes and least efficient parallel to the axes. 10.2 EMPIRICAL STUDIES Two goodnessoffit criteria were used to determine the distance function parameters. The first is the minimization of a sum of absolute deviations given by: i 1 v AD, = X S  Figure 10.1 Contours for = <i\ + = L (Reprinted by permission, "Mathematical Models of Road Travel Distances," Love and Morris, Management Science. Vol. 25, No. 2, February, 1979. Copyright (1979), The Institute of Management Sciences, 290 Weslminster Road, Providence, RI 02903.) where Aj, is the actual travel distance between points Uj and a„ and v is the number of points chosen for consideration. The implication of using this criterion is that the distance function whose parameters are being determined will estimate the greater actual distances more accurately than the shorter distances in the sample. The second criterion is the minimization of a sum of squares given by: JI ( = /h Division by Aj, normalizes the squared deviation and makes this criterion more sensitive than the first to large errors in relation to Aj,. The parameters of the functions were chosen as those best fitting the given criterion. This was done by performing exhaustive searches over intervals, as described in the Appendix, mathematical note 10.1. The differing estimating accuracy of the functions was associated with statistical significance using the Mest for matched pairs. The test is unaffected by the lack of independence of the error in the distance functions. Urban and Rural Road Distances Results of fitting the functions to urban and rural road distances are given in Tables 10.1 to 10.5. Ohio distances (in miles) were supplied by the Ohio Department of Transportation. Wisconsin road distances were measured directly from published road maps (in inches). Each sample set Table 10.1 Minimizing Parameters of d,. Sample        Canton  123.05  1.04   14.11  1.03    (136.71)"  (0.95)   (17.86)  (0.96)   Columbus  86.097  1.02   10.53  1.01    (88.35)  (0.99)   (10.91)  (1.00)   Madison  1015.96  0.98   157.49  0.98    (1057.83)  (0.97)   (163.42)  (0.98)   Milwaukee  628,95  0.95   66.86  0.94    (727.10)  (0,95)   (87.88)  (0.94)   Rural Wis.  508.42  1.05   94.95  1.04    (515.81)  (1.05)   (96.38)  (1.04)   Toledo  99.73  0.97   14.47  0.98    (114.83)  (0.99)   (18.30)  (0.98)   Toledo Sub.  69.60  0.97   11.42  0.97    (76.12)  (0.94)   (13.98)  (0.94)  
"Values in parentheses correspond to the original axes.
Table 10.2  Minimizing Parameters of di    Sample      Canton  78.39  1.20  7.42  1.22  Columbus  80.20  1.27  8.78  1.28  Madison  869.40  1.25  125.73  1.25  Milwaukee  714.83  1.16  72.17  1.18  Rural Wis.  480.19  1.35  72.90  1.34  Toledo  74.13  1.21  8.70  1.24  Toledo Sub.  49.17  1.21  7.18  1.23  Table 10.3  Minimizing Parameters of c/,.    Sample      Canton  65.62  1.16,1.49  6.03"  1.18,1.56  Columbus  68.54  1.18,1.47  6.86  1.18,1.45  Madison  830.34  1.16,1.48  113.24  1.18,1.56  Milwaukee  508.81"  1.03,1.30  46.31"  1.07,1.35  Rural Wis.  452.74»  1.24,1.45  70.45  1.29,1.68  Toledo  71.69  1.18,1.74  8.41"  1.20,1.73  Toledo Sub.  48.13"  1.18,1.78  7.07  1.21,1.81  "Result is for axes rotated through the angle given in Table 10.1; unmarked results  correspond to the original axes.     Table 10.4 Minimizing Parameters of dt    Sample   k,p,s   k,p,S  Canton  60.34  1.43,1.66,1.78  4.10  1.53,1.81,1.99  Columbus  53.76  1.45,1.38,1.52  4.39  1.49,1.41,1.57  Madison  824.20  1.01,1.56,1.51  113.20  1.14,1.54,1.53  Milwaukee  508.81"  1.06,1.30,1.30  46.29"  1.03,1.34,1.33  Rural Wis.  447.10  1.56,1.50,1.59  70.24  1.37,1.65,1.68  Toledo  54.90  1.56,1.68,1.90  4.95  1.56,1.70,1.92  Toledo Sub.  34.08*  1.55,1.88,2.12  2.69  1.64,2.35,2.70  Same as for Table 10.3. 
Table 10.5  Minimizing Parameters of ds.     Sample       Canton  63.37  1.47,  0.41,2.02  5.17  1.47,   0.29,1.92  Columbus  70.99  1.46,  0.09,1.78  7.52  1.50,   0.04,1.81  Madison  855.30  1.65,  0.01,1.46  120.00  1.64,   0.00,1.44  Milwaukee  657.70  1.64,  0.09,1.27  62.05  1.56,   0.11,1.31  Rural Wis.  430.60  1.82,  0.48,1.90  66.80  1.81,  0.36,1.85  Toledo  67.90  1.52,  0.23,1.37  8.01  1.57,  0.21,1.47  Toledo Sub.  47.54  1.58,  0.09,1.43  6.77  1.70,   0.08,1.44 
consisted of 105 road distances corresponding to distances between all pairs of 15 randomly selected points. To explore the effects of a natural barrier, a Toledo subarea sample (Toledo Sub.) was established. This consisted of 15 randomly chosen points west of the Maumee River, which divides the city of Toledo. The other Toledo sample set included points on both sides of the river. The cities were chosen to provide a spectrum of road network regularity and varying degrees of accommodations to physical obstructions. The goodnessoflit of di to the actual road distances was superior to that of di, the other threeparameter function, except for the rural Wisconsin data. Statistical significance (at the 1% level) accompanied this superiority in the Columbus, Milwaukee, Toledo, and Toledo Sub. samples, di includes d„ di, and d, as special cases and therefore estimates distance at least as accurately. This structural superiority led to statistical significance at every opportunity except for d and d in the Madison and rural Wisconsin samples. The fitted parameters of d do not satisfy convexity conditions for any but the Madison and Milwaukee data sets. This would complicate the use of d in facilities location objective functions that are part of urban location models. The fitted parameter values for di and d satisfy the respective convexity conditions in every case. The rectangular distance function d, does not fare well compared to the alternatives. This inferiority was usually supported by clear statistical significance. Only for the Milwaukee sample did the accuracy of u?, surpass even that for the Euclidean function dj, and this was not statistically significant, dj, which models road systems as having no directional bias, emerges as preferable to d, on the basis of accuracy and convenience. Only the inflation factor must be estimated. If an optimal coordinate systei were not incorporated into d, via the angle , the dominance of u?2 over di would be complete. A travel network must have a particularly strong rectangular bias for u?, to come to the fore. Then why has so much of the analysis in previous chapters addressed this metric? Settings such as plant floors where travel is along aisles laid out in a rectangular grid surely satisfy the rectangular bias requirement (see below). Many central sections of city street networks approximate a rectangular grid. Moreover, di possesses structure that often leads to eflftcient solution algorithms for solving location problems. The median weight property of Section 2.2 is an example. The twoparameter function dy is not decidedly inferior to either di or Js, each of which requires the estimation of three parameters. Values of p do not appear to be consistently close to unity, further substantiating the lack of a strong rectangular bias in the road networks of the chosen study areas.
The "elliptical" function proves to be relatively accurate. The absolute value of wii is typically close to zero, but because m, , the locus of points satisfying diqfi) = 1 is indeed an ellipse. The rural sample produces a unique parameter pattern. For this sample, m, = and the magnitude of exceeds that found in other samples. And ds best estimates the rural distances under both criteria. The rural distances sample leads to the greatest value of in u?„ ?2, and u?3. Values of p show no tendency toward 1 or 2 in dy, whereas the values of p and 5 that are similar for d also fall almost midway between 1 and 2. Intercity Road Distances Two additional data sets were constructed to study the effect of length of trip on the parameters and on relative estimating accuracy. The points were 12 cities from the mileage panels of a Wisconsin road map and a Rand McNally Road Atlas. Results are reported in Tables 10.6 and 10.7. Perhaps not surprisingly, u?, proves to be relatively inaccurate. This is most pronounced in the U.S. sample, / is relatively accurate, even outperforming di for the Wisconsin sample. The values of p and 5 in u?4 were approximately equal for each sample and according to each fit criterion. This adds to the appeal of d, which sets p and 5 equal. The convexity condition p/s > 1 of ?4 is satisfied for the U.S. sample and pjs is just slightly less than unity for the Wisconsin sample. This is in opposition to the typical result found above. We observe that parameters differ substantially over the various data sets. Analysts should therefore determine their own "fudge factors" and not generalize from the problemspecific experience reported here. Table 10.6 Minimizing Parameters Under ADf. Values of ADf  Minimizing Parameters  f Wis.  U.S.  Wis.  U.S.  1 625.3  5192.4  A:=0.86, =3  A:=0.90, 617  1» 638.7  5574.7  A:=0.87  A;=0.91  2 416.5  2172.2  A:=1.16  A:=1.18  3 370.3  1943.0  A:= 1.1 l,p= 1.69  fc=1.15, p=1.78  4 354.0  1633.2  fc=1.18, p=1.50  k=M2, p=1.73    i = 1.53  .5=1.68  5 398.1  1825.5  ra, = 1.28, m2=0.09,  m, = 1.36, m;0.15,    , =1.35  =1.50  These results are for the  original axes, ic   0.  
Table 10.7 Minimizing Parameters Under SDf. Values of D,  Minimizing Parameters  / Wis.  U.S.  Wis.  U.S.  1 54.6  525.0  k=OM, e=o  k=0.92, «=19  1" 54.6  607.4  it=0.88  A:=0.92  2 29.3  110.0  /t=1.16  it=1.18  3 19.8  95.7  A:= 1.09, p= 1.57  k=l.]5,p=\JS  4 19.2  73.5  k=l.l6, p=l.55  A:=0.93, p=1.75    s=\.57  i=1.70  5 27.0  79.0  w, = 1.30, , 2=0.11,  w, = 1.35, mj=0.13.    , ,= 1.35  =1.48  These results are for the  original axes.  ie, e = 0.  
Rectangular Distances with "Doubling Back" The distances that are inherent in many industrial office or street grids are often rectangular in nature. In practice, however, the distances between point pairs are, on average, greater than would be indicated by the j?, metric. This extra travel distance is caused by the necessity to "double back." Doubling back may occur due to the existence of blocks or bays into which a floor area is divided, or the necessity to travel between closed areas. For example, a closed area might be a walledoff production cell with an entrance on one side. Such areas may necessitate travel in directions that are different from those which would be traveled if the assumption of perfect rectangular distances prevailed. Doubling back may also occur on street grids due to oneway streets or other restrictive traffic rulest In order to examine the effect of doubling back on rectangular distances. Love and Dowling (1985) fitted dj to a sample of layout patterns which were basically rectangular. However, actual travel in each distance sample was greater than rectangular due to factors which caused doubling back. The results of this study showed that the increased travel distance arising from doubling back on rectangular grids will be accounted for by the value in d, rather than the value of p. As the amount of doubling back increased in successive layouts, the value of also increased while the value of p tended to stay close to unity. Several conclusions were drawn from the study. In situations of this type, practitioners can assume p = 1 with no cause for concern that they are using an invalid model. Furthermore, if the metric is being used in a location model and the only resuh that is important is that of obtaining optimal facility location(s), it is not necessary to know the correct value of A; for that particular situation (ie, set = 1). However, if the objective
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