Impact of proximity-adjusted preferences on rank-order stability in ...
J Geogr Syst (2012) 14:167–187 DOI 10.1007/s10109-010-0140-6 ORIGINAL ARTICLE
Impact of proximity-adjusted preferences on rankorder stability in geographical multicriteria decision analysis Arika Ligmann-Zielinska • Piotr Jankowski
Received: 26 September 2009 / Accepted: 27 September 2010 / Published online: 10 October 2010 Springer-Verlag 2010
Abstract This paper presents a new approach to deriving preferences assigned to evaluation criteria in geographical multicriteria decision analysis. In this approach, the preferences, expressed by numeric weights, are adjusted by distance measures derived from the explicit consideration of a locational structure. The structure is given by locations of decision options and high importance reference objects. The approach is demonstrated on the example of a house selection case study in San Diego, California. The results show that proximity-adjusted preferences for the evaluation criteria can alter significantly the rank order of decision options. Consequently, the explicit modeling of spatial preference variability may be needed in order to better account for decision-maker’s preferences. Keywords Geographical multicriteria decision analysis Spatial decision support systems Sensitivity analysis GIS Choice preferences JEL Classification
D81 C12 C15 C21
1 Introduction The concept of choice preference is well established in philosophy, decision theory, and by extension in Multicriteria Decision Analysis (MCDA). Preferences in
A. Ligmann-Zielinska (&) Department of Geography and Environmental Science and Policy Program, Michigan State University, 130 Geography Bldg, East Lansing, MI 48824-1117, USA e-mail: [email protected] P. Jankowski Department of Geography, San Diego State University, San Diego, CA, USA
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MCDA typically concern either decision objectives or evaluation criteria and are commonly expressed by cardinal weights indicating the importance of an objective/ criterion relative to other objectives/criteria under consideration. The notion conveyed by weights is that the larger the weight value, the more important a given objective/criterion is. It is conventional to normalize the weights such that their sum for n objectives/criteria equals one, which can intuitively be interpreted as the total (100%) of one’s preferences. In the most common approach to expressing choice preferences in MCDA, the weights are linear. This means that a preference afforded to a given criterion is constant irrespective of changes in criterion outcomes (Jankowski 1995; Malczewski 2000; Steele et al. 2009). For example, consider cost as a criterion. Given the range of cost values representative of location alternatives, the weight expressing the relative preference of cost would be constant for all locations under consideration. Linear weights can account, implicitly though, for differences in the ranges of criterion outcomes. Consider, for example, the cost of environmental restoration ranging from $1 million to $8 million per location site, and the percentage of mitigating environmental injury ranging from 85 to 100% (at the least the cleanup will result in 85% restoration and at the most in 100% restoration). Which of the two criteria would you deem as more important? The general rule is that one is concerned with the perceived advantage of changing from the minimum level to the maximum level of criterion outcome (Malczewski 2000). Since the cost values cover a wider range than the percentage of mitigating environmental injury, the cost criterion might likely be deemed as more important, and hence receive a higher weight. A less common approach to expressing choice preferences is to use non-linear weights. Weights are non-linear when a preference afforded to a criterion changes along with criterion outcomes. In such a case, for the same criterion, different weights are assigned to different choice alternatives according to the logic that the more attractive a given criterion outcome (corresponding to a choice alternative), the higher the weight (Voogd 1983). Such a differentiation of weights is not incongruent with the concept of weights used in a geographical approach to multicriteria decision analysis (G-MCDA) (Malczewski 2006; Carver 1991). In G-MCDA, some locations are more important than others, therefore decision-makers can become more inclined toward choosing alternatives that are closer to these locations (or further away in case of detractors like contaminated sites). We argue that spatial bias plays a role in the final choice among decision options. In many cases, it is a major driver in selecting infrastructure improvement projects (transportation, water resources), service locations (schools, fire stations, health clinics, bank branches, stores, etc.), and other location-based resource allocations. Spatial weighting of evaluation criteria involves assigning a non-uniform importance to locations (Feick and Hall 2004) and thus providing a way of articulating an individual’s frame of reference, like a daily activity route or a particular destination place (work, home etc.), which impacts the perception and evaluation of spatial choice alternatives. Spatially explicit weighting involves both topological and non-topological relations reflecting the proximal and distal characteristics of spatial problems. One or multiple spatial relations can be used
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to express the variability of criterion preferences over a geographical decision space. Using spatial relations, we can quantitatively vary the importance of criteria over space and calculate spatial bias. In this article, we propose to express spatial bias by using proximity as the basis for quantifying spatial preference heterogeneity. We call this approach a proximityadjusted preference (PAP). In order to examine the influence of PAP on the results of G-MCDA, we compare the stability of option rank order obtained with traditional MCDA approach, in which preferences assigned to criteria do not have any spatial bias, with G-MCDA modified by the PAP approach, in which the spatial bias is represented by spatial weights (Herwijnen and Rietveld 1999; Feick and Hall 2004; Ligmann-Zielinska and Jankowski 2008). In the remainder of the paper, we formally define PAP and present an application, in which both approaches (traditional and PAP-based) are compared.
2 Proximity-adjusted preferences In the traditional MCDA approach, preferences relate to criteria rather than options themselves. For example, an individual can decide that the price of a house is twice as important as its architectural style. In this case, one specifies the preference for price in relation to architectural style, disregarding the location of the house and its characteristics, some of which may be measurable and some intangible but nevertheless included in the decision-making calculus (e.g. the quality of the neighborhood). Defined in this way, weights are homogeneous over the geographical space and no attention is given to how the decision criteria are perceived spatially. This homogeneous treatment of criterion preferences within a geographical decision space may result in solutions that only partially reflect the decisionmaker’s preferences. Further, this approach leads to a loss of information, which may be critical in solving spatially explicit problems and may result in choice recommendations that are subsequently rejected due to their incongruence with the preference structure of the decision-maker. To overcome this deficiency, modelers have been using geographically explicit criteria based on spatial relationships such as, for example, ‘proximity to work’, ‘containment within a particular school district’, or ‘density (clustering) of shopping outlets in the neighborhood’. While such spatial criteria are an important component of a geographically scoped decision problem, the representation of criteria preferences is still treated in a traditional—aspatial—way. Rinner and Heppleston (2006) proposed a method of incorporating spatial bias by adjusting final composite evaluation scores of decision options using distances to the favored options. This approach works as a way of spatial smoothing (de Smith et al. 2009; Rinner and Heppleston 2006) and is based on the premise that the proximate options are perceived as similar, and thus their final evaluation scores influence each other. While this post hoc adjustment (Rinner and Heppleston 2006) is an interesting method of neighboring option substitution, it is nevertheless limited to the options themselves and does not account for exogenous points of reference. Below we propose an alternative method of incorporating spatiality into G-MCDA
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by adjusting the individual criteria weights rather than the final composite evaluation scores. We introduce an additional component to the multicriteria model of decision situation in the form of a reference location, which acts as a benchmark shaping the importance of criteria. The proposed approach explicitly introduces spatial heterogeneity of preferences in calculating the attractiveness and the subsequent ranking of decision options. We argue that the decision-makers not only place a particular importance on a given criterion, but also pay attention to how an option is situated with respect to this criterion. For example, when choosing a trip accommodation in a particular locality, we may be inclined toward hotels that are cheaper and thus assign the highest preference to hotel price. Moreover, when considering price, we may also prefer a cluster of cheap hotels in case we need to substitute one hotel with another in the same neighborhood. Such a case imposes the necessity of introducing a spatially heterogeneous preference layer in place of a uniform criterion weight, which does not account for the relative benefit of price similarity in one area versus another. Compared with the approach proposed by Rinner and Heppleston (2006), our methodology goes one step further in representing compound spatial preferences for decision criteria. Research on information integration theory (Anderson 1971; Anderson 1981) posits that in discriminating among different choice alternatives individual decision-makers employ a variety of functional forms, which can be modeled as algebraic rules of varying complexity in order to arrive at a judgment. Numerous empirical experiments (Anderson 1991) lend a considerable support to assertion that people do combine different decision factors (criteria) into more complex constructs in order to arrive at an overall appraisal of a choice. The methodology presented in this paper allows for assigning spatially heterogeneous weights to different criteria. Continuing the example of choosing trip accommodation, we may look for a cluster of cheap hotels as well as hotels that are also close to public transit service offering multiple lines to various destinations and, at the same time, affording a fast commute to a conference center. The latter reflects a spatial bias toward a particular location. In this case, we can define a more comprehensive public transit criterion composed of (a) proximity to low- and highfrequency lines and (b) commuting distance to the conference center, which is encapsulated in the criterion weight. As a result, our approach recognizes the fact that diverse aspects of spatial options can differently contribute toward the achievement of different decision objectives. 2.1 Total weight map Consider a decision situation depicted in Fig. 1. There are five spatial options a (circles A through E) and two reference locations (the black squares L1 and L2). Using the traditional aspatial weighting, for a given criterion C, all options a are assigned the same decision weight WC (Fig. 1a). This is representative of a typical weighting scheme, in which space does not influence criteria weights. Assuming a simple Manhattan distance as the proximity measure, we can also calculate both raw and standardized distances to the reference locations L1 and L2 (Table 1).
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Fig. 1 A hypothetical weight (preference) map that depicts the importance of a given criterion uniformly (a), and in a spatially heterogeneous manner (b, c). Circles are options, their radii are constant in (1a), and proportional to weight values in cases (1b) and (1c), and the reference locations are squares
Table 1 Distances from options to reference locations for the example in Fig. 1
Options
Distance to L1
Distance to L2
Manhattan
Standardized
Manhattan
Standardized
A
6
0.33
4
0.75
B
3
0.67
7
0.43
C
5
0.40
3
1.00
D
2
1.00
4
0.75
E
4
0.50
8
0.38
Avg
4
0.58
5.2
0.66
Min
2
0.33
3
0.38
To account for spatially explicit weighting, we introduce the concept of a total weight TW derived from the set N = 1,…, n of options a, which, for a given criterion C, is defined as follows: n X TW C ¼ WC ð1Þ a¼1
In an aspatial case, TWC = n * WC, where n is the number of options. What follows is that TWC can be redistributed among decision options in a non-uniform fashion (Fig. 1b, c). Therefore, to account for spatial bias toward selected locations, we extend the idea of preferences from ‘relative criterion importance’ to ‘relative criterion and option importance’. The maps [b] and [c] in Fig. 1 illustrate this concept. 2.2 Definition of proximity-adjusted preferences One way to redistribute TWC among the options is to use proximity to the reference locations (L) as a spatial adjustment of criteria preferences. Consider the following formula:
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PAPCa ¼ WC
da ld
ð2Þ
where PAPCa is a proximity-adjusted preference (weight) for criterion C and option a, WC is an aspatial (traditional) preference (weight) for criterion C, da is a standardized distance of option a to the reference location L (i.e. spatial bias), ld is a mean of the standardized distances of all options a to L. Definition A proximity-adjusted preference (PAP) is a weight assigned to each pair (C, a) obtained by modifying the criterion weight WC with the distance from option a to the reference location L normalized by the mean distance of all options to L. In other words, the average PAP-adjusted weight value is the same as the aspatial weight value, whereas the specific PAP-adjusted weight value (a weight that corresponds to an individual decision option) deviates from this average. Observe that: n X PAPCa ¼ TW C ð3Þ a¼1
Proof PAPC1 þ PAPC2 þ þ PAPC n ¼
WC ðd1 þ d2 þ þ dn Þ ¼ n WC ¼ TW C ld
Given the hypothetical example (Fig. 1a and Table 1), we can now calculate PAP for the two reference locations L assuming that WC = 3. The results are presented in Table 2. Note that, although WC varies among the options in maps [b] and [c] as opposed to the initial map [a], TWC remains unchanged. 2.3 Extended PAP example using decision rules To illustrate the influence of the PAP concept on final evaluation scores and consequently on the option rank order, we used four different decision rules (aggregation functions) summarized in Table 3. Table 2 Proximity-adjusted preferences for the example in Fig. 1
Options
Aspatial
PAP for L1
PAP for L2
A
3
1.72
3.41
B
3
3.45
1.95
C
3
2.07
4.54
D
3
5.17
3.41
E
3
2.59
1.70
15
15.00
15.00
Sum (TWc)
123
Preferences
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Table 3 Decision rules (aggregation functions) used in the hypothetical example Aggregation function
Description
Simple additive weighting—SAW(Malczewski 1999) P ea = C PC * SCa 1. Criteria outcomes are standardized using the score range procedure or the linear scale transformation (Malczewski 1999) 2. The preferences are then multiplied by the standardized criteria outcomes and summed up to generate a composite option score Rank order—RO (Jankowski et al. 1997) P ea = C PC * SCa 1. Criteria outcomes are standardized using the rank-order standardization as follows. where Ca SCa ¼ LC RR C
For every criterion C: a. Rank every option a (RCa) b. Determine the lowest rank for a given criterion LC c. Calculate the range of ranks (RC) using RC = LC - 1 d. Calculate the standardized criterion outcome (SCa) 2. The preferences are then multiplied by the standardized criteria outcomes and summed up to generate a composite option score
Multiplicative—M (Cobb and Douglas 1928; Chiang 1984) Q 1. Criteria outcomes are standardized using the score range ea ¼ C SPCaC procedure or the linear scale transformation (Malczewski 1999) 2. For each criterion, the standardized criterion outcome is raised to the power of the criterion preference, expressed using [0.0, 1.0] range 3. The results of exponentiation of all criteria are then multiplied to generate a composite option score Ideal Point—IP (Hwang and Yoon 1981) S^
a ea ¼ S^ þS a
a
1. Criteria outcomes are standardized using the score range procedure or the linear scale transformation (Malczewski 1999)
where X 0:5 2. For every criterion C, the worst outcome value (nadir: C^) and the 2 best outcome value (ideal: C*) are established Sa ¼ ð P ð S C Þ Þ c Ca c X 0:5 3. For every criterion-option pair, the difference between the 2 criterion outcome and the criterion ideal is multiplied by the S^a ¼ ðPc ðSCa C ^ ÞÞ c criterion preference and the result is squared. Then, for every option a, these squared results are summed up and raised to the power of , constituting the separation from ideal (S*a). For every criterion-option pair, a difference between the criterion outcome and the criterion nadir is multiplied by the criterion preference, and the result is squared. Then, for every option, the squared results are summed up and raised to the power of , constituting the separation from nadir (S^a ) 4. The composite option score is derived by dividing the separation from nadir by the sum of the separation from ideal and the separation from nadir Common notation: ea is the final evaluation score for option a; PC is the preference (expressed by either PAPCa or WC) for criterion C; SCa is the standardized outcome of criterion C for option a
Suppose that the problem in Fig. 1 is quantified using two decision criteria: access to public transit (C1) and hotel accommodation price (C2). Assume that criterion C1 is associated with the reference location L1 (a conference center),
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Table 4 A hypothetical decision matrix with standardized criteria outcomes and proximity-adjusted weights Options
C1 outcome
C1 preference (%) Aspatial
PAP for L1
C2 outcome
C2 preference (%) Aspatial
PAP for L2
A
0.1
40
23.0
0.5
60
68.1
B
0.2
40
46.0
0.2
60
38.9
C
0.4
40
27.6
0.3
60
90.8
D
0.5
40
69.0
0.3
60
68.1
E
0.9
40
34.5
0.2
60
34.1
whereas criterion C2 is tied to the location L2 (the center of a cluster of cheap hotels). Using equation 2 and distances in Table 1, we can now calculate PAP for both criteria. The results are presented as a decision matrix in Table 4. Using the four decision rules presented in Table 3, we can calculate the final option evaluation scores (ea) using both the aspatial weights and the PAP-based weights. Figure 2 shows the results of these computations. The hypothetical example clearly demonstrates the influence of spatially heterogeneous preferences. Observe that there is a considerable variability among the four decision rules. SAW and M produce rather flat distributions of scores (aspatial range = 28 for SAW and 17 for M, respectively) as opposed to IP and RO, which are quite varied (aspatial range = 53 for IP and 60 for RO, respectively). Furthermore, we can observe that PAPs have a discriminating effect on the final evaluation scores. For every decision rule, the range between the minimum and maximum score increased from the aspatial to the spatial preference quantification. For example, in case of the IP decision rule, the range increased from 60 to 86. Despite PAPs extending the ranges of final evaluation scores, the option rank-order trends obtained with aspatial weights were mostly preserved with PAP-based weights. This is a plausible characteristic of the proposed approach, by which the PAPs have a balancing rather than overwhelming impact on the final rank order of options. Interestingly, the four decision rules result in different winners and different runners-up. The only constant is the worst score of option B in all eight configurations. Introducing spatially heterogeneous preferences with the IP aggregation function did not affect the position of the winning option E. Unlike the IP-generated ranking, the winner (E) and the runner-up (D) switched their ranks in the case of SAW aggregation function after introducing the PAPs as criteria weights. If D and E represent satisfying choices, the decision-maker is likely to select an option closer to his or her area of interest. As this example indicates, the PAP procedure allows for representing such a spatial bias in a formal way.
3 Application In this section, we introduce an example of a real-world decision case to demonstrate and evaluate the proposed PAP concept. The decision problem is
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Fig. 2 Option evaluation scores based on different decision rules and different representations of criteria preferences: a simple additive weighting, b rank order, c multiplicative, and d ideal point
situated in San Diego, California and involves a young family with children in the preschool age. The family needs to choose a house for purchase out of 45 houses available on the market. The stay-at-home dad would like a house which is closer to
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parks with playgrounds, but he is also more inclined toward a few selected adventure parks: Balboa Park and San Diego Zoo, Sea World, and Belmont Park (Fig. 3). The working mum would like to minimize driving to her work place. The house database contains nine attributes, used in the case study as the evaluation criteria, characterizing houses and their location attributes (Table 5). The data was collected during 2003–2005 through fieldwork, census, and local surveys (source: SANDAG http://www.sandag.org/). Driving distances were derived using online mapping software (Google Maps http://maps.google.com). For this example, a few technical assumptions are required. We assume that criteria scores are error-free and, thus, we concentrate on the uncertainty associated with criteria preferences. We standardize most of the criteria using the linear scale transformation (Malczewski 1999) to maintain the proportionality of standardized scores to raw data and minimize the influence of standardization on calculating composite option scores. Only the YEAR (Table 5) was standardized using the
Fig. 3 Location of 45 houses in San Diego, California in relation to the place of work and adventure parks
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Table 5 Decision criteria used in the case study PRICE
BED
YEAR
LOT
LIVG
MDVA
PARK_ ACCESS
DIST_ WORK
DIST_ ADVPK
AVG
282.65
3.4
1958
6,313
273
203,115
0.3
4.7
33.8
MED
275
3
1958
6,090
264
205,723
0.3
5.3
32.1
10
1,109
67
22,458
0.2
2.5
2,000
120
97,300
0.1
0.5
STDEV
42.23
0.6
8.7
MIN
196.25
2
1939
18
MAX
359.25
5
2004
9,330
420
237,388
1.0
9.6
50.5
RANGE
162.99
3
65
7,330
300
140,088
0.9
9.1
32.5
PRICE $ per sq ft, BED number of bedrooms, YEAR year house was built, LOT house lot area (sq ft), LIVG living room area (sq ft), MDVA median housing unit value in the census block group (including condominium apartments) ($), PARK_ACCESS accessibility to parks measured as normalized cumulative Euclidean distance weighed by park area, DIST_WORK driving distance to work (miles), DIST_ADVPK cumulative driving distance to three adventure parks (Balboa Park and San Diego Zoo, Sea World, and Belmont Park) (miles)
score range transformation (Malczewski 1999) due to its interval measurement scale. 3.1 Computational experiments The objective of this case study is to compare the traditional MCDA (where the spatial dimension is represented with distance criteria like distance to work) and the proposed PAP-based MCDA (where the distance criteria are coupled with selected aspatial criteria). We start from performing the traditional (aspatial) prioritization of houses (Experiment 1). Since the exact preferences for the decision criteria are unknown, we use Monte Carlo sampling from predefined random uniform distributions (Table 6, ‘Aspatial Experiment’ column) to calculate the composite evaluation scores and to repeatedly simulate the prioritization of houses (Butler et al. 1997). We adopt an economical perspective, with the strongest preference for affordability and accessibility to parks and with lesser priority given to the other criteria. The next step (Experiment 2) involves house prioritization using spatially adjusted weights. We select two criteria for PAP modification: PRICE and PARK_ACCESS. The former is adjusted with distance to work (DIST_WK) resulting in a new criterion called PRICE_PAP. The latter is modified with distance Table 6 Random uniform probability density functions (PDF) used to simulate criteria preferences Criteria
Aspatial experiment
PAP-based experiment –
PRICE, PARK_ACCESS
Min = 0.15, max = 0.45
PRICE_PAP,PARK_ACCESS_PAP
–
Min = 0.30, max = 0.60
BED, YEAR, LOT
Min = 0.10, max = 0.40
Min = 0.10, max = 0.40
LIVG, MDVA
Min = 0.05, max = 0.35
Min = 0.05, max = 0.35
DIST_WK,DIST_ADVPK
Min = 0.00, max = 0.30
–
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to adventure parks (DIST_ADVPK), and the generated criterion is named PARK_ACCESS_PAP. Thus, instead of having distances to work and adventure parks represented as two additional decision criteria, we use these two criteria to adjust house price and park accessibility. In particular, PRICE_PAP represents the overall monetary cost criterion that combines a proxy for the tangible housing costs (mortgage, real estate taxes, and insurance) with the less tangible everyday commuting costs, which depend on the driving distance to work. On the other hand, PARK_ACCESS_PAP combines the spatial bias toward adventure parks with the area-weighed distance to all parks. The rationale behind constructing PARK_ACCESS_PAP and PRICE_PAP as described above is based on empirical research on real estate purchase choices. Results of empirical studies on the influence of decision criteria in choice making concerning residential housing point out the importance of distance as a location choice criterion, though, they also show that the importance of proximity differs among various socio-demographic groups (Freiden and Bible 1982; Reed and Mills 2007; Gibler et al. 2009). Different decision-makers are characterized by different cognitive styles when valuing the distance from their housing locations to other reference locations. For example, young professionals may value higher the distance from housing to cafe´s and other eating-out establishments than to medical facilities, which may be just the opposite for senior citizens (Freiden and Bible 1982; Reed and Mills 2007; Gibler et al. 2009). To account for the loss of DIST_WK and DIST_ADVPK in the cumulative distribution of weights, we increase the minimum and maximum weight values for the random uniform probability density functions of PRICE_PAP and PARK_ ACCESS_PAP (Table 6, ‘PAP-based Experiment’ column). Specifically, the midpoint value (0.15) of the distance criteria distributions is added to both the minimum and maximum values of PRICE and PARK_ACCESS weight ranges. In terms of individual preferences, the rest of the criteria are assumed to be spatially invariant. Both aspatial and PAP-based experiments were executed using 5000 renormalized weight vectors based on the selected aggregation functions, which are analyzed separately. Out of the four aggregation functions presented in Table 3, we used SAW, IP, and RO to arrive at rank orders of candidate houses, which amounted to six experiments overall. The multiplicative decision rule was excluded due to its highly constraining nature that eliminates an option whenever any of its criteria is assigned a score of zero, which potentially reduces the variability of rank shifts. All computations were performed using customized Python scripting (http://www. python.org/), which is available from the authors.
4 Results and analysis We focused our analysis on determining the statistical significance of the differences between aspatial and PAP-based option rankings. We also tested the impact of various decision rules on the variability of option evaluation scores. Finally, we mapped the shift in ranks for the two conceptualizations of weights within the study area, followed by a recommendation of the decision options.
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Impact of proximity-adjusted preferences on rank-order stability Table 7 t-test results for average shift in ranks (ASR) from the equal weight case
179
Aspatial experiment
PAP-based experiment
Avg
4.3
11.06
Var
1.26
0.56
df
8,683
t-stat
354.5
Cohen’s d
7.1
Simple additive weighting
Ideal point Avg
5.8
7.0
Var
1.43
0.75
df
9,101
t-stat
54.6
Cohen’s d
1.1
Rank order
Comparisons between aspatial scenario and PAP-based scenario (a = 0.05)
Avg
4.46
8.67
Var
1.51
1.81
df
9,918
t-stat
163.3
Cohen’s d
3.26
4.1 The significance of spatially heterogeneous weights for determining house ranks We started the analysis from calculating the importance of spatial bias in determining option rank order. The objective was to determine whether the PAPderived rankings were significantly different, in the statistical sense, from the aspatial rankings. We began from summarizing house rankings using a descriptive statistic called an average shift in ranks (ASR) (Saisana et al. 2005): n 1X ASR ¼ ja rankref a rankj ð4Þ n a¼1 where ASR is the average shift in ranks, a_rankref is the rank of option A in the reference ranking (e.g. equal weight case), a_rank is the current rank of option A. ASR captures the relative shift in the position of the entire set of options and quantifies it as the sum of absolute differences between the current option rank (a_rank) and the reference rank (a_rankref), divided by the number of all options. In the first step of the analysis, we selected equal weight aspatial case as the reference ranking, which was calculated for SAW, IP, and RO, respectively. We used equal weights as the bases for the reference ranking to establish the baseline case without the effect of any preference bias on option scoring and ranking. Using sensitivity analysis to compare the magnitudes of departure from the baseline ranking allows us to measure the significance of different conceptions of criteria weights.
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A two-sample test assuming unequal variances was conducted to compare the ASR of aspatial rankings with the proximity-adjusted rankings. The results are presented in Table 7. To analyze the results, we used two-tail P-values from the t-tests. In all cases, we recorded P \ 0.00005. Since the P-value is considerably less than a, we can conclude that there are statistically significant differences between the aspatial scenario and the PAP-based scenario. For every aggregation function formulation (SAW, IP, and RO), there was a significant difference in ASR for aspatial and spatially heterogeneous preferences. These results suggest that PAP affects the prioritization of houses in the presented case study. Specifically, when we apply PAPs, the average shift in ranks noticeably increases (e.g. by 7 ranks from the aspatial to PAP scenario using the SAW decision rule). Due to the large samples (each scenario is composed of 5,000 simulations), we augmented t-tests with the effect size Cohen’s d statistics (Cohen 1988). For all three decision rules, we observed large values of Cohen’s d statistic indicating that the magnitude of differences is substantial, that is, not only statistically significant but also substantively important. 4.2 Visual comparison of aspatial and PAP-based house ranking scenarios The solution recommendation for the decision problem requires finding highscoring options. First, for each prioritization scenario, we calculated rank summary statistics including minimum, maximum, mean, and variance over all simulations (Andrienko and Andrienko 2005) and created uncertainty plots for each option. Figure 4 illustrates the results for SAW decision rule and the aspatial versus the PAP-based scenarios. With these statistics, we can determine which houses on average rank the highest, which houses have the most or the least sensitive position, and whether any house is dominated by other houses. To establish high-ranked and low-ranked options, we consider the following four cases: 1. 2. 3. 4.
High average rank and low-rank variance signify the winning options and place high confidence in their prioritization (i.e. a robust solution) Low average rank and low-rank variance represent the losing options that may be disregarded. High average rank and high-rank variance characterize options that are potentially good but need further study due to the sensitivity of ranking. Low average rank and high-rank variance designate options that may be dropped from the analysis but with lesser confidence when compared to case 2.
In Fig. 5, we rendered the results on rank stability maps (Jankowski et al. 1997), using as an example the SAW decision rule. Other decision functions used in the study (IP and RO) yielded similar results. The maps depict not only average ranks but also their variances. As noted above, we seek robust options that on average rank high and have a relatively low-rank variance. Such options are depicted with large and medium gray circles. Using rank and rank variance as the selection criteria, we can conclude that no decision rule identifies clear winners in the aspatial scenario, since the high-scoring options (Table 8) are characterized by relatively
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181
Fig. 4 House ranks summarized over all simulations for the simple additive weighting decision rule. Top: aspatial preferences, bottom: PAP-based scenario. Houses ordered by average rank from min (best) to max (worst). Legend: bars—rank range, barbells (error bars)—standard deviation of ranks, dot— average rank
moderate to high standard deviations (see Fig. 4-top, 5-top). Contrary to the aspatial results, the PAP-based simulations provide well-established winners (predominantly houses 31, and 32, and 33 under all three decision rules—Table 8), which have the best rank and the lowest rank variance. This observation is also confirmed by the small error bars in Fig. 5-bottom (std \ 0.5 for top three ranks). Observe that, for SAW, the highest scoring options in the aspatial case are scattered throughout the
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Fig. 5 Rank stability maps of house prioritizations using aspatial simple additive weighting (SAW) scenario (top) and PAP-based SAW scenario (bottom). The size of the circle represents the average house rank, whereas the shade of gray indicates the rank standard deviation. For comparative purposes, both size and color classifications were fixed among the maps
area, whereas the PAP-based scenario results in a well-established southern cluster of high-scoring options, which is closer to the place of work (Fig. 5). RO and IP exhibit the same patterning. Consequently, the high-scoring options for the aspatial preferences are different from the high-scoring options for the proximity-adjusted preferences. The latter options (houses 31, 32, and 33) maintained their high ranks
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Table 8 Six top winning houses recorded by House ID (avg rank; std) for three decision rules representing aspatial and PAP-based scenarios: SAW—simple additive weighting, IP—ideal point, RO—rank order SAW
IP
RO
H19 (3.7; 3.7)
H45 (1.3; 0.6)
Aspatial top rank houses 1
H32 (2.2; 1.8)
2
H19 (3.4; 2.4)
H32 (3.9; 3.7)
H1 (3.4; 1.9)
3
H45 (4.3; 2.1)
H44 (4.7; 4.2)
H6 (3.4; 1.4)
4
H44 (4.7; 3.0)
H13 (5.0; 3.2)
H25 (3.8; 1.9)
5
H13 (5.9; 2.4)
H45 (8.2; 3.4)
H13 (5.6; 2.2)
6
H33 (6.9; 5.4)
H1 (8.5; 4.3)
H44 (7.2; 3.3) H33 (1.3; 0.4)
PAP-based top rank houses 1
H31 (1.3; 0.5)
H32 (1.4; 0.5)
2
H33 (1.7; 0.5)
H19 (2.1; 1.2)
H31 (1.7; 0.4)
3
H32 (3.0; 0.0)
H13 (4.0; 1.3)
H32 (3.0; 0.0)
4
H39 (5.2; 1.4)
H31 (4.1; 1.4)
H45 (4.6; 0.8)
5
H30 (5.8; 1.2)
H44 (4.8; 2.0)
H25 (6.8; 1.5)
6
H37 (5.9; 1.1)
H33 (7.7; 1.7)
H6 (7.6; 1.6)
Bolded houses are common among all three decision rules
regardless of the applied decision rule (Table 8). When the PAPs are applied, some of the high-scoring houses become dominated by others in terms of their rankings. For example, houses 31, 32, and 33 (Fig. 4) dominate houses 39, 30, and 37 since their worst rank is still higher than the best rank of any of the latter houses. In general, given the sizes of the bars in Fig. 5, the aspatial scenarios are much more sensitive to changes in preference values, whereas PAPs have a balancing effect on the stability of house ranking. 4.3 Rank shift maps To further test the spatiality of ranking obtained with proximity-adjusted preferences, we summarized the PAP-based simulations by calculating a shift in average house rank from the aspatial case to the PAP-based case. The shift from the aspatial to the PAP scenario is mapped in Fig. 6 for IP (top) and RO (bottom). SAW rendered a very similar result to RO, and we do not report it here. The maps provide two pieces of information. First, they depict houses that either gained or lost their average ranking positions. Note that the PAP-based gain (improvement) in house ranking occurs when its aspatial rank is higher in value (i.e. worse) than its PAP rank, resulting in a positive difference. Secondly, the maps validate the observation about the general clustering of high-scoring houses south and east of the work place. For threshold distance of 1 mile, the trend is spatially autocorrelated with significant clusters (SAW: Moran’s I = 0.75, Expected I = -0.02, Variance = 0.008, Z-Score = 8.38; IP: Moran’s I = 0.57, Expected I = -0.02,
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Fig. 6 Shift in rank per house from the aspatial scenario to the PAP-based scenario. Top: IP decision rule, bottom: RO decision rule
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Variance = 0.009, Z-Score = 6.34; RO: Moran’s I = 0.65, Expected I = -0.02, Variance = 0.008, Z-Score = 7.41). The west cluster lost its overall ranking position when moving from the aspatial scenario to the PAP-based prioritization. The east cluster also lost its position in the ranking but the loss is less pronounced, that is, not observed for IP (Fig. 6). 4.4 Discussion of results Our general observation is that PAPs have a significant impact on option prioritization. Within the reported experiments, the proposed method of quantifying spatial bias generates a directional shift of high-ranked sites and results in proximal option score smoothing (Fig. 5). When the distance criteria (DIST_WK and DIST_ADVPK) are incorporated into the PAP of price and park accessibility (PRICE_PAP and PARK_ACCESS_PAP), their effect on house ranking is different. In the aspatial case, both DIST_WK and DIST_ADVPK have an equivalent influence on house prioritization, resulting is a scattered distribution of high-rank options, which reside in all three clusters (Fig. 5-top). In contrast with the aspatial case, the PAP-based scenarios gather all high-ranked houses together in the south cluster (Fig. 5-bottom) suggesting that the work distance–adjusted price criterion becomes more influential than the adventure park distance–adjusted park accessibility criterion. For example, by integrating PRICE with DIST_WK, we increase the interactivity among these two decision criteria and decrease their independent additive role in house scoring and ranking. As a result, the impact of PRICE_PAP and PARK_ACCESS_PAP is more pronounced for options that are close to work, moderately priced, and in proximity to many (albeit small) parks. Going back to Fig. 5-bottom, one can observe that both W and E clusters are comparable in terms of house ranking. While the former is located closer to adventure parks, it is also further away from work when the driving distance is considered. Hence, the two distance adjustments cancel out, resulting in similar rank clusters. We can conclude that PAPs have an integrative and non-linear effect on option ranking. By incorporating spatial bias in the form of variable weights, which are functionally dependent on distance between each locational decision option and the reference location(s), the proposed approach provides an alternative to representing complex spatial criteria preferences and, at the same time, affords a reduction in the decision space (from 9 to 7 evaluation criteria in the presented case study, Table 6).
5 Conclusion The main idea behind the work presented in this paper is that the relative importance given to criteria in a spatial multicriteria evaluation problem is not necessarily constant over the geographical space but in fact it may vary. This variability in preferences, represented by spatially adjusted weights, reflects the influence of spatial structures and patterns on the perceived importance of evaluation criteria. The spatial adjustment can result from considering one or multiple spatial relations and computing their values for the decision options. Such values are in effect the
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measures of spatial structure and pattern and can be used as ‘‘spatial’’ criteria in multicriteria evaluation. This idea is not new and has been presented by others. The contribution of this paper is in recognizing that spatial criteria give rise to spatially variable preferences that affect the relative importance assigned to aspatial criteria. The paper also presents an analytical approach to calculating the effect of spatially adjusted weights on the rank order of decision options. The analysis method and the subsequent case study deal with only one spatial relation—proximity. Moreover, the case study presents a simplified situation of distance-adjusted weights under the presence of two groups of reference locations: the place of work and the selected adventure parks. While we have shown that the PAP approach results in significantly different rankings of options, we have not determined which approach yields a ranking that better matches the view of a decision-maker. Therefore, future work should include an empirical investigation of the proposed methodology, focusing on the relevance of PAP to solving spatially explicit decision problems. Further extensions of the presented approach should also address multiple spatial relations and spatial preference adjustments resulting from the simultaneous consideration of more than one spatial relation. An example of this is adjacency to specific reference categories, such as type of built-structures or demographic characteristics of people inhabiting these structures. Acknowledgments The authors would like to acknowledge critical and constructive comments from three anonymous reviewers on the previous version of the manuscript.
References Anderson NH (1971) Integration theory and attitude change. Psychol Rev 78(3):171–206. doi: 10.1037/h0030834 Anderson NH (1981) Foundations of information integration theory. Academic Press, San Diego Anderson NH (1991) Contributions to information integration theory, vol 1, 2, 3. Erlbaum, Hillsdale Andrienko N, Andrienko G (2005) Exploratory analysis of spatial and temporal data: a systematic approach. Springer, Berlin Butler J, Jia JM, Dyer J (1997) Simulation techniques for the sensitivity analysis of multi-criteria decision models. Eur J Oper Res 103(3):531–546 Carver SJ (1991) Integrating multi-criteria evaluation with geographical information systems. Int J Geogr Inf Sci 5(3):321–339 Chiang AC (1984) Fundamental methods of mathematical economics. Mcgraw-Hill, Montreal Cobb CW, Douglas PH (1928) A theory of production. Am Econ Rev 18(1):139–165 Cohen J (1988) Statistical power analysis for the behavioral sciences, 2nd edn. Lawrence Erlbaum Associates, New Jersey de Smith MJ, Goodchild MF, Longley PA (2009) Geospatial analysis: a comprehensive guide to principles, techniques and software tools, 3rd edn. Troubador Publishing, Leicester Feick RD, Hall GB (2004) A method for examining the spatial dimension of multi-criteria weight sensitivity. Int J Geogr Inf Sci 18(8):815–840. doi:10.1080/13658810412331280185 Freiden JB, Bible DS (1982) The home purchase process: measurement of evaluative criteria through pairwise measures. J Acad Mark Sci 10(4):359–376. doi:10.1177/009207038201000402 Gibler KM, Taltavull P, Casado-Dı´az JM, Casado-Dı´az MA, Rodriguez V (2009) Examining retirement housing preferences among international retiree migrants. Int Real Estate Rev 12(1):1–22 Herwijnen MV, Rietveld P (1999) Spatial dimensions in multicriteria analysis. In: Thill J-C (ed) Spatial multicriteria decision making and analysis a geographic information sciences approach. Ashgate, London, pp 77–99
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Hwang CL, Yoon K (1981) Multiple attribute decision making methods and applications: a state of the art survey. Springer, Berlin Jankowski P (1995) Integrating geographical information-systems and multiple criteria decision-making methods. Int J Geogr Inf Syst 9(3):251–273 Jankowski P, Nyerges TL, Smith A, Moore TJ, Horvath E (1997) Spatial group choice: a sdss tool for collaborative spatial decision-making. Int J Geogr Inf Sci 11(6):577–602 Ligmann-Zielinska A, Jankowski P (2008) A framework for sensitivity analysis in spatial multiple criteria evaluation. In: Cova TJ, Miller HJ, Beard K, Frank AU, Goodchild MF (eds) Geographic information science proceedings 5th international conference, Giscience 2008, Park city, ut, USA, September 23–26, 2008, vol 5266/2008. Lecture notes in computer science. Springer, Berlin/ Heidelberg, pp 217–233 Malczewski J (1999) Gis and multicriteria decision analysis. Wiley, New York Malczewski J (2000) On the use of weighted linear combination method in gis: common and best practice approaches. Trans GIS 4(1):5 Malczewski J (2006) Gis-based multicriteria decision analysis: a survey of the literature. Int J Geogr Inf Sci 20(7):703–726. doi:10.1080/13658810600661508 Reed R, Mills A (2007) Identifying the drivers behind housing preferences of first-time owners. Prop Manage 25(3):225–241 Rinner C, Heppleston A (2006) The spatial dimensions of multi-criteria evaluation—case study of a home buyer’s spatial decision support system. In: Raubal M, Miller HJ, Frank AU, Goodchild MF (eds) Geographic information science, proceedings, vol 4197, Lecture notes in computer science, pp 338–352 Saisana M, Saltelli A, Tarantola S (2005) Uncertainty and sensitivity analysis techniques as tools for the quality assessment of composite indicators. J R Stat Soc Ser A Stat Soc 168:307–323 Steele K, Carmel Y, Cross J, Wilcox C (2009) Uses and misuses of multicriteria decision analysis (mcda) in environmental decision making. Risk Anal Int J 29(1):26–33. doi:10.1111/j.1539-6924.2008. 01130.x Voogd H (1983) Multicriteria evaluation for urban and regional planning. Pion, London
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Impact of proximity-adjusted preferences on rankorder stability in geographical multicriteria decision analysis Arika Ligmann-Zielinska • Piotr Jankowski
Received: 26 September 2009 / Accepted: 27 September 2010 / Published online: 10 October 2010 Springer-Verlag 2010
Abstract This paper presents a new approach to deriving preferences assigned to evaluation criteria in geographical multicriteria decision analysis. In this approach, the preferences, expressed by numeric weights, are adjusted by distance measures derived from the explicit consideration of a locational structure. The structure is given by locations of decision options and high importance reference objects. The approach is demonstrated on the example of a house selection case study in San Diego, California. The results show that proximity-adjusted preferences for the evaluation criteria can alter significantly the rank order of decision options. Consequently, the explicit modeling of spatial preference variability may be needed in order to better account for decision-maker’s preferences. Keywords Geographical multicriteria decision analysis Spatial decision support systems Sensitivity analysis GIS Choice preferences JEL Classification
D81 C12 C15 C21
1 Introduction The concept of choice preference is well established in philosophy, decision theory, and by extension in Multicriteria Decision Analysis (MCDA). Preferences in
A. Ligmann-Zielinska (&) Department of Geography and Environmental Science and Policy Program, Michigan State University, 130 Geography Bldg, East Lansing, MI 48824-1117, USA e-mail: [email protected] P. Jankowski Department of Geography, San Diego State University, San Diego, CA, USA
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MCDA typically concern either decision objectives or evaluation criteria and are commonly expressed by cardinal weights indicating the importance of an objective/ criterion relative to other objectives/criteria under consideration. The notion conveyed by weights is that the larger the weight value, the more important a given objective/criterion is. It is conventional to normalize the weights such that their sum for n objectives/criteria equals one, which can intuitively be interpreted as the total (100%) of one’s preferences. In the most common approach to expressing choice preferences in MCDA, the weights are linear. This means that a preference afforded to a given criterion is constant irrespective of changes in criterion outcomes (Jankowski 1995; Malczewski 2000; Steele et al. 2009). For example, consider cost as a criterion. Given the range of cost values representative of location alternatives, the weight expressing the relative preference of cost would be constant for all locations under consideration. Linear weights can account, implicitly though, for differences in the ranges of criterion outcomes. Consider, for example, the cost of environmental restoration ranging from $1 million to $8 million per location site, and the percentage of mitigating environmental injury ranging from 85 to 100% (at the least the cleanup will result in 85% restoration and at the most in 100% restoration). Which of the two criteria would you deem as more important? The general rule is that one is concerned with the perceived advantage of changing from the minimum level to the maximum level of criterion outcome (Malczewski 2000). Since the cost values cover a wider range than the percentage of mitigating environmental injury, the cost criterion might likely be deemed as more important, and hence receive a higher weight. A less common approach to expressing choice preferences is to use non-linear weights. Weights are non-linear when a preference afforded to a criterion changes along with criterion outcomes. In such a case, for the same criterion, different weights are assigned to different choice alternatives according to the logic that the more attractive a given criterion outcome (corresponding to a choice alternative), the higher the weight (Voogd 1983). Such a differentiation of weights is not incongruent with the concept of weights used in a geographical approach to multicriteria decision analysis (G-MCDA) (Malczewski 2006; Carver 1991). In G-MCDA, some locations are more important than others, therefore decision-makers can become more inclined toward choosing alternatives that are closer to these locations (or further away in case of detractors like contaminated sites). We argue that spatial bias plays a role in the final choice among decision options. In many cases, it is a major driver in selecting infrastructure improvement projects (transportation, water resources), service locations (schools, fire stations, health clinics, bank branches, stores, etc.), and other location-based resource allocations. Spatial weighting of evaluation criteria involves assigning a non-uniform importance to locations (Feick and Hall 2004) and thus providing a way of articulating an individual’s frame of reference, like a daily activity route or a particular destination place (work, home etc.), which impacts the perception and evaluation of spatial choice alternatives. Spatially explicit weighting involves both topological and non-topological relations reflecting the proximal and distal characteristics of spatial problems. One or multiple spatial relations can be used
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to express the variability of criterion preferences over a geographical decision space. Using spatial relations, we can quantitatively vary the importance of criteria over space and calculate spatial bias. In this article, we propose to express spatial bias by using proximity as the basis for quantifying spatial preference heterogeneity. We call this approach a proximityadjusted preference (PAP). In order to examine the influence of PAP on the results of G-MCDA, we compare the stability of option rank order obtained with traditional MCDA approach, in which preferences assigned to criteria do not have any spatial bias, with G-MCDA modified by the PAP approach, in which the spatial bias is represented by spatial weights (Herwijnen and Rietveld 1999; Feick and Hall 2004; Ligmann-Zielinska and Jankowski 2008). In the remainder of the paper, we formally define PAP and present an application, in which both approaches (traditional and PAP-based) are compared.
2 Proximity-adjusted preferences In the traditional MCDA approach, preferences relate to criteria rather than options themselves. For example, an individual can decide that the price of a house is twice as important as its architectural style. In this case, one specifies the preference for price in relation to architectural style, disregarding the location of the house and its characteristics, some of which may be measurable and some intangible but nevertheless included in the decision-making calculus (e.g. the quality of the neighborhood). Defined in this way, weights are homogeneous over the geographical space and no attention is given to how the decision criteria are perceived spatially. This homogeneous treatment of criterion preferences within a geographical decision space may result in solutions that only partially reflect the decisionmaker’s preferences. Further, this approach leads to a loss of information, which may be critical in solving spatially explicit problems and may result in choice recommendations that are subsequently rejected due to their incongruence with the preference structure of the decision-maker. To overcome this deficiency, modelers have been using geographically explicit criteria based on spatial relationships such as, for example, ‘proximity to work’, ‘containment within a particular school district’, or ‘density (clustering) of shopping outlets in the neighborhood’. While such spatial criteria are an important component of a geographically scoped decision problem, the representation of criteria preferences is still treated in a traditional—aspatial—way. Rinner and Heppleston (2006) proposed a method of incorporating spatial bias by adjusting final composite evaluation scores of decision options using distances to the favored options. This approach works as a way of spatial smoothing (de Smith et al. 2009; Rinner and Heppleston 2006) and is based on the premise that the proximate options are perceived as similar, and thus their final evaluation scores influence each other. While this post hoc adjustment (Rinner and Heppleston 2006) is an interesting method of neighboring option substitution, it is nevertheless limited to the options themselves and does not account for exogenous points of reference. Below we propose an alternative method of incorporating spatiality into G-MCDA
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by adjusting the individual criteria weights rather than the final composite evaluation scores. We introduce an additional component to the multicriteria model of decision situation in the form of a reference location, which acts as a benchmark shaping the importance of criteria. The proposed approach explicitly introduces spatial heterogeneity of preferences in calculating the attractiveness and the subsequent ranking of decision options. We argue that the decision-makers not only place a particular importance on a given criterion, but also pay attention to how an option is situated with respect to this criterion. For example, when choosing a trip accommodation in a particular locality, we may be inclined toward hotels that are cheaper and thus assign the highest preference to hotel price. Moreover, when considering price, we may also prefer a cluster of cheap hotels in case we need to substitute one hotel with another in the same neighborhood. Such a case imposes the necessity of introducing a spatially heterogeneous preference layer in place of a uniform criterion weight, which does not account for the relative benefit of price similarity in one area versus another. Compared with the approach proposed by Rinner and Heppleston (2006), our methodology goes one step further in representing compound spatial preferences for decision criteria. Research on information integration theory (Anderson 1971; Anderson 1981) posits that in discriminating among different choice alternatives individual decision-makers employ a variety of functional forms, which can be modeled as algebraic rules of varying complexity in order to arrive at a judgment. Numerous empirical experiments (Anderson 1991) lend a considerable support to assertion that people do combine different decision factors (criteria) into more complex constructs in order to arrive at an overall appraisal of a choice. The methodology presented in this paper allows for assigning spatially heterogeneous weights to different criteria. Continuing the example of choosing trip accommodation, we may look for a cluster of cheap hotels as well as hotels that are also close to public transit service offering multiple lines to various destinations and, at the same time, affording a fast commute to a conference center. The latter reflects a spatial bias toward a particular location. In this case, we can define a more comprehensive public transit criterion composed of (a) proximity to low- and highfrequency lines and (b) commuting distance to the conference center, which is encapsulated in the criterion weight. As a result, our approach recognizes the fact that diverse aspects of spatial options can differently contribute toward the achievement of different decision objectives. 2.1 Total weight map Consider a decision situation depicted in Fig. 1. There are five spatial options a (circles A through E) and two reference locations (the black squares L1 and L2). Using the traditional aspatial weighting, for a given criterion C, all options a are assigned the same decision weight WC (Fig. 1a). This is representative of a typical weighting scheme, in which space does not influence criteria weights. Assuming a simple Manhattan distance as the proximity measure, we can also calculate both raw and standardized distances to the reference locations L1 and L2 (Table 1).
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Fig. 1 A hypothetical weight (preference) map that depicts the importance of a given criterion uniformly (a), and in a spatially heterogeneous manner (b, c). Circles are options, their radii are constant in (1a), and proportional to weight values in cases (1b) and (1c), and the reference locations are squares
Table 1 Distances from options to reference locations for the example in Fig. 1
Options
Distance to L1
Distance to L2
Manhattan
Standardized
Manhattan
Standardized
A
6
0.33
4
0.75
B
3
0.67
7
0.43
C
5
0.40
3
1.00
D
2
1.00
4
0.75
E
4
0.50
8
0.38
Avg
4
0.58
5.2
0.66
Min
2
0.33
3
0.38
To account for spatially explicit weighting, we introduce the concept of a total weight TW derived from the set N = 1,…, n of options a, which, for a given criterion C, is defined as follows: n X TW C ¼ WC ð1Þ a¼1
In an aspatial case, TWC = n * WC, where n is the number of options. What follows is that TWC can be redistributed among decision options in a non-uniform fashion (Fig. 1b, c). Therefore, to account for spatial bias toward selected locations, we extend the idea of preferences from ‘relative criterion importance’ to ‘relative criterion and option importance’. The maps [b] and [c] in Fig. 1 illustrate this concept. 2.2 Definition of proximity-adjusted preferences One way to redistribute TWC among the options is to use proximity to the reference locations (L) as a spatial adjustment of criteria preferences. Consider the following formula:
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PAPCa ¼ WC
da ld
ð2Þ
where PAPCa is a proximity-adjusted preference (weight) for criterion C and option a, WC is an aspatial (traditional) preference (weight) for criterion C, da is a standardized distance of option a to the reference location L (i.e. spatial bias), ld is a mean of the standardized distances of all options a to L. Definition A proximity-adjusted preference (PAP) is a weight assigned to each pair (C, a) obtained by modifying the criterion weight WC with the distance from option a to the reference location L normalized by the mean distance of all options to L. In other words, the average PAP-adjusted weight value is the same as the aspatial weight value, whereas the specific PAP-adjusted weight value (a weight that corresponds to an individual decision option) deviates from this average. Observe that: n X PAPCa ¼ TW C ð3Þ a¼1
Proof PAPC1 þ PAPC2 þ þ PAPC n ¼
WC ðd1 þ d2 þ þ dn Þ ¼ n WC ¼ TW C ld
Given the hypothetical example (Fig. 1a and Table 1), we can now calculate PAP for the two reference locations L assuming that WC = 3. The results are presented in Table 2. Note that, although WC varies among the options in maps [b] and [c] as opposed to the initial map [a], TWC remains unchanged. 2.3 Extended PAP example using decision rules To illustrate the influence of the PAP concept on final evaluation scores and consequently on the option rank order, we used four different decision rules (aggregation functions) summarized in Table 3. Table 2 Proximity-adjusted preferences for the example in Fig. 1
Options
Aspatial
PAP for L1
PAP for L2
A
3
1.72
3.41
B
3
3.45
1.95
C
3
2.07
4.54
D
3
5.17
3.41
E
3
2.59
1.70
15
15.00
15.00
Sum (TWc)
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Table 3 Decision rules (aggregation functions) used in the hypothetical example Aggregation function
Description
Simple additive weighting—SAW(Malczewski 1999) P ea = C PC * SCa 1. Criteria outcomes are standardized using the score range procedure or the linear scale transformation (Malczewski 1999) 2. The preferences are then multiplied by the standardized criteria outcomes and summed up to generate a composite option score Rank order—RO (Jankowski et al. 1997) P ea = C PC * SCa 1. Criteria outcomes are standardized using the rank-order standardization as follows. where Ca SCa ¼ LC RR C
For every criterion C: a. Rank every option a (RCa) b. Determine the lowest rank for a given criterion LC c. Calculate the range of ranks (RC) using RC = LC - 1 d. Calculate the standardized criterion outcome (SCa) 2. The preferences are then multiplied by the standardized criteria outcomes and summed up to generate a composite option score
Multiplicative—M (Cobb and Douglas 1928; Chiang 1984) Q 1. Criteria outcomes are standardized using the score range ea ¼ C SPCaC procedure or the linear scale transformation (Malczewski 1999) 2. For each criterion, the standardized criterion outcome is raised to the power of the criterion preference, expressed using [0.0, 1.0] range 3. The results of exponentiation of all criteria are then multiplied to generate a composite option score Ideal Point—IP (Hwang and Yoon 1981) S^
a ea ¼ S^ þS a
a
1. Criteria outcomes are standardized using the score range procedure or the linear scale transformation (Malczewski 1999)
where X 0:5 2. For every criterion C, the worst outcome value (nadir: C^) and the 2 best outcome value (ideal: C*) are established Sa ¼ ð P ð S C Þ Þ c Ca c X 0:5 3. For every criterion-option pair, the difference between the 2 criterion outcome and the criterion ideal is multiplied by the S^a ¼ ðPc ðSCa C ^ ÞÞ c criterion preference and the result is squared. Then, for every option a, these squared results are summed up and raised to the power of , constituting the separation from ideal (S*a). For every criterion-option pair, a difference between the criterion outcome and the criterion nadir is multiplied by the criterion preference, and the result is squared. Then, for every option, the squared results are summed up and raised to the power of , constituting the separation from nadir (S^a ) 4. The composite option score is derived by dividing the separation from nadir by the sum of the separation from ideal and the separation from nadir Common notation: ea is the final evaluation score for option a; PC is the preference (expressed by either PAPCa or WC) for criterion C; SCa is the standardized outcome of criterion C for option a
Suppose that the problem in Fig. 1 is quantified using two decision criteria: access to public transit (C1) and hotel accommodation price (C2). Assume that criterion C1 is associated with the reference location L1 (a conference center),
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Table 4 A hypothetical decision matrix with standardized criteria outcomes and proximity-adjusted weights Options
C1 outcome
C1 preference (%) Aspatial
PAP for L1
C2 outcome
C2 preference (%) Aspatial
PAP for L2
A
0.1
40
23.0
0.5
60
68.1
B
0.2
40
46.0
0.2
60
38.9
C
0.4
40
27.6
0.3
60
90.8
D
0.5
40
69.0
0.3
60
68.1
E
0.9
40
34.5
0.2
60
34.1
whereas criterion C2 is tied to the location L2 (the center of a cluster of cheap hotels). Using equation 2 and distances in Table 1, we can now calculate PAP for both criteria. The results are presented as a decision matrix in Table 4. Using the four decision rules presented in Table 3, we can calculate the final option evaluation scores (ea) using both the aspatial weights and the PAP-based weights. Figure 2 shows the results of these computations. The hypothetical example clearly demonstrates the influence of spatially heterogeneous preferences. Observe that there is a considerable variability among the four decision rules. SAW and M produce rather flat distributions of scores (aspatial range = 28 for SAW and 17 for M, respectively) as opposed to IP and RO, which are quite varied (aspatial range = 53 for IP and 60 for RO, respectively). Furthermore, we can observe that PAPs have a discriminating effect on the final evaluation scores. For every decision rule, the range between the minimum and maximum score increased from the aspatial to the spatial preference quantification. For example, in case of the IP decision rule, the range increased from 60 to 86. Despite PAPs extending the ranges of final evaluation scores, the option rank-order trends obtained with aspatial weights were mostly preserved with PAP-based weights. This is a plausible characteristic of the proposed approach, by which the PAPs have a balancing rather than overwhelming impact on the final rank order of options. Interestingly, the four decision rules result in different winners and different runners-up. The only constant is the worst score of option B in all eight configurations. Introducing spatially heterogeneous preferences with the IP aggregation function did not affect the position of the winning option E. Unlike the IP-generated ranking, the winner (E) and the runner-up (D) switched their ranks in the case of SAW aggregation function after introducing the PAPs as criteria weights. If D and E represent satisfying choices, the decision-maker is likely to select an option closer to his or her area of interest. As this example indicates, the PAP procedure allows for representing such a spatial bias in a formal way.
3 Application In this section, we introduce an example of a real-world decision case to demonstrate and evaluate the proposed PAP concept. The decision problem is
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Fig. 2 Option evaluation scores based on different decision rules and different representations of criteria preferences: a simple additive weighting, b rank order, c multiplicative, and d ideal point
situated in San Diego, California and involves a young family with children in the preschool age. The family needs to choose a house for purchase out of 45 houses available on the market. The stay-at-home dad would like a house which is closer to
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parks with playgrounds, but he is also more inclined toward a few selected adventure parks: Balboa Park and San Diego Zoo, Sea World, and Belmont Park (Fig. 3). The working mum would like to minimize driving to her work place. The house database contains nine attributes, used in the case study as the evaluation criteria, characterizing houses and their location attributes (Table 5). The data was collected during 2003–2005 through fieldwork, census, and local surveys (source: SANDAG http://www.sandag.org/). Driving distances were derived using online mapping software (Google Maps http://maps.google.com). For this example, a few technical assumptions are required. We assume that criteria scores are error-free and, thus, we concentrate on the uncertainty associated with criteria preferences. We standardize most of the criteria using the linear scale transformation (Malczewski 1999) to maintain the proportionality of standardized scores to raw data and minimize the influence of standardization on calculating composite option scores. Only the YEAR (Table 5) was standardized using the
Fig. 3 Location of 45 houses in San Diego, California in relation to the place of work and adventure parks
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Table 5 Decision criteria used in the case study PRICE
BED
YEAR
LOT
LIVG
MDVA
PARK_ ACCESS
DIST_ WORK
DIST_ ADVPK
AVG
282.65
3.4
1958
6,313
273
203,115
0.3
4.7
33.8
MED
275
3
1958
6,090
264
205,723
0.3
5.3
32.1
10
1,109
67
22,458
0.2
2.5
2,000
120
97,300
0.1
0.5
STDEV
42.23
0.6
8.7
MIN
196.25
2
1939
18
MAX
359.25
5
2004
9,330
420
237,388
1.0
9.6
50.5
RANGE
162.99
3
65
7,330
300
140,088
0.9
9.1
32.5
PRICE $ per sq ft, BED number of bedrooms, YEAR year house was built, LOT house lot area (sq ft), LIVG living room area (sq ft), MDVA median housing unit value in the census block group (including condominium apartments) ($), PARK_ACCESS accessibility to parks measured as normalized cumulative Euclidean distance weighed by park area, DIST_WORK driving distance to work (miles), DIST_ADVPK cumulative driving distance to three adventure parks (Balboa Park and San Diego Zoo, Sea World, and Belmont Park) (miles)
score range transformation (Malczewski 1999) due to its interval measurement scale. 3.1 Computational experiments The objective of this case study is to compare the traditional MCDA (where the spatial dimension is represented with distance criteria like distance to work) and the proposed PAP-based MCDA (where the distance criteria are coupled with selected aspatial criteria). We start from performing the traditional (aspatial) prioritization of houses (Experiment 1). Since the exact preferences for the decision criteria are unknown, we use Monte Carlo sampling from predefined random uniform distributions (Table 6, ‘Aspatial Experiment’ column) to calculate the composite evaluation scores and to repeatedly simulate the prioritization of houses (Butler et al. 1997). We adopt an economical perspective, with the strongest preference for affordability and accessibility to parks and with lesser priority given to the other criteria. The next step (Experiment 2) involves house prioritization using spatially adjusted weights. We select two criteria for PAP modification: PRICE and PARK_ACCESS. The former is adjusted with distance to work (DIST_WK) resulting in a new criterion called PRICE_PAP. The latter is modified with distance Table 6 Random uniform probability density functions (PDF) used to simulate criteria preferences Criteria
Aspatial experiment
PAP-based experiment –
PRICE, PARK_ACCESS
Min = 0.15, max = 0.45
PRICE_PAP,PARK_ACCESS_PAP
–
Min = 0.30, max = 0.60
BED, YEAR, LOT
Min = 0.10, max = 0.40
Min = 0.10, max = 0.40
LIVG, MDVA
Min = 0.05, max = 0.35
Min = 0.05, max = 0.35
DIST_WK,DIST_ADVPK
Min = 0.00, max = 0.30
–
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to adventure parks (DIST_ADVPK), and the generated criterion is named PARK_ACCESS_PAP. Thus, instead of having distances to work and adventure parks represented as two additional decision criteria, we use these two criteria to adjust house price and park accessibility. In particular, PRICE_PAP represents the overall monetary cost criterion that combines a proxy for the tangible housing costs (mortgage, real estate taxes, and insurance) with the less tangible everyday commuting costs, which depend on the driving distance to work. On the other hand, PARK_ACCESS_PAP combines the spatial bias toward adventure parks with the area-weighed distance to all parks. The rationale behind constructing PARK_ACCESS_PAP and PRICE_PAP as described above is based on empirical research on real estate purchase choices. Results of empirical studies on the influence of decision criteria in choice making concerning residential housing point out the importance of distance as a location choice criterion, though, they also show that the importance of proximity differs among various socio-demographic groups (Freiden and Bible 1982; Reed and Mills 2007; Gibler et al. 2009). Different decision-makers are characterized by different cognitive styles when valuing the distance from their housing locations to other reference locations. For example, young professionals may value higher the distance from housing to cafe´s and other eating-out establishments than to medical facilities, which may be just the opposite for senior citizens (Freiden and Bible 1982; Reed and Mills 2007; Gibler et al. 2009). To account for the loss of DIST_WK and DIST_ADVPK in the cumulative distribution of weights, we increase the minimum and maximum weight values for the random uniform probability density functions of PRICE_PAP and PARK_ ACCESS_PAP (Table 6, ‘PAP-based Experiment’ column). Specifically, the midpoint value (0.15) of the distance criteria distributions is added to both the minimum and maximum values of PRICE and PARK_ACCESS weight ranges. In terms of individual preferences, the rest of the criteria are assumed to be spatially invariant. Both aspatial and PAP-based experiments were executed using 5000 renormalized weight vectors based on the selected aggregation functions, which are analyzed separately. Out of the four aggregation functions presented in Table 3, we used SAW, IP, and RO to arrive at rank orders of candidate houses, which amounted to six experiments overall. The multiplicative decision rule was excluded due to its highly constraining nature that eliminates an option whenever any of its criteria is assigned a score of zero, which potentially reduces the variability of rank shifts. All computations were performed using customized Python scripting (http://www. python.org/), which is available from the authors.
4 Results and analysis We focused our analysis on determining the statistical significance of the differences between aspatial and PAP-based option rankings. We also tested the impact of various decision rules on the variability of option evaluation scores. Finally, we mapped the shift in ranks for the two conceptualizations of weights within the study area, followed by a recommendation of the decision options.
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Impact of proximity-adjusted preferences on rank-order stability Table 7 t-test results for average shift in ranks (ASR) from the equal weight case
179
Aspatial experiment
PAP-based experiment
Avg
4.3
11.06
Var
1.26
0.56
df
8,683
t-stat
354.5
Cohen’s d
7.1
Simple additive weighting
Ideal point Avg
5.8
7.0
Var
1.43
0.75
df
9,101
t-stat
54.6
Cohen’s d
1.1
Rank order
Comparisons between aspatial scenario and PAP-based scenario (a = 0.05)
Avg
4.46
8.67
Var
1.51
1.81
df
9,918
t-stat
163.3
Cohen’s d
3.26
4.1 The significance of spatially heterogeneous weights for determining house ranks We started the analysis from calculating the importance of spatial bias in determining option rank order. The objective was to determine whether the PAPderived rankings were significantly different, in the statistical sense, from the aspatial rankings. We began from summarizing house rankings using a descriptive statistic called an average shift in ranks (ASR) (Saisana et al. 2005): n 1X ASR ¼ ja rankref a rankj ð4Þ n a¼1 where ASR is the average shift in ranks, a_rankref is the rank of option A in the reference ranking (e.g. equal weight case), a_rank is the current rank of option A. ASR captures the relative shift in the position of the entire set of options and quantifies it as the sum of absolute differences between the current option rank (a_rank) and the reference rank (a_rankref), divided by the number of all options. In the first step of the analysis, we selected equal weight aspatial case as the reference ranking, which was calculated for SAW, IP, and RO, respectively. We used equal weights as the bases for the reference ranking to establish the baseline case without the effect of any preference bias on option scoring and ranking. Using sensitivity analysis to compare the magnitudes of departure from the baseline ranking allows us to measure the significance of different conceptions of criteria weights.
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A two-sample test assuming unequal variances was conducted to compare the ASR of aspatial rankings with the proximity-adjusted rankings. The results are presented in Table 7. To analyze the results, we used two-tail P-values from the t-tests. In all cases, we recorded P \ 0.00005. Since the P-value is considerably less than a, we can conclude that there are statistically significant differences between the aspatial scenario and the PAP-based scenario. For every aggregation function formulation (SAW, IP, and RO), there was a significant difference in ASR for aspatial and spatially heterogeneous preferences. These results suggest that PAP affects the prioritization of houses in the presented case study. Specifically, when we apply PAPs, the average shift in ranks noticeably increases (e.g. by 7 ranks from the aspatial to PAP scenario using the SAW decision rule). Due to the large samples (each scenario is composed of 5,000 simulations), we augmented t-tests with the effect size Cohen’s d statistics (Cohen 1988). For all three decision rules, we observed large values of Cohen’s d statistic indicating that the magnitude of differences is substantial, that is, not only statistically significant but also substantively important. 4.2 Visual comparison of aspatial and PAP-based house ranking scenarios The solution recommendation for the decision problem requires finding highscoring options. First, for each prioritization scenario, we calculated rank summary statistics including minimum, maximum, mean, and variance over all simulations (Andrienko and Andrienko 2005) and created uncertainty plots for each option. Figure 4 illustrates the results for SAW decision rule and the aspatial versus the PAP-based scenarios. With these statistics, we can determine which houses on average rank the highest, which houses have the most or the least sensitive position, and whether any house is dominated by other houses. To establish high-ranked and low-ranked options, we consider the following four cases: 1. 2. 3. 4.
High average rank and low-rank variance signify the winning options and place high confidence in their prioritization (i.e. a robust solution) Low average rank and low-rank variance represent the losing options that may be disregarded. High average rank and high-rank variance characterize options that are potentially good but need further study due to the sensitivity of ranking. Low average rank and high-rank variance designate options that may be dropped from the analysis but with lesser confidence when compared to case 2.
In Fig. 5, we rendered the results on rank stability maps (Jankowski et al. 1997), using as an example the SAW decision rule. Other decision functions used in the study (IP and RO) yielded similar results. The maps depict not only average ranks but also their variances. As noted above, we seek robust options that on average rank high and have a relatively low-rank variance. Such options are depicted with large and medium gray circles. Using rank and rank variance as the selection criteria, we can conclude that no decision rule identifies clear winners in the aspatial scenario, since the high-scoring options (Table 8) are characterized by relatively
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181
Fig. 4 House ranks summarized over all simulations for the simple additive weighting decision rule. Top: aspatial preferences, bottom: PAP-based scenario. Houses ordered by average rank from min (best) to max (worst). Legend: bars—rank range, barbells (error bars)—standard deviation of ranks, dot— average rank
moderate to high standard deviations (see Fig. 4-top, 5-top). Contrary to the aspatial results, the PAP-based simulations provide well-established winners (predominantly houses 31, and 32, and 33 under all three decision rules—Table 8), which have the best rank and the lowest rank variance. This observation is also confirmed by the small error bars in Fig. 5-bottom (std \ 0.5 for top three ranks). Observe that, for SAW, the highest scoring options in the aspatial case are scattered throughout the
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Fig. 5 Rank stability maps of house prioritizations using aspatial simple additive weighting (SAW) scenario (top) and PAP-based SAW scenario (bottom). The size of the circle represents the average house rank, whereas the shade of gray indicates the rank standard deviation. For comparative purposes, both size and color classifications were fixed among the maps
area, whereas the PAP-based scenario results in a well-established southern cluster of high-scoring options, which is closer to the place of work (Fig. 5). RO and IP exhibit the same patterning. Consequently, the high-scoring options for the aspatial preferences are different from the high-scoring options for the proximity-adjusted preferences. The latter options (houses 31, 32, and 33) maintained their high ranks
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Table 8 Six top winning houses recorded by House ID (avg rank; std) for three decision rules representing aspatial and PAP-based scenarios: SAW—simple additive weighting, IP—ideal point, RO—rank order SAW
IP
RO
H19 (3.7; 3.7)
H45 (1.3; 0.6)
Aspatial top rank houses 1
H32 (2.2; 1.8)
2
H19 (3.4; 2.4)
H32 (3.9; 3.7)
H1 (3.4; 1.9)
3
H45 (4.3; 2.1)
H44 (4.7; 4.2)
H6 (3.4; 1.4)
4
H44 (4.7; 3.0)
H13 (5.0; 3.2)
H25 (3.8; 1.9)
5
H13 (5.9; 2.4)
H45 (8.2; 3.4)
H13 (5.6; 2.2)
6
H33 (6.9; 5.4)
H1 (8.5; 4.3)
H44 (7.2; 3.3) H33 (1.3; 0.4)
PAP-based top rank houses 1
H31 (1.3; 0.5)
H32 (1.4; 0.5)
2
H33 (1.7; 0.5)
H19 (2.1; 1.2)
H31 (1.7; 0.4)
3
H32 (3.0; 0.0)
H13 (4.0; 1.3)
H32 (3.0; 0.0)
4
H39 (5.2; 1.4)
H31 (4.1; 1.4)
H45 (4.6; 0.8)
5
H30 (5.8; 1.2)
H44 (4.8; 2.0)
H25 (6.8; 1.5)
6
H37 (5.9; 1.1)
H33 (7.7; 1.7)
H6 (7.6; 1.6)
Bolded houses are common among all three decision rules
regardless of the applied decision rule (Table 8). When the PAPs are applied, some of the high-scoring houses become dominated by others in terms of their rankings. For example, houses 31, 32, and 33 (Fig. 4) dominate houses 39, 30, and 37 since their worst rank is still higher than the best rank of any of the latter houses. In general, given the sizes of the bars in Fig. 5, the aspatial scenarios are much more sensitive to changes in preference values, whereas PAPs have a balancing effect on the stability of house ranking. 4.3 Rank shift maps To further test the spatiality of ranking obtained with proximity-adjusted preferences, we summarized the PAP-based simulations by calculating a shift in average house rank from the aspatial case to the PAP-based case. The shift from the aspatial to the PAP scenario is mapped in Fig. 6 for IP (top) and RO (bottom). SAW rendered a very similar result to RO, and we do not report it here. The maps provide two pieces of information. First, they depict houses that either gained or lost their average ranking positions. Note that the PAP-based gain (improvement) in house ranking occurs when its aspatial rank is higher in value (i.e. worse) than its PAP rank, resulting in a positive difference. Secondly, the maps validate the observation about the general clustering of high-scoring houses south and east of the work place. For threshold distance of 1 mile, the trend is spatially autocorrelated with significant clusters (SAW: Moran’s I = 0.75, Expected I = -0.02, Variance = 0.008, Z-Score = 8.38; IP: Moran’s I = 0.57, Expected I = -0.02,
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Fig. 6 Shift in rank per house from the aspatial scenario to the PAP-based scenario. Top: IP decision rule, bottom: RO decision rule
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185
Variance = 0.009, Z-Score = 6.34; RO: Moran’s I = 0.65, Expected I = -0.02, Variance = 0.008, Z-Score = 7.41). The west cluster lost its overall ranking position when moving from the aspatial scenario to the PAP-based prioritization. The east cluster also lost its position in the ranking but the loss is less pronounced, that is, not observed for IP (Fig. 6). 4.4 Discussion of results Our general observation is that PAPs have a significant impact on option prioritization. Within the reported experiments, the proposed method of quantifying spatial bias generates a directional shift of high-ranked sites and results in proximal option score smoothing (Fig. 5). When the distance criteria (DIST_WK and DIST_ADVPK) are incorporated into the PAP of price and park accessibility (PRICE_PAP and PARK_ACCESS_PAP), their effect on house ranking is different. In the aspatial case, both DIST_WK and DIST_ADVPK have an equivalent influence on house prioritization, resulting is a scattered distribution of high-rank options, which reside in all three clusters (Fig. 5-top). In contrast with the aspatial case, the PAP-based scenarios gather all high-ranked houses together in the south cluster (Fig. 5-bottom) suggesting that the work distance–adjusted price criterion becomes more influential than the adventure park distance–adjusted park accessibility criterion. For example, by integrating PRICE with DIST_WK, we increase the interactivity among these two decision criteria and decrease their independent additive role in house scoring and ranking. As a result, the impact of PRICE_PAP and PARK_ACCESS_PAP is more pronounced for options that are close to work, moderately priced, and in proximity to many (albeit small) parks. Going back to Fig. 5-bottom, one can observe that both W and E clusters are comparable in terms of house ranking. While the former is located closer to adventure parks, it is also further away from work when the driving distance is considered. Hence, the two distance adjustments cancel out, resulting in similar rank clusters. We can conclude that PAPs have an integrative and non-linear effect on option ranking. By incorporating spatial bias in the form of variable weights, which are functionally dependent on distance between each locational decision option and the reference location(s), the proposed approach provides an alternative to representing complex spatial criteria preferences and, at the same time, affords a reduction in the decision space (from 9 to 7 evaluation criteria in the presented case study, Table 6).
5 Conclusion The main idea behind the work presented in this paper is that the relative importance given to criteria in a spatial multicriteria evaluation problem is not necessarily constant over the geographical space but in fact it may vary. This variability in preferences, represented by spatially adjusted weights, reflects the influence of spatial structures and patterns on the perceived importance of evaluation criteria. The spatial adjustment can result from considering one or multiple spatial relations and computing their values for the decision options. Such values are in effect the
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measures of spatial structure and pattern and can be used as ‘‘spatial’’ criteria in multicriteria evaluation. This idea is not new and has been presented by others. The contribution of this paper is in recognizing that spatial criteria give rise to spatially variable preferences that affect the relative importance assigned to aspatial criteria. The paper also presents an analytical approach to calculating the effect of spatially adjusted weights on the rank order of decision options. The analysis method and the subsequent case study deal with only one spatial relation—proximity. Moreover, the case study presents a simplified situation of distance-adjusted weights under the presence of two groups of reference locations: the place of work and the selected adventure parks. While we have shown that the PAP approach results in significantly different rankings of options, we have not determined which approach yields a ranking that better matches the view of a decision-maker. Therefore, future work should include an empirical investigation of the proposed methodology, focusing on the relevance of PAP to solving spatially explicit decision problems. Further extensions of the presented approach should also address multiple spatial relations and spatial preference adjustments resulting from the simultaneous consideration of more than one spatial relation. An example of this is adjacency to specific reference categories, such as type of built-structures or demographic characteristics of people inhabiting these structures. Acknowledgments The authors would like to acknowledge critical and constructive comments from three anonymous reviewers on the previous version of the manuscript.
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