ANALYTICAL HIERARCHY PROCESS (AHP)

Applying AHP in Weighting GIS Model Criteria

The following extends the brief description in the Beyond Mapping column on “Calibrating and Weighting GIS Model Criteria” by Joseph K. Berry appearing in GeoWorld, September, 2003.  The columns in this series are posted at— http://www.innovativegis.com/basis/MapAnalysis/Default.html, select Topic 19, Routing and Optimal Paths.

A companion discussion on applying the Delphi Process for “calibrating GIS model criteria” is posted at— http://www.innovativegis.com/basis/bm/Beyond.htm#BM_supplements.

What is AHP?

Analytical Hierarchy process (AHP) is a quantitative method for ranking decision alternatives by developing a numerical score to rank each decision alternative based on how well each alternative meets the decision maker’s criteria (Operations Management, 4th Edition, 2003, by R. Russell and B. Taylor, Prentice-Hall).

What kind of information is gained?

AHP can be used to determine the relative weights among decision elements for GIS-based Suitability and Routing models.  In its many other applications AHP is used to select the best single alternative that best matches decision criteria (decision-making models).

What is involved in the process?

The process involves 1) identifying the decision elements (map layers), 2) recording relative importance of those elements, 3) construction of an importance table and 4) implementing a simple mathematical solution.

For example, the routing of an electric transmission line described in the Beyond Mapping series (see above reference) is used to demonstrate the process in the following paragraphs.

How are the decision elements identified?

The decision elements are identified by group interaction and discussion.  The Delphi Process can be useful in identifying and calibrating the decision elements.

In the transmission line routing example, the objectives and map layers include avoiding locations that 1) have high Visual Exposure (VE= Visual_Exposure_rating), 2) are close to Sensitive Areas (SA= SA_Proximity_rating), 3) are far from Roads (R= R_Proximity_rating) and 4) have high Housing Density (HD= H_Density_rating).

How are the pairwise comparisons recorded?

A scale from 1= equally important through 9= extremely important is used to record the relative level of importance for the pairwise combinations of the decision elements.  Each member of the group first orders the decision elements to be compared so the statement “<element A> is preferred over <element B>”is correct, and then records the appropriate rating value (1 to 9) for the strength of the opinion.  The number of pairwise combinations is calculated by #Pairs= (N * (N – 1) / 2), where N is the number of decision elements.

For example, a group member might respond to the question “In routing electric transmission lines…”

(VE vs. SA)— avoiding locations of high Visual Exposure is extremely more important (rating= 9) than avoiding locations close to Sensitive Areas.

(VE vs. R)— avoiding locations of high Visual Exposure is strongly more important (rating= 5) than avoiding locations close to Roads.

(VE vs. HD)— avoiding locations of high Visual Exposure is equally important (rating= 1) to avoiding locations of high Housing Density.

(SA vs. R)— avoiding locations close to Roads is strongly to very strongly more important (rating= 6) than avoiding locations close to Sensitive Areas.

(SA vs. HD)— avoiding locations of high Housing Density is very strongly to extremely more important (rating= 8) than avoiding locations close to Sensitive Areas.

(R vs. HD)— avoiding locations of high Housing Density is strongly more important (rating= 5) than avoiding locations close to Roads.

How is an individual importance table constructed?

The responses from the pairwise comparisons are entered into the importance table (comparison matrix) a row at a time.  The order of statement determines where in the table the importance value is placed.  The reciprocal is computed and placed in the reverse statement position.

For example, the first response noted above identifies that VE is extremely (9) more important than SA so the value 9 is placed in position row 2, column 3.  The reciprocal value of 1/9 (.111) is placed in position row 3, column 2.  The last response identifies that HD is strongly (5) more important than R so the value 5 is placed in row 5, column 4; the reciprocal 1/5 (.200) is placed in row 4, column 5.

How are the weights calculated?

Enter pairwise responses into the importance table.

See above.

Step 1: Complete the table by calculating the reciprocal values.

The reciprocal values are converted to decimal values.

Step 3: Sum the column values.

The column sum of the values in the table is calculated (e.g., 1 + 0.111 + 0.200 + 1 = 2.311 for column 2 (VE)).

Step 4: Normalize the table values by dividing by the column sums.

Each value in the table is divided by its corresponding column sum (e.g., 0.111 / 2.311 = 0.05 for table position row 3, column 2 (SA vs. VE)).

Step 5: Sum the normalized row values.

The row sum of the normalized values in the table is calculated (e.g., 0.43 + 0.38 + 0.45 + 0.43 = 1.69 for row 1 (VE)).

***Note: Complete Steps 1 through 4 for each participant and average the weight sets, then determine the minimum value before proceeding to Step 6.

Step 6: Divide the row sums by minimum value for relative weights.

Dividing the calculated average weights by the minimum weight generates a relative scale of influence for each of the map layers.

Importance of avoiding locations—

…of high Visual Exposure (VE) is 10.64 times more important than avoiding locations near Sensitive Areas (SA)

…far from Roads (R) is 3.23 times more important than avoiding locations near Sensitive Areas (SA)

…of high Housing Density (HD) is 10.38 times more important than avoiding locations near Sensitive Areas (SA).

How are the derived weights utilized in a GIS model?

(Click image to enlarge)  The weights are used in “weight-averaging” the map layers to identify an overall suitability map.  In implementing the relative weights in a GIS model, the map layers are multiplied by their respective weights; the products are summed and then divided by the sum of the weights to calculate the weighted average for each map location.  In the example in the accompanying figure, the AHP calculated weights are applied to the calibrated map layers to derive a 3.62 average preference for the identified map location.  The weighted average map contains the overall preferences for each map location in a project area.

(Click image to enlarge)  AHP can be used to derive different weight sets that reflect different preferences of individuals or groups.  In the accompanying figure a “Community” and an “Environmental” perspective is mapped.  Note the dramatic differences in opinion—the community perspective believes avoiding locations of high Housing Density (HD= 10.38) and high Visual Exposure (VE= 10.64) are extremely important in routing a powerline.  The environmental perspective, on the other hand, views avoiding locations near to Sensitive Areas (SA= 10.56) as extremely important while HD and VE considerations are weighted very low.

What information is gained?

(Click image to enlarge)  Subtracting the two suitability maps based on the two AHP-derived weight sets identifies areas of agreement and disagreement in the appropriateness for routing a powerline at each map location.  In the accompanying figure, the green tones represent areas that the Environmental perspective views as more suitable than the Community perspective; opposite interpretation for the red areas.  The grey are indicates no difference in their views.  The difference map can be useful in discussions seeking consensus.

(Click image to enlarge)  Alternative routes can be generated using the two different weight sets and compared both graphically and statistically.  At the top of the accompanying figure, the two derived routes for the powerline are superimposed on the Community and Environmental suitability surfaces.  The red ellipses highlight the areas where the routes encounter locations of very low preference as identified on the opposing suitability surface.  Note how the corresponding routes bend around areas of relatively low preference (green to red).

The tables at the bottom of the figure summarize the data from the criteria maps.  Note the average values for Visual Exposure (VE= 9.1 vs. 3.6), proximity to Sensitive Areas (SA= 11.7 vs. 5.6), proximity to Roads (R= 8.4 vs. 11.7) and Housing Density (HD= 18.7 to 3.0).  The statistics confirm that the Community route tends to avoid high visual exposure and housing density, while the Environmental route tends to avoid being close to sensitive areas.  Both routes have similar proximity to roads values.