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TEACHING GEOG 385.02/GTECH 785.02 |
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Multi-Criteria Evaluation (MCE)
By the end of this lecture you should be able to:
- Discuss two major limitations to the Boolean approach to multiple attribute query.
- Define what is meant by "standardization" and "aggregation" in MCE problems.
- Differentiate between constraints and factors as criteria for MCE.
- Explain what fuzzy sets are and how they can be represented in Idrisi.
- Describe the steps required in the aggregation technique of Weighted Linear Combination.
Boolean Approaches for Decision Making
Multi-criteria evaluation is the solving of a suitability problem (best location or most likely location of some phenomena) using multiple layers of information. Each layer contributing to the solution is a criteria.
You have done several multi-criteria evaluations up to this point. They have all been Boolean approaches.
Exercise 6 from the student tutorial is a good example of a standard Boolean approach to MCE. Review Cartographic model from tutorial.
In this example data for different criteria are brought together for a single solution. Layers with different kinds of data (e.g. slope values, distance values, and landcover categories) are first made comparable – they are standardized to some common scale of values. In this case they are standardized to a scale with just two possible values: 0 and 1.
This scale demands a "crisp" or "hard" decision between what is suitable (i.e. the locations we are interested in) and what is not. There is no "in-between."
In Boolean analysis, once criteria are reduced to 0s and 1s they are combined or aggregated. In Boolean, aggregation is typically multiplication (logical AND); however, it could be other operations such as maximum (logical OR).
These procedures assume that all criteria should be considered equally as important to the solution. There is no way to give different weights to different criteria when the aggregation method is Boolean.
From Exercise 6:
- Slopes less than 2.5 degrees (2.6 is not suitable, 2.4 is as suitable as 0.5).
- Distance at least 250 meters from reservoirs (249 is not suitable, 251 is as suitable as 2000).
- Aggregation of these criteria considers them all equally: low slopes are as important as distance from reservoirs and landuse.
Alternatives?
In terms of decision making, a Boolean approach leaves little room for compromise. There is no room to negotiate alternative scenarios to suitability.
However, there are other scales that we can use for standardization that have more than just two values. We could have a scale with many values representing degrees of suitability. When criteria are scaled to represent degrees of suitability they are referred to as factors.
Compare:
Totally unsuitable Totally suitable
Boolean 0 1
Fuzzy 0------------------------------1
There are examples of Boolean criteria that can never be re-scaled to a continuous scale representing degrees of suitability. These criteria are called constraints.
Factors and constraints examples:
- Slope can become a "factor map."
- Distance from reservoir can become a factor.
- Landuse can be a factor.
- Development only in forested areas and no where else would be a constraint.
- No development in a national park is a constraint.
- No development in water areas is a constraint.
In Idrisi factors are re-scaled 0-255 while constraints remain Boolean (0 or 1).
Also, there are different ways to aggregate multiple criteria that allow us to give different weights to different criteria (factors only).
Boolean Hypothetical Weighted Aggregation
Slopes
.33
.30
Res. Dist.
.33
.50
Landuse
.33
.20
Aggregation of constraints (criteria that can only be represented in Boolean terms) remains an OVERLAY function (e.g. logical AND). For factors (criteria represented by the degree of suitability), we need different methods of aggregation.
One analysis may combine both the Boolean aggregation of constraints and alternative forms of aggregation for factors.
Non-Boolean Standardization for Decision Making
Decision making is often more complicated than a Boolean analysis would allow:
- It involves more people or interest groups with different opinions.
- It involves different ideas about appropriate criteria and their relative importance.
- It involves different degrees of suitability even within one criteria.
- It is an iterative process with changes from one "round" to the next.
We want to use tools that allow for the complexity of "real" decision making.
Our alternative to Boolean standardization will use a new scale where some criteria is represented on a scale of suitability 0-255 (degrees of suitability). Could use other scales but 0-255 is what Idrisi uses as a standard.
The values 0-255 represent membership in the "Fuzzy set" of suitable locations (compare to Boolean set membership – in or out).
Original data (e.g. distance values or slope values) are re-scaled or stretched to this new range of data (much in the same way data is re-scaled to 0-255 colors for display).
Both continuous and categorical data can be re-scaled to 0-255.
Continuous/quantitative data
Data can be re-scaled to new range of data using a simple linear function.
Other functions also exist for re-scaling 0-255 (or any other range). Data can be transformed or "stretched" using non-liner functions such as a sigmoid function.
In Idrisi the module FUZZY can be used for both linear and non-linear "stretches." Look at FUZZY Help dialog to see the range of functions available. We’re only interested in linear and sigmoid for this course.
Categorical/qualitative data
Categorical data can be re-scaled to new values (0-255) using RECLASS or Edit/ASSIGN.
In this case there are arbitrary (or intuitive) measures of suitability assigned to old values of landuse or some other categorical data.
From example: Forested areas are most suitable, but other landuse types might be suitable as well.
Non-Boolean Aggregation for Decision Making
Other forms of aggregation (non-Boolean) generally involve some sort of weighting of factors. Remember that constraints are always combined using Boolean methods.
The procedure we’ll explore is called Weighted Linear Combination. It is basically a weighted averaging technique that works with several images cell by cell (it’s just a map algebra procedure).
Factors are assigned different relative weights compared with each other, they are multiplied by these weights, the results are added, the sum is then divided by the number of factors. This is a cell by cell process that uses factors that have been first standardized (0-255).
Cell values Weights First result Sum Final
Factor
1
108
.30
32.4
Factor 2
229
.20
45.8
90.2 30.06
Factor 3
24
.50
12
Weights must add up to 1 for WLC to work. It is clearly an averaging procedure (all three factors are averaged).
- If their weights were equal then it would be an exact averaging.
- When weights are not equal it is a weighted averaging.
Developing Weights for Weighted Linear
Combination
2 methods for developing weights:
1: You can simple come up with the weights yourself such that they sum to 1. This seems easy, but if relative importance is expressed for criteria only in relation to other criteria then it is difficult.
For example:
o Forested areas are more important than distance from reservoirs.
o Distance from reservoirs is more important than low slopes.
o Low slopes are more important than forested areas (oops!).
To develop weights intuitively requires looking at all of them together, this will not allow for negotiation or compromise looking at criteria 2 at a time.
2: Analytical Hierarchy Process (AHP). There are statistical methods that let you compare criteria 2 at a time. The user specifies the relative importance of one criteria compared to another and does this for all possible combinations of criteria.
The procedure will then tell you how consistent are all of your comparisons and it will give develop weights for you for each criteria.
In Idrisi the module WEIGHT will perform this operation.
Summary
What looks like the simple combination of criteria to get one answer can be expanded using non-Boolean procedures such that there are a variety of possible answers to one suitability problem.
Different answers for the same problem (e.g. suitability of development) result from:
o Considering criteria to be a factor or constraint.
o How the factor is standardized (what function, what thresholds)
o How each factor is weighted (what method to develop weights, what weights)