PE&RS February 2016 - page 110

Modeling Decision Strategies
In order to connect two road key points, the values of each
connection criteria are aggregated, considering the importance
weight of each criterion. The decision strategy is a primary
feature in selecting an aggregation function. One extreme is
the situation in which all the criteria are satisfied, and the
other extreme is the state where any of the criteria is met. Dif-
ferent decision strategies are located between the mentioned
two extreme strategies. In the following section an OWA based
aggregation operators is presented.
Ordered Weighted Averaging Operators
The OWA operator was introduced by Yager (1988) and it
provides a parameterized family of aggregation operators
that include the maximum, the minimum and the arithmetic
mean. The n-dimensional OWA is a mapping OWA: RnR with
an associated weighting vector W= [w1, w2, . . . ,wn] in which
wj∈[0, 1] and j
n
= ∑ 1wj =1. Using this operator a sequence A of n
scalar values for each alternative in the data set are ordered
decreasingly and then weighted according to their ordered
position by the weighting vector. If cj represents the j th largest
value in A, Equation 1 is defined:
OWA A w c
j
n
j j ( ) =
= ∑1
. (1)
The main characteristic of the OWA operator is its flexibil-
ity to model a wide range of aggregation strategies (Cornelis et
al., 2010). The main goal of reordering step is distinguishing
between situations where the highest argument is the best re-
sult and situation where the lowest argument is the best result
(Merigo and Gil-Lafuente, 2008). For example, the Minimum,
the Average, and the Maximum can be modeled by means of
OWA operators as follows:
a. Minimum: Wmin={wj}, where w n=1, wj=0, j≠ n
b. Average: W avg={wj}, where wj=1/n, j= 1, …, n
c. Maximum: Wmax={w j}, where w1=1, wj=0, j≠ 1
The Maximum operator provides the type of “ANDing”
aggregation implemented by the “all” requirement. Similarly,
“ORing” aggregation operator is a representative of Minimum
operator which fulfill an “at least one” type of condition
(Yager, 1988).
In this research the weight vector represents two groups of
weights including order weights and criteria weights.
Determination of Order Weights
Assume we want to calculate the cost of each road segment
using three criteria C= {C1, C2, C3}. The relative importance of
each criterion is denoted by a weighting vector W = {w1,w2,w 3}.
By selecting the appropriate weights in W , different argu-
ments can be emphasized based upon their position within the
ordered matrix of criteria. For example, if most of the weights
were placed near the top of W, the criteria with higher scores
are emphasized, and if the weights were placed near the bot-
tom of W, then the criteria with lower scores in the aggregation
process are highlighted. Yager (1988) introduced “ORness” as
a measure to describe the behavior of an OWA operator. This
operator indicates the degree in which an OWA operator be-
haves like an OR operator and is defined in Equation 2:
Orness W n n w
j
n
j j ( ) = − − ( )
= ∑
( / ) ( . ) 1 1
1
. (2)
As the behavior of an aggregation operator goes from mini-
mum to maximum, the ORness degree will range from 0 to 1.
From Equation 2 it is inferred:
a. ORness (0, … , 0,1) = 0 Min operator
b. ORness (1,0, … , 0) =1 Max operator
c. ORness (1/n,1/n, … , 1/n)½ Weighted Linear Combi-
nation (WLC) operator.
It is clear that the actual type of aggregation performed
by an OWA operator depends upon the form of the weighting
vector. Based on the extensive study on the methods of deter-
mining the weighting vector, three categories are recognized:
Some approaches determine the weights based on the mea-
sures which describe the OWA behavior like ORness (Fuller
and Majlender, 2001; Amin and Emrouznejad, 2006). A
number of approaches have been suggested for obtaining the
associated weights by means of linguistic quantifiers (Yager,
1988; Emrouznejad, 2008; Nadi and Delavar, 2011). Other ap-
proaches employ the decision attitude of the decision-maker
to obtain the associated weights through an objective function
(Yager and Filev, 1994; Ahn, 2008). This study utilizes the
first category to determine the OWA weights, which is more
intuitive for the application in hand.
Determining Order Weights Using ORness
In this paper, a decision strategy is defined using appropri-
ate ORness values which indicates the portion of the criteria
necessary for a good solution. Based on Yager (1996) the OWA
order weights are computed using Equation 3:
w j
n
j
n j = − − ( ) ( ) α α 1 (3)
where α is the degree of optimism which indicates the deci-
sion strategy. The value of α is related to the ORness as fol-
lows (Equation 4):
ORness = +
1
1 α α ≥ 0 . (4)
Changes of α represent a continuum of different decision
strategies between the two extreme cases of logic AND or OR.
The ORness=0 (α∞) represents the strategy corresponding
to the MIN operator. On the other hand, the ORness=1 (α=0)
represents the strategy corresponding to the MAX operator.
If ORness = 0.5 (α = 1), then the strategy corresponds to the
conventional WLC , which is situated at the mid-point of the
continuum between MIN and MAX operators.
The strategy associated with the ORness = 0 is referred
to as the extremely pessimistic strategy (see Table 2); Thus,
only the lowest criterion value is considered in the evaluation
process. Conversely, the extremely optimistic strategy corre-
sponds to ORness = 1 (Jelokhani and Malczewski, 2014).
Table 2. The Aggregation/Decision Strategies Corresponding to Specific
ORness Value (α Parameter)
α ORness Aggregation
strategy
Decision
strategy
0 1 Logic OR (Max) Extremely optimistic
0.1 0.9 — Very optimistic
0.4 0.7 — optimistic
1 0.5 WLC Neutral
2 0.3 — Pessimistic
10 0.1 — Very pessimistic
∞ 0 Logic AND (Min) Extremely pessimistic
110 February 2016 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING