Chapter 6 Model 3: Pairwise Comparison Decision Ball Models
6.2 Construction of Pairwise Comparison Decision Ball Models
This section develops a systematical approach for ranking and displaying alternatives.
The approach includes three models: the Adjusting model, the Gower Plot model, and the Decision Ball model.
Given a C = (ci,k) and a R = (ri,j), where R may be incomplete or inconsistent, a model of adjusting R with addition score functions is formulated below:
Model 6.1 (Adjusting model – Additive score functions )
}
Min{
wk M×1 +2
1 =
∑∑
to achieve ordinal consistency by minimizing the number of preferences (i.e., ) being revised. The elements of matrix U, uj
and otherwise ui,j = 1. The second objective (2) is to achieve cardinal consistency by minimizing the αi,j values, i.e. to minimize the difference between
j
1) is more important than cardinal consistency (2), 1 is multiplied by
a large value M in the objective function. Constraints (6.4) and (6.5) come from Notation 3.1.
Constraint (6.6) sets the upper and lower bound of weights.
Model 6.1 is a nonlinear model, which can be converted into the following linear mixed 0-1 program:
Where (6.8), (6.9) and (6.10) are converted from (6.1), (6.2) and (6.3) respectively.
After the weight vector, (w1, w2, …, wn), is found,
∑
calculated and a complete matrix can be obtained as
, (6.11)
The Adjusting model with multiplicative score functions is formulated as follows.
Model 6.2 (Adjusting Model –Multiplicative score functions )
}
Min{
wk M×1 +2
1 =
∑∑
Constraints (6.12), (6.13), and (6.14) correspond to constraints (6.1), (6.2) and (6.3) respectively. Model 6.2 can be linearized as follows.
}
After a complete matrix is obtained, the ordinal Gower Plots can be used to aid in
detecting the causes of any ordinal inconsistency. The concept of Gower Plots refers to Section 2.2. The Gower Plot model with a multiplicative score function is the same as that with an additive score function.
At last, Decision Ball techniques, as proposed in Section 3.3, with additive or multiplicative score functions are adopted to display ranks and similarities among alternatives.
The solution processes are shown in Figure 6.1 and illustrated below:
Step 1 The decision maker specifies a data matrix C = (ci,k), chooses a type of score function, and inputs a preference matrix R = (r ), where R can be an incomplete matrix. i,j
Step 2 Applying Model 6.1 (if an additive score function is selected) or Model 6.2 (if a
multiplicative score function is chosen) to the data and preference matrix yields a set of weights w, and a revised preference matrix R' =(ri', j), where R' is a complete matrix.
Step 3 Applying Gower Plot model toR', the ordinal Gower Plots before and after adjustment are displayed based on R'.
Step 4 Based on the weights w obtained in Step 2, the score of alternatives Si(w) and dissimilarities )δi,j(w among alternatives are calculated.
Step 5 Applying Decision Ball techniques (Model 3.1 if an additive score function is chosen;
or Model 3.2 if a multiplicative score function is chosen) to S (w) and )i δi,j(w yields the
coordinates (x , y , z ) of alternatives on the Decision Ball. The Decision Ball is then i i i
displayed to the decision maker.
Step 6 The decision maker can observe the ranks and similarities among alternatives on the
ball. Other options can be obtained through Step 3 to Step 5 by setting some ui,j = 0. The decision maker can also adjust preferences in R (Step 1) or R' (Step 2) directly based on the information provided by Gower Plots and Decision Balls, and observe the corresponding changes.
6.3 Illustrative Examples
Two examples are used to illustrate the Decision Maker’s problem solving processes.
For simplicity, only additive score functions are illustrated here.
<Example 6.1> Investment in Mutual Funds
The first example is about an investor who would like to invest in mutual funds. The investor has four major decision criteria to fulfill: (c ) a high total return, (c1 2) large fund size (economies of scale), (c3) low risk (β : Beta), and (c ) low turnover. Six alternatives (A4 1, …, A ) are under considerations as listed in the C in Figure 6.2, where c6 1 3 and c4 are cost criteria.
Suppose the investor chooses to use an additive score function and specifies an incomplete preference matrix R = (r ), where r1 i,j 1,6, r2,3, and r3,6 are left blank because it is difficult for the investor to make comparisons between these alternatives. The data set, preference matrix, and the solving process are depicted in Figure 6.2. (Here R can be checked as ordinally 1
inconsistent since r1,2 <1 and r2,4 <1 and but r1,4 >1.)
Applying Model 6.1 to C and R1 1 yields the solution as 1 = 1, 2 = 3.64, u1,4 = 1, (w1, w , w , w ) = (0.38, 0.19, 0.05, 0.38), and (S , S , S2 3 4 1 2 3, S , S4 5, S6) = (0.56, 0.65, 0.41, 0.82, 0.58, 0.35). The values of unspecified preferences can be computed as r
6
to minimize both ordinal and cardinal inconsistencies. This is regarded as Option 1 of adjusting the preferences. The ordinal Gower Plots, with r1,4 >1 and r1,4 <1, are also depicted in Figure 6.2. Examining the Gower Plots before reversal (i.e., when ), the preference
matrix is ordinally inconsistent because A
4 1
, 1 >
r
and A lie off the half circle, which implies A
1 4 1 and
A4 are the alternative causing major ordinal inconsistencies. By following the suggestion of revising r1,4 as r1,4 <1, the Gower Plot will show the preference matrix is ordinally consistent and the alternatives will be ranked as A4f A2f A5f A1f A3f A . 6
Applying a Decision Ball technique (Model 3.1) based on the results of Model 6.1 (Si(w) and ))δi,j(w yield a set of coordinates for A1, …, A6, with Stress = 6.9%. The corresponding
Decision Ball is shown on the left bottom of Figure 6.2. Examining the Decision Ball, the investor can observe that (i) if he reverses r1,4 from larger than 1 to smaller than 1, the ranks of alternatives is A4f A2f A5f A1f A3f A6 from top to down along a latitudinal line (ii) A , A4 2 and A5 have higher similarities because they are close to each other. For diversifying the investment, the investor may avoid investing A , A and A simultaneously. 4 2 5
If the investor does not like to reverse r1,4, another option can be generated by setting u1,4 = 0 in Model 6.1. Applying Model 6.1 (with a new constraint u1,4 = 0) again yields Option
2 where 1 = 2, = 3.57, u2 1,2 = 1, u1,5 = 1, (w1, w , w , w ) = (0.63, 0.10, 0.05, 0.22), (S , S2 3 4 1 2, S , S , S , S3 4 5 6) = (0.73, 0.70, 0.26, 0.72, 0.53, 0.40). The corresponding Gower Plots and Decision Ball (with Stress 5.9%) are shown on the right bottom of Figure 6.2.
If the investor does not like Option 1 and Option 2, he may modify R1 (or ) directly (Option 3) based on the information provided on the Gower Plot about the alternative(s) causing major inconsistencies, or based on the information provided on the Decision Ball about the scores and similarities among alternatives.
R1'
ri,j A1 A2 A3 A4 A5 A6
-0.80 -0.60 -0.40 -0.20 0.00
A4
Figure 6.2 Decision Process of Example 6.1
A4
-0.80 -0.60 -0.40 -0.20 0.00
A4
c1: Return 67.76 57.82 20.56 48.30 43.45 42.12
c2: Size(million) 1,102 1,100 782 2,215 1,522 1,825
c3: Beta 1.43 1.41 0.85 1.08 1.45 0.99
c4: Turnover 305.26 180.97 139.2 127.94 187.48 402.42
A4
<Example 2> Selection of universities
Consider a student who needs to enroll in a university. He would like to choose from a list of eight candidate universities. The student sets four criteria for choosing a university: (c1) rich campus life estimated by size, (c ) high average salary after graduation, (c2 3) high entrance score, and (c4) low tuition. Low tuition, c4, is considered to be a cost criterion. An additive score function is used to rank the alternatives. The data set C2 and an incomplete preference matrix R are listed as in Figure 6.3. 2
After applying Model 6.1 to the example, three possible ordinally consistent solutions are found. Those are, u1,7 = 1 (Option 1), u3,7 = 1 (Option 2), and u1,3 = 1 (Option 3). The corresponding Gower Plots and Decision Balls (with Stress 7.2%, 6.7%, and 4.5%
respectively) are depicted in Figure 6.3.
Option 1 yields 1 = 1, 2 = 3.51, u1,7 = 1, (w , w , w , w ) = (0.31, 0.59, 0.05, 0.05), (S1 2 3 4 1, S , S2 3, S , S , S , S , S4 5 6 7 8) = (0.62, 0.37, 0.52, 0.12, 0.20, 0.34, 0.45, 0.75). Examining Gower Plot (a) where r1,7 < 1 to know it is ordinally inconsistent because there are some angles between consecutive points not equal to 180/n degrees. Alternatives A , A1 3, and A7 are the ordinally inconsistent alternatives. Reversing r1,7 as r1,7 >1 (means A better than A1 7) generates an ordianlly consistent situation with A8f A1 fA3 fA7 fA2 fA6 fA5 fA4. The related Decision Ball (b) illustrates that considering A8, A , and A , A1 3 8f A1 fA3. However, A3
is more similar to A . Therefore, if the student is not accepted by A , A8 8 3 may be a better choice than A . 1
Suppose the student chooses to reverse r3,7 as r3,7 <1 (means A better than A7 3) (Option 2), the related Decision Ball (d) illustrates the ranks of alternatives are A8f A7f A1f A3f
A2f A 6 f A5f A4. A and A are very close. Thus, if university A1 7 8 is impossible to candidate for enrollment then A as well as A could be a good choice. 1 7
-0.6
graduation 32000 32000 38000 25000 28000 30000 28000 42000 C3 :
Entrance
Score 510 520 550 440 450 500 530 600
C4 : Tuition 32000 45000 40000 56000 55000 44000 38000 35000 Alternative
Criteria
Figure 6.3 Decision Process of Example 6.2
6.4 Summary
This study develops Pairwise Comparison Decision Ball models to help a decision maker improve the quality of decision-making by iteratively reducing ordinal and cardinal inconsistencies. Gower Plots are adopted to detect the alternatives causing ordinal inconsistencies. An optimization model is then developed to provide suggestions for adjusting these inconsistencies conveniently.
Decision Ball techniques are used to provide a useful visual representation of ranks and similarities among alternatives, which are more flexible and easier to observe than traditional 2-dimensional plane and 3-dimensional cube models respectively. In addition, suggestions about how to improve inconsistencies are also shown on the Decision Ball. The decision maker can observe the suggested solutions and choose the most acceptable change to reduce inconsistencies and to rank alternatives more confidently.
The proposed approach assists a decision maker make a more reliable decision by improving inconsistencies. The improvements in inconsistency can be measured by the consistency ratio (CR)(Saaty, 1980), which is briefly illustrated as follows. Given a matrix R of rank n, the consistency index (CI) is first calculated to measure the deviation from a consistent matrix:
) 1 /(
) (λmax −n n−
CI = ,
λmax
where is the maximal eigenvalue of R. Then, the consistency ratio (CR) is computed as the
ratio of the CI to the so-called random index (RI) which is a CI of randomly generated matrices:
CR = CI/RI.
The CR = 0 indicates perfect consistency. The CR before and after adjustments for Example 6.1 and 6.2 are listed in Table 6.1. The CR in both examples can be improved significantly. For instance, in Option 1 of Example 6.1, the CR can be significantly improved from 0.087 to 0.047. In Option 1 of Example 6.2, the CR can be improved from 0.064 to 0.049.
Examples Options CR before
Adjustment
CR after Adjustment
Option 1 0.087 0.047
Example 6.1
Investment in Mutual Funds Option 2 0.078 0.053
Option 1 0.064 0.049
Option 2 0.070 0.053
Example 6.2
Selection of Universities
Option 3 0.060 0.055
Table 6.1 Improvements in inconsistency measured by consistency ratio (CR)