Consistency issues of interval pairwise comparison matrices

Abstract

In this paper, we investigate the consistency issues of interval pairwise comparison matrices in detail. Using logarithmic Manhattan distance to define the deviation degree of a pairwise comparison matrix to consistent pairwise comparison matrices, we propose a new consistency index of pairwise comparison matrices. Based on this consistency index of pairwise comparison matrices, we develop a consistency index of interval pairwise comparison matrices. Several desired properties of the proposed consistency indexes are presented. Furthermore, linear programming (LP) models are developed to compute the consistency indexes. Then, we propose a LP-based consistency improving model, which optimally preserves original pairwise comparison information in improving consistency. Meanwhile, considering the uncertainty plays an important role in the consistency index of interval pairwise comparison matrices, this consistency improving model is extended to simultaneously manage the uncertain degree in interval pairwise comparison matrices. Finally, we discuss the consistency-based prioritization method, and propose the strong consistency index of interval pairwise comparison matrices.

Appendices

Appendix A: Proofs

The proof of Proposition 1.

Proof

Without loss of generality, let $w^{*}\!=\!(w^{*}_{1}, w^{*}_{2},\ldots ,w^{*}_{n})^{T}$ $\in Q_{n}$, and

$$\begin{aligned} \mathrm{GCI}(A)=\frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n} (\log (a_{ij})-\log (w^{*}_{i})+\log (w^{*}_{j}))^{2}.\nonumber \\ \end{aligned}$$

(77)

Let $W^{*}=(w^{*}_{ij})_{n\times n}=(\frac{w^{*}_{i}}{w^{*}_{j}})$, we easily find that $W^{*}\in M_{n}$. From (3), we obtain

$$\begin{aligned} \mathrm{GCI}(A)&= \frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n} (\log (a_{ij})-\log (w_{ij}^{*}))^{2}\nonumber \\&\ge \frac{1}{n^{2}}\underset{P\in M_{n}}{\min }\sum _{i=1}^{n}\sum _{j=1}^{n} (\log (a_{ij})-\log (p_{ij}))^{2}\nonumber \\&= \mathrm{CI}(A). \end{aligned}$$

(78)

Without loss of generality, let $\overline{P}=(\overline{p_{ij}})_{n\times n}\in M_{n}$, satisfying

$$\begin{aligned} \mathrm{CI}(A)=\frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n} (\log (a_{ij})-\log (\overline{p_{ij}}))^{2}. \end{aligned}$$

(79)

Let $\overline{w}=(\overline{w_{1}}, \overline{w_{2}},\ldots ,\overline{w_{n}})^{T}\in Q_{n}$ be the priorities, obtained from $\overline{P}$ using the row geometric mean prioritization method. Since $\overline{P}\in M_{n}$, according to the row geometric mean prioritization method, we have $\overline{p_{ij}}=\frac{\overline{w_{i}}}{\overline{w_{j}}}$. Consequently,

$$\begin{aligned} \mathrm{CI}(A)&= \frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n} (\log (a_{ij})-\log (\overline{w_{i}})+\log (\overline{w_{j}}))^{2}\nonumber \\&\ge \frac{1}{n^{2}}\underset{w\in Q_{n}}{\min }\sum _{i=1}^{n}\sum _{j=1}^{n} (\log (a_{ij})-\log (w_{i})+\log (w_{j}))^{2}\nonumber \\&= \mathrm{GCI}(A). \end{aligned}$$

(80)

Based on (78) and (80), we have $\mathrm{CI}(A)=\mathrm{GCI}(A)$ when the distance between two pairwise comparison matrices uses the logarithmic Euclidean distance metric.

$\square$

The proof of Property 2

If $\{w^{*}_{1}, w^{*}_{2},\ldots ,w^{*}_{n}\}\in Q_{n}$ satisfies $v^{-}_{ij}\le \frac{w^{*}_{i}}{w^{*}_{j}}\le v^{+}_{ij}$, for $i,j=1,2,\ldots ,n$. Let $W^{*}=(w^{*}_{ij})_{n\times n}=(\frac{w_{i}^{*}}{w_{j}^{*}})_{n\times n}$. Then, we have that $W^{*}\in M_{n}$ and $W^{*}\in N_{\widetilde{V}}$. According to Definition 9, we obtain $\mathrm{ICI}(\widetilde{V})=0$. Similarly, if $\mathrm{ICI}(\widetilde{V})=\mathrm{CI}(W^{*})=0$, we can prove that $\exists \{w_{1}, w_{2},\ldots ,w_{n}\}\in Q_{n}$ satisfy $v^{-}_{ij}\le \frac{w_{i}}{w_{j}}\le v^{+}_{ij}$, for $i,j=1,2,\ldots ,n$. This completes the proof of Property 2.   $\square$

The proof of Property 3

According to Definition 5, we easily find that $\widetilde{C}$ is an interval pairwise comparison matrix. Let $A^{*}=(a_{ij}^{*})_{n\times n}\in N_{\widetilde{V}}$, $P^{*}=(p_{ij}^{*})_{n\times n}\in M_{n}$, and

$$\begin{aligned} \mathrm{ICI}(\widetilde{V})=\frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}|\log (a_{ij}^{*})-\log (p_{ij}^{*})|. \end{aligned}$$

(81)

Let $A^{**}=(a_{ij}^{**})_{n\times n}\in N_{\widetilde{F}}$, $P^{**}=(p_{ij}^{**})_{n\times n}\in M_{n}$, and

$$\begin{aligned} \mathrm{ICI}(\widetilde{F})=\frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}|\log (a_{ij}^{**})-\log (p_{ij}^{**})|. \end{aligned}$$

(82)

Based on $A^{*}$ and $A^{**}$, we construct a new pairwise comparison matrix $A^{c}=(a_{ij}^{c})_{n\times n}$, where $a_{ij}^{c}=(a_{ij}^{*})^{\alpha }\times (a_{ij}^{**})^{(1-\alpha )}$. We easily prove that $A^{c}\in N_{\widetilde{C}}$. Based on $P^{*}$ and $P^{**}$, we construct a new pairwise comparison matrix $P^{c}=(p_{ij}^{c})_{n\times n}$, where $p_{ij}^{c}=(p_{ij}^{*})^{\alpha }\times (p_{ij}^{**})^{(1-\alpha )}$. We easily prove that $P^{c}\in M_{n}$. According to Definition 9, we have

$$\begin{aligned} \mathrm{ICI}(\widetilde{C})&\le \frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}|\log (a_{ij}^{c})-\log (p_{ij}^{c})|\nonumber \\&= \frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}|\alpha (\log (a_{ij}^{*})-\log (p_{ij}^{*}))\nonumber \\&\quad +(1-\alpha )(\log (a_{ij}^{**})-\log (p_{ij}^{**}))| \end{aligned}$$

(83)

From (81)–(83), we have that

$$\begin{aligned} \mathrm{ICI}(\widetilde{C})&\le \alpha \mathrm{ICI}(\widetilde{V})+(1-\alpha )\mathrm{ICI}(\widetilde{F})\nonumber \\&\le \max \{\mathrm{ICI}(\widetilde{V}), \mathrm{ICI}( \widetilde{F})\}, \end{aligned}$$

(84)

which completes the proof of Property 3. $\square$

The proof of Property 4

For any $A\in N_{\widetilde{V}}$, we find $A\in N_{\widetilde{F}}$ under the condition that $f_{ij}^{-}\le v_{ij}^{-}$ and $f_{ij}^{+}\ge v_{ij}^{+}$. Thus, $N_{\widetilde{V}}\subseteq N_{\widetilde{F}}$. According to Definition 9, we have $\mathrm{ICI}(\widetilde{F})\le \mathrm{ICI}(\widetilde{V}).$ This completes the proof of Property 4. $\square$

The proof of Proposition 3

In $P_{1}$, constraints (13)–(14) guarantee that $P=(p_{ij})_{n\times n}\in M_{n}$. Constraints (15)–(17) enforce that $d_{ij}\ge |c_{ij}|=|\log (a_{ij})-\log (p_{ij})|$. According to the objective function (i.e., Eq. (12)), any feasible solutions with $d_{ij}>|c_{ij}|$ are not the optimal solution to $P_{1}$. Thus, constraints (15)–(17) guarantee that $d_{ij}=|c_{ij}|=|\log (a_{ij})-\log (p_{ij})|$ and $\mathrm{CI}(A)=\min \frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n} d_{ij}$. As a result, model (9) can be equivalently described as $P_{1}$. This completes the proof of Proposition 3. $\square$

The proof of Proposition 4

In $P_{2}$, constraints (19)–(21) guarantee that $A=(a_{ij})_{n\times n}\in N_{\widetilde{V}}$, and constraints (22)–(23) guarantee that $P=(p_{ij})_{n\times n}\in M_{n}$. Constraints (24)–(26) enforce that $d_{ij}\ge |c_{ij}|=|\log (a_{ij})-\log (p_{ij})|$. According to the objective function (i.e., Eq. (18)), any feasible solutions with $d_{ij}>|c_{ij}|$ are not the optimal solution to $P_{2}$. Thus, constraints (24)–(26) guarantee that $d_{ij}=|c_{ij}|=|\log (a_{ij})-\log (p_{ij})|$ and $\mathrm{ICI}(\widetilde{V})=\min \frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n} d_{ij}$. As a result, model (11) can be equivalently described as $P_{2}$. This completes the proof of Proposition 4. $\square$

The proof of Lemma 1

Constraints (41)–(43) enforce that $d_{ij}\ge |c_{ij}|=|\log (a_{ij})-\log (p_{ij})|$. According to Definition 9, $\mathrm{ICI}(\widetilde{V^{*}})\le \frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n} d_{ij}$. Thus, constraint (40) guarantees $\mathrm{ICI}(\widetilde{V^{*}})\le \overline{\mathrm{ICI}}$. This completes the proof of Lemma 1. $\square$

The proof of Proposition 5

In $P_{3}$, constraints (33)–(34) guarantee that $\widetilde{V^{*}}$ is an interval pairwise comparison matrix. Constraints (35)–(37) guarantee that $A=(a_{ij})_{n\times n}\in N_{\widetilde{V^{*}}}$, and constraints (38)–(39) guarantee that $P=(p_{ij})_{n\times n}\in M_{n}$. From Lemma 1, constraints (40)–(43) guarantee $\mathrm{ICI}(\widetilde{V^{*}})\le \overline{\mathrm{ICI}}$. Moreover, constraints (44)–(45) guarantee $\mathrm{UI}(\widetilde{V^{*}})\le L$, and constraints (46)–(48) enforce that $h_{ij}\ge |g_{ij}|=|\log (v^{+}_{ij})-\log (v^{*+}_{ij})|$. According to the objective function (i.e., Eq. (32)), any feasible solutions with $h_{ij}>|g_{ij}|$ are not the optimal solution to $P_{3}$. Thus, constraints (46)–(48) guarantee that $h_{ij}=|g_{ij}|=|\log (v^{+}_{ij})-\log (v^{*+}_{ij})|$. Similarly, constraints (49)–(51) guarantee that $m_{ij}=|l_{ij}|=|\log (v^{-}_{ij})-\log (v^{*-}_{ij})|$. As a result,

$$\begin{aligned} \frac{1}{2n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}(h_{ij}+m_{ij})&= \frac{1}{2n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}( |\log (v^{+}_{ij})-\log (v^{*+}_{ij})|\nonumber \\&\quad +|\log (v^{-}_{ij})-\log (v^{*-}_{ij})|). \end{aligned}$$

(85)

Thus, model (31) can be equivalently described as $P_{3}$. This completes the proof of Proposition 5. $\square$

The proof of Proposition 6

From constraints (44)–(45), $\widetilde{V^{*}}$ degenerates to a pairwise comparison matrix when $L=0$. According to constraints (35)–(37), we have $\widetilde{V^{*}}=A=(a_{ij})_{n\times n}$, i.e., $v_{ij}^{*-}=v_{ij}^{*+}=a_{ij}$. When $\overline{\mathrm{ICI}}=0$, constraints (40)–(43) guarantee that $\widetilde{V^{*}}$ is a consistent pairwise comparison matrix, that is $\widetilde{V^{*}}=P=(p_{ij})_{n\times n}$. Further, according to constraints (46)–(48) and the objective function (i.e., Eq. (32)), we have $h_{ij}=|g_{ij}|=|\log (v^{+}_{ij})-\log (p_{ij})|$. According to constraints (49)–(51) and the objective function (i.e., Eq. (32)), we also have $m_{ij}=|l_{ij}|=|\log (v^{-}_{ij})-\log (p_{ij})|$. Since $\widetilde{V}$ is a pairwise comparison matrix, we have $v_{ij}^{-}=v_{ij}^{+}$, and $h_{ij}=m_{ij}$. As a result, $\mathrm{CI}(\widetilde{V})=\min \frac{1}{2n^{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}(h_{ij}+m_{ij})$. In this way, $P_{3}$ degenerates to $P_{1}$. This completes the proof of Proposition 6. $\square$

The proof of Proposition 8

We only prove the existence of the optimal solution to $P_{1}$. The existence of the optimal solutions to $P_{2}$, $P_{3}$ and $P_{4}$ can be similarly proved. Any $n\times n$ consistent pairwise comparison matrices satisfy all the constraints of $P_{1}$, and therefore represent feasible solutions. So, $P_{1}$ has a non-empty feasible region. A closed bounded feasible region for $P_{1}$ would satisfy the assumption of Weierstrass theorem. This could be achieved, by introducing an upper bound for $d_{ij}$. Let $\overline{w}=(\overline{w_{1}}, \overline{w_{2}},\ldots ,\overline{w_{n}})^{T}\in Q_{n}$ be the priorities, obtained from $A$ using the row geometric mean prioritization method, and $\overline{P}=(\overline{p_{ij}})_{n\times n}=(\frac{\overline{w_{i}}}{\overline{w_{j}}})_{n\times n}$. According to the row geometric mean prioritization method, we have that $\overline{p_{ij}}=\frac{\root n \of {\prod _{k=1}^{n}a_{ik}}}{\root n \of {\prod _{k=1}^{n}a_{jk}}}$, and $\overline{P}\in M_{n}$ satisfies all the constraints of $P_{1}$. Since $c_{ij}=\log (a_{ij})-\log (p_{ij})$, a suitable inequality that does not affect the optimal solution could be

$$\begin{aligned} d_{ij}&\le |\log (a_{ij})-\log (\overline{p_{ij}})|\nonumber \\&= \left| \log (a_{ij})-\frac{1}{n}\sum _{k=1}^{n} \log (a_{ik})-\log (a_{jk})\right| . \end{aligned}$$

(86)

According to Weierstrass theorem, we prove the existence of the optimal solution to $P_{1}$, which completes the proof of Proposition 8. $\square$

Appendix B: Establishing the ICI thresholds based on Golden and Wang’s approach

Property 4 shows that it is easier to obtain consistency, with the uncertain degree of interval pairwise comparison matrices increasing. In the following, we establish the ICI thresholds of interval pairwise comparison matrices, according to different uncertain indexes. When the uncertain index is $L$, the ICI threshold is denoted as $\overline{\mathrm{ICI}(L)}$. As mentioned earlier, establishing the thresholds of the AHP consistency indexes is an open question (Apostolou and Hassell 2002; Chu anad Liu 2002), and there is disagreement about how to establish these thresholds (e.g., Golden and Wang 1989; Lane and Verdini 1989), so the proposed $\overline{\mathrm{ICI}(L)}$ should only be considered as a decision aid which decision makers use as a reference to obtain the acceptable consistency.

The main idea of establishing $\overline{\mathrm{ICI}(L)}$ is inspired by the work of Golden and Wang (1989). When using Saaty’s consistency ratio rule (i.e., $\le 0.1$), Golden and Wang (1989) have found that the ratio of the acceptably consistent ones among random pairwise comparison matrices is over $20\,\%$ for $n=3$, and about $4\,\%$ for $n>3$. It is impossible to obtain a matrix consistency for $n>5$. The detailed analysis also can be found in Peláez and Lamata (2003).

Here, the established thresholds $\overline{\mathrm{ICI}(L)}$ guarantee the ratio of the acceptably consistent ones among random interval pairwise comparison matrices with the uncertain index $L$ is constant and small, for different matrix sizes.

The procedure of establishing ICI thresholds is described as follows.

The procedure of establishing the ICI thresholds

Input: The matrix size $n$, the number of random matrices $m$, the established uncertain index $L\in [0, 2\log (9)]$, and the parameters $\alpha$

Output: The ICI thresholds, $\overline{\mathrm{ICI}(L)}$

Step 1: For each $n$, we, respectively, generate $m$ pairwise comparison matrices $\{A^{(1)},\ldots A^{(k)}, \ldots ,A^{(m)}\}$, whose entries are uniformly randomly generated using 1–9 scale.

Step 2: Based on $A^{(k)}=(a_{ij}^{(k)})_{n\times n}$, we generate an interval pairwise comparison matrix $\widetilde{V^{(k)}}=(\widetilde{v_{ij}^{k}})_{n\times n}$, where $\widetilde{v_{ij}^{k}}=[v_{ij}^{k-}, v_{ij}^{k+}]$. We consider three cases:

Case A: $\log (9)-\log (a^{(k)}_{ij})\ge L$ and $\log (a^{(k)}_{ij})-\log (1/9)\ge L$. In this case, let $t_{1}=a^{(k)}_{ij}$, and $t_{2}$ is uniformly randomly selected from $\{e^{L+\log (a^{(k)}_{ij})}, e^{\log (a^{(k)}_{ij})-L}\}$, where $e$ is the base of natural logarithms.

Case B: $\log (9)-\log (a^{(k)}_{ij})\ge L$ and $\log (a^{(k)}_{ij})-\log (1/9)<L$. In this case, let $t_{1}=a^{(k)}_{ij}$, and $t_{2}=e^{L+\log (a^{(k)}_{ij})}$.

Case C: $\log (9)-\log (a^{(k)}_{ij})<L$ and $\log (a^{(k)}_{ij})-\log (1/9)\ge L$. In this case, let $t_{1}=a^{(k)}_{ij}$, and $t_{2}=e^{\log (a^{(k)}_{ij})-L}$.

Then, let $v_{ij}^{k-}=\min \{t_{1}, t_{2}\}$ and $v_{ij}^{k+}=\max \{t_{1}, t_{2}\}$.

Step 3: Compute $\mathrm{ICI}(\widetilde{V^{(k)}}), k=1,2,\ldots ,m$, using $P_{2}$.

Step 4: By sorting $\{\mathrm{ICI}(\widetilde{V^{(1)}}),\ldots ,\mathrm{ICI}(\widetilde{V^{(m)}})\}$ in ascending order, we obtain $\{b^{(1)},\ldots b^{(k)},$ $\ldots ,b^{(m)}\}$, where $b^{(k)}$ is the $k$th smallest element in $\{\mathrm{ICI}(\widetilde{V^{(1)}}),\ldots , \mathrm{ICI}(\widetilde{V^{(m)}})\}$.

Step 5: Let $k^{*}=round(m\times \alpha )$, where round is usual round operation. Output the threshold $\overline{\mathrm{ICI}(L)}=b^{(k^{*})}$.

Note 6. In this procedure, Step 2 ensures $\log (v_{ij}^{k+})-\log (v_{ij}^{k-})=L$, which is a stronger condition than $\mathrm{UI}(\widetilde{V^{(k)}})=L$ ($k=1,2,\ldots ,m$). Step 4 ensures that the ratio of the acceptably consistent interval pairwise comparison matrices among $\{\widetilde{V^{(1)}},\ldots ,\widetilde{V^{(m)}}\}$ is approximately $\alpha$. We argue that this ratio is small, so we suggest setting $\alpha =0.05$ according to the principle of small-probability events.

When setting $\alpha =0.05, m=5{,}000$, different matrix size $n$ and different $L$, we run this procedure to obtain the values of $\overline{\mathrm{ICI}(L)}$, which are shown in Table 1.

We conclude that the interval pairwise comparison matrix $\widetilde{V}$ is of acceptable consistency, when $\mathrm{ICI}(\widetilde{V})\le \overline{\mathrm{ICI}(\mathrm{UI}(\widetilde{V}))}$. Otherwise, we conclude that $\widetilde{V}$ is of unacceptable consistency. For example, if the matrix size of $\widetilde{V}$ is 4 and $UI(\widetilde{V})=0.2$, then $\widetilde{V}$ is of acceptable consistency when $\mathrm{ICI}(\widetilde{V})\le \overline{\mathrm{ICI}(0.2)}=0.1887$. Meanwhile, for explanatory convenience, $\overline{\mathrm{ICI}(0)}$ is denoted by $\overline{\mathrm{CI}}$ in this paper.

When using the ICI thresholds, the decision maker first sets a $L$ value, and then the ICI threshold is determined by $\overline{\mathrm{ICL}(L)}$.