Two Hesitant Multiplicative DecisionMaking Algorithms and Their Application to FogHaze Factor Assessment Problem
Abstract
:1. Introduction
2. Preliminaries
3. Hesitant Multiplicative Preference Relations and Consistency Index
4. Consistency Repaired Methods for an HMPR
Algorithm 1: The consistency adjusting process of HMPR based on logarithm least squares model 
Step 1.$\mathrm{Suppose}\mathrm{that}{P}^{(t)}={({p}_{ij}^{(t)})}_{n\times n}=P={({p}_{ij})}_{n\times n}$ and $t=0$, and preset the threshold ${\delta}_{0}$, the controlling parameter $\theta $ and the maximum number of iterations ${t}_{\mathrm{max}}$; 
Step 2. Derive the priority vector ${\tilde{w}}^{(t)}={({\tilde{w}}_{1}^{(t)},{\tilde{w}}_{2}^{(t)},\cdots ,{\tilde{w}}_{n}^{(t)})}^{\mathrm{T}}$ by Equation (6); 
Step 3.$\mathrm{Determine}\mathrm{the}\mathrm{consistency}\mathrm{index}CI({P}^{(t)})$ by using Equation (3); 
Step 4.$\mathrm{If}CI({P}^{(t)})\le {\delta}_{0}$$\mathrm{or}t{t}_{\mathrm{max}}$, then go to Step 7. Otherwise, go to Step 5; 
Step 5.$\mathrm{Let}{P}^{(t+1)}={({p}_{ij}^{(t+1)})}_{n\times n}$, where 
${p}_{ij}^{(t+1)}=\left\{{\gamma}_{ij,k}^{(t+1)}k=1,2,\cdots ,\left{p}_{ij}^{(t+1)}\right\right\},{\gamma}_{ij,k}^{(t+1)}={\left({\gamma}_{ij,k}^{(t)}\right)}^{1\theta}\cdot {\left({\tilde{w}}_{i}^{(t)}/{\tilde{w}}_{j}^{(t)}\right)}^{\theta}$ 
Step 6. Let $t=t+1$ and return to Step 2; 
Step 7.$\mathrm{Output}{P}^{(t)}$, ${\tilde{w}}^{(t)}$, $CI({P}^{(t)})$$\mathrm{and}t$; 
Step 8. End. 
Algorithm 2: The consistency adjusting process of HMPR based on linear optimization model 
Step 1’. See Algorithm 1; 
Step 2’.$\mathrm{According}\mathrm{to}\mathrm{model}(\mathbf{M}4),\mathrm{we}\mathrm{get}\mathrm{the}\mathrm{optimal}\mathrm{nonzero}\mathrm{deviation}\mathrm{values}{\tilde{d}}_{ij}^{(t)}$$\mathrm{and}{\tilde{d}}_{ij}^{+(t)},i,j\in N$, $\mathrm{and}\mathrm{the}\mathrm{priority}\mathrm{weight}\mathrm{vector}{\tilde{w}}^{(t)}={({\tilde{w}}_{1}^{(t)},{\tilde{w}}_{2}^{(t)},\cdots ,{\tilde{w}}_{n}^{(t)})}^{\mathrm{T}}$; 
Step 3’–4’. See Algorithm 1; 
Step 4.$\mathrm{If}CI({P}^{(t)})\le {\delta}_{0}$$\mathrm{or}t{t}_{\mathrm{max}}$, then go to Step 7. Otherwise, go to Step 5; 
Step 5’.$\mathrm{Let}{P}^{(t+1)}={({p}_{ij}^{(t+1)})}_{n\times n}$, where ${p}_{ij}^{(t+1)}=\left\{{\gamma}_{ij,k}^{(t+1)}k=1,2,\cdots ,\left{p}_{ij}^{(t+1)}\right\right\},$ 
${\gamma}_{ij,k}^{(t+1)}={\left({w}_{i}^{(t)}/{w}_{j}^{(t)}\right)}^{\theta}\cdot {\left({\gamma}_{ij,k}^{(t)}\right)}^{1\theta}={\left({\gamma}_{ij,k}^{(t)}\cdot \mathrm{exp}({\tilde{d}}_{ij}^{(t)}{\tilde{d}}_{ij}^{+(t)})\right)}^{\theta}\cdot {\left({\gamma}_{ij,k}^{(t)}\right)}^{1\theta}={\gamma}_{ij,k}^{(t)}\cdot \mathrm{exp}(\theta ({\tilde{d}}_{ij}^{(t)}{\tilde{d}}_{ij}^{+(t)}))$ 
Step 6’–8’. See Algorithm 1. 
5. Illustrative Example Results and Discussion
5.1. Numerical Example
Algorithm 3: The consistency adjusting process of HMPR based on logarithm least squares model 
Step 1.$\mathrm{Let}t=0$,${P}^{(t)}=P$, ${\delta}_{0}=0.3$$\mathrm{and}\theta =0.1$ 
Step 2. By Equation (6), we obtain the priority vector:
$${\tilde{w}}^{(0)}={(0.0934,0.5512,0.1090,0.2464)}^{\mathrm{T}}.$$

Step 3. By Equation (3), we determine $\mathrm{the}\mathrm{consistency}\mathrm{index}CI({P}^{(0)})=0.7573$ 
Step 4. As $CI({P}^{(0)})=0.7573>{\delta}_{0}$, then we utilize Step 5 in Algorithm 1 $\mathrm{to}\mathrm{repair}\mathrm{the}\mathrm{consistency}\mathrm{of}\mathrm{the}\mathrm{HMPR}{P}^{(0)}$ $\mathrm{and}\mathrm{derive}\mathrm{the}\mathrm{new}\mathrm{HMPR}{P}^{(1)}$: ${P}^{(1)}=\left(\begin{array}{cccc}\left\{1\right\}& \left\{0.1453,0.1669,0.1967\right\}& \{0.9847,2.6467\}& \left\{0.1397,0.2939\right\}\\ \{5.0839,5.9916,6.8823\}& \left\{1\right\}& \{1.1759,2.1944\}& \{6.2452,7.8302\}\\ \left\{0.3778,1.0155\right\}& \left\{0.4557,0.8504\right\}& \left\{1\right\}& \left\{0.2647\right\}\\ \left\{3.4025,7.1581\right\}& \left\{0.1277,0.1601\right\}& \left\{3.7779\right\}& \left\{1\right\}\end{array}\right)$ 
Step 5. By Equation (3), we determine $\mathrm{the}\mathrm{consistency}\mathrm{index}CI({P}^{(0)})=0.7573$. $\mathrm{Because}CI({P}^{(1)})=0.6133{\delta}_{0}$, then by using Step 5 in Algorithm 1, we have ${P}^{(2)}=\left(\begin{array}{cccc}\left\{1\right\}& \left\{0.1476,0.1672,0.1938\right\}& \{0.9711,2.3645\}& \left\{0.1544,0.3015\right\}\\ \{5.1601,5.9824,6.7772\}& \left\{1\right\}& \{1.3606,2.3855\}& \{5.6358,6.9081\}\\ \left\{0.4229,1.0297\right\}& \left\{0.4192,0.7350\right\}& \left\{1\right\}& \left\{0.2786\right\}\\ \left\{3.3170,6.4781\right\}& \left\{0.1448,0.1774\right\}& \left\{3.5888\right\}& \left\{1\right\}\end{array}\right)$ 
Step 6. By Equation (3), we obtain $\mathrm{the}\mathrm{consistency}\mathrm{index}CI({P}^{(2)})=0.4968$. $\mathrm{As}CI({P}^{(2)})=$$0.4968>{\delta}_{0}$, then according to Step 5 in Algorithm 1, we have ${P}^{(3)}=\left(\begin{array}{cccc}\left\{1\right\}& \left\{0.1497,0.1674,0.1912\right\}& \{0.9591,2.1363\}& \left\{0.1689,0.3089\right\}\\ \{5.2296,5.9727,6.6820\}& \left\{1\right\}& \{1.5515,2.5716\}& \{5.1383,6.1714\}\\ \left\{0.4681,1.0427\right\}& \left\{0.3889,0.6445\right\}& \left\{1\right\}& \left\{0.2918\right\}\\ \left\{3.2416,5.9202\right\}& \left\{0.1620,0.1946\right\}& \left\{3.4272\right\}& \left\{1\right\}\end{array}\right)$ 
Applying Equation (6), the priority vector of $\mathrm{HMPR}{P}^{(3)}$ $\mathrm{can}\mathrm{be}\mathrm{determined}:{\tilde{w}}^{(3)}=(0.0883,0.4437,$ $0.1908,0.2772{)}^{\mathrm{T}}$. Step 7. $\mathrm{We}\mathrm{calculate}\mathrm{the}\mathrm{consistency}\mathrm{index}CI({P}^{(3)})=0.2923$. Step 8. $\mathrm{Sin}\mathrm{ce}CI({P}^{(3)}){\delta}_{0}$, $\mathrm{then}\mathrm{the}\mathrm{iteration}\mathrm{stops},\mathrm{and}{P}^{(3)}$ is acceptable consistent HMPR. Step 9. $\mathrm{Output}{\tilde{w}}^{(3)}={(0.0883,0.4437,0.1908,0.2772)}^{\mathrm{T}}$. $\mathrm{As}{w}_{2}^{(3)}{w}_{4}^{(3)}{w}_{1}^{(3)}{w}_{3}^{(3)}$, $\mathrm{then}\mathrm{we}\mathrm{have}{x}_{2}\succ {x}_{4}\succ {x}_{1}\succ {x}_{3}$. $\mathrm{Therefore},\mathrm{the}\mathrm{most}\mathrm{important}\mathrm{factor}\mathrm{for}\mathrm{fog}\mathrm{haze}\mathrm{is}{x}_{2}$. 
Algorithm 4: The consistency adjusting process of HMPR based on linear optimization model 
Step 1’. Let $t=0$,${\tilde{P}}^{(t)}=P$,${\delta}_{0}=0.3$ and $\theta =0.1$. 
Step 2’. Using a model (M4), we get the optimal deviation values ${\tilde{d}}_{13}^{+(0)}=0.8124,{\tilde{d}}_{23}^{(0)}=$$1.0245,{\tilde{d}}_{24}^{+(0)}=1.4958,{\tilde{d}}_{12}^{(0)}={\tilde{d}}_{12}^{+(0)}={\tilde{d}}_{13}^{(0)}={\tilde{d}}_{14}^{(0)}={\tilde{d}}_{14}^{+(0)}={\tilde{d}}_{23}^{+(0)}={\tilde{d}}_{24}^{(0)}={\tilde{d}}_{34}^{(0)}={\tilde{d}}_{34}^{+(0)}=0$, and the priority weight vector can be obtained as follows:
$${\tilde{w}}^{(0)}={(0.0745,0.4695,0.0994,0.3566)}^{\mathrm{T}}.$$

Step 3’. Utilizing Equation (3) to get the consistency index $CI({\tilde{P}}^{(0)})=0.7033$. Step 4’. As $CI({\tilde{P}}^{(0)})>{\delta}_{0}$, then we apply Step 5’ in Algorithm 2 to adjust the consistency of HMPR ${\tilde{P}}^{(0)}$, and one can obtain a new HMPR ${\tilde{P}}^{(1)}$ as follows: ${\tilde{P}}^{(1)}=\left(\begin{array}{cccc}\left\{1\right\}& \left\{0.1478,0.1672,0.1935\right\}& \{0.9696,2.3351\}& \left\{0.1561,0.3023\right\}\\ \{5.1658,5.9809,6.7659\}& \left\{1\right\}& \{1.3823,2.4077\}& \{5.5718,6.8125\}\\ \left\{0.4282,1.0314\right\}& \left\{0.4153,0.7234\right\}& \left\{1\right\}& \left\{0.2802\right\}\\ \left\{3.3080,6.4061\right\}& \left\{0.1468,0.1795\right\}& \left\{3.5689\right\}& \left\{1\right\}\end{array}\right)$ 
Step 5’. Using model (M4), we get the optimal deviation values ${\tilde{d}}_{13}^{+(1)}=0.3288,{\tilde{d}}_{12}^{(1)}=$$0.1329,{\tilde{d}}_{14}^{+(1)}=0.6202,{\tilde{d}}_{23}^{+(1)}=0.8904,{\tilde{d}}_{24}^{(1)}=1.3359,{\tilde{d}}_{34}^{+(1)}=0.3337.{\tilde{d}}_{23}^{(1)}={\tilde{d}}_{24}^{+(1)}={\tilde{d}}_{12}^{+(1)}={\tilde{d}}_{13}^{(1)}=$ ${\tilde{d}}_{14}^{(1)}={\tilde{d}}_{34}^{(1)}=0.$ By using Equation (3), we have $CI({\tilde{P}}^{(1)})=0.5016>{\delta}_{0}$. Thus, by Step 5’ in Algorithm 2, one can obtain ${\tilde{P}}^{(2)}=\left(\begin{array}{cccc}\left\{1\right\}& \left\{0.1503,0.1702,0.1966\right\}& \{0.9349,2.1205\}& \left\{0.1701,0.3141\right\}\\ \{5.0865,5.8754,6.6534\}& \left\{1\right\}& \{1.6102,2.6441\}& \{5.1098,6.1204\}\\ \left\{0.4716,1.0696\right\}& \left\{0.3768,0.6210\right\}& \left\{1\right\}& \left\{0.3112\right\}\\ \left\{3.1837,5.8789\right\}& \left\{0.1634,0.1957\right\}& \left\{3.2134\right\}& \left\{1\right\}\end{array}\right)$ 
Step 6’. Using model (M4), we determine the priority weight vector ${\tilde{w}}^{(2)}=(0.1107,0.3910,$$0.2056,0.3027{)}^{\mathrm{T}}$. 
Step 7’. By using Equation (3), we have $CI({\tilde{P}}^{(2)})=0.2855$. As $CI({\tilde{P}}^{(2)})<{\delta}_{0}$, then the iteration stops, and ${\tilde{P}}^{(2)}$ is acceptable consistent HMPR. 
Step 8’. Output ${\tilde{w}}^{(2)}={(0.1107,0.3910,0.2056,0.3027)}^{\mathrm{T}}$. 
Step 9’. As ${\tilde{w}}_{2}^{(2)}>{\tilde{w}}_{4}^{(2)}>{\tilde{w}}_{1}^{(2)}>{\tilde{w}}_{3}^{(2)}$, then we have ${x}_{2}\succ {x}_{4}\succ {x}_{1}\succ {x}_{3}$, and the most important factor for foghaze is ${x}_{2}$. 
5.2. Discussions
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Pei, L.; Jin, F. Two Hesitant Multiplicative DecisionMaking Algorithms and Their Application to FogHaze Factor Assessment Problem. Algorithms 2018, 11, 154. https://doi.org/10.3390/a11100154
Pei L, Jin F. Two Hesitant Multiplicative DecisionMaking Algorithms and Their Application to FogHaze Factor Assessment Problem. Algorithms. 2018; 11(10):154. https://doi.org/10.3390/a11100154
Chicago/Turabian StylePei, Lidan, and Feifei Jin. 2018. "Two Hesitant Multiplicative DecisionMaking Algorithms and Their Application to FogHaze Factor Assessment Problem" Algorithms 11, no. 10: 154. https://doi.org/10.3390/a11100154