## 6 A Practical Example for GDM with IVIFPRs
### A Practical Example of Enterprise Innovation Partner Selection  
Owing to the limitations of resources, high risk of innovation expenditure and long commercialization cycles, an enterprise is unable to carry out innovation activities independently. In this scenario, innovation activities are generated among partners, who can share resources, technologies and critical competitiveness, by which a win-win situation among partners can be achieved. Therefore, selecting innovation partners has become a realistic choice for many firms. There exists a company wants to select and evaluate an appropriate innovation partner. Four alternatives \(\{x_{1},x_{2},x_{3},x_{4}\}\) intend to bid on it. To choose the best partner, the company employs three peer review DMs/experts \(\{d_{1},d_{2},d_{3}\}\) with the different professional knowledge to constitute an evaluation committee. Three DMs/experts construct one pairwise comparison judgment matrix with IVIFPRs by comparing any two partners, DMs furnish their IVIFPRs as  
\[\widetilde{R}^{1}=\left( \begin{array}{cccc}([0.50,0.50],[0.50,0.50])&([0.15,0.30],[0.65,0.70])&([0.30,0.35],[0.40,0.45])&([0.50,0.60],[0.10,0.15])\\ ([0.65,0.70],[0.15,0.30])&([0.50,0.50],[0.50,0.50])&([0.70,0.80],[0.05,0.10] )&([0.60,0.75],[0.10,0.20])\\ ([0.40,0.45],[0.30,0.35])&([0.05,0.10],[0.70,0.80])&([0.50,0.50],[0.50,0.50] )&([0.55,0.70],[0.10,0.20])\\ ([0.10,0.15],[0.50,0.60])&([0.10,0.20],[0.60,0.75])&([0.10,0.20],[0.55,0.70] )&([0.50,0.50],[0.50,0.50])\end{array} \right)\]  
\[\widetilde{R}^{2}=\left( \begin{array}{cccc}([0.50,0.50],[0.50,0.50])&([0.25, 0.40],[0.40,0.50])&([0.20,0.35],[0.60,0.65])&([0.20,0.35],[0.30,0.50])\\ ([0.40,0.50],[0.25,0.40])&([0.50,0.50],[0.50,0.50])&([0.50,0.55],[0.30,0.40])&( [0.55,0.70],[0.10,0.20])\\ ([0.60,0.65],[0.20,0.35])&([0.30,0.40],[0.50,0.55])&([0.50,0.50],[0.50,0.50])&( [0.40,0.65],[0.10,0.20])\\ ([0.30,0.50],[0.20,0.35])&([0.10,0.20],[0.55,0.70])&([0.10,0.20],[0.40,0.65] )&([0.50,0.50],[0.50,0.50])\end{array} \right)\]  
\[\widetilde{R}^{3}=\left( \begin{array}{cccc}([0.50,0.50],[0.50,0.50])&([0.15, 0.30],[0.40,0.50])&([0.10,0.30],[0.45,0.65])&([0.60,0.85],[0.10,0.10])\\ ([0.40,0.50],[0.15,0.30])&([0.50,0.50],[0.50,0.50])&([0.75,0.80],[0.15,0.20])&( [0.55,0.70],[0.10,0.15])\\ ([0.45,0.65],[0.10,0.30])&([0.15,0.20],[0.75,0.80])&([0.50,0.50],[0.50,0.50])&( [0.45,0.85],[0.10,0.15])\\ ([0.10,0.10],[0.60,0.85])&([0.10,0.15],[0.55,0.70])&([0.10,0.15],[0.45,0.85])&( [0.50,0.50],[0.50,0.50])\end{array} \right)\]  
In what follows, the proposed algorithm in this paper is employed to solve the above problem. 1. Predefine consistency threshold \(\mathrm{MCI}_{0}=0.1\) and set the value of parameter \(\varphi=0.5\). 2. Calculate the multiplicative consistency index \(\mathrm{MCI}\Big{(}\widetilde{R}^{t}\Big{)}\) for individual IVIFPR \(\widetilde{R}^{t}(t=1,2,\ldots,m)\) using Equation (16), the results of the calculation are as follows: \[\mathrm{MCI}\Big{(}\widetilde{R}^{1}\Big{)}=1.9679,\ \mathrm{MCI}\Big{(}\widetilde{R}^{2}\Big{)}=1.2745,\ \mathrm{MCI}\Big{(}\widetilde{R}^{3}\Big{)}=\ 2.1035.\]
3. Determine DMs' weights \(\eta_{t}(t=1,2,\ldots,m)\) using Equation (35). \[\eta_{1}=0.2874;\ \eta_{2}=0.4437;\ \eta_{3}=0.2689.\]
4. Using formula \(\mathrm{MCI}\Big{(}\widetilde{R}^{t}\Big{)}\leq\mathrm{MCI}_{0}\), all the individual IVIFPRs \(\widetilde{R}^{t}\) are unacceptable multiplicative consistent. 5. Solving Model (18), the optimal solutions \(\underline{\hat{\mu}}_{ij}\), \(\overline{\hat{\mu}}_{ij}\), \(\underline{\hat{\mu}}_{ij}\) and \(\overline{\hat{\sigma}}_{ij}\) for all \(i,j=1,2,\ldots n\) and \(i<j\) are derived from unacceptable multiplicative consistent IVIFPRs \(\widetilde{R}^{t}(t=1,2,3)\).