## 1 Introduction
[9] and [PERSON] et al. [11] constructed optimization models to derive priority weights by using the associated matrices of IVFPR or IFPR form an IVIFPR. [PERSON] et al. [25] developed a logarithm least optimal model to derive interval priority weights of IFPR and then proposed a two-stage resolution process for GDM with IVIFPRs. The achievements, as mentioned, have significantly advanced the research on GDM with IVIFPRs. Nevertheless, there still exist some drawbacks, as stated below:  
1. While checking the consistency degree of a given IVIFPR, existing research [17; 18] has proposed the definitions of a consistent IVIFPR, which can be used to judge whether an IVIFPR is consistent or not. However, none of them can measure the consistency degree of an inconsistent IVIFPR. Other literature (References [10; 12; 19]) has also defined the consistency of an IVIFPR through extracting associated preference matrices from original IVIFPR, which cannot fully attach the initial preference information and needs to be constructed by the numerous computational efforts. Therefore, it is necessary to find a convenient solution to check the consistency degrees of IVIFPRs. 2. When a given IVIFPR is unacceptably consistent, only [PERSON] et al. [18] and [PERSON] et al. [12] proposed some iterative algorithms to improve its consistency degree. Nevertheless, it may be to repeat these algorithms several times to repair this unacceptably consistent IVIFPR until the acceptable consistency is achieved. This is time consuming and requires heavy workload. Furthermore, it is unknown how much of the repaired IVIFPR reserves preference information for the initial IVIFPR. 3. As for the determination of priority weights of alternatives for IVIFPRs, the existing studies [15; 22; 24] have directly adopted the formulas to rank the alternatives. However, such formulas are only applicable to consistent IVIFPRs and would be ineffective for inconsistent IVIFPRs. Other methods (References [9; 11; 25]) have established some optimizing or linear models to determine the priority weights, noting that these optimizing or linear models only minimize the deviation between the associated matrices form original IVIFPR and the converted consistent one to the most extent. However, for an extremely unacceptable consistent IVIFPR, the priority weights obtained by such models are unreasonable and cannot be accepted in decision making. To fill the literature review gap, this paper concentrates on developing a novel algorithm for IVIFPRs in GDM problems based on acceptability measurement and priority weight determination. Firstly, following the work of [PERSON] et al. [18], the multiplicative consistency index of an IVIFPR is defined considering DM's risk attitude. Then an optimization model is built to improve the consistency degree of an unacceptable multiplicative consistent IVIFPR and obtain an acceptable multiplicative consistent one. With regard to the determination of priority weights of an IVIFPR, inspired by [PERSON] [26], an error-analysis-based extension method is proposed. In GDM problems, DM' weights are generated by the defined multiplicative consistency index. Subsequently, the collective IVIFPR is obtained through using an IVIF weighted averaging operator. Finally, a step-by-step algorithm for GDM with IVIFPRs is developed. An example of enterprise innovation partner selection is analyzed to demonstrate its practicability and effectiveness. Some significant features of this paper are outlined as follows:  
1. [PERSON] et al. [18] defined the multiplicative consistency of IVIFPR to adjust or repair the inconsistent IVIFPRs. The consistent degree of an inconsistent IVIFPR cannot be measured by [18]. To measure the consistency degree of an IVIFPR, according to the multiplicative consistency definition in Reference [18], this paper proposes a new concept of multiplicative consistency index of an IVIFPR by considering various decision-making principles (i.e., the majority and minority principles).