SIMPLIFIED NEUTROSOPHIC INDETERMINATE DECISION MAKING METHOD WITH DECISION MAKERS’ INDETERMINATE RANGES

. There exists the indeterminate situations of truth, falsity, indeterminacy degrees due to the uncertainty and inconsistency of decision makers’ arguments in a complicated decision making (DM) problem. Then, existing neutrosophic set cannot describe the indeterminate information of truth, falsity, indeterminacy degrees. It is noted that the simplified neutrosophic set (SNS) is depicted by truth, falsity, indeterminacy degrees, while a neutrosophic number (NN) can be flexibly depicted by its determinate part and its indeterminate part. Regarding the indeterminate situations of truth, falsity, indeterminacy degrees in indeterminate DM problems, this study first presents a simplified neutrosophic indeterminate set (SNIS) to express the hybrid information of SNS and NN and defines the score, accuracy, and certainty functions of simplified neutrosophic indeterminate elements (SNIEs) with indeterminate ranges to compare SNIEs. Then, we introduce a

As another branch of neutrosophic theory, a neutrosophic number (NN) (Smarandache, 1998(Smarandache, , 2013b(Smarandache, , 2014 was proposed under indeterminate environment and represented as e = a + aI for a, a ∈ ℜ and I ∈ [I -, I + ], where a is a certain term and aI is an indeterminate term along with the indeterminate coefficient a and indeterminacy I ∈ [I -, I + ]. NN indicates a family of interval numbers corresponding to different indeterminate ranges of I ∈ [I -, I + ], which demonstrates its flexibility and convenience in expressing indeterminate information. Therefore, NNs have been widely applied in many areas. For example, mechanical fault diagnosis (Ye, 2016), DM (P. D. Liu & X. Liu, 2018), rock mechanics , optimization programming (Ye, 2018), and so on.
Then, there may exist the indeterminacy of the truth, falsity, indeterminacy degrees given by a group of decision makers due to the indeterminacy and inconsistency of decision makers' cognitions regarding object complexity and variability evaluated in the real DM problem. It is noted that the indeterminacy information of the truth, falsity, and indeterminacy degrees contains the hybrid information of SNS and NN, which cannot be expressed only by the neutrosophic set or NN. Since NN can flexibly depict such an indeterminacy with a changeable interval number (e = [a +aI -, a + aI + ]) or a changeable single value (e = a +aI) depending on specified indeterminate ranges of I ∈ [I L , I U ] or specified single values of I ∈ [I L , I U ], which shows its main highlight in an expression of indeterminate information. Then, SNS (IvSS and SvNS) can depict the truth, falsity, and indeterminacy degrees, but cannot depict such indeterminacy with a changeable interval number/single value of the truth/falsity/indeterminacy degree in indeterminate situations. Obviously, existing neutrosophic DM methods cannot handle such a DM problem with both the indeterminate information of the truth, falsity, indeterminacy arguments and the decision makers' indeterminate ranges/cognitions in indeterminate DM applications. If SNS is combined with NN based on an information expression advantage of both, we can present the new set concept and DM method based on the hybrid information of SNS and NN to carry out the aforementioned issues. Motivated by the new set concept and DM method, this study firstly proposes simplified neutrosophic indeterminate sets (SNISs) to express changeable IvNSs/ SvNSs corresponding to different indeterminate ranges/ values of I ∈ [I -, I + ] and weighted aggregation operators of simplified neutrosophic indeterminate elements (SNIEs), and then establishes a multi-attribute DM method with decision makers' indeterminate ranges in indeterminate DM situations.
To the best of our knowledge, there exists no study regarding the proposed issues in existing literature. Hence, the main contributions of this study are: (1) to present SNIS and a ranking method of SNIEs, (2) to introduce a SNIE weighted arithmetic averaging (SNIEWAA) operator and a SNIE weighted geometric averaging (SNIEWGA) operator, (3) to establish a multi-attribute DM approach with decision makers' indeterminate ranges regarding the SNIEWAA and SNIEWGA operators in SNIS setting, and (4) to apply the proposed DM approach to an indeterminate DM example on choosing a suitable slope design scheme for an open pit mine in SNIS setting for indicating its flexibility and suitability under the indeterminate DM environment.
To realize this study, the rest of the article is constructed by the following parts. Section 1 introduces some preliminaries of SNSs and NNs. Section 2 presents a SNIS concept to depict the indeterminacy information of the truth, falsity, and indeterminacy degrees, and then defines the score, accuracy, and certainty functions of SNIEs with I ∈ [I -, I + ] for ranking SNIEs. In Section 3, the SNIEWAA and SNIEWGA operators are proposed to aggregate SNIEs. For Section 4, a multi-attribute DM approach with decision makers' indeterminate ranges regarding the SNIEWAA and SNIEWGA operators is established in SNIS setting. Then, Section 5 applies the proposed DM approach to an indeterminate DM example on choosing a suitable slope design scheme for an open pit mine in SNIS setting for indicating its flexibility and effectiveness. Lastly, the conclusions and further research are indicated.

SNISs and ranking method
Based on the hybrid concept of both SNS and NN, we can give the definition of a SNIS as the generalization of a SNS concept in indeterminate and inconsistent situations. Definition 1. Set X = {x 1 , x 2 , …, x n } as a universe set. A SNIS Z is defined as the following expression: Obviously, SNIS shows the advantages of its convenience and flexibility in the indeterminate information expressions regarding different indeterminate ranges/values of I ∈ [I -, I + ].

Weighted aggregation operators of SNIEs
Based on the aggregation operators of Eqns (1)-(4) (Zhang et al., 2014;Peng et al., 2016) and the above NN operational relations, we can extend them to the two weighted aggregation operators of SNIEs in this section.
Step 3: The alternatives are ranked based on the ranking method in Definition 3 and the best one is selected.

Indeterminate DM example on choosing a suitable open pit mine slope design scheme
Open pit mine slope design is a fundamental issue in the process of mine design and operation to provide an optimal excavation configuration in the context of safety, ore recovery and financial return (Read & Stacey, 2009). Hence, investors and operators firstly ensure the open pit mine slope stability for preventing the potential risks caused by slope failure (Yong et al., 2019). Then, the economic benefit of mining needs to be considered and ore recovery must be maximized to meet the economic needs of owners. Moreover, open pit mines in most countries generally have mining regulations that specify environmental requirements. It is obvious that the safety, economic and environmental factors should be considered as main assessment indices in the open pit mine slope design. Let us consider a multi-attribute DM problem on choosing a suitable slope design scheme (alternative) for an open pit mine. Suppose that there is a set of four potential alternatives M = {M 1 , M 2 , M 3 , M 4 } for the open pit mine, which must be satisfactorily assessed by the three indices (attributes): the safety factor (R 1 ), the economic factor (R 2 ), and the environmental factor (R 3 ). Then the weight vector of the three attributes is specified as w = (0.36, 0.3, 0.34) by experts/decision makers.
Then, experts/decision makers are required to give the satisfactory assessment of each alternative M j (j = 1, 2, 3, 4) over the attributes R k (k = 1, 2, 3) by the assessment information of the truth, falsity, and indeterminacy NNs  I  I  I  I  I  I  I  I  I Thus, the developed approach is utilized for the indeterminate DM problem with I ∈ [0, 1.5] and described by the following decision process: First, the aggregation values of SNIEs z jk for M j (j = 1, 2, 3, 4) are calculated by Eqns (10) Table 1 and Table 2.
Then, the values of the score function S(z j , I) are calculated by Eqn (5). Consequently, all the decision results regarding the SNIEWAA and SNIEWGA operators are shown in Table 3 and Table 4, respectively.
In Tables 1 and 2, the ranking orders of alternatives and the best slope design schemes regarding the SNIEWAA and SNIEWGA operators are identical when the indeterminate ranges are I = [0, 0], [0, 0.5], while the ranking orders and the best ones regarding the SNIEWAA and SNIEWGA operators indicate some difference when the indeterminate ranges are I = [0, 1], [0, 1.5]. Clearly, the different indeterminate ranges can affect the ranking orders of alternatives. Then the final decision result depends on the indeterminate range of I ∈ [I -, I + ] specified by the decision makers, which demonstrate the effectiveness and flexibility of the proposed DM method in simplified neutrosophic indeterminate setting.
Especially when I = [0, 0] = 0 in Tables 1 and 2, the proposed simplified neutrosophic indeterminate DM method is reduced to the DM methods based on the SvNEWAA and SvNEWGA operators (Zhang et al., 2014;Peng et al., 2016), while when I = [0, 0.5], [0, 1], [0, 1.5] in Tables 1 and 2, the proposed simplified neutrosophic indeterminate DM method is reduced to the DM methods based on the IvNEWAA and IvNEWGA operators (Zhang et al., 2014;Peng et al., 2016). Obviously, the proposed simplified neutrosophic indeterminate DM method contains single-valued and interval neutrosophic DM methods (Zhang et al., 2014;Peng et al., 2016) because SNIS contains its SNS family (SvNS family or IvNS family) depending on the indeterminate values/ranges of I ∈ [I -, I + ]. Therefore, the proposed DM method is the generalization of existing simplified neutrosophic DM methods (Zhang et al., 2014;Peng et al., 2016), while existing simplified neutrosophic DM methods (Zhang et al., 2014;Peng et al., 2016) are only the special cases of the proposed DM method with the specified indeterminate value/range of I ∈ [I -, I + ]. Since the proposed DM method indicates the advantage of its flexibility and generalization by comparison with existing simplified neutrosophic DM methods (Zhang et al., 2014;Peng et al., 2016), the proposed DM method is superior to existing ones (Zhang et al., 2014;Peng et al., 2016).
However, existing various neutrosophic DM methods cannot handle such a DM problem with the hybrid information of SNS and NN (the SNIS information) and decision makers' indeterminate ranges/cognitions in indeterminate DM applications, while this original study not only can present the SNIS information by describing various indeterminate degrees of the truth, falsity, indeterminacy as a generalization of SNS (SvNS and IvNS), but also can demonstrate the superiority of flexible DM in indeterminate DM applications corresponding to decision makers' indeterminate degrees/cognitions for I ∈ [I -, I + ]. Therefore, this study indicates the convenient and flexible advantages in the indeterminate information expression and processing in indeterminate DM problems.

Conclusions
This study proposed the SNIS concept for the first time to depict the hybrid information of both SNS and NN in indeterminate and inconsistent setting, and then presented the score, accuracy, and certainty functions of SNIEs for ranking SNIEs and the SNIEWAA and SNIEWGA operators for aggregating SNIEs. Next, a simplified neutrosophic indeterminate multi-attribute DM approach regarding the SNIEWAA and SNIEWGA operators was put forward along with decision makers' indeterminate ranges to deal with indeterminate DM problems in SNIS setting. Eventually, the developed multi-attribute DM approach was applied in an indeterminate DM example on choosing a suitable slope design scheme for an open pit mine in SNIS setting. By the DM example and comparative analysis, we discuss how the different indeterminate ranges affect the ranking orders of alternatives, and then the decision results show the flexibility and effectiveness of the established multi-attribute DM approach in various indeterminate situations of decision makers, which indicate the main superiority in this study. In the future, this study will be further generalized to pattern recognition, medical diagnosis, and image processing in SNIS setting.