Developing a TOPSIS algorithm for Q-rung orthopair Z-numbers with applications in decision making

No decision can be made without first taking the decision-making process into consideration. Multi-criteria decision-making (MCDM) seeks to identify the optimal option by taking into account multiple criteria during the selection process. Numerous tools and techniques from MCDM can be used in a variety of sectors, including engineering design and finance. Recently, because of the generalization ability and more flexibility of Q-rung orthopair fuzzy sets (Q-ROFS), it has been used widely to solve MCDM problems. However, under uncertainty, evaluating an alternative in the form of Q-ROFS with a full reliability is not always possible. Therefore, in this paper, we have defined the notion of Q-rung orthopair Z-numbers, in which the reliabilities of an evaluated Q-ROFS have also been considered to make decision-making more comprehensive regarding the uncertainty. Mathematically, a Q-rung orthopair Z-number $$Z_Q=(C_Q, ~R_Q)$$ Z Q = ( C Q , R Q ) defined on a non-empty finite set X is a pair of two Q-ROFS, where $$C_Q$$ C Q is Q-ROFS defined on X and $$R_Q$$ R Q is the reliability of the $$C_Q$$ C Q in the form of Q-ROFS. Further, arithmetic operations on these Z-numbers have been introduced. Moreover, a series of dice similarity measures have been defined for Q-rung orthopair Z-numbers. To solve the MCDM problems, a TOPSIS method based on these distance measures has also been discussed. Further, an application of the proposed method has been studied to solve an industrial problem. Furthermore, a sensitivity analysis of the involved parameters has been done to illustrate the stability and efficiency of the proposed approach. Finally, a case study has been conducted investigating the allocation of healthcare resources, along with a comparative analysis of the outcomes attained using the proposed technique. This analysis intends to demonstrate the proposed method’s validity and superiority. Further, it has been shown by numerical experiments that decision-making with Q-ROZN is more suitable since it gives the decision-maker additional freedom in terms of the reliabilities of evaluated Q-ROFS.