Whereas papers, cited as references [7] [A. Yu. Karlovich et al., Integral Equations Oper. Theory 70, No. 4, 451–483 (2011; Zbl 1236.47010)] and [8] [A. Yu. Karlovich et al., Integral Equations Oper. Theory 71, No. 1, 29–53 (2011; Zbl 1239.47040)], are in the context of sufficient and necessary conditions for Fredholmness (that is, a Fredholm criterion) of singular integral operators with shifts (diffeomorphisms) and slowly oscillating data, by the authors of this paper in 2011, the present paper is a continuation of the same by investigating the completion of the Fredholm theory for a certain operator \(N\) (that is defined on page 936 of this paper) computation of indices of simplest singular integral operators with shifts, given by Equation (1.1). It is shown how the Fredholm theory for Mellin pseudo-differential operators can be used to study operators beyond that class of the form cited in Equation (1.1). Definitions of the terms and words, used in the paper, are clarified in Section 1. Necessary details on slowly oscillating functions and shifts and on certain algebra generated by operators I and S, are discussed in Section 2. In Section 4, it is proved that certain operators for all values of \(y\) in (\(1, \infty\)), can be realized as Mellin pseudo-differential operators with symbols in algebra up to compact summands. Further, it is established that the pseudo-differential operator is Fredholm of index zero by invoking the concept from Section 3. Theorem 1.1, on page 936 of the paper, is claimed to be the main result.