Evaluation of benchmarking attribute for service quality using multi attitude decision making approach

Abstract

The essence of benchmarking is the process of identifying the highest standards of superiority for products, services or processes and then make the essential improvements to reach those standards. Today, assessment of service is a main management concern for industries tends to convert benchmark in its respective field. In this study, an endeavor has been made by the authors to evaluate the performance of benchmark model in service industries by using Fuzzy MADM approach where linguistic labels are used to describe attributes of benchmarking which are identified through literature and experts opinion. Afterwards overall numerical index has been computed by using graph theoretic approach, which helps to compare different alternatives of benchmarking for industries’ existence. The results show that responsiveness and reliability is found best alternatives among all the alternatives and Tangibles system is considered as nastiest for the study. This study imposed managers to select significant attributes and alternatives for enhancement of industries at global level.

Appendix

$$\begin{aligned} {\text{Per}}\left( {\text{A}} \right) & = \prod\limits_{1}^{12} {V_{i} } \\ & \quad + \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{ji} )} .D_{k} .D_{i} .D_{m} .D_{n} .D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} \, \\ & \quad + \,\sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{ki} )} \,.D_{l.} D_{m} .D_{n} .D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} \\ & \quad + \,\left\{ \begin{aligned} \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{kl} .D_{li} ).D_{m} .D_{n} .D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ + \,\sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{ji} ).(D_{kl} .D_{lk} ).D_{m} .D_{n} .D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \, \hfill \\ \end{aligned} \right\} \\ & \quad + \left\{ \begin{aligned} \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{ki} )\,(D_{lm} .D_{ml} )} \,D_{n} .D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} \hfill \\ + \,\sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{kl} .D_{lm} .D_{mi)} .D_{n} .D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ \end{aligned} \right\} \\ & \quad + \left\{ \begin{aligned} \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{ji} )\,(D_{kl} .D_{lk} ).(D_{mn} .D_{nm} ).D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ + \,\sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{ji} ).(D_{kl} .D_{lm} .D_{mn} .D_{nk} ).D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ + \,\sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{ki} )(D_{lm} .D_{mn} .D_{nl} ).D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ + \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{kl} .D_{lm} .D_{mn} .D_{ni} ).D_{p} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ \end{aligned} \right\} \\ & \quad + \left\{ \begin{aligned} \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{ji} ).(D_{kl} .D_{lk} ).(D_{mn} .D_{np} .D_{pm} )} .D_{q} .D_{r} .D_{s} .D_{t} .D_{u} \hfill \\ + \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{ji)} .(D_{kl} .D_{lm} .D_{mn} .D_{np} .D_{pk} ).D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ + \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{ki} )(D_{lm} .D_{mn} .D_{np} .D_{pl} ).D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ + \sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{kl} .D_{lm} .D_{mn} .D_{np} .D_{pi} ).D_{q} .D_{r} .D_{s} .D_{t} .D_{u} } \hfill \\ \end{aligned} \right\} \\ & \quad + \,{\text{up}}\,{\text{to}}\, 2 1 {\text{st}}\,{\text{subgrouping}}\,{\text{of13th}}\,{\text{grouping}}\,{\text{i}} . {\text{e}} .\\ & \quad { + }\sum\limits_{i}^{{}} {} \sum\limits_{j} {} \sum\limits_{k} {} \sum\limits_{l} {} \sum\limits_{m} {} \sum\limits_{n} {} \sum\limits_{p} {} \sum\limits_{q} {} \sum\limits_{r} {} \sum\limits_{s} {} \sum\limits_{t} {} \sum\limits_{u} {(D_{ij} .D_{jk} .D_{kl} .D_{lm} .D_{mn} .D_{np} .D_{pq} .D_{qr} .D_{rs} .D_{tu} .D_{ui} } \\ {\text{all}}\,{\text{random}}\,{\text{counts}}\,{\text{i,}}\,{\text{j,}}\,{\text{k,}}\,{\text{l,}}\,{\text{m,}}\,{\text{n,}}\,{\text{p,}}\,{\text{q,}}\,{\text{r,}}\,{\text{s,}}\,{\text{t,}}\,{\text{u}}\,{\text{are}}\,{\text{integers}} \\ \end{aligned}$$

The Eq. (2) gives permanent function for the evaluation. Here the terms are arranged in (12 + 1) groupings. These groupings give the measure of all the attributes and the relative importance between them. The first term (grouping) represents a set of twelve independent subsystem characteristics as D1, D2, D3,…. D12. As system has no ‘self-loop’, so second term or grouping is missing.

Each term of the third grouping represents a set of two elements attribute loops (i.e. Dij.Dji) and is the resultant dependence of attribute i and j and the evaluation measure of N-2 connected terms. Each term of the fourth grouping represents a set of three elements attribute loops (Dij.Djk.Dki or its pair Dik.Dkj.Dji) and the evaluation measure of N-3 unconnected elements or attributes within the system. The fifth grouping contains two subgroups. The terms of first subgrouping consists of four element attribute loops (i.e. Dij.Djk.Dkl.Dli) and the subsystem evaluation index component (Dm.Dn…… Du). The terms of the second grouping are the product of two elements attributes loops (Dij.Dji) (Dkl.Dlk)) and the subsystem evaluation index component (Dm.Dn…… Du).

The terms of the sixth grouping are arranged in two subgroupings. The terms of the first subgroupings are of five element attribute loop (i.e. Dij.Djk.Dkl.Dlm.Dmi) or its pair (Dim.Dml.Dlk.Dkj.Dji) and the subsystem evaluation index component (Dn.Dp…… Du). The second subgrouping consists of a product of two attributes loops (i.e. Dij.Dji) and a three attribute loop (i.e. Dkl.Dlm.Dmk) or its pair (i.e. Dkm.Dml.Dlk) and the subsystem evaluation index component (Dn.Dp…… Du). The terms of seventh groupings are arranged in four subgroupings. The first subgrouping of the seventh grouping is a set of 3- two element attribute loops (i.e. Dij.Dji, Dkl.Dlk, Dmn.Dnm) and the subsystem evaluation index component (Dp.Dq…… Du). The terms of second subgrouping of seventh grouping are of two element attribute loop (i.e. Dij.Dji) and four element attribute loop (i.e. Dkl.Dlm.Dmn.Dnk) with the subsystem evaluation index component (Dp.Dq…… Du). The terms of the third subgrouping of the seventh grouping are of 2- three element attribute loops (i.e. Dij.Dji.Dki and Dlm.Dmn.Dnl) with the subsystem evaluation index component (Dp.Dq…… Du). The terms of fourth subgrouping of seventh grouping are of six elemental attribute loop(i.e. Dij.Djk.Dkl.Dlm.Dmn.Dni) and one subsystem evaluation index component (Dp.Dq…… Du).

The terms of eighth grouping are arranged in four subgroupings. The first subgrouping of the eighth grouping is a set of three element attribute loop (i.e. Dmn.Dnp.Dpm), two element structured as (Dij.Dji) and (Dkl.Dlk) and the subsystem evaluation index component (Dq.Dr… Du). The second subgrouping is a set of a two element diad (Dij.Dji), a five element attribute loop (i.e. Dkl.Dlm.Dmn.Dnp.Dpk) and the subsystem evaluation index component (Dq.Dr… Du). The third subgrouping consists of a three element attribute loop (i.e. Dij.Djk.Dki), a four element attribute loop (i.e. Dlm.Dmn.Dnp.Dpl) and the subsystem evaluation index component (Dq.Dr… Du). Similarly, the fourth subgrouping of the eighth grouping is a seven elemental attribute loop (i.e. Dij.Djk.Dkl.Dlm.Dmn.Dnp.Dpi) and the subsystem evaluation index component (Dq.Dr… Du).

Similarly, other terms of the expression are defined up to the thirteenths grouping. Each term of the grouping as well as the subgroupings have their own independent identities which are useful for the designers and the development analysts for one-to-one evaluation of Advanced Manufacturing Technology.

$$\left[ \begin{aligned} \left( {J_{1}^{T} /J_{2}^{T} /J_{3}^{T} /J_{4}^{T} /J_{51}^{T} /J_{52}^{T} /J_{61}^{T} /J_{62}^{T} /J_{71}^{T} /J_{72}^{T} /J_{73}^{T} /J_{74}^{T} /J_{81}^{T} /J_{82}^{T} /J_{83}^{T} /J_{84}^{T} / \cdots J_{13,18}^{T} /J_{13,19}^{T} /J_{13,20}^{T} /J_{13,21}^{T} } \right) \hfill \\ \times \left( {V_{1}^{T} /V_{2}^{T} /V_{3}^{T} /V_{4}^{T} /V_{51}^{T} /V_{52}^{T} /V_{61}^{T} /V_{62}^{T} /V_{71}^{T} /V_{72}^{T} /V_{73}^{T} /V_{74}^{T} /V_{81}^{T} /V_{82}^{T} /V_{83}^{T} /V_{84}^{T} / \cdots V_{13,18}^{T} /V_{13,19}^{T} /V_{13,20}^{T} /V_{13,21}^{T} } \right) \hfill \\ \end{aligned} \right]$$