A new integrated rough multi-criteria decision-making model for enterprise resource planning software selection

Rough BWM

In this study, the weights of the ERP selection criteria will be determined using the rough BWM method. The steps of this method are summarized below (Lo et al., 2019; Chang et al., 2019).

Step 1: ERP selection criteria are identified.

Step 2: The worst and the best criteria are determined.

Step 3: Best-to-Others and Others-to-Worst Vectors are identified.

Each decision-maker assesses the relative importance $F_{BEj}^{\left(r\right)}$ of the best criterion BE to other criteria j to determine the best-to-others (BEO) vector with using 1–9 scale. (10)${\displaystyle F_{BEj}^{\left(r\right)}=\left(f_{BE1}^{\left(r\right)},f_{BE2}^{\left(r\right)},\dots ,f_{BEBE}^{\left(r\right)},\dots ,f_{BEn}^{\left(r\right)}\right)}$

Likewise, each decision-maker assesses the relative importance $F_{jW}^{\left(r\right)}$ of other criteria j to the worst criterion W to obtain the others-to-worst (OW) vector with using 1–9 scale. (11)${\displaystyle F_{jW}^{\left(r\right)}=\left(f_{1W}^{\left(r\right)},f_{2W}^{\left(r\right)},\dots ,f_{WW}^{\left(r\right)},\dots ,f_{nW}^{\left(r\right)}\right)}$

In Eqs. (10) and (11), $f_{BEBE}^{\left(r\right)}$ and $f_{WW}^{\left(r\right)}$ equal to 1.

Rough BEO and OW vectors are calculated by using Eqs. (3) and (4). In other words, the BEO and OW vectors of each decision maker are combined with Eqs. (3) and (4). These rough vectors are shown in Eqs. (12) and (14). (12)${\displaystyle RN\left(F_{BEj}\right)=\left[f_{BEj}^l,f_{BEj}^u\right]=\left(\left[f_{BE1}^l,f_{BE1}^u\right],\left[f_{BE2}^l,f_{BE2}^u\right],\dots ,\left[f_{BEBE}^l,f_{BEBE}^u\right],\dots ,\left[f_{BEn}^l,f_{BEn}^u\right]\right)}$ (13)${\displaystyle RN\left(F_{jW}\right)=\left[f_{jW}^l,f_{jW}^u\right]=\left(\left[f_{1W}^l,f_{1W}^u\right],\left[f_{2W}^l,f_{2W}^u\right],\dots ,\left[f_{WW}^l,f_{WW}^u\right],\dots ,\left[f_{nW}^l,f_{nW}^u\right]\right)}$

Step 4: The rough optimal weights ($RN\left(w_j^{\mathrm{\ast }}\right)=\left(\left[w_1^{l\mathrm{\ast }},w_1^{u\mathrm{\ast }}\right],\left[w_2^{l\mathrm{\ast }},w_2^{u\mathrm{\ast }}\right],\dots ,\left[w_n^{l\mathrm{\ast }},w_n^{u\mathrm{\ast }}\right]\right)$) are computed by Eq. (14). (14)${\displaystyle Min{\varepsilon }^{\mathrm{\ast }}s.t.\left\{\begin{array}{c}{w}_{BE}^l-f_{BEj}^l.w_j^u\le {\varepsilon }^{\ast }.w_j^u\hfill \\ w_{BE}^l-f_{BEj}^l.w_j^u\ge -{\varepsilon }^{\ast }.w_j^u\hfill \\ w_{BE}^u-f_{BEj}^u.w_j^l\le {\varepsilon }^{\ast }.w_j^l\hfill \\ w_{BE}^u-f_{BEj}^u.w_j^l\ge -{\varepsilon }^{\ast }.w_j^l\hfill \\ w_j^l-f_{jW}^l.w_W^u\le {\varepsilon }^{\ast }.w_W^u\hfill \\ w_j^l-f_{jW}^l.w_W^u\ge -{\varepsilon }^{\ast }.w_W^u\hfill \\ w_j^u-f_{jW}^u.w_W^l\le {\varepsilon }^{\ast }.w_W^l\hfill \\ w_j^u-f_{jW}^u.w_W^l\ge -{\varepsilon }^{\ast }.w_W^l\hfill \\ \sum _{j=1}^n\left(\frac{w_j^l+w_j^u}{2}\right)=1\hfill \\ w_j^u\ge w_j^l\ge 0\hfill \end{array}\right.}$

In Eq. (14), ɛ^∗ indicates the consistency ratio (CR) of the pairwise comparison matrix. If this value is near 0, that means the pairwise comparison matrix is consistent.

Rough WISP

In this study, the rough WISP method is used to rank ERP software. The steps of rough WISP are shown below.

Step 1: Each decision-makers assess alternatives based on selection criteria with using 1–9 scale. A rough decision matrix (RN(G)) is created by using Eqs. (3) and (4). RN(G) is shown in Eq. (15). (15)${\displaystyle RN\left(G\right)={\left[g_{ij}^l,g_{ij}^u\right]}_{m\times n}}$

Step 2: This rough decision matrix is normalized by Eq. (16) to structure rough normalized matrix ($RN\left(H\right)=\left[h_{ij}^l,h_{ij}^u\right]$). (16)${\displaystyle RN\left(H\right)=\left[h_{ij}^l,h_{ij}^u\right]=\left[\frac{h_{ij}^l}{max\left(h_{ij}^u\right)},\frac{h_{ij}^u}{max\left(h_{ij}^u\right)}\right]}$

Step 3: The rough weighted sum and weighted product of normalized values are computed for each alternative with respect to non-beneficial (NBL) and beneficial (BFL) criteria. (17)${\displaystyle RN\left(K^{+}\right)=\left[k_{ij}^{l+},k_{ij}^{u+}\right]=\sum _{j\in BFL}\left[h_{ij}^l\times w_j^l,h_{ij}^u\times w_j^u\right]}$ (18)${\displaystyle RN\left(K^{-}\right)=\left[k_{ij}^{l-},k_{ij}^{u-}\right]=\sum _{j\in NBL}\left[h_{ij}^l\times w_j^l,h_{ij}^u\times w_j^u\right]}$ (19)${\displaystyle RNRN\left(P^{+}\right)=\left[p_{ij}^{l+},p_{ij}^{u+}\right]=\prod _{j\in BFL}\left[h_{ij}^l\times w_j^l,h_{ij}^u\times w_j^u\right]}$ (20)${\displaystyle RN\left(P^{-}\right)=\left[p_{ij}^{l-},p_{ij}^{u-}\right]=\prod _{j\in NBL}\left[h_{ij}^l\times w_j^l,h_{ij}^u\times w_j^u\right]}$

Step 4: The rough utility measures are obtained as follows. (21)${\displaystyle RN\left(Y_i^{sd}\right)=\left[y_i^{sdl},y_i^{sdu}\right]=\left[k_{ij}^{l+}-k_{ij}^{u-},k_{ij}^{u+}-k_{ij}^{l-}\right]}$ (22)${\displaystyle RN\left(Y_i^{td}\right)=\left[y_i^{tdl},y_i^{tdu}\right]=\left[p_{ij}^{l+}-p_{ij}^{u-},p_{ij}^{u+}-p_{ij}^{l-}\right]}$ (23)${\displaystyle RN\left(Y_i^{sr}\right)=\left[y_i^{srl},y_i^{sru}\right]=\left[\frac{k_{ij}^{l+}}{k_{ij}^{u-}},\frac{k_{ij}^{u+}}{k_{ij}^{l-}}\right]}$ (24)${\displaystyle RN\left(Y_i^{tr}\right)=\left[y_i^{trl},y_i^{tru}\right]=\left[\frac{p_{ij}^{l+}}{p_{ij}^{u-}},\frac{p_{ij}^{u+}}{p_{ij}^{l-}}\right]}$

Step 5: The rough utility measures are re-computed as follows. (25)${\displaystyle RN\left({\underset{¯}{Y}}_i^{sd}\right)=\left[{\underset{¯}{y}}_i^{sdl},{\underset{¯}{y}}_i^{sdu}\right]=\left[\frac{1+y_i^{sdl}}{1+max\left(y_i^{sdu}\right)},\frac{1+y_i^{sdu}}{1+max\left(y_i^{sdu}\right)}\right]}$ (26)${\displaystyle RN\left({\underset{¯}{Y}}_i^{td}\right)=\left[{\underset{¯}{y}}_i^{tdl},{\underset{¯}{y}}_i^{tdu}\right]=\left[\frac{1+y_i^{tdl}}{1+max\left(y_i^{tdu}\right)},\frac{1+y_i^{tdu}}{1+max\left(y_i^{tdu}\right)}\right]}$ (27)${\displaystyle RN\left({\underset{¯}{Y}}_i^{sr}\right)=\left[{\underset{¯}{y}}_i^{srl},{\underset{¯}{y}}_i^{sru}\right]=\left[\frac{1+y_i^{srl}}{1+max\left(y_i^{sru}\right)},\frac{1+y_i^{sru}}{1+max\left(y_i^{sru}\right)}\right]}$ (28)${\displaystyle RN\left({\underset{¯}{Y}}_i^{tr}\right)=\left[{\underset{¯}{y}}_i^{trl},{\underset{¯}{y}}_i^{tru}\right]=\left[\frac{1+y_i^{trl}}{1+max\left(y_i^{tru}\right)},\frac{1+y_i^{tru}}{1+max\left(y_i^{tru}\right)}\right]}$

Step 6: The final rough utility value is calculated for each alternative by Eq. (29). (29)${\displaystyle RN\left(Y_i\right)=\left[y_i^l,y_i^u\right]=\frac{1}{4}\times \left(\left[\left({\underset{¯}{y}}_i^{sdl}+{\underset{¯}{y}}_i^{tdl}+{\underset{¯}{y}}_i^{srl}+{\underset{¯}{y}}_i^{trl}\right),\left({\underset{¯}{y}}_i^{sdu}+{\underset{¯}{y}}_i^{tdu}+{\underset{¯}{y}}_i^{sru}+{\underset{¯}{y}}_i^{tru}\right)\right]\right)}$

Step 7: The final rough utility values are transformed into crisp utility values by Eq. (30). (30)${\displaystyle Y_i=\frac{y_i^l+y_i^u}{2}}$

Sensitivity analysis

In this analysis, a total of thirty scenarios were arranged by reducing the weights of the three criteria (PC, EU and MSC) with the highest weights using the following method. Equation (31), which is used to arrange the scenarios, is presented below (Huskanović, Stević & Simić, 2023; Badi & Elghoul, 2023). (31)${\displaystyle W_{n\gamma }=\left(1-W_{n\theta }\right)\frac{W_{\gamma }}{\left(1-W_n\right)}}$

In Eq. (31), W_nγ is a new value of the weight of criterion, additionally, W_γ indicates original value of criterion. Besides, W_nθ presents the reduced criterion weight, and W_n is the original weight of the criterion with a reduced value (Tešić et al., 2023; Wieckowski et al., 2023). Figure 2 indicates the results of the sensitivity analysis.

Criteria weights were changed with 30 scenarios. SFT-3 coded ERP software kept its 5th place in all scenarios. Other ERP software has changed places at least once. ERP software with SFT-1 code was ranked 4th only in S10 scenario. In the remaining scenarios, it took the 6th place. ERP software with SFT-2 code was ranked 3rd in S3 and S4, 2nd in S5 and S6, and 1st in S7-S10. In other scenarios, it maintained its place in the 4th place. SFT-4 coded ERP software was ranked 2nd in S7-0-S10 and ranked 1st in other scenarios. While the SFT-5 coded ERP software was in the 3rd place in S5–S10 and S17–S20 scenarios, it kept its second place in other scenarios. In addition, in other scenarios, it kept its 3rd place. It can be said that the rough WISP method developed according to the results is sensitive to the change in the weights of the criteria.

WS (Sałabun & Urbaniak, 2020) and SCC (Wieckowski et al., 2023) coefficients can be calculated if there are changes in the results verification analysis. Calculated SCC (Table 11) and WS (Table 12) coefficients for comparative analysis and for sensitivity analysis (Fig. 3) are as follows:

All calculated statistical correlation coefficients show high and very high correlations between initial rank and others obtained through comparative and sensitivity analysis.