Profitability analysis using IDEA–DA framework

Abstract

The purpose of this paper is to propose a framework of imprecise data envelopment analysis–discriminant analysis (IDEA–DA) to deal with the profitability analysis of 31 chain stores belonging to a hairdressing company, known as company $L$, in central Taiwan. Return on investment (ROI) is used as the measure of profitability. Five predictor variables are adopted to establish the discriminant function. Five classification types (i.e., 2-group, 3-group, 4-group, 5-group and 6-group) are included in the experiments. The results show that the proposed discriminant function is a robust one with respect to the rankings obtained under different classification types; moreover, the 4-group type is suitable to classify the ROI and it acquires a satisfactory apparent rate. Hence, the 4-group type is suggested in practice work. The study shows the proposed IDEA–DA is an effective tool for company $L$ to conduct the profitability analysis. By firstly discriminating the ROI data collected, consequently showing the aspects for improving ROI, and then use this tool at certain check points in the next year to predict the ROI levels of the stores for the upcoming whole year. For the store which is predicted to be group member with a low or very low ROI level, timely reforms could be made for improving the ROI level of the whole year. The proposed model can also be used to deal with the similar problems for the other chain-store systems which possess the analogue characteristic.

Appendix

In conventional DEA based on the CCR model (Charnes et al. 1978) for the four inputs and one output with exact data, the aggregate efficiency of the target store $k$ is calculated as follows:

$$\begin{aligned} E_k^A&=\hbox { Max}\,\, u_1 y_{1k} \nonumber \\&\hbox {s}.\hbox {t}.\sum _{i=1}^4 {v_i x_{ik} } =1, \\&\quad u_1 y_{1j} -\sum _{i=1}^4 {v_i x_{ij} } \le 0,\quad j=1,\ldots ,31, \nonumber \\&\quad u_1 ,v_1 ,v_2 ,v_3 ,v_4 \ge \varepsilon >0. \nonumber \end{aligned}$$

(a1)

By employing the representative format $x_{4j} =\sum _{\ell =1}^2 {x_4 (\ell )\cdot \delta _j (\ell )}$ elaborated in Sect. 2.2.2 to tackle the imprecise data $x_{4j}$, model (a1) is rewritten as the following IDEA based on the CCR model:

$$\begin{aligned} E_k^A&=\hbox { Max}\,\, u_1 y_{1k} \nonumber \\&\hbox {s}.\hbox {t}.\sum _{i=1}^3 {v_i x_{ik} } +v_4 \sum _{\ell =1}^2 {x_4 (\ell )\cdot \delta _k (\ell )} =1, \nonumber \\&\quad u_1 y_{1j} -\left( \sum _{i=1}^3 {v_i x_{ij} } +v_4 \sum _{\ell =1}^2 {x_4 (\ell )\cdot \delta _j (\ell )} \right) \le 0,\quad j=1,\ldots ,31, \\&\quad \pi _1 x_4 (1)\le x_4 (2)\le \pi _2 x_4 (1) \nonumber \\&\quad a_1 \le x_4 (1)\le a_2 \nonumber \\&\quad u_1 ,v_1 ,v_2 ,v_3 ,v_4 \ge \varepsilon >0. \nonumber \end{aligned}$$

(a2)

Let $\varpi _{4\ell } =v_4 x_4 (\ell )$, and model (a2) is rewritten as model (a3).

$$\begin{aligned} E_k^A&=\hbox { Max}\,\, u_1 y_{1k} \nonumber \\&\hbox {s}.\hbox {t}.\sum _{i=1}^3 {v_i x_{ik} } +\sum _{\ell =1}^2 {\varpi _{4\ell } \cdot \delta _k (\ell )} =1, \nonumber \\&\quad u_1 y_{1j} -(\sum _{i=1}^3 {v_i x_{ij} } +\sum _{\ell =1}^2 {\varpi _{4\ell } \cdot \delta _j (\ell )} )\le 0,\quad j=1,\ldots ,31, \\&\quad \pi _1 \varpi _{41} \le \varpi _{42} \le \pi _2 \varpi _{41} \nonumber \\&\quad a_1 v_4 \le \varpi _{41} \le a_2 v_4 \nonumber \\&\quad u_1 ,v_1 ,v_2 ,v_3 ,v_4 \ge \varepsilon >0. \nonumber \end{aligned}$$

(a3)