A multiple attribute group decision making framework for the evaluation of lean practices at logistics distribution centers

Abstract

In recent years, to adapt rapidly to changing market environments and outdo the competition more companies and organizations have adopted lean management practices. One problem that has arisen in these companies and organizations is the need to develop methods to accurately evaluate the lean practices performance. This study proposes a multiple attribute group decision making (MAGDM) framework to facilitate such evaluations. It deals with the consensus process and selection process for MAGDM problems based on the 2-tuple linguistic computation model. The similarity degree and consensus for the linguistic decision matrix are defined using an Euclidian distance function. An algorithm describing the consensus reaching process is presented and its properties analyzed. The entropy method is generalized to a linguistic setting to derive the importance weights for the attributes. One of the main ideas behind the entropy method is that attributes with quite different values are considered more important and therefore have higher weights. Finally, the developed MAGDM framework is applied to a lean practices evaluation problem for a commercial tobacco company’s logistics distribution centers in China.

Appendix

Proof of Theorem 1.

Proof

According to the strategy in Algorithm 1, we have

$$\begin{aligned} R_{p,h+1}=\left( r_{ij,h+1}^{(p)}\right) _{n\times m} , \end{aligned}$$

where

$$\begin{aligned} r_{ij,h+1}^{(p)}=\gamma r_{ij,h}^{(p)} \oplus (1-\gamma )r_{ij,h}. \end{aligned}$$

Furthermore, we have

$$\begin{aligned} \triangle ^{-1}(r_{ij,h})-\triangle ^{-1}(r_{ij,h+1})&= \sum \limits _{l=1}^t {\lambda _l \triangle ^{-1}\left( r_{ij,h}^{(l)} \right) } -\sum \limits _{l=1}^t {\lambda _l \triangle ^{-1}\left( r_{ij,h+1}^{(l)} \right) }\nonumber \\&= \lambda _p \left( \triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}\left( r_{ij,h+1}^{(p)} \right) \right) \nonumber \\&= \lambda _p \left( \triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\left( \gamma \triangle ^{-1}\left( r_{ij,h}^{(p)} \right) +(1-\gamma )\triangle ^{-1}(r_{ij,h} )\right) \right) \nonumber \\&= \lambda _p (1-\gamma )\left( \triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} )\right) . \end{aligned}$$

(15)

Therefore

$$\begin{aligned}&\left| {\triangle ^{-1}\left( r_{ij,h+1}^{(p)} \right) -\triangle ^{-1}(r_{ij,h+1} )} \right| \\&\quad =\left| {\gamma \triangle ^{-1}\left( r_{ij,h}^{(p)} \right) +(1-\gamma )\triangle ^{-1}(r_{ij,h} )-\triangle ^{-1}(r_{ij,h+1} )} \right| \\&\quad =\left| {\gamma (\triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} ))+(\triangle ^{-1}(r_{ij,h} )-\triangle ^{-1}(r_{ij,h+1} ))} \right| \\&\quad =\left| {\gamma (\triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} ))+\lambda _p (1-\gamma )(\triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} ))} \right| \\&\quad =\left| {(\gamma +\lambda _p (1-\gamma ))\left( \triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} )\right) } \right| \\&\quad <\left| {\triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} )} \right| . \\ \end{aligned}$$

Since

$$\begin{aligned} d\left( r_{ij,h}^{(p)},r_{ij,h}\right) =\frac{\left| {\triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} )}\right| }{g}, \end{aligned}$$

we have

$$\begin{aligned} d\left( r_{ij,h+1}^{(p)},r_{ij,h+1} \right) <d\left( r_{ij,h}^{(p)},r_{ij,h}\right) . \end{aligned}$$

(16)

Consequently,

$$\begin{aligned} SD(R_{p,h+1},R_{h+1})<SD(R_{p,h},R_{h}). \end{aligned}$$

That is,

$$\begin{aligned} { GCI }(R_{p,h+1})>{ GCI }(R_{p,h}). \end{aligned}$$

This completes the proof for Theorem 1.$\square$

Proof of Theorem 2.

Proof

Suppose expert $p$ has the minimum consensus index in the $h$th iteration,

$$\begin{aligned} { GCI }(R_{p,h} )=\min \limits _l \{{ GCI }(R_{l,h} )\}. \end{aligned}$$

We discuss two cases.

Case A: $l=p$. According to Theorem 1,

$$\begin{aligned} { GCI }(R_{p,h+1})>{ GCI }(R_{p,h}). \end{aligned}$$

(17)

It follows that

$$\begin{aligned} { GCI }(R_{p,h+1})>\min \limits _l \{{ GCI }(R_{l,h} )\}. \end{aligned}$$

(18)

Case B: $l\ne p$. In this case, we have ${ GCI }(R_{l,h})>{ GCI }(R_{p,h} )$. This means that $SD(R_{l,h},R_h)<SD(R_{p,h},R_h)$. That is, $\exists \gamma _{l,h}$, $0<\gamma _{l,h}<1$, such that

$$\begin{aligned} SD(R_{l,h},R_h )=\gamma _{l,h} SD(R_{p,h},R_h). \end{aligned}$$

Let $\gamma _h =\max \limits _{l\ne p} \{\gamma _{l,h}\}$. From (15), we have

$$\begin{aligned}&(\triangle ^{-1}(r_{ij,h+1}^{(l)} )-\triangle ^{-1}(r_{ij,h+1} ))^2\nonumber \\&\quad ={\left[ \left( \triangle ^{-1}\left( r_{ij,h}^{(l)} \right) -\triangle ^{-1}(r_{ij,h}))+ \lambda _p (1-\gamma _h )(\triangle ^{-1}\left( r_{ij,h}^{(p)}\right) -\triangle ^{-1}(r_{ij,h})\right) \right] ^2}. \end{aligned}$$

(19)

Since $R_{l,h+1} =R_{l,h}$, for $l\ne p$, we have from the Minkowski’s inequality (Bachman and Narici 2000; Wu and Xu 2012) and (19)

$$\begin{aligned}&\sqrt{\frac{1}{nm}\sum \limits _{i=1}^{n} {\sum \limits _{j=1}^m {\frac{1}{g^2}\left( \triangle ^{-1}\left( r_{ij,h+1}^{(l)} \right) -\triangle ^{-1}(r_{ij,h+1} )\right) ^2} } } \\&\quad \le \sqrt{\frac{1}{nmg^2}\sum \limits _{i=1}^{n} {\sum \limits _{j=1}^m {\left( \triangle ^{-1}\left( r_{ij,h}^{(l)} \right) -\triangle ^{-1}(r_{ij,h} )\right) ^2} } }\\&\qquad +\sqrt{\frac{1}{nmg^2}(\lambda _p (1-\gamma _h ))^2\sum \limits _{i=1}^{n} {\sum \limits _{j=1}^m {\left( \triangle ^{-1}\left( r_{ij,h}^{(p)} \right) -\triangle ^{-1}(r_{ij,h} )\right) ^2} } } \\&\quad =SD(R_{l,h},R_h)+\lambda _p (1-\gamma _h )SD(R_{p,h},R_h )\\&\quad \le \gamma _h SD(R_{p,h},R_h )+\lambda _p (1-\gamma _h )SD(R_{p,h},R_h )<SD(R_{p,h},R_h). \end{aligned}$$

Therefore,

$$\begin{aligned} SD(R_{l,h+1},R_{h+1} )<SD(R_{p,h},R_h )=\max \limits _l \{SD(R_{l,h},R_h )\}. \end{aligned}$$

It follows that

$$\begin{aligned} { GCI }(R_{l,h+1})>{ GCI }(R_{p,h} )=\min \limits _l \{{ GCI }(R_{l,h} )\}. \end{aligned}$$

Summarizing both A and B cases, we have

$$\begin{aligned} \min \limits _l \{{ GCI }(R_{l,h+1} )\}>\min \limits _l \{{ GCI }(R_{l,h} )\},\quad \forall l=1,2,\ldots ,t. \end{aligned}$$

This completes the proof for Theorem 2.$\square$