*5.1. Background*

The ABC Limited Company is a passenger car manufacturer in China. To improve the competitiveness of products and reduce production costs, ABC decides to choose a third-party logistics service provider for logistics outsourcing. Through preliminary selection, five possible logistics providers *H* = {*<sup>H</sup>*1, *H*2, *H*3, *H*4, *<sup>H</sup>*5} are provided for further evaluation with respect to the following four criteria: service (*C*1), relationship (*C*2), quality (*C*3), and equipment systems (*C*4). Furthermore, assume that the weight vector of criteria *Cj* (*j* =1, 2, 3, 4) is ω = (0.4, 0.3, 0.2, 0.1). Three experts with different backgrounds are invited by the company to evaluate the TPLSP. Since these criteria are all qualitative, it is suitable for the experts to express their views in linguistic term sets. The ABC Company uses a LTS of seven terms to evaluate the TPLSP, which can be expressed by *S* = {*<sup>s</sup>*0 : *very poor*,*s*<sup>1</sup> : *poor*, *s*2 : *slightly poor*,*s*3 : *f air*,*s*<sup>4</sup> : *slightly good*, *s*5 : *good*,*s*<sup>6</sup> : *very good*}. The final judgment of the five providers with the hesitant fuzzy linguistic decision matrix *H* = (*HijS* )<sup>5</sup>×<sup>4</sup> are given in Table 4.

To verify the feasibility and effectiveness of the decision method proposed in Section 4, at first, we assume *f* = *f*1(*si*) = *i*2*t* (*t* = <sup>3</sup>).


**Table 4.** The hesitant fuzzy linguistic decision matrix provided by experts.

**Step 1.** Normalize the hesitant fuzzy linguistic decision matrix.

It is already known the criteria *C*1, *C*2, *C*3, *C*4 are all benefit-type criteria, and thus we do not need to do anything.

**Step 2.** According to the score function in Theorem 1, we can calculate the HFLPIS H<sup>+</sup> and the HFLNIS H<sup>−</sup>, which are given as follows:

$$H^{+} = \{ \{ \mathbf{s}\_5 \} , \{ \mathbf{s}\_{44} \mathbf{s}\_6 \} , \{ \mathbf{s}\_{44} \mathbf{s}\_6 \} , \{ \mathbf{s}\_{1'} \mathbf{s}\_4 \} \}$$
 
$$H^{-} = \{ \{ \mathbf{s}\_{0'} \mathbf{s}\_1 \} , \{ \mathbf{s}\_{1'} \mathbf{s}\_3 \} , \{ \mathbf{s}\_{0'} \mathbf{s}\_{1'} \mathbf{s}\_3 \} , \{ \mathbf{s}\_{0'} \mathbf{s}\_2 \} \}$$

**Step 3.** Calculate the distance measure *<sup>D</sup>*<sup>∗</sup>*ωHFL*(*HijS* , *H*+) and *<sup>D</sup>*<sup>∗</sup>*ωHFL*(*HijS* , *<sup>H</sup>*−) for different alternative *Hi*(*i* = 1, 2, 3, 4, 5) respectively, which are given in Table 5.

**Table 5.** The distance measure of each alternative.


**Step 4.** Calculate the closeness coefficient Φ*i*of each alternative *Hi*; they are obtained in Table 6.

 **Table 6.** The closeness coefficient of each alternative.


**Step 5.** Rank the alternatives *Hi* and utilize Φ*i* (*i* = 1, 2, 3, 4, <sup>5</sup>).

It is already known that *H*4 *H*5 *H*1 *H*2 *H*3, which means that the best choice is *H*4. In order to illustrate the impact of the linguistic scale function *f* on MCDM, we use the different linguistic scale functions *f* = *f*2(*si*) (*a* = 1.4, *t* = 3) and *f* = *f*3(*si*) (*α* = *β* = 0.8) to calculate the distance measure between HFLTSs, the results are given in Table 7.

**Table 7.** Results obtained by different linguistic scale functions.


The results between the different linguistic scale functions are shown in Figure 3.

**Figure 3.** The differences between the different linguistic scale functions.
