Sometimes, decision makers give fuzzy evaluation information that cannot be expressed exactly. Pang et al. (2016) came up with PLTS, which have different weights and probabilities. Definition 1

Let L = { ζ ƛ | ƛ = − ∂ , … , − 2 , − 1 , 0 , 1 , 2 , … , ∂ } be an LTS, and the linguistic terms ζ ƛ can be denoted as the value information χ by transformation function κ, and the formula is as follows: (1) κ : [ ζ − ƛ , ζ ƛ ] → [ 0 , 1 ] , κ ( ζ ƛ ) = ƛ + ∂ 2 ∂ = χ . χ can also be translated into linguistic terms ζ ƛ by the shifting function κ − 1 : (2) κ − 1 : [ 0 , 1 ] → [ ζ − ƛ , ζ ƛ ] , κ − 1 ( χ ) = ζ ( 2 χ − 1 ) ∂ = ζ ƛ .

In the cumulative prospect theory (CPT) (Tversky and Kahneman, 1992), DMs will go through two stages when they are faced with choices and make decisions: editing stage and evaluation stage. In the editing stage, the decision-makers will collect information and do some preprocessing to find out the reference point; in the evaluation stage, the decision makers will evaluate the prospect which has been pretreated and choose the best prospect based on a value function λ ( x j ) and a weight function w ( p i ). Therefore, the specific form of the total value function of prospect theory is as follows Tversky and Kahneman (1992): (7) Λ ( x j ) = ∑ j = 1 m λ ( x j ) w ( p j ) , where m is the number of attributes for alternatives; j expresses the jth attribute; λ ( x j ) reflects the DMs’ subjective feeling value according to the actual gains or the losses. The formula of λ ( x j ) is defined as follows Tversky and Kahneman (1992): (8) λ ( x j ) = ( Δ x ) τ , if Δ x ⩾ 0 , − θ ( − Δ x ) ς , if Δ x < 0 , 0 < τ , ς < 1 .

In this formula, x j signifies the value of the jth attribute. Δ > 00$]]> signifies the gain, Δ x < 0 signifies the loss and Δ x = 0 signifies no gain or loss. τ and ς represent the parameters of risk attitudes. These depict the sensitivity of decision makers to gains and losses. θ depicts the coefficient of loss aversion, and θ > 11$]]>. The higher the value of θ, the more risk averse the decision maker is. Tversky and the others obtained the parameters of the value function in CPT with the method of linear regression when the parameters were τ = ς = 0.88, θ = 2.25. It was more consistent with the empirical data. And it is worth mentioning that we take the probabilistic linguistic border approximation area as the reference point to gains and losses in this paper.

See Fig. 1.

Step 1. Transform cost attributes into beneficial ones.

Given an LTS L = { ζ ƛ | ƛ = − ∂ , … , − 1 , 0 , 1 , … ∂ }, and transform the cost attribute ζ ƛ into the beneficial attribute ζ − ƛ .

Step 2. Shift the value of evaluation information L = { ζ i j k | k = 1 , 2 , … , q , i = 1 , 2 , … , m , j = 1 , 2 , … , n } into PLTSs PL = { ζ i j ( γ ) ( p i j ( γ ) ) | γ = 1 , 2 , … , # L i j ( p ) }. Build the evaluation matrix M = ( PL i j ( p ) ) m × n , PL i j ( p ) = { ζ i j ( γ ) ( p i j ( γ ) ) | γ = 1 , 2 , … , # L i j ( p ) } ( i = 1 , 2 , … , m , j = 1 , 2 , … , n ).

Step 3. Obtain the normalized decision matrix NM = ( NPL i j ( p ˜ ) ) m × n with PLTSs, NPL i j ( p ˜ ) = { ζ i j ( γ ) ( p ˜ i j ( γ ) ) | γ = 1 , 2 , … , # L i j ( p ˜ ) , ∑ γ = 1 # L i j ( p ˜ ) p ˜ i j ( γ ) = 1 } ( i = 1 , 2 , … , m , j = 1 , 2 , … , n ).

Step 4. Figure up the combined weight of attributes.

We acquire the objective weight by the entropy method and get the subjective weight given by decision makers.

First, we introduce the specific procedure of the entropy method. It should be noted that when calculating entropy, we use the mean value of the attribute as the reference point to calculate its distance from the normalized attribute value.

Then, we calculate the combined weights by using the following equation. The advantage of using combined weight is that the influence of subjective weight and objective weight can be considered comprehensively.

Decision makers gave the subjective weights w s j = ( w s 1 , w s 2 , … , w s n ), where w s j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n w s j = 1. The objective weight w o j = ( w o 1 , w o 2 , … , w o n ) is calculated by using Eq. (12) directly, where w o j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n w o j = 1. Therefore, the combined weights of attributes w c j = ( w c 1 , w c 2 , … , w c n ) could be defined: (13) w c j = w o j ∗ w s j ∑ j = 1 n w o j ∗ w s j , where w c j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n w c j = 1.

Step 5. Figure out the probabilistic linguistic border approximation area (PLBAA) matrix PLBAA = ( PLBAA j ) 1 × n . The PLBAA could be obtained according to Eqs. (14)–(16). (14) PLBAA = ( PLBAA j ) 1 × n , (15) PLBAA j = { ζ ↔ j ( γ ) ( p ↔ j ( γ ) ) | γ = 1 , 2 , … , # NL i j ( p ↔ ) } , (16) ζ ↔ j ( γ ) ( p ↔ j ( γ ) ) = κ − 1 ( ∏ i = 1 m κ ( ζ i j ( γ ) ) m ) ( ∏ i = 1 m p i j ( γ ) m ) .

Step 6. Figure up the Hamming distance from PLBAA by Eq. (17) and the cumulative prospect distance matrix by using Eq. (18). (17) d ( NL i j ( p ˜ ) , PLBAA j ) = ( ∑ γ = 1 # NL i j ( p ˜ ) | κ ( ζ i j ( γ ) ) ( p i j ( γ ) ) − κ ( ζ ↔ j ( γ ) ) ( p ↔ j ( γ ) ) | ) / # NL i j ( p ˜ ) , (18) Λ i j = [ d ( NL i j ( p ˜ ) , PLBAA j ) ] τ w j , if ( ∑ γ = 1 # NL i j ( p ˜ ) κ ( ζ i j ( γ ) ) ( p i j ( γ ) ) − κ ( ζ ↔ j ( γ ) ) ( p ↔ j ( γ ) ) ) / # NL i j ( p ˜ ) > 0 , 0 , if ( ∑ γ = 1 # NL i j ( p ˜ ) κ ( ζ i j ( γ ) ) ( p i j ( γ ) ) − κ ( ζ ↔ j ( γ ) ) ( p ↔ j ( γ ) ) ) / # NL i j ( p ˜ ) = 0 , − θ [ d ( NL i j ( p ˜ ) , PLBAA j ) ] ς w j , if ( ∑ γ = 1 # NL i j ( p ˜ ) κ ( ζ i j ( γ ) ) ( p i j ( γ ) ) − κ ( ζ ↔ j ( γ ) ) ( p ↔ j ( γ ) ) ) / # NL i j ( p ˜ ) < 0 . 0,\\ {} 0,\\ {} \hspace{1em}\text{if}\hspace{2.5pt}\big({\textstyle\textstyle\sum _{\gamma =1}^{\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})}}\kappa \big({\zeta _{ij}^{(\gamma )}}\big)\big({p_{ij}^{(\gamma )}}\big)-\kappa \big({\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}\big)\big({\stackrel{\leftrightarrow }{p}_{j}^{(\gamma )}}\big)\big)\big/\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})=0,\\ {} -\theta {[d({\textit{NL}_{ij}}(\tilde{p}),{\textit{PLBAA}_{j}})]^{\varsigma }}{w_{j}},\\ {} \hspace{1em}\text{if}\hspace{2.5pt}\big({\textstyle\textstyle\sum _{\gamma =1}^{\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})}}\kappa \big({\zeta _{ij}^{(\gamma )}}\big)\big({p_{ij}^{(\gamma )}}\big)-\kappa \big({\stackrel{\leftrightarrow }{\zeta }_{j}^{(\gamma )}}\big)\big({\stackrel{\leftrightarrow }{p}_{j}^{(\gamma )}}\big)\big)\big/\mathrm{\# }{\textit{NL}_{ij}}(\tilde{p})<0.\end{array}\right.\end{aligned}\]]]>

We take PLBAA j as reference point and the parameters in value function are τ = ς = 0.88, θ = 2.25.

Step 7. Calculate the probabilistic linguistic total prospect value. (19) Λ i ∗ = ∑ j = 1 n Λ i j , i = 1 , 2 , … , m .

Step 8. Rank the value of Λ i ∗ ( i = 1 , 2 , … , m ) to obtain the best alternative.

In the background of e-commerce, online shopping has become a very common way of consumption in people’s lives. The recent epidemic situation also makes people more accustomed to using online shopping for consumption. Therefore, express delivery has become a matter of great concern. Reasonable and effective site selection can improve the service quality and win the favour of customers, so as to achieve the all-win goal of Express Distribution Centre, businesses and consumers. The express industry is the product of rapid economic development. It means that the consumption capacity of a region has a crucial impact on the express business volume. How to choose a proper express distribution centre is of great importance to both express companies and consumers. Generally speaking, the more developed the region is, the more its express business volume will be, and the number of distribution centres will be more and more centralized. Additionally, the site selection also needs to consider the local population and demand. For example, the closer to the central business district, the denser the population, the more demand for express delivery. Besides, for the areas where e-commerce self-employed households are concentrated, the demand for express delivery is very large, which should be considered as the key object. Moreover, to choose a reasonable address for the express delivery centre, we must consider the problem of cost minimization. And convenient transportation can effectively ensure the timeliness of express delivery. Otherwise, in order to ensure service quality, express enterprises can only add more outlets or send more vehicles, no matter which way it will lead to increased costs. We know that choosing the best address of Express Distribution Centre is a classic MADM or MAGDM problem. In this case, we gave five alternative sites ℜ i ( i = 1 , 2 , 3 , 4 , 5 ) to choose from. The experts selected four attributes to consider five alternative addresses: (1) ℑ 1 is the degree of economic development; (2) ℑ 2 is the construction and transportation cost; (3) ℑ 3 is the population status and demand; and (4) ℑ 4 is the degree of convenient transportation. These five potential addresses ℜ i ( i = 1 , 2 , 3 , 4 , 5 ) must be evaluated by using the LTSs: { ζ − 3 = deeply inapposite ( DI ) , ζ − 2 = so inapposite ( SI ) , ζ − 1 = inapposite ( I ) ζ 0 = middling ( M ) , ζ 1 = apposite ( A ) , ζ 2 = so apposite ( SA ) , ζ 3 = deeply apposite ( DA ) } . by the five DMs within the above four beneficial attributes. The evaluation information by using the LTSs is listed in Tables 1–5.

For the convenience of calculation, we first express the evaluation information given by experts by using linguistic term sets (Tables 6–10).

Then, we choose the most appropriate site fot the logistics distribution centre by using CPT-PL-MABAC method.

Step 1. Transform the cost attribute ℑ 2 into a beneficial one. If the cost attribute value is ζ λ , then transform it into beneficial attribute value ζ − λ (see Tables 11–15).