A parts supplier selection framework of mechanical manufacturing enterprise based on D-S evidence theory

Abstract

Parts supplier selection (PSS) is an important part of supply chain management of manufacturing enterprise. In the PSS process, the values of evaluation indicators are often uncertain and incomplete and the importance degrees of evaluation indicators are often instable. To solve this problem, a PSS framework of manufacturing enterprise based on D-S evidence theory is proposed. The indicator system for PSS is established, and the indicators are divided into three categories: quantitative, comprehensive qualitative and direct qualitative. The initial indicator values are processed by membership grade method to obtain the tendency degree. A two-order weighted D-S evidence theory model is constructed to evaluate the screened candidate suppliers. A manufacturing enterprise application case is given finally to illustrate the correctness and feasibility of the proposed framework.

Keywords

Parts supplier selection D-S evidence theory manufacturing enterprise membership grade tendency degree

Introduction

As an important content of raw material purchasing decision of manufacturing enterprise, parts supplier selection (PSS) is a key link in the supply chain management. For most manufacturing enterprises, more than 60% of sales are used to purchase raw materials, spare parts and accessories, etc. Moreover, the procurement cost of a product accounts for more than 70% of the total.^1–3 Based on PSS, manufacturing enterprise can establish a long-term close relationship with several excellent parts suppliers. As a result, procurement risk can be effectively reduced and enterprise profit can be maximized.

There are significant works on PSS as seen in the literature. Much of the researches mainly focus on two aspects as follows. The first is the standard and criterion of PSS. A review of researches on the PSS system, which is made by Boer et al.⁴ covers all stages of PSS including prequalification, evaluation criteria construction and final evaluation.⁴ Shahadat designed an empirical study scheme for PSS. In this scheme, the frequently used evaluation criteria for developing countries in the PSS process are presented based on the data analysis result.⁵ Ho et al.⁶ made an in-depth analysis of 78 studies on multi-criteria PSS problem, summarized the effectiveness and shortcomings of the multi-criteria decision-making method, and explored the evaluation criteria with high attention. The second is the approach and model of PSS. The researches on this aspect mainly focus on multi-criteria decision-making,^7,8 cost-based approach,⁹ empirical conceptual approach,¹⁰ fuzzy set theory,^11,12 mathematical programming,^13,14 artificial intelligence method^15,16 and combination of above methods.^17,18

Reasonable selection of parts supplier will directly affect the production cost and competitiveness of manufacturing enterprise.^19–21 Because of the uncertainty of the market, the information asymmetry among cooperative enterprises and other random factors, manufacturing enterprise is facing great risks in the process of parts supplier evaluation. The indicators involved in parts supplier evaluation of manufacturing enterprise are very extensive. On the one hand, due to the limitations in the decision-maker’s understanding of the parts supplier market, the value of evaluation indicator obtained by the decision-maker is often uncertain and incomplete. On the other, the importance degree of evaluation indicator is clearly not fixed in the case of different needs or different preferences.

D-S evidence theory, also referred to as evidence theory or theory of belief functions, is a general framework for reasoning with uncertainty. First introduced by Dempster²² in the context of statistical inference, the theory was later developed by Glenn Shafer into a general framework for modeling epistemic uncertainty—a mathematical theory of evidence.^23,24 D-S evidence theory is powerful in quantifying the uncertain information in many systems. The research on D-S evidence theory has been developing rapidly in recent years. Beynon et al.²⁵ described the potential offered by D-S evidence theory as a promising improvement on “traditional” approaches to decision analysis. Zhang et al.²⁶ proposed a four-step reliability analysis reliability framework using evidential network to deal with the estimation problem of the system reliability from text-based records. Wu²⁷ used grey-related analysis and D-S evidence theory to deal with the fuzzy group decision making problem of supplier selection. Merigo et al.²⁸ developed a new approach for decision making with D-S evidence theory by using linguistic information. Zhang et al.²⁹ proposed a supplier selection method combining analytic network process with D-S evidence theory. It has been proved that a satisfactory result can be achieved and the uncertainty of decision-making can be reduced by the D-S evidence theory-based approach.^25–29 In summary, D-S evidence theory can effectively deal with multi-criteria decision-making problems with uncertain and incomplete information. Considering the preference of the decision makers to the evaluation indicator, the weight of the evaluation indicator should be introduced into the evidence theory.

Therefore, aiming to the problems of uncertain and incomplete indicator value and instable indicator importance degree, a PSS framework of manufacturing enterprise based on D-S evidence theory and support vector machine is proposed in this paper. Firstly, we establish the indicator system for PSS and divide all indicators into quantitative type, comprehensive qualitative type and direct qualitative type. Secondly, we convert the indicator values from initial values to tendency degrees by membership grade method. Lastly, we construct a two-order weighted D-S evidence theory model to evaluate the candidate parts suppliers.

PSS indicator system

A scientific, holonomic and comprehensive indicator system must be established in PSS. For a parts supplier, its product is the capability representation. Meanwhile, the diathesis of parts supplier can provide strong support to its product. Here, the diathesis of parts supplier mainly includes internal competitiveness, external competitiveness and cooperation ability. Internal competitiveness could be subdivided into innovation ability, manufacturing capacity and agility. Furthermore, a parts supplier is not isolated and is inevitably restricted by the external competitiveness. The external competitiveness mainly includes economic environment, geographical environment, social environment and legal environment. The cooperation ability between manufacturing enterprises and parts suppliers is also affected by technical compatibility degree, cultural compatibility degree, information platform compatibility degree and reputation.

Based on the above analysis, we establish the PSS indicator system shown in Figure 1. The total target of PSS includes four criteria: product competitiveness (I_ProComp), internal competitiveness (I_InComp), external competitiveness (I_ExComp) and cooperation ability (I_CoopAbil). I_InComp, I_ExComp and I_CoopAbil belong to the quantitative type and their value can be obtained by expert scoring after comprehensive consideration of refined factors. The value of I_ProComp is difficult to determine, so it belongs to comprehensive qualitative type. We decompose I_ProComp into four sub-criteria: price level (I_Price), quality level (I_Quality), service level (I_Service) and flexibility level (I_Flexibility). I_Price and I_Quality belong to quantitative type, and I_Service and I_Flexibility belong to direct qualitative type. Therefore, the PSS indicator system is represented as: IndSys = {I_ProComp, I_InComp, I_ExComp, I_CoopAbil} and I_ProComp = {I_Price, I_Quality, I_Service, I_Flexibility}.

Figure 1.

The PSS indicator system. PSS: parts supplier selection.

PSS framework based on D-S evidence theory

PSS decision rules based on D-S evidence theory

PSS is an important part of supply chain management of manufacturing enterprise.

According to D-S evidence theory, the set of candidate parts suppliers to be evaluated in PSS process is defined as the identification framework $Θ = {x_{1}, x_{2}, …, x_{i}, …, x_{N}}$ . All possible sets in $Θ$ are represented by the power set $2^{Θ}$ . When the number of elements in $Θ$ is N and the elements in $Θ$ are incompatible with each other, the number of elements in the power set $2^{Θ}$ is $2^{N}$ . Then we define a set function $m : 2^{Θ} \to [0, 1]$ which satisfies $m (φ) = 0$ and $\sum_{A \subset Θ} m (A) = 1$ . The function $m$ is described as the basic probability distribution function on the PSS identification framework $Θ$ . Here, A is a PSS scheme and $m (A)$ is the basic probability distribution value of A which represents the degree of trust in A. The PSS schemes satisfying the condition $m (A) > 0$ are called the focal elements. For $\forall A \subseteq Θ$ , the fusion rule of finite basic probability distribution functions on the PSS identification framework $Θ$ is

\begin{array}{l} (m_{1} \oplus m_{2} \oplus … \oplus m_{n}) (A) \\ = \frac{1}{K} \sum_{A_{1} \cap A_{2} \cap … \cap A_{n} = A} m_{1} (A_{1}) \cdot m_{2} (A_{2}) \cdot … \cdot m_{n} (A_{n}) \end{array}

(1)

where

K

is the normalization constant

\begin{array}{l} K = \sum_{A_{1} \cap A_{2} \cap … \cap A_{n} \neq φ} m_{1} (A_{1}) \cdot m_{2} (A_{2}) \cdot … \cdot m_{n} (A_{n}) \\ = 1 - \sum_{A_{1} \cap A_{2} \cap … \cap A_{n} = φ} m_{1} (A_{1}) \cdot m_{2} (A_{2}) \cdot … \cdot m_{n} (A_{n}) \end{array}

(2)

Based on D-S evidence theory, the related concepts of decision rules in PSS process are defined as follows:

Definition 1: Belief function. On the PSS identification framework $Θ$ , the total trust degree of A can be expressed as the belief function $B e l (A)$

B e l (A) = \sum_{B \subseteq A} m (B)

(3)

where $B \subset Θ$ and $B e l (A)$ represent the sum of the possibility measurement for all subsets of A.

Definition 2: Plausible function. On the PSS identification framework $Θ$ , the uncertainty degree of A can be expressed as the plausible function $P l (A)$

P l (A) = \sum_{B \cap A \neq φ} m (B)

(4)

where $B \subset Θ$ and $P l (A)$ represent the sum of the uncertainty measurement for all subsets of A.

Definition 3: Trust interval. For a PSS scheme $A$ on the PSS identification framework $Θ$ , the belief function $B e l (A)$ and the plausible function $P l (A)$ are obtained according to equations (3) and (4). The confirmation degree of the PSS scheme $A$ is represented by the trust interval $[B e l (A), P l (A)]$

According to Definitions 1 and 2, $B e l (A)$ reflects the sum of exact reliability of the evidence which support the PSS scheme $A$ and $P l (A)$ reflects the sum of reliability of the evidence which does not negate the PSS scheme $A$ . Therefore, $[B e l (A), P l (A)]$ forms the trust interval in which the evidence supports the PSS scheme $A$ . $B e l (A)$ and $P l (A)$ can be considered as the minimum and maximum probability bounds, respectively, as shown in Figure 2.

Figure 2.

The meaning of trust interval.

The trust interval comprehensively reflects the support degree to the PSS scheme of the belief function and the plausible function. It is more reliable to evaluate the PSS scheme by trust interval method than maximum belief function decision method or maximum plausible function decision method.^30,31 So, we use the multi-attribute decision making based on trust interval to evaluate the PSS scheme.

Since the trust interval of a PSS scheme is an interval number, how to rank the PSS schemes by their interval numbers becomes a key problem. A simple yet practical and more rational preference ranking method of interval numbers is developed which makes no assumption and makes no use of the midpoints of interval numbers. Let a=[a₁, a₂] and b= [b₁, b₂] be two interval numbers, whose possible relationships are shown in Figure 3.

Figure 3.

Possible relationships between two interval numbers.

The degree of one interval number utility being greater than another one is defined as the preference degree. Accordingly, the preference degree of a over b (P(a > b)) is

P (a > b) = \frac{\max (0, a_{2} - b_{1}) - \max (0, a_{1} - b_{2})}{(a_{2} - a_{1}) + (b_{2} - b_{1})}

(5)

In the same way, the preference degree of b over a (P(b > a)) is

P (b > a) = \frac{\max (0, b_{2} - a_{1}) - \max (0, b_{1} - a_{2})}{(a_{2} - a_{1}) + (b_{2} - b_{1})}

(6)

It is obvious that P(a > b)+ P(b > a)=1 and $P (a > b) = P (b > a) \equiv 0.5$ when a = b.

If P(a > b)>P(b > a), then a is said to be superior to b to the degree of P(a > b), which is denoted by $a \overset{P (a > b)}{≻} b$ ; if P(a > b)=P(b > a)=0.5, then a is said to be equal to b, which is denoted by $a \sim b$ ; if P(b > a)>P(a > b), then a is said to be inferior to b to the degree of P(b > a), which is denoted by $a \overset{P (b > a)}{≺} b$ .

According to the property of possible relationships between two interval numbers mentioned above, the preference ranking rules based on trust interval to evaluate the PSS scheme are as follows.

We define that the PSS scheme $A_{i}$ is superior to the PSS scheme $A_{j}$ with a degree of $P (A_{i} > A_{j})$ . The trust intervals of $A_{i}$ and $A_{j}$ are $[B e l (A_{i}), P l (A_{i})]$ and $[B e l (A_{j}), P l (A_{j})]$ respectively, so $P (A_{i} > A_{j})$ can be represented as

P (A_{i} > A_{j}) = \frac{{\begin{matrix} \max [0, P l (A_{i}) - B e l (A_{j})] \\ - \max [0, B e l (A_{i}) - P l (A_{j})] \end{matrix}}}{[P l (A_{i}) - B e l (A_{i})] + [P l (A_{j}) - B e l (A_{j})]}

(7)

where $P (A_{i} > A_{j}) \in [0, 1]$ .

2. The preference ranking rules of PSS schemes are defined as: If $P (A_{i} > A_{j}) > 0.5$ , $A_{i}$ is superior to $A_{j}$ , which is denoted by $A_{i} ≻ A_{j}$ ; if $P (A_{i} > A_{j}) = 0.5$ , $A_{i}$ is equal to $A_{j}$ , which is denoted by $A_{i} \sim A_{j}$ ; if $P (A_{i} > A_{j}) < 0.5$ , $A_{i}$ is inferior to $A_{j}$ , which is denoted by $A_{i} ≺ A_{j}$ ; if $P (A_{i} > A_{j}) > 0.5$ and $P (A_{j} > A_{k}) > 0.5$ , $A_{i}$ is superior to $A_{k}$ , which is denoted by $A_{i} > A_{j} > A_{k}$ .

Processing of evidence

The acquirement of evidence is the key link of applying D-S evidence theory to evaluate the PSS schemes and make decision. The weighted basic probability distribution value ${\tilde{m}}_{t} (A_{i})$ of the focal element $A_{i}$ ( $i = 1, 2, …, l$ and $l < 2^{N}$ ) under the indicator $t$ ( $t \in I n d S y s$ ) is used as the evidence input of PSS. The calculation methods for the weighted basic probability distribution values of each focal element are given below.

After examining the actual situation of each candidate parts supplier, the decision-maker gives the initial value of the quantitative type and direct qualitative type indicators. The definite quantitative type and direct qualitative type indicators are given the exact value, the relatively fuzzy quantitative type indicator is assigned a value interval, and the null value is given to the completely unknown indicator. The tendency degree of the initial indicator value is calculated by the method of membership degree. For the initial indicator value, 5-level comment is set as: { $G_{1}$ , $G_{2}$ , $G_{3}$ , $G_{4}$ , $G_{5}$ }={very bad, bad, medium, good, very good}. Here, $G_{1}$ and $G_{5}$ are, respectively, the comment grade corresponding to the lowest initial value $D_{1}$ and the highest initial value $D_{5}$ of an indicator. Specially, if the value of I_Price is lower, it is better, so the correspondence is exactly reversed.

It is assumed that the corresponding numerical numbers for each rating level are $E (G_{1}) = 0$ , $E (G_{2}) = 0.25$ , $E (G_{3}) = 0.5$ , $E (G_{4}) = 0.75$ and $E (G_{5}) = 1$ . $β_{h}$ is the membership degree of the initial indicator value to the rating level $G_{h}$ . The tendency degree of the focal element $A_{i}$ under the indicator $t$ is represented as $P_{t} (A_{i})$ , and it can be determined according to two different conditions:

Quantitative type indicator. If the initial indicator value of the focal element $A_{i}$ under the indicator $t$ is an exact value, the tendency degree of the initial indicator value can be directly determined. If the initial indicator value of the focal element $A_{i}$ under the indicator $t$ is a value interval $[a, b]$ , there are three cases: when $D_{h} \leq a \leq b \leq D_{h + 1}$ , $P_{t} (A_{i}) = β_{h} \cdot E (G_{h}) + β_{h + 1} \cdot E (G_{h + 1})$ ; when $D_{h} \leq a \leq D_{h + 1}$ and $D_{h + 1} \leq b \leq D_{h + 2}$ , $P_{t} (A_{i}) = β_{h} \cdot E (G_{h}) + β_{h + 1} \cdot E (G_{h + 1}) + β_{h + 2} \cdot E (G_{h + 2})$ ; when $D_{h} \leq a \leq D_{h + 1}$ and $D_{j} \leq b \leq D_{j + 1}$ , $P_{t} (A_{i}) = β_{h} \cdot E (G_{h}) + ··· + β_{j} \cdot E (G_{j + 1})$ .

Direct qualitative type indicator. $P_{t} (A_{i}) = E (G_{i})$ .

By the above definition and method, the tendency degree of every focal element except $Θ$ under the PSS indicator can be solved. Here, $Θ$ can indicate the uncertainty of an indicator by decision makers. If it is not considered, the decision problem is a simple probability distribution problem and the advantages of D-S evidence theory in dealing with multi-indicator decision making problems are not demonstrated.

Meanwhile, the trust degree of decision-maker in each indicator is different and the uncertainty of indicator is reflected by the probability distribution. Thus, the probability distribution value of $Θ$ under different indicators should also be treated differently. Taking PSS problem as an example, the degree of importance of the evaluation indicators is obviously not fixed in the case of different requirements. When costs need to be reduced, I_Price is more important than other indicators and the degree of trust to it must be higher than other indicators. The basic probability distribution value of $Θ$ under I_Price should be smaller than under other indicators. Based on this, the weight of each indicator is introduced to adjust the preference of different decision attributes and solve the probability distribution problem of $Θ$ under different indicators. So the weighted basic probability distribution value of each focal element under any indicator is obtained as ${\tilde{m}}_{t} (A_{i})$ .

The weight of the indicator $t$ ( $t \in I n d S y s$ ) is assumed as $ω_{t}$ ( $ω_{t} \in (0, 1)$ ) which depends on the preference of the decision maker. The bigger the weight value is, the higher the preference degree of the decision maker to this indicator is. Thus, we take a weighted normalization for the basic probability distribution value of all focal elements as follows

{\begin{array}{l} {\tilde{m}}_{t} (A_{i}) = \frac{ω_{t} P_{t} (A_{i})}{\sum_{i = 1}^{l - 1} P_{t} (A_{i})} & A_{i} \neq Θ \\ {\tilde{m}}_{t} (A_{i}) = 1 - ω_{t} & A_{i} = Θ \end{array}

(8)

PSS process

The PSS indicator system shown in Figure 1 mainly has two layers: criteria layer {I_ProComp, I_InComp, I_ExComp, I_CoopAbil} and sub-criteria layer {I_Price, I_Quality, I_Service, I_Flexibility}. Therefore, we construct a two-order weighted D-S evidence theory model to evaluate the candidate parts suppliers. The evidences of the sub-criteria {I_Price, I_Quality, I_Service, I_Flexibility} of I_ProComp are fused and processed, and the weighted basic probability distribution value of I_ProComp is obtained as ${\tilde{m}}_{{P r o C o m}_{p}} (A_{i})$ . Then, we carry out a secondary fusion which includes the weighted basic probability distribution value of I_ProComp, I_InComp, I_ExComp and I_CoopAbil, and the evaluation result of candidate parts suppliers can be obtained. The PSS process is as follows:

Step 1 Construct the set of candidate parts suppliers and give the initial value of the quantitative type and direct qualitative type indicators.

Step 2 Determine the weights of indicators according to the actual demand and the degree of preference.

Step 3 Set up the indicator evaluation level and its reference indicator value based on the initial indicator value.

Step 4 Calculate the membership degree of indicator value corresponding to the indicator evaluation level, and calculate the tendency degree $P_{t} (A_{i})$ of each focal element $A_{i}$ under each indicator $t$ , then calculate the weighted basic probability distribution value of candidate parts suppliers and $Θ$ under each indicator by equation (8).

Step 5 Take the weighted basic probability distribution value ${\tilde{m}}_{P r i c e} (A_{i})$ , ${\tilde{m}}_{Q u a l i t y} (A_{i})$ , ${\tilde{m}}_{S e r v i c e} (A_{i})$ and ${\tilde{m}}_{F l e x i b i l i t y} (A_{i})$ of the four sub-criteria under the criteria I_ProComp as the input evidences, carry out the first evidence fusion by equation (1), and calculate the basic probability distribution value $m_{P r o C o m p} (A_{i})$ of candidate parts suppliers and $Θ$ under the indicator I_ProComp.

Step 6 Normalize the basic probability distribution value $m_{P r o C o m p} (A_{i})$ of candidate parts suppliers and $Θ$ under the indicator I_ProComp and add the weight of I_ProComp, then obtain the weighted basic probability distribution value ${\tilde{m}}_{P r o C o m p} (A_{i})$ . Here, regard the basic probability distribution value $m_{P r o C o m p} (Θ)$ of the focal element $Θ$ obtained by the first evidence fusion as a basic probability distribution value of a $non - Θ$ focal element, and assume $m_{P r o C o m p} (Θ) = m_{P r o C o m p} (A_{l})$ . The way of normalizing and weighting is as follows

{\begin{cases} {\tilde{m}}_{P r o C o m p} (A_{i}) = \frac{ω_{P r o C o m p} m_{P r o C o m p} (A_{i})}{\sum_{i = 1}^{l} m_{P r o C o m p} (A_{i})} \\ {\tilde{m}}_{P r o C o m p} (Θ) = 1 - ω_{P r o C o m p} + {\tilde{m}}_{P r o C o m p} (A_{l}) \end{cases}

(9)

Step 7 Take ${\tilde{m}}_{P r o C o m p} (A_{i})$ , ${\tilde{m}}_{I n C o m p} (A_{i})$ , ${\tilde{m}}_{E x C o m p} (A_{i})$ and ${\tilde{m}}_{C o o p A b i l} (A_{i})$ as the input evidences, carry out the second evidence fusion by equation (1), and calculate the comprehensive basic probability distribution value $m (A_{i})$ of candidate parts suppliers.

Step 8 Calculate the belief function $B e l (A_{i})$ and plausible function $P l (A_{i})$ of all candidate parts suppliers, construct the trust interval $[B e l (A_{i}), P l (A_{i})]$ .

Step 9 Use the decision rules based on the trust interval and select the best parts suppliers.

Case study

Application of a bearing cage supplier selection

A bearing manufacturing enterprise need to select the best bearing cage supplier from three candidate suppliers. After examining the actual situation, the decision maker gives the initial indicator value of the candidate bearing cage suppliers as shown in Table 1.

Table 1.

The initial indicator value of candidate bearing cage suppliers.

Candidate bearing cage suppliers	I_ProComp				I_InComp/full score is 17	I_OutComp/full score is 9	I_CoopAbil/full score is 1
Candidate bearing cage suppliers	I_Price/Yuan	I_Quality/error value (mm)	I_Service	I_Flexibility	I_InComp/full score is 17	I_OutComp/full score is 9	I_CoopAbil/full score is 1
1	64	0.01	good	very bad	15	[8.5, 9]	0.9848
2	1900	0.01	very good	good	[15, 16]	[8.5, 9]	1
3	28000	0.03	very bad	good	[5, 9]	/	1

For the indicators of quantitative type I_Price, I_Quality, I_InComp, I_OutComp and I_CoopAbil, the reference indicator values corresponding to the rating level { $G_{1}$ , $G_{2}$ , $G_{3}$ , $G_{4}$ , $G_{5}$ }={very bad, bad, medium, good, very good} are as follows

$G^{P r i c e}$ ={100000, 10000, 1000, 100, 10}, $G^{Q u a l i t y}$ ={0.05, 0.04, 0.03, 0.02, 0.01}, $G^{I n C o m p}$ ={17, 13, 9, 5, 1}, $G^{O u t C o m p}$ ={1, 3, 5, 7, 9}, $G^{C o o p A b i l}$ ={0, 0.25, 0.5, 0.75, 1}.

The membership degree of each initial indicator value to each rating level is calculated and then the information in Table 1 can be converted into the form of rating level-membership degree as shown in Table 2.

Table 2.

The form of rating level-membership degree.

Candidate bearing cage suppliers	I_ProComp				I_InComp/full score is 17	I_OutComp/full score is 9	I_CoopAbil/full score is 1
Candidate bearing cage suppliers	I_Price/Yuan	I_Quality/error value (mm)	I_Service	I_Flexibility	I_InComp/full score is 17	I_OutComp/full score is 9	I_CoopAbil/full score is 1
1	$\begin{array}{l} G_{4}^{P r i c e}, 0.6 \\ G_{5}^{P r i c e}, 0.4 \end{array}$	$G_{5}^{Q u a l i t y}, 1$	$G_{4}^{S e r v i c e}, 1$	$G_{1}^{F l e x i b i l i t y}, 1$	$\begin{array}{l} G_{4}^{I n C o m p}, 0.5 \\ G_{5}^{I n C o m p}, 0.5 \end{array}$	$\begin{array}{l} G_{4}^{O u t C o m p}, 0.125 \\ G_{5}^{O u t C o m p}, 0.875 \end{array}$	$\begin{array}{l} G_{4}^{C o o p A b i l}, 0.061 \\ G_{5}^{C o o p A b i l}, 0.939 \end{array}$
2	$\begin{array}{l} G_{2}^{P r i c e}, 0.1 \\ G_{3}^{P r i c e}, 0.9 \end{array}$	$G_{5}^{Q u a l i t y}, 1$	$G_{5}^{S e r v i c e}, 1$	$G_{4}^{F l e x i b i l i t y}, 1$	$\begin{array}{l} G_{4}^{I n C o m p}, 0.625 \\ G_{5}^{I n C o m p}, 0.375 \end{array}$	$\begin{array}{l} G_{4}^{O u t C o m p}, 0.125 \\ G_{5}^{O u t C o m p}, 0.875 \end{array}$	$G_{5}^{C o o p A b i l}, 1$
3	$\begin{array}{l} G_{1}^{P r i c e}, 0.2 \\ G_{2}^{P r i c e}, 0.8 \end{array}$	$G_{3}^{Q u a l i t y}, 1$	$G_{1}^{S e r v i c e}, 1$	$G_{4}^{F l e x i b i l i t y}, 1$	$\begin{array}{l} G_{2}^{I n C o m p}, 0.5 \\ G_{3}^{I n C o m p}, 0.5 \end{array}$	/	$G_{5}^{C o o p A b i l}, 1$

Then the tendency degree is calculated as shown in Table 3.

Table 3.

The tendency degree.

Candidate bearing cage suppliers	I_ProComp				I_InComp/full score is 17	I_OutComp/full score is 9	I_CoopAbil/full score is 1
Candidate bearing cage suppliers	I_Price/Yuan	I_Quality/error value (mm)	I_Service	I_Flexibility	I_InComp/full score is 17	I_OutComp/full score is 9	I_CoopAbil/full score is 1
1	0.850	1.000	0.750	0	0.875	0.969	0.985
2	0.475	1.000	1.000	0.750	0.881	0.969	1
3	0.200	0.300	0	0.750	0.375	/	1

The set of candidate parts suppliers is defined as the identification framework: $Θ = {x_{1}, x_{2}, x_{3}}$ in which $x_{1}, x_{2}, x_{3}$ successively indicate the candidate bearing cage supplier 1, 2 and 3.

Considering the actual demand and the preference degree of decision maker, the weight vector of each indicator is determined as follows: ( $ω_{P r i c e}$ , $ω_{Q u a l i t y}$ , $ω_{S e r v i c e}$ , $ω_{F l e x i b i l i t y}$ )=(0.7, 0.9, 0.8, 0.9) and ( $ω_{P r o C o m p}$ , $ω_{I n C o m p}$ , $ω_{O u t C o m p}$ , $ω_{C o o p A b i l}$ )=(0.8, 0.8, 0.9, 0.7).

For the four sub-criteria I_Cost, I_Quality, I_Service and I_Flexibility under the criteria I_ProComp, the weighted basic probability distribution values of all focal elements are calculated based on the tendency degree of initial indicator values of $x_{1}, x_{2}, x_{3}$ and the weight vector ( $ω_{P r i c e}$ , $ω_{Q u a l i t y}$ , $ω_{S e r v i c e}$ , $ω_{F l e x i b i l i t y}$ )=(0.7, 0.9, 0.8, 0.9). The calculation result is as follows

I_Price: ${\tilde{m}}_{P r i c e} (x_{1}) = 0.390$ , ${\tilde{m}}_{P r i c e} (x_{2}) = 0.218$ , ${\tilde{m}}_{P r i c e} (x_{3}) = 0.092$ , ${\tilde{m}}_{P r i c e} (Θ)$ =0.300.

I_Quality: ${\tilde{m}}_{Q u a l i t y} (x_{1}, x_{2})$ =0.600, ${\tilde{m}}_{Q u a l i t y} (x_{3}) = 0.300,$ ${\tilde{m}}_{Q u a l i t y} (Θ)$ =0.100.

I_Service: ${\tilde{m}}_{S e r v i c e} (x_{1})$ =0.343, ${\tilde{m}}_{S e r v i c e} (x_{2})$ =0.457, ${\tilde{m}}_{S e r v i c e} (x_{3}) = 0$ , ${\tilde{m}}_{S e r v i c e} (Θ)$ =0.200.

I_Flexibility: ${\tilde{m}}_{F l e x i b i l i t y} (x_{1})$ =0, ${\tilde{m}}_{F l e x i b i l i t y} (x_{2}, x_{3}) = 0.900$ , ${\tilde{m}}_{F l e x i b i l i t y} (Θ)$ =0.100.

For the criteria I_InComp, I_OutComp and I_CoopAbil, the weighted basic probability distribution values of all focal elements are calculated based on the tendency degree of initial indicator values of $x_{1}, x_{2}, x_{3}$ and the weight vector ( $ω_{I n C o m p}$ , $ω_{O u t C o m p}$ , $ω_{C o o p A b i l}$ )=(0.8, 0.9, 0.7). The calculation result is as follows

I_InComp: ${\tilde{m}}_{I n C o m p} (x_{1})$ =0.334, ${\tilde{m}}_{I n C o m p} (x_{2})$ =0.323, ${\tilde{m}}_{I n C o m p} (x_{3})$ =0.143, ${\tilde{m}}_{I n C o m p} (Θ)$ =0.2000.

I_OutComp: ${\tilde{m}}_{O u t C o m p} (x_{1}, x_{2})$ =0.800, ${\tilde{m}}_{O u t C o m p} (Θ) = 0.200$ .

I_CoopAbil: ${\tilde{m}}_{C o o p A b i l} (y_{1})$ =0.347, ${\tilde{m}}_{C o o p A b i l} (x_{2}, x_{3}) = 0.353$ , ${\tilde{m}}_{C o o p A b i l} (Θ)$ =0.300.

Taking ${\tilde{m}}_{P r i c e} (A_{i})$ , ${\tilde{m}}_{Q u a l i t y} (A_{i})$ , ${\tilde{m}}_{S e r v i c e} (A_{i})$ and ${\tilde{m}}_{F l e x i b i l i t y} (A_{i})$ as the evidence input and carrying out the first evidence fusion, the basic probability distribution values $m_{P r o C o m p} (A_{i})$ of all focal elements are calculated as follows: $m_{P r o C o m p} (x_{1})$ =0.077, $m_{P r o C o m p} (x_{2}) = 0.801$ , $m_{P r o C o m p} (x_{3})$ =0.089, $m_{P r o C o m p} (x_{1}, x_{2})$ =0.012, $m_{P r o C o m p} (x_{2}, x_{3})$ =0.019 and $m_{P r o C o m p} (Θ)$ =0.002.

Normalizing the basic probability distribution values $m_{P r o C o m p} (A_{i})$ of candidate parts suppliers and $Θ$ under the indicator I_ProComp and adding the weight of I_ProComp, the weighted basic probability distribution values ${\tilde{m}}_{P r o C o m p} (A_{i})$ are calculated as follows: ${\tilde{m}}_{P r o C o m p} (x_{1}) = 0.069$ , ${\tilde{m}}_{P r o C o m p} (x_{2}) = 0.721$ , ${\tilde{m}}_{P r o C o m p} (x_{3}) = 0.080$ , ${\tilde{m}}_{P r o C o m p} (x_{1}, x_{2})$ =0.011, ${\tilde{m}}_{P r o C o m p} (x_{2}, x_{3})$ =0.017 and ${\tilde{m}}_{P r o C o m p} (Θ)$ =0.102.

Taking ${\tilde{m}}_{P r o C o m p} (A_{i})$ , ${\tilde{m}}_{I n C o m p} (A_{i})$ , ${\tilde{m}}_{O u t C o m p} (A_{i})$ and ${\tilde{m}}_{C o o p A b i l} (A_{i})$ as the evidence input and carrying out the second evidence fusion, the comprehensive basic probability distribution values $m (A_{i})$ of all focal elements are calculated as follows: $m (x_{1})$ =0.160, $m (x_{2})$ =0.798, $m (x_{3})$ =0.017, $m (x_{1}, x_{2})$ =0.016, $m (x_{2}, x_{3})$ =0.005 and $m (Θ)$ =0.004.

The belief function $B e l (A_{i})$ and plausible function $P l (A_{i})$ of all candidate parts suppliers are calculated and the trust interval $[B e l (A_{i}), P l (A_{i})]$ is determined as follows: $x_{1}$ : [0.160, 0.180], $x_{2}$ : [0.798, 0.823] and $x_{3}$ : [0.017, 0.026]. According to the multiple attribute decision rule based on trust interval, the evaluation result is $x_{2} ≻ x_{1} ≻ x_{3}$ and the best bearing cage supplier is the candidate bearing cage supplier 2.

Discussions

The manufacturing enterprise needs to consider many factors in the process of PSS. Especially in the context of low carbon green supply chain management, the selection and evaluation of parts suppliers are more dependent on the experience and knowledge of decision makers, making it more difficult for them to master the attributes of parts suppliers, so it is difficult to form a fixed experience mode. At the same time, the attribute values of the evaluation indicators are often uncertain and incomplete because of the lack of experience and the limitations of understanding. In addition, the decision maker has a preference for the evaluation indicator of the parts supplier, so the weight of the evaluation indicator should be considered.

D-S evidence theory has a strong ability to deal with uncertain and incomplete information decision problem. It uses a trust function rather than a probability as a measure and the trust function is established by restricting the probability of the event. This method does not have to accurately describe the probability of an event that is difficult to obtain. Based on the advantages of D-S evidence theory model and the analysis of PSS, we proposes a two-order weighted D-S evidence theory model in view of the characteristics of the two layer structure of the PSS indicator system.

For PSS, the decision-makers cannot fully grasp the attributes of the parts supplier, so they cannot give a complete and accurate evaluation indicator value. For indicators that can be directly determined, qualitative or quantitative attribute values can be given; for relatively vague quantitative attribute values, attribute value intervals can be set; for completely unknown attributes, they are completely empty. The model constructed in this paper does not need to give specific and accurate attribute values for each evaluation indicator. The bearing cage supplier selection application proves that the selection result can be obtained using this model without giving complete and accurate evaluation indicator information.

Compared with the model constructed in this paper, the classical fuzzy comprehensive evaluation method³² needs to evaluate each evaluation indicator separately, so as to determine the membership degree of each evaluation indicator to the rating level. This kind of evaluation method requires the decision-maker to make a comprehensive grasp of the attributes of the part supplier, which does not apply to PSS in reality under the background of low carbon green supply chain. After investing a lot of time and cost resources, the specific and accurate attribute values of the evaluation indicator are obtained. Then PSS problem is dealt with based on the classical fuzzy comprehensive evaluation method, and the result is the same as the model constructed in this paper. Therefore, the correctness and feasibility of the model established in this paper have been verified.

Conclusions

Under the uncertain and incomplete information background, a PSS framework of manufacturing enterprise based on D-S evidence theory is proposed aiming at the problem of uncertain and incomplete indicator value and instable indicator importance degree. Compared with traditional PSS mainly depending on decision makers’ experience, the proposed framework establishes the PSS indicator system with the overall consideration of the main factors influencing PSS. The scientificalness and rationality of PSS are improved by D-S evidence theory. The case study shows that mechanical manufacturing enterprise can select the best parts supplier by the proposed PSS framework. By comparing with the classical fuzzy comprehensive evaluation method, it can be found that the proposed PSS framework of manufacturing enterprise can get the correct PSS result when the evaluation indicator information is incomplete and inaccurate, which can save time and cost resources for manufacturing enterprises.

The proposed PSS framework of manufacturing enterprise can provide a solution for similar multi attribute decision making problems with incomplete and inaccurate evaluation indicator information. The establishment of PSS indicator system has a decisive influence on the PSS result. We can do further study under the background of green supply chain, supply chain globalization, intelligent manufacturing, etc. With the fully consideration of PSS characteristics, the PSS indicator system can be modified and ultimately improve the credibility of PSS result.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Natural Science Foundation of Ningxia (NZ17113).

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