# Location Decision-Making of Equipment Manufacturing Enterprise under Dual-Channel Purchase and Sale Mode

Authors: Wei Xu; Lina Liu; Qingshan Zhang; Peng Liu
Journal: Complexity
 (2018)

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## Abstract

Based on the current status of the development of equipment manufacturing enterprises, the main influencing factors and main evaluation indicators of the equipment manufacturing enterprises in the context of dual-channel marketing are analyzed and determined. According to the organizational layout and transaction form of equipment manufacturing enterprises, the problem of equipment manufacturing enterprise location decision-making with online channel and offline channel trading mode is divided into two categories. One is the single-plant location decision problem, and the other is the multiplant location decision problem. In the single-plant location decision problem, we have established the conceptual model and mathematical model. Based on prospect theory, we use multicriteria decision-making method to quantify and standardize the evaluation index and determine the reference value, and the priority of the scheme is determined according to the final comprehensive value. In the multiplant location decision problem, we have established the conceptual model and the logical model and established the mathematical model with the goal of minimizing the cost and time. Finally, an example is given to solve the location problem of single plant and multiple plants. The feasibility of the model is proved by the solution and verification.


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## Body

## 1. Introduction

The rapid change of information technology promotes the intelligent development of manufacturing industry. For example, the United States proposes the “Industrial Internet", the EU plans to build “Future Factory", Germany is leading revolution “Industry 4.0", and China plans for “Made in China 2025". In the past decade, China’s manufacturing industry has made remarkable progress in high-speed railways, construction equipment, communications, and drones. At the same time, it has made a leading position in the global market. As the main force for national industrial development and the development of economy, technology, and national defense, equipment manufacturing is the foundation for the country to become stronger. At present, the marketing concept has changed from “market-centered" to “customer-centered". Equipment manufacturing enterprises must be transformed and upgraded in time to meet the diversified demand of customers. In addition to the appearance, configuration, and other aspects of the product, the diversified demands of customers also have different requirements for service quality, such as product delivery methods and delivery time. For equipment manufacturing enterprises, the online and offline purchasing and sales models will become a new trading channel because of its accurate and convenient characteristics, so as to meet the needs of the customer’s personalized transaction. Therefore, considering the difference between the dual-channel trading mode and the traditional transaction model, we put forward the problem of the location decision of the equipment manufacturing enterprise with dual-channel purchase and marketing. The purpose is to make the enterprise better configure the resources to meet and satisfy the new needs of the customers, and to ensure that the customers in each area can receive the products on time and completely.


## 2. Literature Review

The problem and theory of location decision originated from 1909. After more than 100 years of research and excavation by many scholars, now the location theory can be divided into the classical location theory such as the covering location theory [1], the center location theory [2], and the medium location theory [3]. The research on location has also gradually extended to the progressive coverage location problem [4], standby coverage location problem [5], stratified facilities location problem [6], and competitive location problem [7]. Scholars proposed specific solutions to different location problems, expanded the scope of application of the site selection theory, and guided practical problems, such as the location of factories, warehouses, and logistics distribution centers and the location of public facilities, hospitals, and emergency service stations. At present, the location theory has gradually matured, but scholars’ enthusiasm for location problem has not diminished. In particular, the current world structure is changing with each passing day, and the factors of production are more and more inclined to free flow in a wide range. With the gradual development of industrial 4, cloud services, and intelligent manufacturing, other advanced industrial revolutions have begun. Online and offline channels of mutual integration of trading mode gradually began to be adopted by many equipment manufacturing enterprises.In this paper, we combine the prospect theory with location theory to study the location decision-making of manufacturers with dual-channel procurement and sales. The prospect theory, which was proposed by Kahneman and Tversky in 1979, describes the risk decision-making behavior of people in two different situations, namely, “gain" and “lost" [8]. Unlike traditional expectation theory and expected utility theory, prospect theory is a multiple-attribute decision-making theory that incorporates psychology. It can better describe people’s decision-making behavior. Kahneman won the Nobel Prize in economics in 2002 for his theory of prospect. At present, prospect theory is used to solve all kinds of decision-making problems. Ding Q L. et al. [9] studied the interference management model of the logistics distribution system under the uncertain events, such as delivery address, customer time window, or demand, and determined the method of disturbance measurement based on the prospect theory. Gao S. et al. [10] applies the prospect theory to the decision-making of the power grid planning scheme. By constructing the index system, the value function, and the weight function, the comprehensive prospect value of each scheme is determined and the optimal scheme is selected. Considering the participation of competing products, Jiang Y P. et al. [11] decided to develop new product development plan based on prospect theory. And Chu H R. et. al. [12] studied the newsvendor model under three conditions: buyback contract, shortage penalty, and repurchase and penalty. Gao J W. et. al. [13] combined the prospect theory with interval valued intuitionistic fuzzy numbers and designed an interval valued intuitionistic fuzzy multiple criteria decision-making method based on prospect theory. Qinghe Chen et. al. [14] established the location decision model, according to the characteristics of the old-age health club, combined with the maximum coverage requirements of public facilities. And the satisfaction function was determined by using the foreground theory. Finally, they give an example and get a satisfactory solution through genetic algorithm. That paper is based on the satisfaction of the elderly, with the goal of unilateral satisfaction of one subject. We considered location decision-making on the view of suppliers, producers, online consumers, and offline consumers; there are four participants as decision-makers to get satisfaction. The location decision of the elderly health club is only considered about the cost of facility establishment, which is a static decision. In this paper, transportation costs of both suppliers and distributors are included, which is a dynamic decision.


## 3. Problems Description

### 3.1. Conceptual Model

The location of the factory is an important long-term decision. It not only directly determines the construction cost of the company, but also affects the expenses incurred in the future production and operation of the company, as well as the market development. And, once the decision is determined, it is difficult to change. There are two main issues that need to be resolved in the problem of factory location. The first is selection, which is to choose an area to build the factory. The second is addressing. That is, after selecting a region, specifically choose where to build a factory in the region. When making specific decisions, it is necessary to actually consider three kinds of situations, including the location of a new plant, the relocation of the original site, and the addition of several factories to expand production on the basis of the existing plant. In this paper, we assume that the entity logistics through the online exchange is a third-party logistics carrier. Businesses procure raw materials and sell customer order products through the cloud or website. The two parties implement the payment process through the Internet platform and determine the related responsibilities such as returning goods for replacement. The location of the plant is determined based on the company’s development status, expected goals, the degree of development of the dual-channel marketing model, and the location and purchase or sales volume of the identified and estimated suppliers and demand points. According to the organizational layout and transaction forms of the existing factories, the concept models for single-plant site selection and multiplant site selection were established, as shown in Figures1 and 2.Figure 1
Conceptual model of single-plant location.Figure 2
Conceptual model of multiplant location.


### 3.2. Model Parameters

Symbolic definition and parameter description:S: S=Si∣i=1,2,⋯,l: A set of l suppliers, where Si is the i-th supplier.P: P=Pj∣j=1,2,⋯,m: A set of m alternative plants, where Pj is the j-th alternative point.G: G=Gk∣k=1,2,⋯,n: A set of n demand points, where Gk is the k-th demand point.C P r: Production costs. CPr(i) is the production cost of the i-th plant. It is related to the production scale of the factory.U: U=Uj∣j=1,2,⋯,m: A set of production scales of m plants, where Uj is the production scale of the j-th plant.C T r: Transportation cost. It includes procurement transportation costs and sales transportation costs. Transportation costs are related to shipping rates and distances.C R t: Return cost. It includes returns losses that occur during the purchase and sale process. The return cost is related to the return rate.C: Total cost. It includes production costs, transportation costs, and return costs.Q i j: The amount of raw material provided by supplier Si to plant Pj.Q j k: The amount of products provided by plant Pj to demand point Gk.d i j: The distance between supplier Si and plant Pj.d j k: The distance between Pj and demand point Gk.D j 1: The maximum coverage radius for the j-th factory offline service.D j 2: The maximum coverage radius for the j-th factory online service. The service interval for online transactions is Dj1,Dj2.w: Freight rate. w1off,w2off,w1on,w2on, respectively, represent the freight rates for online purchase, online sale, offline purchase, and offline sale.y: Return probability (return rate). y1off,y2off,y1on,y2on, respectively, represent the return probability for online purchase, online sale, offline purchase, and offline sale.τ: Transportation time per unit distance. τ1off,τ2off,τ1on,τ2on, respectively, represent the transportation time per unit distance for online purchase, online sale, offline purchase, and offline sale.ζ: Unit return cost. ζ1off,ζ2off,ζ1on,ζ2on, respectively, represent the unit return cost for online purchase, online sale, offline purchase, and offline sale.z: Maximum number of factories.F ∗: F∗=F∗i∣i=1,2,⋯,n: A set of n first-level attributes. Fi is the i-th attributes, where F1,F2,⋯,Fn are independent of each other. We set F∗1=C1,C2,⋯,Cn1, F∗2=A1,A2,⋯,An2, F∗3=T1,T2,⋯,Tn3, F∗4=Q1,Q2,⋯,Qn4, F∗5=B1,B2,⋯,Bn5.F: A set of second-level attributes to F, F=F∗1∪F∗2∪F∗3∪F∗4∪F∗5=Fγ∣γ=1,2,⋯,k. Let Fjγ be the value of the γ-th attribute about the j-th alternative, and F′=Fjγ∣j=1,2,⋯,m;γ=1,2,⋯,k.W ∗: W∗=W∗i∣i=1,2,⋯,n: A set of the weight vectors of n first-level attributes, among which, W∗i is the weight of i-th first-level attribute. It satisfies W∗i≥0, ∑i=1nW∗i=1. W∗1=W∗C1,W∗C2,⋯,W∗Cn1 is a set the weight vectors of C1,C2,⋯,Cn1. The weights of A1,A2,⋯,An2, T1,T2,⋯,Tn3, Q1,Q2,⋯,Qn4, B1,B2,⋯,Bn5  are(1)W∗2=W∗T1,W∗T2,⋯,W∗Tn3,W∗3=W∗A1,W∗A2,⋯,W∗An2,W∗4=W∗Q1,W∗Q2,⋯,W∗Qn4,and∑i=1n1W∗Ci=1,∑i=1n2W∗Ti=1,∑i=1n3W∗Ai=1,∑i=1n4W∗Qi=1,∑i=1n5W∗Bi=1.W: W=W∗C∪W∗T∪W∗A∪W∗Q∪W∗B=wγ∣γ=1,2,⋯,k: A set of weights of k second-level attributes.At the same time, define the decision variables as follows.(2)xj:xj=1,alternativepointsareselectedxj=0,alternativepointsarenotselectedaij:aij=1,0≤dij≤Dj1aij=0,orelsebij:bij=1,Dj1≤dij≤Dj2bij=0,orelse


## 3.1. Conceptual Model

The location of the factory is an important long-term decision. It not only directly determines the construction cost of the company, but also affects the expenses incurred in the future production and operation of the company, as well as the market development. And, once the decision is determined, it is difficult to change. There are two main issues that need to be resolved in the problem of factory location. The first is selection, which is to choose an area to build the factory. The second is addressing. That is, after selecting a region, specifically choose where to build a factory in the region. When making specific decisions, it is necessary to actually consider three kinds of situations, including the location of a new plant, the relocation of the original site, and the addition of several factories to expand production on the basis of the existing plant. In this paper, we assume that the entity logistics through the online exchange is a third-party logistics carrier. Businesses procure raw materials and sell customer order products through the cloud or website. The two parties implement the payment process through the Internet platform and determine the related responsibilities such as returning goods for replacement. The location of the plant is determined based on the company’s development status, expected goals, the degree of development of the dual-channel marketing model, and the location and purchase or sales volume of the identified and estimated suppliers and demand points. According to the organizational layout and transaction forms of the existing factories, the concept models for single-plant site selection and multiplant site selection were established, as shown in Figures1 and 2.Figure 1
Conceptual model of single-plant location.Figure 2
Conceptual model of multiplant location.


## 3.2. Model Parameters

Symbolic definition and parameter description:S: S=Si∣i=1,2,⋯,l: A set of l suppliers, where Si is the i-th supplier.P: P=Pj∣j=1,2,⋯,m: A set of m alternative plants, where Pj is the j-th alternative point.G: G=Gk∣k=1,2,⋯,n: A set of n demand points, where Gk is the k-th demand point.C P r: Production costs. CPr(i) is the production cost of the i-th plant. It is related to the production scale of the factory.U: U=Uj∣j=1,2,⋯,m: A set of production scales of m plants, where Uj is the production scale of the j-th plant.C T r: Transportation cost. It includes procurement transportation costs and sales transportation costs. Transportation costs are related to shipping rates and distances.C R t: Return cost. It includes returns losses that occur during the purchase and sale process. The return cost is related to the return rate.C: Total cost. It includes production costs, transportation costs, and return costs.Q i j: The amount of raw material provided by supplier Si to plant Pj.Q j k: The amount of products provided by plant Pj to demand point Gk.d i j: The distance between supplier Si and plant Pj.d j k: The distance between Pj and demand point Gk.D j 1: The maximum coverage radius for the j-th factory offline service.D j 2: The maximum coverage radius for the j-th factory online service. The service interval for online transactions is Dj1,Dj2.w: Freight rate. w1off,w2off,w1on,w2on, respectively, represent the freight rates for online purchase, online sale, offline purchase, and offline sale.y: Return probability (return rate). y1off,y2off,y1on,y2on, respectively, represent the return probability for online purchase, online sale, offline purchase, and offline sale.τ: Transportation time per unit distance. τ1off,τ2off,τ1on,τ2on, respectively, represent the transportation time per unit distance for online purchase, online sale, offline purchase, and offline sale.ζ: Unit return cost. ζ1off,ζ2off,ζ1on,ζ2on, respectively, represent the unit return cost for online purchase, online sale, offline purchase, and offline sale.z: Maximum number of factories.F ∗: F∗=F∗i∣i=1,2,⋯,n: A set of n first-level attributes. Fi is the i-th attributes, where F1,F2,⋯,Fn are independent of each other. We set F∗1=C1,C2,⋯,Cn1, F∗2=A1,A2,⋯,An2, F∗3=T1,T2,⋯,Tn3, F∗4=Q1,Q2,⋯,Qn4, F∗5=B1,B2,⋯,Bn5.F: A set of second-level attributes to F, F=F∗1∪F∗2∪F∗3∪F∗4∪F∗5=Fγ∣γ=1,2,⋯,k. Let Fjγ be the value of the γ-th attribute about the j-th alternative, and F′=Fjγ∣j=1,2,⋯,m;γ=1,2,⋯,k.W ∗: W∗=W∗i∣i=1,2,⋯,n: A set of the weight vectors of n first-level attributes, among which, W∗i is the weight of i-th first-level attribute. It satisfies W∗i≥0, ∑i=1nW∗i=1. W∗1=W∗C1,W∗C2,⋯,W∗Cn1 is a set the weight vectors of C1,C2,⋯,Cn1. The weights of A1,A2,⋯,An2, T1,T2,⋯,Tn3, Q1,Q2,⋯,Qn4, B1,B2,⋯,Bn5  are(1)W∗2=W∗T1,W∗T2,⋯,W∗Tn3,W∗3=W∗A1,W∗A2,⋯,W∗An2,W∗4=W∗Q1,W∗Q2,⋯,W∗Qn4,and∑i=1n1W∗Ci=1,∑i=1n2W∗Ti=1,∑i=1n3W∗Ai=1,∑i=1n4W∗Qi=1,∑i=1n5W∗Bi=1.W: W=W∗C∪W∗T∪W∗A∪W∗Q∪W∗B=wγ∣γ=1,2,⋯,k: A set of weights of k second-level attributes.At the same time, define the decision variables as follows.(2)xj:xj=1,alternativepointsareselectedxj=0,alternativepointsarenotselectedaij:aij=1,0≤dij≤Dj1aij=0,orelsebij:bij=1,Dj1≤dij≤Dj2bij=0,orelse


## 4. The Decision-Making Model of Single-Plant Location

### 4.1. Determining the Evaluation Index

According to the characteristics of procurement, production, transportation, and sales of equipment manufacturing enterprises, the traditional operating model of equipment manufacturing enterprises is integrated with the Internet.In this context, based on the consensus of the previous literature research and the actual investigation of the manufacturers, we classified the factors affecting the location of equipment manufacturing enterprises as five types of influencing factors. In accordance with these five dimensions, 13 appraisal indicators with good applicability were determined, and the hierarchical structure model of the decision-making factors of equipment manufacturing enterprises was constructed, as shown in Figure3.Figure 3
The structural model of the location decision-making influencing factors.(1) Cost Factors. The main considerations are construction costs, knowledge costs, and production and operation costs. The construction cost includes land cost and construction cost; the knowledge cost includes network cost and information cost; and the production and operation cost mainly includes transportation expenses and warehousing expenses incurred during the production and operation process.(2) Time Factor. It mainly considers the transportation time of raw materials and the delivery time of products in the dual-channel procurement and sales process to ensure timely arrival of raw materials and timely delivery of products.(3) Regional Condition Factors. It mainly examines the geographical conditions, industrial conditions, and logistics conditions of the alternative areas. The geographical conditions mainly include road conditions and distances from airports and ports; the industrial conditions mainly examine whether the surrounding industrial environment is agglomeration and how the competition is; and the logistics conditions mainly include the degree of perfection of the logistics industry and whether it can be competent for the product distribution tasks of the manufacturers.(4) Quality of Service Factors. It mainly considers the quality of the rapid feedback of the manufacturers to the supplier in the alternative area and the quality of the rapid response to the customer.(5) External Environmental Factors. It mainly examines whether the regional policy conditions in the candidate areas are superior, whether the information environment is perfect and popular, and whether it destroys the natural environment and does not comply with green manufacturing.


### 4.2. Modeling

#### 4.2.1. Prospect Theory

Prospect theory is a theory describing and predicting the behavior that people are inconsistent with traditional expectation theory and expected utility theory in the process of facing risk decision-making. It is found that the risk preference behavior of people in the face of gains and losses is inconsistent. It becomes a risk pursuit in the face of “loss"; while facing “gain", the risk evasion can be avoided. The establishment and change of reference points affect people’s gain and loss feelings and then influence people’s decision-making. Figure4 is a value function diagram.Figure 4
Value function diagram.According to this value function, people show loss aversion; that is to say, people are more sensitive to losses than to gain. Kahneman and Tversky did such an experiment. They gave the participants a chance to flip a coin: Once the coin was turned up, the subject got 20 dollars, and if they face it up, the subject lost 10 dollars. Because of people’s loss aversion, most of the participants refused to participate in this game, even if they had an average expected return of 5 USD per coin toss (based on expected utility theory). People tend to avoid risk when they are faced with gain and have risk appetites when faced with losses. Kahneman and Tversky found that most people would choose a certain $800 gain instead of the 85% chance to get $1000, i.e., risk aversion; at the same time, when the participants face two choices: sure loss of $800 or 85% probability loss of $1000, most of the participants chose the latter, namely, risk appetite. Ossenbruggen and Hardman et al. [14] considered that the value function lacks the research logic and belongs to the phenomenological model; Tetlock [15] introduces social and emotional frameworks to the value function; Kahneman et al. [16] integrate Simon’s heuristic strategy research results and think that the heuristic strategies people used in decision-making are mainly representative heuristics, easy-to-get heuristics, and anchoring and adjustment heuristics.


#### 4.2.2. Determining the Objective of the Function

In this paper, we address the problem of single-plant site selection. Therefore, the influencing factors mentioned above are set as the main attributes affecting the satisfaction of the enterprise, suppliers, and customers, which is transformed into a multiattribute decision-making problem. We know that the evaluation language is different when evaluating different factors. According to different types of influencing factors, we adopt different methods to quantify the evaluation results. According to the classification of the evaluation results, the attributes of the influential factors affecting the satisfaction can be divided into phrase attributes and clear numerical attributes. For example, regional geographic conditions, regional industrial conditions, regional logistics conditions, service quality, and environmental factors can all be considered phrase-like attributes; construction costs, knowledge costs, and operating costs, as well as transit time, can be categorized as clear numerical attributes. The objective function based on the prospect theory can be expressed as follows.(3)max⁡VFjγ(4)VFjγ=∑j=1m∑γ=1kVFjγ+wFjγ+∑j=1m∑γ=1kVFjγ-wFjγ


#### 4.2.3. Solution Method

Step 1.
Assume that the location of suppliers and demand points is known, and select relatively satisfactory candidate points so that they can better cover the surrounding demand points. According to online channel satisfaction and offline channel satisfaction, each attribute value is calculated for each candidate point, and a program attribute value matrix table is listed, as shown in Table1.
(a) For numerical attributes, the evaluation result is the actual measured value.
(b) For the language-type attribute, this article uses the commonly used seven-level division method, phrase evaluation grade setV=Vv∣v=1,2,⋯,7, among which, V1=(extremelysatisfactory), V2=(greatersatisfaction), V3=(satisfactory), V4=(neutral), V5=(unsatisfaction), V6=(greaterunsatisfaction), and V7=(extremelyunsatisfactory) scales are used to evaluate the factors that manufacturers use to influence the phrase-type attributes of the candidate sites, through fuzzy trigonometric method to numerical illustration. The evaluation phrase can be converted into a numerical value by a triangular fuzzy formula. The calculation formula is as follows:(5)V^v=max⁡v-17,0,v7,min⁡v+17,1
whereVv is the level of the v-th phrase and V^v is the trigonometric fuzzy number of the v-th phrase level. Then there are(6)V^1=0,17,27,V^2=17,27,37,V^3=27,37,47,V^4=37,47,57,V^5=47,57,67,V^6=57,67,1,V^7=67,1,1.
(c) For interval-type attributes, the attribute value is represented by(a1,a2) or a1,a2.
It is assumed that there is an ordered property of fuzzy numbers for each factor [17]. For any Vv∈V,Vv-1∈V, if Vv>Vv-1, there is V^v≻Vv-1. Among them, “≻” means “bebetterthan”.Table 1
Offline channel, attribute evaluation value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 1575 1294 1159 1234 1468 F2 247 366 401 385 260 F3 117 149 230 166 150 F4 (0,12) (36,48) (12,24) (36,48) (0,12) F5 (0,12) (36,48) (36,38) (36,48) (12,24) F6 (6/7,1,1) (3/7,4/7,5/7) (2/7,3/7,4/7) (3/7,4/7,5/7) (5/7,6/7,1) F7 (5/7,6/7,1) (4/7,5/7,6/7) (2/7,3/7,4/7) (2/7,3/7,4/7) (5/7,6/7,1) F8 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F9 (5/7,6/7,1) (3/7,4/7,5/7) (2/7,3/7,4/7) (3/7,4/7,5/7) (5/7,6/7,1) F10 (5/7,6/7,1) (4/7,5/7,6/7) (2/7,3/7,4/7) (2/7,3/7,4/7) (5/7,6/7,1) F11 (3/7,4/7,5/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (3/7,4/7,5/7) F12 (5/7,6/7,1) (4/7,5/7,6/7) (3/7,4/7,5/7) (2/7,3/7,4/7) (5/7,6/7,1) F13 (2/7,3/7,4/7) (3/7,4/7,5/7) (4/7,5/7,6/7) (3/7,4/7,5/7) (3/7,4/7,5/7)Step 2 (select the reference point for each attribute).
The reference point can be set according to the goal of the decision maker, or it can be interpreted as the reference point is the expectation of the decision-makers for each attribute. Therefore, it can be established as(7)E∗c=e∗c1,e∗c2,⋯,e∗cn1,E∗t=e∗t1,e∗t2,⋯,e∗tn1,E∗a=e∗a1,e∗a2,⋯,e∗an1,E∗q=e∗q1,e∗q2,⋯,e∗qn1,E∗b=e∗b1,e∗b2,⋯,e∗bn1,E=E∗c∪E∗t∪E∗a∪E∗q∪E∗b=eγ∣γ=1,2,⋯,k,eγ is the reference point of the γ-th second-level attributes.Step 3 (standardized processing).
The dimensioning of each attribute value and reference vector is performed. The normalization method is as follows.
Normalize reference pointE=eγ∣γ=1,2,⋯,k to Q=qγ∣γ=1,2,⋯,k. Set the set of all secondary attribute values to F′=Fjγ∣j=1,2,⋯,m;γ=1,2,⋯,k. Then the normalized set is V=vjγ∣j=1,2,⋯,m;γ=1,2,⋯,k. The normalized rules are as follows [18].
(a) For the numeric attributes, the normalized rule is as follows:
normalized reference point as shown in formula (8):(8)qγ′=eγ-z1-z1+-z1-,forprofit-typeindexesz1+-eγz1+-z1-,forcost-typeindexes
normalized attributes as shown in formula (9):(9)vjγ′=Fjγ-z1-z1+-z1-,forbenifit-typeindexesz1+-Fjγz1+-z1-,forcost-typeindexes
among which,(10)z1+=max⁡max1≤j≤mFjγ,eγz1-=min⁡min1≤j≤mFjγ,eγ.
(b) For the interval-type attributes, the normalized rule is as follows:
normalized reference point as shown in formula (11):(11)qγ′′low,qγ′′up=eγlow-zlow-zup+-zlow-,eγup-zlow-zup+-zlow-forbenifit-typeindexeszlow+-eγlowzup+-zlow-,zup+-eγupzup+-zlow-forcost-typeindexes
normalized attributes as shown in formula (12):(12)vjγ′′low,vjγ′′up=vjrup-zlow-zup+-zlow-,vjrup-zlow-zup+-zlow-forbenifit-typeindexeszup+-vjrlowzup+-zlow-,zup+-vjrupzup+-zlow-forcost-typeindexes
among which,(13)zup+=max⁡max1≤j≤mvjrup,eγupzlow-=min⁡min1≤j≤mvjrlow,eγlow.
(c) For the statement-type attributes, the normalized rule is as follows:
normalized reference point as shown in formula (14):(14)qγ′′′=eγ,forbenifit-typeattributesnegeγ,forcost-typeattributes
normalized attributes as shown in formula (15):(15)vjγ′′′=v^jγ,forprofit-typeattributesnegv^jγ,forcost-typeattributes
among which,neg is an inverse operator.Step 4.
Calculate the distance between each attribute and the reference point.
(a) For numeric-type attributes, distance is calculated according to formula (16).(16)djγ′=vjγ′-qjγ′
(b) For interval-type attributes, distance is calculated according to formula (17).(17)djγ′′=12vjγ′′low-qγ′′low2+vjγ′′up-qγ′′up2
(c) For statement-type attributes, distance is calculated according to formula (18).(18)djγ′′′=13vjγ′′′1-qjγ′′′12+vjγ′′′2-qjγ′′′22+vjγ′′′3-qjγ′′′32Step 5 (calculation of profit and loss value).
The profit and loss value indicate that the measurement point is satisfaction and dissatisfaction relative to the reference point. The calculation formula of the values of profit and loss is shown as (19).(19)SFjγ=dij,vjγ≥qγ-dij,vjγ<qγ
Among them, whenvjγ≥qγ, we consider S(Fjγ) to be the satisfaction of attribute value Fjγ with respect to reference point qγ; when vjγ<qγ, we consider S(Fjγ) to be the dissatisfaction of attribute value Fjγ with respect to reference point qγ.Step 6 (calculate the prospect value).
According to the prospect theory, the prospect value of attributeFjγ is determined by the profit and loss value. The formula is as follows.(20)VFjγ+=SFjγαvjγ≥qγVFjγ-=-θ-SFjγβvjγ<qγ
Among them,α is the degree of concavity of a satisfactory function, and β is the degree of convexity of the dissatisfied function. α∈0,1,β∈1,0 indicate that the sensitivity of decision-makers to changes in values is diminishing; θ>1 indicates that the curve of the dissatisfied area is steeper than the curve of the satisfactory area. According to the previous experimental results, we take the value of α=β=0.88,θ=2.25.Step 7 (calculate comprehensive satisfaction).
According to the simple weighting principle, the comprehensive satisfaction degree of each candidate point is calculated according to formula (21).(21)VFjγ=∑j=1m∑γ=1kVFjγ+wFjγ+∑j=1m∑γ=1kVFjγ-wFjγ


#### 4.2.4. Example Analysis

The decision maker of an equipment manufacturing enterprise A is to establish an independent new plant in the city of U. According to the investigation and survey of the surrounding area of the city, there are five options available for selection. Evaluate and evaluate the attributes of candidate points from both online and offline channels. The weight of each attribute is as follows: because the evaluation results include three different evaluation results: numeric, language, and interval, the results of language evaluation can be converted into triangular fuzzy numbers. The evaluation values are shown in Tables1 and 2.Table 2
Online channel, attribute evaluation values.
Attributes Alternatives P1 P2 P3 P4 P5 F1 1575 1294 1159 1234 1468 F2 247 366 401 385 260 F3 117 149 230 166 150 F4 (48,60) (84,96) (120,132) (96,108) (60,72) F5 (60,72) (84,96) (108,120) (84,96) (72,84) F6 (6/7,1,1) (3/7,4/7,5/7) (3/7,4/7,5/7) (3/7,4/7,5/7) (5/7,6/7,1) F7 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F8 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F9 (5/7,6/7,1) (4/7,5/7,6/7) (3/7,4/7,5/7) (3/7,4/7,5/7) (5/7,6/7,1) F10 (5/7,6/7,1) (6/7,1,1) (1/7,2/7,3/7) (3/7,4/7,5/7) (2/7,3/7,4/7) F11 (4/7,5/7,6/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F12 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F13 (4/7,5/7,6/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1)According to Steps 1-7, through the calculation steps such as quantifying attribute values, determining expected values, normalizing, calculating profit and loss, etc., the prospect values of each attribute in online and offline channels are calculated, as shown in Tables3 and 4. The positive and negative values indicate the degree of “better” and “better than”, respectively, relative to the ideal point.Table 3
Offline channels, property prospect value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 -1.24 0.21 0.53 0.36 -0.66  F2 0.59 -0.59 -1.11 -0.89 0.51 F3 0.37 0.06 -1.6 -0.32 0.05 F4 0.49 -0.8 0.23 -0.8 0.49 F5 0.54 -0.66 -0.66 -0.66 0.3 F6 0 -0.98 -1.29 -0.98 -0.35 F7 0 -0.4 -1.07 -1.07 0 F8 0 -0.76 -1.37 -0.76 0 F9 0 -0.76 -1.07 -0.76 0 F10 0 -0.4 -1.07 -1.07 0 F11 -0.76 -0.76 -1.37 -0.76 0 F12 0 -0.4 -0.76 -1.07 0 F13 -1.07 -0.76 -0.4 -0.76 0Table 4
Online channels, property prospect value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 -1.24 0.21 0.53 0.36 -0.66 F2 0.59 -0.59 -1.11 -0.89 0.51 F3 0.37 0.06 -1.6 -0.32 0.05 F4 0.54 0.12 -0.94 -0.27 0.4 F5 0.57 0.17 -0.73 0.17 -0.35 F6 0 -0.98 -0.98 -0.98 -0.35 F7 0 -0.76 -1.37 -0.76 0 F8 0 -0.76 -1.37 -0.76 0 F9 0 -0.4 -0.76 -0.76 0 F10 -0.35 0 -1.58 -0.98 -1.29 F11 -0.4 -0.76 -1.37 -0.76 0 F12 0 -0.76 -1.37 -0.76 0 F13 -0.4 -0.76 -1.37 -0.76 0Through calculation, the comprehensive prospect values of each candidate point are obtained as below.(22)VFjγ=0.06,-0.76,-1.57,-1.06,0.04The comprehensive prospect represents the satisfaction of the decision-makers to each scheme. It is known that the decision maker has the highest degree of satisfaction to the alternative point 1 and the priority of the 5 alternatives is 1 to 5.(23)P1≻P5≻P2≻P4≻P3With the highest degree of satisfaction as the goal, we should choose 1 plant at the alternative point.


## 4.1. Determining the Evaluation Index

According to the characteristics of procurement, production, transportation, and sales of equipment manufacturing enterprises, the traditional operating model of equipment manufacturing enterprises is integrated with the Internet.In this context, based on the consensus of the previous literature research and the actual investigation of the manufacturers, we classified the factors affecting the location of equipment manufacturing enterprises as five types of influencing factors. In accordance with these five dimensions, 13 appraisal indicators with good applicability were determined, and the hierarchical structure model of the decision-making factors of equipment manufacturing enterprises was constructed, as shown in Figure3.Figure 3
The structural model of the location decision-making influencing factors.(1) Cost Factors. The main considerations are construction costs, knowledge costs, and production and operation costs. The construction cost includes land cost and construction cost; the knowledge cost includes network cost and information cost; and the production and operation cost mainly includes transportation expenses and warehousing expenses incurred during the production and operation process.(2) Time Factor. It mainly considers the transportation time of raw materials and the delivery time of products in the dual-channel procurement and sales process to ensure timely arrival of raw materials and timely delivery of products.(3) Regional Condition Factors. It mainly examines the geographical conditions, industrial conditions, and logistics conditions of the alternative areas. The geographical conditions mainly include road conditions and distances from airports and ports; the industrial conditions mainly examine whether the surrounding industrial environment is agglomeration and how the competition is; and the logistics conditions mainly include the degree of perfection of the logistics industry and whether it can be competent for the product distribution tasks of the manufacturers.(4) Quality of Service Factors. It mainly considers the quality of the rapid feedback of the manufacturers to the supplier in the alternative area and the quality of the rapid response to the customer.(5) External Environmental Factors. It mainly examines whether the regional policy conditions in the candidate areas are superior, whether the information environment is perfect and popular, and whether it destroys the natural environment and does not comply with green manufacturing.


## 4.2. Modeling

### 4.2.1. Prospect Theory

Prospect theory is a theory describing and predicting the behavior that people are inconsistent with traditional expectation theory and expected utility theory in the process of facing risk decision-making. It is found that the risk preference behavior of people in the face of gains and losses is inconsistent. It becomes a risk pursuit in the face of “loss"; while facing “gain", the risk evasion can be avoided. The establishment and change of reference points affect people’s gain and loss feelings and then influence people’s decision-making. Figure4 is a value function diagram.Figure 4
Value function diagram.According to this value function, people show loss aversion; that is to say, people are more sensitive to losses than to gain. Kahneman and Tversky did such an experiment. They gave the participants a chance to flip a coin: Once the coin was turned up, the subject got 20 dollars, and if they face it up, the subject lost 10 dollars. Because of people’s loss aversion, most of the participants refused to participate in this game, even if they had an average expected return of 5 USD per coin toss (based on expected utility theory). People tend to avoid risk when they are faced with gain and have risk appetites when faced with losses. Kahneman and Tversky found that most people would choose a certain $800 gain instead of the 85% chance to get $1000, i.e., risk aversion; at the same time, when the participants face two choices: sure loss of $800 or 85% probability loss of $1000, most of the participants chose the latter, namely, risk appetite. Ossenbruggen and Hardman et al. [14] considered that the value function lacks the research logic and belongs to the phenomenological model; Tetlock [15] introduces social and emotional frameworks to the value function; Kahneman et al. [16] integrate Simon’s heuristic strategy research results and think that the heuristic strategies people used in decision-making are mainly representative heuristics, easy-to-get heuristics, and anchoring and adjustment heuristics.


### 4.2.2. Determining the Objective of the Function

In this paper, we address the problem of single-plant site selection. Therefore, the influencing factors mentioned above are set as the main attributes affecting the satisfaction of the enterprise, suppliers, and customers, which is transformed into a multiattribute decision-making problem. We know that the evaluation language is different when evaluating different factors. According to different types of influencing factors, we adopt different methods to quantify the evaluation results. According to the classification of the evaluation results, the attributes of the influential factors affecting the satisfaction can be divided into phrase attributes and clear numerical attributes. For example, regional geographic conditions, regional industrial conditions, regional logistics conditions, service quality, and environmental factors can all be considered phrase-like attributes; construction costs, knowledge costs, and operating costs, as well as transit time, can be categorized as clear numerical attributes. The objective function based on the prospect theory can be expressed as follows.(3)max⁡VFjγ(4)VFjγ=∑j=1m∑γ=1kVFjγ+wFjγ+∑j=1m∑γ=1kVFjγ-wFjγ


### 4.2.3. Solution Method

Step 1.
Assume that the location of suppliers and demand points is known, and select relatively satisfactory candidate points so that they can better cover the surrounding demand points. According to online channel satisfaction and offline channel satisfaction, each attribute value is calculated for each candidate point, and a program attribute value matrix table is listed, as shown in Table1.
(a) For numerical attributes, the evaluation result is the actual measured value.
(b) For the language-type attribute, this article uses the commonly used seven-level division method, phrase evaluation grade setV=Vv∣v=1,2,⋯,7, among which, V1=(extremelysatisfactory), V2=(greatersatisfaction), V3=(satisfactory), V4=(neutral), V5=(unsatisfaction), V6=(greaterunsatisfaction), and V7=(extremelyunsatisfactory) scales are used to evaluate the factors that manufacturers use to influence the phrase-type attributes of the candidate sites, through fuzzy trigonometric method to numerical illustration. The evaluation phrase can be converted into a numerical value by a triangular fuzzy formula. The calculation formula is as follows:(5)V^v=max⁡v-17,0,v7,min⁡v+17,1
whereVv is the level of the v-th phrase and V^v is the trigonometric fuzzy number of the v-th phrase level. Then there are(6)V^1=0,17,27,V^2=17,27,37,V^3=27,37,47,V^4=37,47,57,V^5=47,57,67,V^6=57,67,1,V^7=67,1,1.
(c) For interval-type attributes, the attribute value is represented by(a1,a2) or a1,a2.
It is assumed that there is an ordered property of fuzzy numbers for each factor [17]. For any Vv∈V,Vv-1∈V, if Vv>Vv-1, there is V^v≻Vv-1. Among them, “≻” means “bebetterthan”.Table 1
Offline channel, attribute evaluation value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 1575 1294 1159 1234 1468 F2 247 366 401 385 260 F3 117 149 230 166 150 F4 (0,12) (36,48) (12,24) (36,48) (0,12) F5 (0,12) (36,48) (36,38) (36,48) (12,24) F6 (6/7,1,1) (3/7,4/7,5/7) (2/7,3/7,4/7) (3/7,4/7,5/7) (5/7,6/7,1) F7 (5/7,6/7,1) (4/7,5/7,6/7) (2/7,3/7,4/7) (2/7,3/7,4/7) (5/7,6/7,1) F8 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F9 (5/7,6/7,1) (3/7,4/7,5/7) (2/7,3/7,4/7) (3/7,4/7,5/7) (5/7,6/7,1) F10 (5/7,6/7,1) (4/7,5/7,6/7) (2/7,3/7,4/7) (2/7,3/7,4/7) (5/7,6/7,1) F11 (3/7,4/7,5/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (3/7,4/7,5/7) F12 (5/7,6/7,1) (4/7,5/7,6/7) (3/7,4/7,5/7) (2/7,3/7,4/7) (5/7,6/7,1) F13 (2/7,3/7,4/7) (3/7,4/7,5/7) (4/7,5/7,6/7) (3/7,4/7,5/7) (3/7,4/7,5/7)Step 2 (select the reference point for each attribute).
The reference point can be set according to the goal of the decision maker, or it can be interpreted as the reference point is the expectation of the decision-makers for each attribute. Therefore, it can be established as(7)E∗c=e∗c1,e∗c2,⋯,e∗cn1,E∗t=e∗t1,e∗t2,⋯,e∗tn1,E∗a=e∗a1,e∗a2,⋯,e∗an1,E∗q=e∗q1,e∗q2,⋯,e∗qn1,E∗b=e∗b1,e∗b2,⋯,e∗bn1,E=E∗c∪E∗t∪E∗a∪E∗q∪E∗b=eγ∣γ=1,2,⋯,k,eγ is the reference point of the γ-th second-level attributes.Step 3 (standardized processing).
The dimensioning of each attribute value and reference vector is performed. The normalization method is as follows.
Normalize reference pointE=eγ∣γ=1,2,⋯,k to Q=qγ∣γ=1,2,⋯,k. Set the set of all secondary attribute values to F′=Fjγ∣j=1,2,⋯,m;γ=1,2,⋯,k. Then the normalized set is V=vjγ∣j=1,2,⋯,m;γ=1,2,⋯,k. The normalized rules are as follows [18].
(a) For the numeric attributes, the normalized rule is as follows:
normalized reference point as shown in formula (8):(8)qγ′=eγ-z1-z1+-z1-,forprofit-typeindexesz1+-eγz1+-z1-,forcost-typeindexes
normalized attributes as shown in formula (9):(9)vjγ′=Fjγ-z1-z1+-z1-,forbenifit-typeindexesz1+-Fjγz1+-z1-,forcost-typeindexes
among which,(10)z1+=max⁡max1≤j≤mFjγ,eγz1-=min⁡min1≤j≤mFjγ,eγ.
(b) For the interval-type attributes, the normalized rule is as follows:
normalized reference point as shown in formula (11):(11)qγ′′low,qγ′′up=eγlow-zlow-zup+-zlow-,eγup-zlow-zup+-zlow-forbenifit-typeindexeszlow+-eγlowzup+-zlow-,zup+-eγupzup+-zlow-forcost-typeindexes
normalized attributes as shown in formula (12):(12)vjγ′′low,vjγ′′up=vjrup-zlow-zup+-zlow-,vjrup-zlow-zup+-zlow-forbenifit-typeindexeszup+-vjrlowzup+-zlow-,zup+-vjrupzup+-zlow-forcost-typeindexes
among which,(13)zup+=max⁡max1≤j≤mvjrup,eγupzlow-=min⁡min1≤j≤mvjrlow,eγlow.
(c) For the statement-type attributes, the normalized rule is as follows:
normalized reference point as shown in formula (14):(14)qγ′′′=eγ,forbenifit-typeattributesnegeγ,forcost-typeattributes
normalized attributes as shown in formula (15):(15)vjγ′′′=v^jγ,forprofit-typeattributesnegv^jγ,forcost-typeattributes
among which,neg is an inverse operator.Step 4.
Calculate the distance between each attribute and the reference point.
(a) For numeric-type attributes, distance is calculated according to formula (16).(16)djγ′=vjγ′-qjγ′
(b) For interval-type attributes, distance is calculated according to formula (17).(17)djγ′′=12vjγ′′low-qγ′′low2+vjγ′′up-qγ′′up2
(c) For statement-type attributes, distance is calculated according to formula (18).(18)djγ′′′=13vjγ′′′1-qjγ′′′12+vjγ′′′2-qjγ′′′22+vjγ′′′3-qjγ′′′32Step 5 (calculation of profit and loss value).
The profit and loss value indicate that the measurement point is satisfaction and dissatisfaction relative to the reference point. The calculation formula of the values of profit and loss is shown as (19).(19)SFjγ=dij,vjγ≥qγ-dij,vjγ<qγ
Among them, whenvjγ≥qγ, we consider S(Fjγ) to be the satisfaction of attribute value Fjγ with respect to reference point qγ; when vjγ<qγ, we consider S(Fjγ) to be the dissatisfaction of attribute value Fjγ with respect to reference point qγ.Step 6 (calculate the prospect value).
According to the prospect theory, the prospect value of attributeFjγ is determined by the profit and loss value. The formula is as follows.(20)VFjγ+=SFjγαvjγ≥qγVFjγ-=-θ-SFjγβvjγ<qγ
Among them,α is the degree of concavity of a satisfactory function, and β is the degree of convexity of the dissatisfied function. α∈0,1,β∈1,0 indicate that the sensitivity of decision-makers to changes in values is diminishing; θ>1 indicates that the curve of the dissatisfied area is steeper than the curve of the satisfactory area. According to the previous experimental results, we take the value of α=β=0.88,θ=2.25.Step 7 (calculate comprehensive satisfaction).
According to the simple weighting principle, the comprehensive satisfaction degree of each candidate point is calculated according to formula (21).(21)VFjγ=∑j=1m∑γ=1kVFjγ+wFjγ+∑j=1m∑γ=1kVFjγ-wFjγ


### 4.2.4. Example Analysis

The decision maker of an equipment manufacturing enterprise A is to establish an independent new plant in the city of U. According to the investigation and survey of the surrounding area of the city, there are five options available for selection. Evaluate and evaluate the attributes of candidate points from both online and offline channels. The weight of each attribute is as follows: because the evaluation results include three different evaluation results: numeric, language, and interval, the results of language evaluation can be converted into triangular fuzzy numbers. The evaluation values are shown in Tables1 and 2.Table 2
Online channel, attribute evaluation values.
Attributes Alternatives P1 P2 P3 P4 P5 F1 1575 1294 1159 1234 1468 F2 247 366 401 385 260 F3 117 149 230 166 150 F4 (48,60) (84,96) (120,132) (96,108) (60,72) F5 (60,72) (84,96) (108,120) (84,96) (72,84) F6 (6/7,1,1) (3/7,4/7,5/7) (3/7,4/7,5/7) (3/7,4/7,5/7) (5/7,6/7,1) F7 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F8 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F9 (5/7,6/7,1) (4/7,5/7,6/7) (3/7,4/7,5/7) (3/7,4/7,5/7) (5/7,6/7,1) F10 (5/7,6/7,1) (6/7,1,1) (1/7,2/7,3/7) (3/7,4/7,5/7) (2/7,3/7,4/7) F11 (4/7,5/7,6/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F12 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F13 (4/7,5/7,6/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1)According to Steps 1-7, through the calculation steps such as quantifying attribute values, determining expected values, normalizing, calculating profit and loss, etc., the prospect values of each attribute in online and offline channels are calculated, as shown in Tables3 and 4. The positive and negative values indicate the degree of “better” and “better than”, respectively, relative to the ideal point.Table 3
Offline channels, property prospect value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 -1.24 0.21 0.53 0.36 -0.66  F2 0.59 -0.59 -1.11 -0.89 0.51 F3 0.37 0.06 -1.6 -0.32 0.05 F4 0.49 -0.8 0.23 -0.8 0.49 F5 0.54 -0.66 -0.66 -0.66 0.3 F6 0 -0.98 -1.29 -0.98 -0.35 F7 0 -0.4 -1.07 -1.07 0 F8 0 -0.76 -1.37 -0.76 0 F9 0 -0.76 -1.07 -0.76 0 F10 0 -0.4 -1.07 -1.07 0 F11 -0.76 -0.76 -1.37 -0.76 0 F12 0 -0.4 -0.76 -1.07 0 F13 -1.07 -0.76 -0.4 -0.76 0Table 4
Online channels, property prospect value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 -1.24 0.21 0.53 0.36 -0.66 F2 0.59 -0.59 -1.11 -0.89 0.51 F3 0.37 0.06 -1.6 -0.32 0.05 F4 0.54 0.12 -0.94 -0.27 0.4 F5 0.57 0.17 -0.73 0.17 -0.35 F6 0 -0.98 -0.98 -0.98 -0.35 F7 0 -0.76 -1.37 -0.76 0 F8 0 -0.76 -1.37 -0.76 0 F9 0 -0.4 -0.76 -0.76 0 F10 -0.35 0 -1.58 -0.98 -1.29 F11 -0.4 -0.76 -1.37 -0.76 0 F12 0 -0.76 -1.37 -0.76 0 F13 -0.4 -0.76 -1.37 -0.76 0Through calculation, the comprehensive prospect values of each candidate point are obtained as below.(22)VFjγ=0.06,-0.76,-1.57,-1.06,0.04The comprehensive prospect represents the satisfaction of the decision-makers to each scheme. It is known that the decision maker has the highest degree of satisfaction to the alternative point 1 and the priority of the 5 alternatives is 1 to 5.(23)P1≻P5≻P2≻P4≻P3With the highest degree of satisfaction as the goal, we should choose 1 plant at the alternative point.


## 4.2.1. Prospect Theory

Prospect theory is a theory describing and predicting the behavior that people are inconsistent with traditional expectation theory and expected utility theory in the process of facing risk decision-making. It is found that the risk preference behavior of people in the face of gains and losses is inconsistent. It becomes a risk pursuit in the face of “loss"; while facing “gain", the risk evasion can be avoided. The establishment and change of reference points affect people’s gain and loss feelings and then influence people’s decision-making. Figure4 is a value function diagram.Figure 4
Value function diagram.According to this value function, people show loss aversion; that is to say, people are more sensitive to losses than to gain. Kahneman and Tversky did such an experiment. They gave the participants a chance to flip a coin: Once the coin was turned up, the subject got 20 dollars, and if they face it up, the subject lost 10 dollars. Because of people’s loss aversion, most of the participants refused to participate in this game, even if they had an average expected return of 5 USD per coin toss (based on expected utility theory). People tend to avoid risk when they are faced with gain and have risk appetites when faced with losses. Kahneman and Tversky found that most people would choose a certain $800 gain instead of the 85% chance to get $1000, i.e., risk aversion; at the same time, when the participants face two choices: sure loss of $800 or 85% probability loss of $1000, most of the participants chose the latter, namely, risk appetite. Ossenbruggen and Hardman et al. [14] considered that the value function lacks the research logic and belongs to the phenomenological model; Tetlock [15] introduces social and emotional frameworks to the value function; Kahneman et al. [16] integrate Simon’s heuristic strategy research results and think that the heuristic strategies people used in decision-making are mainly representative heuristics, easy-to-get heuristics, and anchoring and adjustment heuristics.


## 4.2.2. Determining the Objective of the Function

In this paper, we address the problem of single-plant site selection. Therefore, the influencing factors mentioned above are set as the main attributes affecting the satisfaction of the enterprise, suppliers, and customers, which is transformed into a multiattribute decision-making problem. We know that the evaluation language is different when evaluating different factors. According to different types of influencing factors, we adopt different methods to quantify the evaluation results. According to the classification of the evaluation results, the attributes of the influential factors affecting the satisfaction can be divided into phrase attributes and clear numerical attributes. For example, regional geographic conditions, regional industrial conditions, regional logistics conditions, service quality, and environmental factors can all be considered phrase-like attributes; construction costs, knowledge costs, and operating costs, as well as transit time, can be categorized as clear numerical attributes. The objective function based on the prospect theory can be expressed as follows.(3)max⁡VFjγ(4)VFjγ=∑j=1m∑γ=1kVFjγ+wFjγ+∑j=1m∑γ=1kVFjγ-wFjγ


## 4.2.3. Solution Method

Step 1.
Assume that the location of suppliers and demand points is known, and select relatively satisfactory candidate points so that they can better cover the surrounding demand points. According to online channel satisfaction and offline channel satisfaction, each attribute value is calculated for each candidate point, and a program attribute value matrix table is listed, as shown in Table1.
(a) For numerical attributes, the evaluation result is the actual measured value.
(b) For the language-type attribute, this article uses the commonly used seven-level division method, phrase evaluation grade setV=Vv∣v=1,2,⋯,7, among which, V1=(extremelysatisfactory), V2=(greatersatisfaction), V3=(satisfactory), V4=(neutral), V5=(unsatisfaction), V6=(greaterunsatisfaction), and V7=(extremelyunsatisfactory) scales are used to evaluate the factors that manufacturers use to influence the phrase-type attributes of the candidate sites, through fuzzy trigonometric method to numerical illustration. The evaluation phrase can be converted into a numerical value by a triangular fuzzy formula. The calculation formula is as follows:(5)V^v=max⁡v-17,0,v7,min⁡v+17,1
whereVv is the level of the v-th phrase and V^v is the trigonometric fuzzy number of the v-th phrase level. Then there are(6)V^1=0,17,27,V^2=17,27,37,V^3=27,37,47,V^4=37,47,57,V^5=47,57,67,V^6=57,67,1,V^7=67,1,1.
(c) For interval-type attributes, the attribute value is represented by(a1,a2) or a1,a2.
It is assumed that there is an ordered property of fuzzy numbers for each factor [17]. For any Vv∈V,Vv-1∈V, if Vv>Vv-1, there is V^v≻Vv-1. Among them, “≻” means “bebetterthan”.Table 1
Offline channel, attribute evaluation value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 1575 1294 1159 1234 1468 F2 247 366 401 385 260 F3 117 149 230 166 150 F4 (0,12) (36,48) (12,24) (36,48) (0,12) F5 (0,12) (36,48) (36,38) (36,48) (12,24) F6 (6/7,1,1) (3/7,4/7,5/7) (2/7,3/7,4/7) (3/7,4/7,5/7) (5/7,6/7,1) F7 (5/7,6/7,1) (4/7,5/7,6/7) (2/7,3/7,4/7) (2/7,3/7,4/7) (5/7,6/7,1) F8 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F9 (5/7,6/7,1) (3/7,4/7,5/7) (2/7,3/7,4/7) (3/7,4/7,5/7) (5/7,6/7,1) F10 (5/7,6/7,1) (4/7,5/7,6/7) (2/7,3/7,4/7) (2/7,3/7,4/7) (5/7,6/7,1) F11 (3/7,4/7,5/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (3/7,4/7,5/7) F12 (5/7,6/7,1) (4/7,5/7,6/7) (3/7,4/7,5/7) (2/7,3/7,4/7) (5/7,6/7,1) F13 (2/7,3/7,4/7) (3/7,4/7,5/7) (4/7,5/7,6/7) (3/7,4/7,5/7) (3/7,4/7,5/7)Step 2 (select the reference point for each attribute).
The reference point can be set according to the goal of the decision maker, or it can be interpreted as the reference point is the expectation of the decision-makers for each attribute. Therefore, it can be established as(7)E∗c=e∗c1,e∗c2,⋯,e∗cn1,E∗t=e∗t1,e∗t2,⋯,e∗tn1,E∗a=e∗a1,e∗a2,⋯,e∗an1,E∗q=e∗q1,e∗q2,⋯,e∗qn1,E∗b=e∗b1,e∗b2,⋯,e∗bn1,E=E∗c∪E∗t∪E∗a∪E∗q∪E∗b=eγ∣γ=1,2,⋯,k,eγ is the reference point of the γ-th second-level attributes.Step 3 (standardized processing).
The dimensioning of each attribute value and reference vector is performed. The normalization method is as follows.
Normalize reference pointE=eγ∣γ=1,2,⋯,k to Q=qγ∣γ=1,2,⋯,k. Set the set of all secondary attribute values to F′=Fjγ∣j=1,2,⋯,m;γ=1,2,⋯,k. Then the normalized set is V=vjγ∣j=1,2,⋯,m;γ=1,2,⋯,k. The normalized rules are as follows [18].
(a) For the numeric attributes, the normalized rule is as follows:
normalized reference point as shown in formula (8):(8)qγ′=eγ-z1-z1+-z1-,forprofit-typeindexesz1+-eγz1+-z1-,forcost-typeindexes
normalized attributes as shown in formula (9):(9)vjγ′=Fjγ-z1-z1+-z1-,forbenifit-typeindexesz1+-Fjγz1+-z1-,forcost-typeindexes
among which,(10)z1+=max⁡max1≤j≤mFjγ,eγz1-=min⁡min1≤j≤mFjγ,eγ.
(b) For the interval-type attributes, the normalized rule is as follows:
normalized reference point as shown in formula (11):(11)qγ′′low,qγ′′up=eγlow-zlow-zup+-zlow-,eγup-zlow-zup+-zlow-forbenifit-typeindexeszlow+-eγlowzup+-zlow-,zup+-eγupzup+-zlow-forcost-typeindexes
normalized attributes as shown in formula (12):(12)vjγ′′low,vjγ′′up=vjrup-zlow-zup+-zlow-,vjrup-zlow-zup+-zlow-forbenifit-typeindexeszup+-vjrlowzup+-zlow-,zup+-vjrupzup+-zlow-forcost-typeindexes
among which,(13)zup+=max⁡max1≤j≤mvjrup,eγupzlow-=min⁡min1≤j≤mvjrlow,eγlow.
(c) For the statement-type attributes, the normalized rule is as follows:
normalized reference point as shown in formula (14):(14)qγ′′′=eγ,forbenifit-typeattributesnegeγ,forcost-typeattributes
normalized attributes as shown in formula (15):(15)vjγ′′′=v^jγ,forprofit-typeattributesnegv^jγ,forcost-typeattributes
among which,neg is an inverse operator.Step 4.
Calculate the distance between each attribute and the reference point.
(a) For numeric-type attributes, distance is calculated according to formula (16).(16)djγ′=vjγ′-qjγ′
(b) For interval-type attributes, distance is calculated according to formula (17).(17)djγ′′=12vjγ′′low-qγ′′low2+vjγ′′up-qγ′′up2
(c) For statement-type attributes, distance is calculated according to formula (18).(18)djγ′′′=13vjγ′′′1-qjγ′′′12+vjγ′′′2-qjγ′′′22+vjγ′′′3-qjγ′′′32Step 5 (calculation of profit and loss value).
The profit and loss value indicate that the measurement point is satisfaction and dissatisfaction relative to the reference point. The calculation formula of the values of profit and loss is shown as (19).(19)SFjγ=dij,vjγ≥qγ-dij,vjγ<qγ
Among them, whenvjγ≥qγ, we consider S(Fjγ) to be the satisfaction of attribute value Fjγ with respect to reference point qγ; when vjγ<qγ, we consider S(Fjγ) to be the dissatisfaction of attribute value Fjγ with respect to reference point qγ.Step 6 (calculate the prospect value).
According to the prospect theory, the prospect value of attributeFjγ is determined by the profit and loss value. The formula is as follows.(20)VFjγ+=SFjγαvjγ≥qγVFjγ-=-θ-SFjγβvjγ<qγ
Among them,α is the degree of concavity of a satisfactory function, and β is the degree of convexity of the dissatisfied function. α∈0,1,β∈1,0 indicate that the sensitivity of decision-makers to changes in values is diminishing; θ>1 indicates that the curve of the dissatisfied area is steeper than the curve of the satisfactory area. According to the previous experimental results, we take the value of α=β=0.88,θ=2.25.Step 7 (calculate comprehensive satisfaction).
According to the simple weighting principle, the comprehensive satisfaction degree of each candidate point is calculated according to formula (21).(21)VFjγ=∑j=1m∑γ=1kVFjγ+wFjγ+∑j=1m∑γ=1kVFjγ-wFjγ


## 4.2.4. Example Analysis

The decision maker of an equipment manufacturing enterprise A is to establish an independent new plant in the city of U. According to the investigation and survey of the surrounding area of the city, there are five options available for selection. Evaluate and evaluate the attributes of candidate points from both online and offline channels. The weight of each attribute is as follows: because the evaluation results include three different evaluation results: numeric, language, and interval, the results of language evaluation can be converted into triangular fuzzy numbers. The evaluation values are shown in Tables1 and 2.Table 2
Online channel, attribute evaluation values.
Attributes Alternatives P1 P2 P3 P4 P5 F1 1575 1294 1159 1234 1468 F2 247 366 401 385 260 F3 117 149 230 166 150 F4 (48,60) (84,96) (120,132) (96,108) (60,72) F5 (60,72) (84,96) (108,120) (84,96) (72,84) F6 (6/7,1,1) (3/7,4/7,5/7) (3/7,4/7,5/7) (3/7,4/7,5/7) (5/7,6/7,1) F7 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F8 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F9 (5/7,6/7,1) (4/7,5/7,6/7) (3/7,4/7,5/7) (3/7,4/7,5/7) (5/7,6/7,1) F10 (5/7,6/7,1) (6/7,1,1) (1/7,2/7,3/7) (3/7,4/7,5/7) (2/7,3/7,4/7) F11 (4/7,5/7,6/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F12 (5/7,6/7,1) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1) F13 (4/7,5/7,6/7) (3/7,4/7,5/7) (1/7,2/7,3/7) (3/7,4/7,5/7) (5/7,6/7,1)According to Steps 1-7, through the calculation steps such as quantifying attribute values, determining expected values, normalizing, calculating profit and loss, etc., the prospect values of each attribute in online and offline channels are calculated, as shown in Tables3 and 4. The positive and negative values indicate the degree of “better” and “better than”, respectively, relative to the ideal point.Table 3
Offline channels, property prospect value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 -1.24 0.21 0.53 0.36 -0.66  F2 0.59 -0.59 -1.11 -0.89 0.51 F3 0.37 0.06 -1.6 -0.32 0.05 F4 0.49 -0.8 0.23 -0.8 0.49 F5 0.54 -0.66 -0.66 -0.66 0.3 F6 0 -0.98 -1.29 -0.98 -0.35 F7 0 -0.4 -1.07 -1.07 0 F8 0 -0.76 -1.37 -0.76 0 F9 0 -0.76 -1.07 -0.76 0 F10 0 -0.4 -1.07 -1.07 0 F11 -0.76 -0.76 -1.37 -0.76 0 F12 0 -0.4 -0.76 -1.07 0 F13 -1.07 -0.76 -0.4 -0.76 0Table 4
Online channels, property prospect value.
Attributes Alternatives P1 P2 P3 P4 P5 F1 -1.24 0.21 0.53 0.36 -0.66 F2 0.59 -0.59 -1.11 -0.89 0.51 F3 0.37 0.06 -1.6 -0.32 0.05 F4 0.54 0.12 -0.94 -0.27 0.4 F5 0.57 0.17 -0.73 0.17 -0.35 F6 0 -0.98 -0.98 -0.98 -0.35 F7 0 -0.76 -1.37 -0.76 0 F8 0 -0.76 -1.37 -0.76 0 F9 0 -0.4 -0.76 -0.76 0 F10 -0.35 0 -1.58 -0.98 -1.29 F11 -0.4 -0.76 -1.37 -0.76 0 F12 0 -0.76 -1.37 -0.76 0 F13 -0.4 -0.76 -1.37 -0.76 0Through calculation, the comprehensive prospect values of each candidate point are obtained as below.(22)VFjγ=0.06,-0.76,-1.57,-1.06,0.04The comprehensive prospect represents the satisfaction of the decision-makers to each scheme. It is known that the decision maker has the highest degree of satisfaction to the alternative point 1 and the priority of the 5 alternatives is 1 to 5.(23)P1≻P5≻P2≻P4≻P3With the highest degree of satisfaction as the goal, we should choose 1 plant at the alternative point.


## 5. The Decision-Making Model of Multiplant Location

### 5.1. Logical Model

On the basis of the location theory of the maximum coverage, we will explore and study the multiplant location decision problem under the condition of stratified coverage. Based on factors such as the number of demand points and suppliers, the number and location of suppliers, the channels for realizing transactions with upstream and downstream enterprises, and the scale of production of factories, we have established an effective model for practical problems. Among them, the logical model is shown in Figure5. The establishment of a well-adapted multiplant site selection model can ensure the company’s effective interests while fully utilizing the resources’ utilization efficiency. It is of great significance to ensure social fairness and promote social development.Figure 5
Logical model of multiplant location.


### 5.2. Mathematical Model

The mathematical model was established with the goal of minimum cost and shortest time as follows.(24)
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0,1Among them, (24) represents the lowest cost target, and the specific target comprises three parts: raw material transportation cost, product transportation cost, and product production cost; (25) is the offline raw material transportation cost and online raw material transportation cost; (26) indicates the cost of transportation under the product line and the cost of online transportation; (27) represents the production cost of the product, including fixed costs and total variable costs; (28) represents the minimum time target and specific time objectives, including transportation time and return time; (29) means transport time, including raw material transport time and product transport time, online raw material transport time, and product transport time; (30) represents return time, including offline raw material return time and product return time, online raw material return time, and product return time; (31) indicates that the offline transport distance is less than the online transport distance and is greater than zero (because it is a ring load cover); (32) represents the number of built factories; (33) means that the total amount of transport is equal to the amount of transport multiplied by the number of shipments; (34) means total fixed cost constraint; (35) means that the plant can be purchased; (36) means that the plant can be sold; (37) means that the raw material resources are constrained; (38) means that the market capacity is limited; (39) means that the offline and online return costs are less than the variable cost. If the return cost is greater than the variable production cost, no return is required. (40) indicates that the production capacity is greater than a certain capacity. (41), (42), and (43) represent the range of values for the decision variable.


### 5.3. Algorithm Design

For solving coverage problems, we use genetic algorithms to get a feasible solution with high satisfaction. Genetic algorithm as a classical algorithm is an open source for researchers, which is a computational model that simulates the natural evolution of biological evolution theory and the processes of genetic mechanism. The specific steps are as follows.Step 1 (coding).
Binary coding is the most commonly used coding method in genetic algorithms. It includes advantages of (a) being simple and easy, (b) conforming to the minimum character set coding principle, and (c) being convenient for analyzing with the pattern theorem. If intervala,b is a domain and δ is the required precision, we can assume that the length of the chromosome is l, shown as follows.(44)l=log⁡b-aδ+1Step 2 (generating the initial population).
A subset of individuals (the same individual can exist) is generated by random numbers, called populationP0.The size of the population is measured by N. We assume that the first generation subset data is the initial value and starts iterating. Under the premise of ensuring the diversity of the solution, in order to improve the speed of convergence and get a satisfactory solution as soon as possible, we let the population size be N=100. The language of MATLAB is popsize=100.Step 3 (definition of fitness function).
We use boundary structure method to construct fitness function in this model. If the function value is as big as possible, then(45)FitfX=cmax-fx,fx<cmax0,others.
And if the function value is as small as possible, then(46)FitfX=fx-cmin,fx>cmin0,others.
Among them,cmax is the maximum estimate of f(x), and cmin is the minimum estimate of f(x).Step 4 (generate newborn populations).
Crossover and mutation operations and determination of operating parameters: This step mainly selects cross-mutation activities between chromosomes with better fitness values. Its purpose is to get the chromosomes with higher fitness values. At the same time it determines the relevant operating parameters, such as crossover rate, mutation rate, and population size. In this paper, in order to fully cross each generation, we selectpc=0.8 (generally 0.6-1.0) as crossover rate and pm=0.01 (generally 0.01-0.1) as mutation rate.Step 5 (matching facilities and demand points).
(1) from the view of plantQij, Qjk, dij, and djk, effective facility allocation ensures that facilities are built to the maximum demand point coverage.
(2) Determine whether the demand covered by the same factory is greater than the factory’s maximum production capacity. If not, then continue to runi+1; if yes, return (1) to redistribute.
Evaluate whether the demand covered by a plant exceeds the maximum capacity of the plant. If not, iterative procedure continues to runi+1; if yes, return (1) to redistribute.Step 6.
Based on the allocation scheme in step 5, the target value of the initial population can be found.Step 7 (recalculate the target value).
On the basis of the parameters in step 6, the single-point crossover, and single-point mutation on the chromosome, with high fitness in the initial population, regeneration of theN chromosome, and formation of a new population P1, we can get the new target value.Step 8 (optimal selection).
Combining the initial populationP0 and the newest species group P1, the newest population of the better N chromosome is still selected to form a new population P2 according to the target value. After the crossover and variation of N times, a satisfactory solution was selected.


### 5.4. Example Analysis

SY City is an important central city, advanced equipment manufacturing base, and national historical and cultural city of the provincial capital, subprovincial city, and megacities approved by the State Council and approved by the State Council. The city administers 10 districts, 2 counties, and 1 county level. The city has an area of 12,900 square kilometers, resident population of 8,291,000, and urbanization rate of 80.55%. At present, a 10 × 10 area Sino-German manufacturing industry park is planned to be calculated, and 50 supply points are provided. The specific coordinates and supply quantities are shown in Table5. The demand points are 100, and the specific coordinates and demand are shown in Table 6. At present, after the analysis of the basic conditions, there are 25 locations where the proposed site can be constructed. The specific coordinates and output are shown in Table 7.Table 5
Coordinates and quantity of suppliers for raw materials.
No Coordinates Supply quantity No Coordinates Supply quantity No Coordinates Supply quantity 1 58,2 741 18 39,42 705 35 3,92 448 2 29,91 656 19 20,92 657 36 33,73 682 3 83,80 632 20 84,16 322 37 42,51 479 4 20,78 543 21 50,67 372 38 15,15 331 5 21,70 619 22 1,42 394 39 84,55 534 6 71,37 330 23 43,69 371 40 37,97 341 7 41,11 494 24 10,82 576 41 87,4 371 8 6,1 676 25 62,98 764 42 14,50 503 9 55,59 392 26 18,55 787 43 52,78 494 10 22,14 405 27 7,34 610 44 97,63 557 11 26,52 479 28 38,46 631 45 35,21 723 12 9,5 518 29 9,98 576 46 93,15 462 13 10,10 504 30 88,57 671 47 15,52 723 14 96,48 541 31 41,97 459 48 90,44 749 15 20,82 433 32 65,3 564 49 59,63 752 16 3,22 797 33 70,53 537 50 7,25 611 17 56,85 655 34 89,10 468Table 6
Coordinates and demand quantity of demand point.
No Coordinates Demand quantity No Coordinates Demand quantity No Coordinates Demand quantity 1 99,23 489 35 62,68 468 68 38,44 430 2 23,50 942 36 65,98 747 69 58,21 799 3 24,83 447 37 5,95 472 70 96,47 593 4 2,76 565 38 42,84 699 71 99,25 841 5 47,99 902 39 88,41 797 72 24,25 854 6 62,99 714 40 93,9 782 73 5,63 710 7 52,29 833 41 96,82 894 74 3,85 405 8 27,25 690 42 35,75 534 75 25,25 559 9 56,8 486 43 68,96 465 76 35,73 901 10 14,24 732 44 3,27 450 77 88,99 608 11 54,82 946 45 77,52 775 78 73,44 782 12 37,10 475 46 1,78 568 79 96,23 614 13 11,59 810 47 48,10 892 80 56,63 843 14 23,16 643 48 32,90 755 81 85,22 618 15 5,95 477 49 65,17 413 82 39,87 736 16 66,24 953 50 21,73 769 83 15,62 625 17 40,61 736 51 76,11 561 84 2,5 604 18 32,26 704 52 47,68 706 85 1,58 424 19 67,12 414 53 32,69 724 86 48,17 910 20 15,38 518 54 92,7 840 87 20,38 979 21 80,100 791 55 24,4 825 88 76,33 789 22 25,36 468 56 61,25 740 89 57,44 614 23 93,86 927 57 34,0 746 90 69,32 582 24 4,98 756 58 14,50 780 91 56,85 787 25 95,94 482 59 86,46 585 92 64,47 648 26 75,35 839 60 76,19 726 93 96,8 692 27 35,35 560 61 38,82 825 94 29,66 418 28 71,54 797 62 37,20 759 95 77,50 794 29 1,2 535 63 82,52 680 96 73,17 600 30 26,46 478 64 47,14 537 97 31,80 970 31 30,52 691 65 53,11 593 98 92,15 906 32 94,4 888 66 40,55 790 99 68,56 473 33 91,53 849 67 42,36 856 100 52,69 856 34 98,63 663Table 7
Coordinates and quantity of plants.
No Coordinates Output No Coordinates Output No Coordinates Output 1 48 23 2620 9 41 90 3730 17 80 46 2910 2 6 95 3770 10 77 62 2810 18 14 13 3280 3 98 31 2780 11 36 66 3430 19 85 17 3960 4 28 79 3170 12 63 38 2870 20 24 3 2790 5 75 51 3620 13 72 58 3310 21 69 42 3200 6 92 47 2650 14 86 73 2590 22 91 64 2950 7 78 87 2760 15 43 34 2560 23 35 52 3800 8 40 89 2900 16 37 57 3380 24 5 97 2550 25 21 25 3840The values of other parameters are shown as follows.(47)w1off=1.5,w2off=1.8,w1on=3.0,w2on=3.5y1off=1%,y2off=2%,y1on=4%,y2on=5%τ1off=3d,τ2off=4d,τ1on=7d,τ2on=8dDj1,Dj2=1,3Then, we have the following.(48)R1=1,R2=3max⁡z=20There are three types of cars, the volume of transportation is as follows.(49)q1=100,q2=200,q3=300Cf=100000Cv=5ζ2off=1.2,ζ2on=1.5∑Qij≤10000∑Qjk≤15000The parameters of the genetic algorithm are set: the population size is 100, the crossover rate is 0.8, the variation rate is 0.01, and the iteration number is 200. As shown in Figure6Figure 6
The relationship between the number of iterations and the value of the objective function.From Figure6, as the number of iterations increases, the first target value with the minimum cost as the target is gradually reduced, while the second target value with the minimum time as the target is gradually increased, and the two targets constrain each other, so the target values are back against the law. When the number of iterations is 131 generations, the two target values are convergent. It can be seen that the genetic algorithm adopted in this section has good convergence for solving this model. Thus, the most suitable location and allocation of multiple plants can be determined. 3 approximate Pareto optimal solutions are randomly selected from the optimization front, and the plant number and target value are shown in Table 8.Table 8
Approximate to the target value of the Pareto optimal solution.
Scheme Numbers of plants Total cost Total time 1 11 1617.91 333.75 2 11 1628.62 329.74 3 12 1638.01 324.54In this case, we can know from the calculation results that the optimal number of plants to be built is 11 or 12 plants. Due to the different construction plans, the construction costs and transportation time will be different. There are three situations. In the first, the system cost is the smallest, and the system will spend more time to deal with the demand. In the second, the system cost is the largest, but the system will spend less time to deal with business processes such as purchase, sale, and return. The third situation is between the above two situations. Therefore, when choosing a plan, the decision-makers should consider the two objectives comprehensively, not only pay attention to the minimum cost or the shortest time. According to the actual conditions, including the external environmental factors and policy conditions of the enterprises, we make targeted decisions. In this case, we have chosen the second scheme in consideration of the dual-channel marketing model, the surrounding industrial conditions of the alternative sites, and the local government’s regional policies for the industry. The distribution of alternative points and demand points is shown in Tables9–11.Table 9
The relationship between location and demand distribution of scheme 1.
Scheme 1 No Alternative points Demand and supply points 1 10 3, 30, 47, 98, 51, 2, 78, 5 2 14 35, 55, 99, 1, 7, 48, 12, 100 3 20 63, 58, 19, 69, 84, 16, 80 4 7 74, 62, 95, 65, 60 5 22 15, 29, 10, 33, 14, 90, 59, 8 6 11 24, 64, 52, 36, 73, 4 7 4 50, 34, 43, 18, 11, 83, 39 8 23 97, 53, 57, 28, 54, 87, 72 9 17 9, 6, 26, 92, 86, 44, 81, 17 10 6 25, 77, 88, 46, 38, 42, 49, 85 11 2 67, 91, 20, 45, 79, 94, 75Table 10
The relationship between location and demand distribution of scheme 2.
Scheme 2 No Alternative points Demand and supply points 1 8 25,92,96,68,55,4,23 2 3 60,45,10,28,87 3 12 51,52,30,54,18,66,9 4 25 17,13,88,37 5 15 15,50,3,47,11,29,26,85 6 17 89,76,58,24,94,81 7 9 74,90,35,71,48,16,98,31,78 8 16 97,53,31,61,70,62,93 9 21 39,72,86,46,27,7,5,19,12,84 10 14 32,59,22,21,44,36,38,67,8 11 18 69,65,20,41,57,73,2,49Table 11
The relationship between location and demand distribution of scheme 3.
Scheme 3 No Alternative points Demand and supply points 1 5 32,50,72,55,38,62 2 24 71,51,53,97,3,66,74,96,89 3 17 34,20,94,80,76,2 4 20 58,56,30,100,60,26,54 5 11 52,25,10,85,84,7,28,86 6 12 92,19,37,75,4,82,42,79,87 7 8 14,95,65,99,6,31,93,16,46 8 4 70,1,73,44,27,35,33 9 23 43,48,98,49,5,45,40 10 19 21,69,36,77,83,91,18 11 16 81,15,64,9,22,11,47 12 18 61,63,24,23,67,59


## 5.1. Logical Model

On the basis of the location theory of the maximum coverage, we will explore and study the multiplant location decision problem under the condition of stratified coverage. Based on factors such as the number of demand points and suppliers, the number and location of suppliers, the channels for realizing transactions with upstream and downstream enterprises, and the scale of production of factories, we have established an effective model for practical problems. Among them, the logical model is shown in Figure5. The establishment of a well-adapted multiplant site selection model can ensure the company’s effective interests while fully utilizing the resources’ utilization efficiency. It is of great significance to ensure social fairness and promote social development.Figure 5
Logical model of multiplant location.


## 5.2. Mathematical Model

The mathematical model was established with the goal of minimum cost and shortest time as follows.(24)
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0,1Among them, (24) represents the lowest cost target, and the specific target comprises three parts: raw material transportation cost, product transportation cost, and product production cost; (25) is the offline raw material transportation cost and online raw material transportation cost; (26) indicates the cost of transportation under the product line and the cost of online transportation; (27) represents the production cost of the product, including fixed costs and total variable costs; (28) represents the minimum time target and specific time objectives, including transportation time and return time; (29) means transport time, including raw material transport time and product transport time, online raw material transport time, and product transport time; (30) represents return time, including offline raw material return time and product return time, online raw material return time, and product return time; (31) indicates that the offline transport distance is less than the online transport distance and is greater than zero (because it is a ring load cover); (32) represents the number of built factories; (33) means that the total amount of transport is equal to the amount of transport multiplied by the number of shipments; (34) means total fixed cost constraint; (35) means that the plant can be purchased; (36) means that the plant can be sold; (37) means that the raw material resources are constrained; (38) means that the market capacity is limited; (39) means that the offline and online return costs are less than the variable cost. If the return cost is greater than the variable production cost, no return is required. (40) indicates that the production capacity is greater than a certain capacity. (41), (42), and (43) represent the range of values for the decision variable.


## 5.3. Algorithm Design

For solving coverage problems, we use genetic algorithms to get a feasible solution with high satisfaction. Genetic algorithm as a classical algorithm is an open source for researchers, which is a computational model that simulates the natural evolution of biological evolution theory and the processes of genetic mechanism. The specific steps are as follows.Step 1 (coding).
Binary coding is the most commonly used coding method in genetic algorithms. It includes advantages of (a) being simple and easy, (b) conforming to the minimum character set coding principle, and (c) being convenient for analyzing with the pattern theorem. If intervala,b is a domain and δ is the required precision, we can assume that the length of the chromosome is l, shown as follows.(44)l=log⁡b-aδ+1Step 2 (generating the initial population).
A subset of individuals (the same individual can exist) is generated by random numbers, called populationP0.The size of the population is measured by N. We assume that the first generation subset data is the initial value and starts iterating. Under the premise of ensuring the diversity of the solution, in order to improve the speed of convergence and get a satisfactory solution as soon as possible, we let the population size be N=100. The language of MATLAB is popsize=100.Step 3 (definition of fitness function).
We use boundary structure method to construct fitness function in this model. If the function value is as big as possible, then(45)FitfX=cmax-fx,fx<cmax0,others.
And if the function value is as small as possible, then(46)FitfX=fx-cmin,fx>cmin0,others.
Among them,cmax is the maximum estimate of f(x), and cmin is the minimum estimate of f(x).Step 4 (generate newborn populations).
Crossover and mutation operations and determination of operating parameters: This step mainly selects cross-mutation activities between chromosomes with better fitness values. Its purpose is to get the chromosomes with higher fitness values. At the same time it determines the relevant operating parameters, such as crossover rate, mutation rate, and population size. In this paper, in order to fully cross each generation, we selectpc=0.8 (generally 0.6-1.0) as crossover rate and pm=0.01 (generally 0.01-0.1) as mutation rate.Step 5 (matching facilities and demand points).
(1) from the view of plantQij, Qjk, dij, and djk, effective facility allocation ensures that facilities are built to the maximum demand point coverage.
(2) Determine whether the demand covered by the same factory is greater than the factory’s maximum production capacity. If not, then continue to runi+1; if yes, return (1) to redistribute.
Evaluate whether the demand covered by a plant exceeds the maximum capacity of the plant. If not, iterative procedure continues to runi+1; if yes, return (1) to redistribute.Step 6.
Based on the allocation scheme in step 5, the target value of the initial population can be found.Step 7 (recalculate the target value).
On the basis of the parameters in step 6, the single-point crossover, and single-point mutation on the chromosome, with high fitness in the initial population, regeneration of theN chromosome, and formation of a new population P1, we can get the new target value.Step 8 (optimal selection).
Combining the initial populationP0 and the newest species group P1, the newest population of the better N chromosome is still selected to form a new population P2 according to the target value. After the crossover and variation of N times, a satisfactory solution was selected.


## 5.4. Example Analysis

SY City is an important central city, advanced equipment manufacturing base, and national historical and cultural city of the provincial capital, subprovincial city, and megacities approved by the State Council and approved by the State Council. The city administers 10 districts, 2 counties, and 1 county level. The city has an area of 12,900 square kilometers, resident population of 8,291,000, and urbanization rate of 80.55%. At present, a 10 × 10 area Sino-German manufacturing industry park is planned to be calculated, and 50 supply points are provided. The specific coordinates and supply quantities are shown in Table5. The demand points are 100, and the specific coordinates and demand are shown in Table 6. At present, after the analysis of the basic conditions, there are 25 locations where the proposed site can be constructed. The specific coordinates and output are shown in Table 7.Table 5
Coordinates and quantity of suppliers for raw materials.
No Coordinates Supply quantity No Coordinates Supply quantity No Coordinates Supply quantity 1 58,2 741 18 39,42 705 35 3,92 448 2 29,91 656 19 20,92 657 36 33,73 682 3 83,80 632 20 84,16 322 37 42,51 479 4 20,78 543 21 50,67 372 38 15,15 331 5 21,70 619 22 1,42 394 39 84,55 534 6 71,37 330 23 43,69 371 40 37,97 341 7 41,11 494 24 10,82 576 41 87,4 371 8 6,1 676 25 62,98 764 42 14,50 503 9 55,59 392 26 18,55 787 43 52,78 494 10 22,14 405 27 7,34 610 44 97,63 557 11 26,52 479 28 38,46 631 45 35,21 723 12 9,5 518 29 9,98 576 46 93,15 462 13 10,10 504 30 88,57 671 47 15,52 723 14 96,48 541 31 41,97 459 48 90,44 749 15 20,82 433 32 65,3 564 49 59,63 752 16 3,22 797 33 70,53 537 50 7,25 611 17 56,85 655 34 89,10 468Table 6
Coordinates and demand quantity of demand point.
No Coordinates Demand quantity No Coordinates Demand quantity No Coordinates Demand quantity 1 99,23 489 35 62,68 468 68 38,44 430 2 23,50 942 36 65,98 747 69 58,21 799 3 24,83 447 37 5,95 472 70 96,47 593 4 2,76 565 38 42,84 699 71 99,25 841 5 47,99 902 39 88,41 797 72 24,25 854 6 62,99 714 40 93,9 782 73 5,63 710 7 52,29 833 41 96,82 894 74 3,85 405 8 27,25 690 42 35,75 534 75 25,25 559 9 56,8 486 43 68,96 465 76 35,73 901 10 14,24 732 44 3,27 450 77 88,99 608 11 54,82 946 45 77,52 775 78 73,44 782 12 37,10 475 46 1,78 568 79 96,23 614 13 11,59 810 47 48,10 892 80 56,63 843 14 23,16 643 48 32,90 755 81 85,22 618 15 5,95 477 49 65,17 413 82 39,87 736 16 66,24 953 50 21,73 769 83 15,62 625 17 40,61 736 51 76,11 561 84 2,5 604 18 32,26 704 52 47,68 706 85 1,58 424 19 67,12 414 53 32,69 724 86 48,17 910 20 15,38 518 54 92,7 840 87 20,38 979 21 80,100 791 55 24,4 825 88 76,33 789 22 25,36 468 56 61,25 740 89 57,44 614 23 93,86 927 57 34,0 746 90 69,32 582 24 4,98 756 58 14,50 780 91 56,85 787 25 95,94 482 59 86,46 585 92 64,47 648 26 75,35 839 60 76,19 726 93 96,8 692 27 35,35 560 61 38,82 825 94 29,66 418 28 71,54 797 62 37,20 759 95 77,50 794 29 1,2 535 63 82,52 680 96 73,17 600 30 26,46 478 64 47,14 537 97 31,80 970 31 30,52 691 65 53,11 593 98 92,15 906 32 94,4 888 66 40,55 790 99 68,56 473 33 91,53 849 67 42,36 856 100 52,69 856 34 98,63 663Table 7
Coordinates and quantity of plants.
No Coordinates Output No Coordinates Output No Coordinates Output 1 48 23 2620 9 41 90 3730 17 80 46 2910 2 6 95 3770 10 77 62 2810 18 14 13 3280 3 98 31 2780 11 36 66 3430 19 85 17 3960 4 28 79 3170 12 63 38 2870 20 24 3 2790 5 75 51 3620 13 72 58 3310 21 69 42 3200 6 92 47 2650 14 86 73 2590 22 91 64 2950 7 78 87 2760 15 43 34 2560 23 35 52 3800 8 40 89 2900 16 37 57 3380 24 5 97 2550 25 21 25 3840The values of other parameters are shown as follows.(47)w1off=1.5,w2off=1.8,w1on=3.0,w2on=3.5y1off=1%,y2off=2%,y1on=4%,y2on=5%τ1off=3d,τ2off=4d,τ1on=7d,τ2on=8dDj1,Dj2=1,3Then, we have the following.(48)R1=1,R2=3max⁡z=20There are three types of cars, the volume of transportation is as follows.(49)q1=100,q2=200,q3=300Cf=100000Cv=5ζ2off=1.2,ζ2on=1.5∑Qij≤10000∑Qjk≤15000The parameters of the genetic algorithm are set: the population size is 100, the crossover rate is 0.8, the variation rate is 0.01, and the iteration number is 200. As shown in Figure6Figure 6
The relationship between the number of iterations and the value of the objective function.From Figure6, as the number of iterations increases, the first target value with the minimum cost as the target is gradually reduced, while the second target value with the minimum time as the target is gradually increased, and the two targets constrain each other, so the target values are back against the law. When the number of iterations is 131 generations, the two target values are convergent. It can be seen that the genetic algorithm adopted in this section has good convergence for solving this model. Thus, the most suitable location and allocation of multiple plants can be determined. 3 approximate Pareto optimal solutions are randomly selected from the optimization front, and the plant number and target value are shown in Table 8.Table 8
Approximate to the target value of the Pareto optimal solution.
Scheme Numbers of plants Total cost Total time 1 11 1617.91 333.75 2 11 1628.62 329.74 3 12 1638.01 324.54In this case, we can know from the calculation results that the optimal number of plants to be built is 11 or 12 plants. Due to the different construction plans, the construction costs and transportation time will be different. There are three situations. In the first, the system cost is the smallest, and the system will spend more time to deal with the demand. In the second, the system cost is the largest, but the system will spend less time to deal with business processes such as purchase, sale, and return. The third situation is between the above two situations. Therefore, when choosing a plan, the decision-makers should consider the two objectives comprehensively, not only pay attention to the minimum cost or the shortest time. According to the actual conditions, including the external environmental factors and policy conditions of the enterprises, we make targeted decisions. In this case, we have chosen the second scheme in consideration of the dual-channel marketing model, the surrounding industrial conditions of the alternative sites, and the local government’s regional policies for the industry. The distribution of alternative points and demand points is shown in Tables9–11.Table 9
The relationship between location and demand distribution of scheme 1.
Scheme 1 No Alternative points Demand and supply points 1 10 3, 30, 47, 98, 51, 2, 78, 5 2 14 35, 55, 99, 1, 7, 48, 12, 100 3 20 63, 58, 19, 69, 84, 16, 80 4 7 74, 62, 95, 65, 60 5 22 15, 29, 10, 33, 14, 90, 59, 8 6 11 24, 64, 52, 36, 73, 4 7 4 50, 34, 43, 18, 11, 83, 39 8 23 97, 53, 57, 28, 54, 87, 72 9 17 9, 6, 26, 92, 86, 44, 81, 17 10 6 25, 77, 88, 46, 38, 42, 49, 85 11 2 67, 91, 20, 45, 79, 94, 75Table 10
The relationship between location and demand distribution of scheme 2.
Scheme 2 No Alternative points Demand and supply points 1 8 25,92,96,68,55,4,23 2 3 60,45,10,28,87 3 12 51,52,30,54,18,66,9 4 25 17,13,88,37 5 15 15,50,3,47,11,29,26,85 6 17 89,76,58,24,94,81 7 9 74,90,35,71,48,16,98,31,78 8 16 97,53,31,61,70,62,93 9 21 39,72,86,46,27,7,5,19,12,84 10 14 32,59,22,21,44,36,38,67,8 11 18 69,65,20,41,57,73,2,49Table 11
The relationship between location and demand distribution of scheme 3.
Scheme 3 No Alternative points Demand and supply points 1 5 32,50,72,55,38,62 2 24 71,51,53,97,3,66,74,96,89 3 17 34,20,94,80,76,2 4 20 58,56,30,100,60,26,54 5 11 52,25,10,85,84,7,28,86 6 12 92,19,37,75,4,82,42,79,87 7 8 14,95,65,99,6,31,93,16,46 8 4 70,1,73,44,27,35,33 9 23 43,48,98,49,5,45,40 10 19 21,69,36,77,83,91,18 11 16 81,15,64,9,22,11,47 12 18 61,63,24,23,67,59


## 6. Discussion

In this paper, we have a positive guiding significance for the site selection of manufacturing enterprises in the e-commerce environment. At the same time, due to constraints of energy, ability, and time, this paper has some limitations. For example, the main influencing factors and evaluation indicators identified in this paper come from the authors’ team’s consensus on past research and actual investigations of some enterprises. Due to the limitations of the regional sample, the determination of the influencing factors and the weights will be biased. Therefore, interested readers can further explore such site selection issues. In particular, in the model of multiplant location, we only consider two objectives of cost and time. But in the practical problems, the location of the enterprise usually has more objective constraints, such as service quality, service efficiency, production scale, production flexibility, and green environmental protection. Secondly, we do not consider the relationship between transport rates, unit distance, and transportation time. In the process of analyzing the example, many variables are changed into parameters in order to avoid heavy and complex calculation process. The location problem is the NP-hard problem, and it cannot find the optimal solution in the process of solving the problem, so the results provided in this paper are only a relatively satisfactory result. We hope that in the future research and work, we will have the opportunity to continue the discussion, and we also hope that more scholars who work on the location problem can continue excavating such problems.


## 7. Conclusion

Based on the research of location theory and classical location problems, combined with the current status of dual-channel purchase and sales of current equipment manufacturing enterprises, we propose the problem of single-plant and multiplant location under dual-channel purchase and sale model. Through the verification and analysis of the examples, we draw the following conclusions: (1) Equipment manufacturing enterprises with a dual-channel marketing model are a major feature of current industrial transformation and industrial revolution and are also the current trend of industrial manufacturing plants. (2) Relative to the traditional plant site selection decisions, with the dual-channel procurement and sales model, the location of the equipment manufacturing company’s factories is more focused on the degree of intelligence around the candidate sites and the development status of the surrounding industries. (3) The construction cost and operating cost are the main factors that need to be considered in the process of site selection. We cannot blindly pursue cost savings, but also pay attention to the development opportunities of enterprises, supporting the development of upstream and downstream enterprises.



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*Source: 3797131-2018-12-09.xml*