With the rapid development and popularisation of web service technologies, there are many
In recent years, WS selection issues (WSSIs) have caught the attention of many scholars [2], and the core issues are the description and evaluation of quality criteria. At present,
To meet the need to objectively describe uncertain QoS criteria and be well adaptive to interval characteristics of criteria through setting preference regions, recent sampling data set is used to describe uncertain QoS criteria and to evaluate comprehensive criteria based on
A WSSI is a multiobjective optimisation process [4], in which, WS quality variables play a positive role on customers’ satisfaction, the extent of which is different according to web types with different QoS [5]. UDDI, as criteria, helps users to find suitable WSs. Curbera et al. [6] introduced SOAP, WSDL and UDDI to unravel the WS web. Typical QoS properties include
A quality constraint tree [11] and a probability theory [12, 13] could be used to build different methods or models to address or measure QoS. Collaborative filtering algorithms are also used to select WSs [14, 15]. Other methods based on genetic algorithm (GA) were introduced in WSS [4, 16, 17]. There are still some othertheorybased approaches used for service selection and reuse, for example, QoS ontology [18] and group decision [19]. Some studies integrated multiple methods to improve selection effectiveness. For a ubiquitous web environment, Wang et al. [20] proposed a service selection approach involving three methods that are fuzzy logic control, mean particle swarm optimisation algorithm and local selection. Purohit and Kumar [21] used classification technique for WSS. Purohit and Kumar [22] discussed a trip planning case study to highlight the role of WSs in a smart city and described six learning models. Although these mentioned approaches could be effective to select suitable services that meet customers’ requirements, they could be too complex and it is necessary to build a simpler approach that can be easily conducted for customers to select WSs.
WSS means to choose a WS candidate from a WS set that could best meet users’ requests. Composite services have more than one QoS criterion, which means WSS is a multipleattributedecisionmaking (MADM) issue. As a method most commonly used to assess composite service QoS, SAWM has two main limitations, namey (1) it cannot show the extent of the users’ satisfaction and provide clear guidance to setting the right weights, and (2) it often has to conduct more iterations taxing to computational resources to have better weights. So, it is most necessary to clarify QoS factors affecting customers’ satisfaction.
Messac [25] and Messac et al. [26] discussed the limitations of SAWM and designed a new effective approach named physical programming (PP) that aims to cut down the computational intensity of massive problems and to put the design process into a more flexible framework. With PP, the value of the objective is divided into some continuous regions to uncover preference for each criterion. Moreover, preference functions are obtained from piecewise spline segment interpolation. After that, all preference functions are integrated into an aggregate preference function which is the objective function. The option with the minimum value will be optimum. Among the linear programming methods, the LPP method is one of the typical preferencebased algorithms that is often used with dynamic weight for multiobjective collaborative optimisation [27]. The LPP method has been applied in more environments [28,29,30] or integrated into other methods [31,32,33,34]. Therefore, in this paper, we use LPP to analyse QoS to solve WSSIs by simplifying the process of WSS.
QoS can be modelled based on different criteria that have been defined in a wide range [35]. From the perspective of stability, the criteria can be categorised into static and dynamic. Static criteria are those that remain the same in a certain stage, such as service price, while dynamic criteria are those of a more dynamic nature due to network conditions, host performance, location, periods and so on. For example, the
In this article, to simulate a true state, sampling data sets are used to describe a dynamic criterion. A sample size will be set according to potential candidates and is decided by a true environment. For instance,
QoS of WS (WS1)
Price ($)  1.5 
[3.5, 3.4, 3.3, 3.7, 3.6]  
[0.88, 0.85, 0.90, 0.84, 0.83]  
[7.6, 7.8, 7.5, 8.0, 7.9] 
Messac [25] proposed
In this article, the objective
There are six preference regions used to express both qualitative and quantitative depictions. The six preference regions of the 1S preference function are shown in Figure 1.
The One vs. Other criteria rule (OVO rule) will be applied to operating PP [37]. There are two options for calculating the values of preference functions.
For the OVO rule, Option 1 is preferred than the Option 2, which means that the worst candidate will always be helped first.
A preference function has the following properties:
A lower value of the preference function is preferred than a higher value.
The function is strictly positive.
The function is continuous, piecewise linear and convex.
The values of the function at given regions are the same for different types.
The magnitude of the function's vertical excursion across any region must satisfy the OVO rule.
The five properties above can be turned into a set of mathematical formulas as follows:
Finally, the preference function takes the form as shown in the following:
Then, the form of the aggregate preference function is shown in the following form:
There are two effects of the
For the
By simplifying the above inequality equation, a new inequality system can be got as the following shows:
Messac et al. [26] built a preference function, like Eq. (3) rather than a simpler form because
There are many WSs with same functional properties but different QoS, and a WSSI means a process during which users select a proper WS.
QoS of these WSs is described by a set of QoS:
At present, a conventional solution to WSSIs under uncertain QoS is using a
In the QoS matrix of a WS set shown in Eq. (6), the
For each QoS criterion, its value is also a kind of set. We can calculate the value by using its preferred function for each element and then take the sum of these values as the evaluation value of this criterion. The sum of evaluation values of all criteria is the comprehensive assessment value of QoS of the WS. When we get all comprehensive assessment values of all WSs, the minimum value should be selected from these comprehensive assessment values, for the WS that the minimum value corresponds to is the one that best serves the users’ preferences.
There are 50 WS candidates with same functional properties but different
QoS matrix of WS set
WS_{1}  [3.0,3.7,3.5,3.6,4.0]  [0.78,0.85,0.79,0.89,0.81]  [7.1,7.4,7.7,7.9,8.0] 
WS_{2}  [4.0,3.6,3.1,4.1,4.5]  [0.69,0.71,0.77,0.79,0.80]  [6.8,6.9,7.7,7.1,7.5] 
WS_{3}  [2.8,2.9,3.4,5.4,3.0]  [0.80,0.79.0.90,0.88,0.85]  [7.5,7.9,8.0,8.5.8.3] 
…  …  …  … 
WS_{50}  [2.8,4.5,4.1,3.7,3.9]  [0.77,0.75,0.80,0.71,0.68]  [7.9,6.9,7.2,7.4,7.5] 
Five steps are designed to provide a solution to the above case.
Preference regions for
Criterion 






2.5  3.0  3.5  4.2  5.5 
Both
Preference regions for
Criterion 






0.90  0.85  0.80  0.75  0.69  
8.3  8.0  7.5  7.0  6.4 
With the solving process of LPP, based on both preference regions and the
Set the initial value:
Calculate
Figure out
Work out the preference functions of the three criteria:
Preference functions of all dynamic criteria can be solved after parameters are calculated. All the functions are as follows:
The preference function for
The preference function for
The preference function for
Based on the preference functions of the three criteria, evaluation values of all criteria can be obtained for the comprehensive values of QoS of each candidate. Among these comprehensive values, QoS (WS23)=14.528 is the minimum. Thus, the minimum value corresponds to WS23, which is the final WS. As a result, WS23 best suits users’ preferences.
Setting boundaries for preference region,
Setting of dynamic weight,
As a result, it is necessary to consider users’ specific needs and the psychology of different user groups when the evaluation standards of WSs are set. For different user groups, it is necessary to set a unique preference region boundary,
LPP is an optimisation algorithm, which entirely releases the decisionmakers from the process of choosing weights [26]. Specifically, LPP avoids subjectivity when determining physically meaningless weights and ratings, to calculate the total scores of the decision. Thus, in contrast with other methods, LPP costs less time to make the criteria design as well as lessens the risk of costly design changes [38]. Additionally, in a multiple criteria environment, LPP can allow decisionmakers to employ a rational manner to express their preference for each objective of interest [39], and it brings an efficient and practical decisionmaking process into practice.
Recent sampling data sets are used to describe uncertain QoS criteria, which indicates that it could be better to follow the true state of uncertain criteria. Because of the limitations of the
To improve the computational efficiency, an improved LPP is conducted, with which a proper initial value of convexity parameters is determined without iteration, and the form of a preference function can be simplified. A WS with a minimum comprehensive assessment value is the optimal selection. The computational example proves that the LPPbased method with a general description is feasible and effective to evaluate aggregate QoS values of composite services.
In fact, the
QoS of WS (WS1)
Price ($)  1.5 
[3.5, 3.4, 3.3, 3.7, 3.6]  
[0.88, 0.85, 0.90, 0.84, 0.83]  
[7.6, 7.8, 7.5, 8.0, 7.9] 
Preference regions for Response Time
Criterion 






2.5  3.0  3.5  4.2  5.5 
QoS matrix of WS set
WS_{1}  [3.0,3.7,3.5,3.6,4.0]  [0.78,0.85,0.79,0.89,0.81]  [7.1,7.4,7.7,7.9,8.0] 
WS_{2}  [4.0,3.6,3.1,4.1,4.5]  [0.69,0.71,0.77,0.79,0.80]  [6.8,6.9,7.7,7.1,7.5] 
WS_{3}  [2.8,2.9,3.4,5.4,3.0]  [0.80,0.79.0.90,0.88,0.85]  [7.5,7.9,8.0,8.5.8.3] 
…  …  …  … 
WS_{50}  [2.8,4.5,4.1,3.7,3.9]  [0.77,0.75,0.80,0.71,0.68]  [7.9,6.9,7.2,7.4,7.5] 
Preference regions for Reliability and Credibility
Criterion 






0.90  0.85  0.80  0.75  0.69  
8.3  8.0  7.5  7.0  6.4 
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Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leaderfollower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on ecommerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long shortterm memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSRAHP model A study of local smoothnessinformed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H_{2}N_{2} atmosphere Algorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Crossborder Ecommerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model ChineseEnglish Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model