Fuzzy Evidential Reasoning Approach for LCMS Competitiveness Evaluation under Incomplete Information

In order to evaluate scientifically low carbon manufacturing system (LCMS) competitiveness under incomplete information, this paper, using fuzzy mathematics theory and evidential reasoning approach, proposes an evaluation method. Firstly, through fuzzy evaluation the influence factors of LCMS competitiveness are characterized by a set of evaluation grades. Secondly, the analytical evidential theory algorithms are used to aggregate the evaluation grades of multiple influence factors, and the assessed values of LCMS competitiveness are obtained under each evaluation index and the system overall goal. According to the evaluation value, the key influence factors of LCMS competitiveness, which need to be improved and enhanced, are found. Lastly, a numerical example is provided.


Introduction
Manufacturing enterprises, as a high discharge and high pollution industry, also have high energy consumption, which occupies a larger proportion in the national energy consumption [1].Low carbon level of manufacturing systems has turned into a competitive power and become a new important index to measure national and regional competitiveness.Consequently, the evaluation of low carbon manufacturing system (LCMS) competitiveness is increasingly important and has become a new research direction for the future manufacturing system theory.
As the operation process for LCMS has the dynamic characteristics, decision makers in the evaluation of LCMS competitiveness often face some uncertainty situations such as lack of decision data and incomplete information.Fuzzy evidence reasoning theory, as a reasoning method dealing effectively with such uncertainty information as randomness, and fuzziness, interval, now has become a powerful tool for expressing and synthesizing uncertainty information in fault diagnosis, target recognition, data fusion, and reliability analysis of mechanical system [2,3].This paper applies fuzzy evidence reasoning theory to the evaluation of LCMS competitiveness under incomplete information and then proposes an evaluation method of LCMS competitiveness based on fuzzy evidence reasoning model.Section 2 reviews the literatures related to this research.Section 3 designs the evaluation index system, the indicator system.Section 4 gives the evaluation model based on fuzzy evidential reasoning.Section 5 gives a numerical example.Finally, this paper is concluded in Section 6.

Literature Review
In recent years, with the global climate change, low carbon economy has got more and more attention, and low carbon manufacturing starts to become a new research hotspot.Current researches can be sorted into three aspects.
(1) Theories and methods for low carbon manufacturing and optimization and design of LCMS: Tridech and Cheng defined low carbon manufacturing and analyzed how LCMS can effectively use energy and resources when reducing carbon emissions intensity in the manufacturing process [4].Rothenberg et al. investigated energy saving and emission through lean workshop and in productive process and focus on the analysis of energy flow tracking problem for manufacturing system [5].Linton et al. researched into the production optimization decisions of a single enterprise under permits and trading mechanism as well as the dynamic evaluation of production system operation [6].Benjaafar et al. investigated how the quick response thought reduces energy use efficiency and increases carbon emissions [7].Venkat and Wakeland, through an example, analyzed energy consumption and carbon emissions of supply chain under lean production structure and unlean production structure [8].Tapio et al. examined how "zero inventory" in lean production, as a management mode, is advantageous to the improvement of the efficiency of production process, but frequent transportation would cause a decline in energy efficiency and an increase in carbon emission [9].Flapper et al., under the situation of improving energy efficiency through products recycling, proposed an optimization decision method for the production operation system of supply chain enterprises [10].
(2) Carbon flow and energy consumption calculation for LCMS: Ball et al. put forward a concept of "zero carbon manufacturing" and establish a system framework to analyze the relationship between the input and output of carbon flow and discuss the possibility of zero carbon manufacturing [11].Matthews et al., based on life cycle evaluation, proposed an evaluation analysis method of supply chain carbon emissions and carbon footprint [12].Jeswiet and Kara put forward a carbon emission index and a calculating method for carbon emission of the power consumption of parts manufacturing in power plants [13].Song and Lee presented a design method of low carbon product system based on BOM of emissions for embedded greenhouse gas [14].Tridech and Cheng, based on the analysis of characteristics of LCMS, provided some key factors and the calculating method affecting the energy consumption law in the machining process [15].Vijayaraghavan and Dornfeld researched into a real-time monitoring technology of energy consumption of machine tools and raised an energy data processing method of manufacturing system based on event stream processing technology [16].Gutowski et al. put forward an enterprise energy consumption calculation method based on power demand side management [17].
(3) LCMS competitiveness evaluation and carbon emission reduction strategies: Mckinnon, from the perspective of supply chain, pointed out the difficulties and the problems of calculating carbon footprint and implementing carbon label and analyzes the potential benefits for carbon footprint and carbon labeling [18].Perry et al. pointed out that the whole process integration system technology for process industries and local energy industries is more effective in reducing carbon footprint than in improving energy efficiency [19].Demailly and Quirion, taking the iron and steel industry as an example, discussed the impact of European carbon emissions trading scheme upon the industry production and income [20].
For global climate change's pressure, various countries propose in succession specific targets for carbon emissions reduction.Under this background, many issues such as the optimization of LCMS, the carbon flow characteristic analysis for manufacturing system, carbon emissions (carbon footprint) calculation, and carbon emissions reduction strategies and methods have got special attention.However, the problem of LCMS competitiveness evaluation has rarely been mentioned.In real life, the manufacturing system is a very complex uncertainty system.Under the uncertain environment, the research on competitiveness evaluation of LCMS is of great significance, both in theory and in practice.

The Design of the Evaluation Index System
There exists some difference between low carbon manufacturing and such concepts as green manufacturing, environmentally benign manufacturing, sustainable manufacturing, and so forth.Low carbon manufacturing, integrating the whole life cycle and considering comprehensively resources and environment efficiency, regards the carbon emissions reduction as the main line or its highlight, and so it belongs to the category of green manufacturing and sustainable manufacturing.LCMS is an input and output system, which shifts all kinds of manufacturing resources and energy into products, to ensure economic effectiveness and achieve carbon emissions reduction.It involves the whole life cycle of products (including market analysis, product design, process planning, machining process, transportation, product sales and after-sales service, and recycling) or partial link [21].LCMS competitiveness puts more emphasis on the coordinated and sustainable development between economy and environment based on traditional manufacturing system competitiveness.
There are many factors affecting the LCMS competitiveness.Based on the TQCF model of making decision attribute (Chryssolouris, 1992) [22], this paper takes system energy consumption and carbon emission into consideration.In other words, major competitive factors influencing LCMS are time, quality, cost, and carbon emission.
(1) Time (): it means the reaction speed of LCMS to the market, which mainly includes  1 from the starting of the market demand to product design time, from product design to design completion time  2 , and from product design completion to the time that customer demand is satisfied.
(2) Quality (): it means the quality of the products made of LCMS, which mainly includes the product design quality  1 , manufacturing quality  2 , and using quality  3 .
(  (𝐴) shows that evaluation grades of evaluation object on the index    are confidence degree of   ; namely,  , (   ()) = {(  ,  , ()),  = 1, 2, . . ., } (0 ≤  , () ≤ 1); the bigger value of  , () means the bigger degrees of belief given by decision makers.If ∑  =1  , () = 1, it means that the degree of belief given by decision makers is 100%, which is a complete evaluation.If ∑  =1  , () < 1, it means that the degree of belief given by decision makers is less than 1, which is an incomplete evaluation.If ∑  =1  , () = 0, it means that the degree of belief given by decision makers is equal to 0, which shows that decision makers cannot make any evaluation of object  due to the unknown [23].

Fuzzy Evidential Reasoning Model.
For an evaluation of LCMS competitiveness in uncertain environment, by using fuzzy evaluation technology, the paper gives the evaluation grades and belief degree of influence index for LCMS competitiveness.Then different evaluation grades are synthesized according to evidential reasoning algorithm (Yang et al., 2002).Finally, the competitive evaluation values of different index and total evaluation of LCMS are concluded [23].The specific evaluation process is as follows.
Step 1.Judge the degree of belief on the competitiveness of the bottom index.Based on historical data, knowledge, and experience, decision maker judges the belief degree affecting the bottom index    ( = 1, 2, . . ., ;  = 1, 2, . . ., ) of LCMS competitiveness and gives different evaluation grade.
Step 2. Construct the basic probability assignment (bpa) function.All indexes for each level where  , denotes the basic probability mass of the LCMS competitiveness being assessed to grade   on the index    and   is unassigned basic probability mass, including two parts:  , and m, , where  , is caused by the index weights and m, by the incompleteness of evaluation on indexes for the LCMS competitiveness.
Step 3. Apply the hierarchical recursion algorithm of evidential reasoning theory to synthesizing of all kinds of evidence.According to Steps 1 and 2, decision maker constructs the basic probability assignment function of bottom indexes.According to Dempster's rule of combination, the comprehensive evaluation grade and belief degrees of LCMS competitiveness are obtained by using the following calculation formula [23]: where   represents the belief degrees of the LCMS competitiveness being assessed to grade   and   is the degrees of belief unassigned to any individual evaluation grade after all the  basic attributes have been assessed.It denotes the degree of incompleteness in the evaluation generated.
With lots of generality, as shown in Table 1, suppose the highest preferred grade having the highest utility is  1 and the least preferred grade having the lowest utility is  5 .Then the maximum, minimum, and average utilities of object  are given by The evaluation value of LCMS competitiveness is as follows.

A Numerical Example
In this section, we give an example to illustrate the evaluation of LCMS competitiveness based on fuzzy evidential reasoning model.According to Section 3, we construct an index system of LCMS competitiveness (see Table 2).
(2) With the application of ( 2), the values of belief degree of each layer are synthesized from the bottom index (second index).Obtain the evaluation belief values of the first level indexes by synthesizing the evaluation belief values of the second level (see Table 2).Similarly, the belief values of evaluation grade for first level indexes , , , and  are synthesized, and then we get the values of belief degree for the total target evaluation grade (see Table 3).
Obviously, according to (3), competitive evaluation value of object  on indexes  and  is a point value, that is, (); other indexes and total goal are interval value, that is, [ min (),  max ()].Namely, the assessed values of the competitiveness are obtained under each index and total goal as follows.
: [ min (),  max ()] = [77.97,80.037], and the average assessed value of the competitiveness evaluation value for evaluation object  is 79.0035, and the degree of uncertainty is 2.067.(3) Analysis of the results is as follows.According to the above data, the general evaluation value of LCMS is 72.8408.The result shows that the system competitiveness level is the general grade, but in general it is still low, which indicates that the system competitiveness remains to be further improved.At the same time, we also see that there are some uncertainties in the evaluation of systemic competitiveness; that is, the degree of uncertainty for the system is  max () −  min () = 2.4635.The more the information and data related to the system competitiveness acquired, the smaller the uncertainty of evaluation process.
According to competitiveness evaluation value in Table 2 and , , , and , the competitiveness of LCMS evaluation index  is the weakest, and  is weaker, but  is the strongest in competitiveness.It shows that  is the key factor which results in the fact that system competitiveness is weak.Therefore, in order to improve the competitiveness of LCMS, the primary work is to improve the competitiveness of system  and then .

Conclusions
Evaluation of LCMS competitiveness under uncertain environment is a very complicated job, in which there are many uncertainty factors, and the information and data related to the evaluation are often ambiguous and incomplete.Therefore, it is difficult to use some traditional evaluation models based on probability theory and mathematical statistics.
Comparing our method with the others, we have found that it uses the belief degree, instead of a single numerical grading, to express the distribution of the evaluation grade.Secondly, it can reveal uncertain information.Moreover, with the method, the model construction difficulties due to the incomplete information and the lack of data can also be avoided.Meanwhile, experts' experience and knowledge are considered in the model construction process.Lastly, the numerical example shows that this method is able to reflect properly the behavior of LCMS and helps to find out the key factors that need improving so as to enhance the system competitiveness.In a word, our research offers a reference to the formulation of LCMS competitiveness strategy and so the improvement in LCMS competitiveness.
3) Cost (): it means the economic consumption of products made of LCMS, which mainly includes the design cost  1 , the raw materials cost  2 , energy consumption cost  3 , manufacturing producing cost  4 , and logistics cost  5 .  ( = 1, 2, . . ., ) are the evaluation grades of each index given by decision makers; namely, ( 1 ,  2 ,  3 ,  4 ,  5 ) = (stronger, strong, ordinary, weak, weaker).If evaluation grade for competitiveness evaluation index    is  1 , it denotes that competitiveness of object  in the aspect of attribute indexes is stronger.If evaluation grade for the index    is  3 , it denotes that competitiveness is ordinary, and so on. ,

Table 1 :
Evaluation grade of LCMS competitiveness.

Table 2 :
The index system of LCMS competitiveness.

Table 3 :
Belief degrees and evaluation grade for LCMS competitiveness and their factors.