Research Paper
Journal of Intelligent & Fuzzy Systems 32 (2017) 249–256 DOI:10.3233/JIFS-151494 IOS Press
249
Models for evaluating the technological innovation capability of small and micro enterprises with hesitant fuzzy information
Song-Qiang Wu∗ College of Economics and Management, NanJing Tech University, Jiangsu, China
Abstract. The enhancement of small and micro enterprises’ technological innovation capability not only helps enterprise increase its core competence to adapt to the changeful environment, but also helps our country to increase its national competence. In order to enhance small and micro enterprises’ technological innovation capability, enterprises should build a well technological innovation system, increase the technology innovation input and be active in technological innovation activities, and at the same time, it is essential for the government to play a role of leading, make effective policies and build good environment. Therefore, it is necessary to collect and study on small and micro enterprises’ technological innovation capability related data, cases and policies in order to know the current situation and trend of small and micro enterprises’ technological innovation capability. In order to make the above-mentioned work more convenient and more efficient, in this paper, we investigate the multiple attribute decision making for evaluating the technological innovation capability of small and micro enterprises with hesitant fuzzy information. Motivated by the ideal of dependent aggregation, we develop the dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator, in which the associated weights only depend on the aggregated hesitant fuzzy arguments and can relieve the influence of unfair hesitant fuzzy arguments on the aggregated results by assigning low weights to those “false” and “biased” ones and then apply them to develop an approach for multiple attribute decision making with hesitant fuzzy information. Finally, an illustrative example for evaluating the technological innovation capability of small and micro enterprises is given to verify the developed approach.
Keywords: Multiple attribute decision making, hesitant fuzzy information, dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator, technological innovation capability, small and micro enterprises
1. Introduction
With the development of the globalization of the world economy and the upsurge of knowledge economy, a new technological revolution will set off across the world. The characteristics of this worldwide trend embody the transformation from conventional industries to knowledge based indus- tries, the closer combination of technology and
∗Corresponding author. Song-Qiang Wu, Professor, Col- lege of Economics and Management, NanJing Tech University, Jiangsu 210009, China. Tel./Fax: +86 02558139565; E-mail: wusqnjut@njtech.edu.cn.
economy, the shortening of product life cycle and the emergence of high technology one after another. The accumulation of knowledge and innovation of technology are becoming the keys to the success of enterprises in this global competition, and significant factors to measure the science and technology com- petitiveness of a country. After China joined WTO, the world has experienced an unprecedented fierce competition. The innovative capacities and activi- ties of technology have become the impetus and source which determine the survival and development of enterprises. Technological innovation capacity has been considered publicly as a crucial factor of
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250 S.-Q. Wu / Evaluating the technological innovation capability of small and micro enterprises
improving the small and micro enterprises’s com- petitiveness [1–3]. Small and micro enterprises is the pillar of national economy and the main part of technological innovation, which indicates that the level of technological innovation capacity concerns the overall development of national economy. How to evaluate the technological innovation capacity in small and micro enterprises objectively, scientifically and effectively with combination of relative theoret- ical analysis of its own development characteristics will be particularly important to those corporations that compete to comprehend their own technological innovation capacity in the industries with appropri- ate strategies, enhance their competitive advantages and gain the best economic and social benefits [4–6]. Sarkar [7] had studied market uncertainty and cor- porate investment relationship in consideration of system risk conditions, and he thinks that increasing the uncertainty may increase the probability of invest- ment of enterprises to some low growth and low risk of investment project. Weeds [8] described if research and development is successful as a Poisson process, and it is used to describe the uncertainty of technology and of Hershey’s bad market opportunity arrival tim- ing. Hamilton [9] point out that due to R&D plays an important role in creating competitive advantage and, therefore, should be treated from a strategic perspec- tive, discussing in detail the multiple features of the R&D strategic options. Lint and Pennings [10, 11] mainly studied the innovation of real option in the process of marketization and pointed out that there were two choices, disposable rapid advancing, and slow advancing and related options opportunities and options value hid in slow advancing. Huchzermeier and Loch [12] proposed a decision model of multi- ple stages and considered in each phase of this model that managers had three solutions: continue to invest in the project, improve the project, and give up the project.
With the coming of the era of knowledge economy, a variety of new knowledge, technology innova- tion, market competition is increasingly fierce, on the basis of knowledge innovation of the technol- ogy innovation become the fundamental driving force of enterprise development. Therefore, how to establish an objective, standardized and feasible evaluation index system to improve the ability of tech- nological innovation, technological innovation and development to seek appropriate ways have impor- tant theoretical and practical significance. At present, research is through the establishment of evaluation system of combining qualitative and quantitative
indicators, and the use of multivariate statistical anal- ysis, analytic hierarchy process, BP neural network method for the whole process of technology innova- tion or all elements of a full of evaluation studies. However, these are more or less related to subjective evaluation indicators, impact of the data collection and evaluation results. Hesitant fuzzy set [13–30] is a useful tool for deal with fuzzy information for evaluating the technological innovation capabil- ity of small and micro enterprises. Thus, in this paper, we investigate the multiple attribute decision making for evaluating the technological innova- tion capability of small and micro enterprises with hesitant fuzzy information. Motivated by the ideal of dependent aggregation [31–33], we develop the dependent hesitant fuzzy Hamacher weighted geo- metric (DHFHWG) operator, in which the associated weights only depend on the aggregated hesitant fuzzy arguments and can relieve the influence of unfair hes- itant fuzzy arguments on the aggregated results by assigning low weights to those “false” and “biased” ones and then apply them to develop this approach for multiple attribute decision making with hesitant fuzzy information. In the next section, we introduce some basic concepts related to hesitant fuzzy sets. In Section 3 we propose the dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) opera- tor. In Section 4 we introduce the MADM problem with hesitant fuzzy information based on the depen- dent hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator. In Section 5, an illustrative example for evaluating the technological innovation capability of small and micro enterprises is pointed out. In Section 6 we conclude the paper and give some remarks.
2. Preliminaries
In the following, we introduce some basic concepts related to hesitant fuzzy sets.
Definition 1. [13] Given a fixed set X, then a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0, 1].
To be easily understood, Xia and Xu [14] express the HFS by mathematical symbol:
E = (〈x, hE (x)〉 |x ∈ X ) , (1) where hE (x) is a set of some values in [0, 1], denoting the possible membership degree of the element x ∈ X to the set E. For convenience, Xia and Xu [14] call
S.-Q. Wu / Evaluating the technological innovation capability of small and micro enterprises 251
h = hE (x) a hesitant fuzzy element(HFE) and H the set of all HFEs.
Definition 2. [14] For a HFE h, s (h) = 1#h ∑
γ∈h γ is called the score function of h, where #h is the number of the elements in h. For two HFEs h1 and h2, if s (h1) > s (h2), then h1 > h2; if s (h1) = s (h2), then h1 = h2.
In the following, Tan et al. [34] proposed a series of Hamacher geometric aggregation operators for HFEs based on the Hamacher operations [35–38] and Hamacher aggregation operators for HFE [39, 40].
Definition 3. [34] Let hj (j = 1, 2, · · · , n) be a col- lection of HFEs, then their aggregated value by using the HFHWG operator is also a HFE, and
HFHWGω (h1, h2, · · · , hn) = n⊗
j=1 ( hj
)ωj = ∪γ1∈h1,γ2∈h2,··· ,γn∈hn⎧⎪⎪⎪⎨ ⎪⎪⎪⎩
γ n∏
j=1 γ
ωj j
n∏ j=1
( 1 + (γ − 1)
( 1 − γj
))ωj + (γ − 1) n∏ j=1
γ ωj j
⎫⎪⎪⎪⎬ ⎪⎪⎪⎭ (2)
where ω = (ω1, ω2, · · · , ωn)T be the weight vector of hj (j = 1, 2, · · · , n), and ωj > 0,
∑n j=1 ωj = 1.
Definition 4. [34] Let hj (j = 1, 2, · · · , n) be a col- lection of HFEs, then their aggregated value by using the HFHOWG operator is also a HFE, and
HFHOWGw (h1, h2, · · · , hn) = n
⊗ j=1
( hσ(j)
)wj = ∪γσ(1)∈hσ(1),γσ(2)∈hσ(2),··· ,γσ(n)∈hσ(n)⎧⎪⎪⎨ ⎪⎪⎩
γ n∏
j=1 γ
wj σ(j)
n∏ j=1
( 1 + (γ − 1)
( 1 − γσ(j)
))wj + (γ − 1) n∏ j=1
γ wj σ(j)
⎫⎪⎪⎬ ⎪⎪⎭ (3)
where (σ (1) , σ (2) , · · · , σ (n)) is a permutation of (1, 2, · · · , n), such that hσ(j−1) ≥ hσ(j) for all j = 2, · · · , n, and w = (w1, w2, · · · , wn)T is the aggregation-associated weight vector such that wj ∈ [0, 1] and
∑n j=1 wj = 1.
Definition 5. [34] Let hj (j = 1, 2, · · · , n) be a col- lection of HFEs, then their aggregated value by using the HFHHG operator is also a HFE, and
HFHHGw,ω (h1, h2, · · · , hn)
= n
⊗ j=1
( ḣσ(j)
)wj = ∪γ̇σ(1)∈ḣσ(1),γ̇σ(2)∈ḣσ(2),··· ,γ̇σ(n)∈ḣσ(n)⎧⎪⎪⎨ ⎪⎪⎩
γ n∏
j=1 γ̇
wj σ(j)
n∏ j=1
( 1 + (γ − 1)
( 1 − γ̇σ(j)
))wj +(γ − 1) n∏ j=1
γ̇ wj σ(j)
⎫⎪⎪⎬ ⎪⎪⎭ (4)
where w = (w1, w2, · · · , wn) is the associated weighting vector, with wj ∈ [0, 1],
∑n j=1 wj
= 1, and ḣσ(j) is the j-th largest element of the hesitant fuzzy arguments ḣσ(j)(ḣσ(j) =
( hj
)nωj ,
j = 1, 2, · · · , n), ω = (ω1, ω2, · · · , ωn) is the weighting vector of hesitant fuzzy arguments hj (j = 1, 2, · · · , n), with ωj ∈ [0, 1],
∑n j=1
ωj = 1, and n is the balancing coefficient.
3. Dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator
Based on the hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator, then, we shall develop the dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator, in which the associated weights only depend on the aggregated hesitant fuzzy arguments and can relieve the influence of unfair hesitant fuzzy arguments on the aggregated results by assigning low weights to those “false” and “biased” ones.
Definition 6. Let hj (j = 1, 2, · · · , n) be a collection of HFEs, the geometric average value of the score function of hj (j = 1, 2, · · · , n) is defined as follows:
s ( h̄ )
= n √√√√ n∏
j=1 s ( hj
) (5)
Definition 7. Let h1 and h2 be two HFSs on X = {x1, x2, · · · , xn}, then the distance measure between h1 and h2 is defined as
‖h1 − h2‖ = |s (h1) − s (h2)| (6) Definition 8. Let hj (j = 1, 2, · · · , n) be a group of HFEs, and s
( h̄ )
the geometric mean of these hesitant fuzzy values, then we define the deviation of these hesitant fuzzy values as
252 S.-Q. Wu / Evaluating the technological innovation capability of small and micro enterprises
σ 2 = 1
n
n∑ j=1
∣∣s (hj) − s (h̄)∣∣2 (7)
Definition 9. Let hj (j = 1, 2, · · · , n) be a collection of HFEs, then we call
s ( s ( hj
) , s
( h̄ ))
= 1 − ∣∣s (hj) − s (h̄)∣∣
n∑ j=1
∣∣s (hj) − s (h̄)∣∣ ,
j = 1, 2, · · · , n. (8) the degree of similarity between the hesitant fuzzy values s
( hj
) and the mean s
( h̄ ) .
In real-life situations, the hesitant fuzzy numbers hj (j = 1, 2, · · · , n) usually take the form of a collec- tion of n hesitant fuzzy numbers which are provided by n different individuals or experts. Some individ- uals or experts may assign unduly high or unduly low preference values to their preferred or repug- nant objects. In this case, we shall give very low weights to these “false” or “biased” opinions, that is to say, the closer a preference value is to the mid one(s), the more the weight. As a result, based on (2), we define the hesitant fuzzy Hamacher weighted geometric(HFHWG) operator weights as follws:
wj = s ( s ( hj
) , s
( h̄ ))
n∑ j=1
s ( s ( hj
) , s
( h̄ )) , j = 1, 2, · · · , n. (9)
Obviously, wj ≥ 0, j = 1, 2, · · · , n and∑n j=1 wj = 1.
By the hesitant fuzzy Hamacher weighted geomet- ric(HFHWG) operator, we can derive the following definition:
DHFHWG (h1, h2, · · · , hn) = n
⊗ j=1
( hj
)s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄)) = ∪γ1 ∈h1 ,γ2 ∈h2 ,··· ,γn∈hn
⎧⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩
γ n∏
j=1
( γj
)s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄))
n∏ j=1
( 1 + (γ − 1)
( 1 − γj
))s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄)) + (γ − 1)
n∏ j=1
( γj
)s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄))
⎫⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎭
(10)
We call the Equation (10) the dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator. Consider that the aggregated value of the DHFHWG operator is independent of the ordering, thus it is also a neat operator.
Now, we can discuss some special cases of the DHFHWG operator with respect to the parameter γ.
• When γ = 1, DHFHWG operator reduces to the dependent hesitant fuzzy weighted geometric (DHFWG) operator as follows:
DHFWG (h1, h2, · · · , hn) = ∪γ1 ∈h1 ,γ2 ∈h2 ,··· ,γn∈hn⎧⎪⎪⎨ ⎪⎪⎩
n∏ j=1
( γj
)s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄))
⎫⎪⎪⎬ ⎪⎪⎭
(11)
• When γ = 2, DHFHWG operator reduces to the dependent hesitant fuzzy Einstein weighted geo- metric (DHFEWG) operator as follows:
DHFEWG (h1, h2, · · · , hn) = ∪γ1 ∈h1 ,γ2 ∈h2 ,··· ,γn∈hn⎧⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩
2 n∏
j=1
( γj
)s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄))
n∏ j=1
( 2 − γj
)s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄)) +
n∏ j=1
( γj
)s(s(hj ),s(h̄)) /
n∑ j=1
s(s(hj ),s(h̄))
⎫⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎭
(12)
4. An approach to multiple attribute decision making with hesitant fuzzy information
Let A = {A1, A2, · · · , Am} be a discrete set of alternatives and G = {G1, G2, · · · , Gn} be a set of attributes. If the decision makers provide several val-
S.-Q. Wu / Evaluating the technological innovation capability of small and micro enterprises 253
ues for the alternative Ai under the state of nature Gj with anonymity, these values can be considered as a hesitant fuzzy element hij . In the case where two deci- sion makers provide the same value, then the value emerges only once in hij . Suppose that the decision matrix H =
( hij
) m×n is the hesitant fuzzy decision
matrix, where hij (i = 1, 2, · · · , m, j = 1, 2, · · · , n) are in the form of HFEs.
Then, we develop the method to solve the MADM problems with hesitant fuzzy information. The method involves the following steps:
Step 1. Utilize the DHFHWG operator:
hi = DHFHWG (hi1, hi2, · · · , hin)
= n
⊗ j=1
( hij
)s(s(hij ),s(h̄i)) /
n∑ j=1
s(s(hij ),s(h̄i))
= ∪γi1 ∈hi1 ,γi2 ∈hi2 ,··· ,γin∈hin⎧⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩
γ n∏
j=1
( γij
)s(s(hij ),s(h̄i)) /
n∑ j=1
s(s(hij ),s(h̄i))
n∏ j=1
( 1 + (γ − 1)
( 1 − γij
))s(s(hij ),s(h̄i)) /
n∑ j=1
s(s(hij ),s(h̄i)) + (γ − 1)
n∏ j=1
( γij
)s(s(hij ),s(h̄i)) /
n∑ j=1
s(s(hij ),s(h̄i))
⎫⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎭
(13)
to derive the overall values hi (i = 1, 2, · · · , m) of the alternative Ai, where s
( h̄i
) = 1
n
∑n j=1 s
( hij
) .
Step 2. Calculate the scores S ( h̃i
) of the overall hes-
itant fuzzy preference value h̃i (i = 1, 2, · · · , m) to rank all the alternatives Ai (i = 1, 2, · · · , m) and then to select the best one(s).
Step 3. Rank all the alternatives Ai (i = 1, 2, · · · , m) and select the best one(s) in accordance with S
( h̃i
) (i = 1, 2, · · · , m).
5. Numerical example
With the acceleration of the economic globaliza- tion, the technical innovation which is as an important spark of the development of the world economy has been shown an unprecedented upsurge. All countries in the world are adjusting their economic structure, perfecting and developing their national innovation system in order to occupy the highest spot in the global economic competition. Because of the low car- bon, green, people-oriented peculiarity of the small and micro enterprises, it is important to advance
the new change of the function of regional devel- opment, ways of production and life and become the backbone small and micro enterprises. Then, the small and micro enterprises clusters play an important role in this process. They generate powerful strength through their own characteristics and influence the regional even the national innovation more and more significantly. So, the small and micro enterprises clus- ters become the common concerned theory in both practice and institute. While the technical innovation ability of the small and micro enterprises which is as the most crucial and active subject of the indus- trial clusters, also dramatically reflects the cluster technical innovation. Therefore, the research about the influence factors of the cluster- small and micro enterprises’ technical innovation ability not only has the important practical significance, also can indicate the direction of development of the cluster. Thus, in this section we shall present a numerical example for
evaluating the technological innovation capability of small and micro enterprises with hesitant fuzzy information in order to illustrate the method proposed in this paper. There is a panel with five possible small and micro enterprises Ai (i = 1, 2, 3, 4, 5) to evaluate. The experts selects four attribute to evaluate the five possible small and micro enterprises: ①G1 is the funding on technological innovation; ②G2 is the knowledge management & organizational learning; ③G3 is the enterprise system; ④G4 is the innovative culture to technological innovation human. In order to avoid influence each other, the decision makers are required to evaluate the five possible small and micro enterprises Ai (i = 1, 2, 3, 4, 5) under four attributes in anonymity and the decision matrix H =
( hij
) 4×4 is presented in Table 1, where
Table 1 Hesitant fuzzy decision matrix
G1 G2 G3 G4 A1 (0.3,0.4,0.6) (0.4,0.7) (0.5,0.6,0.8) (0.4,0.5) A2 (0.5,0.6,0.7) (0.4,0.5) (0.3,0.5,0.6) (0.3,0.4) A3 (0.5,0.7) (0.1,0.2,0.4) (0.3,0.5) (0.4,0.6) A4 (0.4,0.5,0.6) (0.3,0.4) (0.7,0.8) (0.2,0.3,0.4) A5 (0.5,0.6) (0.4,0.6) (0.6,0.7) (0.1,0.2,0.3)
254 S.-Q. Wu / Evaluating the technological innovation capability of small and micro enterprises
hij (i = 1, 2, 3, 4, j = 1, 2, 3, 4) are in the form of HFEs.
Based on the DHFHWG operator, then, in order to select the most desirable small and micro enterprise, we can develop an approach to multiple attribute decision making problems for evaluating the tech- nological innovation capability of small and micro enterprises with hesitant fuzzy information, which can be described as following:
Step 1. Aggregate all hesitant fuzzy value hij (j = 1, 2, 3, 4) by using the dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) oper- ator to derive the overall hesitant fuzzy values hi (i = 1, 2, · · · , 5) of the alternative Ai and calculate the scores s (hi) (i = 1, 2, 3, 4, 5) of the overall hes- itant fuzzy values hi (i = 1, 2, 3, 4, 5) of the small and micro enterprises (suppose that γ = 3):
s (h1) = 0.5305, s (h2) = 0.4849
s (h3) = 0.4127, s (h4) = 0.5344
s (h5) = 0.5284
Step 2. Rank all the small and micro enterprises Ai (i = 1, 2, 3, 4, 5) in accordance with the scores s (hi) (i = 1, 2, 3, 4, 5) of the overall hesitant fuzzy values hi (i = 1, 2, · · · , 5): A4 A1 A5 A2 A3 and thus the most desirable small and micro enter- prises is A4.
6. Conclusion
With the continuous growth of the world economic integration and the entry into China of the overseas enterprises, the small and micro enterprises which play a role in social and political market and improve scientifical technology must improve their technol- ogy innovation capability. Technological innovation capability as an important part of technological inno- vation management is of great significance. However, improving technological innovation capability of small and micro enterprises must study on evaluation index system and models of technology innovation capability. In this paper, we investigate the mul- tiple attribute decision making for evaluating the technological innovation capability of small and micro enterprises with hesitant fuzzy information. Motivated by the ideal of dependent aggregation, we develop the dependent hesitant fuzzy Hamacher weighted geometric (DHFHWG) operator, in which the associated weights only depend on the aggregated
hesitant fuzzy arguments and can relieve the influence of unfair hesitant fuzzy arguments on the aggregated results by assigning low weights to those “false” and “biased” ones and then apply them to develop an approach for multiple attribute decision making with hesitant fuzzy information. Finally, an illustrative example for evaluating the technological innovation capability of small and micro enterprises is given to verify the developed approach. In the future, we shall continue to develop more and more models and approaches for evaluating the technological innova- tion capability of small and micro enterprises based on the different decision analysis tecniques [41–50].
Acknowledgments
The work was supported by the National Social Science Foundation of China under Grant No. 13CGL044, “the 6th installment” of Special Program of China Postdoctoral Foundation (2013T60527), the 52 Batch of National Post Doctoral Fund (2012M521055), National Natural Science Founda- tion of China (71473120), Jiangsu province colleges “ Blue Project” young Key Teacher Project (No. 1,2014) and the 2014 Jiangsu provincial government scholarship studying abroad.
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