Grey Language Hesitant Fuzzy Group Decision Making Method Based on Kernel and Grey Scale
|
|
- Bennett Ward
- 5 years ago
- Views:
Transcription
1 Internationa Journa of Environmenta Research and Pubic Heath Artice Grey Language Hesitant Fuzzy Group Decision Making Method Based on Kerne and Grey Scae Qingsheng Li * ID, Yuzhu Diao, Zaiwu Gong ID and Aqin Hu Schoo of Business, Linyi University, Linyi , China; diaoyuzhu@126.com (Y.D.); zwgong26@163.com (Z.G.); huaqin@yu.edu.cn (A.H.) * Correspondence: iqingshengguani@163.com; Te.: Received: 29 January 2018; Accepted: 27 February 2018; Pubished: 2 March 2018 Abstract: Based on grey anguage muti-attribute group decision making, a kerne and grey scae scoring function is put forward according to the definition of grey anguage and the meaning of the kerne and grey scae. The function introduces grey scae into the decision-making method to avoid information distortion. This method is appied to the grey anguage hesitant fuzzy group decision making, and the grey correation degree is used to sort the schemes. The effectiveness and practicabiity of the decision-making method are further verified by the industry chain sustainabe deveopment abiity evauation exampe of a circuar economy. Moreover, its simpicity and feasibiity are verified by comparing it with the traditiona grey anguage decision-making method and the grey anguage hesitant fuzzy weighted arithmetic averaging (GLHWAA) operator integration method after determining the index weight based on the grey correation. Keywords: kerne and grey scae; grey anguage; hesitant fuzzy; grey correation 1. Introduction The deepening interdependence and mutua infuence of the word economy and the appication of the new generation of Internet information technoogy have significanty infuenced the decisions of the state and enterprises. The compexity, uncertainty and coectivization of decision-making information in poitics, the economy and miitary have become the new norma of decision making. A compex decision-making environment and the imited cognitive abiity of decision makers require overcoming the constraints of traditiona sets. In 1965, the American cybernetics expert Zadeh [1] first proposed the concept of the fuzzy set, which expresses the idea of uncertainty through membership degree and has been widey used in various fieds, such as management decisions, medica diagnosis and engineering contro. The Bugarian schoar Atanassov [2] argued that the membership degree of traditiona fuzzy sets can ony be the numbers between 0 and 1, which cannot dig deeper into the fuzzy degree in actua decision making. Based on the above consideration, Atanassov studied three dimensions of membership degree, non-membership degree and hesitation degree, and gave the reationa expression, which was further deveoped into an intuitive fuzzy set. Since Zadeh proposed fuzzy sets, many schoars have studied their extension forms, such as the intuitionistic fuzzy set (IFS) [3], the type 2 fuzzy set [4,5], the fuzzy mutipe set, and [6,7] the hesitant fuzzy set. This incudes group decision making based on information integration and the measurement of hesitant fuzzy sets. Based on Archimedes T-modue and S-modue, Xia Meimei gave the rues of hesitant fuzzy operation and the information integration operator of hesitant fuzzy decision; defined the distance, simiarity, correation, entropy and cross-entropy of hesitant fuzzy sets; and discussed their properties. Moreover, their correation was studied in order to appy this to group decision-making probems [8]. These studies have aid a foundation for the study of hesitant fuzzy group decision making based on the kerne and gray eve. Int. J. Environ. Res. Pubic Heath 2018, 15, 436; doi: /ijerph
2 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 Increasingy compex decision information requires experts to make group decisions in more cases. Experts need to face the compex environment with both compete and incompete information, both definite information and uncertain information. The compex group decision-making environment has aso attracted the attention of domestic schoars. In 1982, Deng Juong, a Chinese schoar, pubished the first paper about the grey system entited The Contro Probems of Grey Systems in Systems and Contro Letters [9]. In the same year, Deng Juong pubished the first Chinese grey system paper Grey Contro System in the Journa of the Huazhong Institute of Technoogy [10]. These two innovative cassica papers have aroused wide attention and successfuy initiated an emerging trans-discipinary science grey system theory [11]. The grey muti-attribute decision-making method was studied by [12 15]. Youchen et a. studied the inverse order probem of the grey decision mode [16]. Zhang Na et a. estabished the muti-stage grey situation group decision mode based on the Orness measure constraint [17]. Liu Sifeng et a. expored the two-stage grey comprehensive measurement decision mode and the improvement of the trianguar whitening weight function [18]. Luo Dang and Liu Sifeng studied the grey reationa decision method of the grey reationa decision method and an incompete information system [19,20]. Dang Yaoguo et a. studied the grey reationa decision mode based on a dynamic muti-index [21]. Jiang Shiquan et a. put forward a grey reationa decision mode based on area [22]. Xie Ming et a. studied the grey reationa anaysis method of muti-attribute group decision making based on the anaytic hierarchy process (AHP) and its appication [23]. Chen Xiaoxin and Liu Sifeng put forward a grey reationa decision method with partia weight information and a preference to scheme [24]. The rapid deveopment of the grey reationa group decision-making method provides the research foundation for the fuzzy grey reationa decision making in this paper. The differences of knowedge eve, judgment experience and preference of decision experts wi inevitaby ead to the deviation of information interpretation, which further causes uncertain decision-making probems accompanied by fuzzy and grey characteristics, namey, the grey fuzzy muti-attribute group decision-making probem. The intuitionistic fuzzy decision making method was studied by [25 29] on the basis of grey correation combined with evidence reasoning and the expert system uncertainty factor. Hu Lifang et a. organicay combined grey with fuzzy, and put forward the grey fuzzy muti-attribute decision making method under the frame of the cosed word [30]. Li Peng et a. put forward the combination method of grey reation and Dempster-Shafer evidence theory, which was appied to mutipe-criteria decision making (MCDM) based on the intuitionistic fuzzy set (IFS) [31]. Some schoars from Nanjing University of Aeronautics and Astronautics combined grey system with fuzzy theory, and put forward reevant decision methods. For exampe, Li Peng, Liu Sifeng and Zhu Jianjun studied the combination of GR, DS and MYCIN, which was appied to IFS decision-making [32,33]. Liu Yong and Liu Sifeng studied the dynamic muti-attribute grey reationa decision making method based on interva intuitionistic fuzzy [26]. In [34], the grey correation method and non-inear programming mode were used to obtain the optima weight and study the reevant decision methods. The study of grey system and fuzzy combination promotes the fusion and promotion of the uncertainty decision method. Research in the fied of grey anguage muti-attribute decision-making is reativey scarce. Wu Jianwen studied the grey anguage muti-criteria decision method and its appication [35]. Wang Jiang et a. put forward a muti-criteria decision method based on interva grey uncertain anguage [36]. Liu Peide et a. proposed a muti-attribute group decision-making method based on a geometric weighted integration operator of interva grey anguage variabes [37]. Wang Hehua studied the information aggregation mode of anguage evauation based on grey decision and its appication [38]. Li Qingsheng et a. [39 41] put forward the grey hesitant fuzzy set and extended it to the grey anguage form, namey, the grey anguage hesitant fuzzy set, which can convenienty use the specia decision method of a grey system and effectivey sove the probem of hesitant fuzzy decision making. How to take the advantages of a grey system to make grey anguage hesitant fuzzy decision
3 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 expand the appication scope of a grey system is significant for enriching and perfecting grey system theory and the organic combination of uncertain decision making. This paper proposes a kerne and grey scae-scoring function of the grey anguage hesitant fuzzy set which uses the grey correation degree to make decisions. 2. Preiminaries 2.1. Grey Language Set The concepts of anguage vaue and grey scae are generay combined in some practica decision-making processes, so that the decision information is more in ine with the decision habit, which is more ike the description of kerne and grey scae based on a grey system. Wu Jianwen and Wang Jianjiang put forward the grey anguage set by studying the muti-criteria decision and grey system of anguage, and discussed the probem of muti-attribute decision-making based on grey anguage. For exampe, a coege counseor gives the evauation resuts of exceent, good, fair, quaified and faiing. If a coege student is rated as good, the membership degree of the comprehensive evauation vaue of good is 1. However, the anguage evauation vaue of membership degree of 1 is often impractica, as it requires counseors to grasp comprehensive information about the student, incuding earning, interpersona reations, comprehensive abiity in the famiy, society, cassroom and dormitory, etc. Obviousy, this is impossibe in reaity. The decision maker s imited rationaity principe proposed by Simon is coser to reaity. If the counseor can give the membership degree and incompete degree of information (grey scae), it wi be very hepfu for the scientific evauation of coege students. Therefore, evauation vaues in the foowing form can be obtained: A = {x, H i, µ i, v i x X } = {x, good, 0.8, 0.4}, which represents the membership degree to which the coege student is evauated as good at 0.8, and the information uncertainty at Definition of Grey Language Set Definition 1. Set the discourse domain X, and anguage phase set H = {H 0, H 1, H 2, H T 1, H T }. For x X, there is a corresponding anguage vaue H i (H i H). If the membership degree of x to H i is the grey number in [0, 1], its point grey scae is v(x), A = {x, H i, µ i, v i x X } is a grey anguage set (GLS) on X. µ Hi (x) is the membership degree, which can be understood as the cose degree of x to H i [38]. Different from intuitionistic fuzzy sets,v(x) represents the extent to which the decision maker's information is incompete, namey, grey scae, instead of non-membership degree. Athough there is no necessary correation between grey scae v(x) and membership degree µ Hi (x), the smaer the vaue of µ i + v i 1, the better the reiabiity. In genera, the grey anguage set A is abbreviated as A = {H(x), µ(x), v(x) x X }. If a = {H(x), µ(x), v(x)} is GLS, any specific grey anguage eement a in A can be abbreviated as GLE. If the vaues of membership degree and grey scae in a is µ(x) = 1, v(x) = 0, a does not have incompete information degree, and retreats into the genera anguage number Agorithm of Grey Language Number (1) Set a i = {H i, µ i, v i } and a j = { } H j, µ j, v j as arbitrary GLE, according to the foowing rues [35]. a i a j = (H i H j, i µ i+j µ j i+j, max(v i, v j )), when H i and H j are not equa to H 0 simutaneousy. a i a j = (H i H j, max(µ i, µ j ), max(v i, v j )), when H i and H j are equa to H 0. In the expression and cacuation process of grey anguage in this paper, if there is no specia expanation, it is operated as H i and H j are not equa to H 0 simutaneousy. (2) ra i = {rh i, µ i, v i } [35]. (3) a i a j = (H i H j, µ i µ j, v i + v j v i v j ) [35]. (4) a λ i = (H λ i, µλ i, 1 (1 v i) λ ) [35].
4 Int. J. Environ. Res. Pubic Heath 2018, 15, of Properties of Grey Language Number Agorithm Assume that a i = {H i, µ i, v i }, a j = { } H j, µ j, v j and ak = {H k, µ k, v k } are any three grey anguage numbers, and the arithmetic operation has the foowing properties: (1) a i a j = a j a i. (2) (a i a j )a k = a i (a j a k ). (3) a i a j = a j a i. (4) (a i a j )a k = a i (a j a k ). (5) λa i λa j = λ(a i a j ), λ > 0. (6) λ 1 a i λ 2 a i = (λ 1 + λ 2 )a i, λ 1 > 0, λ 2 > 0. (7) (a i a j ) λ = a λ i a λ j, λ 0. (8) a i λ 1 a i λ 2 = a i (λ 1 +λ 2 ), λ 1 0, λ Size of Grey Language Numbers Definition 2. Assume a i = {H i, µ i, v i } and a j = { H j, µ j, v j } are any two grey anguage numbers [38]. If (1 v i ) µ i IH i < (1 v j ) µ j IH j, a i is smaer than a j, denoted as a i < a j ; If (1 v i ) µ i IH i > (1 v j ) µ j IH j, a i is greater than a j, denoted as a i > a j ; If (1 v i ) µ i IH i = (1 v j ) µ j IH j, and H i = H j, µ i = µ j, v i = v j, a i is strongy equa to a j, denoted as a i = a j ; If (1 v i ) µ i IH i = (1 v j ) µ j IH j, and at east one of H i = H j, µ i = µ j, v i = v j, is tenabe, a i is weaky equa to a j, denoted as a i a j. Where, I is subscript operator Hesitant Fuzzy Set When the attribute of a given scheme beongs to the membership degree of a certain set, it is not necessariy a random interva nor does it obey a certain distribution. The performance wi be more prominent in group decision making. The membership degree given by different groups may be uncoordinated, and sometimes it is not necessary to reach consistent possibe vaues. To this end, Torra [42,43] proposed the hesitant fuzzy set which can better meet the needs of group decision making Definitions Reated to Hesitant Fuzzy Sets Definition 3. Assume X is a given set, and hesitant fuzzy set is the projection from X [0,1], abbreviated as A = { x, h(x) x X }. Where, h(x) is the set of severa hesitant fuzzy numbers (HFN) in [0,1]. HFN is the membership degree vaue of x X beonging to set A. Obviousy, there is 0 HFN 1 [42,43]. The hesitant fuzzy eement contains the possibe hesitant fuzzy number in the set. The ength of data in each hesitant fuzzy eement is ikey to vary. The foowing method is used to determine the size of the hesitant fuzzy eement. Definition 4. Any hesitation fuzzy eement h, s(h) = 1/ 1 γ hγ is score function, where is the number of eements in h. There are two hesitant fuzzy eements h 1, h 2. If s(h 1 ) > s(h 2 ), h 1 > h 2. If s(h 1 ) = s(h 2 ), h 1 h 2. Torra beieved that membership degree at both ends of hesitant fuzzy can be understood as an intuitionistic fuzzy number. Definition 5. The enveope of any hesitant fuzzy number h can be converted to intuitionistic fuzzy number through the foowing formua: A env (h) = (h, (1 h + )), where h = minγ, h + = maxγ γ h [42,43]. Based on the connotation of hesitant fuzzy enveope reation [8] studied the enveop reation set operation. (1) A env (h c ) = (A env (h)) c.
5 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 (2) A env (h 1 h 2 ) = A env (h 1 ) A env (h 2 ). (3) A env (h 1 h 2 ) = A env (h 1 ) A env (h 2 ). (4) A env (h λ ) = (A env (h)) λ. (5) A env (λh) = λa env (h). (6) A env (h 1 h 2 ) = A env (h 1 ) A env (h 2 ). (7) A env (h 1 + h 2 ) = A env (h 1 ) + A env (h 2 ) Agorithms of Hesitant Fuzzy Set Assume any three hesitant fuzzy numbers h, h 1, h 2, and Torra [42,43] studied their agorithms: (1) h c = γ h {1 γ}. (2) h 1 h 2 = γ1 h 1,γ 2 h 2 max{γ 1, γ 2 }. (3) h 1 h 2 = γ1 h 1,γ 2 h 2 min{γ 1, γ 2 }. On the basis of studying the operation of Archimedes T-modue and S-modue [8] studied the agorithm of hesitant fuzzy operation. There are arbitrary hesitant fuzzy eements h, h 1, h 2. (1) h λ = γ h { k 1 (λk(γ)) }. (2) λh = γ h { 1 (λ(γ)) }. (3) h 1 h 2 = γ1 h 1,γ 2 h 2 { k 1 (k(γ 1 ) + k(γ 2 )) }. (4) h 1 + h 2 = γ1 h 1,γ 2 h 2 { 1 ((γ 1 ) + (γ 2 )) }. where (t) = k(1 t), and k : [0, 1] [0, + ) is a stricty decreasing function. The hesitant fuzzy operation rue can be defined as beow: Definition 6. There are three hesitant fuzzy numbers h, h 1, h 2. (1) (h c ) λ = (λh) c. (2) λ(h c ) = (h λ ) c. (3) h c 1 hc 2 = (h 1 + h 2 )c. (4) h c 1 + hc 2 = (h 1 h 2 )c. where, h c = γ h {1 γ} [8]. Definition 7. Assume h i (i = 1, 2,, n) as a set of hesitant fuzzy numbers, ω = (ω 1, ω 2,, ω n ) T satisfie, n 1 ω i = 1, ω i 0, (i = 1, 2,, n), ATS HFWA(h 1, h 2,, h n ) = n ω i h i. i=1 ATS HFWA is a hesitant fuzzy weighted arithmetic averaging operator based on Archimedes T-Modue and S-Modue. ATS HFWG(h 1, h 2,, h n ) = ATS HFWG is a hesitant fuzzy weighted geometric averaging operator based on Archimedes T-Modue and S-Modue [8] Kerne and Grey Scae Definitions of Kerne and Grey Scae The study of grey number operation in grey system theory has attracted attention for a ong period of time, but no satisfactory resut has been achieved. Definition 8. Let the grey number [a, a], a < a in the absence of grey number vaue distribution information [44]: If is a continuous grey number, ˆ = 1 2 (a + a) is the kerne of grey number. n i=1 h i ω i.
6 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 If is discrete grey number, a i [a, a], (i = 1, 2,, n) is the possibe vaue of grey number. Its average vaue is the kerne of grey number, namey, ˆ = 1 n n i=1 a i is the kerne of grey number. Definition 9. If is a random grey number, and the mathematica expectation E( ) can be cacuated, E( ) is the kerne of random and can expressed as ˆ = E( ) [44]. Definition 10. Set the background or a of the grey number is Ω, µ( ) as the measure of interva grey number taking the number fied. g o ( ) = µ( )/µ(ω) is the grey eve of grey number, which can be understood as the ratio of a possibe membership degree to background, abbreviated as g o [44]. The properties of Ω and the measure show that the grey scae satisfies the foowing properties: grey scae satisfies the criterion, namey, 0 g o ( ) 1. When g o = 0, the uncertainty is 0, which is the white number for compete certainty; When g o = 1, the uncertainty is 1, which is the back number for compete uncertainty; When 0 < g o ( ) < 1, it is an uncertain grey number. The grey scae refects the uncertainty degree of the things described by the grey number. The white number is a competey certain number, and the back number is competey unknown. When g o is coser to 0, the uncertainty of the grey number is smaer. When g o is coser 1, the uncertainty of the vaue range is greater. Definition 11. ˆ is the kerne of grey number, and g o is the grey scae. ˆ g o is the simpified form of grey numbers. ˆ g o incudes a information of vaue, and has one-to-one correspondence [44]. On the one hand, ˆ g o can be cacuated according to the definition of the kerne and grey scae. On the other hand, the range of can aso be cacuated according to ˆ g o [45] proposed the agorithm based on ˆ and g o and the grey scae non-decreasing axiom Agorithms of Kerne and Grey Scae Axiom 1. Grey scae non-decreasing axiom: grey scae is equa to the maximum grey eve of a grey numbers invoved in the operation, and it wi not decrease [44]. According to Axiom 1, the simpified form k (gk o ) based on the grey number can obtain the grey data agebraic agorithm [45]. (1) k (gk o) + m (gm) o = ( k + m )(gk o( go m). (2) k (gk o) m (gm) o = ( k m )(gk o go m). (3) k (gk o) m (gm) o = ( k m )(gk o( go m). (4) k (gk o) m (gm) o = ( k m )(gk o( go m) ( m = 0). (5) m k (gk o) = (m k )(gk o). (6) ( k (gk o))r = ( k ) r (gk o). Grey heterogeneous data may be incuded by the interva grey number, discrete grey number, rea number or other grey information. A the different data structures and grey information characteristics beong to the same grey category, have a common attribute kerne and grey scae (rea is a specia grey number whose grey scae is 0 ). So the agebraic agorithm between grey heterogeneous data can be studied through the kerne and grey scae. Definition 12. Assume F( ) is a grey heterogeneous data set. For arbitrary i and j, i + j, i j, i j and i j ( j = 0) beong to F( ), and is grey heterogeneous data fied [46].
7 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 Assume F( ) as the grey heterogeneous data set. i, j and k F( ). The kerne and grey scae are i (gi o), j (g o j ) and k (gk o ). Grey heterogeneous data agebraic agorithms have the foowing properties [45]: (1) i (gi o) + j (g o j ) = j (g o j ) + i (gi o). (2) ( i (g o i ) + j (g o j )) + k (g o k ) = i (g o i ) + ( j (g o j ) + k (g o k )). (3) 0 F( ), Make i (gi o) + 0 = i (gi o). (4) For s F( ), s F( ) and make s + s = 0. (5) i (gi o) ( j (g o j ) k (gk o)) = ( i (gi o) j (g o j )) k (gk o). (6) There is unit eement 1 F( ) which makes 1 i (gi o) = i (gi o) 1 = i (gi o). (7) ( i (gi o) + j (g o j )) k = i (gi o) k (gk o) + j (g o j ) k (gk o). (8) i (g o i ) ( j (g o j ) + k (g o k )) = i (g o i ) j (g o j ) + i (g o i ) k (g o k ). 3. Grey Language Hesitant Fuzzy Information Aggregation and Kerne and Grey Scae Scoring Function 3.1. Information Aggregation of Grey Language Hesitant Fuzzy Decision Since the evauation criteria are given in the form of grey anguage, the number of decision experts is the ength of the data, which is denoted as r ij = {(H σ1, µ 1, v 1 ), (H σ2, µ 2, v 2 ), (H σ, µ, v )}, where σk is the IH vaue of the kth item. The data are aggregated by the agorithm of the grey anguage fuzzy number to obtain z ij. z ij = (H 1 (, σ1 µ 1 + σ2 µ 2 + σ µ, max(v σk) 1, v 2 v )) (1) σk Definition 13. Assume A = {a k k = 1, 2, 3 } as the grey anguage set, a k = {H k, µ k, v k } and g(a 1, a 2, a 3, a ) = w k a k. Where, w k is the weight vector corresponding to the grey anguage array a k.w k [0, 1] and w k = 1, g is grey anguage weighted average operator, abbreviated as the GLHFWAA operator. GLHFWAA(a 1, a 2, a ) = w k a k = (w k H k, µ k, v k ) (2) If w = ( 1, 1,, 1 ), GLHFWAA degenerates to grey anguage averaging operator GLAA, GLHFAA(a 1, a 2, a ) = 3.2. Grey Language Hesitant Fuzzy Score Function w k a k = ( 1 H k, µ k, v k ). Definition 14. Assume a i = {H i, µ i, v i } as any grey anguage number, and S(a k ) is the score function of a k. S(a k ) = (1 v k ) µ k IH k (3) Assume gh GLH, GLH = {(H 1, µ 1, v 1 ), (H 2, µ 2, v 2 ), (H, µ, v )} are grey anguage decision data given by experts.
8 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 S(GLH) = 1 S(GLH) is the score function of GLH. S(a k ) k = 1, 2 (4) Exampe. Let GLH = {(3, 0.6, 0.2), (4, 0.6, 0.3), (4, 0.7, 0.2)}, S(GLH) = 1 S(a k ) = 1 3 [(1 0.2) (1 0.3) (1 0.2) 0.7 4] = The above definition transforms uncertain grey anguage information into a score function and becomes a definite vaue, which is theoreticay inadequate. In this paper, it is considered that the score function shoud be a grey number [45] beieved that the kerne ˆ of the grey number is an important representative of the vaue of the grey scae. Kerne is used in a kinds of synthesis operations, and has an important vaue. Aso, the grey agebra operation properties of ˆ g o are discussed, and the reevant agorithm is proved. According to the gradation function of grey anguage and the definition of the kerne and grey scae, the grey anguage scoring function based on the kerne and grey scae is studied in this paper. Definition 15. Assume gh GLH, GLH = {(H 1, µ 1, v 1 ), (H 2, µ 2, v 2 ), (H, µ, v )} S( ˆ g o) = ( 1 ˆ k, max(g 0 k )) = ( 1 µ i IH k, max(v k )) k = 1, 2 (5) S( ˆ g o) is the score function of the grey anguage based on the kerne and grey scae. 4. Kerne and Grey Scae Group Decision Making Method Based on Grey Language Hesitant Fuzzy Information 4.1. Mode Construction Assume that there are m feasibe schemes A 1, A 2,, A m and n evauation criteria of a hesitant fuzzy group decision making probem based on grey anguage. w = (w 1, w 2,, w n ) T is weight vector corresponding to the evauation criteria, where w i represents the weight of P i. Assume the ordered anguage evauation set H = (H 0, H 1,, HT ). According to the evauation set, experts evauate various scheme indexes. In the grey anguage hesitant fuzzy set, beongs to the discrete grey numbers. gh 11 gh gh 1n gh GLH = 21 gh gh 2n gh m1 gh m2... gh mn 4.2. Decision-Making Steps Step 1. In this step, the decision matrix P m n = (p ij ) m n is normaized as R m n = (r ij ) m n. It is necessary to eiminate the infuence of different physica quantities before making a decision and to normaize different types of index vaues. Generay speaking, the cost index vaue shoud be converted to the benefit index. In specia cases, the benefit index vaue aso needs to be converted to the cost index vaue. Cost index vaue (H i, µ, v) is converted to benefit index vaue. Benefit index vaue (H i, µ, v) is converted to cost index vaue: r ij = (H T i, µ, v) (6) r ij = (H T i, µ, v) (7)
9 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 Step 2. Data suppement. Torra studied the hesitancy faced by experts in group decision making and proposed a hesitant fuzzy set suitabe for group decision making. Add the maximum membership scae to the hesitant fuzzy eement when the decision maker has risk preference. Add the minimum membership scae to the hesitant fuzzy eement when the decision maker is risk-averse and pessimistic. Step 3. Transform grey anguage data into a score function of kerne and grey scae through the Formua (4). Step 4. Cacuate grey correation degree and grey scae. Definition 16. ˆ is the kerne part of score function of grey anguage eement based on kerne and grey scae. The grey correation coefficient ζ i (j) and grey correation degree γ(j) between the empirica judgment vaue and the reference weight vaue of each evauation index are obtained by the foowing formua: Make D = ˆ i (j) ˆ 0 (j), and ξ i (j) = min min i j D + ρ max max D i j D + ρ max max D i j (8) γ(i) = 1 n n j=1 ξ i (j), 1 i m (9) where, ρ is the resoution ratio, ρ (0, 1), generay, ρ = 0.5. The correation degree of each sequence directy refects the degree to which each evauation is cose to the optima idea scheme. Definition 17. Assume g 0 (j max ) as the maximum grey scae of kerne vaue in j coumn and g 0 (i max ) as the maximum grey scae of kerne vaue in i row. g 0 (i) = max(g 0 (j max ), g 0 (i max )) 1 j n (10) γ g o(i) = (γ (g1 o )(1), γ (g2 o )(2),, γ (gn o ) (m))t Where, 1 i m (11) Step 5. The schemes are sorted according to the grey correation degree and grey scae. The foowing principes shoud be foowed: (1) When γ(i) > γ(m), i m; (2) When γ(i) = γ(m), and g 0 (i) > g 0 (m), i m; (3) When γ(i) = γ(m), and g 0 (i) = g 0 (m), i m. 5. Exampe Anaysis As an economic deveopment mode harmonious with the environment, the circuar economy has become an important way to promote the comprehensive deveopment abiity of an industria chain. Many industria chains have carried out ecoogica transformations under the guidance of circuar economy idea. The sustainabe deveopment of an industria chain based on the circuar economy is a compicated evauation process, with uncertainty and mutipe objectives. Evauation of the sustainabe deveopment abiity of an industria chain based on the circuar economy aims to study the optimization function of the circuar economy on the design and reconstruction of the industria chain, and to anayze the comprehensive deveopment eve and potentia of the industria chain from the perspective of the circuar economy. The evauation system of the sustainabe deveopment abiity of an industria chain based on the characteristics of the circuar economy can provide scientific and effective means for the operation, management and improvement of the industria chain. Therefore, it is necessary to estabish an industria chain sustainabe deveopment capacity evauation index system covering
10 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 four aspects of economic deveopment: (P 1 ), resource utiization (P 2 ), environmenta protection (P 3 ) and green management (P 4 ), so as to scientificay and objectivey refect the sustainabe deveopment capacity of an industria chain [46]. Four industries A 1, A 2, A 3, A 4 are assumed to be compared in the sustainabe deveopment { of the circuar economy industry chain in a certain area. Three } experts H evauate and seect H = 0 = very poor, H 1 = poor, H 2 = reativey poor, H 3 = genera as the H 4 = good, H 5 = reativey good, H 6 = very good anguage evauation term set. The evauation vaue is represented by grey anguage vaue, forming the grey anguage hesitant fuzzy decision matrix in Tabe 1. Tabe 1. The vaues of the grey anguage hesitant fuzzy decision matrix. Industries Attribute (P 1 ) Attribute (P 2 ) A 1 {(H 3, 0.6, 0.3), (H 4, 0.6, 0.4), (H 4, 0.7, 0.4)} {(H 3, 0.7, 0.4), (H 3, 0.8, 0.4), (H 3, 0.9, 0.3),} A 2 {(H 3, 0.7, 0.4), (H 3, 0.8, 0.4), (H 4, 0.9, 0.1)} {(H 3, 0.9, 0.1), (H 4, 0.7, 0.4), (H 4, 0.8, 0.2)} A 3 {(H 4, 0.7, 0.2), (H 5, 0.6, 0.2), (H 5, 0.8, 0.1)} {(H 4, 0.7, 0.2), (H 5, 0.6, 0.2), (H 6, 0.9, 0.2)} A 4 {(H 3, 0.6, 0.2), (H 4, 0.8, 0.4), (H 5, 0.9, 0.3)} {(H 3, 0.7, 0.4), (H 4, 0.6, 0.4), (H 5, 0.7, 0.4)} Industries Attribute (P 3 ) Attribute (P 4 ) A 1 {(H 2, 0.8, 0.5), (H 3, 0.6, 0.4), (H 4, 0.7, 0.4)} {(H 2, 0.8, 0.3), (H 2, 0.6, 0.4), (H 2, 0.7, 0.4)} A 2 {(H 3, 0.6, 0.2), (H 3, 0.7, 0.4), (H 3, 0.8, 0.3)} {(H 3, 0.7, 0.3), (H 3, 0.9, 0.4), (H 4, 0.8, 0.3)} A 3 {(H 4, 0.8, 0.3), (H 4, 0.8, 0.2), (H 5, 0.7, 0.2)} {(H 4, 0.9, 0.3), (H 5, 0.8, 0.2), (H 6, 0.8, 0.1)} A 4 {(H 3, 0.8, 0.1), (H 3, 0.6, 0.2), (H 4, 0.8, 0.1),} {(H 3, 0.9, 0.2), (H 4, 0.9, 0.3), (H 5, 0.8, 0.2)} 5.1. Method 1 Step 1. Normaize decision making matrix P m n = (p ij ) m n as R m n = (r ij ) m n. A indexes in this exampe are benefit indexes, so they are omitted. Step 2. Suppement data. Since there are three experts in this exampe, none of whom has abstained or given the same score, it is not necessary to suppement the data, and this step is omitted. Step 3. Transform grey anguage data into the score function of the kerne and grey scae through Formua (5), and Tabe 2 is obtained. Tabe 2. ˆ g o form of grey hesitant fuzzy decision matrix. Industries Attribute (P 1 ) Attribute (P 2 ) A (0.40) (0.40) A (0.40) (0.40) A (0.20) (0.20) A (0.40) (0.40) Industries Attribute (P 3 ) Attribute (P 4 ) A (0.50) (0.40) A (0.30) (0.40) A (0.30) (0.30) A (0.20) (0.30) Step 4. Cacuate grey correation degree and grey scae. ξ i (j) = γ(i) = (0.4898, , , ) T
11 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14. γ g o(i) = ( (0.50), (0.40), (0.30), (0.40) ) T. Step 5. Rank the schemes according to the grey correation and grey scae of each industry. Obviousy, A 3 A 4 A 2 A Comparison Method 2 The foowing steps can be obtained according to the traditiona decision making method in grey anguage. Step 1. Normaize decision making matrix P m n = (p ij ) m n as R m n = (r ij ) m n. A indexes in this exampe are benefit indexes, so they are omitted. Step 2. Aggregate data r ij through Formua (1) to obtain z ij, as shown in Tabe 3. Tabe 3. z ij of grey anguage decision matrix. Industries Attribute (P 1 ) Attribute (P 2 ) A 1 {(H , , 0.4)} {(H 3, 0.8, 0.4)} A 2 {(H , 0.89, 0.4)} {(H , , 0.4)} A 3 {(H , 0.7, 0.2)} {(H 5, , 0.2)} A 4 {(H 4, , 0.4)} {(H 4, , 0.4)} Industries Attribute (P 3 ) Attribute (P 4 ) A 1 {(H 3, , 0.5)} {(H 2, 0.7, 0.4)} A 2 {(H 3, 0.7, 0.4)} {(H , 0.8, 0.4)} A 3 {(H 5, , 0.3)} {(H 2, 0.7, 0.3)} A 4 {(H , 0.74, 0.2)} {(H 4, , 0.3)} Step 3. Cacuate the score function S(a ij ) of z ij through Formua (2), as shown in Tabe 4. Tabe 4. S(a ij ) decision matrix of grey anguage hesitation fuzzy. Industries Attribute (P 1 ) Attribute (P 2 ) Attribute (P 3 ) Attribute (P 4 ) A A A A Step 4. Cacuate grey correation coefficient based on S(a i ) to obtain index weight w(j). ξ i (j) = w = (0.2729, , , ) T Step 5. Cacuate the comprehensive criterion vaues of each scheme through the GLHFWAA operator. Z 1 = (H , , 0.4) (H 3, 0.8, 0.4) (H 3, , 0.5) (H 2, 0.7, 0.4) = (H , , 0.5)
12 Int. J. Environ. Res. Pubic Heath 2018, 15, of 14 Simiary, Z 2 = (H , , 0.4), Z 3 = (H , , 0.3), Z 4 = (H , , 0.4). Step 6. Compare Z i (i = 1, 2, 3, 4) and sort the schemes. IH (1 0.5) < IH (1 0.4) < IH (1 0.4) < IH (1 0.3) Z 1 < Z 2 < Z 4 < Z 3. Therefore, industry A 3 is the optima, foowed by industry A 4 and industry A 1 is the worst Comparison Method 3 Step 5 can aso be used to continue the decision making after decision step 4 in method 2. Step 5. Seect the grey anguage S(a ij ) decision matrix and grey correation weight w(j) by using the mutipe inear weighting method. The comprehensive criterion vaue of each scheme is cacuated through the foowing formua: n Z i = S(a ij ) w(j) j= = = [ ] T (0.2729, , , )T Z 1 < Z 2 < Z 4 < Z 3, which is consistent with the sorting obtained through methods 1 and 2. By comparison, the grey scoring function used in method 1 is more reasonabe and simpe. Method 3 is simper than method 2. On the whoe, it can be concuded that method 1 method 3 method Concusions This paper studied the grey anguage decision making method, and defined the scoring function of grey anguage based on the kerne and grey scae which is more suitabe for uncertain decision making. Moreover, a group decision-making method based on the grey correation degree of the kerne and grey scae was proposed. In comparison, the method is more reasonabe than traditiona methods by introducing the kerne and grey scae to avoid adding subjective data, thereby reducing information distortion; in addition, the grey reationa degree decision method does not need to cacuate the index weight, and hence to some extent this method is more convenient. Of course, it is hard to avoid the ack of depth ony through a comparison of three different decision methods of the same exampe, so we wi conduct systematic comparative research on data simuation in future research so as to improve the credibiity of the comparison. Acknowedgments: Nationa Science Foundation of China under Grants ( ); Manufacturing Deveopment Research Institute of China (SK , SK ); Aeronautica Science Fund of China (2016ZG52068). Author Contributions: Qingsheng Li designed the structure and improved the manuscript; Zaiwu Gong proposed the idea; Yuzhu Diao performed the cacuations; Aqin Hu wrote the paper. Conficts of Interest: The authors decare no confict of interest. References 1. Zadeh, L.A. Fuzzy sets. Inf. Contro 1965, 8, [CrossRef] 2. Nassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20,
13 Int. J. Environ. Res. Pubic Heath 2018, 15, of Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1995, 75, [CrossRef] 4. Dubois, D.; Prade, H. Fuzzy Sets and Systems: Theory and Appications; Academic Press: New York, NY, USA, Yamoto, S. Remarks on basics of fuzzy sets and fuzzy mutisets. Fuzzy Sets Syst. 2005, 156, Yager, R.R. On the theory of bags. Int. J. Gen. Syst. 1986, 13, [CrossRef] 7. Miyamoto, S. Mutisets and fuzzy mutisets. In Soft Computing and Human-Centered Machines; Springer: Berin, Germany, Xia, M. Research on Fuzzy Decision Information Integration and Its Measure; Southeast China University: Nanjing, China, Deng, J.L. Contro probems of grey systems. Syst. Contro Lett. 1982, 1, Deng, J. Grey contro system. J. Huazhong Univ. Techno. 1982, 3, Liu, S. Advances in grey system research ( ). J. Nanjing Univ. Aeronaut. Astronaut. 2015, 47, Luo, D. Study of Anaysis Method of Grey Decision Probem; Nanjing University of Aeronautics and Astronautics: Nanjing, China, Chen, X. Study of Grey Muti-Attribute Decision-Making Method; Nanjing University of Aeronautics and Astronautics: Nanjing, China, Song, J. Research on Grey Decision Method and Its Appication; Nanjing University of Aeronautics and Astronautics: Nanjing, China, Hu, L.; Guan, X.; He, Y. A new approach for grey muti-attribute decision making. Contro Decision 2012, 27, , You, C.; Wu, L. Research on inverse order probem of Grey decision Mode. Stat. Decision 2013, 22, Zhang, N.; Fang, Z.; Zhu, J. Muti-Stage grey situation group decision-making mode based on Orness. Contro Decision 2015, 7, Liu, S.; Fang, Z.; Yang, Y. Two stages decision mode with grey synthetic measure and a betterment of trianguar whitenization weight function. Contro Decision 2014, 7, Luo, D.; Liu, S. Study on the method for grey incidence decision-making. Chin. J. Manag. Sci. 2005, 1, Luo, D.; Liu, S. Grey incidence decision-making with incompete information. J. App. Sci. 2005, 4, Dang, Y.; Liu, S.; Liu, B. Study on grey incidence decision mode of the dynamic mutipe-attribute. Eng. Sci. 2005, 2, Jiang, S.; Liu, S.; Liu, Z.; Fang, Z. Grey incidence decision making mode based on area. Contro Decision 2015, 4, Xie, M.; Xiao, X. Grey Reationa Anaysis of Muti-Attribute Group Decision Making Based on AHP and its Appication. In Proceedings of the 16th Nationa Academic Conference on Grey System, China Higher Science and Technoogy Center, Nanjing, China, November Chen, X.; Liu, S. Grey incidence decision-making method with partia weight information but with preference information on aternatives. Syst. Eng. Eectron. 2007, 11, Li, P.; Liu, S. Interva-Vaued intuitionistic fuzzy numbers decision-making method based on grey incidence anaysis and d-s theory of evidence. Acta Auomat. Sin. 2011, 8, Liu, Y.; Jeffrey, F.; Liu, S.; Zhao, H.; Jian, L. Dynamic mutipe attribute grey incidence decision making method based on interva vaued intuitionisitc fuzzy umber. Contro Decision 2013, 9, Li, P.; Zhu, J. Intuitionistic fuzzy custer modes based on cased-based reasoning and grey incidence anaysis. Syst. Eng.-Theor. Pract. 2015, 9, Li, P. Research on Intuitionistic Fuzzy Information Decision-Making Method Based on Grey Reation and Evidence Reasoning; Nanjing University of Aeronautics and Astronautics: Nanjing, China, Li, P.; Liu, S.; Fang, Z. Interva-Vaued intuitionistic fuzzy numbers decision-making method based on grey incidence anaysis and MYCIN certainty factor. Contro Decision 2012, 7, Hu, L.; Wang, C.; Zhu, J.; He, Y. Approach for gery fuzzy MADA in cosed word. Contro Decision 2014, 2, Li, P.; Liu, S.; Fang, Z. Intuitionistic fuzzy numbers decision-making method based on grey incidence anaysis and MYCIN certainty factor. Contro Decision 2011, 26, Li, P.; Liu, S.; Zhu, J. Intuitionistic fuzzy stochastic muti-criteria decision-making methods based on MYCIN certainty factor and prospect theory. Syst. Eng.-Theor. Pract. 2013, 33,
14 Int. J. Environ. Res. Pubic Heath 2018, 15, of Li, P.; Zhu, J.; Liu, S. Intuitionistic fuzzy decision-making methods based on case-based reasoning. Chin. Manag. Manag. Sci. 2015, 23, Wei, G.W. Grey reationa anaysis method for intuitionistic fuzzy mutipe attribute decision making. Expert Syst. App. 2011, 38, [CrossRef] 35. Wu, J. Study on Grey Language Muti-Criteria Decision-Making Method and Its Appication; Centra South University: Changsha, China, Wang, J.; Wu, J. Muti-Criteria decision-making approach based on the interva grey uncertain inguistic. Chin. J. Manag. Sci. 2010, 3, Liu, P.; Zhang, X. Muti-Attribute group decision making method based on interva grey inguistic variabes weighted geometric aggregation operator. Contro Decision 2011, 5, Wang, H. Research on the Mode and Appication of Linguistic Evauation Information Aggregation Based on Grey Decision Making; Nanjing University of Aeronautics & Astronautics: Nanjing, China, Liu, S.; Li, Q.; Zhao, N. Grey correation decision making method of grey hesitant fuzzy sets based on kerne and grey degree. J. Nanjing Univ. Aeronaut. Astronaut. 2016, 48, Li, Q.; Liu, S. Grey hesitant fuzzy sets and its decision making based on grey reation and TOPSIS. J. Jiangsu Univ. Sci. Techno. 2015, 29, , Li, Q.; Liu, S. Grey hesitant fuzzy set and its decision making based on grey reation and vikor method. Math. Pract. Theor. 2015, 45, Torra, V. Hesitant fuzzy sets. Int. J. Inte. Syst. 2010, 25, [CrossRef] 43. Torra, V.; Narukawa, Y. On Hesitant Fuzzy Sets and Decision. In Proceedings of the 18th IEEE Internationa Conference on Fuzzy Systems, Jeju Isand, Korea, August Liu, S.; Fang, Z.; Xie, N. Agorithm rues of interva grey numbers based on the Kerne and the degree of greyness of grey numbers. Syst. Eng. Eectron. 2010, 32, Zeng, B. Agebra agorithm rues of grey isomerism data based on kerne and grey degree and its appication. Stat. Inf. Forum 2013, 29, Liu, L. Evauation of the sustainabiity of the industria chain based on circuar economy. J. Shandong Jianzhu Univ. 2012, 27, 88 91, by the authors. Licensee MDPI, Base, Switzerand. This artice is an open access artice distributed under the terms and conditions of the Creative Commons Attribution (CC BY) icense (
Research of Data Fusion Method of Multi-Sensor Based on Correlation Coefficient of Confidence Distance
Send Orders for Reprints to reprints@benthamscience.ae 340 The Open Cybernetics & Systemics Journa, 015, 9, 340-344 Open Access Research of Data Fusion Method of Muti-Sensor Based on Correation Coefficient
More informationBP neural network-based sports performance prediction model applied research
Avaiabe onine www.jocpr.com Journa of Chemica and Pharmaceutica Research, 204, 6(7:93-936 Research Artice ISSN : 0975-7384 CODEN(USA : JCPRC5 BP neura networ-based sports performance prediction mode appied
More informationConsistent linguistic fuzzy preference relation with multi-granular uncertain linguistic information for solving decision making problems
Consistent inguistic fuzzy preference reation with muti-granuar uncertain inguistic information for soving decision making probems Siti mnah Binti Mohd Ridzuan, and Daud Mohamad Citation: IP Conference
More informationConverting Z-number to Fuzzy Number using. Fuzzy Expected Value
ISSN 1746-7659, Engand, UK Journa of Information and Computing Science Vo. 1, No. 4, 017, pp.91-303 Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue Mahdieh Akhbari * Department of Industria
More informationCombining reaction kinetics to the multi-phase Gibbs energy calculation
7 th European Symposium on Computer Aided Process Engineering ESCAPE7 V. Pesu and P.S. Agachi (Editors) 2007 Esevier B.V. A rights reserved. Combining reaction inetics to the muti-phase Gibbs energy cacuation
More informationNumerical simulation of javelin best throwing angle based on biomechanical model
ISSN : 0974-7435 Voume 8 Issue 8 Numerica simuation of javein best throwing ange based on biomechanica mode Xia Zeng*, Xiongwei Zuo Department of Physica Education, Changsha Medica University, Changsha
More informationMathematical Scheme Comparing of. the Three-Level Economical Systems
Appied Mathematica Sciences, Vo. 11, 2017, no. 15, 703-709 IKAI td, www.m-hikari.com https://doi.org/10.12988/ams.2017.7252 Mathematica Scheme Comparing of the Three-eve Economica Systems S.M. Brykaov
More informationAn Approach to Decision Making with Interval-Valued Intuitionistic Hesitant Fuzzy Information Based on the 2-Additive Shapley Function
INFORMATICA, 2018, Vo. 29, No. 1, 157 185 157 2018 Vinius University DOI: http://dx.doi.org/10.15388/informatica.2018.162 An Approach to Decision Making with Interva-Vaued Intuitionistic Hesitant Fuzzy
More informationSome Measures for Asymmetry of Distributions
Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester
More informationMARKOV CHAINS AND MARKOV DECISION THEORY. Contents
MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After
More informationOn Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1 x 2 ) and Their Byproducts
Commun. Theor. Phys. 66 (216) 369 373 Vo. 66, No. 4, October 1, 216 On Integras Invoving Universa Associated Legendre Poynomias and Powers of the Factor (1 x 2 ) and Their Byproducts Dong-Sheng Sun ( 孙东升
More informationA Solution to the 4-bit Parity Problem with a Single Quaternary Neuron
Neura Information Processing - Letters and Reviews Vo. 5, No. 2, November 2004 LETTER A Soution to the 4-bit Parity Probem with a Singe Quaternary Neuron Tohru Nitta Nationa Institute of Advanced Industria
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationAdjustment of automatic control systems of production facilities at coal processing plants using multivariant physico- mathematical models
IO Conference Series: Earth and Environmenta Science AER OEN ACCESS Adjustment of automatic contro systems of production faciities at coa processing pants using mutivariant physico- mathematica modes To
More informationAn Algorithm for Pruning Redundant Modules in Min-Max Modular Network
An Agorithm for Pruning Redundant Modues in Min-Max Moduar Network Hui-Cheng Lian and Bao-Liang Lu Department of Computer Science and Engineering, Shanghai Jiao Tong University 1954 Hua Shan Rd., Shanghai
More informationDISTRIBUTION OF TEMPERATURE IN A SPATIALLY ONE- DIMENSIONAL OBJECT AS A RESULT OF THE ACTIVE POINT SOURCE
DISTRIBUTION OF TEMPERATURE IN A SPATIALLY ONE- DIMENSIONAL OBJECT AS A RESULT OF THE ACTIVE POINT SOURCE Yury Iyushin and Anton Mokeev Saint-Petersburg Mining University, Vasiievsky Isand, 1 st ine, Saint-Petersburg,
More informationUniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete
Uniprocessor Feasibiity of Sporadic Tasks with Constrained Deadines is Strongy conp-compete Pontus Ekberg and Wang Yi Uppsaa University, Sweden Emai: {pontus.ekberg yi}@it.uu.se Abstract Deciding the feasibiity
More informationSafety Evaluation Model of Chemical Logistics Park Operation Based on Back Propagation Neural Network
1513 A pubication of CHEMICAL ENGINEERING TRANSACTIONS VOL. 6, 017 Guest Editors: Fei Song, Haibo Wang, Fang He Copyright 017, AIDIC Servizi S.r.. ISBN 978-88-95608-60-0; ISSN 83-916 The Itaian Association
More informationOn Some Basic Properties of Geometric Real Sequences
On Some Basic Properties of eometric Rea Sequences Khirod Boruah Research Schoar, Department of Mathematics, Rajiv andhi University Rono His, Doimukh-791112, Arunacha Pradesh, India Abstract Objective
More informationMULTI-PERIOD MODEL FOR PART FAMILY/MACHINE CELL FORMATION. Objectives included in the multi-period formulation
ationa Institute of Technoogy aicut Department of echanica Engineering ULTI-PERIOD ODEL FOR PART FAILY/AHIE ELL FORATIO Given a set of parts, processing requirements, and avaiabe resources The objective
More informationA Simple and Efficient Algorithm of 3-D Single-Source Localization with Uniform Cross Array Bing Xue 1 2 a) * Guangyou Fang 1 2 b and Yicai Ji 1 2 c)
A Simpe Efficient Agorithm of 3-D Singe-Source Locaization with Uniform Cross Array Bing Xue a * Guangyou Fang b Yicai Ji c Key Laboratory of Eectromagnetic Radiation Sensing Technoogy, Institute of Eectronics,
More informationImproving the Reliability of a Series-Parallel System Using Modified Weibull Distribution
Internationa Mathematica Forum, Vo. 12, 217, no. 6, 257-269 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/imf.217.611155 Improving the Reiabiity of a Series-Parae System Using Modified Weibu Distribution
More informationCONSTRUCTION AND APPLICATION BASED ON COMPRESSING DEPICTION IN PROFILE HIDDEN MARKOV MODEL
CONSRUCON AND APPCAON BASED ON COPRESSNG DEPCON N PROFE HDDEN ARKOV ODE Zhijian Zhou, 1, Daoiang i, 3, 3, i i,3,*, Zetian Fu 1 Coege of Science, China Agricuture University, Beijing, China, 100083 Key
More informationMultivariate Rational Response Surface Approximation of Nodal Displacements of Truss Structures
https://doi.org/10.1186/s10033-018-0219-4 Chinese Journa of Mechanica Engineering ORIGINAL ARTICLE Open Access Mutivariate Rationa Response Surface Approximation of Noda Dispacements of Truss Structures
More informationEfficiently Generating Random Bits from Finite State Markov Chains
1 Efficienty Generating Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown
More informationA simple reliability block diagram method for safety integrity verification
Reiabiity Engineering and System Safety 92 (2007) 1267 1273 www.esevier.com/ocate/ress A simpe reiabiity bock diagram method for safety integrity verification Haitao Guo, Xianhui Yang epartment of Automation,
More informationGeneralized multigranulation rough sets and optimal granularity selection
Granu. Comput. DOI 10.1007/s41066-017-0042-9 ORIGINAL PAPER Generaized mutigranuation rough sets and optima granuarity seection Weihua Xu 1 Wentao Li 2 Xiantao Zhang 1 Received: 27 September 2016 / Accepted:
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationMinimizing Total Weighted Completion Time on Uniform Machines with Unbounded Batch
The Eighth Internationa Symposium on Operations Research and Its Appications (ISORA 09) Zhangiaie, China, September 20 22, 2009 Copyright 2009 ORSC & APORC, pp. 402 408 Minimizing Tota Weighted Competion
More informationIntuitionistic Fuzzy Optimization Technique for Nash Equilibrium Solution of Multi-objective Bi-Matrix Games
Journa of Uncertain Systems Vo.5, No.4, pp.27-285, 20 Onine at: www.jus.org.u Intuitionistic Fuzzy Optimization Technique for Nash Equiibrium Soution of Muti-objective Bi-Matri Games Prasun Kumar Naya,,
More informationhttps://doi.org/ /epjconf/
HOW TO APPLY THE OPTIMAL ESTIMATION METHOD TO YOUR LIDAR MEASUREMENTS FOR IMPROVED RETRIEVALS OF TEMPERATURE AND COMPOSITION R. J. Sica 1,2,*, A. Haefee 2,1, A. Jaai 1, S. Gamage 1 and G. Farhani 1 1 Department
More informationA Novel Learning Method for Elman Neural Network Using Local Search
Neura Information Processing Letters and Reviews Vo. 11, No. 8, August 2007 LETTER A Nove Learning Method for Eman Neura Networ Using Loca Search Facuty of Engineering, Toyama University, Gofuu 3190 Toyama
More informationStructural health monitoring of concrete dams using least squares support vector machines
Structura heath monitoring of concrete dams using east squares support vector machines *Fei Kang ), Junjie Li ), Shouju Li 3) and Jia Liu ) ), ), ) Schoo of Hydrauic Engineering, Daian University of echnoogy,
More informationStatistical Learning Theory: A Primer
Internationa Journa of Computer Vision 38(), 9 3, 2000 c 2000 uwer Academic Pubishers. Manufactured in The Netherands. Statistica Learning Theory: A Primer THEODOROS EVGENIOU, MASSIMILIANO PONTIL AND TOMASO
More informationEfficient Generation of Random Bits from Finite State Markov Chains
Efficient Generation of Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown
More informationThe influence of temperature of photovoltaic modules on performance of solar power plant
IOSR Journa of Engineering (IOSRJEN) ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vo. 05, Issue 04 (Apri. 2015), V1 PP 09-15 www.iosrjen.org The infuence of temperature of photovotaic modues on performance
More informationRobust Sensitivity Analysis for Linear Programming with Ellipsoidal Perturbation
Robust Sensitivity Anaysis for Linear Programming with Eipsoida Perturbation Ruotian Gao and Wenxun Xing Department of Mathematica Sciences Tsinghua University, Beijing, China, 100084 September 27, 2017
More informationRelated Topics Maxwell s equations, electrical eddy field, magnetic field of coils, coil, magnetic flux, induced voltage
Magnetic induction TEP Reated Topics Maxwe s equations, eectrica eddy fied, magnetic fied of cois, coi, magnetic fux, induced votage Principe A magnetic fied of variabe frequency and varying strength is
More informationSteepest Descent Adaptation of Min-Max Fuzzy If-Then Rules 1
Steepest Descent Adaptation of Min-Max Fuzzy If-Then Rues 1 R.J. Marks II, S. Oh, P. Arabshahi Λ, T.P. Caude, J.J. Choi, B.G. Song Λ Λ Dept. of Eectrica Engineering Boeing Computer Services University
More informationGeneral Certificate of Education Advanced Level Examination June 2010
Genera Certificate of Education Advanced Leve Examination June 2010 Human Bioogy HBI6T/Q10/task Unit 6T A2 Investigative Skis Assignment Task Sheet The effect of using one or two eyes on the perception
More informationA. Distribution of the test statistic
A. Distribution of the test statistic In the sequentia test, we first compute the test statistic from a mini-batch of size m. If a decision cannot be made with this statistic, we keep increasing the mini-batch
More informationConvergence Property of the Iri-Imai Algorithm for Some Smooth Convex Programming Problems
Convergence Property of the Iri-Imai Agorithm for Some Smooth Convex Programming Probems S. Zhang Communicated by Z.Q. Luo Assistant Professor, Department of Econometrics, University of Groningen, Groningen,
More informationA Fuzzy Approach to Co-Ordinated and Non Co-Ordinated Two Stage Supply Chain
Internationa Journa of Pure and Appied Mathematica Sciences. ISS 097-988 Voume 0, umber (07), pp. -3 Research India Pubications http://www.ripubication.com A Fuzzy Approach to Co-Ordinated and on Co-Ordinated
More informationIn-plane shear stiffness of bare steel deck through shell finite element models. G. Bian, B.W. Schafer. June 2017
In-pane shear stiffness of bare stee deck through she finite eement modes G. Bian, B.W. Schafer June 7 COLD-FORMED STEEL RESEARCH CONSORTIUM REPORT SERIES CFSRC R-7- SDII Stee Diaphragm Innovation Initiative
More informationStatistical Inference, Econometric Analysis and Matrix Algebra
Statistica Inference, Econometric Anaysis and Matrix Agebra Bernhard Schipp Water Krämer Editors Statistica Inference, Econometric Anaysis and Matrix Agebra Festschrift in Honour of Götz Trenker Physica-Verag
More information(This is a sample cover image for this issue. The actual cover is not yet available at this time.)
(This is a sampe cover image for this issue The actua cover is not yet avaiabe at this time) This artice appeared in a journa pubished by Esevier The attached copy is furnished to the author for interna
More informationNumerical solution of one dimensional contaminant transport equation with variable coefficient (temporal) by using Haar wavelet
Goba Journa of Pure and Appied Mathematics. ISSN 973-1768 Voume 1, Number (16), pp. 183-19 Research India Pubications http://www.ripubication.com Numerica soution of one dimensiona contaminant transport
More informationMelodic contour estimation with B-spline models using a MDL criterion
Meodic contour estimation with B-spine modes using a MDL criterion Damien Loive, Ney Barbot, Oivier Boeffard IRISA / University of Rennes 1 - ENSSAT 6 rue de Kerampont, B.P. 80518, F-305 Lannion Cedex
More informationSTA 216 Project: Spline Approach to Discrete Survival Analysis
: Spine Approach to Discrete Surviva Anaysis November 4, 005 1 Introduction Athough continuous surviva anaysis differs much from the discrete surviva anaysis, there is certain ink between the two modeing
More informationWavelet Galerkin Solution for Boundary Value Problems
Internationa Journa of Engineering Research and Deveopment e-issn: 2278-67X, p-issn: 2278-8X, www.ijerd.com Voume, Issue 5 (May 24), PP.2-3 Waveet Gaerkin Soution for Boundary Vaue Probems D. Pate, M.K.
More informationSolution of Wave Equation by the Method of Separation of Variables Using the Foss Tools Maxima
Internationa Journa of Pure and Appied Mathematics Voume 117 No. 14 2017, 167-174 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-ine version) ur: http://www.ijpam.eu Specia Issue ijpam.eu Soution
More informationConstruction of Supersaturated Design with Large Number of Factors by the Complementary Design Method
Acta Mathematicae Appicatae Sinica, Engish Series Vo. 29, No. 2 (2013) 253 262 DOI: 10.1007/s10255-013-0214-6 http://www.appmath.com.cn & www.springerlink.com Acta Mathema cae Appicatae Sinica, Engish
More informationPartial permutation decoding for MacDonald codes
Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics
More informationReliability: Theory & Applications No.3, September 2006
REDUNDANCY AND RENEWAL OF SERVERS IN OPENED QUEUING NETWORKS G. Sh. Tsitsiashvii M.A. Osipova Vadivosto, Russia 1 An opened queuing networ with a redundancy and a renewa of servers is considered. To cacuate
More informationExplicit overall risk minimization transductive bound
1 Expicit overa risk minimization transductive bound Sergio Decherchi, Paoo Gastado, Sandro Ridea, Rodofo Zunino Dept. of Biophysica and Eectronic Engineering (DIBE), Genoa University Via Opera Pia 11a,
More informationORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION
J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,
More informationStudy on Fusion Algorithm of Multi-source Image Based on Sensor and Computer Image Processing Technology
Sensors & Transducers 3 by IFSA http://www.sensorsporta.com Study on Fusion Agorithm of Muti-source Image Based on Sensor and Computer Image Processing Technoogy Yao NAN, Wang KAISHENG, 3 Yu JIN The Information
More informationProcess Capability Proposal. with Polynomial Profile
Contemporary Engineering Sciences, Vo. 11, 2018, no. 85, 4227-4236 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.88467 Process Capabiity Proposa with Poynomia Profie Roberto José Herrera
More informationThe Symmetric and Antipersymmetric Solutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B A l X l B l = C and Its Optimal Approximation
The Symmetric Antipersymmetric Soutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B 2 + + A X B C Its Optima Approximation Ying Zhang Member IAENG Abstract A matrix A (a ij) R n n is said to be symmetric
More informationNumerical Simulation for Optimizing Temperature Gradients during Single Crystal Casting Process
ISIJ Internationa Vo 54 (2014) No 2 pp 254 258 Numerica Simuation for Optimizing Temperature Gradients during Singe Crysta Casting Process Aeksandr Aeksandrovich INOZEMTSEV 1) Aeksandra Sergeevna DUBROVSKAYA
More informationA Comparison Study of the Test for Right Censored and Grouped Data
Communications for Statistica Appications and Methods 2015, Vo. 22, No. 4, 313 320 DOI: http://dx.doi.org/10.5351/csam.2015.22.4.313 Print ISSN 2287-7843 / Onine ISSN 2383-4757 A Comparison Study of the
More informationFormulas for Angular-Momentum Barrier Factors Version II
BNL PREPRINT BNL-QGS-06-101 brfactor1.tex Formuas for Anguar-Momentum Barrier Factors Version II S. U. Chung Physics Department, Brookhaven Nationa Laboratory, Upton, NY 11973 March 19, 2015 abstract A
More informationComponentwise Determination of the Interval Hull Solution for Linear Interval Parameter Systems
Componentwise Determination of the Interva Hu Soution for Linear Interva Parameter Systems L. V. Koev Dept. of Theoretica Eectrotechnics, Facuty of Automatics, Technica University of Sofia, 1000 Sofia,
More informationTheory of Generalized k-difference Operator and Its Application in Number Theory
Internationa Journa of Mathematica Anaysis Vo. 9, 2015, no. 19, 955-964 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ijma.2015.5389 Theory of Generaized -Difference Operator and Its Appication
More informationApproach to Identifying Raindrop Vibration Signal Detected by Optical Fiber
Sensors & Transducers, o. 6, Issue, December 3, pp. 85-9 Sensors & Transducers 3 by IFSA http://www.sensorsporta.com Approach to Identifying Raindrop ibration Signa Detected by Optica Fiber ongquan QU,
More informationDistributed average consensus: Beyond the realm of linearity
Distributed average consensus: Beyond the ream of inearity Usman A. Khan, Soummya Kar, and José M. F. Moura Department of Eectrica and Computer Engineering Carnegie Meon University 5 Forbes Ave, Pittsburgh,
More informationIdentification of macro and micro parameters in solidification model
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vo. 55, No. 1, 27 Identification of macro and micro parameters in soidification mode B. MOCHNACKI 1 and E. MAJCHRZAK 2,1 1 Czestochowa University
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationData Mining Technology for Failure Prognostic of Avionics
IEEE Transactions on Aerospace and Eectronic Systems. Voume 38, #, pp.388-403, 00. Data Mining Technoogy for Faiure Prognostic of Avionics V.A. Skormin, Binghamton University, Binghamton, NY, 1390, USA
More informationParagraph Topic Classification
Paragraph Topic Cassification Eugene Nho Graduate Schoo of Business Stanford University Stanford, CA 94305 enho@stanford.edu Edward Ng Department of Eectrica Engineering Stanford University Stanford, CA
More informationCollective organization in an adaptative mixture of experts
Coective organization in an adaptative mixture of experts Vincent Vigneron, Christine Fuchen, Jean-Marc Martinez To cite this version: Vincent Vigneron, Christine Fuchen, Jean-Marc Martinez. Coective organization
More informationTwo-sample inference for normal mean vectors based on monotone missing data
Journa of Mutivariate Anaysis 97 (006 6 76 wwweseviercom/ocate/jmva Two-sampe inference for norma mean vectors based on monotone missing data Jianqi Yu a, K Krishnamoorthy a,, Maruthy K Pannaa b a Department
More informationGeneral Certificate of Education Advanced Level Examination June 2010
Genera Certificate of Education Advanced Leve Examination June 2010 Human Bioogy HBI6T/P10/task Unit 6T A2 Investigative Skis Assignment Task Sheet The effect of temperature on the rate of photosynthesis
More informationAnalysis method of feeder partition capacity considering power supply security and distributed generation
The 6th Internationa Conference on Renewabe Power Generation (RPG) 19 20 October 2017 Anaysis method of feeder capacity considering power suppy security and distributed generation Weifu Wang 1, Zhaojing
More informationFRST Multivariate Statistics. Multivariate Discriminant Analysis (MDA)
1 FRST 531 -- Mutivariate Statistics Mutivariate Discriminant Anaysis (MDA) Purpose: 1. To predict which group (Y) an observation beongs to based on the characteristics of p predictor (X) variabes, using
More informationGauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law
Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s
More informationSpring Gravity Compensation Using the Noncircular Pulley and Cable For the Less-Spring Design
The 14th IFToMM Word Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.PS3.010 Spring Gravity Compensation Using the Noncircuar Puey and Cabe For the Less-Spring Design M.C.
More informationExpectation-Maximization for Estimating Parameters for a Mixture of Poissons
Expectation-Maximization for Estimating Parameters for a Mixture of Poissons Brandon Maone Department of Computer Science University of Hesini February 18, 2014 Abstract This document derives, in excrutiating
More informationResearch Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation
Appied Mathematics and Stochastic Anaysis Voume 007, Artice ID 74191, 8 pages doi:10.1155/007/74191 Research Artice On the Lower Bound for the Number of Rea Roots of a Random Agebraic Equation Takashi
More informationTwo Kinds of Parabolic Equation algorithms in the Computational Electromagnetics
Avaiabe onine at www.sciencedirect.com Procedia Engineering 9 (0) 45 49 0 Internationa Workshop on Information and Eectronics Engineering (IWIEE) Two Kinds of Paraboic Equation agorithms in the Computationa
More informationOn the evaluation of saving-consumption plans
On the evauation of saving-consumption pans Steven Vanduffe Jan Dhaene Marc Goovaerts Juy 13, 2004 Abstract Knowedge of the distribution function of the stochasticay compounded vaue of a series of future
More informationProblem set 6 The Perron Frobenius theorem.
Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator
More information8 Digifl'.11 Cth:uits and devices
8 Digif'. Cth:uits and devices 8. Introduction In anaog eectronics, votage is a continuous variabe. This is usefu because most physica quantities we encounter are continuous: sound eves, ight intensity,
More informationNew Efficiency Results for Makespan Cost Sharing
New Efficiency Resuts for Makespan Cost Sharing Yvonne Beischwitz a, Forian Schoppmann a, a University of Paderborn, Department of Computer Science Fürstenaee, 3302 Paderborn, Germany Abstract In the context
More informationCryptanalysis of PKP: A New Approach
Cryptanaysis of PKP: A New Approach Éiane Jaumes and Antoine Joux DCSSI 18, rue du Dr. Zamenhoff F-92131 Issy-es-Mx Cedex France eiane.jaumes@wanadoo.fr Antoine.Joux@ens.fr Abstract. Quite recenty, in
More informationStochastic Complement Analysis of Multi-Server Threshold Queues. with Hysteresis. Abstract
Stochastic Compement Anaysis of Muti-Server Threshod Queues with Hysteresis John C.S. Lui The Dept. of Computer Science & Engineering The Chinese University of Hong Kong Leana Goubchik Dept. of Computer
More informationNEW DEVELOPMENT OF OPTIMAL COMPUTING BUDGET ALLOCATION FOR DISCRETE EVENT SIMULATION
NEW DEVELOPMENT OF OPTIMAL COMPUTING BUDGET ALLOCATION FOR DISCRETE EVENT SIMULATION Hsiao-Chang Chen Dept. of Systems Engineering University of Pennsyvania Phiadephia, PA 904-635, U.S.A. Chun-Hung Chen
More informationApproximation and Fast Calculation of Non-local Boundary Conditions for the Time-dependent Schrödinger Equation
Approximation and Fast Cacuation of Non-oca Boundary Conditions for the Time-dependent Schrödinger Equation Anton Arnod, Matthias Ehrhardt 2, and Ivan Sofronov 3 Universität Münster, Institut für Numerische
More informationCONCHOID OF NICOMEDES AND LIMACON OF PASCAL AS ELECTRODE OF STATIC FIELD AND AS WAVEGUIDE OF HIGH FREQUENCY WAVE
Progress In Eectromagnetics Research, PIER 30, 73 84, 001 CONCHOID OF NICOMEDES AND LIMACON OF PASCAL AS ELECTRODE OF STATIC FIELD AND AS WAVEGUIDE OF HIGH FREQUENCY WAVE W. Lin and Z. Yu University of
More informationc 2007 Society for Industrial and Applied Mathematics
SIAM REVIEW Vo. 49,No. 1,pp. 111 1 c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971),
More informationarxiv: v3 [math.sp] 2 Nov 2017
A Fast Agorithm for Computing the Fourier Spectrum of a Fractiona Period arxiv:1604.01589v3 [math.sp] 2 Nov 2017 Jiasong Wang 1, Changchuan Yin 2, 1.Department of Mathematics, Nanjing University, Nanjing,
More informationAnalysis of the problem of intervention control in the economy on the basis of solving the problem of tuning
Anaysis of the probem of intervention contro in the economy on the basis of soving the probem of tuning arxiv:1811.10993v1 q-fin.gn] 9 Nov 2018 1 st eter Shnurov Nationa Research University Higher Schoo
More informationUI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE
UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE Juan Huang, Ronghui Wang and Tao Tang Coege of Traffic and Communications, South China University of Technoogy, Guangzhou, Guangdong 51641,
More informationPYTHAGOREAN FUZZY INDUCED GENERALIZED OWA OPERATOR AND ITS APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING
International Journal of Innovative Computing, Information and Control ICIC International c 2017 ISSN 1349-4198 Volume 13, Number 5, October 2017 pp. 1527 1536 PYTHAGOREAN FUZZY INDUCED GENERALIZED OWA
More informationCompression Ratio Expansion for Complementary Code Set Compressing a Signal to a Width of Several Sub-pulses
Compression Ratio Expansion for Compementary Code Set Compressing a Signa to a Width of Severa Sub-puses Reiji Sato 2nd Research Center, Technica R and D Institute Japan Defense Agency 1-2-24 Ikejiri,
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationTracking Control of Multiple Mobile Robots
Proceedings of the 2001 IEEE Internationa Conference on Robotics & Automation Seou, Korea May 21-26, 2001 Tracking Contro of Mutipe Mobie Robots A Case Study of Inter-Robot Coision-Free Probem Jurachart
More informationSimple Algebraic Proofs of Fermat s Last Theorem. Samuel Bonaya Buya*
Avaiabe onine at www.peagiaresearchibrary.com eagia Research Library Advances in Appied Science Research, 017, 8(3:60-6 ISSN : 0976-8610 CODEN (USA: AASRFC Simpe Agebraic roofs of Fermat s Last Theorem
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationHigher dimensional PDEs and multidimensional eigenvalue problems
Higher dimensiona PEs and mutidimensiona eigenvaue probems 1 Probems with three independent variabes Consider the prototypica equations u t = u (iffusion) u tt = u (W ave) u zz = u (Lapace) where u = u
More information