Assessing The Players’ Performance In The Game Of Bridge .

2y ago
30 Views
2 Downloads
368.58 KB
6 Pages
Last View : 2d ago
Last Download : 2m ago
Upload by : Jayda Dunning
Transcription

American Journal of Applied Mathematics and Statistics, 2014, Vol. 2, No. 3, 115-120Available online at http://pubs.sciepub.com/ajams/2/3/5 Science and Education PublishingDOI:10.12691/ajams-2-3-5Assessing the Players’ Performance in the Game ofBridge: A Fuzzy Logic ApproachMichael Gr. Voskoglou*School of Technological Applications, Graduate Technological Educational Institute (T. E. I.) of Western Greece, Patras, Greece*Corresponding author: mvosk@hol.grReceived March 29, 2014; Revised April 24, 2014; Accepted April 24, 2014Abstract Contract bridge occupies nowadays a position of great prestige being, together with chess, the only mindsports officially recognized by the International Olympic Committee. In the present paper an innovative method forassessing the total performance of bridge-players’ belonging to groups of special interest (e.g. different bridge clubsduring a tournament, men and women, new and old players, etc) is introduced, which is based on principles of fuzzylogic. For this, the cohorts under assessment are represented as fuzzy subsets of a set of linguistic labelscharacterizing their performance and the centroid defuzzification method is used to convert the fuzzy data collectedfrom the game to a crisp number. This new method of assessment could be used informally as a complement of theofficial bridge-scoring methods for statistical and other obvious reasons. Two real applications related tosimultaneous tournaments with pre-dealt boards, organized by the Hellenic Bridge Federation, are also presented,illustrating the importance of our results in practice.Keywords: contract bridge, fuzzy sets, centroid defuzzification methodCite This Article: Michael Gr. Voskoglou, “Assessing the Players’ Performance in the Game of Bridge: AFuzzy Logic Approach.” American Journal of Applied Mathematics and Statistics, vol. 2, no. 3 (2014): 115-120.doi: 10.12691/ajams-2-3-5.1. IntroductionIn this section we shall give a brief description of thefundamentals of the game of contract bridge and we shalldiscuss the advantages of using principles of fuzzy logicin the assessment procedures in general.1.1. The Game of BridgeBridge is a card game belonging to the family of tricktaking games. It is a development of Whist, which hadbecome the dominant such game enjoying a loyalfollowing for centuries.In 1904 Auction Bridge was developed, in which theplayers bid in a competitive auction to decide the contractand declarer. The object became to make at least as manytricks as were contracted for and penalties were introducedfor failing to do so.The modern game of Contract Bridge was the result ofinnovations to the scoring of auction bridge suggested byHarold Stirling Vanderbilt (USA, 1925) and others.Within a few years contract bridge had so supplanted theother forms of the game that "bridge" becamesynonymous with "contract bridge."Rubber Bridge s the basic form of contract bridge,played by four players. Informal social bridge games areoften played this way. Duplicate Bridge is the gameusually played in clubs, tournaments and matches. Thegame is basically the same with the rubber bridge, but theluck element is reduced by having the same deals replayedby different sets of players. At least eight players (in twotables) are required for this. There are also somesignificant differences in the scoring.Bridge occupies nowadays a position of great prestigebeing, together with chess, the only mind sports (i.e.games or skills where the mental component is moresignificant than the physical one) officially recognized bythe International Olympic Committee. Millions of peopleplay bridge worldwide, not only in clubs, tournaments andchampionships, but also on line (e.g. [1]) and with friendsat home, making it one of the world’s most popular cardgames. The World Bridge Federation (WBF) is theinternational governing body of contract bridge. WBF wasformatted in August 1958 by delegates from Europe,North and South America and its membership nowcomprises 123 National Bridge Organizations, with about700000 affiliated members.In the standard 52-card deck used in bridge, the ace isranked highest followed by the king, queen, and jack (allthe above cards called honours) and the spot-cards fromten down through to the two. Suit denominations alsohave a rank order with no trump (NT) being highestfollowed by spades (SP), hearts (H), diamonds (D) andclubs (CL).There are four players in each table, in two fixedpartnerships. Partners sit facing each other. It is traditionalto refer to the players according to their position at thetable as North (N), East (E), South (S) and West (W). SoN and S are partners against E and W.An almost essential tool for playing bridge is the boardcontaining four pockets, one for each player, marked by N,

American Journal of Applied Mathematics and StatisticsE, W and S respectively; 13 play cards are placed in eachpocket. Each board carries a number to identify it and hasmarks showing the dealer (i.e. the player who starts thebidding) and whether each of the two playing sides isvulnerable or not. A side which is vulnerable is subject tohigher bonuses and penalties than one that is not.At the beginning of the game the cards are shuffled,dealt and placed in the pockets of each board. In somecompetitions boards are pre-dealt prior to the competition,especially if the same hands are to be played at manylocations; for example in a large national or internationaltournament. Mechanical dealing machines or specialcomputer software are usually used for this purpose.Each session (hand) of the game is progressing throughthe following phases: Bidding (or auction), play of thecards and scoring the results.Bidding is based on the premise that the lowestavailable to bidders starts with the proposition to takeseven tricks, i.e. one cannot contract to make less thanseven tricks. Given this, the bidding is said to start at theone-level when contracting for a total of seven tricks, atthe two-level for eight tricks and so on to the seven-levelto contract to take all thirteen tricks. Thus, there are 35possible contracts, five at each of the seven levels. Thedealer begins the bidding, and the turn to speak passesclockwise. At each turn a player may either make a bid orpass. It is also possible to ‘double’ an opponent’s bid, orto ‘redouble’ the opponent’s ‘double’, thus increasing thescore of the bid when won, and the penalties, when lost. Ifsomeone then bids higher, any previous ‘double’ or‘redouble’ are cancelled. If all four players pass on theirfirst turn to speak the hand is said to be passed out. Thecards are thrown in and the next board is played. If anyonebids, then the auction continues until there are three passesin succession, and then stops. In this case the last bidbecomes the contract.The team who made the final bid will now try to makethe contract. The first player of this team who mentionedthe denomination (suit or no trumps) of the contractbecomes the declarer. The declarer's partner is known asthe dummy. The player to the left of the declarer leads tothe first trick and may play any card. Immediately afterthis opening lead, the dummy's cards are exposed. Playproceeds clockwise. Each of the other three players in turnmust, if possible, play a card of the same suit that theleader played. A player with no card of the suit led mayplay any card. A trick consists of four cards, one fromeach player, and is won by the highest trump in it, or if notrumps were played by the highest card of the suit led. Thewinner of a trick leads to the next, and may lead any card.Dummy takes no active part in the play of the hand.Whenever it is dummy's turn to play, the declarer must saywhich of dummy's cards is to be played. When dummywins a trick, the declarer specifies which card dummyshould lead to the next trick.When the play ends, the score is determined bycomparing the number of tricks taken by the declaringside to the number required to satisfy the contract.A match can be played among teams (two or more) offour players (two partnerships). At the end of the match inthis case the result is the difference in International MatchPoints (IMPs) between the competing teams and thenthere is a further conversion, in which some fixed numberof Victory Points (VPs) is appointed between the teams. It116is worth to notice that the table converting IMPs to VPshas been obtained through a rigorous mathematicalmanipulation [4].However, the game usually played in tournaments isamong fixed partnerships or pairs. For a pairs event aminimum of three tables (6 pairs, 12 players) is needed,but it works better with more players. Generally you playtwo or three boards at a table - this is called a round - andthen one or both pairs move to another table and playother boards against other opponents. The score for eachhand is recorded to a score sheet, which is kept folded in aspecial pocket of the board provided for this purpose, sothat previous scores could not be read before the board hasbeen played. North is then responsible for entering theresult and showing the completed sheet to East-West tocheck that it has been done correctly. Each pair has anidentity number, which must also be entered on the scoresheet, to show whose result it is. At the end of the gameeach score sheet will contain the results of all the pairswho have played that board. The score sheets are thencollected by the organisers and the scores are compared.The usual method of scoring in a pairs’ competition is inmatch points. Each pair is awarded two match points foreach pair who scored worse than them on that board, andone match point for each pair who scored equally. Thetotal number of match points scored by each pair over allthe boards is calculated and it is converted to a percentage.The pair succeeding the highest percentage wins the game.However, IMPs are also used as a method of scoring inspecial cases, in which the difference of each pair’s IMPsis usually calculated with respect to the mean number ofIMPs of all pairs.There are also several conventions that can be playedbetween the partners. However, a full description of therules and techniques of bridge is out of the purposes of thepresent paper.There are very many books written about bridge, themost famous being probably the book [6] of Edgar Kaplan(1925-1997), who was an American bridge player and oneof the principal contributors to the game. Kaplan’s bookwas translated in many languages and was reprinted manytimes since its first edition in 1964; for instance, [7] is oneof the recent unabridged republications of it. There is alsoa fair amount of bridge-related information on the Internet.For the history, the fundamentals and a detaileddescription of the rules of the game the reader may look atthe web sites [2,3], etc.1.2. Fuzzy Logic as a Tool in AssessmentProceduresThere used to be a tradition in science and engineeringof turning to probability theory when one is faced with aproblem in which uncertainty plays a significant role. Thistransition was justified when there were no alternativetools for dealing with the uncertainty. Today this is nolonger the case. Fuzzy logic, which is based on fuzzy setstheory introduced by Zadeh [19] in 1965, provides a richand meaningful addition to standard logic. A real test ofthe effectiveness of an approach to uncertainty is thecapability to solve problems which involve different facetsof uncertainty. Fuzzy logic has a much higher problemsolving capability than standard probability theory. Mostimportantly, it opens the door to construction of

117American Journal of Applied Mathematics and Statisticsmathematical solutions of computational problems whichare stated in a natural language.The applications which may be generated from oradapted to fuzzy logic are wide-ranging and provide theopportunity for modelling under conditions which areinherently imprecisely defined, despite the concerns ofclassical logicians (e.g. see Chapter 6 of [8,11,12] and itsrelevant references, [13,14,15,17], etc).The methods of assessing the individuals’ performanceusually applied in practice are based on principles of thebivalent logic (yes-no). However these methods are notprobably the most suitable ones. In fact, fuzzy logic, dueto its nature of including multiple values, offers a widerand richer field of resources for this purpose. This gave usseveral times in the past the impulsion to introduceprinciples of fuzzy logic in assessing the performance ofstudent groups in learning mathematics and problemsolving (e.g. see [10,12,13,16,17,18], etc). In this paperwe shall use fuzzy logic in assessing the total performanceof bridge players’ belonging to sets of special interest (e.g.different bridge clubs during a tournament, men andwomen, new and old players, etc).The rest of the paper is organized as follows: In thenext section we develop our new assessment method,which is based on principles of fuzzy logic. In sectionthree we present two real applications illustrating theimportance of our method in practice. Finally the lastsection is devoted to conclusions and discussion on futureperspectives of research on this area.For general facts on fuzzy sets we refer freely to thebook [8].2. The Assessment MethodAs we have already seen in the previous section, in agame of duplicate bridge the performance of each element(pair or team) is characterized by using either matchpoints or IMPs. However, apart from the above officialscoring methods, it is useful sometimes, for statistical orother reasons, to assess the total performance of certainsets of playing elements (single players, pairs, or teams)appearing to have a special interest. For example, thishappens, when one wants to compare the performance oftwo or more clubs participating in a big tournament, theperformance of male and female players or of old andyoung players, etc.One way to do this is by calculating the means of theofficial scores obtained by the elements of thecorresponding sets (mean performance). Here, we shalluse principles of fuzzy logic in developing an alternativemethod of assessment, according to which the higher is anelement’s performance the more its “contribution” to thecorresponding set’s total performance (weightedperformance).For this, we consider as set of the discourse the set U {A, B, C, D, F} of linguistic labels characterizing theplaying elements’ performance, where A characterizes anexcellent performance, B a very good, C a good, D amediocre and F an unsatisfactory performancerespectively. Obviously, the above characterizations arefuzzy depending on the user’s personal criteria, whichhowever must be compatible to the common logic, inorder to model the real situation in a worthy of credit way.In case of a pairs’ competition, for example, with matchpoints as the scoring method and according to the usualstandards of duplicate bridge, we can characterize thepairs’ (or the players’ individually) performance,according to the percentage of success, say p, achieved bythem, as follows:Excellent (A), if p 65%.Very good (B), if 55% p 65%.Good (C), if 48% p 55%.Mediocre (D), if 40% p 48%.Unsatisfactory (F), if p 40 %.In an analogous way one could characterize the teams’(or pairs’) performance with respect to the VPs, gained inbridge games played with IMPs.Assume now that one wants to assess the totalperformance of a special set, say S, of n playing pairs (orplayers'), where n is an integer, n 2. We are going torepresent S as a fuzzy subset of U. For this, if nA, nB. nC,nD and nF denote the number of pairs/players of S that haddemonstrated an excellent, very good, good, mediocre andunsatisfactory performance respectively at the game, wedefine the membership functionm: U [0, 1] in terms of the frequencies, i.e. bynm(x) x , for each x in U. Then S can be written as annfuzzy subset of U in the form: S {(x, x ): x U}.nIn converting the fuzzy data collected from the gamewe shall make use of the defuzzification technique knownas the centroid method. According to this method, thecentre of gravity of the graph of the membership functioninvolved provides an alternative measure of the system’sperformance. The application of the centroid method inpractice is simple and evident and, in contrast to otherdefuzzification techniques in use, like the measures ofuncertainty (for example see [12] and its relevantreferences, or [15]), needs no complicated calculations inits final step. The techniques that we shall apply here havebeen also used earlier in [9,14,16], etc.The first step in applying the centroid method is tocorrespond to each x U an interval of values from aprefixed numerical distribution, which actually means thatwe replace U with a set of real intervals. Then, weconstruct the graph, say G, of the membership functiony m(x). There is a commonly used in fuzzy logicapproach to measure performance with the coordinates (xc,yc) of the centre of gravity (centoid), say Fc, of the graphG, which we can calculate using the following wellknown from Mechanics formulas: xc xdxdy , ycdxdy FF ydxdyF dxdyF(1)In our case we characterize a pair’s performance asunsatisfactory (F), if x [0, 1), as mediocre (D), if x [1,2), as good (C), if x [2, 3), as very good (B), if x [3, 4)and as excellent (A), if x [4, 5] respectively.In other words, if x [0, 1), then y1 m(x) m(F) nFn, if x [1, 2), then y2 m(x) m(D) D , etc. nn

American Journal of Applied Mathematics and StatisticsTherefore in our case the graph G of the membershipfunction attached to S is the bar graph of Figure 1consisting of five rectangles, say Gi, i 1,2,3, 4, 5, whosesides lying on the X axis have length 1. In this case dxdy is the area of G which is equal toFnF nD nC nB n A 1n5 yii 1Also5 xdxdy xdxdy5 yi dy i 1 i 1 0FFii55 i xdxi 11 yi xdx 2 (2i 1) yi , i 1 i 1i 1(1a)and5 yi5i ydxdy ydy dx ydxdy i 1 i 1 0FFin yi1 n 2 ydyyi2i 1 i 1 0 i 1 Therefore, using the relations (1a), formulas (1) aretransformed into the following form:1( y1 3 y2 5 y3 7 y4 9 y5 ) ,21 22222yc y1 y2 y3 y4 y52xc )(2)But, 0 (y1-y2) y1 y2 -2y1y2, therefore y1 y22 2y1y2, with the equality holding if, and only if, y1 y2. Inthe same way one finds that y12 y32 2y1y3, and so on.Hence it is easy to check that (y1 y2 y3 y4 y5)2 5(y12 y22 y32 y42 y52), with the equality holding if, andonly if, y1 y2 y3 y4 y5. But y1 y2 y3 y4 y5 1, therefore1 5(y12 y22 y32 y42 y52) (3), with the equality holding1if, and only if, y1 y2 y3 y4 y5 .52225. Further,2combining the inequality (3) with the second of formulas(2), one finds that 1 10yc, or yc 1 Therefore the10 .unique minimum for yc corresponds to the centre of5 1gravity Fm ( , ).2 10The ideal case is when y1 y2 y3 y4 0 and y5 1. Then9andfrom formulas (2) we get that xc 21yc .Therefore the centre of gravity in this case is the29 1point Fi ( , ).2 2On the other hand, in the worst case y1 1 andy2 y3 y4 y5 0. Then by formulas (2), we find that the1 1centre of gravity is the point Fw ( , ).2 2Therefore the “area” where the centre of gravity Fc liesis represented by the triangle Fw Fm Fi of Figure 2.Then the first of formulas (2) gives that xc Figure 2. Graphical representation of the “area” of the centre of gravityFigure 1. Bar graphical data representation(1182Then from elementary geometric considerations itfollows that the greater is the value of xc the better is thecorresponding group’s performance. Also, for two groupswith the same xc 2,5, the group having the centre ofgravity which is situated closer to Fi is the group with thehigher yc; and for two groups with the same xc 2.5 thegroup having the centre of gravity which is situated fartherto Fw is the group with the lower yc. Based on the aboveconsiderations it is logical to formulate our criterion forcomparing the groups’ performances in the followingform:1. Among two or more groups the group with the higherxc performs better.2. If two or more groups have the same xc 2.5, thenthe group with the higher yc performs better.3. If two or more groups have the same xc 2.5, thenthe group with the lower yc performs better.

119American Journal of Applied Mathematics and Statistics3. Real ApplicationsThe Hellenic Bridge Federation (HBF) organizes, on aregular basis, simultaneous bridge tournaments (pairevents) with pre-dealt boards, played by the local clubs inseveral cities of Greece. Each of these tournamentsconsists of six in total events, played in a particular day ofthe week (e.g. Wednesday), for six successive weeks. Ineach of these events there is a local scoring table (matchpoints) for each participating club, as well as a centralscoring table, based on the local results of all participatingclubs, which are compared to each other. At the end of thetournament it is also formed a total scoring table in eachclub, for each player individually. In this table eachplayer’s score equals to the mean of the scores obtained byhim/her in the five of the six in total events of thetournament. If a player has participated in all the events,then his/her worst score is dropped out. On the contrary, ifhe/she has participated in less than five events, his/hername is not included in this table and no possible extrabonuses are awarded to him/her.In this section and in order to illustrate the importanceof our results obtained in the previous section, we shallpresent two real applications connected to the abovesimultaneous tournaments.The first application concerns the third event of such asimultaneous tournament played on Wednesday, March 12,2014, in which participated 17 in total clubs from severalcities of Greece (see results in [5]). Among those clubswere included the two bridge clubs, let’s call them C1 andC2 respectively, of the city of Patras. Nine in total pairsfrom club C1 played in this event obtaining the followingscores in the central scoring table: 62.67%, 57.94%,56.04%, 55.28%, 50.43%, 46%, 44.75%, 39.91% and36.16%. Eight in total pairs from club C2 played also inthe same event obtaining the following scores: 63.14%,57.64%, 56.86%, 50.17%, 50.13%, 43.28%, 42.11% and36.63%. The above scores give an average percentage49.909% for the first and 49.995% for the second club.This means that the second club demonstrated a slightlybetter mean performance than the first one, but thedifference was marginal; only 0.086%.The above results are summarized in Table 1.Table 1. Results of the two bridge clubs of PatrasFirst club (C1)% ScalePerformanceAmount of pairs 65%A055-65%B448-55%C140-48%D2 40%F2Total9Second club (C2)% ScalePerformanceAmount of pairs 65%A055-65%B348-55%C240-48%D2 40%F1Total8m(x)04/91/92/92/9M(x)03/82/82/81/8Then, using the first of formulas (2) of the previoussectiononefindsthat1 221441xc ( 3. 5. 7. ) 2.278 for the first2 9999 181 1223 38club, and xc ( 3. 5. 7. ) 2.375 for2 8888 16the second club. Therefore, according to our criterion (firstcase) stated in the previous section, the second clubdemonstrated a better weighted performance than the firstone, but the difference is small again; just 0.097 units.The second application is related to the total scoringtable of the players of club C1, who participated in at leastfive of the six in total events of another simultaneoustournament organized by the HBF, which ended onFebruary 19, 2014 (see results in [5]). Nine men and fivewomen players are included in this table, who obtained thefollowing scores. Men: 57.22%, 54.77%, 54.77%, 54.35%,54.08%, 50.82 %, 50.82%, 49.61%, 47.82%. Women:59.48%, 54.08%, 53.45%, 53.45%, 47.39%. The aboveresults give a mean percentage of approximately 52.696%for the men and 53.57% for the women players. Thereforethe women demonstrated a slightly better meanperformance than the men players, their difference being0.874%.The above results are summarized in Table 2.Table 2. Total scoring of the men and women playersMen% ScalePerformanceAmount of playersm(x) 65%A0055-65%B11/948-55%C77/940-48%D11/9 40%F00Total9Women% ScalePerformanceAmount of playersm(x) 65%A0055-65%B11/548-55%C33/540-48%D11/5 40%F00Total5Thus, according to the first of formulas (2) of theprevioussection,wefindthat1 17145xc (3. 5. 7. ) 2.5 for the men players,2 999 181 13125and xc (3. 5. 7. ) 2.5 for the women2 555 10players. Further, the second of formulas (2) gives yc 1 1 2 7 2 1 251[( ) ( ) ( ) ] 0.315 for the men and yc 2 9991621 1 2 3 2 1 211[( ) ( ) ( ) ] 0.22 for the women players.2 55550Thus, according to our criterion (second case) and incontrast to the mean performance, the men demonstrated ahigher weighted performance than the women players.4. Conclusions and DiscussionIn the present paper we developed a new method forassessing the total performance of certain groups of pairsor teams or of bridge players individually, appearing tohave a special interest. In developing the above methodwe represented each of the groups of players’ underassessment as a fuzzy subset of a set U of linguistic labelscharacterizing the bridge players’ performance and weused the centroid defuzzification technique in converting

American Journal of Applied Mathematics and Statisticsthe fuzzy data collected from the game to a crisp number.According to the above assessment method the higher isan element’s performance the more its “contribution” tothe corresponding set’s total performance (weightedperformance). Thus, in contrast to the mean of the scoresof all set’s elements, which is connected to the meangroup’s performance, our method is connected somehowto the group’s quality performance. As a result, when theabove two different assessment methods are used incomparing the performance of two or more groups ofpairs/teams of bridge players, the results obtained maydiffer to each other in certain cases, where there aremarginal differences in the groups’ performance.Two real applications were also presented, related tosimultaneous tournaments (pair events) organized by theHBF. In the first of these applications we compared thetotal performance of the two bridge clubs of the city ofPatras in a particular event of a recent such tournament,while in the second one we compared the performance ofthe men and women players of one of the above clubs,based on their total scoring in the six events of anothersimultaneous tournament.In general, our method is suitable to be applied inparallel with the official bridge scoring methods (matchpoints or IMPs) for statistical and other obvious reasons.Our future plans for further research on the subject aimat applying our new assessment method in more realsituations, including also bridge games (pairs or teams)played with IMPs, in order to get statistically safer andmore solid conclusions about its applicability andusefulness. In a wider basis, since our method is actually ageneral assessment method, it could be extended to coverother sectors of the human activity as well, apart from thestudents’ (e.g. see [10,12,13,16,17,18], etc) and the bridgeplayers’ assessment (in this paper), where we have alreadyapplied 2][13][14][15][16][17]AcknowledgementThe author wishes to thank Dr. Athanasios Tsevis,Chemical Engineer, Hellenic Open University, Patras,Greece, for introducing him to the principles andtechniques of the game of bridge.120[18][19]“Bridge Base Online (BBO)”, available on the Web atwww.bridgebase.com/index.php?d y“Bridge rules and variations”, available on the Web iki/contract bridgeD’ Orsi, E. (chair) et al., “The New WBF IMP to VP Scales”,Technical Report of WBF Scoring Panel, available on the Web /regulations/WBFAlgorithms.pdf“Hellas Bridge Federation” (in Greek), available on the Web atwww.hellasbridge.orgKaplan, E., Winning Contract Bridge Complete, Fleet PublishingCorporation, N. Y., 1964.Kaplan, E., Winning Contract Bridge, Courier Dover Publications,N. Y., 2010.Klir, G. J. & Folger, T. A., Fuzzy Sets, Uncertainty andInformation, Prentice-Hall, London, 1988.Subbotin, I. Ya. Badkoobehi, H. & Bilotckii, N. N., “Applicationof fuzzy logic to learning assessment”, Didactics of Mathematics:Problems and Investigations, 22, 38-41, 2004.Voskoglou, M. Gr., “The process of learning mathematics: Afuzzy set approach”, Heuristic and Didactics of Exact Sciences,,10, 9-13, 1999.Voskoglou, M. Gr., “Fuzzy Sets in Case-Based Reasoning”, FuzzySystems and Knowledge Discovery, IEEE Computer Society, Vol.6, 252-256, 2009.Voskoglou, M. Gr., Stochastic and fuzzy models in MathematicsEducation, Artificial Intelligence and Management, LambertAcademic Publishing, Saarbrucken, Germany, 2011.Voskoglou, M. Gr., “Fuzzy Logic and Uncertainty in MathematicsEducation”, International Journal of Applications of Fuzzy Setsand Artificial Intelligence, 1, 45-64, 2011.Voskoglou, M. Gr. & Subbotin, I. Ya., “Fuzzy Models forAnalogical Reasoning”, International Journal of Applications ofFuzzy Sets and Artificial Intelligence, 2, 19-38, 2012.Voskoglou, M. Gr., “A study on fuzzy systems”, AmericanJournal of Computational and Applied Mathematics, 2 (5), 232240, 2012.Voskoglou, M. Gr., “Assessing Students’ Individual ProblemSolving Skills”, International Journal of Applications of FuzzySets and Artificial Intelligence, 3, 39-49, 2013.Voskoglou, M. Gr & Subbotin, I. Ya,, “Dealing with the Fuzzinessof Human Reasoning”, International Journal of Applications ofFuzzy Sets and Artificial Intelligence, 3, 91-106, 2013.Voskoglou, M. Gr, “Fuzzy Logic in the APOS/ACE InstructionalTreatment for Mathematics”, American Journal of AppliedMathematics and Statistics, 2014.Zadeh, L. A., “Fuzzy Sets”, Information and Control, 8, 338- 353,1965.

Contract Bridge. was the result of innovations to the scoring of auction bridge suggested . by. Harold Stirling Vanderbilt (USA, 1925) and others. Within a few years contract bridge had so supplanted the other forms of the game that "bridge" became synonymous with "contract bridge." Rubber Bridge

Related Documents:

May 02, 2018 · D. Program Evaluation ͟The organization has provided a description of the framework for how each program will be evaluated. The framework should include all the elements below: ͟The evaluation methods are cost-effective for the organization ͟Quantitative and qualitative data is being collected (at Basics tier, data collection must have begun)

Silat is a combative art of self-defense and survival rooted from Matay archipelago. It was traced at thé early of Langkasuka Kingdom (2nd century CE) till thé reign of Melaka (Malaysia) Sultanate era (13th century). Silat has now evolved to become part of social culture and tradition with thé appearance of a fine physical and spiritual .

On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.

̶The leading indicator of employee engagement is based on the quality of the relationship between employee and supervisor Empower your managers! ̶Help them understand the impact on the organization ̶Share important changes, plan options, tasks, and deadlines ̶Provide key messages and talking points ̶Prepare them to answer employee questions

Dr. Sunita Bharatwal** Dr. Pawan Garga*** Abstract Customer satisfaction is derived from thè functionalities and values, a product or Service can provide. The current study aims to segregate thè dimensions of ordine Service quality and gather insights on its impact on web shopping. The trends of purchases have

Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được

Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. Crawford M., Marsh D. The driving force : food in human evolution and the future.

Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.