Best Optimal Stable Matching
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1 Applied Mathematical Scieces, Vol., 0, o. 7, 7-7 Best Optimal Stable Matchig T. Ramachadra Departmet of Mathematics Govermet Arts College(Autoomous) Karur-6900, Tamiladu, Idia K. Velusamy Departmet of Mathematics Chettiad College of Egieerig ad Techology Karur-69, Tamiladu,Idia T. Selvakumar Departmet of Mathematics Chettiad College of Egieerig ad Techology Karur-69, Tamiladu,Idia Abstract Gale ad Shapely proposed a GS algorithm to solve stable marriage problem. This algorithm results a ma optimal / woma optimal matchig. I this paper, the term best optimal stable matchig, preferece value ad satisfactory level are defied ad usig it a ew algorithm, called Best Optimal Stable Matchig Algorithm (BOSMA), is developed to fid best optimal stable matchig. Keywords: Preferece Value,Satisfactory Matrix,Satisfactory value,best Optimal Matchig Itroductio The stable marriage problem (SM) is a well-kow problem of matchig the elemets of two sets []. It is called the stable marriage problem sice the stadard formulatio is i terms of me ad wome, ad the matchig is
2 7 T. Ramachadra, K. Velusamy ad T. Selvakumar iterpreted i terms of a set of marriages. Give me ad wome, where each perso expresses a strict orderig over the members of the opposite sex [], the problem is to match the me to the wome so that there are o two people of opposite sex who would both rather be matched with each other tha their curret parters. If there are o such people, all the marriages are said to be stable. Gale ad Shapley proved that it is always possible to fid a matchig that makes all marriages stable, ad provided a polyomial time algorithm which ca be used to fid oe of two extreme stable marriages, the so-called ma-optimal or woma-optimal solutios []. I the ma-optimal stable matchig, each ma is matched with his best possible parter, while each woma gets her worst possible, amog all stable matchig []. Next, a algorithm is itroduced to fid best optimal stable matchig, usig preferece value, based o satisfactory level ad which is best amog the ma ad woma optimal matchig. Related Defiitios I order to describe the ew algorithm to fid the best matchig usig assigmet techique, we eed to itroduce the followig defiitios.. Defiitio:Preferece value I a SM istace, each member is assiged with a value,o the basis of the order of preferece with respect to ma/woma.this value is called Preferece value.. Defiitio: Me s ad Wome s Preferece value The preferece value assiged to each woma with respect to a ma s preferece list is called me s preferece value ad the preferece value assiged to each ma with respect to woma s preferece list is called wome s preferece value. For a istace cosider a SM ivolvig me ad wome, each of whom prefer all members of the opposite sex i the strict order of preferece. I each ma s preferece list, each woma fids preferece value i which first woma gets preferece value, secod woma, third woma ad so o, the th woma gets preferece value. Similarly i each woma s preferece list, each ma fids preferece value where the first ma gets preferece value secod ma, third ma ad so o, the th ma gets preferece value
3 Best optimal stable matchig 7. Defiitio:Me s preferece value matrix (PM M ) Matrix Represetatio of Me s preferece values form a x matrix,there row represets me ad colum represet wome ad it is deoted as PM M =[w ij ], where w ij is the preferece value of j th woma with respect to i th ma ad it is equal to (j ).. Defiitio: Wome s preferece value matrix (PM W ) Wome s preferece value matrix is similar to that of Me s preferece value matrix except that the rows are represeted by wome ad colums are represeted by me ad it is deoted PM W =[m ij ],where m ij is the preferece value of j th ma with respect to i th woma ad it is equal to (j ). Note:. The preferece value of ay idividual i matchig is maximum, miimum.. Preferece value idicates the highest satisfactory level, whereas idicates the least possible satisfactio level of idividual.. I both PM M ad PM W,the row total should be equal to +.. Defiitio: Satisfactory value matrix (SM M/W ) The satisfactory value matrix with respect to me SM M is defied by sum of PM M ad (PM W ) T. The satisfactory value matrix with respect to wome SM W is defied by sum of (PM M ) T ad PM W. Note:. SM M ad SM W are trasposig of each other.. To fid best matchig, we ca use either SM M or SM W..6 Defiitio:Satisfactory level For a give matchig, satisfactory level for ma / woma is the ratio of sum of the satisfactory value of all me / wome ad sum of satisfactory value of all me ad wome.
4 76 T. Ramachadra, K. Velusamy ad T. Selvakumar.7 Defiitio:Best optimal stable matchig I the stable matchig, the lower value amog the me s satisfactory level ad wome s satisfactory level is cosidered as the Best Optimal Stable Matchig. Example:. A SM istace with three me m,m,m ad three wome w, w,w is cosidered here.the preferece lists of me ad wome are give below i the order of preferece. m :w w w w : m m m m : w w w w : m m m m : w w w w : m m m The preferece value of w, w, w with respect to m is, ad respectively, preferece value of w,w, w with respect to m is, ad respectively ad preferece value of w, w, w with respect to m is, ad respectively. Now the me s preferece value matrix is w w w PM M = m m m Now the wome s preferece value matrix is m m m PM W = w w w The satisfactory value matrix is, w w w
5 Best optimal stable matchig 77 SM M = 6 m m m m m m SM W = 6 w w w It is observed that w w w SM M = 6 m m m = SM W T Best Optimal Stable Matchig Algorithm(BOSMA). Get umber of me (m) or wome (w), say. Get the preferece lists of all me ad wome. Assig preferece value for each w j with respect to each m i, accordig to me s preferece list ad costruct PM M. Assig preferece value for each m i with respect to each w j, accordig to wome s preferece list ad costruct PM W
6 78 T. Ramachadra, K. Velusamy ad T. Selvakumar. Form a satisfactory value matrix SM M/W,addig PM M to the traspose of PM W (or PM W ad traspose of PM M ) 6. Write the row/colum maximum for each row/colum at the right/bottom of the matrix SM M/W. 7. Ecircle the cell, where the correspodig row maximum ad colum maximum meet. 8. If there is a tie betwee cells, the select the cell o the basis of the order of Preferece. 9. Write the matchig correspodig to the ecircled cells. 0. Write a ew matrix by elimiatig the correspodig row ad colum ad repeat step 6. Oly if the matrix exists. The above stated algorithm is applied i five differet sets of preferece lists of me ad wome. A SM istace with three me m,m,m ad three wome w, w,w is cosidered to fid best optimal stable matchig with maximum satisfactory level usig the give algorithm. Case:(The ma optimal ad woma optimal matchig are same) m :w w w w : m m m m : w w w w : m m m m : w w w w : m m m The ma-optimal stable matchig by GS algorithm for the above problem is (m,w ),(m,w ) ad (m,w ) ad thier satisfactory levels with respect to me ad wome are.8% ad 6.%. Similarly the Woma-optimal stable matchig by GS algorithm for the above problem is (m,w ),(m,w ) ad (m,w ) ad thier satisfactory levels with respect to me ad wome are.8% ad 6.%. The best optimal stable matchig by the above metioed algorithm is (m,w ),(m,w ) ad (m,w ). Case:(All me (ma optimal) ad wome (woma optimal) are matched to thier first choices)
7 Best optimal stable matchig 79 m :w w w w : m m m m : w w w w : m m m m : w w w w : m m m Ma-Optimal Woma-Optimal Best Optimal Stable Matchig Matchig (M.S.) (W.S.) Matchig (M.S.) (W.S.) Matchig (m,w ) (m,w ) (m,w ) (m,w ) 60% 0% (m,w ) % 7% (m,w ) (m,w ) (m,w ) (m,w ) M.S-Me s Satisfactory Level W.S-Wome s Satisfactory Level The differece betwee me s ad wome s satisfactory level is less i ma optimal matchig compared to woma optimal matchig. So ma optimal matchig is the best optimal matchig, which is give by the algorithm. Case:(Oe ma (ma optimal) ad oe woma (woma optimal) are matched to the secod ad the rest to thier first choices) m :w w w w : m m m m : w w w w : m m m m : w w w w : m m m Ma-Optimal Woma-Optimal Best Optimal Stable Matchig Matchig (M.S.) (W.S.) Matchig (M.S.) (W.S.) Matchig (m,w ) (m,w ) (m,w ) (m,w ) 6.% 8.% (m,w ) 8.% 6.% (m,w ) (m,w ) (m,w ) (m,w ) M.S-Me s Satisfactory Level W.S-Wome s Satisfactory Level Case:(Oe ma to first ad other me (ma optimal) to secod choices ad all wome (woma optimal)are matched to their first choices) m :w w w w : m m m
8 70 T. Ramachadra, K. Velusamy ad T. Selvakumar m : w w w w : m m m m : w w w w : m m m Ma-Optimal Woma-Optimal Best Optimal Stable Matchig Matchig (M.S.) (W.S.) Matchig (M.S.) (W.S.) Matchig (m,w ) (m,w ) (m,w ) (m,w ) 8.%.7% (m,w ) 0.8% 69.% (m,w ) (m,w ) (m,w ) (m,w ) M.S-Me s Satisfactory Level W.S-Wome s Satisfactory Level Case:(All me (ma optimal) ad all wome (woma optimal) are ot matched to thier first choices) m :w w w w : m m m m : w w w w : m m m m : w w w w : m m m Ma-Optimal Woma-Optimal Best Optimal Stable Matchig Matchig (M.S.) (W.S.) Matchig (M.S.) (W.S.) Matchig (m,w ) (m,w ) (m,w ) (m,w ) 8.%.7% (m,w ).% 66.7% (m,w ) (m,w ) (m,w ) (m,w ) M.S-Me s Satisfactory Level W.S-Wome s Satisfactory Level Coclusio: The matchig obtaied by Gale-Shapley algorithm is ma-optimal, that is, the matchig is preferable for me but upreferable for wome, (or, if we exchage the role of me ad wome, the resultig matchig is woma-optimal). The BOSMA gives Maximum possible satisfactio for both me ad wome. That is, if we icrease the satisfactory level of me, by chagig the matchig, automatically the satisfactory level of wome will decrease.the above discussed algorithm is capable of givig the best of two optimal matchigs(maoptimal ad Woma-optimal),without kowig ma optimal ad woma optimal matchig ad its satisfactory levels.so this algorithm is the most suitable oe to fid best optimal stable matchig,
9 Best optimal stable matchig 7 Refereces [] D. Gale ad L.S. Shapley. College admissios ad the stability of marriage. America Mathematical Mothly, 69(96), pp 9-., [] Ismel Brito ad Pedro Meseguer,Distributed Stable Marriage Problem, Priciples ad Practice of Costrait Programmig - CP 00 th Iteratioal Coferece, CP 00, Sitges, Spai, October -, 00. Proceedigs. [] R.W. Irvig, Ma-Exchage Stable Marriage, Uiversity of Glasgow, Computig Sciece Departmet Research Report, TR-00-77, August 00. [] Kazuo Iwama,Suchi Miyazaki,Hiroki Yagagisawa.Approximatio algorithm for the sex equal stable marriage problem, Algorithms ad Data Structures 0th Iteratioal Workshop, WADS 007, Halifax, Caada, August -7,007. Proceedigs. Received: May, 0
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