Polling Zones Planning Problem

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1 Iteratoal Joural of Egeerg ad Aled Sceces (IJEAS) ISSN: , Volue-3, Issue-3, March 2016 Pollg Zoes Plag Proble Halu Saffet Bala, Nda Kordal Abstract Ths aer troduces a alcato of the use of electroagetc ush-ull echas to the soluto of atheatcal rograg robles. The algorth used s slar to the ethodology used to solve Travellg Salesa Proble usg a oulato based algorth sulatg the ush-ull echas of electroagetc theory, whch was orgally aled to ucostraed otzato robles cotuous sace. The charge (obectve fucto value) of each soluto ot feasble oulato ales attracto ad reulso force o the ots oulato ad oves the obectve fucto towards otalty,.e. to ot wth hghest ull force. Two dfferet versos of TSP based EM echas s used to fd otu oltcal dstrcts, where dstrcts fored ust be coact ad cotguous, whle each zoe have alost equal oulato. The algorths are tested to redraw borders of the ffty ollg zoes of North Cyrus. Idex Ters electroagetc ush-ull echas, eta-heurstcs, oltcal zoes, travellg salesa roble I. INTRODUCTION Electos are udeably oe of the ortat struets deocraces ad they have roouced effects o the socety. The selecto of the ollg zoes s oe of the ortat factors that ay affect the outcoe of a electo. How the borders of a ollg zoes draw should be obectve. I other words, the borders of oltcal zoes should be draw deedet of creatg a advatage or dsadvatage to ay oltcal arty or ay oltcal erso. The followg crtera are usually used for obectvty of oltcal ollg zoes: Oe-Ma-Oe-Vote Prcle (.e. every ollg zoe to have aroxately sae uber of voters), coactess ad beg cotguous. I soe deocraces, oltcal zoes are aulated (gerryaderg) order to favour certa oltcal arty. I order to sto such uethcal ractces, lterature, varous ad eutral rocedures have bee develoed to draw the borders of ollg zoes, geerally solvg a sgle-obectve or ult- obectve otzato odels. The ter Gerryaderg s a ter coed for the delberate chage of the boudares of ollg zoes to aulate the result of electos. It s aed after Elbrdge Gerry, who created a salaader shaed oltcal zoe whch oosto voters were cocetrated (aced) few zoes so that eghbourg zoes oosto s dluted. I ths aer, a ollg zoe la wll be accetable f each zoe geerated s cotguous ad has at ost ±5% oulato devato fro the ea. Subect to these costrats the algorth wll search for coact shaed ollg zoes subect to costrats exlaed above. I Secto 2, the roble of drawg borders of oltcal zoes s descrbed. Secto 3 resets TSP based electroagetc ush-ull echas Secto 4 cotas the algorths used to draw oltcal zoes, ad the results of the uercal exerets for North Cyrus. Secto 5 cocludes the aer ad state ossble future wor. II. THE DETAILS OF THE POLLING ZONES MODEL I suary, a la to create ollg zoes s a allocato of dvsble oulato uts to requred uber of cotguous ollg zoes so that each ollg zoe have coact shae, ad the oulatos of all ollg zoes le wth lus or us 5% of ther ea. The, the crtera ca be suarzed as: Poulato Equalty: All ollg zoes should have alost equal oulatos for far reresetato the assebly; Cotguous: There should a ubroe ath betwee ay two geograhcal ots wth a zoe whch les coletely wth that ollg zoe; Geograhcally coact: deally crcular shae as oosed to a th ad log salaader shaed fgure, ost owerful crtera to revet gerryaderg; Coforty to reset adstratve boudares; Deograhc characterstcs of the oulato. Pollg zoe lag roble, whch s a lear 0-1 rograg roble, s also a subset of cobatoral set arttog robles. I set arttog odels, a oe-to-oe assget o odes to cotguous sets are erfored, based o ode weghts wthout volatg the weght costrat of ay set. Geeral for of odel s (1): Max. cx Subto. T Ax e, X 0 or 1, for 1,..., Ad A s a by bary atrx, e T s a su vector of sze, ad c s a vector of (arbtrary) real ubers (weghts). I ollg zoe lag roble, regos ust be dvded to votg zoes, where every ctze s a eber of oly oe zoe [2]. I recet years, eta-heurstc ethods such as geetc algorths, tabu search ad sulated aealg has bee aled to solve such robles. Rcca &Soe [3] had tested varous algorths cludg tabu search, descet, sulated aealg ad stated that old bachelor accetace roduced the best results the ost of the test robles, esecally whe coactess crtera was used as the obectve. (1) 30

2 Pollg Zoes Plag Proble To solve the Pollg zoe Plag roble other fudaetal hyscs theores have also bee utlzed for solvg cobatoral robles: Potts odel, a statstcal hyscs aroach [4]; ad S syste, whch treats the roble by covertg t to a eergy fucto [5]. A eta-heurstc otzato has bee develoed based o Electroagets (EM), such that a charge s assocated wth ots a ult-desoal soluto sace ad each ot wll exert ull or ush o other ots, rocess ovg the obectve to axzato. Ths rocess wll start a local search Euclda sace a oulato-based fraewor [1]. Although, the cotguty ad oulato lt crtera ca be aled easly, desg of coact zoes s ore colex. Hess, et al [6] used the squared dstace of each erso fro the zoe s cetre as a crtero. The fact that oly the oulato uts are chose as zoe cetres also creates colcatos. Coactess easureet has a drawbac because oly oulato uts ca be zoe cetres. Garfel ad Nehauser [7] roose dstace ad shae ersectves to fd geograhcal coactess. Marl [8] shows that, forulato s arorate whe oulato cetres are ot fxed locatos but t s ot for fxed terrtoral cetres. Youg [9] showed that d 2 test s a easure of coactess, as well as a easure of sze ad, t wll gve a better easure of coactess for a sall urba zoe coared to a large rural zoe of the sae shae. Mehrotra, et al[10], uses a etwor aroach to easure the roxty ters of sallest uber of trast oulato uts betwee two gve oulato uts, s, the uber of edges a shortest ath fro ode to. The obectve becoes zato of the su of dstace cost of oulato uts fro the cetre oulato ut of the zoe. Cha, ad We [11], used fuzzy set theory to geerate ore flexble las for ollg zoes. L, et al, [12] used coactess crtero the obectve fucto of a quadratc rograg odel. The Pollg Zoes Proble, cobatoral roble, s classfed as NP-colete roble ad all the aroaches lsted above are heurstcs. ad exact rocedures. I ths study, a ew heurstc aroach s develoed rather tha usg tradtoal atheatcal rograg techques. III. TSP BASED ELECTROMAGNETIC MECHANISM Electroagetc aroach ales electroagetc theory of hyscs. Pots the soluto sace are cosdered as electrcally charged artcles whch ove towards otalty usg electroagetc ush-ull echas. The charge of each soluto ot s rereseted the obectve fucto value. Soluto ots wth better obectve fucto values have ore charge tha other ots. As the alcato of echas, ots wth ore charge (better obectve fucto value) attract other ots, ad ots wth less charge (worse obectve fucto value) reulse other ots. As the charge value creases, sze of attracto creases too. Fally, a total force vector exerted to a ot s calculated by sug all ush-ull force vectors for ovg the ot the drecto of total force aled o t [1]. The orgal electroagetc aroach ales o otzato robles wth cotuous varables the followg for, as gve eqs. (2) ad (3). fx t. x s LU, UL :. (2), xrlx, U 2,1,..., (3) Brbl, et al [10] revsed ths forulato to esure coverge ad rove that ther revsed verso eables global covergece wth robablty oe. Electroagetc aroach has bee aled varous cobatoral otzato robles such as schedulg, travellg salesa ad soft coutg whch also dscuss hybrd alcatos.. Geeral schee of Electroagetc-le algorth ay be outled as gve below: Italze () terato <= 1 whle terato < MAXNIT do Local (MLSI, δ) F<= Calculate (F) Move (F) Iterato <= terato + 1 ed whle Algorth has four hases Italzato of algorth, coutato of total force exerted o each artcle, oveet alog the drecto of the force ad local search. A. Italzato: Algorth starts wth a radoly geerated tal soluto of sale ots the feasble rego defed by uer ad lower boudares of desoal hyer-cube. After that, the obectve fucto value of each sale ot s couted; g(x) ad the ot, whch has the best obectve fucto value, selected as best soluto, x b. B. Local Search: Local search s used to rove the curret soluto obectve fucto towards a local ota. Uer ad lower bouds each deso ad rado ste araeter δ Є [0,1] are used to create a feasble rado legth to ove sale ots. I case eghborhood search fds a roved soluto tha the curret ot wth MLSI teratos, udate x as the x b. C. Calculate Force: The charge of each ot q, whch deteres ot s ull or ush ower s calculated accordg to the obectve fucto value of a ot. Pots wth better obectve fucto value have ore charge tha others. The charges are calculated usg: q ex f (4) ı b best x f x / f x f x, 1,2,...,

3 I calculatg the force betwee two ots, the Coulob s law of the electroagetc theory s used, where the force exerted o a ot va other ots s versely roortoal to the square of the dstace betwee the ots ad drectly roortoal to the roduct of ther charges. Pot wth better obectve fucto value attracts other ots, whle a ot wth worse obectve fucto value reels the others. Later a vector of the total force exerted o each ot by other ots s calculated. Ths vector wll detere the oveet drecto of that ot. I the orgal Electroagetc algorth there s a ossblty of beg traed by a local a ad thus the global ota ay ever be reached. Brbl, et al [13] troduced a erturbed ot ad chaged the force coutato, as a result, they show that algorth coverges global otu wth robablty of oe. The ost dstat ot fro the best ot, x b, s selected as the erturbed ot x. The total force value of x aeded, whle others rea the sae (5). I case of erturbed ot x, the cooet forces are erturbed by a rado uber 0< λ <1. The drectos of the cooet forces are reversed accordg to: Iteratoal Joural of Egeerg ad Aled Sceces (IJEAS) ISSN: , Volue-3, Issue-3, March 2016 Fgure 2 Dffereces of AL ad RK fors. Table 1 Illustrato for use AL for the Electroagetc-le algorth. F qq x x, ffx fx xx qq x x, ffx fx 2 xx 2 D. Move Pot Alog the Total Force: Pots are oved the drecto of total force vector by a rado ste legth as equato (6). x F / F RNG, 1,2,..., x (6) Whle orgally Electroagetc-le algorth was aled to real-valued-soluto robles, ollg zoe lag ad Travellg Salesa (TSP) robles are teger rograg robles. Ths study troduces two dfferet aroaches to odel the ollg zoe lag as a TSP roble. Frst s a ethod roosed by Debels, et al [14], for the schedulg roble gve by Wu, et al [15], where ordered real ubers rereset a tour. They use two dfferet reresetatos, ether the actvty-lst (AL) or the rado-ey (RK). The dfferece betwee AL ad RK fors of a schedule s gve Fgure 2. A soluto of 10-cty TSP, where AL s used electroagetc-le ethod, s llustrated table 2. (5) I the secod ethod that was roosed by Keedy, ad Eberhardos, [17] for the dscrete artcle swar otzato ethod dscussed by Javada, et al [16], s used. I ths aroach, a tour s rereseted by dscrete-bary ubers. If a ot reresets a tour of ctes as gve below: X xxxxx 11, 12,..., 1 1,..., x 1,0, where x = 1, RNG uxf F 0 the ste of the tour cty s vsted. I order to aly the 2,1,..., (7) bary tour reresetato, the total force vector calculated (as xl f F 0 Le 5 of Table 1) shows the chace of the bary varable Where u ad l are the uer ad the lower bouds of the tag the value oe the followg ste of the algorth th deso ad λ s the rado ste sze. (Le 6). The greatest ostve cooet of F s selected, ad f the corresodg cooet of X s equal to zero, t s chaged to 1. New Move rocedure s aled to ove the ots ad ae ecessary chages o X to obta a feasble IV. APPLICATION OF EMM FOR TSP TO POLLING tour. Table 2 shows a alcato of these stes o a ZONES MODEL terato of a tour of fve ctes wth the sequece of 1, 3, 4, 2, 5 ad the roved (oved ot) tour whe the greatest force F s f 33. A. Adatato to the roble of Poltcal zoes Plag: Above ethods are used to fd tours TSP. Ths study s usg actvty-lst reresetato for defg tours. Costructed tours are trasfored to a ollg zoe la by ovg alog the tour ad sug the oulatos utl the threshold oulato s reached, where the threshold defed as the Ideal zoe oulato ± ε. Ideal zoe oulato s obtaed as the rato of total uber of voters to uber of zoes to be arttoed ad ε = 5%. Hgh Electoral Coucl of T.R.N.C. also requres ±5% tolerace lts for oulato arty, whle geeratg a la for ollg zoes. 32

4 Pollg Zoes Plag Proble Table 2 Illustrato of Bary for ad oveet. Fgure 3 - Zog la of İsele The la s oulato varace fucto (8) s based o deal zoe oulato, oulato ealty (P) ad axu tolerace allowed (Var = 5% ) Measure (8) POP P 1 Max Po IdDstPo Var * IdDstPo 5%, 0 1 Po IdDstPo * IdDstPo Coactess crtero s also used for evaluato of a ollg zoe la.. As dscussed before there are varous alteratves for easurg coactess. I ths study we are usg a easure based o TSP. For each geerated zoe legth of corresodg TSP tour (Dst ) s calculated whch s the oralzed by the average dstace travelled of all zoes (MeaTSP) as show by the equato (9). Dst MeaTSP Measure COMP 1.. MeaTSP * (9) Poulato ad coactess easures are cobed ad the obectve fucto value for the roble s obtaed (10). I order to balace these two easures, θ s used. ObFucVal Measure c * Measure P. (10) B. Paraeters ad Results The rary araeters ths ethod for the sale soluto gve are lsted Table 3. The algorth s aled to the coutes of T.R.N.C. by usg the Cty ad Adstratve Mas of T.R.N.C. A secal characterstc of Cyrus s ts rregular shae whch causes the algorth to geerate less coact zoe las. Zog las geerated for İsele ad Lefoşa coutes are reseted fgures 3 ad 4, resectvely. İsele couty has a secal shae. Zoe la of İsele, s ot so reasoable because of the rregular shae of the Yeşlöy ad Yeereöy vllages. These vllages have too ay eghbours as coared to the other vllages, ad they are located the ddle of the zoe. Because of ths reaso, coactess easureet s low for the zoe la of İsele. Table 4 - Paraeters used alcato of the ethod. Nae of Paraeter No. sale ots of Max. No. of teratos Max. o. of local search teratos Local search araeter Poulato tolerace Poulato ealty Sybol MAXNIT MLSI δ ε P θ Bala ce coef. Value used Fgure 4 - Zog la of Lefoşa V. CONCLUSION Electroagetc-le algorth reseted ths aer uses TSP odellg ad ecodg to geerate the ollg zoe las. To our best owledge, ths s the frst such algorth used ths roble doa. The fact that coactess easure ad behavour of the obectve fucto drectly affects obtaed zoe las, s a observed dsadvatage. I future the algorth wll be tested o bechar robles for better coarso. To crease the qualty of solutos, local search algorths wll be troduced coucto wth ult-obectve otzato ad dfferet coutatoal ethods. REFERENCES [1] Brbl, S.I.; ad Fag, S.C A electroagets-le echas for global otzato. Joural of Global Otzato 25(3): [2] Balas, E.; ad Padberg M.W Set Parttog: A Survey. SIAM Revew 18(4): [3] Rcca, F; ad Soe, B Local search algorths for oltcal dstrctg. Euroea Joural of Oeratos Research 189: [4] Chou, C.; ad L, S Tag gerryader-statstcal hyscs aroach to oltcal dstrctg roble. Physca A 369:

5 [5] Chou, C.; ad L, S S systes ad oltcal dstrctg roble. Joural of Magets ad Magetc Materals 310: [6] Hess, S.W.; Weaver, J.B.; Segfeldt, H.J.; Whela, J.N.; ad Ztlau, P.A Noartsa oltcal redstrctg by couter. Oeratos Research 13: [7] Garfel, R.S.; ad Nehauser, G.L Otal oltcal dstrctg by lct euerato techques. Maageet Scece 16: [8] Marl, P.G Alcato of the trasortato odel to a large scale dstrctg roble. Couters ad Oeratos Research 8: [9] Youg, H.P Measurg the Coactess of Legslatve Dstrcts. Legslatve Studes Quarterly XIII: [10] Mehrotra, M.; Johso, E.L.; ad Nehauser, G.L A otzato based heurstc for oltcal dstrctg. Maageet Scece 44: 8. [11] Cha, W.Y.; ad We, B.C Coactess easureet usg fuzzy ult-crtera decso ag for redstrctg. I Proceedgs of IEEE Rego 10 Iteratoal Coferece o Electrcal ad Electroc Techology August, Cruse Sh, SuerStar Vrgo. IEEE. [12] L, Z.; Wag, R.S.; ad Wag, Y A quadratc rograg odel for oltcal dstrctg roble, I Proceedgs of 1 st Iteratoal Syosu o Otzato ad Systes Bology August, Beg, Cha. [13] Brbl, S.I.; Fag, S.C.; ad Sheu, R.L O covergece of a oulato-based global otzato algorth. Joural of Global Otzato 30: [14] Debels, D.; De Reyc, B.; Leus,R.; ad Vahouce, M A hybrd scatter search/electroagets eta-heurstc for roect schedulg. Euroea Joural of Oeratos Research 169: [15] Wu, P.; We, N.C.; ad Fag, H.C Revsed electroagets-le echas for the travellg salesa roble. I Proceedg of the 36th It. Cof. o Couters ad Idustral Egeerg, Tae, Tawa. [16] Javada, N.; Alha, M.G.; ad Tavaol-Moghadda, R A dscrete bary verso of the electroagets-le heurstc for solvg travellg salesa roble. I Proceedgs of Fourth Iteratoal Coferece o Itellget Coutg (ICIC 2008), Shagha, Cha, Seteber. Lecture Notes Artfcal Itellgece, Srger-Verlag Berl Hedelberg. [17] Keedy, J.; ad Eberhart, R.C A dscrete bary verso of the artcle swar algorth. I Proceedgs of the IEEE Iteratoal Coferece o Systes, Ma, ad Cyberetcs. Pscatawary, NJ, USA October. IEEE Press. BIOGRAPHIES Iteratoal Joural of Egeerg ad Aled Sceces (IJEAS) ISSN: , Volue-3, Issue-3, March 2016 Halu Saffet Bala He was bor Aara, Turey He had receved hs BS (1966) ad MS (1968) degrees Electrcal Egeerg fro METU, Aara, Türye. He had also eared MS Oeratos Research 1974 at Uversty of Pesylvaa. He s curretly the Dea of Faculty of Busess Adstrato at Uversty Medterraea Uversty Karasa ad hs research terests clude eta-heurstc soluto ethodologes for lear ad o-lear otzato robles ad ther alcatos to dustral robles. Assoc. Prof. Bala s a eber of the Tursh Oeratos Research Socety. Nda Kordal He was bor Lefoşa, T.R.N.C Colete hs B.Sc. ad M.Sc. degrees Easter Medterraea Uversty o Idustral Egeerg year 1999 ad 2002, resectvely. Maor feld study s alcatos of oeratos research. He s curretly dog hs Ph.D. study Easter Medterraea Uversty studyg ult-obectve otzato techques ad alcatos Idustral Egeerg. Curret research terests are ult-obectve otzato, atheatcal odellg, ergoocs ad occuatoal health ad safety aageet systes. Mr. Kordal s a lecturer at CIU ad eber of the Tursh Oeratos Research Socety ad Chaber of Idustral Egeers T.R.N.C