Experimental Comparison of Genetic Algorithm and Ant Colony Optimization to Minimize Energy in Ad-hoc Wireless Networks

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1 Proceedgs of e World Cogress o Egeerg ad Computer Scece 013 Vol II, 3-5 October, 013, Sa Fracsco, USA Expermetal Comparso of Geetc Algorm ad At Coloy Optmato to Mme Eergy Ad-hoc Wreless Networs Ibuuola A. Modupe, Oludayo O. Olugbara, Suday O. Ojo ad Abodu Modupe Abstract Reported s paper are e results of a smulated expermetal comparso of Geetc Algorm (GA) ad At Coloy Optmato (ACO) meta-heurstcs, w regards to er sutablty ad performace addressg e problem of eergy cosumpto mmato ad-hoc wreless etwors. A eergy fucto model based o Geographc Adapte Fdelty (GAF) topology maagemet scheme s used settg e smulato expermet. Results show at GA ad ACO meta-heurstcs are sutable optmato techques for eergy cosumpto mmato ad-hoc wreless etwors, w GA gg e least eergy cosumpto comparso to ACO. Idex Terms Ad-hoc Networs, Meta-heurstcs, Geographc Adapte Fdelty, Geetc Algorm, At Coloy Optmato W I. INTRODUCTION reless ad-hoc etwor s a etwor of odes at are coected togeer wout e use of a cetral base stato. Nodes ad-hoc wreless etwors are usually battery-operated ad are mostly deployed crtcal eromets such as e mltary oes, hostle, haardous, flooded areas ad a emergecy healcare stuato where t s almost mpossble to replesh e batteres. Ths maes t ecessary to cosere battery eergy for e etwor lfespa to be prologed, ad hece e sustaablty of operatos. The rates at whch odes e etwor cosume eergy dffer depedg o wheer e odes are a trasmttg, receg, lsteg or sleepg state [1]. The least eergy s cosumed whe e ode s a sleepg state. Howeer, all odes wll ot always be e sleepg state. The eergy rato betwee odes lsteg, receg ad trasmttg states s dcated as 1:1.05:1.4 ad 1:1.:1.7 [1], []. Equal ad adjustable grds as well as geetc algorm models were descrbed [1] for eergy Mauscrpt receed July, 01; resed August 01, 01. Ibuuola A. Modupe s curretly a master studet w e Departmet of Iformato, Tshwae Uersty of Techology, Pretora, 0001, Sou Afrca (emal: omea7@gmal.com) Oludayo, O. Olugbara s w departmet of Iformato Techology, Durba Uersty of Techology Sou Afrca, where he s curretly a Full Professor of Computer Scece ad Iformato Techology (e-mal: oludayoo@dut.ac.a) Suday, O. Ojo s a dea, Faculty of Faculty of Iformato ad Commucato Techology ad Research Professors at e Tshwae Uersty of Techology's (TUT), Soshague Sou Campus.(emal: ojoso@tut.ac.a) A. Modupe s w e Departmet of Iformato, Tshwae Uersty of Techology, Pretora, 0001, Sou Afrca (emal: modupea@tut,ac,a) optmato ad hoc wreless etwors. The results show at a geetc algorm ehaces more eergy sag e etre etwor compared to equal ad adjustable grd model. Bhodear et al [3] operated o a hgh umber of sesors for GA to geerate ts desg. The uformty of e sesg pots was made optmal ad e commucato eergy cosumpto was mmed w e costrats met. Hussa et al [4] addressed e eergy optmato problem by usg GA to determe e eergy-effcet clusters ad to detfy e cluster heads for e trasmsso of data. The result shows at GA retas small eergy effcet for loger tme durato. Srsaapor et al [5] ad Odem et al [6] propose protocols o At Coloy Optmato (ACO) for eergy optmato wreless ad hoc etwors. Srsaapor et al [5] propose a At-Based Eergy Coserato (ABEC) protocol at coseres eergy ad hoc etwors usg pheromoe tral process at coloy to set up e broadcastg of pacets for e trasmsso of data order to prolog e lfetme of e etwor. Odem et al [6] propose a protocol at uses At Coloy Optmato (ACO) to optme wreless sesor etwors routg pas ad also prode a effecte mult-pa data trasmsso meod to achee a relable commucato. I s study, we descrbe e Geographcal Adapte Fdelty (GAF) eergy model for addressg e eergy cosumpto mmato problem ad-hoc wreless etwors usg two meta-heurstcs techques Geetc Algorm (GA) ad At Coloy Optmato (ACO). These meta-heurstcs are used to geerate smallest amout of eergy bo drecto ( ad ) of e ad-hoc wreless etwors. GAF s descrbed as a topology maagemet scheme at s used to sae more eergy wreless etwor by groupg e odes a etwor to rtual grds. The odes at fall w e same grd are smlar ad resposble for data trasmsso. As a result, oly oe ode ca be acte at a ge tme (at s ode at recees ad trasmts data to e ext grd) whle e remag odes are made to sleep order to sae eergy. The smlartes of ese odes are computed usg locato formato such as Global Postog System (GPS) to partto e etwor area to grds. The cosumpto of eergy ca e be balaced by rotatg e trasmttg data amog e odes each grd to facltate e acte odes to correspod effectely durg e broadcast of data from e source odes to destato ode [1]. Ths paper reports o e results of a study o how eergy cosumpto a wreless Ad-hoc etwor ca be mmed usg e GAF eergy model. The model was smulated ISSN: (Prt); ISSN: (Ole)

2 Proceedgs of e World Cogress o Egeerg ad Computer Scece 013 Vol II, 3-5 October, 013, Sa Fracsco, USA usg GA ad ACO toolboxes MATLAB. These two meods (GA ad ACO) are parts of meta-heurstcs used for solg cooluted optmato problems. A comparso s e made o e mmal eergy geerated by e metaheurstcs toolboxes based o ACO ad e GA model. The remag part of s paper s succctly summared as follows. I Secto II, we descrbe e GAF aalyss of eergy cosumpto formulato. I Secto III, we descrbe e GAF geetc algorm model. I Secto IV, we descrbe e GAF at coloy optmato model. I Secto V we preset expermetal results of e meta-heurstcs techques of geetc algorm ad at coloy optmato. I Secto VI, we ge a cocludg remar ad prode motato for future wor. II. GEOGRAPHIC ADAPTIVE FIDELITY MODEL GAF protocol coseres eergy by parttog e odes e etwor area to rtual grds as llustrated Fg. 1. I s model, ad hoc wreless etwor w a L B rectagular area s dded to grds so at data ca be forwarded grd by grd to e destato ode. Fg. 1 shows e eergy cosumpto model a rectagular GAF model [1]. Fg. 1: Eergy cosumpto adjustable rectagular GAF Model The total eergy cosumed a grd E s e addto of e eergy geerated by e ode whe e lsteg, receg, sleepg ad sleepg states represeted as follows [1]: E = ettt + ertr + eltl + ests (1) where e t s e power geerated whe e ode s trasmttg, er s e power geerated whe e ode s receg, e l s e power geerated whe e ode s lsteg state ad e s s e power geerated by e ode whe a sleepg state. T t, Tr, Ts ad Tl represet e tme durato for e etwor trasmttg, receg, lsteg ad sleepg states, respectely, as compared to e etwor states [7]. et = a + cr er = a + b () el = a es = d a, b, c ad d are costats at are determed by electroc compoets of e ode w a correspodg alue of "a" as 0.083J / S, followed by "b" as 0.017J / S, followed by "c" as 0.000J / S ad "d" as 0.013J / S / m, where '' represet e power dex for commucato pa loss w a alue of 3. Fg.1 shows R represets e omal rage at esures at ay two odes at are adjacet grds ca drectly commucate. The parameters es ad e l are equalet [8] ad d has a ery small alue at s close to ero compared to a ad b. The result of e eergy cosumed by e grd E s computed as e summato of powers trasmttg e t, receg er ad sleepg e s states multpled by tme duratot t ad T r. Howeer, we substtute equato () to equato (1) to obta a model at descrbes e amout of eergy cosumed e etre grd as follows: E = a + bt + cr T (3) r t Where Tt = Dt dr ad T r = D r dr are e tme duratos for trasmttg ad receg e traffc data respectely. The parameter µ s defed as e trasmtted or receed data rate bts per secod w a ge alues of 50ps (lobt per secod). The data traffc demad wreless etwors s usually assumed to be statc, but recet studes dcated at e data traffc demad wreless etwor s hghly dyamc ad upredctable ature [7]. The arable testy of e ad-hoc etwor λ s descrbed as e rato of e traffc data D to e etwor area measure bt/sec. Thus, D = L B λ (4) From equato (4), t s mmedately obous at D s equalet to: D = L B λ (5) Acte odes a grd at are closer to e destato ode wll hae more data to be trasmtted a ose at are far from t. Ths mples at ese odes wll hae a shorter trasmsso rage a ose at are far for eergy effcecy. The trasmtted ad e receed data traffc e grd ca be obtaed by deductg e grd leg for bo trasmssos ad receg e posto of e grd from e leg of e etre etwor. The dstace betwee e odes each grd for trasmttg L ad receg L r data to ad from e destato ode s obtaed by subtractg e summato of e grds leg x from e etwor leg to ge e followg equatos: L L ( x1 + x + L+ x 1) ( x + x + L+ x ) = L t (6) r = L 1 We substtute equato (6) to equato (5) to ge e trasmtted traffc data D ad receed traffc data D for e grd as follows: t t r ISSN: (Prt); ISSN: (Ole)

3 Proceedgs of e World Cogress o Egeerg ad Computer Scece 013 Vol II, 3-5 October, 013, Sa Fracsco, USA 1 D t = L x B λ = Dr = L x B λ = 1 As show Fg. 1, e omal rage R esures e drect commucato betwee e odes e adjacet grds ad by Pyagoras eorem, e omal rage whe e umber of grd s oe s ge as follows: R 1 = X + B (8) Smlarly, e omal rag R for grd e etwor s determed as: ( X + X ) R = 1 + B (9) Equatos (8) ad (9) as well as T t ad T r are substtuted to equato (3) to obta e total eergy cosumed e frst E1 ad grd E of e etwor respectely. Ths s obtaed as follows: a E 1 = + b Dr + c ( X + B ) Dt (10) µ The total eergy erefore cosumed e etre etwor s: a E + b Dr + c x + x + B = 1 D µ = (11) (7) ( ) t III. GAF GENETIC ALGORITH MODEL Ths wor employs Geetc Algorm (GA) metaheurstc techque embedded to GAF eergy model geerated equato (11) to obta mmum eergy cosumpto e etre wreless ad-hoc etwor as show Fg.1. GA s ofte descrbed as oe of e most effecte meta-heurstc techques at are wdely used for solg cooluted optmato problems by mmcg e bologcal eoluto computg model to fd e possble optmal soluto [9]. GA mmes e ftess fucto or e objecte fucto of e optmato model by depedg o ts ma operators. The ree parametrc operators, Selecto, Mutato ad Crossoer, are defed GA algorm as follows. Selecto Ths fucto chooses parets for e ext geerato based o er scaled alues from e ftess scalg fucto. Mutato Ths fucto maes small radom chages betwee e dduals e populato, whch prode geetc dersty ad eable e GA to search a broader soluto space. Crossoer Ths fucto combes two dduals or parets to form a ew ddual or chld for e ext geerato. Thus, e GAF eergy model ge by equato (11) s appled as e ftess fucto so at e GAF/GA-based costrat ISSN: (Prt); ISSN: (Ole) optmato problem becomes: Mme: a r ( ) f = + b D + c X + X 1 + B Dt = µ (1) Subject to e costrat: x1 = L (13) = 1 where s e leg of e etre etwor as show Fg.1 ad each x represets e GA arable. GA geerates e best ftess fucto by choosg e approprate operator as summared Table 1. The total eergy geerated by e GA fuctos shows at stochastc uform, Gaussa ad scattered fuctos geerate e best ftess fucto by meetg e costrat equato (13). TABLE I ENERGY CONSUMPTION GENERATED BY GA OPERATORS Operators Selecto Mutato Crossoer Eergy (J) Fuctos Expermetal Result Is Costra t met? (Y/N) Stochastc Uform Gaussa Scattered Y Remader Uform Uform Sgle Pot N Adapte Feasble Two Pot N Roulette Gaussa Itermedate N Touramet Gaussa Heurstc N The maer whch ese fuctos were used GA MATLAB toolbox to geerate e mmum eergy as well as meetg e costrat of e ftess fucto s descrbed as follows: (A) The selecto fuctos aalable GA toolbox of MATLAB clude: stochastc uform, remader, uform, roulette ad touramet. The stochastc uform geerated e mmum eergy by layg out a le whch each paret correspods to a secto of e le of leg proportoal to ts expectato. The algorm moes alog e le steps of equal se, oe step for each paret. At each step, e algorm allocates a paret from e secto t lads o. The frst step s a uform radom umber less a e step se. (B) The mutato fuctos aalable GA toolbox of MATLAB clude: Gaussa, uform, adapte feasble. Gaussa fucto w e scale ad shr factor of 1 gae e desred mmum eergy by addg a radom umber to each ector etry of a ddual. Ths radom umber s tae from a Gaussa dstrbuto cetered o ero. The arace of s dstrbuto ca be cotrolled w two parameters. The scale parameter determes e arace at e frst geerato. The Shr parameter cotrols how arace shrs as geeratos go by. If e shr

4 Proceedgs of e World Cogress o Egeerg ad Computer Scece 013 Vol II, 3-5 October, 013, Sa Fracsco, USA parameter s 0, e arace s costat. If e shr parameter s 1, e arace shrs to 0 learly as e last geerato s reached. (C) The crossoer fuctos aalable GA toolbox clude: scattered, sgle pot, two pots, termedate, heurstc ad armetc. Scattered fucto geerates e mmum lowest mmum eergy compared to oer fuctos by creatg a radom bary ector. It e selects e gees where e ector s a 1 from e frst paret ad e gees where e ector s a 0 from e secod paret ad combes e gees to form e chld. The applcato of ese operators was demostrated GA toolbox to mplemet Equatos (1) ad (13) to ge e optmal eergy of Joules compared to e optmal of eergy of Joules obtaed [3], we obtaed a lower eergy our desg model. IV. GAF ANT COLONY OPTIMIZATION MODEL At Coloy Optmato (ACO) s a meta-heurstc techque troduced e 1990s by Dorgo et al [7] as a oel ature spred meta-heurstc. ACO s a approxmato algorm for geeratg good optmal solutos to hard combatoral optmato problems a reasoable amout of computatoal tme [8]. ACO ca be successfully used solg optmato problems by sprg e foragg behaor of real ats. Real ats search for food by fdg e shortest pa from er est to e food source wout hag e streg of so but w e exchage of formato by depostg chemcal substace called pheromoe o e groud. The depost of pheromoe by e ats mars e faorable pa at oer ats e coloy wll trael ad t creates pheromoe trals w arous cocetratos whch ls e food source to e est. Fg depcts double brdge expermetal setup ACO research by Deeubourg et al [9]. It mtates e foragg behaor of ad pheromoe tral lyg of at speces. The double brdge coects a est of ats of e Argete at speces. Fg. : Double Brdge Expermetal Setup I s study, eergy cosumpto wreless ad hoc etwors s optmsed usg ACO-GAF by followg e hybrd framewor approach proposed by Schluter et al [10]. Each ode e etwor fds ts correspodg eghborg odes e etwor by trasmttg formato from oe ode e grd to e oer utl t reaches ts destato ode. The choce of a ode e subsequet grd s a probablstc choce based o Probablstc Desty Fucto (PDF). I prcple, ay fucto P ( y) 0 satsfes e property: ISSN: (Prt); ISSN: (Ole) P( y) δ y = 1 (14) It uses s stead of a pheromoe table used e orgal ACO as descrbed Srsaaporphat et al [5]. Gaussa fucto s oe of e examples of fuctos used PDF. It has e adatage of geeratg radom umbers w a fast samplg tme. A sgle Gaussa fucto g (y) s oly able to focus o oe mea ad caot hadle two or more dsjot areas of e search doma. Keepg trac of e beefts of a Gaussa fucto ca oercome s problem. A PDF G (y) cosstg of a weghted sum of seeral oe-dmesoal Gaussa fuctos g (y) s cosdered for eery dmeso y of e orgal doma. The Gaussa fucto s erefore ge as: G where g y) = = 1 w g ( y) ( (15) 1 ( q ω ) ( y) = l (16) σ π σ where: w = The weghts for ddual Gaussa fuctos for PDF σ = Stadard Deatos ω = Correspodg Gaussa fuctos y = y dmeso of e problem ad, = Kerel umbers of e ddual Gaussa fucto The aboe fucto s charactered by w, σ, ω. They gude e sample solutos roughout e search doma ad ey represet e pheromoes. The update of ese pheromoes s drectly coected to e update process of e soluto. The weght w at dcates e mportace of a at s calculated w a lear proporto ad s ge as: w ( 1) = + = j 1 (17) A lear order of prorty s establshed w e fxed dstrbuto of e weghts w e soluto arche. A update e solutos foud meas at a pheromoe update has exsted based o e best solutos foud so far. Each tme a ew soluto (at) s ealuated ad created w terato, ts pealty fucto alue s compared to e solutos saed e arche startg from e best soluto to e worst soluto. If e ew soluto foud s better a e best soluto e arche, e e ew soluto wll tae e posto of e best soluto. Oerwse, e best soluto wll reta ts posto e arche. It eeds to be oted at e solutos saed e arche are e meas ad deatos used for e PDF ad ey etal e prcpal part of e pheromoe ftess. Ths way, updatg e soluto arche w better solutos leads automatcally to a poste pheromoe update. A egate pheromoe update meas e droppg of e least soluto e arche each tme a ew soluto s brought to e soluto arche. ACOm obta e fal eergy fttest alue usg e followg procedures MATLAB:

5 Proceedgs of e World Cogress o Egeerg ad Computer Scece 013 Vol II, 3-5 October, 013, Sa Fracsco, USA (A) ACOm starts by choosg e ftess fucto ealuatos (B) It chooses e feasble soluto to be attaed from e objecte fucto whch seres as e stoppg crtera (C) States e populato se of e at (D) States e umber of PDF erel (E) Prodes e oracle parameter for e pealty meod (F) Loos to ts performace w e call of ts local soler at eery terato usg e best fttest alue as e startg pot ad local soler after e last terato usg e curret best fttest alue as e startg pot (G) Stores e soluto foud to e arche. V. EXPERIMENTAL RESULTS The GA fds ts optmal eergy alue for e optmato problem cosdered by optmg e correspodg arables x of e ftess model ge by equatos (1) ad (13). We started e expermet by determg e approprate populato se for e ftess alue. After seeral rus, a populato se of 350 was acheed. The 350 populato se was ru alogsde w GA operators as show Table 1 to determe e best ftess alue (at s e mmum eergy cosumed). The best ftess alue was obtaed w e selecto of stochastc uform (selecto fucto), followed by Gaussa (mutato fucto) w a scale ad shr factor of 1 ad fally followed by scattered (crossoer fucto). The performaces of ese fuctos yelded e best ftess alue as dscussed Secto III. Fg. 3 shows e mmum eergy obtaed. The result shows at at each geerato, eergy s cosumed. Whe e geerato was at ts tal (betwee 0 ad 3), e eergy cosumpto geerated creased, but as e geerato steadly progresses, eergy cosumpto reaches ts maxmum ad started decreasg as e geerato progresses utl a costat alue s reached whe e geerato was at 5. The eergy cosumpto e bega to fluctuate as e geerato progresses utl t reaches e mmum eergy cosumed. software owed by IIM-CSIC for solg optmato problems usg At Coloy Optmato techque. I ACOm, e ftess of a at s ealuated by e pealty fucto alue, whch seres as e ftess stadard for e objecte fucto. Our mplemetato to geerate e mmum eergy cosumpto ad hoc etwors s acheed usg e detaled formato secto IV ad also applyg e stadard set by Schluter et al [10]. Fg. 4 ad Fg. 5 ge e detals o how ACO-GAF geerated e mmum eergy alue e etwor. There are 18 feasble solutos obtaed ad saed to e soluto arche for e frstree smulato ru MATLAB. Fg. 4: The ftess fucto alue for frst ree rus usg ACO-GAF Fg. 4 dcates at 3030 ealuatos, e umber of at for obtag e optmal alue for e eergy cosumpto creases. The cremetal costructos of ew solutos are geerated Fg. 5 by rug e program two more tme. Ths s doe accordace w e weght fucto defed equato (17) ad mplemeted MATLAB. Fg. 5: The ftess fucto alue for e last two rus usg ACO-GAF Fg. 3: Mmato of e etwor eergy by GA-GAF The objecte fucto was mplemeted equato (11) w ACOm software alogsde w e exteded framewor by Schluter et al [10] MATLAB to obta e mmum eergy cosumed e etwor. ACOm s The result shows at e best eergy cosumpto of 7.65J was geerated. Addtoally, we obsered at as e umber of at creases, e rate at whch e odes cosume eergy reduces, ereby leadg to mmum eergy cosumpto by e whole etwor. We ca deduce from Fg. 4 ad Fg. 5 at as e ealuato performace creases, e outcome of e eergy cosumpto e etwor mproes. VI. CONCLUSION Ths paper descrbes e applcato of GA ad ACO ISSN: (Prt); ISSN: (Ole)

6 Proceedgs of e World Cogress o Egeerg ad Computer Scece 013 Vol II, 3-5 October, 013, Sa Fracsco, USA metaheurstcs techques to obta e mmum eergy cosumpto ad-hoc wreless etwors. Durg e expermet, e approprate populato se was determed ad used w e selected GA operator (stochastc uform, Gaussa ad scattered) to get e best ftess fucto alue. The ACOm software MATLAB s appled to e objecte fucto to geerate e mmum eergy cosumpto alue. Ths was performed order to achee e mmum eergy cosumpto. The mmum eergy cosumpto geerated by e GAF/GA s 7.716J whle GAF/ACO geerated mmum eergy of 7.650J. From ese results, e eergy geerated from bo meta-heurstcs, e mmal eerges geerated by bo models are smaller whe compared to e ftess fucto geerated by We et. al [3]. We ca erefore fer at e selected meta-heurstcs optmato techques are hghly effecte for solg eergy mmato problem wreless ad-hoc etwor. Our result shows at e eergy geerated by GA-GAF model s ot sgfcatly dfferet from e eergy geerated by ACO-GAF model. Ths shows at metaheurstcs based optmato meods are effecte ad useful for mmg eergy cosumpto ad-hoc wreless etwors to prolog battery lfespa. Future study wll be coducted to compare oer meta-heurstcs such as e partcle swarm optmato ad tabu search to erfy f ey ca equally mme eergy cosumpto ad-hoc wreless etwors effectely e way GA ad ACO acheed s study REFERENCES [1] W. Feg, H. Alshaer, ad J.M.N. Elmrgha, Optmato of Eergy Cosumpto Rectagular Ad hoc Wreless Networs, Four Iteratoal Coferece o Commucato ad Networg, Xa, Cha, pp1-5, 009. [] B. Chem, K. Jameso, H. Balarsha, ad R. Morrs, Spa: A Eergy-effcet Coordato Algorm for Topology Mateace Ad hoc Wreless Networs, Mobcom, ol. 8, 00. [3] A.P. Bhodear, R. Vg, M.L. Sgla, C. Ghashyam ad P. Kapur, Geetc Algorm Based Node Placemet Meodology for Wreless Sesor Networs, Proc. Iteratoal MultCoferece of Egeers ad Computer Scetsts (IMEC 09), Hog Kog, 009. [4] S. Hussa, A.W. Mat ad O. Islam, Geetc Algorm for Eergy Effcet Clusters Wreless Sesor Networs, Proc. Iteratoal Coferece o Iformato Techology (ITNG 07), IEEE Computer Socety, 007, do: /itng [5] Srsaaporphat, C. ad She, C.-C. At-based Eergy Coserato for Moble Ad hoc Networs, The 1 Iteratoal Coferece o Computer Commucatos ad Networs (ICCCN 003), DE, USA, pp 3-37, 003 [6] Odem, S. ad Karaboga, D. Routg Wreless Sesor Networs usg a At Coloy Optmato (ACO) Router Chp, Sesors Joural, 9(), pp , 009. [7] Dorgo M, Color A, Maeo V. Iestgato of some Propertes of a At Algorm, Proceedgs of Parallel Problem Solg from Nature, Elseer, pp , 199. [8] Blum C, Rol A. Metaheurstcs Combatoral Optmato: Oerew ad Coceptual Comparso, ACM Computg Surey, pp , 003. [9] Deeubourg J.-L, Aro S, Goss S, Pasteels J.-M. The Self Orgag Exploratory Patter of e Argete, Joural of Isect Behaour, pp , [10] Shuluter M, Egea J.S, Baga J.R. Exteded At Coloy Optmato for No-coex Mxed Iteger Nolear Programmg, Joural of Computers ad Operatoal Research, pp 17-9, 009. ISSN: (Prt); ISSN: (Ole)