# Searching Optimal Buyer Coalition Structure by Ant Colony Optimization

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2 CS * Hece, = ag ax V ( CS ). (2) i= CS L L is ow as a laye, ad the set of all CSs is i L = L. I CFG the value of each coalitio is give by a i chaacteistic fuctio which is siply defied as the su of the values of the coalitios that it cotais. Refeece [2] shows the calculatio of the total ube of coalitio stuctues as follow, a i= Z ( a, i), (3) whee a is the ube of agets. Ad Z ( a, i) is the ube of coalitio stuctues with i coalitios, Z ( a, i) = i * Z ( a, i) + Z ( a, i ), (4) whee Z ( a, a) = Z ( a,) =. If a is sall, a = 4, the ube of coalitio stuctues is 5, see Fig.. Howeve, whe ube of agets iceases liealy, the size of the poble, 2, iceases expoetially [27]. This is such a difficult tas to seach fo optial solutio whe the ube of aget is bigge. So, the algoith called ACO_FBC is poposed to seach fo the optial solutio. The poposed algoith is based o at coloy algoiths (ACO) which ae ispied fo the behavio of eal ats. The ajo shotcoig of the ACO_FBC is that it povides o guaatee to fid the optial solutio because ACO is oe of the heuistic fuctios. Howeve, it sees to wo well i ou pactice. The pupose of this pape is to seach fo the optial buye coalitio stuctue by applyig ACO techique. The pape is divided ito five sectios icludig this itoductio. The est of the pape is ogaized as follows. Sectio 2 descibes basic at coloy optiizatio. I sectio 3, we show the otivated exaple icludig the atheatical details of the poposed algoith. I sectio 4, we show the expeiets. To esue the quality of the algoith, the siulatio esults ae copaed with the GoupPacageStig schee. Fially, the coclusios ad futue wo ae i the last sectio. II. ANT COLONY OPTIMIZATION BACKGROUND At coloy optiizatio (ACO) is a pobabilistic techique fo fidig optial paths i fully coected gaphs though a guided seach, by aig use of the pheooe ifoatio [29]. It is a paadig fo desigig etaheuistic algoith fo cobiatoial optiizatio pobles [5]. The ai idea i at coloy algoiths is to use atificial ats that iteatively costuct solutios to cobiatoial optiizatio pobles. The fist ACO algoith was iitially poposed by Coloi, Doigo ad Maiezzo [2] [23] i 997 which ow as At Syste (AS). Now, thee ae seveal adaptatios of such algoiths to coplex optiizatio pobles [9], [7] [9], [24]. The global ats pefo a siple evaluatio of soe egios defied i the seach space, i ode to update the egios fitess. The ACO was applied to seveal pobles such as the tavelig salesa poble [7] ad the shop schedulig poble ad ixed shop schedulig [8]. I atue, eal ats ae capable of fidig the shotest path fo a food souce to thei est without usig visual cues [20]. The geeal stuctue of ACO algoiths ca be descibed as follows. Step : Iitialize the pheooe tails ad paaetes. Step 2: While (teiatio coditio is ot et) do the followig: - Costuct a solutio; - Ipove the solutio by local seach; - Update the pheooe tail o tail itesities. Step 3: Retu the best solutio foud. I the ACO, the pocess begis by iitiatig copletely ado ats. Theses atificial ats build solutios to a optiizatio poble while updatig pheooe ifoatio o its visited tail. Atificial ats build a feasible solutio by ecuetly applyig a stochastic geedy ule. While ceatig its tou, a at deposits a substace called pheooe o the goud ad follows the path by peviously pheooe deposited by othe ats. Oce all ats have copleted thei tous, the at which foud the best solutio deposits the aout of pheooe o the tou accodig to the pheooe tail update ule. The best solutio foud so fa i the cuet iteatio is used to update the pheooe ifoatio. The pheooe τ, associated with the lie joiig i ad j, is updated as follow: τ ( ρ) τ + τ, (5) = whee ρ is the evapoatio ate which ρ (0, ]. The easo fo this is that old pheooe should ot have too stog a ifluece o the futue. Ad is the aout of pheooe laid o lie (i, j) by at : / Q L τ = 0 τ whee Q is a costat, ad L is the legth of the tou pefoed by at. By costuctig a solutio, it stats fo the statig city to visit a uvisited city. Whe beig at the city i, the at selects the city j to visit though a stochastic echais with a pobability If lie (i, j) is used by at. (6) othewise, p give by: Issue 4, Volue 5,

7 Suppose the cuet at, called at, chooses the aget fo the statig poit. The, the at seaches fo the ext agets based o the pobability p show i (9). If thee agets,, 2 ad 6, have bee chose espectively to be i the sae sub goup deoted as {, 2, 6. I the ACO_CS algoith, the cuet at geeates the path coectig aog selected agets. Theefoe, solid lies ae used to joi thee of these agets, See Fig. 6(a). The utility of the subgoup {, 2, 6 is 575, see the calculatio of Util {,2,6. Agai, the cuet at fids the ext aget with the pobability show i (8). Suppose that the at chooses the path fo vetex 6 via a dotted lie to aget 3, see Fig. 6(b). If the aget ube 3 ad 4 have high possibility to be chose ito the sae sub coalitio, deoted as {3, 4, the the at chooses a solid lie betwee 3 ad 4, see Fig. 6(c). The utility of {3, 4 is 450, see the calculatio of Util {3,4. The last aget, aget ube 5, is joied with o oe. It is a isolated aget, theefoe this aget is coected with the othe by two dotted lies, see Fig. 6(d) ad Fig. 6(e)). Its utility is 330, see the calculatio of Util {5. The, the coalitio ca be witte as {,2,6{3,4{5 with the gad total utility of 355. Fially, the at deposits the aout of pheooe o the tail accodig to the pheooe value i (0). U C = a C a a ( L * ), X i = a = C a a ( L * ), 2 X z a ( * ) 2 * i = a a K X + K Max a = ( ( * )) 3, C Li X K = a C i = =... a a ( * ), L X i = z a C = (3) whee K, K 2 ad K 3 ae the costat, ad X a is the equest X of a aget a. So, = v ( CS). (4) U C C Let s the costat K =00, K 2 = 2, ad K 3 =50. So, utility of each sub coalitio, {, 2, 6, {3, 4, ad {5 ca be calculated as follows. Util {,2,6 =K (2++2+2)+K 2 (4*5)-(4*5+2*0+*5)-K 3 = 00*7+2* = 575 Util {3,4 =K (2++3)+K 2 (2*30)-(2*30+4*5)- K 3 = 00*6+2* = 450 Util {5 =K *4+K 2 (4*0)-K 3 = 00*4+2*40-50 = 330 v ( CS) = U C C = Util {,2,6 + Util {3,4 + Util {5 = = 355 (a) Costuctig {,2,6 (b) Costuctig {,2,6{3 (c) Costuctig (d) Costuctig {,2,6{3,4 {,2,6{3,4{5 (e) The fial gaph of {,2,6{3,4{5 Fig. 6 The ACO_CS gaph of {,2,6{3,4{5 IV. SIMULATION AND EXPERIMENTS This sectio explais the idea of how the ACO_CS algoith by usig a epiical exaple. The, it shows the expeiet esults i detail. I the expeiet, we use the paaeteized fuctio as stated i (3) ad (4), so that we ca egulate soe data ad alte the seach space to obseve ou poposed fuctio i pactice. I ou algoith, we assue that all buyes ae equested to paticipate i costuctig the coalitio stuctue. To test the ACO_FBC algoith, a siulatio was developed usig Java pogaig laguage. The siulatio us o a Petiu(R) D CPU 2.80 GHz, 2 GB of RAM, IBM PC. ACO_FBC paaetes iclude α = 0.5, β =, ad the ube of atificial ats = 000. Seveal expeiets have bee coducted usig a diffeet set of buyes by ado with 5, 0, 20, ad 30 buyes espectively while the pice list is show i Table. The suay esults of ou expeiets ae peseted i Issue 4, Volue 5,

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