HEURISTICS FOR ONE CLASS OF MINIMAL COVERING PROBLEM IN CASE OF LOCATING UNDESIRABLE GOODS

Transcription

1 HEURSTCS FOR ONE CLASS OF MNMAL COVERNG PROBLEM N CASE OF LOCATNG UNDESRABLE GOODS BRANKA DMTRJEVC MLORAD VDOVC The Faculy of Transpor and Traffc Engneerng, The Unversy of Belgrade, Belgrade, Serba and Monenegro Absrac Modern socey has experenced an ncreasng publc awareness of envronmenal ssues. n parcular, problems whn he feld of ransporaon and he locaon of dangerous goods facles have araced consderable publc aenon. n hs paper, he specfc problem of locang undesrable goods has been consdered. Such facles should be locaed n some of he nework nodes belongng o a known dscree se, under condons of mnmal safey dsance, boh beween he warehouse facles hemselves and beween he warehouses and oher neghborng objecs. The objecve here s o maxmze he quany of goods sored whle a he same me respecng mnmal safey dsances. Ths paper presens he problem formulaon and proposes heursc soluon approaches whch are esed on numerous examples. Keywords: undesrable good, locaon, safey dsance, heursc. NTRODUCTON Many facles whch are wdely used and provde a servce are known as undesrable or obnoxous. Examples of undesrable facles nclude sold wase reposores, pollung plans, radoacve wase sorage ses and explosve sorage ses, as well as nose, odor or hea emers. Those facles generae dfferen undesrable effecs whch can be fel over a ceran geographcal space, and makng decsons abou her spaal poson s crucal when comes o mnmzng he envronmenal rsks. Ths s why one of he mos acve research areas whn locaon heory n recen years has been he locaon of undesrable or obnoxous facles. Numerous locaon models n hs area use a sngle objecve funcon ha eher maxmzes he sum of he dsances beween he facles and he demand pons (maxsum creron), or maxmzes he mnmal dsances beween he undesrable facles and he cusomers (maxmn creron). A very comprehensve revew of dsance maxmzaon models for undesrable sngle facles, as well as mulfacly locaons can be found n [5]. A varey of sngle and mul crera mehods have been developed dependng on he problem saemen [], [], [3], [5], [6] and [7]. Anoher area of research s he mnmzaon of he oal populaon affeced by he facly whch leads o a mnmal coverng problem. Fndng he crcle of gven radus R coverng he leas oal (mnsum) or maxmum (mnmax) wegh of he desnaons s suded n []. n [8] s shown how boh of he aforemenoned problems may be solved smulaneously by he bcrera model. However, n spe of fac ha leraure offers numerous dfferen approaches and models devoed o he locaon of undesrable facles, many problems n hs area reman undscovered, and have no ye been analyzed. Ths paper, whch s an exenson of prevous researches [9], [], consders problem of locang facles for sorng dangerous goods whn a smaller closed regon, desned only for hs purpose. More precsely, he area desgnaed for sorng undesrable facles s nsde a

2 regon whch only has nodes whch are canddaes for locang sorage objecs. Ousde he regon are neghborng facly nodes whch despe no beng desned for sorng maerals, mus also be made safe from he undesrable effecs of he sored maerals. The objecve s o maxmze he quany of maeral sored whle a he same me respecng mnmal safey dsances, boh beween he sorage facles hemselves, and beween he sorage facles and any neghborng objecs. The regon self, because of s dmensons whch may be negleced when compared wh he servce regon, could be assumed o be a servce pon for all of he assgned cusomers, and from here cusomer nodes, as well as ransporaon coss may be excluded from any furher analyss. The res of he paper s organzed as follows. Chaper nroduces he problem n more deal and presens he formulaon of he locaon problem. Chaper 3 proposes heursc algorhms and presens numercal example of her applcaon, Chaper presens heursc algorhms performances and Chaper 5 offers some concludng remarks.. PROBLEM DESCRPTON AND FORMULATON Sorng and keepng dangerous goods lke explosves, flammable maerals and compressed gasses s characerzed by he opporuny o ransfer any undesrable effecs o he objecs n he neghborhood hus causng desrucon, serous damage and fre n hese areas. Those effecs spread sphercally from he source, reachng he surroundng objec whn a ceran radus known as he mnmal safey dsance. The mnmal safey dsance usually depends on he quany and characerscs of he acvaed maeral, as well as oher relevan characerscs (buldng consrucon, he muual spaal poson of he donor and accepor objecs...). n hs paper, he relaons beween he safey dsance, he maeral characerscs and he quany used correspond o explosves, alhough smlar relaons may be assumed n cases of oher obnoxous facles. Based on hs assumpon, he mnmal safey dsance may be deermned as [9,]: - consan value, whch depends only on he explosve s characerscs; - he funcon of he ne explosve wegh for quany (Q). n hs case he mnmal safey dsance R may be calculaed by usng R=P Q /3. Here, R s he mnmal safey dsance requred; P s he proecon facor dependng on he degree of rsk assumed or permed, ndependen of he knd of explosve, and Q s he ne explosve wegh for he gven quany of a ceran explosve. n hs paper he locaon problem when he safey dsance does no depend on he quany sored, and whch has a consan value for a ceran class of maerals have been analyzed. Le G=(N,A) be a nework, where N ={,...,, j,..., n} s he se of nodes, and A s he se of arcs (,j). To each arc (,j) A a nonnegave scalar c j s assocaed whch represens he eucldean dsance beween nodes,j. Se N s paroned no wo subses N=N N E, where N represens he nodes nsde he regon whch are he canddaes for sorng maerals, and N E represens he ousde nodes whch are no desgnaed for sorng maerals, bu mus be kep a a safe dsance from he maerals sored n he nsde nodes. Two scalars R, R E are assocaed o maeral sored whch represens he mnmal safey dsances for nernal facles, and neghborng objecs respecvely. To each node N s he assocaed wegh v, whch represens he node's capacy resrcon for maeral sored. Based on prevous noaons, QD (Quany ndependen Dsance) locaon problem, n case when only one undesrable maeral should be sored, analyzed here may be formulaed as he followng nonlnear - programmng problem: Those nodes could be suaed nsde he regon oo, bu because of smplcy s assumed here ha hey are only ousde he regon.

3 Maxmze ZQ= v N x (.) subjec o: x x (c R ), x, x ;, j N (.) c x j j E j k E j R, N, j N (.3) {, }, N (.) The objecve funcon, presened by Eq. (.), s o maxmze he quany of undesrable maeral sored. The Eq. (.) provdes ha each par of nodes where undesrable maerals are sored s muually a he correc safey dsance. The Eq. (.3) provdes ha he all canddae nodes for sorng undesrable maeral are a he correc safey dsance from he facles ousde he regon. The Eq. (.) defnes he values of he decson varable. To solve he problem, QD heursc has been proposed. 3. SOLUTON APPROACH Le Ω N be he se of nodes ncluded n he soluon, Ω Ω be he se of nodes ncluded n he soluon n -h eraon, E N be he se of nodes excluded from he soluon, E N be he se of nodes whch are excluded from he soluon n -h eraon, and le A N be he se of acve nodes n -h eraon. For he arbrary node A, le Π() be he subse of all nodes j A, whch are a he dsance from node, less han he mnmal safey dsance (c j <R ). Tha s: Π ( ) = j c < R, j A (3.) { } j For he arbrary node A he followng measuremens have also been defned: α( ) = v v j j { Π()\} β( ) = v + v j j { A \ Π() } v wα() = v v j v j { Π()\} w β() = v αβ() = v + + wαβ() = v A v A j + v j v j { A \ Π() } v j v j { A \ Π() } j { Π()\} v v A j v j v j { } { } A \ Π() j Π()\ Obvously, α() summarzes he wegh of node N and he negave weghs of all hose nodes whch a he dsance from node N are less han he mnmal safey dsance. Therefore, α() represens uly ganed by sorng maxmum quany of dangerous goods n he node N, whch s calculaed as a resul of subracon of endangered nodes capaces from capacy of he node N. Measuremen β() summarzes he wegh of node N and he weghs of all nodes whch a he dsance from node N are eher equal o or greaer han he mnmal safey dsance. β() represens uly ganed by sorng maxmum quany of dangerous goods n he node N, whch s calculaed as he sum of capacy of he node N and he capacy of nodes whch are no affeced by sorng dangerous maeral n he node N. Measuremen wα() s modfed (weghed) measuremen α() n whch he sum of negave (3.) (3.3) (3.) (3.5) (3.6) (3.7)

4 weghs of all hose nodes whch a he dsance from node N are less han he mnmal safey dsance s mulpled by relave wegh of he node N. Measuremen wβ() s modfed (weghed) measuremen β() n whch he sum of weghs of all nodes whch a he dsance from node N are eher equal o, or greaer han he mnmal safey dsance s mulpled by relave wegh of he node N. The wegh of he node N has been emphaszed whn boh measuremens. Measuremen αβ() represens oal uly calculaed by addng he wegh of he node N, he weghs of all nodes whch a he dsance from he node N are eher equal o or greaer han he mnmal safey dsance and he negave weghs of all hose nodes whch a he dsance from node N are less han he mnmal safey dsance. Measuremen wαβ() s modfed (weghed) measuremen αβ() accordng o he same prncple as wα() and wβ(). Based on hose measuremens, sx varaons of QD heursc algorhm are proposed: QDα, QDβ, wqdα, wqdβ, QDαβ and wqdαβ. Those algorhms dffer from each oher only n he measuremens appled, herefore hey are presened as unfed followng algorhm. 3.. QD HEURSTC ALGORTHMS STEP Se = STEP Se Ω =, E =, A =N STEP 3 For each node A, defne se Π() accordng o Eq. (3.) STEP For each node A, calculae uly: α() accordng o Eq. (3.) for QDα heursc; β() accordng o Eq. (3.3) for QDβ heursc; wα() accordng o Eq. (3.) for wqdα heursc; wβ() accordng o Eq. (3.5) for wqdβ heursc; αβ() accordng o Eq. (3.6) for QDαβ heursc; wαβ() accordng o Eq. (3.7) for wqdαβ heursc STEP 5 Fnd node wh he maxmal uly value of: α ( ) = max [α()] for QDα heursc; β ( ) = max [β()] for QDβ heursc; wα ( ) = max [wα()] for wqdα heursc; wβ ( wαβ ( A ) = max [wβ()] for wqdβ heursc; A A A A A αβ ( ) = max [αβ()] for QDαβ heursc; ) = max [wαβ()] for wqdαβ heursc. f wo or more nodes have he same maxmal uly value of: α(); β(); wα(); wβ(); αβ(); wαβ(), node should be chosen arbrarly. STEP 6 Se = + STEP 7 Updae se Ω : Ω = Ω STEP 8 Updae se E : E =E - {j j Π( )\ } STEP 9 Updae se A : A =A - \{E } STEP f A, proceed wh STEP 3, oherwse go o STEP STEP Se Ω = Ω, E=E STEP Calculae he objecve funcon value ZQ= v Ω STEP 3 End algorhm. 3.. QD HEURSTC APPLCATON EXAMPLE The proposed heursc s demonsraed n he followng example. Se N has been gven by nodes' coordnaes N ={(3,), (6,), (8,), (6,5), (89,77), (8,65), (38,59) and (5,9)}. Node weghs are v j = 79, 7, 56, 55, 67, 33, 56 and. Le R =6. The llusrave problem, he eraons of he QDα heursc algorhm applcaon and he resuls are shown n Fg. PROBLEM s eraon v 5=67 v =55 v 6=6 v 7=56 v 3=56 R =6 v =79 v 8= v =

5 Fgure The QDα applcaon example. QD HEURSTC PERFORMANCES The proposed heursc algorhms were also esed on hree groups of 3 randomly generaed problems. The frs group comprsed of problems wh 8 canddae nodes, he second wh and he hrd wh 6 canddae nodes. n all generaed problems, he value of mnmal safey dsance s he same, R =cons. Each group of 3 problems s dvded no a hree problems subgroups: whn he frs subgroup he weghs of all nodes are equal,.e. v =V, N ; whn he second subgroup he weghs of nodes are generaed n he range from V o V; whn he hrd subgroup he weghs of nodes are generaed n he range from V o V. Furher, each subgroup s dvded no a problems n whch he average node dsance s vared n range from.r o.r. Percenage of he opmal soluons and percenage of he average relave soluon error based on 9 numercal examples are shown n Fg.., Fg... and Fg..3., for he group problems wh 8, and 6 nodes, respecvely. Those performances are also shown for negraed QD ( QD ) heursc, by he same fgures, as a resul of fndng he bes of all generaed problems soluons calculaed for all heurscs.

6 Opmal soluons[%] v =V N R R 3R R R R 3R R Average relave soluon error[%].. z QDβ z wqdβ z QDα z wqdα z QDαβ a wqdαβ z QD Opmal soluons[%] v (V, V) R 3 R 3R R Average relave soluon error[%] 3 R R 3 3R R Opmal soluons[%] 9 8 v (V, V) Average relave soluon error[%] 3 7 R R 3R3 R R R 3R3 R Fg... QD heursc algorhms performances for problems wh 8 canddae nodes

7 Opmal soluons[%] v =V N Average relave soluon error[%] 5 3 zqdβ zwqdβ zqdα zwqdα zqdαβ zwqdαβ zqd 97 R R 3R 3 R R R 3R3 R 5 Opmal soluons[%] v (V, V) Average relave soluon error[%] 3 5 R R 3R3 R R R 3R 3 R 6 Opmal soluons[%] v (V, V) Average relave soluon error[%] 5 R R 3R 3 R R R 3R3 R Fgure. QD heursc algorhms performances for problems wh canddae nodes

8 Opmal soluons[%] v =V N Average relave soluon error[%] zqdβ zwqdβ zqdα zwqdα zqdαβ zwqdαβ zqd 95 R R 3R3 R R R 3R3 R Opmal soluons[%] 8 6 v (V, V) Average relave soluon error[%] 8 6 R R 3R3 R R 3 R 3R R 8 Opmal soluons[%] 8 6 v (V, V) Average relave soluon error[%] 6 R R 3R3 R R R 3R 3 R Fgure.3 QD heursc algorhms performances for problems wh 6 canddae nodes Generally, from Fgure. o Fgure. s noceable ha he wors heurscs performances are n he cases of average node dsance are almos he same wh he value of R.

9 Percenage of opmal soluons, n he cases of equal nodes wegh, s raher unfed for all he heurscs and has no dropped below 95%, and n he cases of QD has no dropped below 97%. Furher o hs problems subgroup, maxmal percenage of relave soluon errors of heursc algorhms applcaons s.5%, as for he QD s less hen %. n oher wo problem subgroups, v =[V-V], v =[V-V], hese performances dffers from prevous subgroup. For example, for problems wh 6 nodes, mnmal percenage of opmal soluons calculaed by QD heursc s 86.3%, for v =[V-V], whle for v =[V-V] s 9.7%. Maxmal percenage of relave soluon errors calculaed by QD heursc for he same problem group s.6% for v =[V-V], as for v =[V-V] s.9%. Relavely small dfferences n beween resuls performances whn hese problem subgroups show subsanal sably of QD heursc n cases of nodes wegh changes. These dfferences are consequence of he ncreased resuls calculaed by weghed heurscs (wqdα, wqdβ, wqdαβ), when nodes wegh range s wder. To compare, for subgroup v =[V-V], of problem group wh 6 nodes, mnmal percenage of opmal soluons calculaed by weghed heurscs s: 55.5%, 7.3% and 7.6%, n case of v =[V-V], for wqdα, wqdβ and wqdαβ respecvely, whle n case of v =[V- V] hs performance akes values as follows: 58%, 58.% and 57%. Heurscs QDβ and QDαβ, n all generaed examples produce resuls wh negleced dfferences. s neresng ha resuls calculaed by hese wo heurscs, for subgroups v =[V-V] and v =[V-V], show sably oward nodes wegh changes. A he same me average resuls calculaed by hese heurscs, comparng wh oher heurscs, are of he leas precson. For esng he heurscs a specal program has been creaed n programmng language DELPH V5. (Buld 5.6..). On PC Penum,.5 GHz wh 5 MB SDRAM and BUS speed MHz, soluon calculaon per one heursc for problem wh 5 nodes akes 8 sec and 56 msec, whch ncludes recordng calculaon resuls no he ougong fle, n hs case 5. MB. To compare, soluon calculaon per one heursc for problem wh 6 nodes akes 6 msec, resulng wh ougong fle of.8 KB, whch ndcaes ha calculaon per one node akes jus a b less hen a msec, whle he res of he me s spen on recordng resuls n ougong fle. 5. CONCLUDNG REMARKS n hs paper, formulaon for he problem of locang undesrable facles when he safey dsance does no depend on quany sored has been consdered. To solve he problem sx effcen and easy o use heursc algorhms have been proposed and her performances were subsequenly esed on numerous examples. The resuls are promsng and encourage furher research. Acknowledgemens Ths research was suppored, n par, by he Mnsry of Scence and Envronmenal Proecon of he Republc of Serba No. TD-78A BBLOGRAPHY [] Brmberg J, Wesolowsky G (995), The reclnear dsance mnsum problem wh mnmum dsance consrans, Locaon Scence 3, 3-5; [] Drezner Z, Wesolowsky G (98), A maxmn locaon problems wh maxmum dsance consrans, AE Transacon, 9-5; [3] Drezner Z, Wesolowsky G (985), Locaon of mulple obnoxous facles, Transporaon scence 9, 93-; [] Drezner Z, Wesolowsky G (99), Fndng he crcle or recangle conanng he mnmum wegh of pons, Locaon Scence, 83-9; [5] Erku E, Neuman S (989), Analycal models for locang undesrable facles, European Journal of Operaonal Research, 75-9;

10 [6] Konfory Y, Tamr A (997), The sngle facly locaon problem wh mnmum dsance consrans, Locaon Scence 5, 7-63; [7] Munoz-Perez J, Saameno-Rodrguez JJ (999), Locaon of an undesrable facly n a polygonal regon wh forbdden zones, European Journal of Operaonal Research, ; [8] Plasra F, Carrzosa E (999), Undesrable facly locaon wh mnmal coverng objecves, European Journal of Operaonal Research 9, 58-8; [9] Vdovć M, Cvec S (996), Locang hazardous maerals when he safey dsance depends on he quany sored, Armed Forces professonal scenfc journal Vol XLV No., 3-3; [] Vdovć M, Dmrjevc B (), Locang hazardous maerals when safey dsance depends on quany sored, Conference book from he Annual nernaonal Conference of he German Operaons Research Socey (GOR) Jonly organzed wh he Neherlands Socey for Operaons Research (NGB), Tlburg, Neherlands.