SAFE MOVEMENT OF HAZARDOUS MATERIALS THROUGH HEURISTIC HYBRID APPROACH: TABU SEARCH AND GAME THEORY APLICATION

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1 P A M U K K A L E Ü Nİ V E R Sİ T E Sİ M Ü H E N Dİ S LİK F A K Ü L T E Sİ P A M U K K A L E U N I V E R S I T Y E N G I N E E R I N G F A C U L T Y M Ü H E N DİS LİK BİLİML E Rİ D E R GİSİ J O U R N A L O F E N G I N E E R I N G S C I E N C E S YIL CİLT SAYI SAYFA : 008 : 4 : : SAFE MOVEMENT OF HAZARDOUS MATERIALS THROUGH HEURISTIC HYBRID APPROACH: TABU SEARCH AND GAME THEORY APLICATION Hakan ASLAN Sakarya Ünverstes, Mühendslk Fakültes, İnşaat Mühendslğ Bölümü, 5487, Sakarya Gelş Tarh : Kabul Tarh : ABSTRACT The safe movement of hazardous materals s recevng ncreased attenton due to growng envronmental awareness of the potental health affects of a release causng ncdent. A novel approach developed n ths paper through a game theory nterpretaton provdes a rsk-averse soluton to the hazardous materals transportaton problem. The dspatcher mnmzes the expected maxmum dsutlty subect to worst possble set of lnk falure probabltes, assumng that one lnk n the network fals. The expected cost at the Nash equlbrum s a useful measure to evaluate the routng strateges for the safe movement of hazardous materals. Key Words : Transportaton of hazardous materals, Routng problem, Game theory. YAKLAŞIMSAL HİBRİD METODLA GÜVENLİ TEHLİKELİ MADDE TAŞIMACILIĞI: TABU ARAŞTIRMA VE OYUN TEORİSİ UYGULAMASI ÖZET Tehlkel madde taşımacılığında oluşacak herhang br kazanın gerek nsan sağlığına gerekse de çevreye olacak olan etklerne karşı oluşan duyarlılığın artmasıyla, bu çeşt taşımacılığın güvenlğne yönelk kaygılar artmıştır. Bu makalede tehlkel madde taşımacılığı oyun teors çerçevesnde en az rsk çerecek çözüm metoduyla modellenmştr. Taşımacı organzasyon şebekedek lnklerden maksmum olumsuz etky oluşturacak olanın üzernde br kazanın olduğu varsayımıyla beklenen maksmum malyet mnmze etmeye çalışarak Nash denges çerçevesnde en güvenl taşıma güzergâhını belrlemektedr. Anahtar Kelmeler : Tehlkel madde taşımacılığı, Araç yönlendrme, Oyun teors.. INTRODUCTION Hazardous materals (hazmats) are part of our every day lfe. As a consequence of ndustral development, huge quanttes of hazardous materals are yearly produced and obvously the producton of them goes together wth ther transportaton. For most members of ndustral socetes lfe wthout hazmats s nconcevable. Hazardous materals are defned as those whch are found to be n a quantty and form that may pose an undesrable rsk to publc health and safety or property when transported (Hobeka et al., 986). In ths sense, hazardous materals or dangerous goods nclude explosves, gases, flammable lquds and solds, oxdsng substances, posonous and nfectous substances, radoactve materals, 79

2 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan corrosve substances and hazardous wastes (Erkut and Verter, 998). In any debate about the transportaton of dangerous goods where safety s of prmary concern, t s very mportant that there s a full understandng of the magntude of the rsks nvolved and the causes and maor contrbutors so that properly nformed decsons can be made. A specal nterest area n hazardous materals transportaton s the determnaton of best routng strateges to reduce the rsk of Low-Probablty, Hgh-Consequence (LPHC) ncdents. Ths study develops a new approach to the soluton of the effectve routng strategy for the transportaton of ths type hazardous materals. The soluton approach s based on the combnaton of the game theory and tabu search. The soluton s a mxed strategy Nash Equlbrum to a -player, noncooperatve, zero-sum game.. STATEMENT OF THE PROBLEM The rsk models for routng of hazardous materals proposed htherto assume that suffcent hstorcal data exst to determne the frequency and consequences of the release ncdents, and that past observatons can adequately be used to estmate future expectatons. Hence, the qualty of the proposed models s heavly depended on the data avalable. However, due to the rare occurrence of ncdents nvolvng hazmat shpments, the qualty and accuracy of the data are open to dscusson... Vehcle Routng Problem The best strategy for routng corresponds to the best path for the shpment. In real lfe, there are problems whch requre the determnaton of the best routes from a central depot to several demand ponts (customers) by a set of vehcle routes, the well known Vehcle Routng Problem (VRP). To llustrate ths, consder the followng problem. Assume that a petrol company has a bg refnery, used as a depot, and a fleet of vehcles located at ths depot whch are used to carry the petrol to the numerous servce statons scattered all over the regon (country). The problem here s to fnd the least expected cost routes for a fleet of vehcles, located at the refnery, n such a way that each servce staton s vsted once and only once by exactly one vehcle, takng nto account the fact that the total demand of any vehcle route must not exceed the capacty of the vehcle servcng that partcular route. The algorthm that developed targets the soluton of ths problem. The vehcle routng problem s an easy problem to descrbe but hard to solve. A problem of sze Y can be solved n polynomal tme f the number of teratons s a polynomal functon of Y. Nondetermnstc polynomal tme problems (abbrevated as NP-problem) are those whose solutons can be checked n polynomal tme (even f t takes exponental tme to fnd the soluton). However, NP-hard problems are not solvable n polynomal tme. Almost all vehcle routng problems are NP-hard and, hence, unlkely to be solvable n polynomal tme (Lenstra et al., 98). A problem s easy f t has an algorthm wth tme complexty O(Y ι ) for a constant ι, where Y s the sze of the problem. Here, O(Y ι ) s a functon such that: O(Y ι ) = a ι Y ι a ι- Y ι- +.+a Y +a 0 () Where, a are constants. Such complexty s sad to be of polynomal order, and the algorthm s called a polynomal tme algorthm. On the other hand, f any algorthm for the problem requres a complexty not bounded by a polynomal functon n Y, t s consdered to be dffcult or ntractable. Typcal orders that are not polynomal are O(Y logy ), O(ιY), and (Y!). Computatonal complexty theory has provded strong evdence that the soluton for some optmsaton problems belongng to the NP-class s lkely to requre a computaton tme that grows exponentally wth problem sze. Thus, the attenton pad to the study of exact algorthms for NP-hard problems n general s lkely to dmnsh as the sze of the problem sze ncreases.. WHY SHOULD WE USE HEURISTIC (APPROXIMATE) ALGORITHMS? As the VRP s an NP-hard problem, the exact algorthms for the soluton of VRP can only solve problems of small sze (Laporte,998). In practce, however, algorthms are requred to solve dffcult problems of large sze and cannot be solved n polynomal tme. It may not be possble to solve these problems exactly usng even the most sophstcated algorthms. In ths sense, the mportance of approxmate (heurstc) algorthms s recognsable. These algorthms can provde nearoptmal solutons for large-szed problems n reasonable amounts of computaton tme and wth reasonable storage space requrements. Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

3 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan Heurstc algorthms are, hence, wdely used to solve such problems n a reasonable amount of tme. Accordngly, for the soluton of real-world vehcle routng problems heurstc algorthms are developed along wth some exact algorthms for relatvely small problems. The sze of the problem whch can be solved usng exact algorthms s severely lmted. A large number of NP-hard combnatoral problems of practcal sze can only be solved effcently by heurstc methods, rather than by exact methods. The soluton of these problems can only be obtaned f all the possble combnatons of the decsons and varables are explored. Heurstcs play an effectve role n such problems by ndcatng a way forward to reduce the number of evaluatons and to obtan solutons wthn reasonable tme constrants. Almost all large-szed nstances of combnatoral optmsaton problems, such as travellng salesman problem and other routng problems, many knds of flow and ob shop schedulng problems, can be solved effectvely by heurstcs... Tabu Search As a local search metaheurstc, ntally ntroduced by (Osman, 99), Tabu Search (TS) s a metastrategy teratve procedure that can gude almost any type of local steepest-descent research to explore ts soluton space more wdely by buldng extended neghbourhood wth partcular emphass on avodng beng caught n a local optmum. A soluton s neghbourhood s the set of solutons that can be reached by a move. A transton from a feasble soluton to a transformed feasble soluton s referred to as a move. Tabu search s based on the general prncple of ntellgent problem solvng and was partly motvated by the observaton that human behavour appears to operate wth a random element that leads to nconsstent behavour gven smlar crcumstances. The resultng tendency to devate from a charted course mght be regretted as a source of error but can also prove to be a source of gan. Tabu search operates n a smlar way wth the excepton that new courses are not chosen randomly. Instead, tabu search proceeds accordng to the supposton that there s no pont n acceptng a new soluton unless t s to avod a path already nvestgated. Ths nsures that new regons of a problem soluton space are nvestgated wth the goal of avodng local mnma and ultmately fndng the desred soluton. Tabu search along wth other metaheurstcs, such as Smulated Annealng (Osman, 99), shares the ablty to gude the search for teratve mprovements. In ths context, tabu search provdes a gudng framework for explorng the soluton space beyond ponts where an embedded heurstc would become trapped n a local mnmum... Game Theoretc Approach for Vehcle Routng Problem In ths research a new approach s proposed to ncorporate the rsk averseness n hazardous materals routng. As gven by (Bell, 000), n order to represent the ratonal behavour n ths stuaton a two-player, zero-sum, non-cooperatve game s vsualsed between a route planner (dspatcher) who tres to fnd the best possble tour(s) for each vehcle located at a sngle depot and a vrtual system spoler (evl-entty, demon) who ams at falng one lnk n the system n order to gve maxmum possble damage (cost) to the operator. The term noncooperatve mples that the operator s not aware of whch lnk s gong to be faled by the evl-entty and conversely the evl-entty has no dea about whch lnks are gong to be chosen by the route planner to transport the hazardous materal requred by the customers. The mxed-strategy Nash equlbrum (Nash, 95) s a good tool to obtan the expected routng cost at equlbrum wth respect to the lnk usage and falure probabltes. Ths equlbrum routng cost hghlghts the outcome of a mental game n whch one of the players s a ratonal route planner wth an extremely pessmstc vew about lnk costs. At the mxed strategy Nash equlbrum, expected trp costs are both mnmsed wth respect to lnk choce probabltes and maxmsed regardng lnk falure probabltes. The game s a Nash-game due to the fact that no player s superor or domnant to the other and no player knows the next acton to be taken by the other player. The routng problem that s specfcally addressed n ths research s related to shpment of hazardous materals from a depot to some customers through a set of vehcles located at a depot. More specfcally, t s assumed that the probablty of any vehcle nvolvng n an ncdent durng the shpment s qute low, but the resultng consequence s extremely hgh. One of the man shortcomngs of ths problem s that avalablty and accuracy of the data related to probablty of the occurrence of ncdents s always questonable. Game theory llustrates how a route planner should determne hs routng strateges when the ncdent probabltes are not known. The extremely pessmstc (hence, rsk averse) nature of the planner, on the other hand, ensures the best way of escapng from the rsk, as the possble consequences are extremely hgh. Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

4 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan 4. MODELLING CONSIDERATIONS OF RISK AVERSE ROUTING OF HAZARDOUS MATERIALS To understand the modellng framework of ths research t s mportant to clarfy the underlyng concept of rsk-averseness. In the context of ths research, rsk-averseness means that the route planner pursues a set of vehcle routes as to mnmse hs maxmum expected loss on the assumpton that unknown lnk falure probabltes wll govern the routng of the vehcles n such a way that they are least favourable to hm. Hence, n ths way the planner can determne hs best routes regardless of the falure probabltes of the lnks. As long as the operator stcks to hs equlbrum lnk choce probabltes, the total expected cost of the system never goes above hs equlbrum expected loss regardless of the lnk falure probabltes. At ths uncture t should be mentoned that, these lnk falure probabltes do not represent the actual falure probabltes (the probabltes of occurrence of ncdents, whch are unknown), rather they characterse the worst case scenaro. Thus, the lnk falure probabltes of the soluton ndcate the relatve mportance of the lnks as they depct the case where the consequences of a falure are expected to be worst. As the data regardng the actual falure probabltes of the lnks are rare, the falure probabltes are qute small and unknown for each lnk. However, one of the man characterstcs of the route planner s to be rsk averse. Accordngly, he chooses hs best set of routes on the bass of the fact that only one lnk s faled at each teraton, whch yelds the worst expected consequence. Ths mples that multple falures are not consdered and, hence, the total lnk falure probabltes are equal to. The equlbrum soluton obtaned analysng gamelke nature of the problem gves the mnmsed and maxmsed expected system cost n terms of lnk choce probabltes and lnk falure probabltes, respectvely. If ether the route operator or the vrtual evl-entty devates from hs equlbrum probabltes, ths wll result n gvng a chance to the other sde to ncrease hs expected beneft. The optmum assgnment of the vehcles to the routes yelds a mxed strategy Nash Equlbrum of a noncoperatve two player, zero-sum game between a route planner seekng the most relable (least cost) set of vehcle routes and a vrtual network tester wth the ablty to fal a lnk. Furthermore, ths equlbrum pont s known as rsk-averse equlbrum as t s based on worst-case lnk falure probabltes. The obectve functon of ths problem can be formulsed as follows: Mn [Max subect to q p q 0 where; = f n p (cq + c ( q ) )] () all 0 all lnks scenaros p s the lnk-usage probablty c s the travel cost on lnk under scenaro when f lnk s faled q s the probablty of scenaro n c s the travel cost on lnk under scenaro when lnk operates normally The obectve functon s maxmzed wth respect to probablty of scenaros and mnmzed wth respect to lnk-usage probabltes. Ths mental game starts wth the assumpton that the operator determnes hs best routes by consderng that each lnk n the system has the same falure probablty. The best routes at ths stage are determned through Savngs Algorthm. Ths heurstc algorthm s one of the earlest and perhaps the most wdely known algorthm for VRP. It should be ponted out that there s always a possblty that there could be sngle-customer routes for some ndvdual customers. The savng values for ths algorthm are based on the expected costs, calculated by usng lnk usage and falure probabltes, rather than the actual costs. The vehcle routes obtaned from Savngs algorthm are mproved by usng Tabu Search, a metastrategy for gudng known local-search heurstcs to overcome local optmalty. It conssts of explorng the search space by dentfyng neghbourhood of a gven soluton, whch contans so-called transformed solutons that can be reached n a sngle teraton. The tabu search algorthm used n the model s nspred from the algorthm developed by (Barbarosoglu and Ozgur, 999). A feasble soluton for the problem that we descrbed above s represented by S=(S,S, S, S m ), where, S defnes unquely the Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

5 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan subset of customers of V=(,,,,k) routed by a vehcle from the set of vehcles M=(,,,..,m) such that, m US = V = S S = Φ dk ρ k k S C(S) = C(S ) M, M M In the lght of above explanatons and formulatons, the steps of the soluton procedure to obtan both lnk usage and scenaro probabltes can be set out as below; Step. Intalse q for all scenaros and τ 0 Step. Set expected lnk costs to n f c ( q ) + c q for all lnks. Step. Fnd the ntal least expected cost routes usng Savngs Algorthm and then employ the Tabu Search to mprove the ntal routes. Step4. Determne the auxlary lnk aux usage probabltes p f lnk s on aux the set of best routes, p 0 otherwse. Step5. Update lnk usage probabltes: τ p + τ aux τ = p + (p p ) for all lnks τ+ Step6. Fnd the scenaro whch maxmses τ p + * c to,, the lnk cost c s equal f c f lnk s faled for scenaro n c otherwse. Step7. Get the auxlary scenaro probabltes. q aux, for all other scenaros, q aux l 0 Step8. Update the lnk falure τ+ τ aux τ probabltes: q = q + (q q ) τ+ for all scenaros. Step9. τ τ+ and go back to Step and repeat all the steps tll a satsfactory convergence s obtaned. What ths procedure bascally provdes s that both the route planner (system operator) and the system spoler (evl-entty) can determne ther best strateges n terms of lnk usage and falure probabltes, respectvely, based on the accumulated hstory of the other player s choce of strateges. It should be ponted out that the followng assumptons preval the whole soluton algorthm descrbed.. The lnks connectng depot and the customers are assumed to be a straght lne. Although there may be many alternatve paths avalable between the customers and depot, t s assumed that the shortest straght lne between the customers and depot s the lnk, whch s used for hazmat transportaton.. The characterstcs of each lnk are assumed to be homogenous wthn each specfc lnk. Accordngly, the probablty of occurrence of an ncdent s assumed to be the same along the lnk. Characterstcs of a road segment ncludng number of lanes, surface qualty, traffc volume may affect the probablty of occurrence of an ncdent whle the vehcle travelng on that lnk.. Although the actual falure probabltes of the lnks are unknown, they are deemed to have qute low values ( mllon per truck mles). The possble consequences of each falure, however, are extremely hgh. Hence, the route operator s thought to be rsk-averse for decdng the vehcle routes through whch hazmats are to be carred. 4. The normal and faled costs of the lnks are homogenous along the lnks. 5. The hazardous materal s assumed to be carred n repeated shpments from depot to the customers. In the lght of explanatons above, the followng Fgure summarse the calculaton steps of the man algorthm developed. 4.. Improvng the Intal Soluton Through Tabu Search The computatonal results ndcate that the ntal soluton obtaned by usng Savng Algorthm s not good enough. Hence, a tabu search procedure has been used to mprove the ntal soluton. Durng ths process, the feasble solutons are mproved by obtanng ther neghbourhoods. In order to ntensfy the soluton mprovement n each route locally we use a generaton mechansm, whch descrbes how a soluton S can be altered to generate another Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

6 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan neghbourng soluton S '. In the model developed, the neghbourhood soluton s generated through a mechansm known as λ-nterchange generaton mechansm, ntroduced by Osman (99). Once the neghbourng solutons are obtaned through ths generaton mechansm, t s well-known that these solutons can be mproved through local moves based on exchangng edges (Caseau and Laburthe, 999). Inserton procedure s employed to determne the best possble poston of the customer to be nserted wthn the route. The opt exchange procedure frst ntroduced by Ln (965) s used to generate a so-called -optmal (-opt) tours. A tour s called -optmal f there s no possblty to shorten the tour by exchangng two lnks. MAIN ALGORITHM Expected Cost Matrx Savng Algorthm (Intal Soluton) Improved Soluton Tabu Search 5. LINEAR PROGRAMMING FORMULATION As stated by Equaton, the obectve functon of the problem s the mnmsaton and maxmsaton of the total expected routng cost n terms of lnk usage probabltes and lnk falure probabltes, respectvely. Equaton can be reformulated wth respect to routes, rather than lnks, as; Max (Mn C =, subect to; h = q = h c h ) () q h 0 for all routes q 0 for all scenaros where; Lnk Falure Procedure: Vrtual evl-entty fals the lnk. Update Procedure for Lnk Falure Probabltes Q Update Update Procedure for Lnk Usage Probabltes P Update C s the total expected routng cost c s the cost of route under scenaro h s the probablty of route s chosen q s the probablty of scenaro The lnear programmng formulaton of ths maxmn problem s gven by (Bell, 000) as follows: Let C * be the soluton to the problem. Then; Fnd new Expected Cost Matrx Check the Convergence NO * c h C for all scenaros YES STOP Fgure. Calculaton steps of proposed algorthm. 4.. Method of Successve Averages Once the fnal soluton set of routes s obtaned, then the lnk usage and falure probabltes are updated based on the Method of Successve Averages (MSA). MSA s a flexble and stable soluton algorthm. But t may also be costly n terms of the number of teratons needed. Bell and Ida (997) provde a comprehensve descrpton of ths method, whch s employed n the algorthm. The followng lnear program s proposed n order to fnd out the optmal mxed routng strategy for vehcles carryng extremely dangerous hazardous materals. Mn M wth respect to h subect to ch M 0 for all scenaros (4) h = h 0 for all routes In the same way, the optmal mxed lnk falure strategy for the evl entty s gven by followng lnear program. Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

7 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan Mn M* wth respect to q subect to * cq M 0 for all scenaros (5) q = q 0 for all scenaros It should be ponted out that the route choce probabltes obtaned from Equaton. 4 and lnk falure probabltes obtaned from Equaton. 5 are dual varables. In other words, the dual varables of Equaton. 4 wth the constrants are equal to the scenaro probabltes q of Equaton.5 wth constrants gven. At the soluton, M s equal to M*. The soluton of ths lnear programmng problem s unque n M,hence M*, but not n general unque n h or q. 6. NUMERICAL CALCULATIONS Some related problems are solved usng the algorthmc model descrbed to llustrate the effectveness of the soluton procedure. Snce, there s no data avalable from the real world cases, the soluton algorthm s tested usng some hypothetcal problems. As the soluton process reflects a mental game between the route planner and vrtual network tester, the obtaned results should be regarded as the possble outcomes of ths mental game. The problems that are dealt wth nvolve one depot and a number of customers wth known demands. The demands are servced by the vehcles wth a fxed capacty located at the depot. The shpment costs of the lnks are represented by two matrces: normal cost matrx, and faled cost matrx. Normal cost matrx llustrates the case when the lnks operate normally. Faled cost matrx, however, ndcates the case when an ncdent occurs on the lnk durng the shpment. These faled costs are assumed to be catastrophc costs,.e. the resultng cost to the communty s hgher than a predetermned acceptable level. In order to llustrate the quck convergence of the expected trp cost, a problem nvolvng 5 customers, one depot and one vehcle s tested usng the proposed algorthm.the cost matrxes of the problem are gven by Table and Table below. As the mnmum cost routes are obtaned by enumeratng all possble routes and selectng the one wth mnmum trp cost, the resultng routes are the absolute mnmum cost tours. The Fgure below llustrates the convergence behavour of the expected trp cost Table. Normal cost matrx Table. Faled cost matrx As the mnmum cost routes are obtaned by enumeratng all possble routes and selectng the one wth mnmum trp cost, the resultng routes are the absolute mnmum cost tours. The Fgure below llustrates the convergence behavour of the expected trp cost. The lnk usage and falure probabltes of ths problem are llustrated by Fgure and Fgure 4, respectvely. In ths problem, there are 5 lnk usage probabltes and also 5 lnk falure probabltes. Those whose usage and falure probabltes are bgger than zero are selected and ther convergence behavour s presented. The Fgure reveals the fact that the sum of the lnk usage probabltes taken over all lnks that enter or leave any node sums to. For example, the values of these probabltes for lnks 0-, -, -4, -5 sum to. On the other hand, the Fgure 4 llustrates the fact that the value of the total lnk falure probabltes s. Convergence of lnk usage probabltes s slow. Ths s beleved to be because of the nature of the non-unqueness of these probabltes. The same equlbrum expected trp cost may be gven by dfferent set of lnk usage and lnk falure probabltes. Another reason could be related to the MSA procedure that we employed to update the lnk-related probabltes. As we mentoned prevously, although MSA procedure s qute smple and effectve, t may cause the convergence of the lnk usage and falure probabltes, expected trp cost as well to some extent, to be slow. Thereby, another soluton method, rather than MSA, could result n a qucker convergence wth regard to these probabltes. The followng fgure llustrate the convergence of the lnk falure probabltes of the gven problem Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

8 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan Expected Cost Iteraton Number Fgure. Convergence of total expected trp cost.. Lnk Usage Probabltes Lnk 0- Lnk 0- Lnk 0-5 Lnk - Lnk -4 Lnk -5 Lnk - Lnk -4 Lnk -5 Lnk -4 Lnk -5 Lnk Iteraton Number Fgure. Convergence of lnk usage probabltes. 0,7 0,6 0,5 Lnk Falure Probabltes 0,4 0, Lnk 0- Lnk -4 Lnk - Lnk -5 Lnk 4-5 0, 0, Iteraton Number Fgure 4. Convergence of lnk falure probabltes. Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

9 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan The above problem was also solved by usng lnear programmng soluton n order to determne the exact values for the lnk usage and lnk falure probabltes. It should, however, be ponted out that ths method s sutable for only small sze problems as t would be an extremely hard ob to determne the all avalable actual routes. Ths s the man motvaton for why heurstc approach s employed for the soluton procedure of the developed algorthm. Ths soluton procedure s mplemented to determne exact values for the lnk usage probabltes n the 5- customer problem, n whch complete 0 dfferent routes are avalable, n order to compare wth the values obtaned by the teratve MSA method and to establsh whether there are multple optma hence whether there are multple optma. In order to employ ths method, let us represent the Equaton 4 n detal for the problem wth 0 dfferent routes. Mn M; CR h + CR h CR0 h0 M CR h + CR h CR0 h0 M CR h + CR h CR0 h0 M CR h + CR h CR0 h0 M.. (6) subect to h + h + h + h h0 = h, h, h, h 4,..., h0 0 (7) In order to solve ths problem we employed E04MFF/E04MFA computer program. Ths program was desgned to solve lnear problems of the form; mnmse N x F T c x, subect to x l { } u Ax where c s n element vector and A s an m L by n matrx. The resultng route choce probabltes, obtaned usng ths program through possble combnaton of dfferent 0 routes, are gven by the Table below. To be more precse, frst route (R ) s, for example, and the second route (R ) s , and so on. Startng from the top left corner towards rght, each cell represents these possble routes. Table. Route usage probabltes: Lnear programmng As can be seen from ths table, there are only dfferent routes out of 0 possble routes to be used n order to make sure that the selected routes provde the operator wth the mnmum total expected routng cost regardless of the lnks to be faled by the evl entty whch tres to maxmse the total travel cost. Table ndcates that Route8, Route 4 and Route4 are used to delver the requred hazardous materals from the sngle depot to the customers. These routes consst of the followng lnks; Route8: h 8 = 0.55 Route4: h 4 = 0.4 Route4: h 4 = 0. The followng Table 4. llustrates the comparson values for the lnk usage probabltes obtaned from both MSA method and Lnear Programmng. Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

10 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan Table 4. Comparson of lnk usage probabltes. Lnear Programmng Heurstc Soluton Lnk Lnk 0- Lnk Lnk Lnk Lnk Lnk Lnk Lnk Lnk Lnk Lnk These values ndcate that the slow convergence of the lnk usage probabltes, although the MSA method resulted n qute close to the optmum values after 00 teratons, s bascally due to the nonunqueness of the lnk usage probabltes. The total expected travel cost yelded by MSA method s unt (See Fgure ). The Lnear programmng soluton gves ths value as 88.7 unt. As a result, we can state that the MSA method s an effectve and relable method for the soluton the problem formulated n the model developed. 7. CONCLUSIONS Convergence of lnk usage and lnk falure probabltes s slow. Ths s beleved to be because of the nature of the non-unqueness of these probabltes. The same equlbrum expected trp cost may be gven by dfferent set of lnk usage and lnk falure probabltes. Another reason could be related to the Method of Successve Averages (MSA) procedure that we employed to update the lnk-related probabltes. Although MSA procedure s qute smple and effectve, t may cause the convergence of the lnk usage and falure probabltes, expected trp cost as well to some extent, to be slow. Thereby, another soluton method, rather than MSA, could result n a qucker convergence wth regard to these probabltes. The obtaned expected trp cost states that as long as the route operator stcks to hs equlbrum lnk usage (choce) probabltes, ths cost would be hgher than ths value under no scenaro. In other words, whatever the lnk falure probabltes are, the total expected trp cost would not be hgher than ths expected trp cost f the lnk usage probabltes at ths equlbrum pont are n use. Thus, the choce of routes s based on the lnk falure probabltes. The correspondng equlbrum expected trp cost, hence, may be nterpreted as the result of best routng strategy based on the worst set of lnk ncdent probabltes assumng that an ncdent occurs. Ths routng strategy clearly reflects the extremely pessmstc and hence rsk-averse nature of the route planner. The convergence behavour of the expected trp cost s also dependent on the randomly selected customers durng the tabu search for the mprovement of the ntal routng soluton. If the customers selected randomly are those whose ncluson or deleton would result n best possble route mprovement, then relatvely better routes are obtaned. Ths, thereby, would cause the convergence to be relatvely qucker to those random customers whose nserton and deleton does not gve such an mprovement. 8. REFERENCES Barbarosoglu, G. and Ozgur, D A tabu search algorthm for the vehcle routng problem, Computers and Operatons Research. (6), Bell, M.G.H. and Ida, Y Transportaton Network Analyss. John Wley and Sons, Chchester, UK. Bell, M.G.H A game theoretc approach to measurng the performance relablty of transport networks, Transportaton Research, Part B. Vol : B Caseau, Y. and Laburthe, F Heurstcs for large constrant vehcle routng problems, Journal of Heurstcs. (5), 8-0. Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89

11 Safe Movement of Hazardous Materals Through Heurstc Hybrd Approach : Tabu Search and Game Theory Aplcaton, H. Aslan Erkut, E. and Verter, V Modellng of transport rsk for hazardous materals, Operatons Research. 46 (5), Hobeka, A.G., Barnham, J. and Santoso, I. B Selecton of preferred hghway routes for the shpment of spent nuclear fuel between Surry and North Anna Power Statons n Vrgna, State of Art Report : Recent advances n hazardous materals, Transportaton Research, Transportaton Research Board Laporte, G Exact algorthms for travellng salesman problem and the vehcle routng problems, Les Cahers du Gerad, G Lenstra, J.K and Kan, A.H.G. 98. Complexty of vehcle routng and schedulng problems. Networks. (), -7. Ln, S Performance analyss of networks wth unrelable components. BSTJ. (44), Nash, J.F. 95. Non-cooperatve games. Annals of Mathematcs. (54), Osman, I. H. 99. Metastrategy smulated annealng and tabu search algorıthms for the vehcle routng problem, Annals of Operatons Research. (4), Mühendslk Blmler Dergs () Journal of Engneerng Scences () 79-89