Vehicle Routing Problem with Time Windows for Reducing Fuel Consumption

Transcription

1 3020 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER 2012 Vehcle Routng Problem wth Tme Wndows for Reducng Fuel Consumpton Jn L School of Computer and Informaton Engneerng, Zhejang Gongshang Unversty, Hangzhou, P.R.Chna Emal: jnl@mal.zjgsu.edu.cn Abstract Most of the studes on vehcle routng problem wth tme wndows (VRPTW) am to mnmze total travel dstance or travel tme. Ths paper presents the VRPTW wth a new objectve functon of mnmzng the total fuel consumpton. A mathematcal model s proposed to formulate ths problem. Then, a novel tabu search algorthm wth a random varable neghborhood descent procedure (RVND) s gven, whch uses an adaptve parallel route constructon heurstc, ntroduces sx neghborhood search methods and employs a random neghborhood orderng and shakng mechansms. Computatonal experments are performed on realstc nstances whch shed lght on the tradeoffs between varous parameters such as total travel dstance, total travel tme, total fuel consumpton, number of vehcles utlzed and total wat tme. The results show that the soluton of mnmzng total fuel consumpton has the potental of savng fuel consumpton contrary to the soluton of the tradtonal VRPTW, and s benefcal to develop envronmental-frendly economes. Index Terms vehcle routng, tme wndows, fuel consumpton, tabu search, transportaton I. INTRODUCTION Transportaton has hazardous mpacts on the envronment, such as resource consumpton, land use, acdfcaton, toxc effects on ecosystems and humans, nose and Greenhouse Gas (GHG) emssons[1]. Among these, fuel consumptons are the most concernng as they not only are the man cost of the companes, but also cause serous polluton whch has drect consequences on human health and envronment. Rsng fuel prces and growng concerns about GHG polluton of transportaton on the envronment call for revsed plannng approaches of road transportaton to reduce fuel consumpton. Our purpose s to ntroduce a new varant vehcle routng where optmzng the VRP by mnmzng the fuel consumpton. The Vehcle Routng Problem (VRP)[2] ams at plannng the routes of a fleet of vehcles on a gven network to serve a set of clents under sde constrants. The lterature on the VRP and ts varants s rch[3][4][5]. A common varant s the VRP wth Tme Wndows (VRPTW) where the mnmum-cost routes s found startng from and returnng to the same depot that vsts a set of clents once, each wth a predefned tme slot. Comprehensve surveys of soluton technques for the VRPTW can be found n Solomon[6] and Cordeau et al.[7]. Fglozz[8] proposed an teratve route constructon and mprovement heurstcs for the VRP wth soft tme wndows, whch penaltes are mposed to the total cost n case the clent delvery tme wndow can t be met. Pang[9] presented a route constructon heurstc wth an adaptve parallel scheme. Yu and Yang[10] developed an mproved ant colony optmzaton (IACO) to solve perod vehcle routng problem wth tme wndows, n whch the plannng perod s extended to several days and each clent must be served wthn a specfed tme wndow. Najera and Bullnara[11] proposed and analyzed a novel mult-objectve evolutonary algorthm, whch ncludes methods for measurng the smlarty of solutons, to solve the multobjectve problem. The tradtonal objectves of the VRPTW focus on mnmzng the total dstance traveled by all vehcles or mnmzng the total cost, whch usually s a lnear functon of dstance. It has been suggested by a number of studes that there are opportuntes for reducng fuel consumpton by extendng the tradtonal VRP objectves to account for wder envronmental and socal mpacts rather than just the economc costs[12]. But to mnmze the vehcle s travel dstance does not necessarly produce the optmal soluton from a fuel-effcency standpont. The reason s that the fuel consumpton of a vehcle s affected not only by the travel dstance, but also by other factors such as vehcle speed and road gradent n each segment [13][14]. Some related studes have taken nto account energy consumpton n vehcle routng from ther own dfferent perspectves. Kara et al.[15] ntroduced a so-called energy-mnmzng vehcle routng problem whch s an extenson of the VRP where a weghted load functon (load multpled by dstance) s mnmzed. Kuo[16] proposed a smulated annealng (SA) algorthm for fndng the tme-dependent vehcle routng wth the lowest total fuel consumpton. Suzuk[17] developed an approach to the tme-constraned, multple-stop, truckroutng problem that mnmzes the fuel consumpton and pollutants emssons. To our knowledge, the above studes have not mentoned VRPTW for reducng the fuel consumpton. The fuel consumpton durng VRPTW routes are affected by many factors, e.g. travel dstance, speed, road gradent, and load. Most studes fal to properly ntegrate these do: /jcp

2 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER factors, especally the fuel consumpton durng the wat tme at clents stes. In VRPTW, the vehcle s often requred to wat at clents stes because vehcle s arrve may precede the start of tme wndow. However, most studes have mplctly assumed that the fuel consumpton durng wat tme s zero. The fact s that a vehcle may consume fuels durng wat tme for varous reasons such as heatng or coolng the drver s compartment. So, we descrbe a comprehensve measurement approach of fuel consumpton takng nto account a broader and comprehensve factors ncludng dstance, speed, load and wat tme, etc. Then, we defne a mnmal-fuel VRPTW model. To solve ths problem, a novel tabu search algorthm wth a Random Varable Neghborhood Descent procedure (RVND) s gven. We also perform analyses usng numercal examples to shed lght on the tradeoffs between varous performance measures of vehcle routng, such as dstance, travel tme and fuel consumpton, assessed through a varety of objectve functons. Ths paper s organzed as follows. The next secton provdes a formulaton of mnmal-fuel VRPTW. Secton 3 descrbes the tabu search algorthm wth RVND for the model. Computatonal experments and analyses are presented n Secton 4. The fnal secton contans our conclusons. II. FORMULATION OF MINIMAL-FUEL VRPTW A. Prolbem Descrpton The mnmal-fuel VRPTW can be defned as follows. Let G = (V, E ) be a complete graph wth V = { 0,1, L, n } as the set of nodes and E = {(, j), j, j} as the set of arcs defned between each par of nodes. Node 0 s the depot. There exsts a homogeneous set of vehcles K = { 1,2, L, m}, each wth capacty Q. Each edge (, j) E has non-negatve travel dstance d j and travel tme t j.every clent V \{0} has demand d, vehcle s arrval tme a, wat tme w, servce tme s, and a request to be served wthn a predefned tme nterval [ e, l ]. The objectve s to fnd the optmal routes of mnmzng the total fuel consumptons whle meetng the overall clent demands. The constrants nclude: Depot constrants: The vehcles start from and returnng the only one depot. Vehcle s capacty constrants: The total load a vehcle carres can t exceed ts capacty. Tme wndow constrants: The vehcle s requred to vst a clent wthn a predefned tme wndow. The vehcle s allowed to arrve before the openng of the tme wndow, and wat untl the clent s avalable. But, the vehcle s not allowed to arrve after the close of the tme wndow. So, ths s a VRP wth hard tme wndow. B. Measurement of Vehcle s Fuel Consumpton We ntroduce a measurement of fuel consumpton smlar to Suzuk[17]. Let MPG j be vehcle s mles per gallon of fuel consumpton n arc (, j), and v j be the average speed n arc (, j). MPG j s expressed as a functon of v j. MPGj = α 0 + α1v j (1) where α 0 0, α 1 0 are the parameters to be estmated. Snce the effect of vehcle speed on mpg based on longtme data, to compute the effect of vehcle speed on mpg for each arc based on the road gradent and load, we adjust (1) as: MPGj = ( α 0 + α1v j ) γ jπ j (2) where γ j > 0 s the parameter of the road gradent factor, whch measures the devaton of a vehcle s mpg n each arc from the standard, flat terran value ( γ j = 1 represents the flat terran, γ j < 1 descrbes the postve gradent, and γ j > 1 s for the negatve gradent), π j > 0 s the parameter of load factor that measures the devaton of a vehcle s mpg n each arc from the average value based on the load. The effect of vehcle load on mpg can be expressed as a lnear functon: mpg = β 0 + β1l, where L s the load, β 0 0 s the mpg of a vehcle when t s empty, β 1 < 0 s the coeffcent measurng the loss of mpg caused by addtonal load. So, we can descrbe π j as: β + β 0 1 Y j π j = (3) β0 + β1μ where Y j V \{0} s the clent set unvsted when the vehcle s travelng on arc (, j) ; and μ s the average load of the vehcle n the long run. Equaton (3) ndcates that when the load s μ, π j = 1; when t s less than μ, π j > 1; when t s larger than μ, π j < 1. Let ρ 0 denote the average amount of fuel consumed per hour whle a vehcle s watng at clent stes. The fuel consumpton at clent durng the wat tme s descrbed as: WGPH = w ρ (4) C. Proposed Formulaton Accordng to the measurement of fuel consumpton from (2) to (4), the formulaton of mnmal-fuel VRPTW s descrbed as the followng: mn dj xjk ( α + v (, j) E 0 α1 j ) γ jπ j s. t. x = 1, j \{0} jk d + w ρ (5) N \ {0} (6)

3 3022 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER 2012 j x x jk = 1, \{0} = 0k 0 jk V \{0} j \{0} x = 1, k K x k = x0 jk = m \{0} j\{0} (7) (8) 0 (9) xjk 1, S V \ {0}, S 2 S j S xjk = x jk, k K, \ {0} j j d xjk Q, k K \{0} j ( a + w + s + t a ) x 0,, j, j, k K j j jk (10) (11) (12) (13) a l, (14) e a + w l, (15) x jk { 0,1},, j, k K (16) where x jk s the set of decson varables, for each arc (, j) and vehcle k, x jk =1 f and only f the optmal soluton, arc (, j) s traversed by vehcle k and equal 0, otherwse. The objectve functon s derved from (5) that measures the total fuel consumpton of vehcle routng. Constrants (6) and (7) are constrants of vstng each clent once. Constrants (8) and (9) are depot constrants. Constrants (10) are used to avod sub-loop of the routes. Balance of flow s descrbed through constrants (11). Constrants (12) are vehcle s capacty constrants. Constrants (13)-(15) are tme wndow constrants. Constrants (16) are the decson-varables constrants. III. TABU SEARCH ALGORITHM A. Framework of Algorthm Tabu Search (TS) has been used to study the classcal VRP and ts varants[7], and presents better performance. So, we choose TS as our framework of algorthm. Compared wth Genetc Algorthm (GA), Smulated Annealng (SA) and other metaheurstcs, TS has faster searchng ablty and hgh qualty of escapng the local optmum. However, TS s dependent on ts ntal soluton. A better ntal soluton wll help TS to search for a better soluton. Meanwhle, gven neghborhood structures usually restrct the stablty and global search capablty of the algorthm. To conquer the shortcomngs of TS, ths paper presents an mproved tabu search wth RVND. Also, we ntroduce an adaptve parallel route constructon heurstc (APRCH) [9] to construct the ntal solutons. Ths algorthm can generate hgh qualty ntal solutons, and have been verfed better than the common constructon algorthms e.g. savng algorthms, nserton algorthms. We adopt random varable neghborhood descent procedure (RVND) whch randomly selects a neghborhood operator to generate canddate solutons to change the fxed search order and strength the ablty of global optmzaton. The detaled steps of the algorthm are shown n the followng. Step 1: Intalzaton. The APRCH s used to construct X. Suppose the maxmum teraton the ntal soluton 0 sze s MaxOuterIter, and maxmum teraton sze of allowng the solutons not to be mproved s MaxInnerIter. Set the neghborhood structures: the nter-route neghborhood set NL 1 ={1-0, 1-1, 2-2, 2-opt*} and the ntra-route neghborhood set NL 2 ={Or-opt, Reverse}. Calculatng TF ( X 0), let X = X 0, where X s the current optmal soluton. Step 2: Local search. Performng the RVND based on current soluton Selectng a neghborhood N NL1 randomly to operate on current soluton, and the best non-tabu soluton X s obtaned If TF ( X ) < TF( X ), the neghborhood operators n NL 2 are executed n sequence to further mprove the soluton; Select the best non-tabu soluton as current soluton, update the current optmal soluton X, reset NL 1, and go to step 2.1; otherwse update NL 1 = NL1 \ { N} If NL 1 = φ, go to step 3, else go to step 2.1. Step 3: Restartng and shakng. If the current optmal soluton s not mproved exceedng a gven teratons sze (MaxInnerIter), restartng s adopted on current optmal soluton and the tabu lst s cleared. To change the current search drecton, three neghborhood operators are selected to execute from 1-0, 1-1, 2-opt*, 2-2, Or-opt, Reverse. Then, go to step 2. Step 4: Evaluaton of algorthm termnaton. If the maxmum teraton sze (MaxOuterIter) s reached, the algorthm s termnated and the optmal soluton X s output, else go to step 2. B. Intal Soluton Constructon The APRCH s desgned to specfcally construct the ntal solutons for VRPTW. So, we construct the ntal soluton based on the dea of APRCH. Ths algorthm balances the effect of fuel consumpton, tme urgency and watng tme on route constructon by assgnng weghts. It can adjust the weghts of these three cost factors adaptvely to gan the best route constructon. When a vehcle k serves clent j travelng through arc (, j), the detal to calculate the cost factors below. (1) Fuel consumpton cost Snce the vehcle load s unknown before the route constructon s completed, the load s not taken nto account n fuel consumpton cost to smplfy the problems. dj f jk = + w jρ (17) ( α0 + α1vj ) γ j (2) Tme urgency cost It s the tme of vehcle k from the arrval tme of clent j to the latest openng servce tme as shown n Fg. 1. u = l a + w + s + t ) (18) jk j ( j

4 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER C. Local Search We employ sx VRP neghborhood structures[9][18], whch are 1-0,1-1,2-2,2-opt*,Or-opt, and Reverse. The neghborhood operators ncludng 1-0,1-1,2-2, and 2-opt* are nter-route neghborhoods, and the rest nvolvng Oropt and Reverse are ntra-route neghborhoods. The route s encoded wth natural numbers where the depot s denoted as 0 and the clents s represented as 1,2,,n. Dfferent wth the tradtonal local search, the clents are operated wth those neghborhoods under the condtons where vehcle s capacty and tme wndows constrants must be met, and the objectve functon s reduced. (1) 1-0, 1-1, 2-2. These are nter-route movements, also called Swap/Shft movements. 1-0, that s Shft (1,0), represents that a clent from one route s nserted nto another route. For example, the clent 3 s moved from the route ( ) to the route (067890) before clent 8, whch generates the route ( ). 1-1, that s Swap (1,1), performs permutaton between a clent from one route and a clent from another route. For example, clent 3 from the route ( ) and clent 8 from the route (067890) are swapped to gan the route ( ). 2-2, that s Swap(2,2), means that two adjacent clents from one route are permutated by another two adjacent clents from other route. For example, the adjacent clents 2 and 3 from the route ( ) are exchanged wth the adjacent clents 7 and 8 from the route (067890) to obtan a route ( ). (2) 2-opt*. It s the extenson of Shft(1,0). Ths operator can realze permutaton between adjacent tal clents from one route and adjacent tal clents from another route. For example, clents 3,4,5 from route ( ) are exchanged wth clents 8,9 from route (067890) to get a route ( ). The operator can also construct one route by connectng the tals of the two routes to reduce the number of vehcles, e.g. the tal of the route ( ) s connected wth the tal of the route (067890) to generate a route ( ). (3) Or-opt. It s one of the ntra-route neghborhood. One, two or three adjacent clents are removed and nserted n another poston of the route. For nstances, the clents 2 and 3 from the route ( ) are re- a w e Fgure 1. Tme urgency cost of clent j. (3) Watng tme cost It s a watng tme when vehcle k arrves at clent j s locaton. w jk = max{ 0, e j ( a + w + s + tj )} (19) We further defne the Ftness measure as a weghted sum of the three factors by the weghts ω d, ω u and ω w assgned to the fuel consumpton, tme urgency and watng tme cost factors respectvely. Ftnessjk = ω d fjk + ωuujk + ωwwjk (20) where ω d + ωu + ωw = 1, ω d 0, ω u 0, ω w 0. The detaled steps of the algorthm are shown below. Step 1: Calculate the lower bound on the number of vehcles requred servng all clents based on total clent demands and vehcle s capacty by the celng functon as follows: d \{0} m = (21) Q Defne the ntal step sze s d, s u, s w [0,1] of the weghts ω d, ω u and ω w, and search ranges [ l d, μd ], [ lu, μu ] and [ l w, μ w ] [0,1]. Set step convergence factor λ 1 [0,1] and nterval convergence factor λ 2 [0,1]. Gven termnaton step sze of the algorthm s mn. Step 2: Intalze m empty routes to the vehcle set R, wthout clent assgned Select the locaton before the fnal node of the current route as the poston the clent s nserted nto. For example, If the current route s (0,1,2,,,0), the poston the clent j s nserted nto s shown n Fg. 2. Judge whether the clents unassgned to the routes are taken as canddate nodes to be assgned accordng to vehcle s tme and capacty constrants. If there are clents whch can t be assgned to any route, a new vehcle must be ntated to serve these clents. Then the route of the generated vehcle s added to R. 0 1 s 2 t j Fgure 2. The poston nserted by clent j Let U be the set of clents unassgned to any route. For the route of vehcle k belongs to R, calculate mn{ Ftness jk }, k R, j U, where the fnal clent of the route s. And Let the best clent to be assgned to the correspondng vehcle. a j u jk l j 0 j t 2.3. Repeat steps untl U = φ. If the fuel consumpton s reduced, update the current optmal soluton whch s the best soluton found from the begnnng to the current of the algorthm. Step 3: Based on the current step szes ( s d, su, sw ) and the search ranges ( [ l d, μd ], [ lu, μu ] and [ l w, μw] ), enumerate all the vald combnatons of the weghts ω d, ωu an ω w. If a new better soluton s found, adjust the search range so that the weghts that gve the best soluton are located n the mddle of the new search range, and recalculate the step sze. If no better soluton s found, reduce both the step sze and the search range of the weghts n terms of the convergence factors λ 1 and λ 2. Repeat steps untl the step sze s smaller than s. mn

5 3024 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER 2012 nserted n the poston before the clent 7 to gan a route ( ). (4) Reverse. Ths s also one of ntra-route neghborhoods. Ths movement reverses the sub-route drecton. For example, the clent 2, 3, and 4 from the route ( ) had ther drecton reversed to form a route ( ). In order to expand the search space, the depots are assgned based on the tme and vehcle s capacty constrants after the neghborhood operator s performed. For example, n the route ( ) whch has been executed by one neghborhood, f the large demands of clent 5 make the vehcle volate ts capacty, the depots can be assgned accordng to the capacty constrants to generate the route ( ). In these sx neghborhood structures, the computatonal complexty of nter-route neghborhoods (1-0, 1-1, 2-2, 2- opt*) and Or-opt s O ( n 2 ) whle the complexty of the neghborhood Reverse s O (n). D. Restartng and Shakng Restartng s to select a new soluton as current soluton to search. In ths paper, the current optmal soluton s selected as the restartng pont. If the current optmal soluton s not mproved n a gven teraton sze, ths optmal soluton s taken as current soluton whle smultaneously clearng the tabu lst to search soluton space along a new drecton. To change the fxed search drecton, shakng s appled to explore more soluton space searchng along a new drecton. Here, shakng s used by performng three neghborhoods selected randomly from the sx neghborhood structures (1-0, 1-1, 2-2, 2-opt*, Or-opt, Reverse). IV. COMPUTATIONAL ANALYSIS Experments were run wth data generated as realstcally as possble. A transportaton company that operatng class-8 trucks plan delvery to clents. We randomly generate hypothetcal, yet realstc, VRPTW nstances. In each nstance the problem specfcatons ( d j, v j, t j, γ j, d, e, l, s )[17][19] are determned by random number operators as shown n table 1. We solve each nstance usng three models, whch are mnmaldstance model (denoted as F D ) wth mnmzng the total travel dstance, mnmal-tme model (denoted as F T ) wth mnmzng the total travel tme, and mnmal-fuel model (denoted as F F ) wth mnmzng the total fuel consumpton. Three classes of problems wth n =10, 60 and 100 nodes are generated, where each class ncludes 10 nstances and the expermental results s the average value of these 10 nstances. We descrbe some of the parameters for TS wth RVND n the followng: s d, su, sw =0.1; [ l d, μd ] = [ l u, μu ] = [ l w, μw] =[0,1]; λ 1 =0.01; λ 2 =0.01; smn = ; MaxOuterIter=100; MaxInnerIter=10. All experments are conducted on a PC wth 1.58G Hz speed and 2Gb RAM. The TS algorthm wth RVND s coded n Matlab 7.0. A common tme-lmt of 2 mnutes was mposed on the soluton tme of all nstances. Parameters TABLE I. EXPERIMENTAL DATAS Values (ranges) Fxed Parameters α 0 (Speed regresson ntercept) α 1 (speed regresson slope) β 0 (load regresson ntercept) β 1 (load regresson slope) ρ (fuel consumed per wt h) gal. μ (average or base load) 33,451 lbs s (servce tme at clent) 0.1 h(all clents) Q (vehcle s capacty) 45,000 lbs Random varables (specfc to each experment) γ j (road gradent factor) e (the earlest start servce tme) 8:00 am to 1:00 pm l (the latest start servce tme) 2-6 h after e d (clent demand) 5,000-10,000 lbs d j (arc dstance) 5-50 mles v j (arc speed) mph t j (arc travel tme) t j = dj / vj h A. Effect of the Varaton n Number of Nodes Ths secton presents the results of analyses n dfferent number of nodes usng the three dfferent models (F D, F T and F F ). Table 2 shows the fve followng measures obtaned by the three models: total travel dstance (TD), total travel tme (TT), total fuel consumpton (TF), the number of vehcles utlzed (m), and total wat tme (wt). All values are standardzed to one for the F D objectve. n TABLE II. RESULTS OF EXPERIMENTS ON VARIOUS NUMBER OF NODES F T TD TT TF m wt TD TT TF m wt F F

6 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER The fgures shown n Table 2 makes t clear that there s no sgnfcant dfference n the solutons yelded by models F T, F D and F F n terms of vehcles utlzed where the average reductons n number of vehcles utlzed usng model F F s up to 0.31% than those produced by F D, and s up to 0.19% than those produce by F T. Ths shows that the TS wth RVND can reduce the number of vehcles utlzed to guarantee no sgnfcant dfference n number of vehcles utlzed. F F acheves an average reducton of up to 7.58% n total wat tme than F D, but an average ncrease of up to % than F T. We can see that the fuel consumpton durng wat tme s one of the factors that affect the total cost of fuel consumed, but does not play a leadng role when ρ =0.7. Some nterestng mplcatons of the results presented n Table 2 are as follows. Although the average total travel tme and total wat tme yelded by F T decrease by 22.40% and 60.65% respectvely over those yelded by F D, the average total travel dstance and total fuel consumpton ncrease by 45.04% and 29.73% respectvely, whch suggests that the fuel consumpton durng wat tme s not a domnatng factor, and F D has better performance n savng fuel than F T. In addton, F T yelds less number of vehcles utlzed compared wth F D ( n =60 savngs up to 0.92%, n =100 up to 0.55%), whch ndcates that F T can save the number of vehcles utlzed n contrast wth F D. B. Effect of the Varaton n Fuel Consumpton durng the Wat Tme Ths secton presents the effect of the varaton n watng-tme fuel consumpton on varous measures. The vehcle s fuel consumpton durng the wat tme s related wth a number of factors such as vehcle types, ar and so on. To ths end, usng a sngle 60-node nstance, we have performed addtonal experments when ρ =0.3, 0.5, 0.7, 0.9 as reported n Table 3. The results presented n Table 3 suggest that as the watng-tme fuel consumpton ncreases, the total fuel consumpton grows, and the rato of watng-tme fuel consumpton to total fuel consumpton s also ncreasng, e.g. the rato s 7.77%, 11.97%, 15.34%, and 18.66% respectvely. The reason s that as growng watng-tme fuel consumpton, the watng-tme fuel consumpton becomes an mportant factor affectng the total cost of fuel consumpton. So, to mnmze the total fuel consumpton, F F searches for the routes reducng the watng-tme fuel consumpton, whch decreases the watng tme. We also note that F F acheves a sgnfcant fuel reducton over F D, and F F has a better fuel-savng performance (fuel-savng range ncreases from 4.23% to 5.89%) wth ncreasng ρ, average fuel savngs up to 5.07%. Snce F T always fnds the mnmal total wat tme, F T can save more fuel wth ncreasng ρ, and the dfference of F T, F F, and F D n fuel consumpton becomes narrow, e.g. the range of fuel savngs yelded by F F gradually decreases n comparson wth those yelded by F T (decreasng from 35.94% to 27.21%). However, F F stll has a average fuel reducton of up to 30.94% over F T. The analyss tells us that F F could obtan the most economcal fuel-savng routes on varous watng-tme fuel consumpton. Takng nto account the number of vehcles utlzed, F F s better than F D (average reducton up to 1.12%). F T can fnd the soluton wth mnmal number of vehcles, especally can reduce more number of vehcles utlzed (decreasng 1.79% n average) than F D. C. Effect of the Varaton n Tme Wndows In ths secton, we present results of computatonal experments to analyze the effects of dfferent tme wndow constrants. For ths purpose, we use the dstance data of a sngle 60-node nstance wth ρ =0.7. Accordng to above results, the average maxmum tme of the tour s 6 hour, and the maxmum nterval of the tme wndows s selected as 6. Tme wndows are ntally qute loose; they are chosen as randomly selected ntervals of 90% of the maxmum tme nterval. From ths nstance, 20 nodes are then randomly selected whle smultaneously narrowng down the correspondng tme wndows by a factor of δ set n the range of 10-90% n ncrements of 20%. We ensure that the tme wndows are not tght enough to generate nfeasble solutons. Results of ths experment are shown n Table 4. ρ TABLE III RESULTS SHOWING THE EFFECT OF THE VARIATION IN WAITING-TIME FUEL CONSUMPTION F T TD TT TF m wt TD TT TF m wt F F

7 3026 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER 2012 TABLE IV. RESULTS OF EXPERIMENTS ON VARIOUS TIME WINDOW CONSTRAINTS δ F T F F TD TT TF m wt TD TT TF m Wt The results presented n Table 4 ndcate that the fuel consumpton ncreases wth tme wndows narroweddown. The fuel savngs wth F F are more apparent under loose tme wndows ( δ = ) where the maxmum fuel savngs acheved by F F s 5.52% when δ =0.1 over those produced by F D, and s 26.15% when δ =0.1 over those yelded by F T. There s no sgnfcant dfference n fuel consumpton for models F T, F D, and F F under tghtened tme wndows ( δ = ). The reason s that the feasble soluton space becomes narrow wth tghtened tme wndows, and the selected routes are very lmted. Moreover, as the tme wndows become tght the number of vehcles utlzed ncreases. It s worth mentonng that the average reductons n number of vehcles usng model F T s 1.09% compared to those produced by F F, and s 1.97% compared to those produced by F D, especally under tghtened tme wndows the reductons n number of vehcles s sgnfcant. These results suggest that model F T can yeld the solutons wth mnmal number of vehcles, and reduce the number of vehcles utlzed as many as possble. V. CONCLUSION Study on VRPTW consderng fuel consumpton s crucal to the dstrbuton operatons of food logstcs and cold chan logstcs. By ntroducng the dea of reducng fuel consumpton and protectng envronment, a mnmalfuel VRPTW model, a varant of the well-known VRP, s proposed. We present a measurement approach of fuel consumpton affected by many factors such as travel dstance, load, speed, and watng tme, etc. Due to the model belongng to NP-hard problems, a novel tabu search algorthm wth random varable neghborhood descent procedure s gven. Ths algorthm uses adaptve parallel algorthm to generate hgh-qualty ntal solutons whle the neghborhood structures adopt random varable neghborhood descent and the restartng and shakng mechansm are also ntroduced. Fnally, a comparatve study on mnmal-dstance model, mnmaltme model and mnmal-fuel model s performed by computatonal experments. Results of the computatonal experments on realstc nstances yelded the followng mportant conclusons: The tradtonal objectve of mnmal travel dstance or mnmal travel tme does not necessarly mply mnmzaton of fuel. A tradeoff exsts among travel dstance, travel tme and fuel cost. The mnmal-fuel model provdes a 6.20% mprovement n fuel consumpton over the mnmal-dstance model, but ncreases the travel dstance by 3.84% on average. And t also reduces fuel consumpton by 27.67% compared to mnmal-tme model, but ncreases travel tme by 21.90% on average. A mnmal-dstance model has a better performance n savng fuel consumpton than mnmal-tme model (average savngs up to 6.63%), but mnmal-tme model can reduce the number of vehcles utlzed, especally for the case wth more nodes and tghtened tme wndows (reducton of vehcles up to 3.51%). The mnmal-fuel soluton can sgnfcantly save the fuel consumpton smultaneously reducng the pollutant emssons, especally when the number of nodes and watng-tme fuel consumpton ncrease. Concernng over global warmng has grown, reducng the carbon emssons has become an mportant ssue for all ndustres. Mnmzng fuel consumpton wll become ncreasngly mportant n comparson wth other crtera e.g. short travel dstance or tme. Also, there are a lot of opportuntes for future research to present new models consderng other factors e.g. heterogeneous vehcles, tme-dependent speeds and random clent demands, or to develop a more structural optmzaton strategy e.g. SA or GA algorthms, and thereby extend the results of the present researches. ACKNOWLEDGMENT The authors wsh to thank the revewers for ther valuable comments. Ths work was supported n part by the Natonal Natural Scence Foundaton of Chna (Grant No ), Humantes and Socal Scences Foundaton of Mnstry of Educaton of Chna (Grant No. 12YJC630091), Zhejang Provncal Natural Scence Foundaton of Chna (Grant No. LQ12G02007), and Zhejang Provncal Commonweal Technology Appled Research Projects of Chna (Grant No.2011C23076). REFERENCES [1] T. Bektas, G. Laporte. The polluton-routng problem, Transportaton Research Part B, vol.45,no.8, pp ,2011.

8 JOURNAL OF COMPUTERS, VOL. 7, NO. 12, DECEMBER [2] G. Dantzg, J. Ramser, The truck dspathng problem, Management Scence, vol.6, no. 1, pp ,1959. [3] F. L, B. Golden, E. Wasl, The open vehcle routng problem: Algorthms, large-scale test problems, and computatonal results, Computers& Operatons Research, vol.34, no.10, pp , [4] C.Y. Ren. Research on sngle and mxed fleet strategy for open vehcle routng problem, Journal of Software, vol.6,no.10,pp ,2011. [5] J.Q. Wang. Effcent ntellgent optmzed algorthm for dynamc vehcle routng problem, Journal of Software, vol. 6, no.11,pp ,2011. [6] M. M. Solomon, Algorthms for the vehcle routng and schedulng problems wth tme wndow constrants, Operatons Research, vol.35, pp , [7] J.F. Cordeau, G. Laporte, and A. Mercer, A unfed tabu search heurstc for vehcle routng problems wth tme wndows, Journal of the Operatonal Research Socety, vol. 52, pp , [8] M.A. Fglozz, An teratve route constructon and mprovement algorthm for the vehcle routng problem wth soft tme wndows, Transportaton Research Part C, vol.18, pp , [9] K.W. Pang, An adaptve parallel route constructon heurstc for the vehcle routng problem wth tme wndows constrants, Expert Systems wth Applcatons, vol. 38, no.9 pp , [10] B. Yu, Z.Z. Yang, An ant colony optmzaton model: The perod vehcle routng problem wth tme wndows, Transportaton Research Part E,vol.47, pp , [11] A.G. Najera, J.A. Bullnara, An mproved mult-objectve evolutonary algorthm for the vehcle routng problem wth tme wndows, Computers & Operatons Research, vol.38, pp , [12] A. Sbh, R.W. Eglese, Combnatoral optmzaton and green logstcs, 4OR: A Quarterly Journal of Operatons Research, vol.5, no. 2, pp , [13] E. Ercsson, Independent drvng pattern factors and ther nfluence on fuel use and exhaust emsson factors, Transportaton Research Part D, vol. 6, no.5, pp ,2001. [14] K. Brundell-Frej, E. Ercsson. Influence of street characterstcs, drver category and car performance on urban drvng patterns, Transportaton Research Part D, vol.10, pp , [15] I. Kara, B.Y. Kara, M.K. Yets, Energy mnmzng vehcle routng problem, In: Dress, A. Xu, Y. Zhu, B. (Eds.), Combnatoral Optmzaton and Applcatons, Lecture Notes n Computer Scence, vol Sprnger, Berln/Hedelberg, pp , [16] Y. Kuo, Usng smulated annealng to mnmze fuel consumpton for the tme-dependent vehcle routng problem, Computers & Operatons Research, vol. 59, no.1, pp , [17] Y. Suzuk, A new truck-routng approach for reducng fuel consumpton and pollutants emsson, Transportaton Research Part D: Transport and Envronment, vol.16, no.1, pp.73-77,2011. [18] A. Subramanan, L.M.A. Drummond, C. Bentes, et al., A parallel heurstc for the vehcle routng problem wth smultaneous pckup and delvery, Computers& Operatons Research, vol.37, pp , [19] US department of energy, Offce of energy effcency and renewable energy, Transportaton Energy Data Book, 28th ed. USDoE, Washngton, DC., Jn L was born n Jangsu provnce, Chna. He receved the B.S. and M.S. degrees from Nanchang Unversty, Nanchang, Jangx Provnce, Chna n 2004 and 2007 respectvely. He receved the Ph.D. degree n management scence and engneerng department, School of management, at Fudan Unversty, Shangha, Chna n From July 2010 untl now he works as an assstant professor n the Department of Management Scence and Engneerng, School of Computer Scence and Informaton Engneerng, Zhejang Gongshang Unversty, Hangzhou, Zhejang Provnce, Chna. He has authored/coauthored more than 20 scentfc papers. Hs research nterests nclude logstcs and supply chan management, system modelng and smulaton, network optmzaton, and emergency management. Dr. L s a fellow of Chna Socety of Logstcs (CSL), Chnese Computer Federaton (CCF), and Chnese Assocaton for System Smulaton(CASS).