Developing an applied algorithm for multi-trip vehicle routing problem with time windows in urban waste collection: A case study

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1 80700WM000.77/ X880700Waste Management & eseachbabaee Tiolaee et al. eseach-aticle208 Open Access Supplement Developing an applied algoithm fo multi-tip vehicle outing poblem with time windows in uban waste collection: A case study Waste Management & eseach 209, Vol. 37() Supplement 4 3 The Autho(s) 208 Aticle euse guidelines: sagepub.com/ounals-pemissions DOI: ounals.sagepub.com/home/wm Efan Babaee Tiolaee,2, Pavin Abbasian 3, Mehdi Soltani 4 and Seyed Ali Ghaffaian 5 Abstact This pape studies a multi-tip vehicle outing poblem with time windows specifically elated to uban waste collection. Uban waste collection is one of the municipal activities with lage costs and has many pactical difficulties. In othe wods, waste collection and disposal is a costly tas due to high opeating expenses (fuel, maintenance, ecycling, manpowe, etc.) and small impovements in this field can esult in temendous savings on municipal expenditue. In the aised poblem, the goal is to minimize total cost including tavesing cost, vehicle employment cost, and exit penalty fom pemissible time windows. In this poblem, the waste is deposited at the points indicating the demand nodes, in which each demand shows the volume of geneated waste. Consideing multiple tips fo vehicles and time windows ae the most citical featues of the poblem, so that the pioities of seving some specific places such as hospitals can be obseved. Since vehicle outing poblems (VP) belongs to NP-had poblems, an efficient simulated annealing (SA) is poposed to solve the poblem. The computational esults show that ou poposed algoithm has a geat pefomance in a shot computational time in compaison with the CPLEX solve. Finally, in ode to demonstate the applicability of the model, a case study is analyzed in Ian, and the optimal policies ae pesented. eywods Vehicle outing poblem, multiple tips, uban waste collection, simulated annealing, time windows, CPLEX solve eceived 8th June 208, accepted 9th Septembe 208 by Associate Edito Albeto Bezama. Intoduction The wold population has gown damatically in the last 0 yeas, with annual gowth ates of.%. In paticula, the total numbe of living human beings on Eath has isen fom 6 billion in 999 to 7 billion now (US Census Bueau ). This damatic gowth was mainly due to the advances in medicine and agicultue. If the population is able to eep gowing at this ate, it is estimated that the wold s population will each about 9 billion by 2040 (US Census Bueau 2 ). These factos illustate why the management of waste has been one of the wold s geatest concens ove the past few decades. Uban aeas poduce the highest amount of waste, so they need an efficient system fo waste collection and disposal. Maintenance of a pope waste management system is vey difficult and costly. Today, the poduction of vaious solid wastes and the occuence of a vaiety of social, economic and envionmental inconveniences associated with them, has caused uban sevice management poblems with collection, tanspotation, pocessing, and disposal of such waste. Since between 60% and 80% of solid waste management costs ae elated to waste collection and tanspotation, evaluation and optimization of this system will have a significant ole in solving and educing poblems in uban sevices management (Tiolaee et al., 208a). Theefoe, the poblem of uban waste collection can be fomulated in vaious Depatment of Industial Engineeing, Mazandaan Univesity of Science and Technology, Babol, Islamic epublic of Ian 2 Young eseaches and Elite Club, Ayatollah Amoli Banch, Islamic Azad Univesity, Amol, Islamic epublic of Ian 3 Faculty of Medicine, Isfahan Univesity of Medical Sciences, Isfahan, Islamic epublic of Ian 4 Depatment of Industial and Mechanical Engineeing, Qazvin Banch, Islamic Azad Univesity, Islamic epublic of Ian 5 Depatment of Industial Engineeing and Management Systems, Amiabi Univesity of Technology, Tehan, Islamic epublic of Ian Coesponding autho: Efan Babaee Tiolaee, Depatment of Industial Engineeing, Mazandaan Univesity of Science and Technology, Danesh 5, No. Sheyh Tabaasi, Babol, Mazandaan , Islamic epublic of Ian. e.babaee@in.iut.ac.i Ceative Commons CC BY: This aticle is distibuted unde the tems of the Ceative Commons Attibution 4.0 License ( which pemits any use, epoduction and distibution of the wo without futhe pemission povided the oiginal wo is attibuted as specified on the SAGE and Open Access pages ( en-us/nam/open-access-at-sage).

2 Babaee Tiolaee et al. 5 vesions of the vehicle outing poblem (VP). VP has been studied to be modeled in a wide ange of common poblems such as tanspotation, supply chain management, poduction planning, and communications (aadimas et al., 2007; Mimohammadi et al., 207; Tiolaee et al., 207). Cuently, inteest in waste management is gowing. Consideing the mao theats to human health and its impact on the envionment, management of hazadous waste equies much moe attention. Howeve, in the liteatue, thee is a lac of specific techniques to optimize such citical activities, which ae chaacteized by moe pecise limitations fo waste management in the uban community. To fill this gap, we povide an appoach to addess outing poblems and scheduling of vehicles fo collecting and disposing of hazadous waste (De Buece et al., 208). The vehicle outing poblem with time windows (VPTW) is a genealized poblem of the VP with capacity constaints in which each custome must be seviced at a paticula time inteval (this inteval is nown as a time window) (Tiolaee et al., 207). Since this poblem emphasizes on time in its appoach, it is moe applicable in pactice and has theefoe attacted moe attention in scientific cicles. nown examples of VPTW include the distibution of cash to ban banches, uban and industial waste collection, fuel delivey to gas stations, school sevices, and so on. By consideing time windows, the pioities of sevice ae defined accoding to the waste types including household waste, wastes of electic and electonic equipment (WEEE), chemical waste, hospital waste, etc. With espect to the hamfulness level of the waste, the pioities of sevice ae defined though had (stict) o soft (light) time windows. The pioity of sevices to some special places such as hospitals and health centes that poduce moe hazadous waste than household waste can be easily consideed by defining time windows. Futhemoe, WEEE is also categoized as a hazadous waste due to its utilization of dangeous substances which leads to difficult challenges duing the disposal and ecovey pocesses (Gambeini et al., 2008). WEE is cuently egaded as being one of the quicest developing waste steams in the Euopean Union with an annual incease ate of 3 5% it needs to teated using an efficient collection system. In the following, we will examine the liteatue elated to the poblem, especially in the context of the waste collection and developed solution methods. Beltami and Bodin (974) intoduced one of the fist poblems of waste collection and the outing poblem in the fom of a classic VP fo New Yo City and Washington Municipalities, along with a vaiety of vehicles (tucs, tugs, towing vessels, and mechanical sweepes). They impoved Clae and Wight s (964) innovative algoithm fo finding optimal outes. The esult of thei wo was applied to the municipalities and ceated significant benefits. Tung and Pinnoi (2000) developed the innovative algoithm poposed by Solomon (987) to solve the VPTW poblem and applied it to the waste collection poblem in Hanoi, Vietnam. In addition to taing into account the classical assumptions of VPTW, they consideed a pactical site fo the disposal of collected waste. They implemented 2-Opt and O-Opt algoithms to impove the quality of the applied algoithm s solutions. im et al. (2006) also studied a VPTW fo waste collection consideing lunch and est peiods fo dives. Thei goal was to minimize the numbe of vehicles and the total tip time. They also implemented an algoithm based on clusteing of aeas to solve thei poblem and used it in the waste management fo some Noth Ameican municipalities. Buhal et al. (202) studied a VPTW associated with the uban waste collection. They tied to find an expedient way to calculate the cost of the optimal oute fo the waste collecting vehicles tansit fom the stat of thei oute at the base to the dischage at the disposal site. A ecycling pocess of WEEE was analyzed efficiently by Gambeini et al. (2008) consideing the thee levels of: etun points; collecting points; and ecycling plants. They designed a mixed-intege linea pogamming (MILP) model to maximize the efficiency of the collection system though minimizing unsatisfied calls coming fom the fist two levels. Finally, they could demonstate the applicability of the poposed model by conducting a case study in the Italian egion of Emilia omagna. Gambeini et al. (200) designed a WEEE tanspotation-optimization netwo model associated with both technical decisions such as optimal outing and scheduling of vehicles and envionmental pefomance. Ma-Otiz et al. (203) intoduced a geedy andomized adaptive seaching pocedue algoithm to solve a multi-tip multi-peiod vehicle outing and scheduling poblem fo the collection of WEEE. Angelelli and Speanza (2002) pesented a peiodic vehicle outing poblem with intemediate facilities. When a vehicle eaches a middle facility, its capacity becomes zeo again. They developed a Tabu seach algoithm to solve thei poblem. Teixeia et al. (2004) developed an innovative method fo solving the peiodic vehicle outing poblem suited fo thee types of waste (plastics/metal, glass, and pape). Thei poposed solution consisted of thee phases: () defining an aea fo each vehicle; (2) defining the type of waste to be collected pe day; and (3) selecting the ode of the points to cove the demands. Efani et al. (207) pesented a novel appoach to detemine optimal bin location and collection outes using a geogaphic infomation system. They suveyed a case study in Ian by thei poposed method and could demonstate that the cuent collection system is not optimal due to its incompatibility with the uban stuctue and population distibution. Babaee Tiolaee et al. (206) also developed a obust mathematical model fo solving the capacitated ac outing poblem (CAP) paticula to the uban waste collection in uncetain conditions. They solved the poblem by developing a hybid simulated annealing (SA) and wee able to pove the effectiveness of thei poposed algoithm in compaison with the CPLEX solve. Tiolaee et al. (208a) developed a MILP model fo the multi-tip CAP in ode to minimize total cost in the scope of the uban waste collection. They consideed sepaate places fo depots and disposal facilities specific to the uban waste collection. They poposed an Impoved Max Min Ant System to solve well-nown test poblems and lage-sized instances. Theefoe, the innovations of this eseach ae descibed below:

3 6 Waste Management & eseach 37() Supplement. Developing a novel waste collection model fo the eal-wold applications by consideing a maximum available time fo vehicles in addition to the possibility of having multiple tips had and soft time windows fo coveing all demand nodes accoding to the pioities of sevice, and sepaate locations fo the disposal site and the depot. 2. Poposing a constuctive efficient method fo geneating initial solutions, and subsequently applying an efficient SA to impove them. 3. Investigating a case study in ode to demonstate the applicability of the poposed model and to detemine the optimal policy. In the following: the poposed mathematical model is pesented in the second section; in the thid section, the poposed solution method is intoduced; in the fouth section, the numeical esults fom the implementation of the algoithm ae pesented; and finally, the conclusions and suggestions fo futue wos ae given in the fifth section. Poblem desciption and modeling This poblem involves obtaining the optimal numbe of used vehicles and the optimal outes of each vehicle in ode to minimize the total cost, which includes the usage cost of vehicles, tavesing cost though netwo edges, and violating penalty cost fom the pemissible time windows of sevices. The vehicles ae located at the depot (node numbe ) and stat thei tip towad the demand nodes (the nodes with waste), and afte filling to thei capacity, they should go to the disposal site (node n) in ode to unload the collected waste, get thei capacity bac, and stat thei next tip fom the disposal site towad the emaining demand nodes if possible. Finally, they will go bac to the depot in ode to complete thei sevice tou. In addition to capacity constaints, the maximum allowable usage time of each vehicle is also impotant; when the vehicles ae close to this maximum available time, they have to etun to the disposal site and eventually etun to the depot afte being unloaded. It is also impotant to conside time windows fo each demand node to uphold optimal uban waste collection policies. The assumptions of this poblem ae as follows: Each demand node is only seviced by one vehicle. Thee is one depot and one disposal site. The vehicles will stat fom the depot, go to the disposal site when filled, and stat again if thee is time. The vehicles ae heteogeneous and the gid is asymmetic. The vehicles will etun to the depot at the end of thei tip. Each vehicle has the maximum time of sevice. Each vehicle may have multiple tips, fist one fom the depot to the disposal site, and thei othe tips will stat fom the disposal site and again, end at the disposal site. The time and cost of a oute is the same fo all vehicles. Each demand node has a had and soft time window of sevice, which means that exit fom a soft time window will have a penalty, and exit fom a had time window is not possible. Figue. Multiple tips in a vehicle tou. Figue depicts the schematic poblem. Index is used to define vehicles multiple tips. If is, it epesents the fist tip, stating fom the depot to the disposal site. If is 2 o geate, the vehicle begins the tip at the disposal site and ends at the disposal site. This is shown clealy in Figue. In Figue, the bold lines demonstate the fist tip, and the dashed lines show the second tip of the vehicle, which ends at the disposal site and eventually etuns to the depot afte completing the tou. Linea Pogamming (LP) models have the ideal possibility of being solved using commecial softwae such as GAMS and LINGO (Mohammadi et al., 207; Tiolaee et al., 207). In the following, the poposed novel MILP model is pesented. Indices and sets NC S i, Paametes Set of nodes with demand (demand node) Set of total nodes Set of vehicles Set of vehicle tips Any optional subset of the demand node set Demand node index Vehicle index Tip index c i Cost of the tavesing edge (i, ) Pe Ealy sevice cost pe demand node Pl Late sevice cost pe demand node e i, l i Had time window of demand node i ee i, ll i Soft time window of demand node i W Vehicle capacity d Demand of node T max Maximum available time fo each vehicle M Optional lage numbe n The node indicating the disposal site and the numbe of total nodes in the netwo with the fist node being the depot ul Unit of loading time fo vehicles at demand nodes uu Unit of dischaging time fo vehicles at the disposal site t i Time fo tavesing edge (i, ) θ Convete coefficient of distance to cost (Ianian ial/m)

4 Babaee Tiolaee et al. 7 Vaiables tt i LT UT Ye i Yl i i x Aival time at demand node i Total loading time of vehicle in tip Total unloading time of vehicle in tip Amount of ealy sevice of the demand node i by vehicle Amount of late sevice of the of the demand node i by vehicle Ifvehicle moves fom node i to node = intip without seving node, 0 Othewise. i y () Ifvehicle seve node in tip = though tavesing edge ( i, ), (2) 0 Othewise. i i = = i= = = LT + UT + t x T, (2) xi S S \{, n}, S, (3) i S S = i n 2 n max x x, (4) + n { } x x 23,,,,, (5) tt = tt + t y NC, (6) i= = = ( i i ) i u = 0 If vehicle isused, Othewise. (3) tti = 0 i =, (7) Mathematical equations including obective function and constaints ae as follows: ei tti li i NC, (8) Minimize Z = θ + = i= NC i= = = = i c x i i i ( PeYe + PlYl ) + cv u = (4) Ye ee tt, i NC, (9) i i i Yl tt ll, i NC, (20) i i i subect to i i x = x i \{, n },,, (5) i = i= = = y NC, (6) NC x u, (2) in = i NC x u, (22) i = i= d y W,, (7) in + i NC { } x M( u ) 23,,,,, (23) i i y x i,,,, (8) i i= = = x Mu, (9) LT = ul d y,, (0) = i= i UT = uu d y,, () = i= i x n M ( u ) + { 23,,, },, (24) NC { } xi, yi, u 0,, Yei, Yli, tti, LT, UT 0 i,,,. (25) Equation (4) defines the obective function of the poblem fo minimizing the total cost including the usage costs of vehicles, tavesing costs, and the penalty costs fo violating fom pemissible time windows (soft time window). Equation (5) denotes the flow balance fo each vehicle. When a vehicle aives at a

5 8 Waste Management & eseach 37() Supplement specific node, it should leave fom that specific node. Equation (6) ensues that each demand node is only seviced by one vehicle. Equation (7) indicates vehicle capacity constaints. Equation (8) indicates that the demand node is seved by the vehicle which has aleady aived at it. In othe wods, a vehicle may coss a node but not seve it. Equation (9) indicates that the vehicle is used when its cost is paid. Equations (0) and () espectively epesent the total loading and unloading time fo each vehicle pe tip. Equation (2) indicates the usage time limitation fo each vehicle. Equation (3) eliminates sub-tous. Equations (4) and (5) ensue that the ode of vehicle tips numbe is fom to + in succession. Equations (6) and (7) calculate the aival time of vehicles at demand node. Equation (8) defines the had time window of sevice fo each demand node. Equations (9) and (20) espectively calculate the amount of ealy o late sevice fo violating soft time windows. Equations (2) and (22) ensue that the vehicle s fist tip begins at the depot and ends at the disposal site. Equations (23) and (24) indicate that, on the second, thid, and subsequent tips, vehicles will begin thei tip fom the disposal site and etun to the disposal site. Othe tips will stat fom the disposal site. Equation (25) specifies the types of the vaiables. Lineaization of Equation (6) i i tt = tt + t y NC, (6) i= = = ( i i ) i f tt i,,,, (26) i i i f M y i,,,, (27) f tt M( y ) i,,,, (28) i i vaious puposes, which indicates the high efficiency of SA fo solving outing poblems (Tiolaee et al., 208a, 208b, 208c). The following sub-sections descibe the mechanism of SA and the innovative algoithm to geneate the appopiate initial solutions. Innovative algoithm to geneate initial solutions An innovative andom geneato algoithm is developed to geneate the initial solutions; the steps to implement this method ae as follows: Step : Choose a vehicle andomly. The fist tip stats fom the depot. Step 2: Choose v nodes accoding to the time window, pioity (eligible) and the least distance to the depot among the emaining demand nodes, choose one on andom and then go to step 3. Step 3: If thee ae still any emaining demand nodes eligible to be sevices and added to the vehicle s tip, go to step 4, othewise, go to step 5. Step 4: The capacity and usage time limitation of the vehicle allowing them to be chosen, select one of them andomly and then go to step 3. If thee ae no eligible demand nodes, go to step 5 (the last consideed node is eligible to be seved only if the vehicle has enough time and capacity to seve it and then move bac to the disposal site). Step 5: Go to the disposal site and then go to step 6. Step 6: If all demand nodes ae coveed, go to step 7, othewise, update vehicle s capacity constaint. If the vehicle s time limit allows fo at least a tip fom the disposal site to a demand node, go to step 3, othewise, choose the next vehicle and go to step 2. Step 7: Stop the algoithm. The numbe of the geneated initial solutions is set to 500, and the best solution is epoted afte implementing the poposed SA. The flowchat of the poposed algoithm to geneate the initial solutions is shown in Figue 2. tt = fi + ti yi NC, (29) i i= = = f 0 i,,,. (30) Poposed solution Due to the complexity of the poblem, it is not efficient to solve it with pecise methods (Solomon, 987). Theefoe, an innovative algoithm fo geneating an initial solution and a SA is poposed to solve the poblem. Many studies have been caied out on using this algoithm to solve optimization poblems with SA The poposed SA is applied to impove the solutions and all the initial solutions ae individually impoved with this algoithm. SA is a local seach algoithm that has the ability to escape fom local optimums. This algoithm is vey efficient at solving nonconvex o discete optimization poblems. As the same way, SA is employed to solve intege pogamming poblems (Glove and ochenbege, 2006). Moeove, the simplicity of the implementation, the convegence popeties, as well as the hill climbing movements to escape the local optimum tap have led to the impovement of the initial solutions geneated in each iteation. The geneal famewo of the algoithm is that it stats with an initial solution the

6 Babaee Tiolaee et al. 9 Figue 2. Flowchat of the innovative algoithm fo geneating the initial solutions. initial paametes of the algoithm include the numbe of the algoithmic iteation at each tempeatue ( M ), the initial tempeatue ( T 0 ), the tempeatue decease ate (α ), the final tempeatue ( T end ), and the Boltzmann constant ( ) that ae initialized befoe the stat of the seach, then a neighbohood is consideed fo the initial solution. If the value of the geneated neighboing obective function is bette than the cuent obective function, the neighboing obective function is substituted as the solution and othewise, a andom numbe in the ange of [0, ] is compaed with the algoithm value if the andom numbe is less than the value of the algoithm, the solution is accepted. At each tempeatue, thee ae a numbe of iteations, and then the tempeatue deceases. The stop condition is to each the final tempeatue. Ultimately, the values of the paametes ae detemined using the tial and eo method, which ae shown as follows: M = 5, α = 098., T = 200, T =, = 08. (3) Local seach 0 The local seach methods fo constucting neighbohoods in the algoithm ae as follows: (A) Displacement: In this method, the location of one of the nodes in the constuctive oute sting (fom the depot to the disposal site and then fom the disposal site to the disposal site) changes (see Figue 3). (B) eplacement: In this method, the locations of two nodes in the constuctive oute sting ae eplaced (see Figue 4). (C) 2-Opt: In this method, the location of a pimay node and a final node on the path edges ae eplaced (see Figue 5. end

7 0 Waste Management & eseach 37() Supplement Figue 3. Displacement. Figue 4. eplacement. Figue 7. Simulated annealing pseudo-code. Table. andomly geneated samples. Figue 5. 2-Opt. Poblem numbe Total numbe of nodes Numbe of vehicles available in each poblem Figue 6. O-Opt. (D) O-opt: In this method, the position of the two nodes is changed simultaneously in the constuctive oute sting (see Figue 6). The implementation of the algoithm is shown by pesenting the pseudo-code in Figue 7. Computational esults In ode to evaluate the efficiency of the poposed algoithm, eight poblems with small and medium dimensions, and seven lage dimension poblems ae solved and the quality of the solutions is analyzed. Table povides infomation about the andomly geneated samples. To solve the poblem, a compute with a Coe i7-4720hq 2.60GHz pocesso and 8.00GB of AM is used, the poposed algoithm is coded in MATLAB softwae, and the poposed mathematical model is coded and un in GAMS softwae (CPLEX solve) as the suggested exact method. The obtained esults ae pesented in Table 2. The aveage values epoted fo the obective function and un time of SA is detemined afte P 7 2 P2 2 3 P3 5 3 P4 8 4 P P P P P P P 60 7 P P P P unning 0 times. Compaing the best solution fom the algoithm with the best solution so fa and calculating the pecentage of gap, it is found that the poposed algoithm geneates suitable esults in a shot computational time. As it is clea, fo the eight examples above, the eo ate in SA elative to the exact method is, on aveage, appoximately equal to %, which indicates the ability of the algoithm to find the appopiate solutions. CPLEX is not able to solve the P9 P5 within applying time limitation of 3600 seconds. In tems of un time, the aveage time equied to solve a poblem in small dimensions fo the exact method is about 205 seconds and fo the poposed algoithm is about 3.5 seconds. This infomation epesents the high speed of the poposed SA. Figue 8 shows this un time compaison.

8 Babaee Tiolaee et al. Table 2. Computational esults. Poblem Numbe of used vehicles CPLEX obective value CPLEX un time (seconds) Simulated annealing (SA) mean obective value SA mean un time (seconds) SA best obective value Gap (%) P P P P P P P P P P P P P P P Figue 8. un time compaison. Case study To study the applicability of the poposed mathematical model, a case study is conducted in a distict of the Sananda city using the poposed algoithm of the eseach. Pimaily, by eviewing the infomation in this distict, including the pocedue of waste collection, outing, and the numbe of equied vehicles, the cuent status of the waste collection system is caefully examined. This occus afte the equied infomation, such as the usage cost of the vehicles (i.e., the wages of dives and thei cew, and fuel cost), distance between netwo nodes, vehicle capacity, demand values of nodes and the time windows given fo specific nodes such as hospitals and clinics, has been povided by the municipality s expets. It should be noted that the distict has two clinics that need to be consideed as the pioity fo the sevice. It also has a 5-ton vehicle fo waste collection. The value of θ is also assumed to be 0 units. The demand fo each nomal node is also estimated at between 00 and 200 ilogams. Figue 9 shows the distict s aeial map. The distict has an aea of 330 squae ilometes and a vehicle paing as the depot, and has 43 nodes of demand. Numbes 23 and 43 indicate the nodes of the clinics, which ae estimated at 0 and 20 ilogams of waste, espectively. These demand nodes and vehicle paing ae specified in Figue 9. Moeove, the netwo gaph associated with this map is shown in Figue 0. Due to the availability of infomation, the distict begins the waste collection sevice fom 7:00 pm to 2:00 pm and all of the waste in the aea is collected by a 5-ton dump tuc. The outing of this vehicle in the cuent situation is in accodance with the ode of demand numbes whee the vehicle moves fom paing to the demand node, and then to the demand node 2, and so on.

9 2 Waste Management & eseach 37() Supplement Figue 9. Aeial map of the distict. Figue. Optimal outing fo the case study poblem. Table 3. The optimal solution obtained fom the case study. Optimal outing Obective value (Ianian ial) un time (seconds) Paing Disposal site Disposal site paing 55,050, Table 4. The compaing status of the cuent and optimal policies. Values Cuent status Optimal status Figue 0. Gaph netwo specifying the access outes fo demand nodes. Thee is no attention to the pioity of seving the clinical nodes in ode to pevent hazadous contamination. In this eseach, we implement ou poposed algoithm consideing the pioities and the othe input paametes. Afte solving the poblem, the esult is given in Figue chaacteized by its pioitization and outing schedule. The pioities of the optimal outing, the obective function, and the un time of the case study ae povided in Table 3. As it is obvious, the tuc can cove all the demands with two tips. As shown in Table 3, nodes 43 and 23 ae espectively the ninth and fifteenth pioities of the sevice. Pioity of node Pioity of node Obective value (Ianian ial) 63,500,000 55,050,000 To compae the obtained esults with the cuent state, Table 4 is pesented. As shown in Table 4, afte obtaining the optimal solution of the case study, the obective function has dopped 8,450,000 Ianian ial, which is equivalent to a 3.3% decease in the total cost. It should be noted that such savings ae obtained while the pioity of the clinics is egaded. It is obvious that the savings would cetainly be geate without these pioities. Conclusions and futue suggestions Optimal outing and vehicle allocating ae one of the impotant decisions of oganizations such as municipalities in the uban

10 Babaee Tiolaee et al. 3 waste collection, since optimal vehicles allocation and thei optimal outing can lead to a significant pecentage eduction of elated costs. In this pape, a MILP model is poposed fo heteogeneous multi-tip vehicle outing poblem with time windows specific to the uban waste collection, which aims to detemine the optimal sevice outes and the optimal numbe of the used vehicles. In ode to solve the poblem, a SA algoithm is developed based on the consideed assumptions in the poblem. The esults obtained by solving the poblem in small and medium sizes indicate that this algoithm offes nea-optimal solutions in compaison with the CPLEX solve. Finally, a case study is investigated in ode to evaluate the applicability of the poposed model in the eal wold. Afte analyzing the case study, it has been demonstated that the poposed model can impove the cuent status by geneating a 3.3% decease in the total cost. Fo futue wo suggestions, the demand paamete can be consideed uncetain in the model and the othe metaheuistics such as ant colony optimization and genetic algoithm can be tested to solve the poblem as a ival fo SA. Declaation of conflicting inteests The authos declaed no potential conflicts of inteest with espect to the eseach, authoship, and/o publication of this aticle. Funding The authos eceived no financial suppot fo the eseach, authoship, and/o publication of this aticle. Notes. See 2. See popgaph.php/ OCID ids Efan Babaee Tiolaee Seyed Ali Ghaffaian efeences Angelelli E and Speanza MG (2002) The peiodic vehicle outing poblem with intemediate facilities. Euopean Jounal of Opeational eseach 37: Babaee Tiolaee E., Alinaghian M, Bahshi Sasi M, et al. (206) Solving a obust capacitated ac outing poblem using a hybid simulated annealing algoithm: A waste collection application. 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