Rollon-Rolloff Vehicle Routing Problem. in the Waste Collection Industry

Transcription

1 Rollon-Rolloff Vehicle Routing Problem in the Waste Collection Industry Juyoung Wy 1, Byung-In Kim 2 Industrial & Management Engineering, POSTECH (Pohang University of Science & Technology) San 31, Hyoja-Dong, Pohang, Kyungbuk, Republic of Korea hwiriric@postech.ac.kr 1, bkim@ postech.ac.kr 2 Seongbae Kim 3, Surya Sahoo 4 Institute of Information Technology, Inc FM 1488, Magnolia, TX 77354, USA skim@e-iit.com 3, surya@e-iit.com 4 Abstract - We introduce the rollon-rolloff vehicle routing problem which is a variant of vehicle routing problem. In this problem, large s are at demand locations such as construction sites. Tractors move a at a time between demand locations and disposal facilities and storage yards. The objective is to minimize the number of required tractors and the total travel time of tractors. The problem considered in this paper has complicated constraints including multiple disposal facilities, multiple continaer storage yard locations, seven service types of demands, time windows of each location, available size, type and lunch break for the tractor drivers. We propose a heuristic approach consisting of several algorithms for the problem and conduct experiments using benchmark data sets of two types. The benchmark data are derived from real world waste collection problems and generated by the authors. The effectiveness of the proposed methods is demonstrated by computational experiments. Keywords: rollon-rolloff, waste collection, vehicle routing problem 1. INTRODUCTION There are three types of waste collection problems: residential, commercial and industrial waste collection (Golden et al., 2002). The residential waste collection involves servicing private homes. Vehicles move along the street to collect and dispose trash. For the commercial waste collection waste problem, vehicles visit restaurants, strip malls and commercial areas to collect and empty full s. The industrial waste collection problem has customers that require large level services such as construction site, downtown areas and large shopping malls having a large amount of garbage. A typical commercial is eight loose yard, while the for industrial waste may range twenty to forty yard. For the reason, a tractor can transport only one at a time (Kim et al., 2006). This paper handles the third problem. A Tractor conveys a between customers (demands), disposal facilities (landfills), storage yards (yards) and depot for satisfying requirements of demands. Each demand requests a service such as ordering an empty to dump garbage, changing size of a and collecting of a full. The industrial waste collection problem has been defined as the skip problem, the rollon-rolloff collection and rollon-rolloff vehicle routing problems (rollon-rolloff VRP) in previous researches. We also call the problem of this paper the rollon-rolloff VRP. The objective is to minimize the number of required tractors and the total travel time of tractors to service all of given demands. For the problem, we present a lower bound calculation method and propose an iterative heuristic method. The remainder of this paper is organized as follows. After we briefly review the related literature in section 2, the problem is described in detail in section 3. A lower bound calculation method and a proposed method are presented in sections 4 and 5, respectively. Some experimental results are given in section 6. Section 7 provides concluding remarks. 2. LITERATURE REVIEWS This section briefly reviews the literature on waste

2 collection problems, particularly related on the rollonrolloff VRP. Golden et al. (2002) introduce the waste collection problem as one of numerous applications of vehicle routing problem. Eisenstein and Iyer (1997) suggest the dynamic algorithm for the residential waste collection problem. They use stochastic models to uncertainty of amount of garbage and service time. Sahoo et al. (2005) propose a heuristic method using GIS and clustering method for the residential and commercial waste collection problems. They could significantly reduce the operational cost of a real waste collection company by the proposed method. Tung and Pinnoi (2000) present mathematical models and a heuristic method for the residential waste collection problem and apply the method to an actual problem. Their method resulted in more efficient routes than the real world practice. Kim et al. (2006) define the commercial waste collection problem as a variant of vehicle routing problem with time window (VRPTW) and use an extended insertion algorithm. They present a clustering-based waste collection VRPTW algorithm for vehicle work load balancing and the visual attractiveness of routes. Meulemeester et al. (1997) use two service types of demands for the rollon-roloff VRP. A landfill for a demand of service type 1 is located at the same place as the depot while a landfill for a service type 2 is far away from the depot. While there is a dedicated landfill for each demand in their problem, there are multiple eligible landfills for each demand in our problem such that we need to select a proper landfill for them. Bodin et al. (2000) study the rollon-roloff VRP which has a landfill and a yard with an infinite number of s. However, there are a lot of yards and the available number of s in each yard are limited in this paper. Therefore, if there is no a necessary in the closest yard from the current location of tractor, the yard cannot be used and we should consider another yard. In addition, Bodin et al. (2000) classify demands by four service types and propose heuristic methods for the problem. Type 1, type 3 and type 4 are defined as asymmetric vehicle routing problem and type 2 is introduced as bin packing problem. They suggest algorithms for each separated problem and integrate the algorithms. Baldacci et al. (2006) research the rollon-rolloff VRP with multiple disposal facilities and multiple yards as our problem. However, they do not consider the type of landfill and the available number of s in each yard and use five types of demands. The problem considered in this paper has complicated constraints including multiple disposal facilities, multiple storage yards, seven service types of demands, time windows of each location, available size, type and lunch break for the tractor drivers. In addition, some service types of demands can be changed to another service types if the solution of changed type is better than those of original type. To the best of our knowledge, no previous works have handled this kind of complicated problem. 3. ROLLON-ROLLOFF VEHICLE ROUTING PROBLEMS (RR-VRP) In this problem, demand locations such as construction sites, large shopping centers, and downtown areas are defined as customers. Each demand requires a type of service. Table 1 shows service types of demands that is defined by a real waste collection company in the U.S. Pattern in the table presents the movement of a tractor for the service type. For instance, to serve a BTY customer, a tractor needs to move an empty from the customer to a yard. A bold part in the pattern, C-Y, means fundamental routes for the service type. Locations of a brace (e.g., ) are available places where the tractor can depart and arrive before and after visiting the fundamental routes. The service type FFY that requires picking up and emptying a full from yard is preferred to be served first because the demand was not completed in the previous day. In addition, some service types of demands can be changed to another service types. E/R can be conducted as S/O if the demand allows changing of service type and the solution by S/O is better than those of E/R. Basically, a tractor pickups a full from the demand location to a landfill, empties it in the landfill and returns it to the demand for the E/R. However, if the demand requires to get a same size not the same, the tractor transports an empty of the same type to the demand and changes with the full one as the demand of S/O. A Tractor transports a between demand locations (demands), disposal facilities (landfills), storage yards (yards) and the depot for satisfying demands of customer. A tractor can carry only one at a time since the size is large. We assume that there are a single depot and multiple landfills and multiple yards. Additionally, types of landfills are given for each demand and the tractor chooses a landfill that has the minimum cost to empty a full but some demands have to use a specific landfill. In addition, the problem has time window constraints of the tractors, demands, landfills, yards and the depot. Each tractor has a variable length of the available work time. A lunch break time within a proper time window need

3 to be considered. However, if a tractor goes back to the depot before the latest time of lunch break time window, the break can be omitted. As shown in Figure 1 (c), the lunch break cannot be taken between the fundamental route for a demand. D Type of service BTY 1 DEL 2 DNR 3 E/R 4 FFY 5 REL 6 S/O 7 E/R C L C (a) A route for a tractor Figure 1: Example of a route Table 1: Service types ID Meaning Pattern Bring to Yard - Bring empty from customer to yard Delivery - Deliver an empty to customer - Need to arrive at the customer with an empty Do Not Return - Remove a full from the customer to a landfill Empty and Return - Pick up a full, empty it in a landfill, return it to the customer Full From Yard - Pick up a full from yard (the demand location is same with yard location), empty it Relocate - Change the location, only have service time Switch Out (Exchange size of ) - Container size change. - Bring an empty and change with the full one - Need to arrive at the customer with an empty ** D: Depot, L: Landfill, C: customer(demand), Y: yard C-Y- (empty)-c- C-L- C-L-C- Y(C)-L- C-C - (empty)-c-l- The aim of this research generates routes to minimize the number of required tractors and the total travel time of S/O Y C L (b) A route to be added S/O Y C L Lunch (c) Impracticable position of lunch break tractors to serve all of given demands satisfying constraints. 4. LOWER BOUND This section describes a lower bound for the rollonrolloff VRP. The lower bound is modified from LB1 of Bodin et al. (2000). LB1 is the sum of total service time and minimum total travel cost of a solution. The minimum total travel cost is obtained by a partial graph (PG) problem. The service time of a demand is the sum of travel time between fundamental routes and loading/unloading time of a in each location of fundamental routes. The total service time means sum of service time of all demands. The travel cost is the sum of travel time between demand locations and the PG problem provides minimum total travel cost in a feasible solution. In our problem, service time for servicing a demand and travel cost between demand locations in a solution may vary according to the service sequence of demands and which landfill or yard is used for the demand. 5. SOLUTION APPROACHES Most previous researches for waste collection problems suggest heuristic algorithms for solution approaches due to complexity of the problem. Because the problem of this paper has more complicated constraints we also propose heuristic methods. 5.1 Construction algorithm Figure 2 shows a procedure of the algorithm for generating an initial solution. The algorithm selects an available tractor randomly and initializes the tractor. In Step3, fundamental routes of the unserved demands are created and service time and travel time for the route are calculated. Candidate list of demands is made by the fundamental routes of unserved demands as shown in Figure 3. If the service type of demand is one of DNR, E/R, FFY, S/O, fundamental routes need a landfill having minimum cost as Figure 1 (b). Also, the time window of all locations should be satisfied. The tractor spends time for loading and unloading a and the time depends on each location. The fitness of demand is calculated according to service time, travel time, waiting time to be required for the demand in Step3. The algorithm uses the roulette method that chooses a demand to be served next from candidate list according to fitness. The fitness values are a weighted sum of waiting time due to the time window of the demand and travel time from current location of the tractor to the next demand (Step4).

4 Figure 2: Generation algorithm for an initial solution Step1. Select and initialize an available tractor. Step2. If a demand which is not served is remained go to Step3, else go Step8. Step3. Generate candidate list of unserved demands. If the list is empty go to Step8, else go to Stop4. Step4. Select a demand to serve from the candidate list and remove the demand from the list. Step5. Check the service feasibility of the selected demand. If the service of the demand is available go to Step6, else go to Step4. Step6. Add the demand to the tractor and update the data of tractor such as travel time and service time. Step7. Check availability of conducting lunch break. If the lunch break can be inserted to the tractor, add the lunch break. Go to Step2. Step8. If there is over a demand which is not served the tractor goes back to depot, selects and initializes an available tractor and go to Step3, else go to Step9. Step9. Terminate the algorithm. In some cases, a yard should be added into the fundamental routes for servicing the next demand. For example, after conducting demand E/R, the tractor needs to visit a yard to get an empty to serve next demand S/O as shown in Figure 1 (a). We set the priority for the FFY since the FFY prefers to be served at the beginning of route. Therefore, it has higher priority to be selected first than others. Some E/R demands can be changed to S/O. We use methods such as l using E/R, 2 using the type with minimum cost between E/R and S/O, 3 using randomly selected type between E/R and S/O, 4 using both E/R and S/O for candidate list, 5 using randomly selected method among above 4 ways (l~4) to generate fundamental routes for E/R which can be changed to S/O. And then we adopt best one among results of 5 cases (l~5). Candidate list candidates Visiting locations C andidate_list BTY (ID :1234) E/R (ID :1243) S/O (ID :1245) D EL (ID :1224) D EL (ID :1224) Figure 3: Example of candidate list Y (ID :1234) L (ID :134) E/R (ID :1243) L (ID :134) E/R (ID :1243) S/O (ID :1245) L (ID :138) 5.2 Improvement algorithms In this paper, several algorithms are proposed to improve solutions Changeable E/R algorithm If a demand E/R can be served as S/O, this improvement algorithm is performed as following. If the demand is served as E/R in the solution, try to change the type to S/O. When the changed solution is improved, update the solution. If the demand is served as S/O in the solution, try to change the type to E/R. When the changed solution is improved, update the solution Inter-route improvement algorithms We use three inter-route improvement algorithms: 1-1, 1-0, and combining two routes algorithm. The first one exchanges a demand from two routes. The second one moves a demand from a route to another. The third one selects a pair of routes and tries to combining two routes to one route when the number of customers is less than a certain number (e.x. ten) Intra-route improvement algorithms We use a two-opt and a searching full enumeration algorithm. The former changes the sequence of demand of a route to improve the solution. The later considers all available sequence of demand of each route to improve the solution. However, this algorithm is conducted when the number of demands which are included in a route is less than a certain number (e.x. ten) since it is time consuming works Integrated algorithm The integrated algorithm consists of an initial solution generation algorithm and solution improvement algorithms as shown in Figure 4. After an initial solution is generated by the generation algorithm, the solution is improved by changeable E/R, inter-route, intra-route improvement algorithms. Searching full enumeration algorithm and combining two routes algorithms are conducted after a certain number of iteration.

5 Figure 4: Integrated algorithm procedure 6. EXPERMENTAL RESULTS 6.1 Experimental methods and Data We test effectiveness of proposed methods using data set of two types, A# and B#. One of them is derived from a real waste collection company in the U.S and the other one is artificially generated. Table 2 shows the summary of data sets. All of data have time window constraints of locations and lunch break constraints and include all of service types of demand. The length of workday of each tractor is different. The number of iteration is limited to 500 for the integrated algorithm. The Experimental environments is MS Windows7, CPU 2.66GHz, 6.00GB RAM and the used program languages is C++. No. Data ID Table 2: Experimental data set demand Start Generation of initial solutions Changeable E/R demand Inter-route: 1-1 Inter-route: 1-0 Intra-route: Changing sequences Termination criterion? Yes Intra-route: Searching full enumeration Inter-route: Combining two routes Terminate yard landfill A1 Nor_no A2 Nor_no A3 Nor_no A4 Nor_no A5 Nor_no A6 Nor_no A7 Nor_no No Description Lunch break A8 ER_no A9 ER_no A10 ER_no E/R change A11 FL_no A12 FL_no Fixed landfill B1 Nor_ B2 Nor_ B3 Nor_ B4 ER_ B5 ER_ B6 ER_ B7 FL_ B8 FL_ B9 FL_ Lunch break E/R change Fixed landfill B10 All_ B11 All_ E/R change, B12 All_ Fixed landfill 6.2 Experimental results Table 3 presents the results of experiments. The first column is lower bound. In each column, tractor means the number of used tractors and total travel time is sum of travel time of total used tractors. Company(A) and Heuristics (B) represent the results by specialists of waste collection company and by the proposed methods, respectively. For data type A, the heuristic method decreases a required tractor of Nor_no1, Nor_no6, FL_no2 and a few of total travel time for all of data. ER_no1 and ER_no2 have the same data with Nor_no1 and Nor_no2, respectively, except that they include some changeable E/R demands. Because the suggested method changes service type from E/R to S/O, the results of ER_no1 and ER_no2 are better than Nor_no1 and Nor_no2, respectively. The real waste collection company provides infeasible solutions for Nor_no7 and ER_no2. It means that generating feasible solutions by specialists is so difficult since the problem includes many complicated constraints such as time windows, landfill type, size and type, the number of s in each yard, lunch break and so on. However, the method of this paper can offer feasible solutions stably, therefore confusion of the tractor drivers by infeasible route will be diminished. We test the proposed method using data set B with large demands. Nor_#, ER_#, FL_# and ALL_# have the same data but with a few difference constraints. ER_# involves changeable E/R and FL_# includes the fixed landfill constraint for some demands. All_# have all of the constraints. When the data involves a demand with the fixed landfill constraint the solution is getting worse since the demand cannot use the cheapest landfill. When the data having changeable E/R obtains better solutions than Nor_# because the changeable E/R option is used when the solutions are improved by the constraint. Since the data type B are generated by ourselves we only compare the results of heuristics to the lower bounds.

6 The last column shows the gap of total travel time between the lower bound and the results of company and heuristic methods. The feasibility of time window constraints is not considered in the lower bound. The lower bound will be tighter if the time window constraint is considered in the lower bound calculation. This is one of our future works. No Data ID Table 3: Experimental results Lower bound (LB) Total travel tractor time (sec) Company (A) Heuristics (B) Gap tractor Total travel time (sec) tractor Total travel time (sec) (A) LB (B) LB A1 Nor_no A2 Nor_no A3 Nor_no A4 Nor_no A5 Nor_no A6 Nor_no A7 Nor_no A8 ER_no A9 ER_no A10 ER_no A11 FL_no A12 FL_no B1 Nor_ B2 ER_ B3 FL_ B4 All_ B5 Nor_ B6 ER_ B7 FL_ B8 All_ B9 Nor_ B10 ER_ B11 FL_ B12 All_ REFERENCES Baldacci, R., Bodin, L., Mingozzi, A. (2006) The multiple disposal facilities and multiple inventory locations rollon rolloff vehicle routing problem, Coumputers & Operations Research, 33, Bodin, L., Mingozzi, A., Baldacci, R., Ball, M. (2000) The rollon-rolloff vehicle routing problem, Transportation Science, 34, Eisenstein, D.D. and Iyer, A.V. (1997) Garbage collection in Chicago: a dynamic scheduling model, Management Science, 43( 7), Golden, B.L., Assad, A.A., Wasil, E.A. (2002) Routing vehicles in the real world: applications in the solid waste, beverage, food, dairy, and newspaper industries. In: P. Toth, D. Vigo, editors. The vehicle routing problem. Philadelphia, PA: SIAM; Kim, B., Kim, S., Sahoo, S. (2006) Waste Collection Vehicle Routing Problem with Time Windows, Computers & Operations Research, 33(12), Meulemeester, L. De., Laporte, G., Louveaux, F.V., Semet, F. (1997) Optimal sequencing of skip collections and deliveries, Journal of the Operational Research Society, 48, Sahoo, S., Kim, S., Kim, B.-I., Kraas, B., Popov Jr, A. (2005) Routing optimization for waste management, Interfaces,35( 1), Tung, D.V. and Pinnoi, A. (2000) Vehicle routingscheduling for waste collection in Hanoi, European Journal of Operational Research, 125, AUTHOR BIOGRAPHIES 7. CONCLUSIONS In this paper, we introduced the rollon-rolloff vehicle routing problem with complicated constraints such as multiple landfills, multiple yards, landfill types, seven types of demand, time window, size and type, lunch break of drivers and so on. Since the complexity of the problems we proposed heuristic approaches and test the heuristic methods using benchmark data set. The proposed methods could generate better solutions than those of specialists of waste collection company. For the future works, we can consider the lower bound considering time window and extending the problem with the on-call constraints. Juyoung Wy is a Ph.D. candidate at the Department of Industrial & Management Engineering (IME), POSTECH (Pohang University of Science and Technology), Republic of Korea. Byung-In Kim is an Associate Professor in IME, POSTECH. He received a Ph.D. from RPI, NY, USA. Seongbae Kim is the Chief Technology Officer of Institute of Information Technology, Inc., Magnolia, Texas, USA. He got a Ph.D. from Texas A&M University, TX, USA. Sahoo Surya is the Chief Executive Officer of Institute of Information Technology, Inc., Magnolia, Texas, USA. He got a Ph.D. from University of Memphis, TN, USA.