A Generic Algorithm for Waste collection Vehicle routing problem with Timewindows and Conflicts

Thai Tieu Minha, Tran Van Hoaib,

aHo Chi Minh City Institute of Physics, Vietnam Academy of Science and Technology, 1 Mac Dinh Chi, Ho Chi Minh City, VietnambDepartment of Computer Science & Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet, Ho Chi Minh City, Vietnam

Abstract

In this paper, we address an application of vehicle routing problem (VRP) in the real life, namely waste collectionproblem. Constraints of waste collection problems considered are not only time windows but also conflicts betweenproperties of the waste. A combination of flow and set partitioning formulation is suggested to model the problemin case of multi-objective optimization. In order to minimize the total traveling time and number of vehicles ofsolution, we propose using a Generic Algorithm (GA) with -interchange mechanism [1]. The -interchange operatoris improved for compatible with new sub-routes construction and the decoding process is also modified to can acceptthe solution after decoding is not the same as the original solution before decoding.

In experiments, we use the set of benchmark data in Kim et al. [2] works with some modification. Values ofwaste properties are attached into the benchmark data, along with a conflict matrix presenting conflict between theproperties. Computational result show that our GA approach can are competitive to best-published results of thevehicle routing problem with time windows. The algorithms become more useful in the vehicle routing problem withtime windows and conflict (VRPTWC) in which the conflicts are important details. Finally, we compare the solutionof GA with others meta-heuristic algorithm and evaluate the influence of the conflict between the waste properties onthe solutions.

Keywords: Waste collection, Vehicle routing problems, Conflicts, Heuristics

1. Introduction

VRPTWC is a new variant of VRP, its constraints area combination of VRP with Time Windows and VRPwith Conflicts. The time windows constraint will limitthe time vehicles visit the stop. The conflict betweenwaste properties will limit the amount of commodityin each vehicle. Even though the weight of waste inthe vehicle is nearly empty, the vehicle cannot collectmore the waste if groups of between the waste are con-flict. These constraints are often encountered in the haz-ardous materials transportation problem or the industri-ous waste collection. In the practice, the problems ismore complex when more detail are added such as thereare more than one landfill locations (disposal facilities)or the lunch break at noon of the drivers. The wastecollection activities in our problem are described below.

A company is in charge of collecting the hazardous

Corresponding author. Tel:Email address: hoai@cse.hcmut.edu.vn (Tran Van Hoai)

industry waste in a city. Infrastructure of company in-clude depot where store vehicles and landfills or treat-ment stations where the waste disposed. Vehicles arebegin working day at the depot and after working daythey return the depot. The waste will be vehicle col-lected at factories (we call factories are stops). Dependon working time at each factory, we have windows timeframe which vehicles must visit on time. Waiting timewill happen if the vehicles visit the stop before the ear-liest time of windows time frame. However we cannotvisit stop after the latest time of windows time frame.Each vehicle, there are the total traveling distance lengthconstraint and the amount of commodity constraint forit.

About waste conflict constraints, base on informa-tion about hazardous waste properties of Decision No.23/2006/QD-BTNMT of Ministry of Natural Resourcesand Environment Vietnam 1, the hazardous waste prop-

1http://www.chatthainguyhai.net/documents/qd_

23-2006-qd-btnmt.pdf

Preprint submitted to Ten tap chi December 22, 2012

Table 1: Properties of waste and matrix conflict between properties

Group Characteristic property Matrix conflict

1 2 3 4 5 6 7

1 Flammability x2 Explosive x3 Oxidation x4 Corrosion x5 Infection x x6 Toxicity x7 Biochemical toxicity x

erties are classified into 7 groups and in each group hasone characteristic property, see Table 1. The waste iscollected in the factories in of the above groups and itcan not store together with others group in the sameroute if conflict. This constraint is released if only ifthe vehicle return landfill and release the waste. Con-flict ruler between types of the waste is described inmatrix conflict in Table 1. Base on the matrix conflictwe can identify the property conflict pairs. Example,the group 1 has flammability property cannot store andtravel along with the waste belong to group 2 has explo-sive property.

In the mathematical viewpoint, the conflict con-straints in the waste collection VRP are difficult to rep-resent formulation. These constraints are not exactlythe same for each sub-route, each route and each solu-tion in waste collection VRP. Belong to the propertiesof the waste collected, the constraints will limit someother properties in sub-route until the waste disposed onthe landfill. We call these constraints are partial con-straints, because they only appear and disappear in eachsub-route. Addition, one landfill may be visited manytimes to dispose the waste. Thus we cannot representthe problem as a model that all vertices have just onein-degree and one out-degree. We can summarize all ofconstraints and objectives of problem following:

Constrains:

Capacitive constrain in the vehicle. Route length constrain in each vehicle. Conflict type of the waste constrain in subroute. Time windows constrains in the stop.

Objectives:

Minimize number of vehicles. Minimize travel time.

Minimize wait time. Minimize toltal route length.In order to solve the problem with multi-objectives

and complex constraints, we propose using a hybridgenetic algorithm. The algorithm was implementedand developed base on combination meta-heuristic tech-niques. First, we improve some basic heuristics forbuilding the set of initial solutions. After that, meta-heuristics technique is used for improving set of solu-tions and exploring the better solutions. In our GA,we combine the global technique and local search tech-nique to create the new hybrid algorithm. Global searchtechnique help to explorer the new potential space insearch space while local search technique help to im-prove the quality of solution.

The paper proceeds as follows. Section 1, we intro-duce about the problem VRPTWC and its constraints.Section 2, we review the existing work relate the prob-lem and Section 3 we represent some definitions, thestructure of the solution and the mathematical model.Detail about generic algorithm are discussed in Section4. Section 5, we compare our experiments with the re-cent studies and Section 6 we provide conclusions.

2. Literature review

In this section, we review the previous studies on ve-hicle routing problem with time windows (VRPTW),vehicle routing problem with conflicts (VRPC), heuris-tic and meta-heuristic methods.

Early research on VRPTW is started by Pullen andWebb [3], Savelsbergh [4], Knight and Hofer [5], Mad-sen [6]. Very limited work has been done on opti-mal approaches in solving VRPTW of practical size,while heuristics have been found to be more effectiveand efficient, Solomon and Desrochers [7]. Solomon[8] proposed benchmark problem sets for the VRPTWand heuristics as savings heuristics, time-oriented andnearest-neighbor heuristics, insertion heuristics andtime-oriented sweep heuristics. These heuristics con-tinue to be used in works [9], [2], [10].

Desrochers et al. [11] introduced mathematical for-mulation for the VRPTW problem with O(n2) variables.Optimization algorithms as branch and bound, dynamicprogramming and set partitioning are used to computelower bound of the problem. Addition, three types ofapproximation algorithms have been widely applied tounconstrained routing problems are construction meth-ods, iterative improvement methods and incomplete op-timization are reviewed.

2

Braysy and Gendreau [10] discuss how heuristicmethods should be evaluated and propose using the con-cept of Pareto optimality in the comparison of differentheuristic approaches in VRPTW. Both traditional so-lution construction and improvement techniques werecomprehensively reviewed. Usually, VRPTW heuristicsare measured against two criteria: solution quality interms of objective function value and speed but in theirresearch, simplicity of implementation, flexibility androbustness are also essential attributes of good heuris-tics.

Tung and Pinnoi [9] apply techniques of VRPTWfor waste collection activities in Hanoi, Vietnam. Theproblem is complicated by several time windows, inter-arrival time constraints at each customer point and mul-tiple landfill locations. They formulate the probleminto a mixed-integer program and propose a heuristicprocedure consisting of construction and improvementphases. Besides that, they modify the Solomons I1heuristic and improve phase combines the power of Or-opt and 2-opt together.

Kim et al. [2] research continue extending theVPRTW for Waste Management. Set of landfill loca-tions are considered in the problem instead of there isonly one landfill location in [9]. Addition, the problemhas the lunch break constraints, vehicles must stop withan hour from 11 AM to 12 PM. For this problem, Kimet al. present a shape metric Sm in order to quantify theroute (or cluster) compactness and develop a new algo-rithm is clustering-based waste collection VRPTW.

Zhu [12] represents a genetic algorithm (GA) to solveVRPTW. In clustered datasets, he uses intuitive integerstring to represent chromosomes and incorporates sev-eral new crossover operations and other techniques suchas hybrid hill-climbing and adaptive mutation schemeusing statistical measures to achieve good results.

The paper of Hamdi-Dhaoui et al. [13] first timedeveloped heuristic method for Vehicle Routing Prob-lem with Conflicts. This study presents a mathemati-cal model, lower bound for the problem. Heuristics areILS (Iterated Local Search), GRASP-ELS (Greedy Ran-domized Adaptive Search Procedure - Evolutionary Lo-cal Search) also introduced in their study.

3. Definitions and model

Considering a complete graph G = (V, E) has V ={0, 1, ..., n + m 1, n + m} is the set of vertices includedepot, landfills and stops and E as the set of edges inV . We note the depot as vertex 0, the set N of stopsare vertices from 1 to n, and the set M of landfills fromvertices n + 1 to n + m. A complete graph is a simple

Figure 1: Represent Manhattan distance and Euclidean distance inGoogle Map.

undirected graph in which every pair of distinct verticesis connected by an unique edge.

Distance between vertices are used is Manhattandistance instead of Euclidean distance. In our opin-ion, Manhattan distance represents more accurately themoving path length that vehicles traveled in the city, seeFig 1. With 2 points i(xi, yi) and j(x j, y j), the distancedi j is:

di j = |xi x j| + |yi y j| (1)We define S is the set of feasible solutions of the prob-

lem in the searching space. A solution S is a set ofroutes = {r1, r2, ..., rk1, rk}, which route ri is the i-throute of the solution and traveled exactly by one vehicleand k = ||s|| is number vehicles used in the solution. Theroutes are starting and ending at the depot. Howeverthey are not Hamiltonian cycles 2 as same as the normalVRPTW because one landfill can be visited many times,see Fig 2.

A route is divided into several sub-routes ri ={i1, i2, ..., ipi }, with ij is sub-route j-th and pi = ||ri||is number sub-routes in the route i-th of the solution. Asub-route ij is a path begins when the vehicle start col-lecting the waste and ends when it release the waste orreturn the depot. Types of structure of sub-route of theroute is illustrated in Fig 3.

2Hamiltonian cycle is a path in an undirected graph that visits eachvertex exactly once aside from the necessary repetition of the start andend vertex.

3

Figure 2: An instance of the solution in watse collection VRPTWC.

Figure 3: Three types of sub-route in waste collecting VRPa) The starting sub-route i1: is the first sub-route in route, it starts atthe depot and ends at the landfill.b) The normal sub-route ij (1 < j < pi): start at the landfill and finishat other landfills. If two landfills in sub-route are the same, we have asimple cycle.c) The ending sub-route ip: is the last sub-route of the route, it startsat the landfill and ends at the depot. Before return the depot vehiclemust visit the landfill to dispose the waste.

At each stop or landfill in the set V of vertices hasone time window frame [ei, li] that represent the earliesttime and latest time vehicles can visit the vertex. At thedepot, time window frame [e0, l0] represent the time thatvehicles start collecting waste and the time that vehiclesmust return to depot.

Let ui denote the demand to collect at the vertex i andsi denote the service duration at the vertex i. Depot andlandfill have not the demand. The di j is the Manhattandistance and ti j is the moving time from vertex i to jcorresponds to edge (i, j) E. K is the set of existingvehicles in depot, W is the amount of commodity limitof the vehicles and L is maximum distance which thevehicle can move in per day. Let ai j be a binary coeffi-cient that takes value 1 if waste property i conflict withwaste property j, otherwise takes value 0. We definesome decision variables:

xki j = 1 if and only if the edge (i, j) is traveled byvehicle k and otherwise xki j = 0.vki is the starting service time at the stop i serviced by

vehicle k.

wki is the amount of commodity in vehicle k after visitstop i.

As described above, objectives of problem are min-imize the total length of routes, the waiting time, themoving time and number vehicles, that means the prob-lem has multi-objectives. There are many ways tosolve this problem in which weighted sum of objectivefunctions method is the most popular approaches. Themethod transforms the global objective function fromthe vector space to the scalar space. The objective ofthe problem:

min(1

kK

(i, j)E

ti jxki j + 2kK

(i, j)E

di jxki j

+ 3kK

(i, j)E

(ski ei)xki j)

(2)

1, 2, 3: the weights assigned to each objective inorder to express the relative importance the correspond-ing objectives.

Constraints: kK

(i, j)E

xki j = 1,k K (3)

Nj=1

xk0 j 1,k K (4)

N+Mj=N+1

xkj0 1,k K (5)

Nj=1

xkj0 = 0,k K (6)

Ni=1

xki j =Ni=1

xkji = 1,k K, j V (7)

N+Mi=0

xki j N+Mi=0

xkji = 0,k K, j V (8)kK

(i, j)E

xki jdi j L (9)

wki + u j wkj (1 xki j)M,k K,(i, j) E (10)

0 < wki W,k K,i N (11)

wki = 0,k K,i {0} M (12)4

E ski L,k K,i {0} M (13)

vki + si + ti j vkj (1 xki j)M,k K,(i, j) E (14)

eiN+Mj=1

xki j ski liN+Mj=1

xki j (15)

Let = {1, 2, ..., q} denote the set of all feasiblesub-routes in G, each element r is sub-route of G.We define the binary variable yrki j be equal to 1 if andonly if edge (i, j) r and traveled by vehicle k. Theconflict constraint:

qr=1

ai jyrki j = 0,k K,(i, j) E (16)

4. Generic algorithm

Our implement GA is divided into five major stepsor phases: initial population, selection, crossover, mu-tation and adaptation, in which first four step as seemas the traditional GA. The evolution of population isdriven by the selection, recombination, mutation phasesrespectively while adaptation phases helps individualsimprove themselves. Thus, the idea of the algorithm notonly base on biological evolution but also base on cul-tural evolution 3.

Algorithm GA(1) INITIAL population pop(2) SET cont = stop condition(3) while cont != true do(4) pop = naturalSelect(pop)(5) pop = recombine(pop)(6) pop = mutate(pop)(7) pop = adapt(pop)(8) cont = Update stop condition(9) FINISH

4.1. Initial population

In order to setup the initial population, we use theideal is proposed by D.V. Tung and A. Pinnoi research.Authors start from Solomons I1 Insert heuristic andmodify it by adding criteria in Eq 17 to Eq 19.

3Cultural evolution, also called sociocultural evolution, the de-velopment of one or more cultures from simpler to more complexforms http://www.britannica.com/EBchecked/topic/146212/cultural-evolution

C1(i, u, j) = 1C11(i, u, j) + 2C12(i, u, j) (17)

C11(i, u, j) = diu + du j di j (18)

C12(i, u, j) = v ju v j (19)

C2 = d0uC11 C1(i, u, j) (20)in which:di j: distance from vertex i to vertex jv ju and v j: starting service time at vertex j after and

before insert vertex u into the sub-route.1: the weight of the distance increase when insert a

new stop into the present sub-route.2: the weight of the waiting time increase when in-

sert a new stop into the present sub-route 1 + 2 = 1.: the weight express the distance between the depot

and the insert stop.: the distance adjusted factor.The stop is selected insert into the route has the

largest value in Eq. 17 without violate any constrains.However, this method is only designed to create justone feasible solution and support for optimizing localsearch algorithm. For the set of feasible solutions, theseheuristics needed improvement.

Algorithm A(1) UNROUTED all stops(2) CREATE new solution S(3) CREATE new route R

SET ref = DepotSET dist = Maximum travel distance

(4) if CAN create new sub-route thenSR = createSubroute(ref, dist) //Algorithm BRepeat

SET stop = The stop has value C2 is maxINSERT stop into SRROUTED stop

Until CANT find the stop insert into SRDELETE Depot in the last of SRINSERT SR into the route RSET ref = The landfill at the end of SRUPDATE dist = The remain travel distanceGOTO (4)

elseif EXIST the sub-route in R then

INSERT the Depot into the last of RINSERT R into the solution SGOTO (3)

elseGOTO (5)

5

(5) FINISH

Algorithm A, in the function createSubroute, a ran-dom stop will be selected and insert into the first posi-tion of the new sub-route. Thus, the feasible solutionsin population are created randomly.

4.2. Representation

An individual is represented as a string of the se-quence of vertices. In the the chromosome represen-tation, the depot and landfills is removed from the se-quence of vertices. That will help us advance in usingcrossover and mutation operator. In decode process, thedepot and landfills are restored. The first, the depot willbe inserted at the beginning and the end position of thechromosome string. The next step, the process of check-ing constraints is executed, when the constraint is vio-lated, the suitable landfill will be inserted into the chro-mosome string. The suitable landfill can be found byscan all landfill in problem and get the landfill with theminimization of distance between two neighbor stop.

4.3. Fitness function

Base on objective function in Eq 2, the fitness func-tion be expressed as:

C = 1Cdist + 2Cmove time + 3Cwait time (21)

in which:Cdist: The total move distance of all vehicles.Cmove time: The total move time of all vehicles.Cwait time: The total wait time of all vehicles.Depend on importance of objectives, the correspond-

ing weights will have great or small value. Our study,we consider the total move distance objective and totalmove time objective is more important than total waittime objective. About number vehicles optimization,this objective is not consider as main component in theglobal objective function, however, it will be consideredin GA search process.

4.4. Operators and heuristics

Next, we discuss about operators are used in GA tosolve for waste collecting VRPTWC. Elements in theinitial population are created by Algorithm A may benot dominant and in the searching process it will beevolved after each generation. The GA try to find dom-inant solutions in the set of Pareto solutions and its op-erators are essential tool help the search process explorethe searching space.

Partial-mapped crossover operator (PMX) has beenproposed by Goldberg and Lingle [14] that is main com-ponent to reproduce next generation. With the twoselected individual, PMX exchanges one or many se-quences of vertices between two chromosomes and re-arranges them when exchange phase finished. Fig. 4 isperformance of steps in crossover process implementedby PMX operator. Two new permutation strings are gen-erated maybe not satisfy constraints of problem. How-ever, encoding process will recorrected them to satisfyconstraints.

Figure 4: Steps in crossover process implemented by PMX operator

Random exchange mutation operator is an operatorused in mutation phase in our GA implement. Themutation operator only executes just one argument isa chromosome. It exchanges two or more genes ran-domly in the chromosome. Fig. 5 illustrates our imple-ment of the random exchange mutation operator with3 crossover point at . At two or more positions of thechromosome is selected, the vertices will be exchanged.In the figure the vertex at position 2 exchange with thevertex at position 6 and similar to vertex 8 and 9.

Figure 5: An example of random exchange mutation operator.

-interchange operator (-opt) is used in adaptationthat is an operator help the local search embed in GA ex-plorer search space. The -interchange was introducedby in Osman and Christofides [15] for the capacitatedclustering problem. However, authors implement it onlyfor the route is the Hamiltonian cycle. Here, we extend-opt for using with the routes are not the simple cycle.

Assuming pi and qj are two sub-routes i-th and j-

th of two routes rp and rq in the solution S and pi and qj are two paths in sub-routes. |||| and |||| and they not store the landfills and thedepot. -interchange generated the new feasible solu-

6

tion by exchange two set vertices are subsets of twosub-routes in two routes of the solution. If pi andqj are denoted two new sub-routes, we have

pi =

(pi \) and qj = (

qj\)

and the new solution

= {r1, ..., rp, ..., rq, ..., rk} with rp = (rp\pi )pi

and rq = (rq\qj )qj are two new routes.

The space of the neighbor of solution is set of theall feasible solution created by -opt is N(). Assum-ing that each route r has ns sub-routes and each sub-route has same number of vertices nv and nv > , weconsider the count of elements in N(s) space.

There are Ck2 pair routes to exchange:

Ck2 =k!

2!(k 2)! (22)

, with route pair (ri, r j), every sub-route ip ri canexchange with ns sub-routes in r j. The count of sub-route pairs can exchange in each route pair is n2s . In eachsub-route pair (ip,

jq), amount of path to exchange can

be chosen in each sub-route are:

nv 1i=1

i = (nv 2 ) (23)

Thus, amount of elements in neighbor N(s) space is:

n2s2(nv 2 )

2 k!2!(k 2)! (24)

, by the product of ns and nv is proportional to N andnumber vertices of graph and k is not too large constant(k

Figure 6: Illustrate decoding process of the chromosome in Fig. ??

individual with the high ranking in population costhas the high probability of mating, while the otherswith the low cost has the low probability of mating.

Mating: is the creation of offspring from the par-ents selected. In here, PMX is used in mating pro-cess.

Mutation: as mentioned above, MA use the ran-dom exchange mutation operator for chromosomeis chosen for mutation. Number of chromosomeswill be chosen with the given rate as input parame-ter of the mutation operator. Except elite individu-als, others may be muted with a certain probability.

Adaptation: that is process improves fitness cost ofeach individual in population by the local search.Hill Climbing (HC) is applied to individuals aftermutation process. With a determined rate, some ofindividuals in population are adapted to improvetheir fitness cost. The neighborhood is exploredwith -opt interchange.

5. Computational results

Our algorithms are implemented in GNU C++ lan-guage in the a CentOS 6.3 machine of 2.2GHz Xeonprocessor with 32 nodes and 128 GB of RAM. Table ??show the set up parameters of the algorithms for eachwaste collection VRPTWC benchmark problem sets.

The benchmark data for waste collection VRPTW isproposed by used Kim et al. [2] in their research. How-ever, these benchmark data need additional informationabout the properties of the waste in order to use for

Table 3: The best solution results of our algorithms for the Wastecollection VRPTWC benchmark problem sets.

Algorithm Value Number stops

102 277 335 444

HC

F -307 -511 -269 -86Tc 6 365 1692 1769Nv 4 3 8 11TD 280 511 269 86Tw 139 0 0 0Sm 1036 1071 651 183

GA

F -294 -527 -263 -91Tc 1 9 44 14Nv 3 3 6 10TD 272 527 263 91Tw 107 0 0 0Sm 979 1144 738 184

MA

F -271 -521 -249 -56Tc 3256 9006 63471 74224Nv 3 3 7 7TD 230 521 249 56Tw 202 0 0 0Sm 1074 1119 747 151

the waste collection VRPTWC. Our benchmark set, theproperty of the waste is a new column and is attachedinto the origin benchmark. The waste properties aredenoted from 1 to 7 as shown in Table 1. We insertthese value sequentially into the waste property columnin each row which store information about the regularstop. For the other rows (rows store information aboutthe landfill and depot), the value 0 is inserted.

In order to compare MA with other algorithms, theresult of Hill-Climbing (HC) and Generic Algorithm(GA) are also presented. The computational results areshown in Table 3 include: fitness value (F), computa-tional time (Tc) in second, number vehicles (Nv), to-tal travel distance (TD) in mile, waiting time (Tw) inminute and shape metric (Sm). The fitness value pro-vides an overview of the quality of solution while othervalues as total travel distance, waiting time, number ve-hicles and shape metric provide detail information. Thecomputational time shows the time cost of the algorithmand the shape metric is a value to quantify the routecompactness, more detail about shape metric see [2].The total travel distance is the total travel distance ofroutes of the solution, the waiting time is the time vehi-cles takes to wait on all routes of the solution.

In case of waste collecting VRPTW, when the conflictcondition is not considered, our results are comparedwith experiments of Kim et al. [2]. In order to optimizecomputational time cost, GA is selected for computeand compare. We set up the population number equalthe stop number in each benchmarks. Table 2 presents

8

Table 2: The solution of the waste collection VRPTW by our experiments and Kim et al. [2]

Problem set Tc TD Sm Nv

GA B.-I. Kim GA B.-I. Kim GA B.-I. Kim GA B.-I. Kim

102 stop 4 3 174 205 637 399 2 3277 stop 111 37 483 521 928 858 3 3335 stop 647 28 206 191 674 410 5 5444 stop 248 372 80 82 180 70 9 10804 stop 103282 92 509 769 4941 2350 4 51051 stop 50714 329 1378 2370 7774 3615 8 181351 stop 95 1039 2145 71599 stop 212 1459 4141 131932 stop 424 1395 3166 172100 stop - 408 1833 5761 16

the computational result for all benchmarks by GA. Wegain the better solutions with the total traveling distancevalue and number vehicles are less than. However, ourimplement is not included break-time lunch constraintas same as [2]. Because the constraint is relaxed, thesearching space is more extend and thus our algorithmis more advantage in the searching process.

6. Conclusion

Phan tam chua viet

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IntroductionLiterature reviewDefinitions and modelGeneric algorithmInitial populationRepresentationFitness functionOperators and heuristicsComponents of the MA

Computational resultsConclusion