Line-Based Optimization of LTL-shipments using aMulti-Step Genetic Algorithm

Christian Tummel∗

IMA / ZLW & IfURWTH Aachen University

Dennewartstrasse 2752068 Aachen, Germany

christian.tummel@ima-zlw-ifu.rwth-aachen.de

Tobias PyttelIMA / ZLW & IfU

RWTH Aachen UniversityDennewartstrasse 27

52068 Aachen, Germanytobias.pyttel@ima-zlw-

ifu.rwth-aachen.de

Philipp WoltersIMA / ZLW & IfU

RWTH Aachen UniversityDennewartstrasse 27

52068 Aachen, Germanyphilipp.wolters@ima-zlw-

ifu.rwth-aachen.deEckart HauckIMA / ZLW & IfU

RWTH Aachen UniversityDennewartstrasse 27

52068 Aachen, Germanyeckart.hauck@ima-zlw-

ifu.rwth-aachen.de

Sabina JeschkeIMA / ZLW & IfU

RWTH Aachen UniversityDennewartstrasse 27

52068 Aachen, Germanysabina.jeschke@ima-zlw-

ifu.rwth-aachen.de

ABSTRACTMotivated by the so-called “CloudLogistic”-concept as aninnovative, line-based way for dealing with less than truckload (LTL) shipments in cooperation networks, this paperintroduces a genetic algorithm as a heuristical approach fordealing with multi-objective optimization problems. Basedon the implied optimization problem - the NP-hard multi-depot heterogeneous fleet vehicle routing problem with timewindows and assignment restrictions (m-VRPTWAR) - fourdifferent optimization goals of the “CloudLogistic”-conceptare introduced and a multi-step approach is motivated.

Therefore, two different optimization steps are presentedand transferred into a genetic algorithm. Additionally, twoinnovative problem-specific genetic operators are introducedby combining a generation-based approach and a usage-based approach in order to create a useful mutation process.A further usage-based approach is used to realize a problem-specific crossover operator. The presented genetic multi-stepapproach is a useful concept for dealing with multi-objectiveoptimization problems without the need of a single combinedfitness function.

Keywordsgenetic algorithm, m-VRPTWAR, multi-objective, multi-step, LTL, generation-based, usage-based, CloudLogistic

∗corresponding author

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.CIPLS 2013, IEEE Symposium on Computational Intelligence in Produc-tion and Logistics Systems 2013 Singapore, SingaporeCopyright 20XX ACM X-XXXXX-XX-X/XX/XX ...$15.00.

1. INTRODUCTIONIn its recent study for traffic linkages in Germany the Fed-

eral Ministry for Transport, Building and Urban Develop-ment (BMVBS) forecasts a substantial growth of traffic vol-ume in road haulage up to the year 2025 [8]. The long-haulroad transportation contributes especially to this trend. Agrowth of transport volume of about 55% and an increasein traffic of about 84% is expected. From the ecological andthe economic point of view, these trends should not onlybe faced with an adjustment of the road network, but alsowith a more efficient use of the existing infrastructure [9].In Germany, every fifth truck in commercial freight traffic isdriving without any load at all [16].

The “CloudLogistic”-concept addresses these challengesand represents an innovative concept for freight cooperationto strengthen the market position of Small and Medium-sized Enterprises (SME) in the long-haul road transporta-tion. The concept tranfers the strengths of already existinggeneral cargo and Full Truck Load (FTL) networks to thearea of Less than Truck Load (LTL) transports.

Usually it is not possible for a Small and Medium-sized Lo-gistic Service Provider (LSP), to transport several LTL ship-ments together in one truck because there are not enoughshipments within similar source and target areas. Hence, fora single LSP, several trucks are required for the transport ofseveral LTL shipments. The “CloudLogistic”-concept bun-dles LTL shipments of several cooperating LSP, via a coop-eration network by combining corresponding LTL shipmentsto generate synergetic effects. Thereby, the concept relies ona line-based logistics model.

On the one hand, the implied optimization problem dealswith the assignment of a set of shipments to a certain setof freight routes in order to minimize unused cargo volumeof the vehicles by observing some given hard constraints(i. e. time-window-constraint, presented in [22]). On theother hand, this assignement additionally has to observesome other optimization goals like the maximization of qual-

70978-1-4673-5905-4/13/$31.00 c©2013 IEEE

ity aspect by choosing higher-ranked LSP for transportationtasks.

After introducing the “CloudLogistic”-concept in detailand after a rough outline about the related work the math-ematic model of this concept is transferred into a geneticapproach. Afterwards the multi-step approach is presentedas an innovative way to deal with the given problem and itsdiffering optimization goals.

2. THE CLOUDLOGISTIC CONCEPTSimilar to the IT term “Cloud Computing”, the so-called

“CloudLogistic”-concept describes the ability and opportu-nity for LSP to share unused resources by participating ina freight cooperation network. This is done by using itsinfrastructure, its resources and scaling them locally whilesharing their own infrastructure and resources with the net-work. Several shipments of different network-partners willinitially be bundled and assigned to previously establishedfreight routes.

Each route is operated by a partner within the coopera-tion. For a combined disposition of LTL shipments, freightroutes are established, composed of the relation between asource and a target area that is served regularly by severaltrucks via point-to-point transportation. Trucks are pro-vided by the cooperating partners. Shipments are collectedin the source area of the route and then carried directly tothe corresponding target area without any turnover at all.In the target area shipments are locally distributed (see Fig-ure 1). The basic aim of the “CloudLogistic” concept is todetermine an assignment of shipments to a certain set offreight routes while decreasing the number of needed trucks,respectively needed FTL capacity. Based on the investi-gations in [24], the CloudLogistic-approach maximizes theecological and the economic benefits of the overall optimiza-tion potential if the main optimization goal consist of theminimization of the needed FTL capacity.

The assignment of shipments to a set of certain freightroutes by minimizing the number of used trucks is classi-fied as a NP-hard optimization problem [24]. The problemis illustrated in Figure 1. First, shipment A is pushed intothe system - which can only be assigned to freight route 2.In contrast, both routes are feasible for the transportationof shipment B. If shipment B is assigned to freight route 1and freight route 2 does not have sufficient unused capac-ity, shipment C may not be delivered because C can alsoonly be assigned to freight route 2. The number of suchcollisions increases, if the source and target areas of multi-ple freight routes overlap and additionally a large number ofcorresponding shipments have to be distributed.

Figure 1: Visualization of the implied assignementproblem of three shipments and two freight routes.

3. RELATED WORKVehicle routing problems have always been the subject of

research since the first works on this issue by [10]. In [24]a multilevel solution method has been introduced for the“Multi-Depot Heterogeneous Fleet Vehicle Routing Prob-lem with Time Windows”. At first, the algorithm deter-mines reasonable clusters of nodes by using a heuristical ap-proach and afterwards distributes them on a couple of trucksin a valid way by solving a Mixed Integer Linear Program(MILP). This method is able to solve problem instances withup to 100 nodes, but the deviations from the optimal solu-tions are at up to 30%. Furthermore, the solution methodneither takes the assignment restrictions into account nor isthe aim of a minimization of unused cargo volume consid-ered.

The same problem class was covered by [11], this workdoes not focus on the development of a solution, but thedevelopment of an efficient mathematical formulation of theproblem model was concerned. At the beginning, a compactMILP is described and extended by specific rules that reducethe size of the MILP (measured by the number of variables).It shows, that this could significantly decrease computationtime. Considering the more general problem class of the“Vehicle Routing Problem with Time Windows” (VRPTW),the papers [6], [7], [4] and [5] may be consulted.

Solving VRPTW problems by using genetic algorithms,has been subject to many approaches in the past. [3] investi-gates genetic algorithms as a heuristic for obtaining near op-timal solutions. Therefore, a collection of new crossover op-erators, summarized under the Cross #1 (MX1) and MergeCross #2 (MX2), are introduced. More conductive optimiz-ing has motivated designing these operators.

Thangiah presents GIDEON in [21], a cluster-first route-second approach for solving the VRPTW within two steps.In a first step a genetic algorithm is used to cluster cus-tomers into sectors with additional post-optimization in asecond step. GENEROUS representative standing for GE-NEtic ROUting System was described in [19]. In detail, therepresentation step makes use of a stochastic selection usinga linear ranking scheme to bias the selection process towardsthe best solution. Further, in the recombination phase, boththe sequence-based and the route-based crossover are de-scribed in connection to the genetic approach solving theVRPTW. In terms of mutation, three types of mutation op-erators, their advantages and their necessity are presented.A“Parallel Hybrid Genetic Algorithm”was developed in [1].The goal of this approach targets is to be fast, cost-effectiveand highly competitive. This new concept of “co-evolutionof two populations”distributes the assignments of minimiza-tion of the total traveled distance and generating feasiblesolutions to two populations. Because the size of the secondpopulation, which is in charge of finding feasible solutions,is one less than the size of the first population, only smallerfeasible populations are transferred to the first populationand these overwrite older versions. Moreover, an insertion-based crossover, a reinsertion crossover operator and a suiteof six mutation operators are proposed. As a continuativereference according to [1], [2] can be consulted.

[12] illustrates a parallelization of a two phase procedu-ral approach, consisting of a (1, λ)-evolution strategy forminimizing the number of vehicles and a tabu search al-gorithm for minimizing the total distance. Related to theparallelization above [13] can be consulted as similar work

2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS) 71

in this direction. K. C. Tan, et al. use a messy genetic algo-rithm (mGA). Besides the motivation of usage of a mGA theCut and Splice Operators are highlighted. Furthermore, theauthors cater to the coding and to compare with competingheuristics. The basics concerning messy genetic algorithmscan be taken out from [20]. A genetic algorithm within ahybrid search algorithm, in cooperation with a variable in-tensive tabu search was presented in [17]. This work consistsof two new genetic operators leaned on the natural phe-nomenon of dominate und recessive existence of genes. In[18] a parallel cooperative multi-search procedure - based onthe solution warehouse approach - was proposed for findinga good solution according to the VRPTW. Therefore, theauthors described important cooperating methods to whichgenetic algorithms belong.

In one of our latest contributions we introduced the m-VRPTWAR ([24]), which was the first contribution in thisspecific direction. This problem deals with an assignmentof a certain set of shipments to a set of freight routes thatminimizes the unused cargo volume of the vehicles. The as-signment of each shipment is restricted to a subset of freightroutes. Furthermore, the shipment has to be delivered in aspecified time window. Thus, it is necessary to determine anorder of the shipments of each freight route that guaranteesthe observance of all time windows. An introduction to thisproblem, including an Integer Linear Program (ILP) formu-lation and first calculation results for an optimal solution tothe problem can be found in [24]. Furthermore, a heuristicapproach for evaluating the feasibility of an instance of them-VRPTWAR was presented in [23].

4. TERMS AND DEFINITIONSTo build the following remarks on a solid mathematical

framework, at first it is necessary to define some fundamen-tal terms according to the definition in [24]. Afterwardsthese can be used to introduce the genetic approach.

Definition 1. A timestamp t ∈ N is a non-negative inte-ger, that references a specific point of time. All timestampsshare a common foundation (e.g. 01.01.1900, 0:00:00) andspecify the time in seconds that has passed since this pointof time. An ordered pair T = 〈t1, t2〉 of timestamps witht1 ≤ t2 is called time window. The difference between thetimestamps t = t2 − t1 ≥ 0 is named time span.

Definition 2. An ordered pair A = 〈λ, ϕ〉, whose com-ponents are geographic coordinates on the globe, is calledgeographic location. For the longitude λ ∈ R and thelatitude ϕ ∈ R the conditions −180◦ ≤ λ ≤ 180◦ and−90◦ ≤ ϕ ≤ 90◦ hold. Furthermore, W denotes the setof all geographic locations.

Definition 3. A function d : W × W → R is called dis-tance function, if for any geographic locations A = B =C ∈ W the following conditions hold:

• d(A,B) ≥ 0The distance is always positive

• d(A,B) = 0 ⇔ A = BA location only has a distance of zero to itself

• d(A,B) ≤ d(A,C) + d(C,B)A detour via location C must not decrease the distance

Definition 4. Let A,B ∈ W be two geographic locations,r ∈ R a real number and d : W×W → R a distance function.Then the set GA,B,r,d = {X ∈ W | d(A,X) + d(X,B) ≤r} ⊂ W is called area. In this case A,B are called focalpoints of this area. Furthermore, let G denote the set ofall areas. The shape of this area is significantly differentby the choice of its distance function. For example, a func-tion which calculates the Euclidean distance between twogeographic locations differs significantly to a function whichdetermines the time that is needed to reach one geographiclocation from another when using the road network. In theCloudLogistic concept, the second type of distance functionis chosen and called dt. So the shape of the area GA,B,r,dt

describes an elliptic isochrone.

Now, we extend the definition of an area according toa multizone-approach. This approach is needed to realizedifferent optimization goals presented in chapter 6.1. Themultizone-approach is illustrated in Figure 1.

Definition 5. Let GA,B,r,d be an area. Further let Z ={z1, z2, . . . , zn} be a set of n distances with 0 < z1 < z2 <. . . < zn = r and n > 1. We call GA,B,r,d,Z a multizone-area.

Definition 6. Let V ∈ W be the source location of a ship-ment, E ∈ W the target location of a shipment, w ∈ R therequired loading metres, m ∈ R the total weight, T = 〈ts, te〉the delivery time window, T = 〈ts, te〉 the time window ofthe transportation deadline and t the time span needed forunloading. The tuple s = 〈V,E,w,m, T, T , t〉 is called ship-ment. The set of all shipments will be denoted as S.

Definition 7. Let Gs ∈ G be the source area, Ge ∈ G thetarget area, c ∈ R the available loading metres, n ∈ R thepayload limit, a ∈ N

+ the total amount of shipments andτ = 〈τs, τe〉 a time window, that determines a specific day.The tuple l = 〈Gs, Ge, c, n, a, τ〉 is called freight route.The route is served on day τ by exactly one truck. If therelation between start and target areas has to be served by ktrucks, each one of these trucks defines a new freight routel1, . . . , lk with its individual vehicle characteristics. Let L

denote the set of all freight routes.

Definition 8. A set of assignments is a function z : S →L with z(si) = lj , if si ∈ S is transported by freight routelj ∈ L with i, j ∈ N

+. A valid set of assignments is givenif the set holds for all given hard constraints. A collectionof relevant hard constraints is presented in [22].

5. A GENETIC APPROACHOne of the significant steps for the developement of a ge-

netic approach is to transfer the mathematical model intogenetic language consisting of genes, chromosomes, and thepopulation. So in computer-aided problem solving, using agenetic algorithm, the problem has to be encoded in computer-usable syntax. In order to get a suitable encoding, two im-portant guidelines should be kept in mind:

• Similar candidates representing possible solutions shouldhave a similar fitness

• and if possible, the solution space (set of all candi-dates representing possible solutions) should be closedin connection with the used genetic operators.

72 2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS)

Figure 2: Example of the genetic representation

The following definition abstracts the above introducedoptimization problem into computer-usable, linked integervalues considering these guidelines.

Definition 9. Let S be the set of all shipments and let Lbe the set of all freight routes. Furthermore, let z : S → L

be a set of assignements according to the previous definitionand the set M = {z | z is valid} the set of all valid setsof assignements. Then z implies explicitly the following setof tuples which contains all pairs of indices of shipments tofreight routes:z′ = {(i, j) | z(si) = lj , z ∈ M, si ∈ S, lj ∈ L} ⊆ (N+ × N

+).The set M ′ = {z′ | z ∈ M implies z′} is called the solutionspace of the genetic approach and represents all valid setsof assignements of shipments to the available freight routes.So a single z′ ∈ M ′ is called chromosome, g ∈ z′ gene andp ⊆ M ′ population.

An alternative approach to describe the scenario with agenetic representation, could have been done by using themap L → P(S) instead of using a representation based onS → L. In this case, a specific truck with its assigned ship-ments is represented as a single gene. By changing only asingle gene of such a representation (for example, in the caseof a simple mutation or doing a crossover), the new chromo-some does not represent a valid solution, because a single orsome shipments will not be assigned or would be assignedtwice to a specific truck.By the use of the above representation, we designed a di-

rect correlation of a shipment to its vehicle. Here the solu-tion space is closed using the specific genetic operators whileobserving some given hard constraints (i. e. capacity con-straints, which has to check all cases). The representationis also conform to the first guideline, because by mutatinga single gene, the concerned chromosome’s fitness will bechanged minimally.

6. MULTI-STEP OPTIMIZATION

6.1 Goals of the OptimizationIn order to specify the genetic approach in more detail (re-

spectively fitness, initialization, selection and reproduction),the main goals of the optimization have to be discussed. Inour approach we try to reach four different goals:

• Minimizing the number of needed trucks to reach thehighest degree of capacity utilization (Overall Struc-tural Goal)

• Maximizing the so-called “fairness” for each truck, ifa truck would be used rarely in the past (FairnessGoal)

• Maximizing the total quality of the whole assignmentby preferring service providers that offered a high qual-ity in the past (Quality Goal)

• Minimizing the zone-based distance of each shipmentto its source and target area focal points (Green Goal)

These goals were chosen and ranked in different work-shops with experts from the logistics segment. Because theOverall Structural Goal is essential for the CloudLogisticapproach, it must be observed that this goal must be reachedwith a maximum deviation of 5% of the best solution thatcould be found. Therefore, we decided to apply a multi-stepapproach by dividing the whole optimization process intotwo seperate steps which are illustrated in figure 3.In a first step (described in detail in Chapter 6.2) the

optimization process searches for the best solution whichminimizes the total number of needed trucks. If the minimalsolution is found or a specific time constraint for the durationof this process is reached, the minimal number of neededtrucks n is used to specify the solution space of the secondoptimization step.By adding an additional restriction in terms of a new hard

constraint (only solutions are allowed that use a maximum ofn+5% of n trucks), the solution space is reduced to these so-lutions which fulfill this condition. Choosing the remainingthree optimization goals as a combined goal with weightedsubgoals, the implied “fitness-landscape” - which representsthe quality of a solution according to the specific goal forthe whole solution space - changes in order to represent thenew goals. The solution space also changes to a subset ofthe possible solutions of step one. All other solutions willbe removed from the solution space. Then, the optimizationprocess will search for a new solution which maximizes thecombined goal of the remaining three optimization goals.The specific weights for the combined optimization goal ofstep two has been set within the workshops to 0.6 (GreenGoal), 0.2 (Quality Goal) and 0.2 (Fairness Goal).The evaluation of the different goals in detail (i. e. the

Fairness Goal) is not the subject of this contribution. Atthis moment, it is only important to know that the respec-tive estimation depends only on static information whichis available without needing further information about thewhole solution.In Chapter 6.2 the first step of this approach will be

defined in detail by introducing a suitable fitness function(6.2.1), by applying corresponding initialization (6.2.2) andselection (6.2.3) processes, by motivating an innovative problem-specific way for mutation and doing crossover operationsfor the reproduction (6.2.4) and additionally by setting theconcerning termination statement (6.2.5). Respectively inChapter 6.3 the second optimization step will be defined inthe same way regarding its specific problem definition.

6.2 Optimization: Step 1

6.2.1 FitnessIn this optimization step, the total number of needed

trucks has to be minimized. In general, a real value f mustbe defined for the genetic algorithm to evaluate the qual-ity of a specific solution in form of a chromosome accord-ing to definition 9. The so-called fitness of a chromosomeis needed to compare chromosomes among each other and

2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS) 73

Figure 3: Multi-step approach for reaching different optimization goals

enables a selection process according to the “survival-of-the-fittest” principle from Charles Darwin. Hence, to describethe fitness of a chromosome for every encoded solution (chro-mosome) a real number out of R has to be defined. Thefunction which gives every chromosome a real number asrepresentation for the quality of the solution is called fitnessfunction. The constitution of the fitness functions of bothoptimization steps are problem-specific and defined accord-ingly to different optimization goals motivated in chapter 6.In the first iteration, the fitness is defined as the quotient ofthe number of all avaiable trucks to the number of loadedtrucks. Thereby, a fitness of a higher value has a betterquality than a fitness of a lower value. The determinationof these values does not take much computational effort. Itis only necessary to count the different truck representativesof each gene for the whole solution. Therefore, a fitness ap-proximation is not needed. The mathematical formulationis given in the following definition.

Definition 10. Let f1(z) be the fitness of a chromosomez′ representing the number of loaded vehicles ∈ L, then thefitness function is defined as f1 : M ′ → N ⊂ R with f1(z

′) =|L|

|{j|(i,j)∈z′}| .

It has to be examined, if this kind of fitness function isapplicable in this genetic approach. It is imaginable thatanother factor i. e. the sum of the load factor of all truck isa better choice to move this approach in the right direction.

6.2.2 InitializationThe initialization needs to be provided with a couple of

solution. In order to develop a suitable IT-solution for theshipment-processing, which is presented in [22], a first pro-cessing step ensures that for every point of time, there willbe a known solution in memory. At the beginning of the op-timization process this is the only known solution which canbe provided. Hence, the pre-processing provides some infor-mation that also speeds up the entire optimization process.For example, the determination of possible routes accordingto a shipment is part of this pre-process. Furthermore, thepriority zones according to a shipment on a route are iden-tified in this process. Because the hard constraints mustnot be violated, this chromosome is just copied popsize− 1time. The size of a population (popsize) has to be empir-ically identified in respect to the total number of receivedshipments.

6.2.3 SelectionThe selection of chromosomes transferred into the tempo-

ral population is done by the roulette-selection presented byGoldberg [15]. For more detailed information, [14] may bereferred to.

Based on the actual population the fitness of each chro-mosome is calculated. So, a part of the survival probabilityfrom each chromosome is built by calculating the averageof all fitnesses, followed by calculating the distance of everychromosome’s fitness to the average. Then every fitness isadded by the greatest negative value of all calculated dis-tances. The survival probabilities are calculated by dividing

74 2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS)

every chromosomes compensated differences by the sum ofall compensated differences. It is obvious that the sum ofthe survival probabilities must be one. So, a set of real num-bers between zero and one is defined and the chromosomesare assigned to this set in the following. Chromosome one isassigned to the number of the above defined set in the rangebetween zero and its survival fitness. Chromosome two isassigned to the numbers of the set in range between the sur-vival probability of previous chromosome and the survivalprobability of previous chromosome plus the survival prob-ability of this chromosome. This must be continued untilevery chromosome is assigned to the above defined set.

To choose a chromosome for the temporal population,which has to be reproduced, a random real number betweenzero and one is created. This number is compared to theabove defined set. The chromosome assigned to this num-ber is chosen for the temporal population. This step mustbe repeated until the wanted temporal population size isreached.

Another selection strategy which has to be evaluated is toselect only a number of popsize − x chromosomes to infectthe selected population by x additional copies of the bestchromosome ever found in all generations.

The advantage of this method is, that good solutions areadopted with a high probability. However, also bad solutionsare included in the search and thus, an appropriate measureto escape from local maximum is given. In the followingchapter the modification of chromosomes, which requires thegeneration of the new population out of the temporal pop-ulation, is described. Therefore, two new problem-specificoperators are defined.

6.2.4 ReproductionThe reproduction of the selected chromosomes is a crit-

ical component of the genetic algorithm and is importantto assure a suitable quality of the chosen approach. How-ever, it should be noticed that the chosen operators complywith the defined second guideline in Chapter 5: “The solu-tion space . . . should be closed in connection with the used. . . operators”.

By applying operators to chromosomes according to Def-inition 9, the length of these chromosomes should not bechanged. Furthermore, it has to be ensured that a shipmentwill not be assigned to a route which cannot handle the ship-ment. This is done by the enrichment of all shipment infor-mation with a linked list of all possible routes through thepre-processing step. Beside this, all given hard constraintshave to be checked before inheriting chromosomes into thenext generation of a population. If these constraints areviolated the reproduction has to be repeated.

According to the defined encoding above, while design-ing the intended genetic operators, the specific optimizationgoals have to be taken into account. Generally, these opera-tors are divided into the mutation and the crossover process.The mutation will be used to ensure the diversity of differentchromosomes in a population. Remembering Figure 3, thismeans that starting with a solution on a “hill” with localmaximum this solution can be mutated in such a way thatthe generated mutations are able to lead to a better evo-lution. In contrast, a crossover operator changes sequencesof a chromosome with the sequence of another chromosome.Thus, this operator implies the “climbing” of the actual hill.

The mutation operator is a genetic operator which ran-

Figure 4: Example for the mutation probability ofa gene g with a generation-based and a usage-basedcomponent

domly modifies single genes of a chromosome. The decisionwhether a gene mutates is made by using a mutation proba-bility. On the one hand, solutions out of the whole solutionspace have to be taken into account to search in the breadthfor good solutions. On the other hand, the optimization ofgood solutions should be improved towards their optimum.So we decided to design a generation-based approach whichreduces the mutation probability of each gene according tothe runtime of the algorithm.

Besides the generation-based approach a usage-based ap-proach, is used for increasing the efficiency of the algorithmby using the knowledge about the structural conditions toreach the optimization goal more rapidly. This approachadditionally manipulates the mutation probability of a gene- described above - by using information about the actualamount of assigned cargo to a specific truck (which is impliedby the gene). This chosen strategy aims to identify geneswhich implied truck is nearly empty and using this informa-tion to re-assign the shipment to another truck. The fol-lowing definition describes the mutation probability of anygene according to the assumptions made above. Figure 4illustrates the two different probability components of thisapproach.

Definition 11. Let gen be the number of the current gen-eration and let p be the current population. Furthermore,let z′ ∈ p be a single chromosome of this population and letg = (i, j) ∈ z′ be a single gene of this chromosome. Addi-tionally, let h(j) be the usage in a percentage of the vehiclej. Then the mutation probability of a gene g is de-fined as: Pg = 1

α∗(1+gen)+ (1 − 1

α∗(1+gen)) ∗ β ∗ h(j) with

(α, gen, h(j)) ∈ R×N×R, α > 0, gen ≥ 0, 0 ≤ h(j) ≤ 1 and0 < β ≤ 1. The maximal mutation probability β of an genshould be in size of 1/m or lesser, where m is the number ofgenes. The constant α is used to adjust the influence of thetime-based approach.

The crossover operator as the second genetic operatoris introduced. We choose a truck-based approach, to identifya useful collection of genes for this operation. In the worstcase, the usage of a single truck is at 50% because manygenes have to be changed to fill the truck with additionalgoods or to empty it completely. So this approach identifiesa collection of genes which represents assignments of ship-ments to a truck with a usage of nearly 50%. By doing this,we ensure that the crossover operation fills a specific truckup with additional shipments.

Figure 5 visualizes this procedure and motivates this ap-proach. In this example, two randomly chosen chromosomes

2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS) 75

Figure 5: Crossover strategy by using the utilizationof trucks to identify a collection of genes

are selected. Truck a is identified as a resource with a usageof nearly 50%. After the crossover procedure, a new chro-mosome is born where the usage of truck a has been grown.According to this approach, we divide the whole procedureof the route-based crossover into the following four steps.These steps will be repeated until every chromosome of thecurrent population are swapped:

1. Select two chromosomes v′1 and v′2 randomly out of thepool of mutated chromosomes.

2. Select from the first chromosome r different truck-ID’sj1, j2, . . . , jr. (Choosing the trucks is also done by us-ing a specific probability deviation which prefers truckswith a usage of 50%.)

3. Swap all genes with (i, j1), (i, j2), . . . , (i, jr) of bothchromosomes for every i by exchanging the genes atthe equivalent position.

4. Verify for the new chromosomes that all the hard con-straints hold. If there is any violation, r has to bereduced and the procedure has to be repeated at thestage of step two. If the verification of the hard con-straints is successful, both created chromosomes aretransferred into the temporal population to evaluatetheir fitness.

6.2.5 TerminationThe termination criteria of both optimization steps is sub-

ject of temporal constraints. The runtime for both steps isbounded by experts from the logistics sector to a maximumof 60 minutes. That time is acceptable concerning all othercorresponding business processes. For a first approach, atime window of 45 minutes for step one and a time windowsof 15 minutes for step two is supposed.

As a result of the optimization step one, a population witha good fitness is found. That means, we found a solutionwith a low number of needed trucks. The best known solu-tion now defines the lowest known number of needed trucksx. For a further optimization, the best known solutions canbe provided beside this interesting index.

6.3 Optimization: Step 2

6.3.1 FitnessThe second step of the optimization process aims for three

hybrid goals: the Fairness Goal, the Quality Goal and the

Green Goal (see Chapter 6.1). After the termination of stepone (see Chapter 6.2.5), the index x is provided to rede-fine the available solution space according to the approachexplained in Chapter 6.1. To ensure that optimization steptwo only finds solutions which respect the lowest found num-ber of trucks x, a new restriction is added in form of a hardconstraint - also described in 6.1.

Definition 12. To define a fitness function for this step,three individual fitness functions for every subgoal have tobe built and normalized. At this time, we assume that suchfunctions for each goal are given. Then the fitness func-tion is defined as function f2(v

′) = 0.6 ∗ fgreen(v′) + 0.2 ∗

fquality(v′)+ 0.2 ∗ ffairness(v

′) using the weights mentionedin chapter 6.1.

6.3.2 InitializationTo initialize optimization step two the solutions of the last

population found by step one are stored after its termina-tion as the initial population for step two. But only thosesolutions of those which even hold the new hard constraintwill be used further. Consequently, the popsize of the newpopulation is lower or equal with the popsize of step one.To reach a specific popsize some chromosomes will be du-plicated randomly. The choice of a suitable popsize will besubject to our further work.

6.3.3 SelectionTo select chromosomes and to transfer them into the tem-

poral population refer the roulette selection by Goldberg,which is described in Chapter 6.2.3.

6.3.4 ReproductionThe hybrid character of the combined approach make it

hard to define useful problem-specific genetic operators. Theindividual goals are too different to create any genetic op-erators which enrich the mutation or crossover process withthe use of additional knowledge. Therefore, we decided touse a simple mutation with help of the introduced rouletteselection by Goldberg. Also the used crossover is a k-point-crossover described in [14]. This operator uses a definedconstant crossover probability which decides the probabilityfor crossover for every gene of a chromosome.

6.3.5 TerminationAs explained in Chapter 6.2.5, the algorithm has to ter-

minate after a runtime of 15 minutes. After reaching thislimit the solution with the best identified fitness is chosenas the best solution for the multi-step approach.

7. CONCLUSION & OUTLOOKIn this paper an innovative way for dealing with multi-

objective optimization problems has been presented. By in-troducing four different goals of the “CloudLogistic”-conceptwith its individual requirements a multi-step approach hasbeen motivated. It has been shown, that the presentedmethod is a suitable approach for dealing with different kindof goals by deviding them into two independent steps. Forthe first step, two innovative problem-specific genetic oper-ators have been created. Therefore, a generation-based ap-proach and a usage-based approach have been combined tocreate a useful mutation procedure. Furthermore, a usage-based approach has been used for realizing a problem-specific

76 2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS)

crossover operator. This step has two outcomes: An upperbound for the single goal and a couple of good solutions forentering the second optimization step. By chosing the inves-tigated upper bound as a new hard constraint, the secondstep has been enabled to search for suitable solutions regard-ing to the other optimization goals while reaching the goalof the first step within its defined boundaries.

The evaluation of this concept has to be subject of fur-ther works. Especially the empirical research for chosingsuitable popsizes and for refining the specific parametersof the generation-based / usage-based approach have to beevaluated. Moreover, this approach has to be transferred toother multi-objective problems with similar requirements toelaborate the flexibility and the suitability for daily use.

8. REFERENCES[1] J. Berger and M. Barkaoui. A hybrid genetic

algorithm for the vehicle routing problem with timewindows. In Advances in Artificial Intelligence. 12thBiennial Conference of the Canadian Society forComputational Studies of Intelligence, pages 44–51.Morgan Kaufmann, 1998.

[2] J. Berger and M. Barkaoui. A route-directed hybridgenetic approach for the vehicle routing problem withtime windows. INFOR, 41:179–194, 2003.

[3] J. L. Blanton, Jr. and R. L. Wainwright. Multiplevehicle routing with time and capacity constraintsusing genetic algorithms. In Proceedings of the 5thInternational Conference on Genetic Algorithms,pages 452–459, San Francisco, CA, USA, 1993.Morgan Kaufmann Publishers Inc.

[4] L. Bodin and B. Golden. Classification in vehiclerouting and scheduling. Networks, 11(2):97–108, 1981.

[5] L. Bodin, B. Golden, A. Assad, and M. Ball. Routingand Scheduling of Vehicles and Crews: The State ofthe Art. Computers and Operations Research,10(2):63–212, 1983.

[6] O. Braysy and M. Gendreau. Vehicle RoutingProblem with Time Windows, Part I: RouteConstruction and Local Search Algorithms.Transportation Science, 39(1):104–118, 2005.

[7] O. Braysy and M. Gendreau. Vehicle RoutingProblem with Time Windows, Part II: Metaheuristics.Transportation Science, 39(1):119–139, 2005.

[8] Bundesministerium fur Verkehr Bau undStadtentwicklung (BMVBS). Prognose derdeutschlandweiten Verkehrsverflechtung 2025, 2007.

[9] Bundesministerium fur Verkehr Bau undStadtentwicklung (BMVBS). Masterplan Guterverkehrund Logistik, 2008.

[10] G. B. Dantzig and J. H. Ramser. The TruckDispatching Problem. Management Science,6(1):80–91, 1959.

[11] R. Dondo, C. A. Mendez, and J. Cerda. An optimalapproach to the multiple-depot heterogeneous vehiclerouting problem with time window and capacityconstraints. Latin American Applied Research,33(2):129–134, 2003.

[12] H. Gehring and J. Homberger. A parallel hybridevolutionary metaheuristic for the vehicle routingproblem with time windows. In Proceedings ofEUROGEN99–Short Course on Evolutionary

Algorithms in Engineering and Computer Science,pages 57–64, 1999.

[13] H. Gehring and J. Homberger. Parallelization of atwo-phase metaheuristic for routing problems withtime windows. Journal of Heuristics, 8(3):251–276,May 2002.

[14] I. Gerdes, F. Klawonn, and R. Kruse. EvolutionareAlgorithmen: Genetische Algorithmen - Strategien undOptimierungsverfahren - Beispielanwendungen.Computational Intelligence. Vieweg+Teubner Verlag,2004.

[15] D. E. Goldberg. Genetic Algorithms in SearchOptimization and Machine Learning. Addison-Wesley,1989.

[16] S. Helmreich and H. Keller. FREIGHTVISION -Sustainable European Freight Transport 2050:Forecast, Vision and Policy Recommendation.Springer-Verlag, Heidelberg, 2011.

[17] W.-K. Ho, J. C. Ang, and A. Lim. A hybrid searchalgorithm for the vehicle routing problem with timewindows. International Journal on ArtificialIntelligence Tools, 10:431–449, 2001.

[18] A. Le Bouthillier and T. G. Crainic. A cooperativeparallel meta-heuristic for the vehicle routing problemwith time windows. Computers and OperationsResearch, 32(7):1685–1708, July 2005.

[19] J.-Y. Potvin and S. Bengio. The vehicle routingproblem with time windows part ii: Genetic search.INFORMS Journal on Computing, 8(2):165–172, 1996.

[20] K. C. Tan, T. H. Lee, K. Ou, and L. H. Lee. A messygenetic algorithm for the vehicle routing problem withtime window constraints. In Proceedings of the 2001Congress on Evolutionary Computation CEC2001,pages 679–686, COEX, World Trade Center, 159Samseong-dong, Gangnam-gu, Seoul, Korea, 27-30May 2001. IEEE Press.

[21] S. R. Thangiah. Vehicle routing with time windowsusing genetic algorithms. Applications Handbook ofGenetic Algorithms: New Frontiers, pages 253–278,1995.

[22] C. Tummel, C. Franzen, P. Friedrichsmeier, N. Vossen,P. Wolters, E. Hauck, and S. Jeschke. CloudLogistic -Line-Based Optimization for the Disposition of LTLShipments. In Proceedings of the 17th InternationalSymposium on Logistics, South Africa, Capetown,2012, to be published.

[23] C. Tummel, C. Franzen, E. Hauck, and S. Jeschke. Anincremental online heuristic for evaluating thefeasibility of the m-VRPTWAR. In Proceedings of The3rd International Conference on Logistics andTransport & The 4th International Conference onOperations and Supply Chain Management, Male,Maldives, pages 647,660, 2011.

[24] C. Tummel, C. Franzen, E. Hauck, and S. Jeschke.The Multi-Depot Heterogeneous Fleet VehicleRouting Problem with Time Windows andAssignment Restrictions (m-VRPTWAR). InProceedings of The 3rd International Conference onLogistics and Transport & The 4th InternationalConference on Operations and Supply ChainManagement, Male, Maldives, pages 661,672, 2011.

2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS) 77