1 Introduction
In industry, especially logistics [17] and 24/7 production [26], many decisions must be made under uncertain knowledge about future events. Production planning problems may include tasks appearing over time, which is often expressed via release dates [30], while logistic problems like the traveling salesperson problem (TSP) may be augmented with dynamically changing traffic [11] or – more abstract – with dynamically changing distance matrices [33].
Here, we consider the scenario that repeated decision requests are not independent from the previous ones; rather, every decision reduces the number of possible actions/decisions in future affecting the quality of the best possible solution that can be achieved.
If optimization methods are used to support the decision process, an optimizer could iterate for short time intervals (eras). After each era, a human decision maker (DM) may decide whether the proposed solution is executed or not. This situation is studied in the field of interactive optimization [23] (headword: humanintheloop).
In analyzing the quality of optimization algorithms it is quite common to replace the human DM by a software agent (automatic decision maker) [2] to speed up experiments or to make it amenable to a theoretical analysis. The software agent may obey some simple guidelines or even a complex set of usersupplied preferences to arrive at a decision. The situation becomes even more complex in case of multiple objectives [10]; if the multiobjective optimizer uses the aposteriori approach, the (automatic) DM must additionally decide, which nondominated solution should be picked from the Paretofront.
Here, we also take the approach of replacing the human by an automatic decision maker in case of a dynamic biobjective vehicle routing problem, where the goal is to minimize both the distance traveled by a single vehicle and at the same time minimize the number of unvisited customers which ask for services over time. This special variant of the TSP problem includes the additional problem of subset selection of serviced customers and is similar to the socalled TSP with profits [1]. However, the problem considered here comprises a dynamically growing set of customers, who request service over time.
It is important to note, that the focus of this work is not on the performance of the optimization algorithm but (1) on the impacts of specific (automatic) decision making rules on the final solution and (2) the visualization of subsequent decisions and solutions as a preliminary step towards an (interactive) decision support system. The dynamically generated solutions are compared to the solution of the socalled clairvoyant optimizer which has complete knowledge about the future, i.e., request times of dynamic customers. In our study we run experiments for all possible combinations of a fixed set of automatic DM decision rules to scrutinize as many aspects of the decision rule as possible.
The work is organized as follows: Section 2 details the dynamic multiobjective problem before describing the dynamic multiobjective evolutionary optimization algorithm supporting the decision maker in Section 3. The experimental setup (including the automatic decision maker) and results are described in Section 4. The conclusion and the prospects of this work towards inclusion in interactive decision support systems are presented in Section 5.
2 Problem Description
We consider a dynamic vehicle routing problem for which the overall goal is to have a single service vehicle visiting customer locations from the set of all customer locations.
The dynamic character of this problem originates in the timedependent appearance of customers from . The set of customers is divided into two disjoint subsets: Mandatory customers are known at time and must be visited by the vehicle while dynamic customers ask for service at request time as time passes by. They can either be visited or not. In a realworld context, we may imagine the vehicle as a customer service vehicle with fixed orders and spare time to handle dynamically emerging service requests.
We assume that the vehicle leaves w.l.o.g. at a start depot and ends at an end depot .^{1}^{1}1This assumption is more general than starting and ending at the same depot, although a circular tour may be the normal case in realworld scenarios. The optimization task is to (1) minimize the overall tour length and at the same time (2) minimize the number of unserved dynamic customers. Undoubtedly, the goals conflict with each other and we are faced with a complex dynamic combinatorial multiobjective optimization problem (MOP), for which we strive to find a set of (near) optimal compromises. Here, we adopt the notion of Paretodominance for a definition of optimal compromise solutions: for two vehicle tours and we say that dominates , if is not worse in any objective and strictly better in at least one objective [12]. The set of all nondominated solutions is termed the Paretoset, its image in the objective space is called the Paretofront. Hence, in our scenario for each point in time a biobjective problem needs to be solved and the problem can be fully described by the sequence of all tradeoff solution sets. Since time is continuous, this approach is infeasible in practice. A common approach is to discretize the time horizon, i. e., the time interval in which dynamic customers pose requests, into a number of phases , socalled eras, of length (see, e. g. [31]).
At the beginning of each era time has already passed and we may consider the set , with being the set of dynamic customers, which asked for service before time , as a static MOP. Tackling this static MOP with the algorithm of our choice results in an approximation of the Paretoset. Finally, a decision maker (DM) is given the resulting set of tradeoff solutions in each era and has to decide on how to guide the vehicle on the road until the beginning of the next era where then, more knowledge about dynamic customers becomes available. A crucial aspect here is that in each era time already passed and consequently the vehicle might already have visited a subset of mandatory and/or optional customer locations. These decisions are irreversible and (1) may have a strong impact on the achievable solution quality (this is because a part of the solution space may become infeasible) and (2) exhibit a strong dependence on the decisions made by the DM in foregoing eras.
Static formulations of biobjective vehicle routing problems or socalled traveling salesperson problems with profits [14] have been addressed by several authors so far, e. g. proposing exact constraint methods [4], approximations schemes [15]
or metaheuristics
[20, 1]. Additionally, dynamic decision making gained some attention in the context of vehicle routing problems in general [29, 25]. However, work on the intersection, i. e. dynamic multiobjective vehicle routing problems is still rare. Braekers et al. [9] show in their extensive literature review that less then 3% of the literature between 2009 and 2015 address dynamic aspects. According to them, mentionable research includes work by [32], [22], [21], [18], and [3]. This list may be extended by mostly evolutionary approaches of [28], [16]. Although of dynamic nature, the problem formulations have very diverse characteristics like moving service time windows, multiple vehicles, or changing structures of the network. Own work addressed a clairvoyant and nondynamic variant of the here discussed problem with an evolutionary multiobjective algorithm (EMOA) [24]. We enhanced the EMOA in a followup work by local search integration into the evolutionary search process [7]. This clairvoyant approach is considered here as reference approach. A sophisticated dynamic variant was presented in [8].3 The Dynamic MultiObjective Evolutionary Algorithm
We adopt the DEMOA introduced in [8]. Note that the focus of this work is on the influence of subsequent decisionmaking. Therefore, and due to space limitations, we omit most implementation details and present the working principles. For detailed pseudocode we refer the interested reader to [8]. Also the implementation is available in a public GitHub repository^{2}^{2}2https://github.com/jakobbossek/dynvrp. We advice the reader to consult Fig. 1 for visual support while reading the following text.
The input for the DEMOA is a problem instance , a time resolution , a number of eras and a population size . The optimization process starts at time and the algorithm treats the problem as a sequence of static MOPs (see Section 2). Note however, that the first era is a special case, since , i. e., no dynamic requests arrived so far, and there is no possibility to vary the second objective. Hence, in the 1st era, a singleobjective Hamiltonian path problem (HPP) on the set has to be solved. An approximate solution is calculated with the stateoftheart solver EAX [27] for the symmetric TravellingSalespersonProblem (TSP) after reducing the HPP to a symmetric TSP problem by a sequence of transformations [19]. Note that in the first era the decisionmaker has no choice as there is just a single solution (see era 1 in Fig. 1). In subsequent eras time already passed, is nonempty and as a consequence the problem turns into a true multiobjective problem. Here, the DEMOA calls a static EMOA whose internals are discussed in the following. The EMOA initializes a multiset of candidate solutions. Each candidate solution
is fully described by three vectors of length
. A binary vector indicates which customers are to be visited by the service vehicle (note that the depots 1 and need to visited in any case and are thus not encoded).^{3}^{3}3We want to point out that the current implementation knows the total number of customers in advance for legacy reasons. However, it only operates on those customers, who asked for service before time . Clearly, it is straight forward to adapt the implementation into a true blackbox scenario, where the number of dynamic requests is not known apriori. Another vector holds a permutation of , i.e., the actual tour where during fitness evaluation only those entries with are considered. Finally, the vectorstores percustomer mutation probabilities. If
, the corresponding customer is fixed and not affected by mutation. While in the second era individuals are generated at random (fixing mandatory customers by setting and for ), in eras more effort is put into the initialization to transfer as much information from the solution set of the preceding era as possible. The challenge here is that once era starts, the vehicle may already have visited dynamic customers with request times (this is illustrated by means of example in Fig. 1 last column. Here, bold edges show the fixed, already driven initial tour) given by the decision at the end of the previous era. Fig. 1). As a consequence, those customers cannot be inactive and hence need to be treated as mandatory customers by the EMOA in all upcoming eras. Moreover, the initial tour, i.e., the part of the tour that has already been driven by the vehicle, needs to be identical for all feasible individuals. Here, the EMOA relies on a sequence of repairing mechanisms.Given the population the algorithm continues by adopting a strategy with NSGAII [13] survival selection. Variation is based on feasibilitypreserving mutation. Here, each bit is flipped independently with probability . Subsequently, swapmutation alters the permutation string : with probability a sequence of exchanges is performed. In addition, every generations the population is boosted towards shorter tours by applying EAX localsearch where the EMOA accounts for the fact that certain nodes have already been visited. Once the stopping condition has been triggered, e.g., a maximum number of generations has been reached, the solution set is presented to a decisionmaker who has to decide on exactly the solution which determines the adaptation of the ongoing vehicle route and which serves as a template for the initialization of the population in the next era.
4 Computational Experiments
4.1 Experimental Setup
In order to gain insights into the decision making process we conducted a twostage study. In a first series of experiments we perform a systematic study of decision making strategies. Subsequent experiments focus on a selected sample of decision making strategies on a broader set of instances in order to confirm the lessons learned.
For the exhaustive experiments we selected three structurally different instances from the pool of instances introduced in [24]: one instance with customer locations spread uniformly at random in the Euclidean plane and two clustered instances with two and three groups of instances respectively. All instances share mandatory customers (including depots) and dynamic customers with customers in total. For details on the generation process we refer to [24]. We fixed the number of eras and considered three different ranking based rules for the decision maker in each era. For ranking, the solutions of the approximation set obtained in era are sorted in ascending order of tour length and therefore in descending order of the number of unvisited customers. Let denote this order. The rank decision maker () then decides for the solution with . Note that small values of favor solutions with short tours whereas values closer to 1 put a higher emphasis on keeping the number of unvisited dynamic customers low. Our setup considers in each era^{4}^{4}4We do not consider the extremal values and for two reasons: (1) to avoid a combinatorial explosion of possible configurations for exhaustive evaluation; (2) extremal decisions are usually unrealistic as they either imply to ignore all optional customers (0) or to accept every optional customer (1).; we use a tuple notation for sequences of decision maker decisions, termed decision paths in the following, e. g., describes the decision path where the DM puts more emphasis on short tours in the first four eras but decides to cover more dynamic requests in the last three eras.
Parameter  Setting 

Generations per era  65 000 
100  
0.6  
Local search at generations  initial, halftime, last 
Time limit for local search  1s 
We run experiments for all decision paths, i. e., we cover all kinds of decision scenarios. At each stage, five independent runs were performed for each of the three instances resulting in a set of experiments in total.
The results of this “exhaustive” experimentation served as starting point for subsequent experiments on a broader set of benchmark instances; all 75 instances with introduced in [24]. However, due to combinatorial explosion only a small subset of four decision paths (with different outcomes in the last era) were considered here for an extensive analysis (for details on the selection process, see Section 4.2). For each combination of problem instance and decision path we run the DEMOA 25 times independently in this series of experiments. The parameter configuration of the DEMOA follows the suggestions in [8] and is listed in Table 1.
4.2 Results
Next, we investigate the influence of considered graph topologies as well as the implications of final and intermediary decision making onto the solution development over time. Therefore, we perform a stepwise narrowing of perspective to focus on interesting insights for our considered instances, topologies, and decision strategies in the context of our exhaustive experimental results.
Tour length  #Dyn. customers  
Type  Last decision  Mean  Std  Mean  Std 
0.25  997.7  59.94  22.495  5.7486  
0.50  1070.3  57.16  15.645  4.3148  
2 clusters  0.75  1174.9  65.14  8.105  2.4266 
0.25  1227.6  28.61  37.654  0.7662  
0.50  1416.4  38.62  24.854  0.9167  
uniform  0.75  1657.1  42.32  12.069  0.8743 
4.2.1 General observations
In Figure 2 we provide a first overview of the results for all decisions in each era and for all decision paths for the uniform instance and the instance with two clusters^{5}^{5}5Results for the third instance with three clusters are omitted here since these are very similar to those of the 2cluster instance.. Note that in order to compare our results to the clairvoyant EMOA approach – either visually or by performance metrics – we transformed the results of all eras to the aposteriori solution space, which covers the whole potential of arbitrary decision paths.^{6}^{6}6We explain this transformation in more in detail here: in the first era, we have zero dynamic requests and consequently zero unvisited dynamic customers. However, in the aposteriori solution space this solution corresponds to
unvisited dynamic customers. Therefore, in order to make solutions comparable, a linear transformation of the second objective to the clairvoyant EMOA solution space is required.
From this highlevel perspective, we can identify an interesting property of solution distribution, which not only depends on the applied decision strategies but is strongly related to the considered topology of the instances: For clustered topologies, tour length tends to expose larger variability in later eras, and – on a first glance – no matter which combination of decisions is used, while in uniform topologies distinct clusters of solutions represent the different weighting of decision strategies.
4.2.2 Topology and final decision making influence
The observed results can be explained by a deeper analysis of the experiments. Here, we focus on two representative topologies consisting of two clusters and a uniformly distributed set of customers, respectively. For both topologies, each instance, and all decision maker strategies, we determine the upper bound of unvisited customers for the last era and compute the mean upper bound as well as standard deviation. More detailed values for mean and standard deviation of the upper bound of both objectives – and split according to the final preference – are available in Table
2. Each upper bound of unvisited customers is determined by the already visited customers on the traveled partial tour, which is the result of decisions made during previous eras.For clustered topologies (ref. to Figure 2, left) we find a low mean upper bound and large standard deviation, while for the uniform topology the mean upper bound of unvisited customers is rather high with little variance. Consequently on average, for clustered topologies, the decision maker preference at the onset of the last era allows only little flexibility for final solutions. This leads to the stronger focus on the lower right area of objective space. For uniform instances, the on average larger upper bound of unvisited customers leads to a larger and less flexible range for diverse forming of (intermediary) solutions over all eras and finally to more distinct clustering solutions in objective space.
The coloring of final solutions with respect to the final decision maker preference in Figure 3 provides additional insights into the partitioning of the exhaustively generated solutions. In fact, we find that in both cases the last decision preference has significant influence on the solution position. For the uniform instances, however, the preferences are more distinct due to more certain planning flexibility in the final era.
From Figure 3 and Table 2 we also conclude that solutions for uniform topologies expose less variance in quality and converge closer to the aposteriori solution fronts determined by application of the clairvoyant EMOA. As a consequence, more solutions of the dynamic approach individually outperform solutions on the aposteriori front.^{7}^{7}7The aposteriori front used here was achieved as nondominated set of the union of ten clairvoyant EMOA runs. This effect is rooted in decreasing complexity of the tour planning component of the biobjective problem under the successive dynamic decision making [8]
. Due to decision making over time eras, partial tours are already completed such that the combinatorial decision space shrinks to the still available customers leaving the tour planning problem with less degrees of freedom. Clearly, this observation holds for clustered instances, too. Here, the mean upper bound is even lower. However, in the clustered setting the service vehicle might need to travel back and forth between clusters in order to fulfill the decision maker preferences which oftentimes might lead to enlarged tour length in particular in late eras and a high preference on the second objective. We will catch up on this important aspect later on.
4.2.3 Intermediate decision making
While so far we analyzed the final decision maker preference for some instances, the following stage of investigation is focused on the influence of intermediary decisions. A subset of decision paths, which led to the nondominated as well as completely dominated solutions in the last era over all considered topologies is selected. For this selection, we detail the effects of decision steps that yielded very good and very bad results, compared on final solutions. For the following discussion, we investigate the results up to specific solution phenotypes, i. e., the development of specific tours over time. We present detailed results for two exemplary but representative out of 75 instances.
Figure 4 provides detailed insight into the development of solutions under different DM strategies. In order to visualize the effect of the permutation of decisions inside a strategy (and also due to space limitations), we restrict ourselves to one uniform topology and one topology with twoclusters again and show results of single representative runs. At the top of the figure we show the nondominated solutions of all eras of these topologies regarding four strategies that follow (a) only preferences, (b) first four times and then three times preferences, (c) the inverse strategy to (b), and (d) only decisions. Below the nondominated solutions, we visualize the development of exemplary tours of the solutions. We omit era 1, where the vehicle has not traveled yet and also omit some intermediate tours to show a second example tour. For each tour, the decision path via intermediate solutions is included into the respective top figure and annotated with the era number.
For the clustered topology, we find a strategysensitive behavior that is related to when (in which era) preferences are used. The overall observation is that preferences, which do not put a strong focus on minimizing the number of unvisited customers (represented by a sequence of only preference) lead to rather short tours (according to the second objective). In these tours, the vehicle transfers to the other cluster only once. Introducing a strong preference for visiting all customers (represented by a sequence of only preferences) forces the vehicle to transfer multiple times between clusters, see Figure 4 bottom right plot. This behavioral changes are also observable for planned tours, when preferences mix, e.g., when a strong preference for visiting many customers is only present at the beginning or the end of the strategy sequence. In many sequences with changing decision preferences (not shown here as figure), we observe that planned transfers in early eras vanish in following eras (due to preferences later). From this behavior we conclude, that intermediary preference ordering can have decisive influence on the solution generation process for clustered instances. With respect to minimization of unvisited customers and dynamic appearance of customers, decision maker preferences have different degrees of greediness: a preference is far less greedy than a strategy and often allows the vehicle during tour planning to remain in the current cluster, as far fewer customers need to be served.
Introducing a more greedy strategy often forces a vehicle transfer to serve the preferred amount of customers. The observations however show, that flexible replanning is still possible as long as the partial tour has not been realized.
Considering the exemplary but representative results from Figure 4, we can conclude, that strategy preferences are less important compared to the clustered case. As mandatory and dynamic customers are uniformly distributed, planned tours do not need to be changed extensively in order to visit or ignore a customer. When we switch preferences from less greedy to more greedy, customers “on the way” can be included. The same way, dynamic customers can be excluded again from a planned tour, often without significant changes in the overall tour length.
The described effects for clustered and uniform topologies can also be observed in the resulting nondominated fronts for the eras. For clustered topologies, the front usually exposes a gap, which corresponds to the additional traveled distance in size. It appears, that solutions cannot be realized without transferring the vehicle multiple times. Such case usually does not happen for uniform topologies such that the approximated efficient front does not expose a gap.
4.2.4 Performance measurement
In order to support our observations from the previous paragraphs we continue with indicatorbased performance assessment of the DEMOA in comparison to the approximation sets calculated by the clairvoyant EMOA. We aim to quantify the quality of the overall final approximation set in the last era. We use the hypervolume indicator [34] to measure the space enclosed by a reference set (nondominated set of the union of clairvoyant EMOA approximation and all front approximations for the problem instance obtained by the DEMOA) and the DEMOA approximation . We restrict our analysis to the DEMOA approximation sets of the final era only. We take account for the upper bound that restricts the possible number of unvisited dynamic customers in the last era as follows: only solutions of the clairvoyant EMOA whose second objective is lower or equal to the maximum upper bound in the last era for each instance over all 25 performed runs are taken into consideration. We want to stress that this comparison – and the one in the next paragraph – is obviously highly unfair, i. e., (1) the clairvoyant EMOA has a clear advantage over the dynamic approach due to its apriori knowledge of request times and (2) the solutions of the clairvoyant might not even be feasible anymore in the last era. Hence, we do not expect the DEMOA to beat the clairvoyant EMOA by any means. Instead, our goal is to learn how close we can approach the clairvoyant solutions with the dynamic approach. Figure 5 shows the distributions of the HVindicator split by instance and the four DMstrategies discussed before. We show results for a random sample of 10 uniform and 10 clustered instances. The plots confirm our previous observations: in the case of customers distributed uniformly at random in the Euclidean plane the final approximation sets are close to the reference set. In contrast, for clustered topologies the situation is different. Here, as the vehicle possibly needs to transfer between clusters multiple times, the oracleperspective of the clairvoyant EMOA is much more advantageous and has a much larger impact. In other words, the HVindicator is less close to the one of the clairvoyant EMOA.
5 Conclusions
For biobjective vehicle routing, problem dynamics have to be efficiently addressed while suitable decision maker strategies accounting for the tradeoff of minimizing overall tour length and maximizing the number of served customers are required simultaneously. We build upon previous work which provides a sophisticated dynamic EMOA hybridized with local search and specifically investigate the influence of respective decision maker preferences and strategies.
As vehicle tours for a given problem instance evolve over the focused time horizon, decision maker preferences regarding both objective functions may change in the course of the day. We assume that the decisions for possibly altering a predefined tour based on new customer requests have to be made at predefined time intervals which of course subsequently impacts optimization algorithm behavior and thus also influences solution selection decisions which have to be made at later stages.
In systematic experiments, we investigated the influence of decision paths, i.e. sequences of (possibly different) decision maker preferences and solution selections. We present a decision support system enhanced by informative figures visualizing the vehicle tour over time and the characteristics of the candidate tradeoff solutions at the points of required decisions.
We confirm the reasonable suspicion that decision making is sensitive to the underlying problem topology. For clustered topologies, intermediate decisions should be considered carefully, as too greedy approaches can lead to multiple vehicle transfers between clusters and massively deteriorate solution quality. For uniform instances, sensitivity is low and the last decision for the optimal tradeoff solution is of major importance for final tour quality. Consequently, it is important for the decision maker to estimate the customer location topology for adjusting the greediness of decision making.
Future work directions are manifold with the most promising being listed below:

We see much room for algorithmic improvements. The insights gained in this paper suggest that biased mutation (e. g., activating customers in the current cluster with higher probability) may have beneficial effects on the solution quality of the DEMOA. Furthermore, utilizing probabilistic models to predict upcoming customer requests can be leveraged to achieve more thoughtful algorithmic tour planing.

Last but not least the major goal is to refine the presented approach in terms of providing toolsupport for informative interactive decision making in this highly dynamic environment. Moreover, we will include predefined agentbased decision maker paths into the algorithm which adapt to problem topology characteristics via automatically extracting problem features and which can be adjusted if needed.
References

[1]
(2018)
Experimental analysis of design elements of scalarizing functionsbased multiobjective evolutionary algorithms
. Soft Computing. External Links: Document Cited by: §1, §2.  [2] (2018) Artificial decision maker driven by PSO: an approach for testing reference point based interactive methods. In Parallel Problem Solving from Nature  PPSN XV  15th International Conference, Proceedings Part I, pp. 274–285. Cited by: §1.
 [3] (2013) An adaptive evolutionary approach for realtime vehicle routing and dispatching. Comput. Oper. Res. 40 (7), pp. 1766–1776. External Links: ISSN 03050548, Document Cited by: §2.
 [4] (2009) An exact [epsilon]constraint method for biobjective combinatorial optimization problems: application to the traveling salesman problem with profits. European Journal of Operational Research 194 (1), pp. 39–50. Cited by: §2.
 [5] (2017) Solving the biobjective traveling thief problem with multiobjective evolutionary algorithms. In 9th International Conference on Evolutionary MultiCriterion Optimization  Volume 10173, EMO 2017, Berlin, Heidelberg, pp. 46–60. External Links: ISBN 9783319541563, Link, Document Cited by: 1st item.

[6]
(2013)
The travelling thief problem: the first step in the transition from theoretical problems to realistic problems.
In
IEEE Congress on Evolutionary Computation
, pp. 1037–1044. Cited by: 1st item.  [7] (2018) Local Search Effects in Biobjective Orienteering. In Proc. of the Genetic and Evolutionary Computation Conference, GECCO ’18, New York, NY, USA, pp. 585–592. External Links: ISBN 9781450356183, Link, Document Cited by: §2.
 [8] (2019) BiObjective Orienteering: Towards a Dynamic MultiObjective Evolutionary Algorithm. In Evolutionary MultiCriterion Optimization, Lecture Notes in Computer Science (LNCS), Vol. 11411, pp. 1–12. Cited by: §2, §3, §4.1, §4.2.2.
 [9] (2016) The vehicle routing problem: State of the art classification and review. Computers & Industrial Engineering 99, pp. 300–313. External Links: ISSN 03608352, Document Cited by: §2.
 [10] J. Branke, K. Deb, K. Miettinen, and R. Slowinski (Eds.) (2008) Multiobjective optimization: interactive and evolutionary approaches. Springer. Cited by: §1.
 [11] (201206) Dynamic traveling salesman problem: value of realtime traffic information. IEEE Transactions on Intelligent Transportation Systems 13 (2), pp. 619–630. External Links: Document, ISSN 15249050 Cited by: §1.
 [12] (2006) Evolutionary Algorithms for Solving MultiObjective Problems (Genetic and Evolutionary Computation). SpringerVerlag New York, Inc., Secaucus, NJ, USA. External Links: ISBN 0387332545 Cited by: §2.

[13]
(2002)
A fast and elitist multiobjective genetic algorithm: NSGA–II
. IEEE Transactions on Evolutionary Computation 6 (2), pp. 182–197. Cited by: §3.  [14] (2005) Traveling salesman problems with profits. Transportation Science 39 (2), pp. 188–205. External Links: Document, Link, https://pubsonline.informs.org/doi/pdf/10.1287/trsc.1030.0079 Cited by: §2.
 [15] (2013) Approximation schemes for biobjective combinatorial optimization and their application to the TSP with profits. Computers & Operations Research 40 (10), pp. 2418–2428. Cited by: §2.
 [16] (2014) A multiobjective vehicle routing and scheduling problem with uncertainty in customers’ request and priority. Journal of Combinatorial Optimization 28, pp. 414–446. Cited by: §2.
 [17] (2017) Supply chain network design under uncertainty: a comprehensive review and future research directions. European Journal of Operational Research 263 (1), pp. 108–141. Cited by: §1.
 [18] (2012) An improved lns algorithm for realtime vehicle routing problem with time windows. Comput. Oper. Res. 39 (2), pp. 151–163. External Links: ISSN 03050548, Document Cited by: §2.
 [19] (198311) Transforming asymmetric into symmetric traveling salesman problems. Oper. Res. Lett. 2 (4), pp. 161–163. Cited by: §3.
 [20] (2008) Multiobjective metaheuristics for the traveling salesman problem with profits. Journal of Mathematical Modelling and Algorithms 7 (2), pp. 177–195. Cited by: §2.
 [21] (2012) A comparative study between dynamic adapted pso and vns for the vehicle routing problem with dynamic requests. Appl. Soft Comput. 12 (4), pp. 1426–1439. External Links: ISSN 15684946, Document Cited by: §2.
 [22] (2011) Online vehicle routing and scheduling with dynamic travel times. Comput. Oper. Res. 38 (7), pp. 1086–1090. External Links: ISSN 03050548, Document Cited by: §2.
 [23] (2015) A review and taxonomy of interactive optimization methods in operations research. ACM Transactions on Interactive Intelligent Systems 5 (3). Cited by: §1.
 [24] (2015) Evaluation of a MultiObjective EA on Benchmark Instances for Dynamic Routing of a Vehicle. In Proc. of the Genetic and Evolutionary Computation Conference, GECCO ’15, New York, NY, USA, pp. 425–432. External Links: ISBN 9781450334723, Link, Document Cited by: §2, §4.1, §4.1.
 [25] (2011) Anticipatory Optimization for Dynamic Decision Making. Operations Research/Computer Science Interfaces Series, Vol. 51, Springer New York. Cited by: §2.
 [26] (2006) Models for production planning under uncertainty: a review. International Journal of Production Economics 103 (1), pp. 271–285. Cited by: §1.
 [27] (2013) A powerful genetic algorithm using edge assembly crossover for the traveling salesman problem. INFORMS Journal on Computing 25 (2), pp. 346–363. Cited by: §3.
 [28] (2018) A framework for solving realtime multiobjective VRP. In Advanced Concepts, Methodologies and Technologies for Transportation and Logistics, J. Zak, Y. Hadas, and R. Rossi (Eds.), Advances in Intelligent Systems and Computing, Vol. 572, pp. 103–120. Cited by: §2.
 [29] (2013) A review of dynamic vehicle routing problems. European Journal of Operational Research 225 (1), pp. 1–11. Cited by: §2.
 [30] (2012) Scheduling: theory, algorithms, and systems. 4th edition, Springer, New York. Cited by: §1.
 [31] (2013) Dynamic multiobjective optimization: a survey of the stateoftheart. In Evolutionary Computation for Dynamic Optimization Problems, S. Yang and X. Yao (Eds.), pp. 85–106. Cited by: §2.
 [32] (2010) The dynamic multiperiod vehicle routing problem. Comput. Oper. Res. 37 (9), pp. 1615–1623. External Links: ISSN 03050548, Document Cited by: §2.

[33]
(200408)
An approach to dynamic traveling salesman problem.
In
Proceedings of 2004 International Conference on Machine Learning and Cybernetics (IEEE Cat. No.04EX826)
, Vol. 4, pp. 2418–2420 vol.4. External Links: Document, ISSN Cited by: §1.  [34] (2000) Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evol. Comput. 8 (2), pp. 173–195. External Links: Document, ISSN 10636560, Link Cited by: §4.2.4.
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