Abstract
In many decision making cases, we may have a hierarchical situation between different optimization tasks. For instance, in production scheduling, the evaluation of the tasks assignment to a machine requires the determination of their optimal sequencing on this machine. Such situation is usually modeled as a BiLevel Optimization Problem (BLOP). The latter consists in optimizing an upperlevel (a leader) task, while having a lowerlevel (a follower) optimization task as a constraint. In this way, the evaluation of any upperlevel solution requires finding its corresponding lowerlevel (near) optimal solution, which makes BLOP resolution very computationally costly. Evolutionary Algorithms (EAs) have proven their strength in solving BLOPs due to their insensitivity to the mathematical features of the objective functions such as nonlinearity, nondifferentiability, and high dimensionality. Moreover, EAs that are based on approximation techniques have proven their strength in solving BLOPs. Nevertheless, their application has been restricted to the continuous case as most approaches are based on approximating the lowerlevel optimum using classical mathematical programming and machine learning techniques. Motivated by this observation, we tackle in this paper the discrete case by proposing a CoEvolutionary MigrationBased Algorithm, called CEMBA, that uses two populations in each level and a migration scheme; with the aim to considerably minimize the number of Function Evaluations (FEs) while ensuring good convergence towards the global optimum of the upperlevel. CEMBA has been validated on a set of bilevel combinatorial productiondistribution planning benchmark instances. The statistical analysis of the obtained results shows the effectiveness and efficiency of CEMBA when compared to existing stateoftheart combinatorial bilevel EAs.
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Appendices
Appendix 1: A comparison between the Das&Dennis and the DSDM methods.
This first appendix illustrates the difference between the Das&Dennis (Das and Dennis 1998), and the DSDM (Chaabani et al. 2015) methods as shown in Fig. 6. For the case of a continuous search space, the Das&Dennis method could be used. However, for the case of a discrete search space, the Das&Dennis method is inapplicable. The DSDM method is a variant of Das&Dennis method and it could be used to generate a set of points in discrete search spaces. We must mention here that the Das&Dennis method generates a set of solutions in the objective space, while the DSDM method works in the decision space. The distribution of the reference points for the Das&Dennis method with 3objective optimization problem (M = 3) and a spacing of \(\delta = 0.2\) (P = 5) is presented in Fig. 6a. Thus, 21 reference points (H = 21) are generated in a normalized hyperplane. We mention here that the reference directions are represented by the lines, which are constructed from the origin to each of these reference points. Figure 6b illustrates the obtained results for the DSDM method with three decision variables where the domains are: \(D_{x_{1}}\)= [0,2,5,13], \(D_{x_{2}}\)= [4,7,9,17], and \(D_{x_{3}}\)= [5,8,11,16]. In order to generate the reference points with the DSDM method in a discrete space, a uniform spacing noted \(\delta _{i}\) is calculated for each decision variable as follow: \(\delta _{i} = max_{i}/P\) (i is the decision variable number, P is a fixed parameter based on the dimension of the problem). Thus, the obtained \(\delta _{i}\) for a P = 3 are: \(\delta _{1} = 13/3=4\), \(\delta _{2} = 17/3=5\), and \(\delta _{3} = 16/5=5\). After that, the range values (\(R_{i}\)) are generated for each decision variable by adding the first value in each range on a set \(R_{i}\), and determining the following \(R_{i}\) members that obey the \(\delta _{i}\). Thus, the obtained range values are as follows: \(R_{1} = [0,5,13]\), \(R_{2} = [4,9,17]\), and \(R_{3} = [5,11,16]\). The obtained solutions for this example are: (0,4,5), (0,9,11), (0,17,16), (5,4,5), (5,9,11), and (13,4,5).
Appendix 2: Illustration of the interaction process between the two levels in a BLOP
This second appendix is devoted to illustrate in details the interaction process between the upperlevel and the lower one in bilevel optimization. Indeed, these two levels are dependent on each other because: (1) the upperlevel solution vector \(x = (x_{u}, x_{l})\) could not be evaluated without optimizing its corresponding solution subvector \(x_{l}\) and (2) the lowerlevel solution \(x_{l}\) could not be evaluated without receiving its corresponding upperlevel solution subvector \(x_{u}\) as fixed parameter (Huang and Wang 2020; Sinha et al. 2020, 2017b). For this reason, to precisely evaluate any upperlevel solution \(x = (x_{u}, x_{l})\), we need to effectively approximate the optimal lowerlevel solution \(x_{l}^{*}\) that corresponds to the upperlevel solution subvector \(x_{u}\), which acts as a fixed parameter within the lowerlevel search process. In the following, we describe the sequence of steps required to well compute the fitness function of an upperlevel solution \(x = (x_{u}, x_{l})\) as illustrated by Fig. 7. First, the subvector of upperlevel decision variables \(x_{u}\) is passed as a fixed parameter to the lowerlevel search algorithm. Second, this latter evolves a population of lowerlevel solutions (i.e., lowerlevel decision variable vectors) for a number of generations with the aim to approximate the optimal follower solution \(x_{l}^{*}\) corresponding to the parameter \(x_{u}\). Third, once the lowerlevel algorithm termination criterion is met, the obtained approximation of \(x_{l}^{*}\) is sent to the upperlevel and thus the upperlevel solution vector becomes \(x = (x_{u}, x_{l}^{*})\). As illustrated by Fig. 7, to evaluate the upperlevel solution \(x^{A} = (x_{u}^{A}, x_{l}^{A})\), \(x_{u}^{A}\) is passed as fixed parameter to the lowerlevel EA. The latter executes an evolutionary process to approximate the optimal lowerlevel solution \({x_{l}^{A}}^{*}\) corresponding to \(x_{u}^{A}\). Based, on Fig. 7, the obtained approximation is the solution vector E. Finally, E is assigned to \(x_{l}^{A}\); and thus the upperlevel solution becomes \(x^{A} = (x_{u}^{A}, E)\). The fitness function of \(x^{A}\) could now be computed. It is important to note that the better the approximation E is, the higher the precision of the fitness evaluation of \(x^{A} = (x_{u}^{A}, x_{l}^{A})\) is. We conclude that the quality of any upperlevel solution vector \(x = (x_{u}, x_{l})\) depends on two main factors that are: (1) the quality of the \(x_{u}\) subvector that depends on the values of its components (upperlevel variables) and (2) the quality of its obtained corresponding optimal lowerlevel \(x_{l}\) solution approximation (lowerlevel variables). For this reason, the lowerlevel algorithm plays a crucial role in the precise evaluation of each upperlevel solution quality (upper fitness value). Differently speaking, the better the quality of \(x_{l}^{*}\) for a particular upperlevel solution \(x = (x_{u}, x_{l})\) is, the higher the precision of the fitness computation of \(x = (x_{u}, x_{l})\). This high precision allows the bilevel algorithm to compute the fitness function values of the upper population with more exactitude and thus detecting promising search directions in the upperlevel search space more effectively and efficiently. This is the key characteristic of the interaction process between the upperlevel algorithm and the lowerlevel one in bilevel optimization. It is important to note that if we consider the case of a poor lowerlevel search process, the upper level fitness values computations will be biased with noise; which will significantly deteriorate the search process of the upperlevel algorithm. For an extremely bad case, a very poor lower level search process could make the upperlevel algorithm behavior acting like a random search. For this reason, there is a need to evaluate the performance of the compared algorithms’ lowerlevels search methods. To do so, Legillon et al. (2012) proposed the direct rationality and the weighted one (defined in Sect. 4.3) as two metrics to evaluate the performance of a lowerlevel algorithm (cf. Algorithm 4). The former expresses the mean number of times the lowerlevel algorithm was able to improve the solutions with respect to the population of the previous generation, while the latter evaluates the mean quantity (magnitude) of the improvements. In summary, we conclude that: (1) the lowerlevel optimality approximation is considered in CEMBA as a necessary constraint that appears as one of the upperlevel constraints and (2) the lowerlevel performance should be evaluated for all compared algorithms to assess the quality of its contribution in the precise evaluation of the upperlevel solutions, and hence in the effective exploration of promising search directions in the upperlevel search space.
To empirically show the importance of the optimization of the lowerlevel objective function (while respecting its constraints), we have conducted a set of experiments using two versions of each of the two following algorithms: CEMBA and CODBACRO:

1.
CEMBAWLO and CODBACROWLO are two variants of CEMBA and CODBACRO (respectively) that do not optimize the lowerlevel objective function (WLO means Without Lowerlevel Optimization); and

2.
The original CEMBA and CODBACRO, which already optimize the lowerlevel objective function.
The obtained results on the 23 bip instances and the 10 bipr ones are presented in Table 8. We observe that the results of the original CEMBA and CODBACRO are extremely far better than those of CEMBAWLO and CODBACROWLO, for all test instances. This could be explained by the importance of the optimization of the lowerlevel objective function. Indeed, when this latter is not optimized (only the upperlevel objective function is optimized while respecting all inequality and equality constraints of both levels), the upperlevel fitness function computation will not be precise at all. More specifically, based on the illustrative example of Fig. 7, the subvector \(x_{l}^{A}\) of the upperlevel solution \(x^{A}\) will be assigned a lowerlevel solution that is much poorer than E; which means that it is far from the corresponding follower optimum. In this way, the solution \(x^{A}\) will not respect the constraint of the optimality of the lowerlevel objective function, which will induce an imprecise fitness computation for the upperlevel solution \(x^{A} = (x_{u}^{A}, x_{l}^{A})\). By doing so for all upperlevel population members, the EA will no more be able to guide the search towards the leader global optimum and its behavior will be equivalent to a random search in the upperlevel search space, which is the case of CEMBAWLO and CODBACROWLO.
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Said, R., Elarbi, M., Bechikh, S. et al. Solving combinatorial bilevel optimization problems using multiple populations and migration schemes. Oper Res Int J (2021). https://doi.org/10.1007/s1235102000616z
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Keywords
 Combinatorial bilevel optimization
 Evolutionary algorithms
 Computational cost
 Population decomposition
 Migration schemes