Evolutionary Multitasking in Permutation-Based Combinatorial Optimization Problems: Realization with TSP, QAP, LOP, and JSP

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1 Evolutionary Multitasking in Permutation-Based Combinatorial Optimization Problems: Realization with TSP, QAP, LOP, and JSP Yuan Yuan, Yew-Soon Ong, Abhishek Gupta, Puay Siew Tan and Hua Xu School of Computer Science and Engineering, Nanyang Technological University, Singapore Singapore Institute of Manufacturing Technology, Singapore Department of Computer Science and Technology, Tsinghua University, Beijing , China Abstract Evolutionary computation (EC) has gained increasing popularity in dealing with permutation-based combinatorial optimization problems (PCOPs). Traditionally, EC focuses on solving a single optimization task at a time. However, in complex multi-echelon supply chain networks (SCNs), there usually exist various kinds of PCOPs at the same time, e.g., travel salesman problem (TSP), job-shop scheduling problem (JSP), etc. So, it is desirable to solve several PCOPs at once with both effectiveness and efficiency. Very recently, a new paradigm in EC, namely, multifactorial optimization (MFO) has been introduced to explore the potential of evolutionary multitasking, which can serve the purpose of simultaneously optimizing multiple PCOPs in SCNs. In this paper, the evolutionary multitasking of PCOPs is studied. In particular, based on a recently proposed multitasking engine known as the multifactorial evolutionary algorithm (MFEA), two novel mechanisms, namely, a new unified representation and a new survivor selection procedure, are introduced to better adapt to PCOPs. Experimental results obtained on well-known benchmark problems not only show the benefits of the two new mechanisms but also verify the promise of evolutionary multitasking for PCOPs. In addition, the results on a test case involving four optimization tasks demonstrate the potential scalability of evolutionary multitasking to many-task environments. I. INTRODUCTION Evolutionary algorithms (EAs) are a class of populationbased metaheuristics inspired by biological evolution. Due to the simplicity of the approach, free from assumptions about the fitness landscape, robust response to changing situations, and many other facets, EAs have been widely applied to various kinds of optimization problems, e.g, single-objective optimization [1], [2], multi or many-objective optimization [3] [5], dynamic optimization [6]. Although EAs have become popular in many fields of science and engineering, it is well known that pure EAs usually perform unsatisfactorily on hard problems, as the no free lunch theorem [7] implies. Hence, it is a common practice to incorporate some knowledge about the underlying optimization problem into the evolutionary-based search so as to enhance the performance of EAs. Memetic algorithms (MAs) [8] realize this by integrating the global search of EAs with problem-dependent local search (LS), and have achieved great success in combinatorial optimization problems (COPs) [9] [12]. The main idea of MAs is to exploit the knowledge of the current problem, whereas some other research efforts [13] [16] exploit problem knowledge in EAs in a different way, whose goal is to exploit the knowledge extracted from one problem to guide the search of EAs on another similar problem. Undoubtedly, such kinds of knowledge transfer in EAs are meaningful and promising. However, how to transfer and what knowledge is to be transferred are among some of the non-trivial challenges of such techniques, requiring careful design and deep understanding of problem characteristics. Moreover, another limitation of these techniques is that they are usually only applicable in the same problem domains. Very recently, a new paradigm in evolutionary computation (EC), namely, multifactorial optimization (MFO) [17], has been introduced to explore the potential of evolutionary multitasking. Unlike EC paradigms performing explicit transfer of knowledge from one task to another similar one [13] [16], evolutionary multitasking encompasses multiple optimization tasks at a time and facilitates the implicit transfer of knowledge across diverse problems via simple genetic transfer, thus achieving intra-domain and/or cross-domain optimization in a seamless manner. Generally, there is no need to have a concrete view of the relationships between the optimization tasks before multitasking, thereby making the paradigm easy to use. Once the unified representation for the tasks to be optimized is identified, these tasks can be directly tackled using a multitasking solver, where information sharing occurs in the unified search space. The multifactorial evolutionary algorithm (MFEA) [17] is one such recently proposed solver, which has been found to harness the genetic complementarity between tasks. In this paper, extending from the first study on evolutionary multitasking [17], our interest and focus is on the class of permutation-based COPs (PCOPs) [18], [19], which exist widely in many real-world scenarios such as supply chain networks (SCNs). The contributions of the present work are delineated as follows:

2 1) A new unified representation scheme is introduced for the evolutionary multitasking of PCOPs, leading to higher search efficiency; 2) A new survivor selection procedure is introduced for MFEA, which significantly reduces the computational complexity of the search; 3) The performance of MFEA on more than three tasks are investigated for the first time; 4) Some new insights about evolutionary multitasking have been derived from the experimental results obtained on PCOPs. The rest of this paper is organized as follows. Section II introduces the background knowledge of this paper. Section III describes the details of the algorithms considered in the experimental studies. Section IV provides the proof-of-concept computational results and discussions. Finally, Section V concludes this paper. A. Multifactorial Optimization II. PRELIMINARIES MFO formally describes an evolutionary multitasking environment where K optimization tasks are to be performed simultaneously. Without loss of generality, all tasks are assumed to be minimization problems. For the j-th task T j, its objective function is defined as f j : Ω j R, where Ω j is the search space of T j. The aim of MFO is to find {x 1, x 2,..., x K } = argmin{f 1 (x), f 2 (x),..., f K (x)}, where x j Ω j. In MFO, all the individuals in a given population P are encoded in a unified search space encompassing Ω 1, Ω 2,..., Ω K. Thus each individual p i, where i = {1, 2,..., P }, in the population P can be decoded into a task-specific solution for each of K optimization tasks. Based on the population P, the following definitions associated with p i are provided. Definition 1: The factorial rank rj i of p i on task T j, where j {1, 2,..., K}, is the index of p i in the list of population members sorted in ascending order with respect to f j. Definition 2: The scalar fitness φ i of p i is given by φ i = 1/ min K j=1{rj i}. Definition 3: The skill factor τ i of p i is given by τ i = argmin j {rj i }, where j {1, 2,..., K}. According to the scalar fitness, we can simply compare population members in a multitasking environment, laying the foundation to the design of EAs for multitasking. B. Multifactorial Evolutionary Algorithm MFEA is inspired by the bio-cultural models of multifactorial inheritance, which well serves the MFO purpose. The basic procedure of MFEA is shown in Algorithm 1. MFEA has four core features, i.e., unified representation, assortative mating, selective evaluation, and scalar fitness based selection. The random key (RK) [20] is used as the unified representation in MFEA. Suppose the dimension of the j-th task T j is given by D j, then the RK representation of an individual in MFEA can be dentoed as y = (y 1, y 2,..., y D ), where D = max D j=1 {D j} and the j-th RK value y j [0, 1]. Algorithm 1 Basic Procedure of MFEA 1: Randomly generate the initial population P with size N. 2: Evaluate each individual on all K optimization tasks. 3: Compute the skill factor (τ) for each individual. 4: while stopping condition is not met do 5: Apply genetic operators on P to get the offspring population P. 6: Evaluate the individuals in P for selected optimization tasks only. 7: Concatenate P and P to get the intermediate population Q. 8: Update the scalar fitness (φ) and skill factor (τ) of each individual in Q. 9: Select N best individuals (in terms of scalar fitness) from Q to form the next population P. 10: end while When addressing the task T j, only the first D j RK values in x are referred. This sort of unification not only avoids the challenge associated with the curse of the dimensionality but also encourages the discovery and implicit transfer of useful genetic material from one task to another. The principle of assortative mating (i.e., Step 5 of Algorithm 1) is that individuals prefer to mate with those belonging to the same cultural background. The procedure is described in Algorithm 2, where rmp is the random mating probability and rand is a random number between [0, 1]. Algorithm 2 Assortative Mating 1: Randomly select two parents p i and p j from P. 2: if τ i = τ j or rand < rmp then 3: Perform crossover on p i and p j to get two offspring individuals c i and c j. 4: else 5: Perform mutation on p i to get an offspring c i. 6: Perform mutation on p j to get an offspring c j. 7: end if The selective evaluation (i.e., Step 6 of Algorithm 1) implies that the offspring is evaluated for only one task instead of every task, making MFEA computationally practical. Moreover, the mechanism is very simple to comprehend and implement: the offspring is evaluated only on the task T τ, where τ is the skill factor of its parent (if the offspring has two parents, just randomly choose one). In addition, it should be noted that the evaluation in MFEA refers to task evaluation, i.e, LS is executed straight after the objective function evaluation. The scalar fitness based selection (i.e., Step 9 of Algorithm 1) follows an elitist strategy. Since every solution in the offspring population P is evaluated on only one task, its objective values with respect to all unevaluated tasks are artificially set to. Therefore, according to Definitions 1 and 2, we can easily assign the scalar fitness and skill factor to the solutions in Q (i.e., Step 8 of Algorithm 1) and then perform an elitist selection in terms of scalar fitness. C. Permutation-Based Combinatorial Optimization Problems The PCOPs refer to a class of COPs, where the natural representation of solutions is in the form of a permutation. In this

3 paper, we consider four kinds of PCOPs, i.e., travel salesman problem (TSP), quadratic assignment problem (QAP), linear ordering problem (LOP), and job-shop scheduling problem (JSP). 1) TSP: Given a list of n cities and n n distance matrix D = [d ij ], where d ij is the distance between city i and city j. The objective of TSP is to find the shortest possible path visiting each city exactly once and returns to the origin city. A solution can be given by a sequence of cities indicated as a permutation σ = (σ 1, σ 2,..., σ n ), and the objective function is formulated as follows n f(σ) = d σi 1σ i + d σnσ 1 (1) i=2 2) QAP: Given the n n flow matrix H = [h i,j ] and distance matrix D = [d ij ], the objective of QAP is to minimize the following function, where σ = (σ 1, σ 2,..., σ n ) is a permutation of {1, 2,..., n} n n f(σ) = h σi σ j d σi σ j (2) i=1 j=1 3) LOP: Given an n n matrix C = [c ij ], the objective of LOP is to determine a simultaneous permutation (σ) of the rows and columns of C such that the sum of the superdiagonal entries is as large as possible. The objective function is as follows n n f(σ) = c σiσ j (3) i=1 j=i Since only minimization problems are considered in this paper, we indeed minimize f(σ) for LOP. 4) JSP: Given n jobs {1, 2,..., n} and m machines {1, 2,..., m}, each job i consists of m i precedence constraint operations O i,1, O i,2,..., O i,mi, where O i,j, j {1, 2,..., m i }, must be processed on a specified machine k with uninterrupted processing time p i,j,k. There are N = n i=1 m i operations in total, and a schedule can be represented as a sequence of N operations (a permutation σ). Let C i the completion time of job i in the schedule σ, then the goal of JSP is to minimize the following function called makespan: f(σ) = max n {C i} (4) i=1 Note that not all the permutations is a feasible schedule of JSP due to the predefined order of operations within one job, which is different from that in TSP, QAP and LOP. If a permutation is infeasible, a repair procedure described in [12] is used to transform it to a feasible one. III. PROPOSED ALGORITHM FOR EVOLUTIONARY MULTITASKING OF PCOPS In Sections III-A and III-B, we introduce two novel mechanisms, namely, a new unified representation and a new survivor selection procedure, as an extension of the basic MFEA proposed in [17], which are designed for evolutionary multitasking of PCOPs. Section III-C then describes the local search procedures considered in the respective PCOPs. A. A Unified Representation The random key representation scheme has broad applicability and can be used both on continuous optimization problems and PCOPs. However, there are two obvious limitations of using a RK representation when dealing with PCOPs. On one hand, the decoding can be inefficient since the transformation from the RK representation to the permutation is required for each fitness evaluation of EAs, leading to the additional computing cost incurred. On the other hand, the decoding process of RK in PCOPs can be highly lossy since only information on relative order is derived, thus many solutions that differ in the genotype can correspond to a single permutation or the same solution in the phenotype. This has the effect of delimiting the explorative capability of EAs. Since the focus of our interests in this paper is on evolutionary multitasking of problems limited to only permutation based combinatorial optimization, it makes sense to consider a permutation based encoding over the more general RK representation. Suppose there are K tasks of PCOPs to be solved and the size of the j-th task T j is D j. Then the unified representation σ = (σ 1, σ 2,..., σ D ) is a permutation of (1, 2,..., D), where D = max K j=1 {D j}. When σ is associated with the j-th task T j, just choose all the integers no larger than D j from σ and keep them in the same relative order as in σ. These D j integers thereby form a permutation of (1, 2,..., D j ) and can be easily interpreted by T j. Fig. 1 depicts this procedure for three PCOP tasks, i.e., two TSP tasks with 3 and 5 cities, respectively, and one QAP task with a size of 4. Fig. 1. A solution (Unified representation) Solution for the TSP task with 3 cities B. Survivor Selection Solution for the TSP task with 5 cities Solution for the QAP task with the size 4 The interpretation of unified representation for each task. In this subsection, we suggest a new survivor selection procedure (replacing Steps 8 and 9 in Algorithm 1 for MFEA), referred as level-based selection (LBS), which is simpler and more computationally efficient. LBS maintains K ordered lists L 1, L 2,..., L K during the evolutionary process of MFEA. The list L j, j {1, 2,..., K}, consists of every solution whose skill factor is j in the current population P, and these solutions are sorted in the ascending order with respect to f j. The assortative mating mechanism described in Algorithm 2 remains intact for the generation of offspring solutions. The subtle difference is that when an offspring is assigned for evaluation on a selected task T j, insert the offspring into the ordered list L j after evaluation and set its skill factor to j. So, after the reproduction process, we have K ordered lists consisting of total 2N (N is the population size) solutions.

4 Then the 2N solutions are partitioned into different levels. Let F i be the i-th level that contains the set of solutions in the i-th position of L 1, L 2,..., L K, and if L j < i, F i consists of no solutions in L j. The LBS procedure, aiming to select N solutions to form the next population, is described in Algorithm 3. Fig. 2 provides an example of this selection process for 3 tasks with N = 7, where a shaded circle indicates a solution is selected for the next population. TABLE I LOCAL SEARCH NEIGHBORHOODS ADOPTED FOR TSP, QAP, LOP AND PFSP. Problem Local search neighborhood TSP 2-opt neighborhood [21] QAP swap neighborhood [22] LOP insert neighbourhood [23] JSP N6 neighborhood [24] L1 L2 s11 s12 s13 s14 s15 s21 s22 s23 s24 will replace the original solution in the population. Fig. 3 depicts an example search process of the LS working on the solution given previously in Fig. 1, which considers a TSP instance with 3 cities. L3 s31 s32 s33 s34 s35 A solution (Unified representation) An improved solution (Unified representation) F1 F2 F3 F4 F Fig. 2. The illustration of level-based selection procedure. In the original MFEA, the computational complexity per search generation is dominated by Step 8 of Algorithm 1. Here, every solution in Q should be ranked for every objective function, thus O(KN log N) computations are incurred. With the LBS procedure, the computational complexity of MFEA per generation is now dominated by the inserting of N offsprings into the K ordered lists, which has an average-case complexity of O(N log(nk 1 )) and worst-case complexity of O(N log N). This translates to a K times improvements in computational complexity per search generation over the originally proposed MFEA. Moreover, it should be mentioned that the effects of the two different selection methods are usually similar. This is because, in the original MFEA, when a solution is evaluated on T j, its objective function values with respect to all unevaluated task are set to, thus its minimum factorial rank is achieved on T j with very high probability. Algorithm 3 Level-Based Selection Procedure 1: Input L 1, L 2,..., L K. 2: P, u 0, l 1. 3: while u + F l N do 4: P P F l. 5: u u + F l. 6: l l : end while 8: Randomly select N u solutions in F l Solution for the TSP task with 3 cities Fig. 3. A. Experimental Setup LS procedure Improved solution for the TSP task with 3 cities An illustration of the LS. IV. EXPERIMENTAL STUDY In our experiments, we investigate three MFEA variants listed in Table II. MFEA denotes the original MFEA algorithm [17]. MFEA-Perm differs from MFEA only in that it employs a permutation encoding over RK as the unified representation. MFEA-Perm-LBS on the other hand, employs a permutationbased unified representation as well as the LBS procedure described in Section III-B. All three MFEA variants are implemented in Java and run on a PC with 3.2 GHz and 16 GB of RAM memory. For each kind of PCOPs, we consider three large-scale benchmark instances, which are summarized in Table III. In each test case, we consider two or more of these instances in a MFEA variant and solve them simultaneously. For each test case, the best, median and worst performances of each algorithm across 20 independent runs are then presented. TABLE II THE THREE MFEA VARIANTS CONSIDERED. C. Local Search Procedures A specific local search may be effective on some kinds of problems but not for others. To ensure the search performance, different LS neighborhoods are adopted in this paper for different kinds of PCOPs, which are listed in Table I. All LS procedures use a first improvement pivoting rule, i.e., once an improved solution is found, it is immediately applied. Moreover, all LS procedures are performed in the spirit of the Lamarckian learning, i.e., the improved solution Algorithm Representation Survivor Selection MFEA Random key Original selection MFEA-Perm MFEA-Perm-LBS Permutation (see Section III-A) Permutation (see Section III-A) Original selection Level-based selection (see Section III-B) The simulated binary crossover (SBX) and polynomial mutation (PM) [25] are used as the genetic operators for the

5 Fig. 4. Comparing convergence trends of f 1 and f 2 in multi-tasking and single-tasking (intra-domain). Fig. 5. Comparing convergence trends of f 1 and f 2 in multi-tasking and single-tasking (cross-domain). PCOPs TABLE III BENCHMARK INSTANCES CONSIDERED IN EXPERIMENTS. Benchmark instances TSP kroa200 [27] lin318 [27] linhp318 [27] QAP sko100a [28] sko100b [28] tai150b [29] LOP N-be75eec 150 [30] N-be75oi 150 [30] N-be75oi 250 [30] JSP la27 [31] la28 [31] la39 [31] TABLE IV PARAMETER SETTINGS OF MFEA VARIANTS. Parameter Value Population size 60 Maximum number of generations 300 Maximum local search iterations 50 random mating probability (rmp) 0.3 Distribution index of SBX 20 Distribution index of PM 20 Probability of PM 1/D RK representation. For the permutation based representation, the ordered crossover and swap mutation [26] are used. The parameter settings used in the experimental studies are summarized in Table IV. B. Effect of the Permutation-Based Unified Representation In this subsection, we demonstrate the effect of the permutation-based unified representation by comparing MFEA with MFEA-Perm. As can be observed from the results reported in Table V, MFEA-Perm shows clear benefits over MFEA on all the three test cases and is always better than MFEA in terms of median results. In terms of the computational time, MFEA incurs 723s for each test case on average, whereas MFEA-Perm takes 440s. From the results, MFEA- Perm is noted to outperform MFEA in both effectiveness and efficiency. TABLE V COMPARISON BETWEEN MFEA AND MFEA-PERM IN TERMS OF BEST, MEDIAN, AND WORST OBJECTIVE FUNCTION VALUES. Cases MFEA MFEA-Perm f 1 f 2 f 1 f 2 kroa lin sko100a tai150b N-be75oi N-be75oi la la f i is the objective of i-th task and the superior f i is marked in bold. C. Effect of the Proposed Survivor Selection Table VI presents the comparison results between MFEA- Perm and MFEA-Perm-LBS. It can be seen that the difference in the average performance of both MFEA variants are relatively small on the test cases comprising two tasks, verifying our claim made in Section III-B. As discussed in Section III-B, the average computational complexity of the proposed survivor selection procedure is O(N log(nk 1 )), in contrast to that of the basic MFEA which is O(KN log N). This implies that

6 TABLE VI COMPARISON BETWEEN MFEA-PERM AND MFEA-PERM-LBS IN TERMS OF BEST, MEDIAN, AND WORST OBJECTIVE FUNCTION VALUES. Cases MFEA-Perm MFEA-Perm-LBS f 1 f 2 f 1 f 2 kroa lin sko100a tai150b N-be75oi N-be75oi la la f i is the objective of i-th task and the superior f i is marked in bold. TABLE VII MULTITASKING VS. SINGLE-TASKING RESULTS (BEST, MEDIAN WORST) IN INTRA-DOMAIN OPTIMIZATION. Cases Multitasking Single-tasking f 1 f 2 f 1 f 2 kroa lin kroa linhp lin linhp sko100a sko100b sko100a tai150b sko100b tai150b N-be75eec N-be75oi N-be75eec N-be75oi N-be75oi N-be75oi la la la la la la f i is the objective of i-th task and the superior f i is marked in bold. TABLE VIII MULTITASKING VS. SINGLE-TASKING RESULTS (BEST, MEDIAN, WORST) IN CROSS-DOMAIN OPTIMIZATION. Cases Multitasking Single-tasking f 1 f 2 f 1 f 2 kroa tai150b lin sko100b linhp sko100a kroa N-be75oi lin N-be75oi linhp N-be75eec kroa la lin la linhp la sko100a N-be75eec sko100b N-be75oi tai150b N-be75oi sko100a la sko100b la tai150b la N-be75eec la N-be75oi la N-be75oi la f i is the objective of i-th task and the superior f i is marked in bold.

7 when K is small, the superiority of the LBS over the original selection procedure in MFEA will not be very apparent. Since the test cases involves only 2 tasks, i.e., K = 2, the highly similar results between MFEA-Perm and MFEA-Perm-LBS make sense due to the low K value. Thus, it makes good sense to conduct additional studies on cases where K is large so as to better understand the benefits of the LBS procedure. Taking the cue, our further experimental study indicates that approximately 33% of compute cost savings can be attained by MFEA-Perm-LBS over MFEA-Perm when K = 10, i.e., 10 multitasks. D. Evolutionary Multitasking vs. Single-tasking In this subsection, we study the search performances of evolutionary multitasking over single-tasking for solving P- COPs. Although the potential benefits of evolutionary multitasking has been demonstrated on continuous and discrete benchmark problems in [17], the coverage on PCOPs therein is preliminary. First, the intra-domain optimization of PCOPs is considered, i.e., the multiple tasks to be solved using MFEA- Perm-LBS belong to the same domain. Table VII presents the experimental results of MFEA-Perm-LBS for intra-domain optimization, which is pitted against those attained via singletasking, i.e., each task or PCOP problem instance is optimized individually. It can be seen that for TSP and QAP, multitasking generally performs better than single-tasking. For the LOP, multi-tasking fares better on the first test case {N-be75eec 150, N-be75oi 150}. On the remain two cases, noteworthy improvements are observed on the larger task albeit at the cost of a minor compromise with respect to the smaller counterpart problem instance/task. At this juncture, it is important to keep in mind that the effort expended on a single problem during single-tasking, is shared between the two problems during multitasking. From this perspective, the overall effectiveness of evolutionary multitasking is more clear. For the JSP, the observation is similar to that of LOP. Note here that there naturally does exist the possibility of some negative transfer [17] during multitasking, as is found to be the case for {la28, la39}, where single-tasking outperforms multitasking. Fig. 4 gives the convergence trends for three test cases in intra-domain multitasking, where MT and ST means multi-tasking and single-tasking respectively. Note that every point in the curve is the median result over 20 runs. Now, we further consider the case of cross-domain optimization and input two different types of PCOPs into MFEA- Perm-LBS. Table VIII shows the detailed comparison results. From Table VIII, it is interesting to see that multitasking exhibits more superiority in cross-domain than in intra-domain. Indeed, multitasking performs better on nearly all the 18 cross-domain test cases. The convergence trends for three cross-domain cases are plotted in Fig. 5. It can be seen that the single-tasking search usually stagnates at the early stage, whereas the multitasking can generally decrease the objective values more quickly and finally converge to a better objective value. To our knowledge, there is no other existing techniques to address cross-domain multitask optimization, thus the advantage of multitasking in this respect is very encouraging. We suspect part of the reason of the success is that different types of PCOPs are more likely to provide distinct search biases and further promote the diversity in the unified search space via the implicit genetic transfer phenomenon. But this is only a preliminary explanation, more investigations are needed in the future. TABLE IX THE MEDIAN RESULTS ON THE TEST CASE WITH FOUR TASKS {KROA200, LIN318, SKO100A, SKO100B}. f 1 f 2 f 3 f 4 Single-tasking Multiasking (300 generations) Multiasking (1200 generations) f i is the objective of i-th task and the best f i is marked in bold. In the above, we only present the performance of multitasking with two tasks. It will be interesting to see the outcome when more than two tasks are put into the MFEA. Here, we consider a test case having four tasks, i.e., {kroa200, lin318, sko100a, sko100b}. Table IX lists the median results obtained by MFEA-Perm-LBS. Note that, if multitasking is allocated the same number of generations as single-tasking, only 1/K computational cost is paid on average for each task compared to single-tasking; where K is the number of tasks. So, under this setting, it is unrealistic to expect that the performance of multitasking can consistently surpass that of single-tasking if K is large. Accordingly, in this situation, more computational efforts are needed to conclude whether multitasking has benefits or not. The worst-case scenario is that the number of generations allocated for multitasking is K times of that for single-tasking. If multitasking happens to outperform single-tasking within this worst case, we can still say that multitasking is beneficial. From Table IX, it can be seen that multitasking (at the 300 generations mark) is worse than single-tasking in terms of f 2. But if we extend multitasking to = 1200 generations, all objectives are improved. Some other cases with many tasks are also tested with similar outcomes, i.e., we find that multitasking is usually better than single-tasking at least at the worst-case scenario, validating the feasibility of multitasking for many tasks. V. CONCLUSION AND FUTURE WORK This paper focuses on the evolutionary multitasking of P- COPs. Four kinds of well-known PCOPs, i.e., TSP, QAP, LOP and JSP are considered. To make MFEA more effective and efficient for PCOPs, a permutation based unified representation is adopted instead of random key representation. Moreover, a new survivor selection is introduced for MFEA, whose computational complexity per search generation is much less than that of the original algorithm. In the experimental s- tudy, we first examine the effect of the two newly proposed

8 mechanisms. Then, through experiments both in intra-domain and cross-domain optimization, we try to figure out whether multitasking has advantages over single-tasking for PCOPs. The main experimental findings are as follows: 1) Multitasking is promising in the intra-domain optimization within TSP and QAP; 2) There exists the possibility of negative transfer, as is observed in some intra-domain cases with JSP. 3) Multitasking has interestingly shown more promise in cross-domain optimization than in intra-domain optimization. 4) For the cases involving many tasks, multitasking can often outperform single-tasking at least at the worst-case scenario. These experimental observations have shown the potential of evolutionary multitasking in intra-domain and cross-domain optimization and its scalability to many task environments. In particular, it is inspiring to see the superior performance of evolutionary multitasking in cross-domain optimization settings, a feature that cannot be readily achieved by any offthe-shelf optimizer. In the future, we should conduct deeper analysis towards why evolutionary multitasking works, especially in cross-domain optimization. To handle many tasks, more effective multitasking solvers need to be developed. In addition, evolutionary multitasking should be applied to more practical engineering problems to further demonstrate its efficacy. ACKNOWLEDGMENT This work was supported in part by the A*Star-TSRP funding, in part by the Singapore Institute of Manufacturing Technology-Nanyang Technological University (SIMTech- NTU) Joint Laboratory and Collaborative Research Programme on Complex Systems, in part by the Computational Intelligence Graduate Laboratory at NTU. 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