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Colored Traveling Salesman Problem Jun Li, Senior Member, IEEE, MengChu Zhou, Fellow, IEEE, Qirui Sun, Xianzhong Dai, and Xiaolong Yu

Abstract—The multiple traveling salesman problem (MTSP) is an important combinatorial optimization problem. It has been widely and successfully applied to the practical cases in which multiple traveling individuals (salesmen) share the common workspace (city set). However, it cannot represent some application problems where multiple traveling individuals not only have their own exclusive tasks but also share a group of tasks with each other. This work proposes a new MTSP called colored traveling salesman problem (CTSP) for handling such cases. Two types of city groups are defined, i.e., each group of exclusive cities of a single color for a salesman to visit and a group of shared cities of multiple colors allowing all salesmen to visit. Evidences show that CTSP is NP-hard and a multidepot MTSP and multiple single traveling salesman problems are its special cases. We present a genetic algorithm (GA) with dualchromosome coding for CTSP and analyze the corresponding solution space. Then, GA is improved by incorporating greedy, hill-climbing (HC), and simulated annealing (SA) operations to achieve better performance. By experiments, the limitation of the exact solution method is revealed and the performance of the presented GAs is compared. The results suggest that SAGA can achieve the best quality of solutions and HCGA should be the choice making good tradeoff between the solution quality and computing time. Index Terms—Genetic algorithm (GA), greedy algorithm, hill-climbing algorithm, modeling, multiple traveling salesman problem (MTSP), simulated annealing (SA) algorithm.

I. I NTRODUCTION ENERALIZED from the traveling salesman problem (TSP), the multiple TSP (MTSP) is a well-known combinatorial optimization problem [1]. It aims to determine a family of tours with minimal total cost for multiple salesmen to visit each city exactly once within a given set and eventually return to their home city. MTSP and TSP arise in a variety of applications that require addressing

G

Manuscript received August 20, 2014; revised November 12, 2014; accepted November 13, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61374069, Grant 61374148, Grant 61175113, and Grant 51105076, in part by the Fundamental Research Funds for the Central Universities of China, and in part by the U.S. National Science Foundation under Grant CMMI-1162482. This paper was recommended by Associate Editor T. Vasilakos. J. Li is with the Ministry of Education Key Laboratory of Measurement and Control of CSE (Complex Systems of Engineering), Southeast University, Nanjing 210096, China, and also with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: [email protected]). M. Zhou is with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102 USA and also with Department of Electrical & Computer Engineering, King Abdulaziz University, Jeddah, Saudi Arabia (e-mail: [email protected]). Q. Sun, X. Dai, and X. Yu are with the Ministry of Education Key Laboratory of Measurement and Control of CSE, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCYB.2014.2371918

scheduling, planning, routing, and/or sequencing issues. TSP applications to machine scheduling and sequencing, and vehicle routing are reported in [15]. They are also found in the areas of circuit wiring [17] and statistical data analysis including ordering and clustering objects, e.g., gene ordering in [32] and protein clustering in [19]. The applications of multiobjective TSP can be significantly prompted due to the most recent advances in the multiobjective optimization algorithms, e.g., a multiobjective evolutionary algorithm using decomposition and ant colony [20], a memetic algorithm based on decomposition [21], a snowdrift game optimization [46], a physarum optimization [26], a coevolutionary technique called multiple populations for multiple objectives [48], Gaussian classifier-based evolutionary strategy [50], particle swarm optimization [51]–[56], and segment-based search to improve evolutionary algorithms [25]. MTSP applications are well summarized in [4]. For example, Gorenstein [14] reports the press print scheduling of a periodical with multieditions. Carter and Ragsdale [5] stress its use in preprint insert advertisement scheduling. Applications in crew scheduling [22], [40], [49], interview scheduling [13], workforce balancing [28], and hot rolling scheduling [41] are reported. Autonomous robot or vehicle motion planning [3], [33], [47] represents an important application area of MTSP. SarielTalay et al. [35] and Toth and Vigo [42] investigate a vehicle routing problem. Cheong and White [7] have investigated how to dynamically determine a tour for TSP based on real-time traffic congestion data. MTSP is utilized recently to plan the visit sequences of desired regions for unmanned aircraft vehicles in [9]. The result shows its significant reduction in the expected total travel cost of realistically sized pickup and delivery in a congested urban environment. In essence, MTSP is an abstraction of the practical problems in which multiple executing individuals (traveling salesmen) are involved and share a common workspace (city set). In other words, every city of MTSP can be accessed by each salesman. However, not all executing individuals have the same workspace in some application problems. Take the scheduling of a typical multimachine engineering system (MES) as shown in Fig. 1 as an example. The workspaces of individual machines are not the same but overlap partially with each other. Thus, each machine has to perform not only the operations independently in its exclusive workspace, but also complete all the tasks with other machine(s) together in the overlapping workspace. Due to the partially overlapping workspace, MTSP cannot be applied to model the scheduling problem of MES. Yet the basic elements of such a problem, i.e., objective, machine individuals, and tasks, can still match the objective, salesmen, and cities of MTSP, respectively. Unfortunately, no counterparts of

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Fig. 1. Dual-bridge waterjet cutting machine tool [29]. It consists of two independent bridges machines with an overlapping workspace for both machines to enter and two exclusive workspaces at both ends of an overlapping workspace for each machine only.

different task groups over exclusive and shared workspaces in MES can be found in MTSP. By abstracting such scheduling problems, we propose a new MTSP with differently colored city sets to refer to the task groups, called Colored TSP (CTSP). It is a general problem that is derived from but not limited to scheduling problems of multimachine systems. It thus deserves a great deal of research in theory and practice. As same as TSP and MTSP, CTSP is NP-hard. Moreover, the restricted visiting makes its solution more difficult and time-consuming than that of MTSP. Many MTSP instances of appropriate sizes can be solved by using some exact methods, e.g., cutting plane methods, branch-and-cut methods, and well-known branch-and-bound procedures. Gavish and Srikanth [11] propose a branch-and-bound algorithm for solving large-scale symmetric MTSP instances. The result shows that the sizes of non-Euclidean problems and Euclidean ones can be up to 500 cities and ten salesmen, and 100 cities and ten salesmen, respectively. It is proven that the heuristics are faster and more efficient than the exact methods to solve large-size MTSP perhaps at expense of solution optimality. They are thus applicable to solve those cases in which satisfactory solutions suffice. Several artificial neural network approaches extended from ones for TSP have also been presented to solve MTSP in [18], [31], [38], and [43]. Genetic algorithms (GAs) [45] represent another direction. Zhang et al. [49] and Tang et al. [41] use GA to solve MTSP arising from hot rolling scheduling. It is converted into a single TSP and solved with a modified GA. Carter and Ragsdale [6] propose a new two-part chromosome and related operators for MTSP. Computational test shows that their approach results in a smaller search space and, in many cases, produces better solutions than those with a single chromosome or two. Ant colony, tabu search, and simulated annealing (SA) algorithms are developed to solve MTSP by Ghafurian and Javadian [12], Ryan et al. [33], and Song et al. [39]. A hybrid estimation of distribution algorithm with decomposition is proposed to solve a multiobjective MSTP by Shim et al. [36]. Sofge et al. [37] provide a comparison among a variety of evolutionary computation algorithms for solving MTSP, including neighborhood attractor scheme, “shrink-wrap” algorithm, particle swarm algorithm, Monte Carlo algorithm, GA, and evolutionary strategies.

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Li et al. [23] call CTSP as MTSP* and present a GA solution. However, CTSP is not formulated in a mathematically rigorous way. The main contributions of this paper are to: 1) give a formal definition of CTSP and reveal the differences among CTSP, MTSP, and the combination of MTSP and TSP by comparing their solution space size; 2) develop three GAs to effectively solve CTSP; and 3) compare the performance of three GAs and the exact solution via LINGO by experiments. The comparison results are extensive and can be used to assist engineers in selecting an appropriate algorithm for a particular CTSP application. Next, Section II defines and formulates CTSP. Section III recalls GA and proposes three improved GAs. Section IV conducts experiments and performance comparison of the proposed algorithms. This paper is concluded in Section V. II. F ORMULATION OF CTSP A. CTSP Formulation Suppose that a CTSP has m salesmen and n cities, where m < n and n, m ∈ Z = {1, 2, 3, . . .}. It can be formulated over a complete digraph G = (ℵ, E), where vertex set ℵ = {0, 1, 2, . . . , n − 1} numbers the cities; and each edge in (i, j) ∈ E, i = j, is associated with a weight ωij representing a visit cost (e.g., distance) between two cities i and j. ℵ is divided into m + 1 disjoint nonempty sets, i.e., shared city set U and exclusive ones Vi , ∀i ∈ Zm = {1, 2, . . . , m}. Vertex di ∈ U ∪ Vi represents the depot where salesmen i departs and returns. We assign salesman i with color i. ∀a ∈ Vi , its color is i and only salesman i can visit it. ∀a ∈ U, its color set is (a) ⊆ Zm with | (a)| > 1 and if i ∈ (a), salesman i can visit a. The objective of CTSP is to determine m Hamiltonian cycles over G with the least total tour cost. Here any vertex of each exclusive set must be visited exactly once by a specified salesman and any vertex of the shared set must be visited by a salesman exactly once. The composition of shared city set U can be various in CTSP. The most common one is that U is shared by all m salesmen, i.e., ∀a ∈ U, (a) = Zm . An example is shown in Fig. 2. The vertices in areas V1 − V3 represent the exclusive cities of salesmen 1–3, respectively. U is the only shared city set among all salesmen. Salesman i can visit Vi ∪ U. The integer programming model of CTSP when di = 0 ∈ U, ∀i ∈ Zm and ∀a ∈ U, (a) = Zm is presented as follows. Binary variable xijk = 1, i = j, i, j ∈ ℵ, and k ∈ Zm , if the kth salesman passes through edge (i, j); and otherwise, xijk = 0. uik is the number of nodes visited on the kth salesman’s tour from the depot up to node i. The objective function of the problem is Minimize f =

m  n−1  n−1 

ωij xijk .

(1)

k=1 i=0 j=0

Subject to the following constraints: first, every salesman starts from and returns to city 0 (the depot), that is: n−1  i=1

x0ik = 1

(2)

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The formation of any disconnected sub-tour among nodes in ℵ\{0} is prohibited by the following equation incorporating with (10): uik − ujk + n × xijk ≤ n − 1, j = i, i, j ∈ ℵ\{0}, k ∈ Zm .

Fig. 2. Example of CTSP with only one shared city set U accessible to all salesmen.

and n−1 

xi0k = 1, k ∈ Zm .

(3)

i=1

Salesman k cannot visit an exclusive city of other salesmen starting from his own exclusive city, and salesmen other than salesman k are forbidden to visit the latter’s exclusive city, as indicated in turn by  xijk = 0 (4) i

and

 i

j

xjik = 0, i ∈ Vk , j ∈ ℵ\ (U ∪ Vk ), k ∈ Zm .

(5)

j

Salesman l (= k) can neither start from an exclusive city of salesman k nor return to it  xijl = 0 (6) i

and

 i

j

xjil = 0, i = j, k = l, i ∈ Vk , j ∈ ℵ, l ∈ Zm .

Theorem 1: CTSP is NP-hard. Proof: By its definition, CTSP is modified from an MTSP by dividing its cities into multiple exclusive city sets and a shared city set with the additional accessibility restrictions formulated by (4)–(7). Let each Vi , i ∈ Zm , contain only one city and remove city 0 from U. The model becomes one for a multidepot MTSP. On the other hand, when U = {0}, the model turns into one for multiple single TSPs with the same depot 0. TSP and MTSP have been proved to be NP-hard [5], [6]. With the recovery of U or Vi , the time complexity of the model will not change due to its operations. Hence, CTSP is NP-hard. From the proof, we have the following result. Corollary 1: A multidepot MTSP and multiple single TSPs are the special cases of CTSP. Consider a combination of TSP and MTSP consisting of several single TSPs each over an exclusive city set and a MTSP(s) over a common city set, given the same multiple salesmen. Can CTSP be converted to such a combination for solution? The answer is not. The former requires the multiple salesmen to visit all shared cities after visiting all exclusive ones or the opposite. In CTSP, salesmen visit different cities including both exclusive and shared ones in any order. Therefore, CTSP has a solution space size different from that of such a combination and MSTP with respect to the same problem size due to additional city visit restrictions (accessibility). Clustered TSP [8], [10], [27] and generalized TSP [16] are two variants of TSP. Both of them are defined on the same graph G as that of TSP. The former aims to determine the least cost Hamiltonian cycle over G in which the vertices of any cluster are contiguous while the latter aims to find the least cost circuit where exactly one vertex from each cluster appears. To the authors’ best knowledge, no work generalizes them with multiple salesmen. CTSP differs from them since it is with multiple salesmen and different clusters of shared and exclusive cities.

(7) III. GA S FOR CTSP

j

Each city except city 0 must be visited by a salesman exactly once n−1  m 

xijk = 1

(8)

xjik = 1, j = i, i ∈ ℵ\{0}.

(9)

j=0 k=1

and n−1  m  j=0 k=1

A salesman may visit a shared city and a pair of entry and exit is required if it is the case   xjlk = xijk , i = j = l, j ∈ U, i, l ∈ Vk ∪ U. (10) l

(11)

i

A. Dual-Chromosome Representation and Solution Space Analysis Due to the combinatorial complexity of CTSP, it is necessary to employ heuristics to solve sizable problems. GAs represent one type of heuristics that many researchers have applied to TSP and MTSP [5], [6], [30]. Li et al. [23] develop a basic GA to solve CTSP. The existing single chromosome and two-part chromosome coding schemes used in [6] are not suitable to CTSP due to its city visit requirement. We represent a solution of CTSP modeled in Section II as dual-chromosomes that are decimally coded, i.e., city and salesman chromosomes with the individual length being n−1. The depot 0 for all salesmen is not coded in the chromosomes and is added into the final solution to meet (2) and (3). The first chromosome has a

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Fig. 3.

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Example of CTSP dual-chromosome coding.

permutation of n−1 cities while the second assigns a salesman to each of the shared and exclusive cities in the corresponding position of the first, following the city-salesman matching relationship represented by (4)–(7) and (10). Given any generation a city can only appear exactly once in an individual in accordance with (8) and (9). The solution coding prevents the formation of a sub-tour as (10) and (11) do. A coding example of CTSP with n = 10 and m = 3 is shown in Fig. 3. Genes 1 and 2 in the city chromosome are the exclusive cities for salesman 1, genes 3 and 4 for salesman 2, and genes 5 and 6 for salesman 3, respectively. It represents the city-salesman matching relationship that must be met. The shared cities are genes 7–10 that can be assigned to any salesman. Cities 2, 8, 7, and 1 (in that order) are visited by salesman 1. Similarly, cities 10, 4, and 3 (in that order) are visited by salesman 2, and cities 9, 5, and 6 (in that order) are visited by salesman 3. Given a CTSP with m salesmen and n cities, the exclusive city count ofsalesman i is ni and the number of the shared cities is n0 , m i=0 ni = n. Theorem 2: The solution space size of CTSP with dualchromosome coding is n!mn0 . Proof: For a city chromosome with n cities regardless of exclusive and shared cities, there are n! permutations. Given the permutation of cities, there is only one assignment to a prescribed salesman for any exclusive city and there are m assignments to all salesmen for each of n0 shared cities due to the city-salesman matching relationship (4)–(9). Hence, for any permutation of cities, there are 1n−n0 × mn0 = mn0 permutations of salesmen. A solution of CTSP is the combination of a city chromosome and a salesman chromosome. Hence, the solution space size of CTSP, i.e., the permutations and combinations of dual-chromosome, must be n!mn0 . Theorem 3: The solution space size of CTSP with the same size and dual-chromosome coding, is smaller than that of MTSP but larger than that of a combination of MTSP and TSPs. Proof: As given in [6], the solution size of MTSP with dual-chromosome coding is n!mn . The solution space size of CTSP n!mn0 is smaller than n!mn . The combination of TSPs and MTSP means that each salesman’s visit of an exclusive city set is treated as several independent TSPs and all salesmen’s visit of a shared city set as an MTSP. The solution is the combination of two sections’ results. For TSP i, its city permutation is ni !. To regard m TSPs as a total, its permutation is m! For the section of MTSP with n0 shared cities, its permutation is n0 !mn0 . Hence, the solution space size  of the combinationn of 0 TSPs and MTSP is the permutation ( m 1 (ni !)×m!)×(n0 !m ). It is smaller than n!mn0 . Therefore, the conclusion holds. It makes CTSP differ from MSTP and the combination of TSPs and MTSP in essence due to their solution space sizes with respect to the same problem size and city visit restrictions. Any salesman can visit any city in the city set of MSTP

Fig. 4.

Example of city crossover.

Fig. 5.

Example of city mutation (CM).

while any salesman can visit his own exclusive cities and shared ones in any order in CTSP. CTSP does not without require that all shared cities must be visited after all his exclusive cities or the opposite, unlike the combination of TSP and MTSP. B. Basic GA Li et al. [23] adopt the combination of a roulette wheel method and elitist strategy as the selection operation. They compare three pairs of compositions of the crossover and mutation operators and the result shows that the combination of city crossover and mutation (CCM) has the best performance. A city crossover operator is a modified partially matched crossover. Fig. 5 shows a crossover process of a dual-chromosome with a single crossover of city chromosomes. First, two segments with the same range are selected at random from the two city individuals of given parents, and then the two segments are swapped, thereby resulting in two new individuals. Second, replace the redundant genes outside the segments in the two city chromosomes by ones meeting the gene swapping relationship between the two mentioned segments until no redundant gene remains. Finally, check if each exclusive city is assigned to a correct salesman and correct the wrong assignments if any. An example of city crossover is shown in Fig. 4. In step 1, the crossover segments in gray are selected into the two city chromosomes and then their genes at the same positions are swapped one by one. The swapping relationship is 8-3, 9-8, 5-2, 4-7, 7-1, and 1-10. The swapping results in two new individuals as shown in step 2. Following this swapping relationship, the redundant genes outside the two segments are replaced stepwise till no redundant gene is found. For example, the repeated gene 3 at the last second position of

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Fig. 6.

5

City graph with distance weights.

the left city chromosome should be replaced by 8 according to the swapping relationship 8-3. However, after replacement, new gene 8 is still repetitive and has to be replaced again by gene 9 according to the swapping relationship 9-8. The resultant exclusive cities 5, 3, 7, 1, and 6 in the left chromosome and cities 2, 5, and 4 in the right one are assigned to the wrong salesmen. Step 4 reassigns them to the correct salesmen and results in two feasible individuals. A city mutation process of a dual-chromosome includes two steps, i.e., first, a pair of genes in a city chromosome are selected for swapping; and second, if needed, correct the two corresponding genes in the salesman chromosome following the city-salesman matching relationship. In Fig. 5, genes 8 and 7 are selected and swapped. They are the shared cities and can be assigned to any salesman. Therefore, the city-salesman matching relationship is satisfied and thus the mutation is over. A fitness function is used to reflect how the CTSP objective is achieved. The greater fitness values the better, but the less total cost the better. Hence, we take the reciprocal of total cost f of solution x as the fitness function F(x) =

1 . 1+f (x)

(12)

The process of basic GA consists of selection, CCM, and fitness calculation. Li et al. [23] indicate that the evolution of GA is slow and it is easy to trap into a local optimum. C. GA With Greedy Initialization The decision made by a greedy algorithm at each step may not reach the best in the global view but the local optimum. However, it can obtain a satisfactory solution rapidly because it avoids the great effort needed to exhaust all possibilities to find the optimal solution. We use it to optimize the individuals of the initial population generated randomly at the first step of GA. An initial population of high quality can accelerate the population evolution of GA to reach a satisfactory solution rapidly. We name this improved algorithm as a GA with greedy initialization, in short, GAG. With regard to CTSP, a city visit needs to meet the proximity. Namely, a salesman at the current city selects the next one that is closest to it. It can optimize a solution by reordering its sequence. For example as shown in Fig. 6, a randomly generated visit sequence is 0 → 2 → 5 → 3 → 1 → 4 → 0. The sum of distances is 25 + 50 + 50 + 45 + 45 + 25 = 240. It can be optimized into 0 → 3 → 4 → 5 → 1 → 2 → 0

by using the greedy algorithm. The new sun of distance is 22 + 25 + 27 + 35 + 25 + 25 = 159, which is much less than the original one. Note that, in some cases, edge weights may not be distance, e.g., traffic flows [44], and then they should be converted into proper distance values. The generation process of the initial population in GAG is as follows. Step 1: Determine if the number of individuals in the current initial population is equal to the set number N or not. If so, exit. Step 2: Generate a city sequence randomly and assign the exclusive cities to the specified salesman and the shared cities to all the salesmen randomly. The resulting sequence is named individual a. Step 3: Reorder the city sequence of a by the shortest distance criterion to minimize the visit cost and obtain individual a . Step 4: Detect if a has already existed in the population or not. If so, go back to step 2; otherwise, insert it into the population and go back to step 1. D. Hill-Climbing GA (HCGA) A hill-climbing algorithm utilizes a neighborhood search technique to search, like hill-climbing, in a single direction so as to improve the quality of a solution [24]. Starting from an existing node, it generates a new solution with a method of neighborhood point selection and compares it with the value of the existing one. If the former is larger, replaces the latter by the former; otherwise, return the latter and set it as the maximum. Repeat the process of climbing upward until the highest point is reached. The local search power of the algorithm is very strong and it is a common method used for the local optimum search. The mentioned GAG as a global search algorithm does not exploit the information of individuals except the individual fitness to guide the search. Its convergence performance is limited. A hill climbing algorithm has a strong ability of local search and its search is led according to the quality of individuals. To improve the local search ability of GA, the best individual of each generation can be selected by using a hill-climbing operation. Specifically, if a better individual is obtained through hill-climbing, it replaces the original one; and otherwise, the original one remains in it. Note that the HCGA adopts the greedy algorithm to optimize the initial population too. The neighborhood point selection greatly impacts the hillclimbing algorithm and this work adopts two-point swapping. Given CTSP with m(≥ 2) salesmen, one should select 2m genes via this selection strategy. The fitness must be recalculated after every time of gene swapping. A hill-climbing operation includes the following steps. Step 1: Determine if the ith salesman performing the current swapping is the mth salesman, i.e., i = m. If so, exit. Step 2: Select two city genes assigned to the ith salesman, from the city chromosome of a. Swap them and obtain individual a . Step 3: Determine if the fitness value of a is greater than that of a. If so, let a = a ; and otherwise, keep a.

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TABLE I CTSP C ASES AND T HEIR S OLUTIONS BY LINGO (U NIT: KM )

Fig. 7. Flowchart of HCGA based on the basic GA with the greedy operation for the population initialization in the first step and the hill climbing operation in the fourth step.

Step 4: Let i = i + 1, and return to step 1. The main procedure of HCGA is summarized in Fig. 7. E. SAGA SA is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space [2]. In particular, during the search, it accepts not only a better solution but also a worse one by following the Metropolis criterion with the probability slowly approaching zero. It can jump out of the regions of local optima and guarantees its convergence performance. SA can be used to improve convergence of the above GAG. The solutions and the optimal one of CTSP are in turn similar to the states of each particle and the lowest energy state of particles in SA. The objective function of CTSP to reach the shortest route corresponds to the SA energy function. By setting an appropriate initial temperature and following some cooling schedule designed, SA can solve CTSP. SA and GA can be combined in two common ways, i.e., to perform SA on: 1) each individual and 2) the best ones of the GA population before and after selection, crossover, and mutation, respectively. The former needs too much time due to the large population and long chromosomes. Instead, this work adopts the latter. The procedure SAGA is given as follows. Step 1: Set initial temperature t0 , and generate initial population of GA, P(0) with initial epoch l = 0. Step 2: If the termination condition of GA is met, go to step 7; otherwise, calculate the fitness of each individual in

population P(l) and perform selection, crossover, and mutation on individuals according to the obtained fitness value to obtain new population P1 . Step 3: Select the best individual in P1 as the initial solution, x0 , of SA. Let current temperature t0 and current cooling count k = 0. Step 4: If it reaches the iteration step length at each temperature, go to step 5. Otherwise, select randomly a feasible neighbor x from the neighborhood of current solution x and determine whether to accept it with the Metropolis criterion. Namely, calculate f = f (x ) − f (x) and if f ≤ 0, x = x ; and otherwise, if exp(−f /tk ) > r, where r is a random number obtained from sampling a uniform distribution over (0, 1), repeat step 4. Step 5: Let k = k + 1 and tk = α • tk where α is a cooling coefficient. Justify whether the termination condition of SA is satisfied. If not, go to step 4. Step 6: Replace the original individual in P1 by the one obtained by SA to obtain new population P2 . Let l = l + 1 and P(l) = P2 . Go to step 2. Step 7: Output results. The flowchart of SAGA is similar to that shown in Fig. 7 where the hill-climbing operation needs to be replaced by SA. IV. E XPERIMENTS AND R ESULTS Experiments are performed to compare the performances of the presented algorithms for solving CTSP. First, LINGO software as a comprehensive tool for integer programming is used to solve exactly the given integer programming model when it is solvable. The inadequacy of exact solution with LINGO is illustrated by a group of experiments. Then, a case study allows one to observe the specific evolution and solution processes of the tested algorithms and subsequently analyze their convergence rates and solution quality. Finally, several CTSP cases with different sizes are used to make a comprehensive statistical analysis and comparison of their solution quality and computational time. A. LINGO Versus GAs In Table I, ten CTSP cases are designed where all city data are from dataset eil51 in the TSP collection, TSPLIB, cited in http://people.sc.fsu.edu/~jburkardt/datasets/tsp/tsp.html. The specific divisions of shared and exclusive city sets are enclosed in the on-line supplementary file. The computer used is Dell Inspiron 620s running Windows 7 (32 bits) with

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Fig. 8.

7

Optimal routes of the case with n = 41 and m = 3. Fig. 9.

CPU Intel Corei3 and 2 GB RAM at 3.30 GHz. The result is shown in Table I. The optimal routes corresponding to the seventh solution is illustrated in Fig. 8 but the latter three cases cannot be solved within two days. The settings that a salesman shuttles between his exclusive cities and shared ones appear in Fig. 8. Note that such solution is impossible to obtain with the combination of TSP and MTSP. The restriction of salesmen on exclusive city visits makes its solution more difficult and time-consuming than that of MTSP. When the size of CTSP is small, e.g., n < 41, LINGO can solve the problems quickly. However, as n and m increase, the time consumption grows sharply. For example, given the two cases of n = 31, when m = 2, the time consumed is only 94 s, while when m = 3, that grows up to 15 min. It is mainly due to the sharp increase in the number of constraints, namely, there are 1937 constraints for m = 2 while 3067 for m = 3, as listed in the fourth column of Table I. It results in a substantial increase in computing time. For the two cases with m = 3, the time consumed is 26 s, 94 s, and 6.4 h for n = 21, 31, and 41, respectively. The number of variables and constraints exponentially grows. With the further increase in the CTSP size, its exact solution becomes more and more difficult. For example, starting from n = 41 and m = 4, its solution cannot be obtained via LINGO within two days. Next, we confirm if GA is faster and more efficient than an exact method in the solution of CTSP. We set the parameters, i.e., the size of a population to be 30, crossover probability 0.7, and mutation probability 0.1. Each algorithm runs ten times and the maximum number of epochs is 2000. The parameters related to the SA operation are set, i.e., the initial temperature to be 100, the total time of cooling 60, the step length at each temperature 30, and annealing coefficient 0.9. Table II lists the results of the ten cases in Table I with the four GAs. With the increase of the case size, the increase of time consumed with GAs is much slower than that with LINGO. For example, the increase of the time consumption with GAs is not more than 40 s while that with LINGO is by ten or more times for the cases with three salesmen and the increasing cities. For the cases with 31 cities, with the increase of salesmen, their time consumed with LINGO increases by

Evolution performance of GAs with 10 min.

about ten and six times but that with GAs takes on a very small fluctuation less than 8 s. GAs can always obtain a solution but not necessarily an optimal solution. By Table II, under the termination condition of 2000 epochs, the solutions with HCGA and SAGA are close to the available exact ones with LINGO. SAGA achieves the best solution quality with the average, maximum and minimum errors being 3.54, 9.8, and 0.56 km. The GAs are applicable to solve those cases in which optimal solutions are too costly but good-quality solutions suffice. B. Case Study for GAs A CTSP with n = 51 and m = 4 is shown in Fig. 10 where U is the shared city set and V1 − V4 are the exclusive city sets (visit areas) of salesmen 1 − 4, respectively. The same computer mentioned before is used. Next, the four algorithms, GA, GAG, HCGA, and SAGA are applied to solve the problem and their performances are compared. By our experience, the termination condition of the algorithms becomes 10 min to ensure the fairness among them while their other parameters remain unchanged. The results are illustrated in Table III. 1) Convergence Rate: To compare their convergence rates, we plot the best convergence processes of individuals obtained by them as shown in Fig. 9. The total tour length of the best initial individuals in GA is about 1200 km. With the greedy operation, the quality of the initial population in GAG, HCGA, and SAGA is greatly improved. They converge more rapidly than basic GA with respect to the same termination condition of time. Four GAs terminated at about the 15 000th, 8500th, 9600th, and 6200th epochs, respectively. However, the latter three GAs’ actual evolution is accomplished earlier at about 8000th, 8500th, and 5000th epochs, respectively. The basic GA’s convergence is extremely slow and the evolution does not seem to end at about the 15 000th epoch. It is clearly much more time-consuming than the others. In addition, SAGA is the best among them while HCGA outperforms GAG in their convergence rate. Without a hillclimbing operation, GAG needs about 2000 epochs to evolve to the result with total tour length of 550 km due to its weakness in local search; while SAGA and HCGA spend only about

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TABLE II C OMPARISON A MONG T OUR L ENGTH BY LINGO AND GA S (U NITS : KM AND S )

TABLE III R ESULTS OF F OUR GA S (U NIT: KM )

local search ability of a hill-climbing operation. But around the 9500th epoch, it stops. It may fall into a local optimum. The mean tour length of the ten times of experiments with SAGA is 501.42. Comparing with GA, GAG, and HCGA, it achieves the tour length reduction by 37, 49, and 21 units, respectively. It indicates that the SA operation can enhance the convergence performance of a GA and the possibility of escaping from the local optima. The total tour length of the best solution with SAGA is 496.70 as given in Fig. 10. In summary: 1) the convergence rate of SAGA is the highest among the three considered GAs due to the introduction of the SA operation into GA and 2) SAGA can more easily escape from the local optima and yields better solutions among the four algorithms. Fig. 10.

Best tours of the case with n = 51 and m = 4 with SAGA.

150 and 300 epochs to achieve the same or better results. After the evolution of HCGA ends at the 8500th epoch with the total tour length of about 520 km, SAGA can continue to evolve until at about the 5000th epoch with the total tour length of about 500 km. 2) Solution Quality: From Fig. 9, we find that both GA and GAG are hard to reach their best result at about the 15 000th and 8500th epochs with the total tour length of about 540 and 550, respectively. It is clear that after reaching the same result of GAG, HCGA can continue to converge to a better solution, as shown in Fig. 9, due to the strong

C. Group Experiment of GAs 1) Experimental Setup: We design three groups, 11 cases of experiments, for the comprehensive analysis of the algorithms’ performance. The three groups of CTSP are assigned in turn with city sizes, 51, 76, and 101. All the city data are from datasets eil51, eil76, and eil101 in TSPLIB, and the shared city set and various exclusive city sets are grouped according to Table IV. The cities’ distribution is shown in the on-line supplementary file. Table IV is designed and optimized by referring to the Taguchi method. In it, different sets of exclusive cities of each case are partitioned averagely as far as possible and the portion of the shared cities out of all cities fluctuates between 0.2 and 0.5. This

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9

TABLE IV D ESIGN OF CTSP E XPERIMENT

can prevent CTSP from being MTSP or multiple individual TSPs. The three groups include various typical cases. The proposed four algorithms are applied to solve these cases. For the fairness of the comparison in the solution quality, the solution with each algorithm is allowed to evolve sufficiently. Therefore, the maximum of epochs of all GAs are set to 10 000. With the increase in the number of cities, the initial population size of GAs should increase accordingly. Each case’s initial population size is set to 40. The crossover and mutation rates of the four algorithms are set to the same 0.9 and 0.1, respectively. For HCGA the number of hillclimbing is set to 50 and for SAGA the initial temperature is set to 100, the times of annealing 50, the iteration count at each temperature 30, and the cooling coefficient 0.9. 2) Solution Quality: Table V gives the results of the group experiment designed in Table IV. Each case is run for ten times by using each algorithm. Table V records the statistics of the best, worst, and mean solutions with the corresponding time consumed from the ten samples of each case. By observing Table V, we find that the quality of the solutions obtained by GA are the worst for all cases regardless of the best, worst, and mean solutions. The reason behind it is its slow convergence and ease to trap into local optima, as shown in Fig. 9. The rest algorithms, GAG, HCGA, and SAGA achieve the better quality of solutions. The initial population of GA becomes better by using a greedy operation. However, the biggest defect of GA remains in GAG, i.e., easy trapping into local optima. HCGA almost outperforms GAG in every best and mean result since it introduces a hill climbing operation to enhance the local search ability of GAG. However, for the cases with 101 cities the former is worse than the latter in solution quality. The reason is that the increase of a case size more likely gives rise to the challenging problems of hill climbing, e.g., the convergence to a local maximum, ridges, and alleys. SAGA almost achieves the best results among them. Its reason is that the SA itself is a global algorithm with the ability to escape from a local optimum while the hill-climbing algorithm is just a local one. Additionally, the growth of the best solutions of SAGA is the steadiest with case sizes. 3) Time Consumption: We attempt to analyze the time consumption of the GAs according to the experimental results given in Table V. All the algorithms’ time consumption increases with the growth of the city size. For example, as

the city count grows from 51 to 101, the basic GA and SAGA consume the average time from 471 to 836 s, and from 1086 to 2365 s, respectively. Meanwhile, the different growth rate of time consumption of the algorithms can be observed from the results. SAGA makes the greatest increase followed in turn by HCGA, GAG, and the basic GA. It shows that the introduction of the greedy, hill climbing, and SA operations leads to more computing time and the bigger the size of a case, the greater the computing time. For the cases with the same size, the time consumption order is GA, GAG, and HCGA, and SAGA from the smallest to largest. For instance, to solve the case with 101 cities and four salesmen, GA consumes 938 s, GAG 1284 s, hillclimbing 1829 s, and SAGA 2168 s. It indicates that the introduction of greedy initialization or hill climbing consumes additional time. Thus, GAG needs more computing time than GA with respect to the same case size. HCGA is more timeconsuming than GAG to reach the same termination epoch. SAGA exploits both the greedy initialization strategy and SA that is more complicate than the hill climbing. Thus, it is more time-consuming than HCGA. Given the cases with the same city count, no consistency can be found concerning the time consumption of the algorithms with the growth of salesmen. For example, the cases with 101 cities, the time consumed by all GAs do not solely increase or decrease. It suggests that for an algorithm, its time consumption in solving CTSP can be influenced by many factors, including the salesmen count, the partition of the shared city set and exclusive city sets. The relationship between the computing time and these factors is very complicated. It is difficult to analyze by the current experiments and we intend to consider it in the future. D. Summary and Discussion CTSP is applicable to some application problems where multiple traveling individuals deal with not only their own exclusive tasks but also a shared group of tasks that allows one of them to do. For example, the scheduling of multiple machine systems containing radially arranged individual machines whose workspaces sharing a single common section can be formulated as CTSP to solve. The scheduling of the MES as shown in Fig. 1 can be converted into a typical case of CTSP with two salesmen. Before problem formulation, the tasks to be performed by multiple machines such as points, lines, and curves, either planar or spatial, should be partitioned and assigned by following some process requirements. Then, the assigned tasks should be abstracted into vertices of a graph. The vertices and individual machines are finally modeled as cities and salesmen in a CTSP, respectively. Specifically, each group of exclusive (shared) jobs is represented as a set of exclusive (shared) cities. The scheduling objective is to search a job sequence for each machine such that the total job execution cost of all machines is the least. It can be formulated by the objective function of the corresponding CTSP with a proper definition of visit cost. According to the above experimental results, SAGA is the best option to solve a realistic size of CTSP if only the solution quality is concerned; while HCGA is recommended if it

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TABLE V E XPERIMENT R ESULT W ITH THE F OUR A LGORITHMS (U NITS : KM AND S )

is necessary to obtain good solution quality in relatively short time. To the authors’ best knowledge, the largest scale MTSP solved by Gavish and Srikanth [11] has 500 cities. Since CTSP needs more time and memory than MTSP for the same problem size due to its restriction of city accessibility and city partition on salesmen, we defer these large-scale cases to our future research.

methods [20], [21], [25], [26], [46], [48], [50]–[62] in terms of their solution quality. The fast methods [64]-[72] can be used for such a purpose. ACKNOWLEDGMENTS The authors would like to thank the anonymous reviewers for their constructive suggestions that have helped us to improve this paper.

V. C ONCLUSION In this paper, we present a new MTSP, called CTSP, where different salesmen have exclusive city sets and share a common set of cities. It arises from the applications where multiple individuals work cooperatively at their workspaces that are not the same but partially overlap with each other. This paper formulates this new problem in a mathematically rigorous way. We propose three GAs, i.e., GA with Greedy Initialization, Hilling-Climbing GA, and Simulated Annealing GAs, improved from the basic one. The solution space size is deduced with dual-chromosome coding in them. The difference among CTSP, MTSP, and the combination of MTSP and TSPs is analyzed. The capacity and limitation of the exact solution method with LINGO software to solve CTSP are revealed by comparing it with the four GAs. By experiments, the performance of the GAs in terms of their evolution rate, solution quality, and time consumption, is comprehensively compared. The results suggest that SAGA can achieve the best solution and HCGA can obtain good-quality solution in relatively short time. The future work intends to deal with large-size CTSPs that cannot be handled by the existing methods and their applications to the optimal operation of MES. The quick evaluation of CTSP’s optimal solution’s lower bound is highly desired and it can be used as an indicator to evaluate and compare different intelligent optimization

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Jun Li (M’07–SM’13) received the M.S. degree in mechanical engineering from Jiangsu University, Zhenjiang, China, and the Ph.D. degree in control theory and control engineering from Southeast University, Nanjing, China, in 2003 and 2007, respectively. From 2008 to 2010, he was a Post-Doctoral Fellow with Southeast University, where he is currently an Associate Professor with the School of Automation. He is a Visiting Scholar of the Jersey Institute of Technology, Newark, NJ, USA. His current research interests include operational research, Petri nets, discrete event systems, and robotics. He has published over 30 journal papers.

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MengChu Zhou (S’88–M’90–SM’93–F’03) received the B.S. degree in control engineering from the Nanjing University of Science and Technology, Nanjing, China, the M.S. degree in automatic control from the Beijing Institute of Technology, Beijing, China, and the Ph.D. degree in computer and systems engineering from Rensselaer Polytechnic Institute, Troy, NY, USA, in 1983, 1986, and 1990, respectively. He joined the New Jersey Institute of Technology, Newark, NJ, USA, in 1990, where he is currently a Distinguished Professor of Electrical and Computer Engineering. He is also a visiting Professor with King Abdulaziz University, Jeddah, Saudi Arabia. His current research interests include Petri nets, sensor networks, web services, big data, semiconductor manufacturing, transportation, and energy systems. He has over 560 publications including 11 books, 260+ journal papers (majority in IEEE Transactions), and 22 book-chapters. Dr. Zhou is the Founding Editor of the IEEE Press Book Series on Systems Science and Engineering, and an Associate Editor of the IEEE T RANSACTIONS ON S YSTEMS , M AN AND C YBERNETICS : S YSTEMS, the IEEE T RANSACTIONS ON I NDUSTRIAL I NFORMATICS, and the IEEE T RANSACTIONS ON I NTELLIGENT T RANSPORTATION S YSTEMS. He is a Life Member of the Chinese Association for Science and Technology-USA, New York, NY, USA, and served as its President in 1999. He is a fellow of the International Federation of Automatic Control and the American Association for the Advancement of Science.

Qirui Sun received the B.S. degree in mechatronics engineering from the China University of Mining Technology, Xuzhou, China, and the M.S. degree in control theory and control engineering from Southeast University, Nanjing, China, in 2010 and 2013, respectively. He is currently an Engineer with Huawei Technologies Company, Ltd., Nanjing, China. His current research interests include artificial intelligence algorithms, discrete event systems, and exchange technology.

Xianzhong Dai received the Ph.D. degree in electrical engineering from Tsinghua University, Beijing, China, in 1986. From 1986 to 1988, he was with Tsinghua University. In 1988, he joined Southeast University, Nanjing, China, where he is currently a Professor with the School of Automation and the Director of the Key Laboratory of Complex Engineering System Measurement and Control, Ministry of Education, Beijing, China. From 1991 to 1993, he was a Post-Doctoral Researcher with Erlangen University, Erlangen, Germany, and from 1999 to 2000, he was a Senior Visiting Researcher with Cornell University, Ithaca, NY, USA. In 2001, he was a Visiting Researcher with Erlangen University. His current research interests include measurement and control, robot control, and complex power system control. Dr. Dai was the recipient of the China’s National Outstanding Youth Foundation Award, the Humboldt Research Award, the National Science and Technology Progress Award (Class III), the National Technical Invention Award (Class II), and the National Teaching Achievement Award (Class II). He is honored with the titles of the National Eminent Teacher and the National Model Teacher.

Xiaolong Yu received the B.S. degree in control engineering from Northeast University, Shenyang, China, in 2011. He is currently a Postgraduate in control theory and control engineering with the School of Automation, Southeast University, Nanjing, China. His current research interests include Petri nets and optimization algorithms.

Colored Traveling Salesman Problem.

The multiple traveling salesman problem (MTSP) is an important combinatorial optimization problem. It has been widely and successfully applied to the ...
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