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Solving the single and multiple asymmetric Traveling Salesmen Problems by generating subtour elimination constraints from integer solutions Maichel M. Aguayo, Subhash C. Sarin & Hanif D. Sherali To cite this article: Maichel M. Aguayo, Subhash C. Sarin & Hanif D. Sherali (2018) Solving the single and multiple asymmetric Traveling Salesmen Problems by generating subtour elimination constraints from integer solutions, IISE Transactions, 50:1, 45-53, DOI: 10.1080/24725854.2017.1374580 To link to this article: https://doi.org/10.1080/24725854.2017.1374580
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Solving the single and multiple asymmetric Traveling Salesmen Problems by generating subtour elimination constraints from integer solutions Maichel M. Aguayoa , Subhash C. Sarinb and Hanif D. Sheralib a Departamento de Ingenieria Industrial, Universidad de Concepcion, Concepcion, Chile; b Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA, USA
ABSTRACT
ARTICLE HISTORY
We present an algorithm to solve single and multiple asymmetric traveling salesmen problems (ATSP and mATSP) by generating violated subtour elimination constraints from specific integer solutions. Computational results for the ATSP reveal that the proposed approach is able to solve 29 out of 33 well-known instances taken from the literature (involving between 100 and 1001 cities) to optimality within an hour of CPU time. Furthermore, the proposed approach is demonstrated to outperform any of the most effective state-of-the-art exact algorithms available in the literature when applied to solve the given ATSP instances via their equivalently transformed symmetric TSP representations. For the mATSP, the proposed approach is able to solve 27 out of 36 instances derived from the ATSP library involving up to 1001 cities to optimality within an hour of CPU time and also outperforms the direct solution by CPLEX, one of the three most effective formulations reported in the literature for this class of problems. The proposed approach is easy to implement and can be used to solve ATSP and mATSP as stand-alone models or can be applied in contexts where they appear as sub-models within some application settings.
Received June Accepted August
1. Introduction The Traveling Salesman Problem (TSP) is a combinational optimization problem that is extensively studied in the literature, with hundred of papers and several books having been devoted to its study. The problem involves determining a minimal cost tour that starts and ends at a given base city after having visited a set of cities exactly once. The TSP can be classified into Symmetric (STSP) or Asymmetric (ATSP) problems, based on whether the distance or cost to travel between any two cities is symmetric or asymmetric, respectively. Practical applications of the TSP include the drilling of printed circuit boards, X-ray crystallography, overhauling gas turbine engines, order-picking in warehouses, computer wiring, clustering of data arrays, vehicle routing, and production scheduling, among others (Reinelt, 1994). An extension of the ATSP, which has received relatively lower levels of attention, is the multiple Asymmetric Traveling Salesmen Problem (mATSP), where multiple salesmen are located at the depot who need to collectively perform the task of touring each city exactly once at a minimal total cost. Some applications of this problem reported in the literature include print press scheduling, crew scheduling, school bus routing, interview scheduling, hot strip-rolling scheduling, and design of global navigation satellite surveillance networks (Bekta¸s, 2006). In this article, we present an algorithmic approach for the solution of the ATSP and mATSP by generating Subtour Elimination Constraints (SECs) from integer solutions. Typically, the generation of SECs is performed from fractional points by solving a min-cut problem (see Padberg and Rinaldi (1990)). The idea of generating SECs from integer points was first used in CONTACT Subhash C. Sarin Copyright © “IISE”
[email protected]
KEYWORDS
Asymmetric traveling salesman problem; multiple asymmetric traveling salesman; subtour elimination constraints
Dantzig et al. (1954) to solve the STSP, where the authors added SECs by visually inspecting a given solution. In Miliotis (1976, 1978), SECs were also generated from integer points but, in contrast with Dantzig et al. (1954), an algorithmic process was proposed to automatically generate SECs. Recently, the STSP has also been solved in Pferschy and Stanˇek (2017) using this idea with the aim of exploiting the effectiveness of current commercial solvers, with instances involving up to 400 cities being solved with reasonable effort. The similarity between this article and these previous studies is that we also generate SECs from integer points; however, our approach differs in the algorithmic process employed to accomplish this task. In Dantzig et al. (1954), Miliotis (1976, 1978), and Pferschy and Stanˇek (2017), whenever new SECs are generated, the current relaxed problem is solved to optimality. On the other hand, in our approach, we generate SECs as lazycuts, which are invoked whenever an integer solution to the problem is identified in any part of the branchand-bound tree during the process of implicitly solving the original problem, with the procedure ultimately terminating with a proven optimal solution. We also differ from Dantzig et al. (1954), Miliotis (1976, 1978), and Pferschy and Stanˇek (2017) in that we solve the ATSP and mATSP instead of the STSP, where the former are relatively more challenging classes of problems. The main contribution of this article is in designing an effective implementation of the idea of generating SECs from integer solutions for solving ATSP and mATSP, with problem instances involving up to 1001 cities, while using off-the-shelf software. Even with this limited algorithmic focus, the approach is demonstrated to be competitive with one of the most effective branch-and-cut algorithms reported in the literature when
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applied to equivalently transformed STSP representations of the given ATSP instances and is able to solve all mATSP instances of size greater than 443 cities to near optimality with a maximum optimality gap of 2.18%. Moreover, this approach can be applied either to solve ATSP or mATSP as stand-alone models or can be used in contexts where they appear as sub-models within some application settings. The remainder of this article is organized as follows. In Sections 2 and 3, we present exponential and polynomial-length formulations as well as describe the proposed approaches to generate SECs from integer points for the ATSP and mATSP, respectively. In Section 4, we present computational results for the proposed approach when applied to large-sized ATSP and mATSP instances involving between 100 and 1001 cities and compare its performance against the most effective formulations reported in the literature for both ATSP and mATP when solved directly by CPLEX. We also compare our results with one of the most effective state-of-the-art exact algorithms that is publicly available for solving STSP (Concorde, by Applegate et al. (2006)) by applying it to equivalently transformed STSP representations of the given ATSP instances. Finally, concluding remarks are presented in Section 5.
2. The ATSP The ATSP can be concisely defined as follows: Given a complete graph with vertex set N in which city 1 denotes the base city and an asymmetric distance matrix [ci j ], i, j ∈ N, determine an optimal tour that starts and ends at the base city after having visited city i exactly once, ∀i ∈ N, while minimizing the total distance traveled. 2.1. Formulations for the ATSP Let xi j be a binary variable that equals one if city i directly precedes city j in the ATSP solution and zero otherwise, ∀i, j ∈ N, i = j. Then, the ATSP can be formulated as follows: ci j x i j , (1) ATSP: Minimize i∈N j∈N, j=i
subject to: x ji = 1, xi j = 1,
∀N ⊂ N, N = ∅.
xi j ≥ 1,
(7)
j∈N−N
Reviews of exact methods for solving the ATSP have been presented in Laporte (1992), Fischetti et al. (2002), D’Ambrosio et al. (2010), and Roberti and Toth (2012). Generally, branchand-bound and branch-and-cut algorithms relax Constraints (6) and generate the SECs as needed, along with several additional facet-defining inequalities. A recent survey of the most effective exact approaches for the ATSP was presented in Roberti and Toth (2012), where the branch-and-bound algorithms of Fischetti and Toth (1992) and Choi et al. (2003) and the branchand-cut methods of Fischetti et al. (2003) and Applegate et al. (2006) (Concorde) are compared on instances involving up to 443 cities. The results exhibit that branch-and-cut algorithms are most effective in solving these problems. A comparative analysis of polynomial-length SEC-based formulations for the ATSP has been presented in Öncan et al. (2009) and in Roberti and Toth (2012). In particular, Roberti and Toth (2012) empirically compared several compact formulations and concluded that the three most effective formulations from the viewpoint of a direct solution by CPLEX are MTZ (Miller et al., 1960), GG (Gavish and Graves, 1978), and DL (Desrochers and Laporte, 1991). These are presented next and will be used as a benchmark for comparison with the proposed approach in the following discussions. Letting ui indicate a real number representing the order in which city i is visited in an optimal tour, ∀i ∈ N − {1}, the MTZSECs can be stated as follows: ui − u j + (|N| − 1)xi j ≤ |N| − 2,
i, j ∈ N − {1}, i = j, (8) ∀i ∈ N − {1}. (9)
1 ≤ ui ≤ |N| − 1,
Letting yi j be a real variable indicating flow in arc (i, j), the GGSECs are given by
∀i ∈ N,
i, j ∈ N, i = j, yi j ≤ (|N| − 1)xi j , y ji − yi j = 1, ∀i ∈ N − {1},
(3)
j∈N, j=i
(10) (11)
j∈N, j=i
yi j ≥ 0,
j∈N, j=i
i, j ∈ N, i = j.
(12)
(4) ∀i, j, ∈ N, i = j.
(5)
The objective function (1) minimizes the total cost, and Constraints (2) and (3), respectively, enforce that each city i is arrived at and departed from once. Constraints (4) are SECs, which we discuss next. Typically, SECs are formulated as either an exponential set of constraints or a polynomial number of restrictions by introducing additional variables. Dantzig et al. (1954) introduced the following exponential number of SECs: xi j ≤ |N | − 1, ∀N ⊂ N, N = ∅. (6) i∈N j∈N , j=i
i∈N
(2)
SECs, xi j ∈ {0, 1},
∀i ∈ N,
j∈N, j=i
By combining Equations (2) and (6), Constraints (6) can be rewritten as
The MTZ-SECs were strengthened by Desrochers and Laporte (1991) as follows (denoted by DL-SECs): ui − u j + (|N| − 1)xi j + (|N| − 3)x ji ≤ |N| − 2, −ui + (|N| − 3)xi1 +
i, j ∈ N − {1}, x ji ≤ −1,
i = j, (13)
j∈N−{1}, j=i
ui + (|N| − 3)x1i +
i ∈ N − {1},
(14)
xi j ≤ |N| − 1, i
j∈N−{1}, j=i
∈ N − {1}.
(15)
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2.2. Proposed approaches to generate SECs from integer points for the ATSP We propose to solve the ATSP by generating SECs from integer points. Toward this end, the approaches described below were implemented using CPLEX’s lazy constraints callback function, where specific violated SECs (if any) are generated whenever an integer solution to the current relaxed problem is identified (see details and examples in Rudin (2012) and IBM (2016)). These approaches are implemented in a traditional branch-and-cut scheme, where the presence of subtours is checked whenever an integer-feasible solution is identified. If this solution does not involve subtours, then we update the incumbent solution. Otherwise, SECs are added to the model, and the Linear Programming (LP) relaxation of the problem is solved. The procedure finally terminates after having identified an optimal solution to the original problem. The identification of all subtours for a given assignment solution satisfying Equations (2), (3), and (5) is straightforward. For any such x-solution, we identify all node subsets Ni , i = 1, . . . , s, which form disjoint connected subgraphs or subtours, where the set N1 contains the base city, {1}, and s represent the total number of such subtour-based node sets in the current solution. We explore several approaches for generating SECs at any step as described below: Approach 1 (A1 ): Append a single cut of type (6) based on a set having the largest cardinality. Approach 2 (A2 ): Generate all s cuts of type (6). Approach 3 (A3 ): Generate all s cuts of type (7). Approach 4 (A4 ): Generate a single cut of type (7) based on the set N1 . Note that Approaches A2 and A3 are identical, except for the structure of the (equivalent) cuts generated. Our motivation for testing both of these procedures is to investigate whether a standard solver performs differently due to the sparsity structure of the type of cuts added at any step. We also append to all formulations and proposed approaches the two-city SECs given by xi j + x ji ≤ 1,
∀i, j ∈ N − {1},
i = j.
(16)
2.3. Transformation of ATSP to an equivalent STSP representation Similar to Fischetti et al. (2002, 2003), we investigated the solution of ATSP instances by transforming them to equivalent STSPs. This can be achieved by using either the two-node or three-node transformation proposed in Jonker and Volgenant (1983) and Karp (1972), respectively. Fischetti et al. (2003) reported that the former method is, in general, more effective; thus, we only considered this technique in our computational experiments. The most effective branch-and-cut algorithm for the STSP currently is the one by Applegate et al. (2006), which is publicly available as Concorde (2005). We used Concorde with default parameters and QSopt (Applegate et al., 2011) as the LP solver following Hoos and Stützle (2014). The related results are presented in Section 4.
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3. The mATSP The mATSP can be concisely defined as follows: Given a complete graph with vertex set N in which city 1 denotes the base city, an asymmetric distance matrix [ci j ], i, j ∈ N, and m salesmen located at the base city, determine m tours that start and end at the base city after collectively having visited city i exactly once, ∀i ∈ N, while minimizing the total distance traveled. Next, we present both exponential and polynomial-length formulations for the mATSP as well as our proposed approaches for generating SECs from integer points as lazycuts. 3.1. Formulations for the mATSP The mATSP can be formulated as follows: mATSP: Minimize ci j x i j ,
(17)
i∈N j∈N, j=i
subject to: x1 j = m,
(18)
j∈N, j=1
x j1 = m,
(19)
j∈N, j=1
x ji = 1,
∀i ∈ N − {1},
(20)
xi j = 1,
∀i ∈ N − {1},
(21)
j∈N, j=i
j∈N, j=i
SECs, xi j ∈ {0, 1},
(22) ∀i, j, ∈ N, i = j.
(23)
Constraints (18) and (19) enforce that m salesmen depart from and return to the base city. Constraints (20) and (21) ensure that each city other than the base is arrived at and departed from once. Constraints (22) are SECs, which we discuss next. It can be shown that Constraints (6) are not valid for the mATSP, and they need some refinement. Consider an instance having five cities (n = 5) and two salesmen (m = 2) with a valid solution given by route 1: x12 = 1, x23 = 1, x31 = 1, and route 2: x14 = 1, x45 = 1, x51 = 1. If we let N = {1, 2, 3}, then Constraints (6) impose x12 + x13 + x21 + x23 + x31 + x32 ≤ 2, which cuts off this feasible solution. Therefore, Constraint (6) needs to be imposed just for subsets of cities not involving city 1 as follows: xi j ≤ |N | − 1, ∀N ⊂ N − {1}, N = ∅. i∈N j∈N , j=i
(24) Exact algorithms for the mATSP have been presented by Laporte and Nobert (1980) (where instances involving up to 70 cities were solved to optimality), Ali and Kennington (1986) (where instances involving up to 100 cities were solved to optimality), and Gromicho et al. (1992) (where instances involving up to 120 cities were solved to optimality). Another exact approach was presented by Gavish and Srikanth (1986) who solved to optimality instances involving up to 500 cities but for the multiple symmetric traveling salesmen problem.
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A review of the polynomial-length formulations for the mATSP appears in Kara and Bekta¸s (2006) and Sarin et al. (2014), where the latter presents a comprehensive review of 32 formulations for the mATSP and demonstrates that the most effective formulations for the mATSP when solved directly by CPLEX are, again, MTZ, GG, and DL as extended by Kara and Bekta¸s (2006) to the mATSP, which are presented next. The MTZ-SECs adapted to the mATSP are as follows: ui − u j + (|N| − m)xi j ≤ |N| − m − 1, 1 ≤ ui ≤ |N| − m,
i, j ∈ N − {1}, i = j, ∀i ∈ N − {1}.
(25) (26)
The GG-SECs adapted to the mATSP are as follows: yi j ≤ (|N| − m)xi j , y ji − yi j = 1, j∈N, j=i
i, j ∈ N, i = j,
(27)
∀i ∈ N − {1}
(28)
i, j ∈ N, i = j.
(29)
j∈N, j=i
yi j ≥ 0,
The DL-SECs adapted to the mATSP by Kara and Bekta¸s (2006) are given by (denoted by KB) ui + (m − |N| − 2)x1i − xi1 ≤ |N| − m − 1,
i ∈ N − {1}, (30)
ui + x1i + xi1 ≥ 2, i ∈ N − {1}, (31) ui + u j + (|N| − m)xi j + (|N| − m − 2)x ji ≤ |N| − m − 1, i, j ∈ N − {1}, i = j. (32) 3.2. Proposed approaches to generate SECs from integer points for the mATSP Similar to the ATSP, we propose to solve the mATSP by generating SECs from integer points. The identification of all subtours for a given set of xi j -values satisfying Constraints (18)–(21)
and (23) is straightforward. For any such x-solution, we identify all node subsets Ni , i = 1, . . . , s, which form disjoint connected subgraphs, with N1 containing {1} and the others representing subtours, where s represents the total number of such node sets in the current solution. We test several approaches for generating SECs at each step as described below: Approach 5 (A5 ): Append a single cut of type (24) based on the set having the largest cardinality from N2 , . . . , Ns . Approach 6 (A6 ): Generate s − 1 cuts of type (24) based on the sets N2 , . . . , Ns . Approach 7 (A7 ): Generate s cuts of type (7) based on the sets N1 , . . . , Ns . Approach 8 (A8 ): Generate a single cut of type (7) based on the set N1 . We also add to all formulations and proposed approaches the two-city SECs (16).
4. Computational results The proposed algorithms were coded using Microsoft Visual Studio Professional 2012, with CPLEX 12.5.1 as the solver with default parameters. The direct formulations were solved using OPL with default parameters. All runs were made on an Intel Xeon E5-2687W 3.1 GHz computer with 1 GB of RAM and running Windows 7, with a maximum permissible CPU time limit of 3600 seconds. Table 1 displays the 51 problem instances used in our computational experiments. We considered all 28 ATSP instances presented in Traveling Salesman Problem Library (TSPLIB) (1997), 18 instances from Cirasella et al. (2001) that are randomly generated to simulate various real-world applications, and five scheduling instances provided by Roberti and Toth (2012). The instances with over 100 cities are the set of large real-world ATSP instances used in Fischetti et al. (2003). The
Table . Well-known ATSP instances taken from the literature. Name br atex ftv ftv ftv p ftv atex ftv ryp ft ftv ftv ft ftv atex Balas ftv kro ftv td. Balas ftv dc Balas ftv
n
Source
Optimal
Name
n
Source
Optimal
TSPLIB () TSPLIB () TSPLIB () TSPLIB () TSPLIB () TSPLIB () TSPLIB () Cirasella et al. () TSPLIB () TSPLIB () TSPLIB () TSPLIB () TSPLIB () TSPLIB () TSPLIB () Cirasella et al. () Roberti and Toth () TSPLIB () TSPLIB () TSPLIB () Cirasella et al. () Roberti and Toth () TSPLIB () Cirasella et al. () Roberti and Toth () TSPLIB ()
dc ftv dc ftv ftv Balas ftv ftv dc dc code Balas code td. rbg rbg rbg rbg dc atex big dc dc dc td.
Cirasella et al. () TSPLIB () Cirasella et al. () TSPLIB () TSPLIB () Roberti and Toth () TSPLIB () TSPLIB () Cirasella et al. () Cirasella et al. () Cirasella et al. () Roberti and Toth () Cirasella et al. () Cirasella et al. () TSPLIB () TSPLIB () TSPLIB () TSPLIB () Cirasella et al. () Cirasella et al. () Cirasella et al. () Cirasella et al. () Cirasella et al. () Cirasella et al. () Cirasella et al. ()
IISE TRANSACTIONS
table columns indicate the instance name (name), the number of cities (n), the source from where the instance was taken (source), and the optimal cost reported in the literature (optimal ATSP cost), respectively. 4.1. ATSP ... Comparison with the most effective existing polynomial-length formulations Table 2 presents a comparison of the results obtained for the proposed approaches (A1 , A2 , A3 , and A4 ) and the direct solution by CPLEX of the polynomial-length formulations for the ATSP (DL, GG, and MTZ) on 18 relatively small-sized instances of up to 91 cities. For each approach, the respective columns indicate the integer solution obtained (ZIP ) and the CPU time in seconds to reach optimality or the integer optimality gap after 3600 seconds (T/gap). If a feasible solution was obtained within the time limit, then we report the value of the achieved optimality gap. All proposed approaches A1 –A4 solved all instances to optimality, but A3 outperformed A1 , A2 , and A4 as well as DL, GG, and MTZ by achieving minimum CPU times in 17 out of 18 problem instances. Also, A2 outperformed DL, GG, and MTZ except for one instance that GG solved faster. GG was the only formulation that could directly solve all problem instances to optimality. However, A2 and A3 outperformed GG, where the minimum, maximum, and average solution times achieved by A2 and A3 were 0.1, 135.1, and 15.7 seconds and 0.0, 426.8, and 24.5 seconds, respectively, whereas those for GG were 0.2, 772.1, and 76.2 seconds, respectively. On average, A2 and A3 solved these instances 4.9 and 3.1 times faster than GG. In this and subsequent tables, we have highlighted the minimum (T/gap) in bold for each instance. In Table 3, we compare the performances of the proposed approaches (A1 , A2 , A3 , and A4 ) and the direct solution by CPLEX of the polynomial-length formulations for the ATSP (DL, GG, and MTZ) on 33 medium- and large-sized problem instances of up to 1001 cities. Note that “—” denotes that a feasible solution was not obtained in the allowed CPU time limit
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of 3600 seconds. Columns are identical to those of Table 2. Approaches A2 and A3 outperformed the other methods by solving 30 out of the 33 ATSP instances to optimality, whereas the DL, GG, and MTZ formulations were able to solve to optimality only 12, 16, and 13 out of the 33 instances, respectively. For the three instances (atex8, dc895, and dc932) that neither A2 nor A3 were able to solve to optimality, the minimum and maximum gaps obtained for the former and latter were 0.05% and 2.52% and 0.12% and 2.87%, respectively. The DL, GG, and MTZ formulations were unable to obtain a feasible solution to 8, 11, and 7 out of 33 cases, respectively, or to generate feasible solutions for instances exceeding 563 cities; only DL obtained a feasible solution to the 600-city problem (atex8), but it yielded an optimality gap of 20.14% at termination.
... Comparison with a state-of-the-art exact algorithm Roberti and Toth (2012) showed that the state-of-the-art exact algorithms to solve the ATSP are the branch-and-bound methods reported in Choi et al. (2003) (CDT) and Fischetti and Toth (1992) (FT) and the branch-and-cut algorithms presented in Fischetti et al. (2003) (FLT) and Applegate et al. (2006) (Concorde). However, Concorde is the only code publicly available, and it is designed to solve STSPs. Therefore, for the sake of interest, we investigate how applying a state-of-the-art algorithm for STSPs to equivalently transformed instances of ATSP would compare against the directly applied proposed approach for the ATSP. To that end, we compared the results of the most effective proposed approach, A2 , with those obtained by Concorde (for Windows/Cygwin) using the two-node transformation on instances with over 100 cities. The obtained results are presented in Table 4, which displays the CPU times required by A2 and Concorde. A2 and Concorde were unable to reach optimality in 3 and 4 out of the 33 problem instances, respectively. Note that for the proposed approach, solutions were obtained with a maximal optimality gap of 2.53% within the allowed time limit; however, this information is not available with Concorde. For the instances where both A2 and Concorde achieved optimality, the
Table . Comparison of the results obtained by the proposed approaches A1 , A2 , A3 , and A4 with those obtained by direct solution of the polynomial-length formulations DL, GG, and MTZ for the ATSP by CPLEX on small-sized problem instances. A1 Name br atex ftv ftv ftv p ftv atex ftv ryp ft ftv ftv ft ftv atex Balas ftv
A2
A3
A4
DL
GG
MTZ
n
ZIP
T/gap
ZIP
T/gap
ZIP
T/gap
ZIP
T/gap
ZIP
T/gap
ZIP
T/gap
ZIP
T/gap
. . . . . . . . . . . . . 0.50 . . . .
. . . . . . . . . . . . . 0.51 . . . .
0.03 0.12 0.09 0.09 0.28 0.72 0.27 7.27 0.33 0.45 0.14 0.37 0.48 . 0.59 426.83 2.12 0.72
. . . . . . . . . . . . . . . . . .
. . . . . .% . .% . . . . . . . .% . .
. . . . . . . . . . . . . . . . . .
. . . . . .% . .% . . . . . .% . .% . .
Note. A bold value is the minimum (T/gap) achieved for the corresponding instance.
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M. M. AGUAYO ET AL.
Table . Comparison of the results obtained by the proposed approaches A1 , A2 , A3 , and A4 with those obtained by direct solution of the polynomial-length formulations DL, GG, and MTZ for the ATSP by CPLEX on medium and large-sized problem instances. A1 Name kro ftv td. Balas ftv dc Balas ftv dc ftv dc ftv ftv Balas ftv ftv dc dc code Balas code td. rbg rbg rbg rbg dc atex big dc dc dc td.
n
ZIP
A2 T/gap
. . . .% . . .% . . . . . . .% . . . . . .% . . . . . . — — — — . .% — — — — .
ZIP
A3 T/gap
1.08 0.69 0.11 19.33 . 1.09 76.82 . 3.78 . 2.53 . . 121.17 2.29 . 10.31 7.36 0.37 74.65 . 6.10 . . 11.43 21.0 337.93 2.52% . 224.95 0.07% 0.05% 24.74
ZIP
A4 ZIP
T/gap
. — — . — —
. . . . . . .% . . . . . . .% . . . . . — . . . . . . — — . . — — .
T/gap
. . . . 0.95 . . 1.36 . 1.22 . 0.87 0.94 . . 4.9 . . . 1267.62 4.41 . 1.06 2.71 . . . .% 39.55 . .% .% .
DL ZIP
GG T/gap
. . .% .% . .% .% . .% . .% . . .% . . .% .% . .% . .% — — .% . — — — — .% — — — — — — — — — —
ZIP
MTZ T/gap
ZIP
T/gap
. . . . . .% . .% . . . .% .% .% . . . .% . . . . . . . . .% .% . . . . .% .% .% .% . . .% .% .% . . .% — — .% — — .% — — . — — .% — — — — — — — — — — — — — — — — — — — — — — — — — — — —
Note. A bold value is the minimum (T/gap) achieved for the corresponding instance.
minimum, maximum, and average solution times for A2 were 0.4, 337.9, and 23.9 seconds, whereas for Concorde these values were 0.4, 373.3, and 56.3 seconds, respectively. On average, A2 solved these instances 1.64 times faster than Concorde, where it outperformed the latter with respect to CPU time for 21 out of the 29 instances. Table . Comparison of the CPU times required by the proposed approach A2 and Concorde (one of the state-of-the-art exact algorithms reported in Roberti and Toth () for solving the ATSP). Name kro ftv td. Balas ftv dc Balas ftv dc ftv dc ftv ftv Balas ftv ftv dc
n
A2
Concorde
Name
n
A2
Concorde
. 0.7 . 19.3 1.2 1.1 76.8 1.5 3.8 1.7 2.5 1.3 . 121.2 2.3 . 10.3
0.7 . 0.4 . . . . . . . . . 1.0 373.3 . 3.9 .
dc code Balas code td. rbg rbg rbg rbg dc atex big dc dc dc td.
7.4 0.4 74.7 6.1 6.1 9.6 11.3 . . . — 44.4 . — — 24.7
. . . . . . . 3.7 5.4 — — . 204.7 — — .
Note. A bold value is the minimum CPU time achieved for the corresponding instance.
4.2. mATSP Instances for the mATSP were derived from those presented in Table 1. For the sake of brevity, we selected a subset of these problem instances with the number of salesmen ranging from two to four. In Table 5, we compare the performances of the proposed approaches (A5 , A6 , A7 , and A8 ) with the direct solution by CPLEX of the most effective compact formulations for the mATSP (KB, GG, and MTZ). The column designations in Table 5 are identical to those in Table 2. We have highlighted the minimum (T/gap) valued in bold for each instance. Approach A6 outperformed the other methods by solving 27 out of the 36 mATSP instances to optimality, whereas KB, GG, and MTZ were able to solve to optimality only 7, 7, and 8 out of the 36 instances, respectively. Note that A7 solved to optimality only 26 out of the 36 mATSP instances, but it achieved the minimum T/gap in the largest number of instances (22 out of the 36 problems). For those nine and ten instances derived from balas200, atex8, dc985, and dc932 that A6 and A7 , respectively, were unable to solve to optimality, the minimum and maximum gaps for the former and the latter were 0.08% and 4.13% and 0.02% and 2.18%, respectively. We point out that KB, GG, and MTZ were unable to solve instances with a size greater than 443 cities except for one instance of atex8 solved by KB to a gap of 17.71%. We also study the impact of increasing the number of salesmen on the proposed approach A6 (see Table 6), which solved
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Table . Comparison of the proposed approaches A5 , A6 , A7 , and A8 with the direct solution of the polynomial-length formulations KB, GG, and MTZ for the mATSP by CPLEX. A5 Name
n
m
ZIP
A6 T/gap
kro
. . . Balas . . . Balas . . . Balas .% .% .% rbg . . . rbg . . .% dc .% .% .% atex .% .% .% big . . . dc — — — — — — dc — — — — — — td. 22.33 47.86 .
ZIP
A7 T/gap
0.89 1.87 . 16.85 8.39 12.28 452.23 . 182.91 2959.10 1471.70 2210.37 . . . . 9.38 . . . . .% .% .% . . . .% .% .% .% .% .% . . .
ZIP
A8 T/gap
ZIP
KB T/gap
. . . . 1.15 . . . . . . . . .% 493.90 .% . . . .% . .% .% .% 3.63 . 1.26 . 4.91 . . 7.47 . . 2.38 . 275.65 .% 228.45 — — 156.09 .% 2.18% — — 1.74% — — 1.88% — — 33.48 . 54.01 .% 26.91 . 0.05% — — 0.02% — — 0.04% — — 0.03% — — 0.04% — — 0.02% — — . . . . 56.18 .
ZIP
GG T/gap
ZIP
T/gap
MTZ ZIP
T/gap
. . . . . . . . . .% .% .% .% .% .% .% .% .% .% .% .% .% .% .% .% . .% .% .% .% .% .% .% .% .% .% .% . . . . . . . . .% — — .% . — — . . — — . — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — .% — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
Note. A bold value is the minimum (T/gap) achieved for the corresponding instance.
the largest number of instances to optimality in Table 5. The number of salesmen, m, in this experiment is given by m =
0.05n + 5i , for i = 0, . . . , 3 for each n. The approach A6 was able to solve 16 out of the 20 instances to optimality, and on those
four instances derived from atex8 for which A6 failed to obtain optimal solutions, the maximum optimality gap was 2.64%. Note that instances derived from atex8 could not be solved to optimality in Table 5 as well. Overall, note that increasing the number of salesmen did not impact the performances of A6 .
Table . Performance of the proposed approach A6 on larger mATSP instances with a varying number of salesmen. Name
n
m
T/gap
Balas
Balas
Balas
dc
atex
. . . . . . . . . . . . . . . . .% .% .% .%
4.3. Discussion Given the results presented above for the ATSP, we can infer the following: 1. The most effective ATSP formulations reported in the literature (DL, GG, and MTZ) could solve problem instances to optimality involving up to 443 cities within 1 hour of CPU time when optimized directly by CPLEX (12.5.1). 2. The most effective of the proposed approaches were A2 and A3 , which were able to solve 30 out of the 33 problem instances to optimality. We notice that generating several cuts to break all subtours at once using either Constraints (6) or (7) is more efficient than only generating a single cut each time SECs are invoked. Table 7 presents the number of SECs generated for each approach for some instances presented in Table 1. For example, in the problem instance dc176, A2 and A3 generated 124 and 253
52
M. M. AGUAYO ET AL.
Table . Number of SECs generated by each of the proposed approach A1 , A2 , A3 , and A4 when solving the ATSP. Name kro Balas Balas dc Balas dc dc Balas code dc atex big dc
A1
A2
A3
A4
SECs, respectively, whereas A1 and A4 added 1720 and 2672 SECs, respectively. This pattern can be observed for the other instances presented in this table as well. 3. The proposed approach (A2 ) outperformed a state-ofthe-art exact algorithm (Concorde) when applied to solve the given ATSP instances via their equivalently transformed STSP representations. 4. The effectiveness of the proposed approach stems from the fact that it efficiently generates a judicious set of tight cuts (SECs), which, as is well known, happen to be facetdefining for the ATSP. In the same vein, the following comments apply for the mATSP: 1. The most effective mATSP formulations reported in the literature (DL, GG, and BK) could solve instances to optimality involving up to 443 cities within 1 hour of CPU time when solved directly by CPLEX (12.5.1). 2. The most effective of the proposed approaches were A6 and A7 . Similar to the ATSP, generating several cuts to break all subtours at once using Constraint (7) or (24) turned out to be more effective than appending only a single cut each time SECs are generated as in approaches A5 or A8 . Furthermore, increasing the number of salesmen did not impact the performances of A6 . 3. The effectiveness of the proposed approaches A6 and A7 is further highlighted by the fact that all mATSP instances of size greater than 443 cities were solved to near optimality with a maximum optimality gap of only 2.18%. In contrast, only one instance of size greater than 443 cities could be solved by one of the most effective formulations reported in the literature (Kara and Bekta¸s, 2006) but with a much larger optimality gap of 17.71%.
5. Concluding remarks In this article, we used the idea introduced in Dantzig et al. (1954) and Miliotis (1976, 1978) to generate SECs solely from integer points to solve large-sized ATSP and mATSP. For the ATSP, our proposed approach outperformed the direct solution by the commercial solver (CPLEX 12.5.1) of the three most effective polynomial-length formulations reported in Roberti and Toth (2012) (MTZ, GG, and DL). In addition, our results indicated that the proposed algorithm is competitive with one
of the most effective branch-and-cut algorithms reported in the literature. An additional advantage of our approach over the most effective existing exact approaches for the ATSP as described in Fischetti and Toth (1992), Choi et al. (2003), Fischetti et al. (2003), and Applegate et al. (2006) is that it can be easily implemented using an off-the-shelf optimization software package that incorporates the generation of lazycuts such as CPLEX and GUROBI, among others. For the mATSP, the proposed approach outperformed the direct solution by a commercial solver (CPLEX 12.5.1) of the three most effective polynomial-length formulations reported in Sarin et al. (2014) (MTZ, GG, and BK), and it solved all mATSP instances of size greater than 443 cities to near optimality with a maximum optimality gap of 2.18%, where the instance of size greater than 443 cities could be solved by only one of the formulations reported in the literature but with a much larger optimality gap of 17.71%. For future research, we intend to study adaptations of the proposed approaches for solving different extensions of the ATSP and mATSP, as well as investigate their application to other routing and scheduling problems where ATSP or mATSP appear as sub-models.
Notes on contributors Maichel M. Aguayo is an Assistant Professor in the Industrial Engineering Department at University of Concepción, Concepción, Chile. He is a graduate of Virginia Tech, where he received his doctoral degree. His research interests are in the areas of logistics, routing problems, and applied mathematical programming. Subhash C. Sarin is the Paul T. Norton Endowed Professor in the Grado Department of Industrial and Systems Engineering at Virginia Polytechnic Institute and State University, Blacksburg, Virginia. His research interests are in the areas of production scheduling, applied mathematical programming, and design and analysis of manufacturing systems. He has published several papers in Industrial Engineering and Operations Research journals and has co-authored three books in the production scheduling area. He is a recipient of several prestigious awards at the university, state, and national levels. He has served on the editorial boards of many journals. He is a fellow of the Institute of Industrial and Systems Engineering and a full member of the Institute for Operations Research and the Management Sciences. Hanif D. Sherali is a University Distinguished Professor Emeritus in the Industrial and Systems Engineering Department at Virginia Polytechnic Institute and State University. His areas of research interest are in mathematical optimization modeling, analysis, and design of algorithms for specially structured linear, nonlinear, and continuous and discrete nonconvex programs, with applications to transportation, location, engineering and network design, production, economics, and energy systems. He has published over 343 refereed articles in various operations research journals and has coauthored nine books, with a total Google Scholar citation count of over 27 025 and an H-index of 62. He is an elected member of the National Academy of Engineering, a fellow of both INFORMS and IIE, and a member of the Virginia Academy of Science Engineering and Medicine.
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