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Combining integer programming with a GRASP heuristic for the prize-collecting
traveling salesman problem
Glaubos Clímacoa,∗, Luidi Simonettia, Isabel Rossetib aUniversidade Federal do Rio de Janeiro, Departamento de Engenharia de Sistemas e Computação, Cidade Universitária, Centro de Tecnologia, Bloco H, 21941-972, Rio de Janeiro-RJ, Brazil bUniversidade Federal Fluminense, Instituto de Computação, Rua Edmundo March, S/N - Campus da Praia Vermelha, Boa Viagem, 24210-330, Niterói-RJ, Brazil
Abstract
We propose a new separation procedure for a well-known mathematical formulation and a hybrid heuristic for the prize-collecting traveling salesman problem. The hybridization consists of using a GRASP-based heuristic to gather information about the search space, reduce it, and then apply an integer linear programming method. To validate our proposals, we conducted experiments on instances from the literature and on a new set of problems presented in this work, which showed the effectiveness of the proposed approaches.
Keywords: prize-collecting, traveling salesman, separation procedure, hybrid heuristic
- Introduction
Few combinatorial optimization problems have as much applicability as the traveling salesman problem (TSP). Practically any situation involving decision making that affects the sequence of tasks or operations has some as- pect of the TSP [3]. Sometimes such problems are eventu- ally formulated as special instances of TSP, or more often, become generalizations of TSP. The problem of the prize- collecting traveling salesman (PCTSP) is a generalization of the classical TSP and was initially proposed by Balas [2], as a model for scheduling the daily operations of a steel rolling mill. The PCTSP can be described as a traveling salesman who must visit some cities, where each one of them has a prize (pi) and an associated penalty (wi), considering travel costs cij between the cities. A prize is collected whenever a city is visited, and a penalty is applied when- ever the city is not visited. The objective is to minimize the sum of travel costs and penalties paid, guaranteeing the collection of a minimum prize (P RIZE). This problem has a lot of practical applications such as daily scheduling of a steel mill, planning helicopter routes for offshore oil rigs, and creating routes for tourists [3]. Figure 1 illustrates a PCTSP solution containing seven vertices, and for visual purposes, consider a Euclidean dis- tance between the vertices. Several authors proposed different approaches to tackle PCTSP. Fischetti and Toth [14] introduced some bounding
∗Corresponding author Email addresses: [email protected] (Glaubos Clímaco), [email protected] (Luidi Simonetti), [email protected] (Isabel Rosseti)
20 5
32
21
13
40
27
4
30
38
7
20
3
pi prize of vertex i
penalty of vertex i
wi Route
Figure 1: A PCTSP solution with seven vertices, cost 179 and meet- ing the PRIZE with value 129.
procedures, based on different relaxations. A branch-and- bound algorithm was also devised and applied to instances with up to 200 nodes. Bienstock et al. [4] developed a new formulation for the PCTSP, in which uses cut-set constraints for sub-cycle eliminations in the route. The authors also presented an approximation algorithm with constant bounds that combines a linear relaxation of the problem, with the Christofides algorithm [17], widely used
Preprint submitted to Operations Research Letters October 23, 2020
for resolution of the TSP. Dell’Amico et al. [11] presented a lagrangian heuris- tic to obtain an upper bound, by relaxing the minimum prize constraint, dualizing it in a lagrangian way, and then applying the sub-gradient method. Awerbuch et al. [1] elaborated an algorithm of complexity (log 2 n) for the Quota TSP, which with some transformations, also in- cludes the PCTSP. Gomes et al. [16] devised a hybrid heuristic that combines a heuristic based on GRASP meta- heuristic [12] with the search heuristic VNS [20]. Torres and Brito [28] presented GRASP heuristics for the orien- teering problem (OP) and the PCTSP, and a new formu- lation for the PCTSP, based on flow constraints for the sub-cycles elimination. Chaves and Lorena [6] proposed: i) a new mathemat- ical formulation by combining the both models of Torres and Brito [28] and Balas [2]; and ii) two heuristics based on clustering search. These heuristics are similar, and the main difference between these two aforementioned heuris- tics is how initial solutions are built. The first heuristic generates solutions making use of an evolutionary process (ECS), while the second one creates initial solutions em- ploying a GRASP heuristic associated with a VNS/VND method (CS*). These same authors still produced a new hybrid heuris- tic by combining GRASP and VND again, but performed their experiments in a new set of 108 instances [7]. Finally, the last work that addresses the PCTSP was carried out by Pedro et al. [23], in which the authors proposed a solution to the problem through a tabu search, which incorporates the GENIUS heuristic to its construction phase, and the 2-opt neighborhood in the local search phase. The contributions of this work can be summarized as: i) a new separation procedure for the cut-sets of Bienstock et al. [4]; ii) a comparison among different mathematical models of the PCTSP; iii) a hybrid heuristic that combines integer linear programming (ILP) with a heuristic based on the GRASP metaheuristic [26]; and iv) a new set of instances. The remainder of this paper is organized as follows. In Section 2, an ILP formulation is described along with a new separation procedure. In Section 3, the proposed heuristic approach is presented. In Section 4, detailed computational results are exhibited illustrating the effec- tiveness of the proposed approaches. Finally, in Section 5, some conclusions and future investigations are drawn.
- Mathematical formulation
Let G = (V, E) be a complete and undirected graph, in which V and E are the sets of nodes and edges, respec- tively. Associated with each edge e = (i, j) ∈ E there is a cost ce satisfying the triangle inequality, and with each vertex i ∈ V a non-negative penalty wi. The formulation requires the integer variables xij ∈ { 0 , 1 } which is equal to one, if edge (i, j) ∈ E is part of the route; or to zero,
otherwise; and yi ∈ { 0 , 1 } if node i ∈ V is visited (yi = 1) or not (yi = 0). For every subset of vertices S, let δ(S) be the set of edges with one end in S and the other in V \ S. The origin node u has a fixed prize pu = 0 and penalty wu = ∞. Then, according to Bienstock et al. [4], the prize collecting traveling salesman problem can be formulated as follows:
M in
e∈E
cexe +
i∈V
wi(1 − yi) (1)
s.a. :
e∈δ(i)
xe = 2yi, ∀i ∈ V (2)
∑
i∈V
piyi ≥ P RIZE (3) ∑
e∈δ(S)
xe ≥ 2 yi, ∀(S ⊂ V \ {u}, i ∈ S)
(4) xe ∈ { 0 , 1 }, ∀e ∈ E (5) yi ∈ { 0 , 1 }, ∀i ∈ V (6)
The objective function (1) aims to minimize the sum of travel costs and penalties. The degree constraints (2) ensure that a feasible solution goes exactly once through each visited vertex. Constraint (3) imposes a minimum bound on the collected prize. The cut-set constraints (4) ensure the connectivity of the route and, along with con- straints (2), are responsible for eliminating sub-cycles in the route. The separation procedure of these cut-sets is described in Section 2.1. Finally, constraints (5) and (6) impose that all variables be 0-1. Note that if an edge e ∈ δ(v) is part of a solution, then vertex v must be visited, and hence we have added the following constraints (7) to the formulation of Bienstock et al. [4]. From now on, we called this modified formulation as Bienstock+.
xe ≤ yi, ∀e ∈ δ(i) (7)
2.1. Separation procedure The model proposed by Bienstock et al. [4] theoreti- cally contains an exponential number of cut-set constraints (4). However, the violation of such constraints can be checked in polynomial time, making a cutting plane algo- rithm an interesting approach. In the following, we de- scribe the separation procedure used to identify the vio- lated cut-set inequalities. First, all cut-set constraints (4) are dropped from the model, so they can be added later dynamically as cutting planes. Then, along with the nodes of the branch-and- bound tree, whenever the solver finds an LP solution, ei- ther integer or fractional, the separation procedure starts. A support graph G∗^ = (V ∗, E∗) is associated with the solution S = (x∗, y∗), obtained through the linear relax- ation of the model, where V ∗^ = {i ∈ V : y∗ i > 0 } and
A vertex is inserted into the RCL if its incremental cost is inferior to the threshold cmin^ + (α × (cmax^ − cmin)), where cmin^ and cmax^ are, respectively, the smallest and the largest incremental costs in CL, and a greedy parameter α ∈ [0, 1]. The incremental cost was calculated by the following greedy function, which considers the insertion of a vertex k between two other vertices i and j.
g(k) = (^) i,jmin∈S:i 6 =j
( cik + ckj − cij − wk
) (8)
where cij is the cost of the edge (i, j), and wk is the penalty for not visiting vertex k. The costs cik and ckj are, respec- tively, for the edges (i, k) and (i, k). After constructing an initial solution, a local search is performed by the RVND heuristic.
3.1.2. Local search phase Proposed by Mladenović and Hansen [19], the Vari- able Neighborhood Descendant (VND) is a local search metaheuristic that uses different neighborhood structures. Given an ordered list of neighborhoods, the VND starts by exploring the first neighborhood of S, N k(S), and if a bet- ter solution is not found, the next neighborhood N (k+1)(S) is explored. Otherwise, it returns to the first neighborhood on the list. The algorithm stops when all neighborhood structures are explored, returning the best solution found. In this work, we have used the Random VND instead of conventional the VND, where there is no fixed sequence of neighborhoods, because they are sorted on each applica- tion of the local search. The RVND is detailed in the pseudo-code of Algorithm 2, and it starts by shuffling the list of neighborhood struc- tures. Then, we can see that the counter k is incremented by one to try the next neighborhood if the current solution is not improved. Otherwise, the local optimal solution for the current neighborhood is obtained, and k is set to 1 so that the loop restarts for the newly accepted solution. When none of the neighbors are able to improve S, i.e., k > kmax, the best current solution is returned.
Algorithm 2: RVND(S, N ). 1: Set a random order for the neighborhood structures in N 2: k ← 1 3: while k ≤ k_max do 4: S′^ ← f irstImprovingSolution(N (k)(S)); 5: if S’ is better than S then 6: S ← S′; 7: k ← 1 ; 8: else 9: k ← k + 1; 10: end if 11: end while 12: return S
For the RVND, we have used the following 15 neighbor- hood structures proposed in Silva [27] and da Silva [10]. This structures are based on classical movements for the
TSP as swap and shift, and heuristics as GENIUS and Cheapest Insertion. Ten of them are intra-route:
- shift: changes the position of a vertex within the route;
- swap: swaps the position of two vertices in the route;
- or-opt: movement similar to shift, but the position of n vertices is changed;
- 2-opt: removes two nonadjacent edges of the solution and inserts two new ones to keep the single cycle;
- 3-opt: removes three nonadjacent edges and inserts three new ones, similar to 2-opt;
- remove_simple_re_insert_cheapest: removes a vertex and reinserts it via cheaper insertion;
- remove_simple_re_insert_genius: removes a vertex and reinsert it via GENIUS method;
- remove_genius_re_insert_cheapest: Removes a vertex using a GENIUS removal method and inserts using lower cost criteria between adjacent vertices;
- remove_genius_re_insert_genius: removes and reinsert a vertex of the route using GENIUS;
- remove_cheapest_re_insert_genius: removes using chea- per insertion criteria and re-enter via GENIUS;
and five of them are extra-route:
- double_remove_simple_insert_cheapest: tries to replace two vertices with a single one that does not belong to the solution;
- remove_simple_insert_cheapest: replaces a vertex of the route with an outside vertex, inserting via cheaper inser- tion;
- remove_simple_insert_genius: replaces a vertex of the route with an outside vertex, inserting via GENIUS;
- remove_genius_insert_genius: replaces a vertex of the route with an outside vertex, by removing and inserting via GENIUS; and
- remove_genius_insert_cheapest: removes a vertex from the solution via GENIUS and inserts a new one via cheap- est insertion. In each RVND call, the neighborhoods are randomly selected, making the sequence of the explored neighbor- hoods random. In this work the first improvement tech- nique [20] was used, so the neighborhood exploration is interrupted as soon as a better-quality solution is reached, returning to the first neighborhood structure. The algo- rithm ends when all the 15 neighborhoods are performed without improvement of the current solution.
3.2. The proposed hybrid heuristic Metaheuristics represent an important class of tech- niques for dealing with difficult combinatorial problems. However, the focus on a single metaheuristic is rather re- strictive for state-of-the-art advancement by addressing practical optimization problems [5]. In this work, we pro- pose a hybrid heuristic (MPO) that combines a GRASP- GENIUS-RVND heuristic with ILP. Following the idea of Clímaco et al. [9], our hybrid heuristic starts with the execution of the GRASP-GENIUS- RVND base heuristic, which provides an upper bound to
the problem (f (S∗)) and generates a set of “unpromising edges” (E^0 ). If in all the heuristic iterations, a certain edge e is never considered part of a local optimal solution, then e belongs to E^0. The next steps are: remove from the ILP model all variables xe : e ∈ E^0 , set the upper bound to the model, and then solve it. The pseudo-code of the proposed heuristic is detailed in Algorithm 3. In line 1, the best solution (S∗), the best solution value (f (S∗)), and the set E^0 are initialized. E^0 starts being the set E itself. From lines 2 to 7, for maxIt iterations, the following steps are performed. In line 3 an initial solution (S) is constructed, and in line 4 a local search is applied over S, yielding in a solution S′. In line 5, if S′^ is a better solution than S∗, then the best solution found so far is updated. After the local search, the set E^0 is updated in line 6, by removing from E^0 all the edges present in S′. Once the heuristic part is finished, in line 8, all variables correspond- ing to edges in E^0 are removed from the ILP model. In line 9, the final solution is obtained by solving the restricted model. Note that we made use of an upper bound value (f (S∗)), so the ILP solver discards all solutions worse than S∗. Finally, the MPO solution is returned in line 10.
Algorithm 3: MPO(maxIt, seed, model, α) 1: S∗^ ← ∅; f (S∗) ← ∞; E^0 ← E; 2: for k = 1 to k ≤ maxIt do 3: S ← GENIUS(seed, α); {Construction phase} 4: S′^ ← RVND(S, seed); {Local search phase} 5: S∗^ ← min(S∗, S′); 6: update(S′, E^0 ); k ← k + 1; 7: end for 8: model ← remove(E^0 , model); 9: S∗^ ← solve(model, f (S∗)); 10: return S∗;
- Computational results
To validate our proposals, experiments were conducted with two groups of instances from the literature [6, 7] and a new set of test-problems. All methods proposed in this paper were implemented in the C++ language and com- piled with g++ 5.4.0. The tests were run on an Intel(R) Core(TM) i7-6700 [email protected] machine, with 16GB of RAM. For the implementation and execution of the math- ematical formulations, the Gurobi solver (version 6.5.2) [21] was used, with its heuristics, cuts, and preprocessing disabled. Besides, a one-hour processing time limit was determined and the parallelization capability of the pro- cessor was not used. For the MPO, we performed tuning experiments with three instances of Chaves and Lorena [6] (v20, v100a, and v500a), which represent, respectively, small, medium, and large problems. The MPO was run ten times for each parameter combination between α ∈ { 0. 2 ,-
- 4 , 0. 6 , 0. 8 } and maxIt ∈ { 50 , 70 , 90 , 100 }, and the set-
tings yielding the best average results was α = 0. 6 and maxIt = 70.
4.1. Instances of Chaves and Lorena [6] The first group of instances consists of problems with |V | ∈ { 10 , 20 , 30 , 50 , 100 , 250 , 500 }, randomly generated at the following intervals. Travel cost between nodes: cij ∈ [50, 1000], and prize associated with each node: wi ∈ [1, 750]. The minimum prize to be collected, P RIZE, represents 75% of the sum of the prizes of all nodes. These test prob- lems are available in http://www.lac.inpe.br/~lorena/ instancias.html.
4.1.1. Mathematical formulations An experiment was carried out with the formulations of Chaves and Lorena [6], Torres and Brito [28] and the modified formulation of Bienstock et al. [4], Bienstock+. All the formulations were implemented and executed in the same computational environment. Also, we remark that in the literature, the formulation of Chaves and Lorena [6] does not present better results than those shown in this paper, even with a time limit greater than 42 hours; and the model of Torres and Brito [28], to the best of our knowledge, has never been tested before on any instance used in this paper. The results of this experiment are presented in Table 1. The first column refers to the name of the tested instances and for each model, the solution value obtained and the time spent (in seconds) are presented. The best results are highlighted in bold, and the “-” symbol indicates that no solution was found within one-hour CPU time. Table 1: Results for the mathematical formulations for the instances of Chaves and Lorena [6].
Bienstock+ (^) Brito [28]Torres & Lorena [6]Chaves & Inst. Sol T(s) Sol T(s) Sol T(s) v10 1765 0.001 1765 0.030 1765 0. v20 2302 0.010 2302 0.700 2302 1. v30a 3582 0.001 3582 2.660 3582 4. v30b 2515 0.001 2515 3.670 2515 6. v30c 3236 0.030 3236 12.630 3236 15. v50a 4328 0.100 4328 314.740 4328 422. v50b 3872 0.120 3872 375.300 3872 688. v100a 6762 0.270 7633 3600.000 7454 3600. v100b 6760 0.170 7290 3600.000 7668 3600. v250a 14083 12.670 - 3600.000 - 3600. v250b 13632 8.350 14935 3600.000 14715 3600. v500a 25848 102.060 - 3600.000 - 3600. v500b 26389 130.170 - 3600.000 - 3600.
From Table 1, one can realize that the Bienstock+ for- mulation is superior to the others in the literature, both in solution quality and processing time. The modified for- mulation of Bienstock et al. [4] along with the proposed separation procedure was able to solve all 13 test problems proposed by Chaves and Lorena [6], proving optimality in 6 cases, in which the optimal solution was unknown until now. On the other hand, the formulations of the literature were able to solve only instances with up to 50 vertices, considering the time limit of one hour. Among the liter- ature models, the formulation of Chaves and Lorena [6]
first group of instances. Hence, the running times consid- ered for the CS are those reported by Chaves and Lorena [7] multiplied by the scale factor 0.30. Table 4 summarizes the results obtained with exper- iments performed between the MPO and CS heuristics, indicating the number of wins and ties for each heuristic, in terms of the best solution, average solution and aver- age time spent. The complete results are available at www- .deinf.ufma.br/˜glaubos/papers/PCTSP-complete-re- sults.pdf
Table 4: Number of wins and ties between MPO and CS.
Best Sol. Avg. Sol. Avg. T.(s) MPO wins 51 54 46 Ties 17 7 0 CS wins 40 47 62
From Table 4, one realizes that concerning the qual- ity of the best solution obtained, the MPO outperformed CS in 47.22% out of 108 instances, reaching an optimal solution in 55 of them; while the CS wins in 37.03% of the cases, reaching an optimal solution in 32. Regarding the average solution, the MPO maintains its superiority, winning in 50%, while the CS wins in 44% of the instances. Concerning the CPU-time spent, the CS proved to be more efficient, being faster in 65.74% of the cases. We believe that this superiority of the CS is due to the math- ematical solver that, when called within the MPO, tries to prove the optimality of the solution obtained, even though it is already an optimal one for the restricted search space. This procedure, inherent to the branch-and-bound mod- ule of the solver, makes MPO spend more processing time without improving the best current solution.
4.3. New instances
Until then, one can observe that the Bienstock+ model is capable of solving all available instances in the literature, in less than 3600 seconds. Therefore, in this paper, we propose a new group of instances, larger than those of Chaves and Lorena [6] and Chaves and Lorena [7]. This new group contains 15 instances, adapted from TSP test-problems available in the TSPLIB repository [25]. As in Chaves and Lorena [6], for both prize pi and penalty wi of each vertex i, random values were assigned in the range [1, 100]; while the P RIZE to be collected corre- sponds to 75% of the total instance prize. The Bienstock+ formulation was not able to obtain a feasible solution for any of the new instances, due to problems related to the lack of memory of the computer used. Therefore, for these instances, experiments were per- formed with the MPO heuristic proposed in this paper, which maintained the same parameter values of the pre- vious experiments. The results of this last experiment are presented in Table 5. From Table 5 one can realize that, for the MPO, the new instances are much more difficult than those in the literature. There are cases in which the problem has 2392
Table 5: Results of the MPO for the new instances
MPO Instances BestSol Avg.Sol Avg.T. (s) d1291 63808 64911.0 20388. d657 50856 51226.0 2258. fl1577 38542 38934.7 60984. gr666 243835 250758.0 2282. nrw1379 65517 66007.2 26864. p654 36525 37047.5 1952. pcb1173 65191 65972.4 15168. pr1002 237275 237727.0 8230. pr2392 371962 376414.0 142900. rat783 15902 16245.5 5168. rl1889 323509 325744.0 50727. u1060 205934 207250.0 5747. u1817 74493 75532.0 53103. vm1084 225980 227398.0 9594. vm1748 317133 319585.0 40351.
vertices (pr2392), requiring an average CPU time of 142900 seconds to be solved by the MPO; and there are also eas- ier problems, such as p654, which even being solved by the MPO in 1952.2 seconds, cannot be solved by the Bi- enstock+ formulation.
- Conclusions
In this paper, the well-known prize-collecting traveling salesman problem (PCTSP) was addressed. Four formu- lations from the literature were implemented and tested. One of them, the formulation of Bienstock et al. [4], has an exponential number of cut-sets constraints. There- fore, a polynomial complexity procedure was proposed for these separations, in addition to another set of constraints that we assumed to strengthen the linear relaxation of the model. In terms of non-exact approaches, a hybrid heuris- tic was proposed following the idea of Clímaco et al. [9], combining a GRASP and an RVND heuristic with ILP. After defining the strategies for the PCTSP, experi- ments were conducted in three sets of instances (one pro- posed in this work). For the first set of instances [6], the Bienstock+ formulation was the best, solving all the cases in a much shorter time than the others. The literature for- mulations were unable to solve instances with more than 50 nodes, using a time limit of 3600 seconds. In this first group of instances, the performance of the proposed heuris- tics was compared with the best heuristic in the literature (CS). The new MPO algorithm outperformed CS, con- cerning the best and average solutions, especially for larger instances (|V | > 100 ). For the second group of instances, the Bienstock+ model solved all cases in up to 300 seconds, while the other lit- erature formulations were not able to solve instances with more than 80 vertices, within an hour timeout. Consid- ering the four subgroups and the performance of all mod- els, the most difficult subgroup was cij ∈ [1, 1000] and wi ∈ [1, 100] with 43 cases where an optimal solution was
reached, while the easiest one was cij ∈ [1, 10000] and wi ∈ [1, 1000], with 58 instances solved. Despite the balance in the number of victories in terms of solution quality, the number of optimal solutions achieved by CS is always lower than with the MPO, for all the four subgroups. Another observation is that the MPO reached the optimal in instances of different sizes, while the CS only reached the optimality for problems with up to 80 vertices. Due to the literature instances were easily solved by the Bienstock+ formulation, in this paper, we proposed 15 larger new instances. When executing the Bienstock+ model, we have had problems related to the lack of memory of the used computer. Thus, we performed experiments with the MPO, which had no difficulty in finding feasible solutions for any of the new instances. In general, the MPO proved to be a robust heuristic, once it was tuned for a small set of instances, and performed well for three different sets of instances. As future investigations, it is intended to: seek alter- natives for a better stop criterion for the solver executed within the MPO; conduct a more in-depth study on the correlation between the E^0 set and the characteristics of the problem; verify the feasibility of applying data mining to our heuristic; and employ the same MPO heuristic to other variants of the TSP, since several of these problems differ only in their objective function or in few constraints.
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