Two difficulties arise when solving the set covering problem (SCP) with metaheuristic approaches: solution infeasibility and set redundancy. In this paper, we first present a review and analysis of the heuristic approaches that have been used in the literature to address these difficulties. We then present a new formulation that can be used to solve the SCP as an unconstrained optimization problem and that eliminates the need to address the infeasibility and set redundancy issues. We show that all local optimums with respect to the new formulation and a 1-flip neighbourhood structure are feasible and free of redundant sets. In addition, we adapt an existing greedy heuristic for the SCP to the new formulation and compare the adapted heuristic to the original heuristic using 88 known test problems for the SCP. Computational results show that the adapted heuristic finds better results than the original heuristic on most of the test problems in shorter computation times.
The set covering problem (SCP) is a popular optimization problem that has been applied to a wide range of industrial applications, including scheduling, manufacturing, service planning, and location problems [ Let Let
Minimize
The objective function (
When solving the model with metaheuristic algorithms, two issues arise: solution infeasibility and set redundancy. A solution to the SCP is considered to be infeasible if one or more of the elements of the universe
In this paper, we first review and analyze the literature to highlight the difficulties in dealing with solution infeasibility and set redundancy when solving the SCP with metaheuristic algorithms (Section
In general, metaheuristic algorithms can be divided into three categories.
In the following sections, we review the literature of solving the SCP with metaheuristic approaches and analyze how each category of metaheuristics addresses
When the SCP is solved with constructive metaheuristics, the local optimums found at the end of each constructive iteration are usually feasible. In fact, the constructive iteration ends when all of the elements are covered. For this reason, these metaheuristics do not have to deal with the infeasibility issue. However, the local optimums are not necessarily free of redundant sets, and a redundancy removal heuristic is needed. Constructive metaheuristics for the SCP includes ant colony optimization [
Evolutionary algorithms for the SCP need to address both infeasibility and set redundancy issues. Most evolutionary algorithms that are used to solve the SCP are based on the genetic algorithm (GA). Most of the GAs use a binary string solution representation where
The first approach uses a repair heuristic to transform infeasible solutions to feasible solutions before the evaluation step of the GA. A greedy-like repair heuristic is usually used [
The second approach involves penalizing the objective value of infeasible solutions to drive the search toward the feasible region. A penalty term that makes infeasible solutions less attractive than feasible ones is added to the objective function. In [
The feasibility constraint makes designing an effective local search metaheuristic for the SCP a difficult task. For this reason, few-local-search only heuristics have been developed for the SCP [
Most local search algorithms for the SCP use a simple 1-flip neighbourhood structure defined by moves that only add (remove) one set at a time to (from) the solution. When a local optimum is reached, which is usually a feasible solution, it is difficult to decide in which direction to continue the search. Two cases arise. If the search space is restricted to the feasible region, only redundant sets are allowed to be removed. If no redundant sets exist in the solution, at least one redundant set must be added before a remove move is allowed to be performed. As a result, the infeasible region of the search space will not be explored and the search will tend to fall into local optimums and cycles. A more complex neighbourhood called If the search space is not restricted to the feasible region, the cost minimization objective drives the search toward the infeasible region, by removing sets from the current configuration (to minimize the cost), and it is unclear when to restore feasibility. In such situations, penalty approaches are usually used to penalize infeasible solutions.
If the penalty weights are too high, neighbors in the feasible region will be preferred over neighbors in the infeasible region, making the infeasible region unreachable. Lower or dynamic penalty weights are usually used to make the search more effective by allowing it to reach infeasible regions.
If the penalty weights are too low, the final solution found is not guaranteed to be feasible. A tabu search heuristic that uses such low penalties is proposed in [
Dynamic penalty approaches, in which the penalty weights are repeatedly adjusted, are used to balance the search between the feasible and infeasible regions without using a repair operator or neighbourhood restrictions [
In this work, we propose a new formulation of the SCP with a maximization objective. The aim of the proposed formulation is to express the real objective of the SCP in the objective function which is to Let Let Let
Maximize
Constraint (
The optimal solution of the proposed formulation is a feasible solution (covers all elements).
Suppose that the optimal solution does not cover all of the elements and has an objective value
The optimal solution of the proposed formulation covers all elements at a minimal cost.
We proved in Claim
From heuristic algorithms perspective, we replaced a constrained optimization problem with an unconstrained optimization problem that has the same optimal solutions. Unconstrained optimization problems are known to be much easier to solve with heuristic algorithms than constrained optimization problems.
Eventhough the proposed formulation is a full mathematical programming formulation for the SCP, it is similar to the existing penalty approaches but with some important differences. The objective function presented in (
The proposed approach is different from high-penalty approaches because some infeasible solutions might have a better objective value than some feasible ones. For instance, let
The proposed approach is different from low-penalty approaches because the penalties are high enough to drive the search toward the feasible region. We showed that the optimal solutions with respect to the new formulation are guaranteed to be feasible. The proof of feasibility of the optimal solution also shows that any infeasible solution can be transformed to a feasible one with a better objective value. For instance, in the previous example, the infeasible solution
The proposed penalty approach is different from dynamic penalty approaches because the penalty weights are static and no adjustment is needed.
When high-penalty approaches are used, the search process of a heuristic algorithm is disturbed by the high penalties and driven immediately to the feasible region. On the other hand, low penalties do not disturb the search but cannot ensure feasibility. The aim of our approach is to choose the lowest possible penalties that avoid disturbing the search process while ensuring feasibility. Ensuring feasibility means that any infeasible solution can be transformed to a feasible one with a better objective value.
The new formulation eliminates all issues related to solution infeasibility and set redundancy that were discussed in the literature review (Section
In this section, we present a simple descent heuristic that is based on the new formulation and that uses a 1-flip neighbourhood structure. We also show that all local optimums with respect to the new formulation and the 1-flip neighbourhood are feasible and free of redundant sets.
The proposed descent heuristic (DH) is an adaptation of the classical greedy heuristic that has been used in the literature for the SCP [
DH starts from a given configuration and performs a sequence of moves on it until the solution is locally optimal. It uses a simple 1-flip neighbourhood structure with two types of moves: add and remove moves. add(
sol
In contrast to the classical greedy heuristic, DH automatically removes the redundant sets from the solution. Let
Consider
We showed that all of the solutions that are found with DH are feasible and free of redundant sets. With respect to the new formulation and the 1-flip neighbourhood structure, these solutions are local optimums. This is also true for all solutions obtained with any descent heuristic that is based on the new formulation and that uses the same neighbourhood structure. As a result, all local optimums with respect to the new formulation and the 1-flip neighbourhood structure are feasible and free of redundant sets.
In this section, we present computational experiments with the proposed descent heuristic that is based on the new formulation. Although we showed in the previous sections that the new formulation provides many advantages over the classical formulation, the final performance of any metaheuristic algorithm depends on the implementation, the tuning of the parameters, and the sophistication of the approach. We do not assume that any metaheuristic approach that is based on the new formulation will outperform all metaheuristic approaches that are based on the classical formulation. In addition, experimenting with all classes of metaheuristics will not prove (or disprove) the superiority of the proposed formulation. Instead, we compare our descent heuristic to the original greedy heuristic that is based on the classical formulation. The aim is to compare the two formulations using similar algorithms. Since greedy heuristics are used for intensification in most of the metaheuristic approaches for the SCP, evaluating the effectiveness of a new descent heuristic that can replace these greedy heuristics provides a good indication of how suitable is the new formulation to metaheuristic approaches.
We compare DH to the classical greedy heuristic (GH) [
OR-Library benchmarks.
Characteristics | Cost | Number of moves | |||||
---|---|---|---|---|---|---|---|
Instance | Size | Density | Best known | DH | GH | DH | GH |
4.1 | 200 |
2% | 429 | 433 | 434 | 77 | 93 |
4.2 | 200 |
2% | 512 | 523 | 552 | 75 | 94 |
4.3 | 200 |
2% | 516 | 531 | 546 | 79 | 96 |
4.4 | 200 |
2% | 494 | 503 | 507 | 70 | 93 |
4.5 | 200 |
2% | 512 | 515 | 518 | 72 | 95 |
4.6 | 200 |
2% | 560 | 575 | 597 | 78 | 83 |
4.7 | 200 |
2% | 430 | 444 | 449 | 74 | 77 |
4.8 | 200 |
2% | 492 | 493 | 525 | 70 | 77 |
4.9 | 200 |
2% | 641 | 672 | 672 | 82 | 99 |
4.10 | 200 |
2% | 514 | 519 | 528 | 71 | 86 |
5.1 | 200 |
2% | 253 | 265 | 273 | 76 | 88 |
5.2 | 200 |
2% | 302 | 314 | 335 | 71 | 82 |
5.3 | 200 |
2% | 226 | 230 | 230 | 66 | 82 |
5.4 | 200 |
2% | 242 | 246 | 254 | 69 | 86 |
5.5 | 200 |
2% | 211 | 214 | 215 | 73 | 87 |
5.6 | 200 |
2% | 213 | 216 | 227 | 69 | 85 |
5.7 | 200 |
2% | 293 | 297 | 305 | 76 | 84 |
5.8 | 200 |
2% | 288 | 297 | 304 | 77 | 85 |
5.9 | 200 |
2% | 279 | 281 | 290 | 68 | 84 |
5.10 | 200 |
2% | 265 | 271 | 274 | 74 | 81 |
6.1 | 200 |
5% | 138 | 149 | 143 | 39 | 56 |
6.2 | 200 |
5% | 146 | 156 | 154 | 44 | 53 |
6.3 | 200 |
5% | 145 | 149 | 157 | 43 | 46 |
6.4 | 200 |
5% | 131 | 134 | 140 | 46 | 51 |
6.5 | 200 |
5% | 161 | 180 | 182 | 47 | 50 |
A.1 | 300 |
2% | 253 | 258 | 269 | 82 | 97 |
A.2 | 300 |
2% | 252 | 262 | 268 | 78 | 93 |
A.3 | 300 |
2% | 232 | 243 | 248 | 80 | 105 |
A.4 | 300 |
2% | 234 | 240 | 243 | 84 | 107 |
A.5 | 300 |
2% | 236 | 240 | 246 | 79 | 107 |
B.1 | 300 |
5% | 69 | 72 | 71 | 41 | 45 |
B.2 | 300 |
5% | 76 | 79 | 78 | 44 | 50 |
B.3 | 300 |
5% | 80 | 84 | 84 | 47 | 46 |
B.4 | 300 |
5% | 79 | 84 | 88 | 44 | 50 |
B.5 | 300 |
5% | 72 | 72 | 75 | 46 | 48 |
C.1 | 400 |
2% | 227 | 237 | 252 | 102 | 110 |
C.2 | 400 |
2% | 219 | 230 | 225 | 93 | 128 |
C.3 | 400 |
2% | 243 | 249 | 258 | 89 | 102 |
C.4 | 400 |
2% | 219 | 229 | 239 | 94 | 115 |
C.5 | 400 |
2% | 215 | 222 | 222 | 93 | 106 |
D.1 | 400 |
5% | 60 | 64 | 66 | 49 | 54 |
D.2 | 400 |
5% | 66 | 68 | 69 | 52 | 50 |
D.3 | 400 |
5% | 72 | 77 | 80 | 54 | 59 |
D.4 | 400 |
5% | 62 | 62 | 66 | 52 | 54 |
D.5 | 400 |
5% | 61 | 65 | 67 | 49 | 61 |
E.1 | 500 |
10% | 29 | 30 | 30 | 30 | 35 |
E.2 | 500 |
10% | 30 | 33 | 35 | 31 | 37 |
E.3 | 500 |
10% | 27 | 29 | 31 | 29 | 31 |
E.4 | 500 |
10% | 28 | 32 | 31 | 32 | 33 |
E.5 | 500 |
10% | 28 | 30 | 30 | 30 | 32 |
F.1 | 500 |
20% | 14 | 16 | 17 | 16 | 17 |
F.2 | 500 |
20% | 15 | 16 | 16 | 15 | 16 |
F.3 | 500 |
20% | 14 | 17 | 15 | 17 | 17 |
F.4 | 500 |
20% | 14 | 17 | 15 | 17 | 14 |
F.5 | 500 |
20% | 13 | 15 | 15 | 17 | 15 |
G.1 | 1000 |
2% | 176 | 186 | 191 | 132 | 146 |
G.2 | 1000 |
2% | 154 | 166 | 176 | 115 | 139 |
G.3 | 1000 |
2% | 166 | 178 | 182 | 126 | 147 |
G.4 | 1000 |
2% | 168 | 178 | 179 | 128 | 138 |
G.5 | 1000 |
2% | 168 | 179 | 182 | 127 | 131 |
H.1 | 1000 |
5% | 63 | 69 | 69 | 68 | 65 |
H.2 | 1000 |
5% | 63 | 70 | 72 | 62 | 67 |
H.3 | 1000 |
5% | 59 | 63 | 66 | 62 | 62 |
H.4 | 1000 |
5% | 58 | 65 | 64 | 65 | 61 |
H.5 | 1000 |
5% | 55 | 60 | 61 | 61 | 60 |
Airline and bus driver crew scheduling problems.
Characteristics | Cost | Number of moves | |||||
---|---|---|---|---|---|---|---|
Instance | Size | Density | Best known | DH | GH | DH | GH |
AA03 | 106 |
4.05% | 33155 | 34637 | 35642 | 48 | 61 |
AA04 | 106 |
4.05% | 34573 | 36153 | 36749 | 45 | 62 |
AA05 | 105 |
4.05% | 31623 | 32249 | 32995 | 45 | 65 |
AA06 | 105 |
4.11% | 37464 | 38043 | 39422 | 43 | 70 |
AA11 | 271 |
2.53% | 35478 | 36965 | 39054 | 76 | 90 |
AA12 | 272 |
2.52% | 30815 | 33663 | 34044 | 77 | 85 |
AA13 | 265 |
2.60% | 33211 | 36337 | 37345 | 77 | 91 |
AA14 | 266 |
2.50% | 33219 | 36048 | 36530 | 77 | 95 |
AA15 | 267 |
2.58% | 34409 | 36269 | 37996 | 73 | 94 |
AA16 | 265 |
2.63% | 32752 | 36185 | 37160 | 79 | 85 |
AA17 | 264 |
2.61% | 31612 | 34326 | 36484 | 69 | 91 |
AA18 | 271 |
2.55% | 36782 | 39594 | 40603 | 84 | 101 |
AA19 | 263 |
2.63% | 32317 | 34749 | 36093 | 71 | 92 |
AA20 | 269 |
2.58% | 34912 | 37047 | 37744 | 82 | 86 |
BUS1 | 454 |
1.89% | 27947 | 28871 | 29673 | 88 | 100 |
BUS2 | 681 |
0.51% | 67760 | 69685 | 70606 | 282 | 280 |
Railway crew scheduling problems.
Characteristics | Cost | Number of moves | |||||
---|---|---|---|---|---|---|---|
Instance | Size | Density | Best known | DH | GH | DH | GH |
RAIL507 | 507 |
1.2% | 174 | 205 | 212 | 150 | 169 |
RAIL516 | 516 |
1.3% | 182 | 202 | 202 | 181 | 186 |
RAIL582 | 582 |
1.2% | 211 | 243 | 251 | 191 | 212 |
RAIL2586 | 2586 |
0.4% | 948 | 1102 | 1185 | 770 | 917 |
RAIL2536 | 2536 |
0.4% | 691 | 828 | 891 | 581 | 660 |
RAIL4284 | 4284 |
0.2% | 1065 | 1303 | 1385 | 997 | 1091 |
RAIL4872 | 4872 |
0.2% | 1534 | 1802 | 1900 | 1339 | 1521 |
Most metaheuristic approaches for the SCP have been exclusively tested on OR-Library benchmarks. Because these benchmarks are relatively small, we experimented with larger problems that have been less frequently used in the literature.
In all presented tables, the name of each instance is given in the first column, the size of each instance is given in the second column (number of elements
Percentage deviation from the best-known solution: OR-Library benchmarks 4.1 to 6.5.
Percentage deviation from the best-known solution: OR-Library benchmarks A.1 to H.5.
Percentage deviation from the best-known solution: airline and bus scheduling problems.
Percentage deviation from the best-known solution: railway scheduling problems.
In both DH and GH, each iteration involves finding the best set to be added (removed) to (from) the solution and updating the underlying data structure after a move is performed. Thus, the algorithmic complexity of each iteration is similar in both heuristics. In practice, the computation times are highly dependent on the implementation and the characteristics of the problem solved (size and density). For instance, finding the best move to be performed in each iteration can be implemented using a loop that iterates over all sets or using a priority-queue-based data structure. Preliminary testing showed that choosing one way or another greatly affects the speed comparison of the discussed heuristics. To avoid an implementation-dependent comparison, and because these aspects of the implementation are out of the scope of this work, we recorded the number of iterations instead.
Both heuristics are deterministic, and only one run is required. The value of
Our descent heuristic performed better than GH by finding better solutions for most of the test problems. For OR-Library benchmarks, DH found better solutions than GH for 47 instances, equal solutions for 10 instances, and worse solutions for 9 instances. For the airline, bus, and railway scheduling problems, DH found better solutions than GH for all problems except one (equal solutions for RAIL516). The percentage deviations presented in Figures
Average number of iterations and percentage deviations.
Problems | Average number of iterations | Average percentage deviation | ||
---|---|---|---|---|
DH | GH | DH | GH | |
OR-Library benchmarks | 64.89 | 74.51 | 5.46 | 7.31 |
Airline and bus problems | 82.25 | 96.75 | 5.99 | 9.14 |
Railway problems | 601.29 | 679.43 | 17.12 | 22.81 |
DH also performed fewer iterations than GH for most of the test problems. For OR-Library benchmarks, DH performed fewer iterations than GH for 56 instances, equal number of iterations for seven instances, and more iterations for only two instances. For the airline, bus, and railway scheduling problems, DH performed fewer iterations than GH for all problems except one (more iterations for BUS2). The average number of iterations performed by DH and GH is presented in Table
As a result, the proposed descent heuristic that is based on the new formulation performs better than the corresponding greedy heuristic that is based on the classical formulation by finding better results for most of the test problems using fewer iterations, which can lead to shorter computation times.
In this paper, we identified two issues that arise when solving the SCP with metaheuristic approaches: solution infeasibility and set redundancy. We highlighted the difficulties of addressing these issues when solving the SCP with the different classes of metaheuristics and proposed a new formulation that overcomes these difficulties. We showed that this formulation is, in fact, a new penalty approach that uses static penalty weights that are low enough to avoid disturbing the search but high enough to ensure the feasibility of the final solution. We also showed that all local optimums with respect to the new formulation and the 1-flip neighbourhood structure are feasible and free of redundant sets. As a result, building metaheuristic approaches for the SCP using the new formulation is straightforward.
To provide a first computational experience using the new formulation, we adapted a known greedy heuristic for the SCP to the new formulation and compared the adapted version to the original version using 88 set covering problems. The adapted version that is based on the new formulation found better solutions than the original version that is based on the classical formulation for 69 tests problems, equal solutions for ten problems, and worse solutions for nine problems. In addition, the adapted version performed fewer iterations than the original version for 78 test problems, equal number of iterations for two problems, and more iterations for eight problems. Thus the adapted version finds better solutions than the original version in potentially shorter computation times. Moreover, the adapted version was easier to implement because we did not need to handle feasibility and set redundancy.
Most current metaheuristic approaches for the SCP incorporate a descent or greedy heuristic that is responsible for the intensification part of the search. Thus, having a more effective descent heuristic can lead to better metaheuristic approaches.