An Improved Ant Colony Optimization Based on Candidate Strategy and Grid Search for the Vehicle Routing Problem with Simultaneous Pickup and Delivery

Abstract

This paper studies a vehicle routing problem with simultaneous pickup and delivery (VRPSPD), which has an important application in logistics and other areas. The problem is that the depot provides both forward supply service and reverse recovery service to customers, and determines the lowest-cost vehicle distribution routes that satisfy the needs of all customers on the basis of considering constraints. To solve this problem, we develop an improved ant colony optimization algorithm based on candidate strategy and grid search (ACO-CS). The candidate strategy of ACO-CS reduces the running cost and speeds up the convergence rate by limiting and reducing the number of unvisited nodes. At the same time, we propose to use the grid search method to tune the parameters to enhance the algorithm’s optimization capability and improve its performance. Three benchmark test problems are selected to verify the effectiveness of the proposed algorithm for solving different types and sizes of instances. The computational results show that the proposed algorithm is competitive in solving the Dethloff (2001) and Montane & Galveo (2006) test problems, and its solution quality, calculation time and algorithm stability are better than the variant algorithms in the literature. Finally, a practical case of logistics distribution is introduced to verify the reliability of the algorithm, and the results show that the ACO-CS can provide a more reasonable and economical solution.

1. Introduction

Logistics and distribution, with the rise of e-commerce, have undergone rapid development, and at the same time also brought about transportation problems. Transportation service is an important part of logistics distribution. Reducing transportation costs and improving transportation efficiency are effective ways to address transportation problems as well as to accelerate the development of the logistics industry. The vehicle routing problem (VRP) is the core problem of transportation optimization. Therefore, the study of the VRP problem is of great significance for solving transportation problems in logistics and distribution.

The VRP problem was first proposed in 1959 by scholars Dantzig and Ramser [1], which optimizes the routes for transporting goods while satisfying the customer’s needs, with the aim of delivering the goods to the destinations at the lowest transportation cost. The traditional VRP problem can only deliver or pick up goods, which is a one-way distribution mode. However, in the actual logistics distribution, there will be the re-transportation of recyclable goods and waste products, which will lead to the phenomenon of empty trucks back and cause the waste of resources. The emergence of the vehicle routing problem with simultaneous pickup and delivery (VRPSPD) can effectively solve the problems caused by one-way distribution. This type of problem provides simultaneous delivery and pickup services to customers, combines forward and reverse logistics, and improves vehicle load factor and resource utilization.

The VRPSPD problem has received extensive attention and research from scholars since it was first proposed by Min [2] in 1989, and a wealth of research results have been achieved. Among them, Dethloff [3] proposed a heuristic construction method and provided randomly generated examples at the same time. Nagy and Salhi [4] proposed an enhanced integrated heuristic algorithm and allowed for the existence of infeasible solutions during the solution process. Chen and Wu [5] proposed a hybrid heuristic method and used an insertion algorithm to generate its initial solution. Wassan et al. [6] proposed a reactive tabu search heuristic algorithm by combining neighborhood search and tabu search. Ai and Kachitvichy anukul [7] combined the cheapest insertion heuristics and 2-opt operation based on particle swarm optimization algorithms to improve the quality of the solution. Gajpal and Abad [8] introduced three local search operators, 2-opt, customer insertion/exchange multipath operator, and sub-path exchange multipath operator to improve the quality of the solution based on the ant colony system. Gajpal and Abad [9] designed four saving heuristics, all of which can create a new route by merging two existing routes. Souza et al. [10] proposed a hybrid heuristic algorithm combining an iterative local search algorithm, variable neighborhood descent and GENIUS. Jun and Kim [11] designed a new heuristic algorithm, which consists of three parts: route construction process, route improvement process and perturbation process. Goksal et al. [12] based on particle swarm optimization and introduced a variable neighborhood descent algorithm for local search while using a simulated annealing strategy to maintain the population diversity of the algorithm. Pan et al. [13] proposed a heuristic algorithm, which first generated the initial solution with the construction algorithm, and then reduced the number of routes with the path elimination algorithm. Kalayci and Kaya [14] introduced variable neighborhood search as local search into the ant colony algorithm to further explore the search space. In recent years, studies on the VRPSPD problem include: Chen and Fang [15] proposed a two-layer particle swarm optimization, where the outer layer is used to solve the group of customers in each vehicle, and the inner layer is to find the optimal visit order of customers. Park et al. [16] introduced an improved genetic algorithm in which roulette is used for selection, a two-point crossover operator for reproduction and a chaotic mutation operator for mutation. Oztas and Tus [17] proposed a hybrid algorithm combining iterative local search, variable neighborhood descent and threshold acceptance meta-heuristics. Praxedes et al. [18] proposed a unified branch reduction algorithm to solve VRPSPD. At present, in addition to the research on VRPSPD, there is an increasing number of studies on variants of VRPSPD [19,20,21,22].

Based on the above brief review of the work on VRPSPD, it is clear that there is a wide variety of methods for solving VRPSPD. Among them, heuristic algorithms and metaheuristic algorithms [23] are the main solution methods. Ant colony optimization (ACO), as a kind of metaheuristic algorithm, is often applied to optimize problems in various fields. Compared with other algorithms, ACO has strong global search ability and robustness because each ant is independent of the others in the process of solving. However, the positive feedback feature of the algorithm makes it easy to fall into a local optimum, and the random selection in the search process tends to slow down its convergence. To address and overcome the shortcomings of the traditional ACO, Liang et al. [24] proposed an enhanced ACO algorithm, which enhances the traditional ACO algorithm by dynamically adjusting the optimization information, pheromone volatility factor, pheromone updating strategy, and the state transfer probability during the iteration process. Du et al. [25] proposed an improved ACO algorithm based on mutation operation and dynamic evaporation coefficients. Xu et al. [26] improved the heuristic function and pheromone update strategy for basic ACO. Wu et al. [27] designed a new ACO variant that introduces a new heuristic mechanism with directional information in the optimization search process, improves the heuristic function and state transfer probabilities, and non-uniformizing the initial pheromone. Zhao et al. [28] proposed an adaptive pheromone strategy based on learning automata to adapt to the different optimization stages of the algorithm in order to improve the efficiency and quality of the ACO. Kurdi [29] proposed a heuristic information with stochasticity, diversity and improvability to improve the search capability of ACO. Liu et al. [30] proposed an improved dynamic adaptive ant colony optimization, which includes heuristic strategies with directional information, adaptive pseudo-random transfer strategies, and improved local and global pheromone updating mechanisms. Martin et al. [31] improved the state transfer probabilities of the basic ACO to improve the exploration performance of the algorithm. Paniri et al. [32] improved the heuristic function and initial pheromone of ACO to accelerate the convergence of the algorithm. The improved ACO proposed by Yang et al. [33] modified the search rules and pheromone release rules of the basic ACO to improve the search ability of the algorithm.

Based on the above, it can be found that in view of the shortcomings of traditional ACO in recent years, the improved methods mainly focus on four aspects: heuristic information/function, state transition rule, pheromone evaporation factor and pheromone update. In addition, there is a correlation between the parameters of ACO, and the combination of the parameters directly affects the performance of the algorithm, while the optimization of ACO parameters is more often used in parameter tuning methods [34]. How to improve the traditional ACO from a new perspective and explore a new method of optimizing parameter combinations to solve the VRPSPD problem is the focus of this paper. Therefore, this paper proposes an improved ant colony algorithm based on candidate strategy and grid search to solve VRPSPD, and its main work is as follows: Firstly, the improvement of ACO is not limited to the above four perspectives; we create a new perspective, consider the improvement of the candidate set for storing unvisited nodes and propose a candidate strategy. Secondly, in most of the existing studies on the parameters of the ACO, the parameter values are provided directly or the parameters are set using tuning and orthogonal methods. For this reason, we provide a grid search method for tuning the parameters; so far in the literature, the use of this method for optimization of the ACO parameters is relatively rare. Finally, a local search operator is introduced to explore the solution space better.

The rest of the paper is organized as follows. Section 2 is the problem description and modeling. Section 3 is the proposed algorithm. Section 4 is numerical experiments and analysis. Section 5 is the case study. Section 6 is the conclusion.

2. Problem Description and Model Establishment

VRPSPD means that given a depot and a set of customers located in different geographic locations and a fleet of vehicles. The vehicles start from and return to the depot after delivery and pickup services for customers, in which the amount of deliveries and pickups by the customers cannot exceed the maximum capacity of the vehicles. And plan the shortest route for each vehicle to satisfy the customer’s delivery and pickup needs.

VRPSPD can be defined as a directed graph $G=(V,E)$, where $V=\{0,1,2,\dots ,n\}$ denotes the set of all nodes, 0 denotes the depot, $V_0=\{0,1,2,\dots ,n\}$ denotes the set of customers, and $E=\{(i,j),i,j\in V_0\}$ denotes the set of edges. The distance between customer i and customer j is denoted by $c_{ij}$. Vehicle $k\in K=\{i,\dots ,m\}$ starts from the depot and returns to the depot, assuming that all vehicles have the same specifications. Each customer i has a delivery demand $d_i$ and a pickup demand $p_i$. $d_i$ denotes the quantity of goods delivered from the depot to customer i. $p_i$ denotes the quantity of goods delivered from customer i to the depot. It is assumed that different kinds of goods are compatible and can be loaded in the same vehicle. There are no delivery and pickup demands at the depot. The customer’s total demand on any route cannot exceed the vehicle’s maximum load capacity C, where $L_{0k}$ denotes the load of the vehicle k leaving the depot, $L_j$ denotes the load of the vehicle after it serves the customer. Figure 1 shows a schematic diagram of VRPSPD.

According to the above description of the VRPSDP problem, the following mathematical model is established:

$$\begin{array}{c}\hfill min\sum _{i\in V}\sum _{j\in V}\sum _{k\in K}{c}_{ij}{x}_{ijk}\end{array}$$

(1)

$$\begin{array}{c}\hfill s.t.\sum _{i\in V}\sum _{k\in K}{x}_{ijk}=1,\forall j\in V_0\end{array}$$

(2)

$$\begin{array}{c}\hfill \sum _{i\in V}{x}_{ihk}=\sum _{j\in V}{x}_{hjk},\forall h\in V_0;\forall k\in K\end{array}$$

(3)

$$\begin{array}{c}\hfill \sum _{j\in V}{x}_{0jk}=1,\forall k\in K\end{array}$$

(4)

$$\begin{array}{c}\hfill \sum _{j\in V_0}{x}_{0jk}=\sum _{i\in V_0}{x}_{i0k},\forall k\in K\end{array}$$

(5)

$$\begin{array}{c}\hfill \sum _{i\in V}{x}_{i0k}=1,\forall k\in K\end{array}$$

(6)

$$\begin{array}{c}\hfill L_{0k}=\sum _{i\in V}\sum _{j\in V_0}{d}_j{x}_{ijk},\forall k\in K\end{array}$$

(7)

$$\begin{array}{c}\hfill L_j\ge L_{0k}-d_j+p_j-M(1-x_{0jk}),\forall j\in V_0;\forall k\in K\end{array}$$

(8)

$$\begin{array}{c}\hfill L_j\ge L_{0k}-d_j+p_j-M(1-\sum _{k\in K}{x}_{ijk}),\forall i\in V_0;\forall j\in V_0;i\ne j\end{array}$$

(9)

$$\begin{array}{c}\hfill L_{0k}\le C,\forall k\in K\end{array}$$

(10)

$$\begin{array}{c}\hfill L_j\le C+M(1-\sum _{i\in V}{x}_{ijk}),\forall j\in V;\forall k\in K\end{array}$$

(11)

$$\begin{array}{c}\hfill x_{ijk}\in \{0,1\},\forall i\in V_0;\forall j\in V_0;\forall k\in K\end{array}$$

(12)

The objective function (1) represents the minimization of distance. Constraint (2) restricts each customer to be assigned to only one route. Constraint (3) ensures that each customer is served by the same vehicle. Constraints (4)–(6) ensure that each vehicle starts at the depot, arrives at a customer and must leave for another customer, and ends up at the depot. Constraint (7) is the initial vehicle load. Constraint (8) represents the vehicle load after serving the first customer. Constraint (9) is the ”en route” vehicle load. Constraints (10) and (11) are vehicle capacity constraints that require the total customer demand not to exceed the maximum capacity of the vehicle. Constraint (12) is the decision variable.

3. Solution

Ant colony optimization algorithm is a metaheuristic algorithm inspired by the foraging principle of ants in nature, which is one of the effective methods for solving discrete problems. Because ants can transmit information according to their unique information mechanism—pheromones—the ant colony can always find the shortest path to reach the food source in different environments. The ant guides its direction during movement by the pheromones it releases on its path and by sensing the pheromones its companions leave behind on the path. The greater the pheromone concentration, the greater the probability that the ants will choose the path, and the more ants will pass on the path, which in turn will again enhance the pheromone concentration on the path, so as to show a positive feedback phenomenon. According to the ant foraging principle, the core of the algorithm is the choice of the ant’s path at the next moment (state transfer) and the increase and evaporation of pheromone on the path (pheromone update).

In this paper, we propose an improved ant colony algorithm based on candidate strategy and grid search, named ACO-CS. On the one hand, in traditional ACO, ants need to calculate the state transfer probability for all unvisited nodes to determine the next visited node, but this method will increase the runtime of the algorithm due to the excessive number of unvisited nodes in the early stage of the algorithm. To address this problem, the candidate strategy reduces the running cost by limiting or reducing the number of unvisited nodes, while improving the quality of the current solution. In addition, the positive feedback characteristic of ACO can guide the algorithm to the direction of the optimal solution, but in the later stage of the algorithm, this characteristic will make the difference of pheromone concentration on the good path and the bad path obvious, and it is easy to make the ACO fall into a local optimum when the suboptimal solution dominates. To address this problem, we use the swap operator as a local search method to avoid the algorithm falling into a local optimum. On the other hand, the setting of ACO parameters has an important impact on the performance of the algorithm, and a reasonable combination of parameters can enhance the algorithm’s optimization-seeking ability. How to determine the optimal parameter combination to make the ACO solution performance better is also one of the core problems in ACO research. To address this problem, we used the grid search method to determine the parameter combinations of the algorithm (given in detail in Section 4). The pseudo-code of the ACO-CS algorithm is shown in Algorithm 1, and each part of the proposed algorithm is explained and described in detail in the following subsections. Section 3.1 describes the candidate strategy proposed in this paper. Section 3.2 describes the process of constructing the solution. Section 3.3 shows the local search method. Section 3.4 describes the pheromone update process.

In Algorithm 1, the relevant data for the problem is entered, including the geographic location of the customers, pickup demands and delivery demands (point 1). Initialize the parameters of the algorithm (point 2). Then enter the main loop and iterate to find the optimal path and optimal solution (points 3–32). Where points 5–15 are the process of constructing the solution, that is, m ants select the node according to the candidate strategy and the pseudo-random ratio rule, and complete the tour. Points 16–28 are the process of calculating the objective function. After each ant constructs the path, calculate the function value $f\left(i\right)$ corresponding to the current solution s, then performs the local search for the current solution s at a certain frequency to obtain the solution $s^{\ast }$ and calculates the corresponding function value $f^{\ast }\left(i\right)$, If $f^{\ast }\left(i\right)<f\left(i\right)$, then $f^{\ast }\left(i\right)$ replaces $f\left(i\right)$. Then update the current optimal path and the optimal distance (point 29), update the pheromone on the path (point 30). Finally, the optimal solution is output.

3.1. Candidate Strategy

The set of all unvisited nodes in the process of building a solution is called a candidate set. The ant selects the next node to be visited from the candidate set according to the state transition probability. At this time, the probability between the current node and all nodes in the candidate set needs to be calculated, which will increase the computing time to a certain extent. Since the goal of the problem solved in this paper is to minimize the distance, we consider reducing the running cost without reducing the quality of the solution. To realize this purpose, this paper proposes a candidate strategy. The main idea of the strategy is that with the current node as the center, some unvisited nodes are selected from all unvisited nodes according to certain rules to form a candidate set. In other words, the running time of the algorithm is reduced by reducing the number of unvisited nodes in the candidate set while maintaining the solution quality.

Algorithm 1 Pseudo-code of the proposed algorithm ACO-CS

1: Input the relevant data of the problem; // input the location of the node, delivery demands, pickup demands   2: Initialization: set the initial value of the iterations Iter=1, the maximum number of iterations Itermax=200; and the other parameters of ACO-CS;   3: while $Iter\le Itermax$ do   4:       /*Build routes for all ants*/   5:      for i=1 to m do   6:            The candidate set is determined by distance, and the candidate region allow is expanded by radius R; //Section 3.1   7:            j=1;   8:            while j ≤ n do   9:                  if ∼ isempty(allow) then 10:                        Calculate the state transition probability P; 11:                        Select the next node based on the state transfer probability and determine if the node satisfies the constraints; //Section 3.2 12:                  end if 13:                  Update candidate set; //Section 3.1 14:            end while 15:       end for 16:       /*Calculate the value of the objective function*/ 17:       f=zero(m,1); 18:       for i=1 to m do 19:            s=table(i,:); 20:            $f\left(i\right)$=fitness(s); 21:            if rem(Iter,nfreq)==0 then 22:                  Perform a local search for s and get $s^{\ast }$; //Section 3.3 23:                  $f^{\ast }\left(i\right)$=fitness($s^{\ast }$); 24:                  if $f^{\ast }\left(i\right)\le f\left(i\right)$ then 25:                        $f\left(i\right)=f^{\ast }\left(i\right)$ 26:                  end if 27:            end if 28:       end for 29:       Update optimal routes and distances; 30:       Update pheromone; //Section 3.4 31:       Iter=Iter+1; 32: end while 33: Output the optimal solution

The rule for selecting some unvisited nodes to form the candidate set is as follows: take the current node i as the center and r as the radius to draw a circle. Judge whether there are unvisited nodes within the range of radius r. If there are, all the nodes within the range constitute the candidate set; if not, expand the search to judge whether there are unvisited nodes within the circle of radius r–2r (2r–3r …). Since we use concentric circles to determine the candidate set based on distance, the unvisited nodes stored in the candidate set are the closest nodes to the current node i within the specified range, thus ensuring that the quality of the understanding is not degraded. To better understand the above process, we give an example of a candidate strategy in Figure 1.

Figure 2 shows an example of selecting some unvisited nodes as a candidate set from the total unvisited nodes. In Figure 2a, node 1 as the center, all unvisited nodes form a candidate set C = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, there are three candidate nodes within radius r, and nodes 2, 3 and 7 constitute the candidate set $C_1$ = {2, 3, 7} of node 1, the range for selecting the next node at node 1 changes from C to $C_1$. In Figure 2b, node 14 as the center, all unvisited nodes form a candidate set C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17}, no unvisited node is found in the range of radius r and thus no candidate set is constructed, so it is necessary to expand the search scope and find 5 nodes in the range of radius 2r, which are nodes 10, 11, 12, 13 and 15 and form the candidate set $C_{14}$ = {10, 11, 12, 13, 15}, the range for selecting the next node at node 14 changes from C to $C_{14}$. In Figure 2c, node 16 as the center, all unvisited nodes form a candidate set C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17}. There are no unvisited nodes in radius r, and no unvisited nodes can be found in radius 2r after expanding the search scope. Continue to expand the search scope, and nodes 3, 4, 11, 12, 15 and 17 can be found in radius 3r, forming the candidate set $C_{16}$ = {3, 4, 11, 12, 15, 17}. Therefore, the range for selecting the next node at node 16 has changed from C to $C_{16}$.

The candidate strategy is based on the current node as the center, and the unvisited nodes found within the radius r(expanded by a certain percentage) constitute the candidate set. This strategy can reduce the number of unvisited nodes in the candidate set, which plays a vital role in the solution construction of the algorithm. In addition, it can also be noticed from Figure 2 that the nodes in the candidate set provide more possibilities to find neighboring nodes on the optimal path, regardless of whether the current node’s location is central (node 1) or remote (node 16). This is because the candidate strategy proposed in this paper is designed based on a distance-based hierarchical screening approach, which effectively prevents high-quality nodes from being prematurely excluded. The rule always prioritizes including the node closest to the current one, and since this study aims to minimize total travel distance, nearby nodes are naturally preferred for constructing high-quality solutions. Moreover, the algorithm does not fixate on selecting only a small number of nodes but instead gradually expands the search range layer by layer, avoiding the direct discarding of potentially high-quality nodes on the periphery. Thus, it reduces computational effort while ensuring that quality nodes are not excluded too early, thereby maintaining solution quality. Below we describe the application of the candidate strategy in the solution construction process.

3.2. Solution Construction

The construction of solutions is also called the construction of ants’ paths. At each step of the construction path, the ant needs to select the next node from the candidate set according to the state transition rule. To avoid ants choosing the same node repeatedly, a path memory vector is used to store the nodes that the ant passes through, which is called the tabu table. Therefore, the probability that ant k chooses the next node j at node i at time t is calculated using Equation (13).

$$\begin{array}{c}\hfill P_{ij}^k\left(t\right)=\left\{\begin{array}{ccc}\hfill {\displaystyle \frac{{\tau }_{ij}^{\alpha }\left(t\right)·{\eta }_{ij}^{\beta }\left(t\right)}{{\sum }_{s\in Allowe{d}_k}{\tau }_{ij}^{\alpha }\left(t\right)·{\eta }_{ij}^{\beta }\left(t\right)}},& & s\in Allowe{d}_k\hfill \\ \hfill 0,& & otherwise\hfill \end{array}\right.\end{array}$$

(13)

In Equation (13), ${\tau }_{ij}\left(t\right)$ denotes the pheromone on the path $\left(ij\right)$ at time t, ${\eta }_{ij}\left(t\right)$ denotes the heuristic function, $\alpha$ denotes the information heuristic factor, and $\beta$ denotes the expectation heuristic factor. $Allowe{d}_k=\{C-tabu\}$ denotes the set of unvisited nodes, where C denotes all nodes, tabu denotes nodes that have been visited.

According to the state transition rule, when the ant selects the next node, it needs to calculate the probability of all nodes in the candidate set, and when there are many unvisited nodes in the early stage, the running cost of the algorithm will be greatly increased. Therefore, the candidate strategy proposed in Section 3.1 is used to improve the running efficiency of the algorithm.

The solution construction process using the candidate strategy is shown in Figure 3. Where red indicates the current node, yellow indicates the candidate unvisited node, blue indicates the visited node, and white indicates the unvisited node. Ant k starts from the depot (0) and obtains candidate set $C_0$ = {1, 2, 3, 5} by candidate strategy. The next visited node 1 is selected by calculating the state transfer probability and determining whether the constraints are satisfied after adding node 1 to the path. If they are satisfied, node 1 is added to the path that ant k is currently constructing and continues to visit the node; otherwise, it returns to the depot and restarts constructing the path. Then, node 1 as the center, the candidate set $C_1$ = {2, 7} is obtained by the candidate strategy, and the next node 2 is obtained by calculating the probability and satisfies the constraint conditions, and continue visiting the node. With node 2 as the center, candidate set $C_2$ = {3} is obtained by candidate strategy, and the next node 3 is visited by calculating the probability and satisfying the constraint conditions, and continues to visit the node.With node 3 as the center, continue to select the next node, but the constraint is not satisfied, the visit is ended and return to the depot, which obtains the ant k path $s_1=\{0-1-2-3-0\}$. Repeat the above process until all nodes are visited. Finally the complete path $S=\{s_1,s_2,s_3\}$ is obtained, where $s_1=\{0-1-2-3-0\}$, $s_2=\{0-4-5-0\}$, and $s3=\{0-7-6-0\}$.

During the path construction process, in order to make the ants more purposeful in the search space, this paper adopts a pseudo-random proportion (14) to select the next node.

$$\begin{array}{c}\hfill j=\left\{\begin{array}{ccc}\hfill argmax{\tau }_{ij}^{\alpha }\left(t\right)·{\eta }_{ij}^{\beta }\left(t\right),& & q\le q_0\hfill \\ \hfill P_{ij},& & q>q_0\hfill \end{array}\right.\end{array}$$

(14)

where $q_0$ is a user-defined value, $q={\displaystyle \frac{iter}{ite{r}_{max}}}$. When $q\le q_0$, the next node j is selected by greedy method; when $q>q_0$, the next node j is selected according to the roulette method, $P_{ij}^k$ is calculated using Equation (13).

3.3. Local Search

The positive feedback feature of the ACO algorithm can make it have good convergence speed, but it will enlarge the difference between good and bad paths with the updating of pheromones. If the initial solution is a sub-optimal solution, then the positive feedback will give the solution an advantage and the algorithm will fall into a local optimum. Therefore, this paper introduces a local search operator in ACO to avoid the above situation.

Local search is the process of neighborhood search to get a better solution. Where the intra-path neighborhood structure obtains the neighborhood solution of the current routing by changing the customer’s location in various ways. The inter-path neighborhood structure is to obtain neighborhood solutions by exchanging customers between two different paths. These two neighbourhood structures aim to improve the solution and to focus the search on promising parts of the search space. In this study, we use the inter-route neighbourhood structure (swap operator) to perform a neighbourhood search for the current solution s according to a certain frequency $nf$. An example of the swap operator is shown in Figure 4.

The swap operator includes three neighborhood structures: swap1-1, swap2-2, and swap2-1. The swap1-1 operation is to randomly select two sub-routes in the current solution and the exchange of one node in each sub-route to form two new paths. The swap2-2 operation is to randomly select two sub-routes in the current solution and exchange two nodes in each sub-route to form two new paths. Swap2-1 exchanges two nodes on one sub-route with one node on the other route. Because the three neighborhood structures exchange nodes on different paths to form a new path, the difference is that the number of nodes exchanged is different. Therefore, swap1-1 is chosen as the local search operator in this study.

3.4. Pheromone Updates

The change of pheromone guides the evolution of ACO towards the global optimal solution. As more ants pass on the path, the more pheromones are on the path, and the more likely the ants are to choose the path, further increasing the pheromone concentration on the path. However, when fewer ants pass on the path, the pheromones on the path evaporate over time, and the ants are less likely to choose the path, further reducing the pheromone concentration on the path. When all ants have completed a tour, the pheromone concentrations on the corresponding paths need to be updated, so as to better guide the ants to find the optimal path. Therefore, the pheromone concentration on the path at time t + 1 is updated according to Equations (15)–(17):

$$\begin{array}{c}\hfill {\tau }_{ij}(t+1)=(1-\rho )\times {\tau }_{ij}\left(t\right)+\Delta {\tau }_{ij}\left(t\right)\end{array}$$

(15)

$$\begin{array}{c}\hfill \Delta {\tau }_{ij}\left(t\right)=\sum \Delta {\tau }_{ij}^k\left(t\right)\end{array}$$

(16)

$$\begin{array}{c}\hfill \Delta {\tau }_{ij}^k\left(t\right)=\left\{\begin{array}{ccc}\hfill {\displaystyle \frac{Q}{L_k}},& & thekthantpassesthroughthepath(i,j)\hfill \\ \hfill 0,& & otherwise\hfill \end{array}\right.\end{array}$$

(17)

where $\rho$ denotes the pheromone evaporation coefficient, $\Delta {\tau }_{ij}\left(t\right)$ denotes the pheromone increment on the path $(i,j)$ in this cycle, $\Delta {\tau }_{ij}^k\left(t\right)$ denotes the pheromone released by the kth ant through the path $(i,j)$ in this cycle, Q is a pheromone constant, $L_k$ denotes the length of the path $(i,j)$ that the kth ant passes through in this cycle.

4. Numerical Experiments and Results

This section gives the test problems, parameter settings, and computational results for each test problem of the ACO-CS algorithm. The proposed algorithm is written in MATLAB2018b and runs on Windows 7 operating system with Intel(R) Core(TM) i5-6200U 2.30 GHz processor and 4.00 GB RAM.

4.1. Test Problems

There are three common benchmark sets for VRPSPD problems, including Salhi & Nagy (1999), Dethloff (2001), and Montane & Galveo (2006). Salhi & Nagy (1999) test set has 28 problem instances in which the customer size is 50–199. Of the 28 CMT instances, half of the instances have path length limitation, while the other half have path length and time limitation. The Dethloff (2001) test set has 40 problem instances, each with customer size of 50. These problem instances are divided into SCA and CON instances. In the SCA instance, customers are randomly and uniformly distributed in the interval [0, 100]. While in the CON instances, half of the customers are randomly and uniformly distributed in the interval [0, 100], and half of the customers are randomly and uniformly distributed in the interval [100/3, 200/3]. Montane & Galveo (2006) consists of 56 problem instances, each with customer size of 100. These problem instances can be divided into three categories based on the geographic distribution of customers: random distribution R, centralised distribution C and mixed distribution RC. In the three benchmark test sets above, we did not find publicly available test problems on Salhi & Nagy (1999) and Dethloff (2001), so this study constructs the corresponding problem instances based on the detailed descriptions of problem instance construction in literature [3], literature [4], and literature [17]. Montane & Galveo (2006) use open test problems.

4.2. Parameter Settings

For metaheuristic algorithms, parameter combination is also one of the factors that affect the performance of the algorithm. A good parameter combination enables the algorithm to obtain high-quality solution in a short time. In this paper, we use the grid search method to adjust the algorithm parameters. Grid search is often applied in the field of machine learning and pattern recognition with the aim of determining the optimal parameters to achieve the best performance [35,36]. The core idea of this method is to traverse all possible parameter combinations, then compare the results obtained according to the evaluation criteria, and finally determine the most efficient parameter combination. At present, there are relatively few studies on ACO parameters using grid search, so we extend the application scope of grid search to determine ACO parameters.

4.2.1. Grid Search Optimization

Grid search is a classic method for parameter optimization. In this study, we first determine the reasonable value ranges for each parameter to be optimized and set the traversal step size, enumerating all parameter combinations within the range; then we statistically determine the optimal parameter combination for each test instance, calculate the frequency of each parameter value in all the optimal combinations; finally, based on this frequency distribution, we select general-purpose parameters that are suitable for all test instances.

The parameter range of this grid search is mainly determined based on two points: Firstly, it refers to the classic literature and general experience values used in the ant colony algorithm to solve the vehicle routing problem, ensuring the rationality of the range; Secondly, it is adapted and optimized in combination with the scale of the example and the characteristics of the problem, eliminating obviously unreasonable ranges and reducing the amount of invalid calculations. This paper focuses on analyzing the three core parameters: the information heuristic factor $\alpha$, the expected heuristic factor $\beta$, and the pheromone evaporation factor $\rho$. Finally, the range of values is determined as: $\alpha \in [1,5]$, $\beta \in [1,5]$, and $\rho \in [0.1,0.9]$.

In order to ensure the accuracy of the algorithm parameters, 10 experiments are performed on different parameter sets within a certain range, and the average value of the 10 experimental results is taken as the experimental results for comparison. Meanwhile, to ensure the search performance of the algorithm in different types of test problems, the CMT1X and CMT1Y of Salhi & Nagy (1999) test set, SCA3-0, SCA8-0, CON3-0 and CON8-0 of Dethloff (2001) test set, C101, C201, R101, R201, RC101 and RC201 of Montane & Galveo (2006) test set, a total of 12 sample instances are used for parameter analysis. Figure 5 shows the results of parameter combinations for 12 sample instances.

In Figure 5, we analyse the relationship between three important parameters in the algorithm, namely the information heuristic factor, the expectation heuristic factor and the pheromone evaporation factor. The depth of the color in the figure indicates the quality of the algorithm to solve the problem.

Table 1. Optimal parameter combination.

Parameter	CMT1X			CMT1Y					SCA3-0			SCA8-0					CON3-0			CON8-0			R101				R201		C101				C201			RC101		RC201
$\alpha$	4	4	3		3	3	4		3	3	4		3	1	3		2	2	3		2	1		5	5	3		4		4	5	5		5	5		5		3	1	4	4
$\beta$	4	1	4		3	4	5		4	4	2		2	2	3		5	3	1		3	4		4	3	1		3		3	2	5		3	4		4		4	4	3	2
$\rho$	0.2	0.8	0.4		0.5	0.6	0.8		0.5	0.6	0.6		0.4	0.5	0.6		0.6	0.5	0.8		0.6	0.8		0.7	0.3	0.3		0.8		0.9	0.7	0.9		0.6	0.7		0.6		0.6	0.6	0.2	0.1

shows the optimal parameter combinations obtained for each sample instance among the numerous parameter combinations of Figure 5.

To apply the parameters to all the test problems, we use 12 sample instances as a baseline to determine the parameter values used in this paper by calculating the frequency.

Table 2. Parameter value frequency of occurrence.

Parameter	$\mathit{\alpha }$					$\mathit{\beta }$						$\mathit{\rho }$
parameter value	1	2	3	4	5		1	2	3	4	5		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
frequency	3/31	3/31	10/31	8/31	7/31		3/31	5/31	9/31	11/31	3/31		1/31	2/31	2/31	2/31	4/31	10/31	3/31	5/31	2/31

shows the frequency of occurrence of each parameter value in the 31 optimal combinations.

According to the frequency values in Table 2, the final parameter values are determined as follows: $\alpha =3,\beta =4,\rho =0.6$. In addition, the other parameter values used in the algorithm are: $q_0=0.2,nf=20,itermax=200$.

4.2.2. Performance Analysis of Parameter Tuning

To verify the effectiveness of parameter tuning, this paper conducts a control experiment. The experiment is divided into an empirical parameter group and a grid search optimization parameter group. Apart from the core parameters, all configurations, such as the number of ants, the maximum number of iterations, the candidate set strategy, and the running environment, are kept consistent in both groups.

The empirical parameter group adopts the standard general values of the ant colony algorithm used in the VRPSPD problem-solving: $\alpha =1,\beta =5,\rho =0.1$. The preferred parameter group uses the optimal parameter combination obtained through grid search in this paper.

The two sets of algorithms independently ran each of the entire test instances 10 times. By comparing the optimal solutions, average values, standard deviations and the running time of a single run, the performance differences were analyzed. The experimental comparison results are shown in

Table 3. Performance comparison of empirical parameters and optimal parameters.

Instance	Group	Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time
CMT1X	empirical group	809.73	818.89	813.99	2.71	138.38
	optimal group	590.07	619.62	608.75	11.54	101.74
CMT1Y	empirical group	809.44	820.85	816.81	3.71	123.43
	optimal group	599.08	616.90	608.02	8.44	99.41
SCA3-0	empirical group	1831.94	1864.69	1847.93	10.19	122.56
	optimal group	726.77	732.22	729.79	15.00	95.04
SCA8-0	empirical group	2109.78	2154.51	2136.95	11.27	130.18
	optimal group	1419.44	1430.14	1424.85	8.19	105.20
CON3-0	empirical group	2043.91	2072.71	2055.27	9.41	142.69
	optimal group	775.87	799.24	789.61	10.99	91.59
CON8-0	empirical group	1399.97	1531.56	1493.02	39.48	138.93
	optimal group	1069.70	1096.34	1084.13	8.03	115.47
R101	empirical group	2853.75	2957.13	2911.03	34.32	296.99
	optimal group	2854.57	2916.60	2884.74	106.66	250.88
R201	empirical group	2812.43	3090.16	2983.78	87.13	252.69
	optimal group	947.27	983.91	966.00	29.73	199.78
C101	empirical group	3510.86	3663.84	3577.89	54.71	297.30
	optimal group	3446.05	3674.61	3584.98	43.59	258.05
C201	empirical group	3527.27	3803.65	3678.58	92.11	394.63
	optimal group	1364.89	1466.54	1415.57	91.56	265.86
RC101	empirical group	3450.59	3707.24	3548.87	98.2	279
	optimal group	3521.54	3621.03	3570.41	148.67	257.08
RC201	empirical group	3461.32	3614.13	3557.56	92.11	279.44
	optimal group	1111.77	1208.62	1137.57	270.60	198.33

below.

From the experimental results in Table 3, it can be seen that compared with the traditional empirical parameter combination, the optimized parameters obtained through grid search perform better overall in terms of solution quality and running efficiency. Moreover, the algorithm stability has been improved in most cases, fully verifying the value of the parameter tuning work.

Firstly, in terms of solution quality: In the vast majority of test instances, such as CMT1X, CMT1Y, SCA3-0, SCA8-0, CON3-0, CON8-0, R201, C201, RC201, etc., the optimal solution, the worst solution, and the average solution of the preferred parameter group are significantly lower than those of the empirical parameter group. The path optimization effect is very remarkable. Taking CON8-0 as an example, the average solution decreased from 1493.02 to 1084.13; the average solution of R201 decreased from 2983.78 to 966.00, and the solution accuracy improved significantly. Only for the three instances R101, C101, and RC101, the results of the two groups were close, and the preferred parameters did not show obvious disadvantages. Overall, parameter tuning effectively exploited the optimization ability of the algorithm and significantly improved the quality of the solution.

Secondly, in terms of algorithm stability (standard deviation), the standard deviation is used to measure the degree of fluctuation in the results of multiple runs. Furthermore, the smaller the value, the stronger the algorithm’s stability. In instances such as SCA8-0, CON8-0, R201, and C101, the standard deviation of the preferred parameter groups has significantly decreased, and the algorithm’s robustness has greatly improved. Among them, the standard deviation of CON8-0 decreased from 39.48 to 8.03, with a particularly remarkable improvement. For some large-scale instances (R101, RC101, RC201), the standard deviation slightly increased, mainly due to the greater difficulty in solving these types of problems and the complexity of the search space, but it still remained within the acceptable range.

Thirdly, in terms of computational efficiency, among all 12 sets of test instances, the single-run time of the optimal parameter group was consistently shorter than that of the empirical parameter group. This indicates that the optimized parameters can reduce the ineffective search during the algorithm’s iterative process, thereby improving the solution quality while further shortening the computing time and achieving a simultaneous optimization of accuracy and efficiency.

In conclusion, the parameters obtained through grid search optimization have significantly improved the solution quality and operational efficiency of the algorithm. The stability of most cases has also been enhanced. This parameter setting scheme is highly reasonable and practical.

4.3. Calculation Results and Comparison

In this subsection, we first give the calculation results of the proposed algorithm on the test problem, and then provide the comparison results with other algorithms.

4.3.1. Results of the Test Problem

The proposed algorithm is run 10 times on each problem instance in the test set introduced in Section 4.1.

Table 4. Results of Salhi & Nagy (1999) test problem.

Instance	ACO					ACO-CS
	Best Sol.	Worse Sol.	Avg Sol.	Avg Time		Best Sol.	Dev	Worse Sol.	Dev	Avg Sol.	Dev	Avg Time	Dev
CMT1X	626.74	665.29	647.92	148.14		590.07	−5.85	619.62	−6.86	608.75	−6.05	101.74	−31.32
CMT1Y	630.01	657.91	642.94	117.18		599.08	−4.91	616.90	−6.23	608.02	−5.43	99.41	−15.16
CMT2X	1032.50	1064.66	1051.70	211.73		967.44	−6.30	983.94	−7.58	974.30	−7.36	176.12	−16.82
CMT2Y	1038.37	1077.36	1050.58	288.42		958.62	−7.68	986.93	−8.39	971.63	−7.51	172.93	−40.04
CMT3X	1196.66	1266.98	1234.47	375.45		1063.85	−11.10	1082.34	−14.57	1070.73	−13.26	238.16	−36.57
CMT3Y	1218.91	1285.76	1240.77	375.97		1057.55	−13.24	1090.03	−15.22	1072.02	−13.60	285.94	−23.95
CMT4X	1439.90	1588.78	1539.78	449.47		1303.92	−9.44	1323.39	−16.70	1312.29	−14.77	403.05	−10.33
CMT4Y	1493.15	1596.06	1518.93	431.04		1263.84	−15.36	1322.53	−17.14	1299.45	−14.45	397.15	−7.86
CMT5X	1906.81	2028.04	1934.68	621.15		1656.61	−13.12	1686.19	−16.86	1666.76	−13.85	595.47	−4.13
CMT5Y	1903.54	1923.16	1910.01	642.18		1640.67	−13.81	1681.47	−12.57	1668.77	−12.63	610.18	−4.98
CMT6X	634.80	664.28	650.42	145.01		598.76	−5.68	610.17	−8.15	606.15	−6.81	101.38	−30.09
CMT6Y	622.42	654.25	639.11	144.47		588.25	−5.49	616.22	−5.81	604.20	−5.46	100.67	−30.32
CMT7X	1016.45	1063.49	1035.07	267.28		949.94	−6.54	980.91	−7.77	970.30	−6.26	168.94	−36.79
CMT7Y	1015.62	1062.79	1043.89	388.40		958.70	−5.60	981.47	−7.65	972.61	−6.83	169.71	−56.31
CMT8X	1117.99	1193.24	1149.98	355.62		978.24	−12.50	1009.06	−15.44	996.20	−13.37	205.25	−42.28
CMT8Y	1118.05	1184.41	1154.82	292.35		986.66	−11.75	1007.08	−14.97	1002.40	−13.20	225.75	−22.78
CMT9X	1511.07	1593.83	1564.70	727.54		1287.60	−14.79	1328.89	−16.62	1309.75	−16.29	394.96	−45.71
CMT9Y	1561.69	1599.55	1575.68	787.12		1285.48	−17.69	1321.37	−17.39	1306.22	−17.10	390.80	−50.35
CMT10X	1993.54	2052.78	2011.02	1262.03		1660.62	−16.70	1680.33	−18.14	1668.06	−17.05	812.10	−35.65
CMT10Y	2005.49	2067.91	2027.96	1442.33		1650.61	−17.70	1678.69	−18.82	1665.87	−17.85	594.76	−58.76
CMT12X	1020.58	1068.03	1047.37	282.14		987.57	−3.23	1030.27	−3.54	1009.29	−3.64	224.57	−20.40
CMT12Y	1032.74	1078.93	1054.99	362.76		990.95	−4.05	1029.70	−4.56	1016.36	−3.66	225.14	−37.94
CMT13X	1232.15	1249.48	1240.59	400.89		1171.35	−4.93	1205.42	−3.53	1195.43	−3.64	307.59	−23.27
CMT13Y	1224.01	1258.23	1240.79	403.29		1189.08	−2.85	1210.97	−3.76	1201.94	−3.13	312.45	−22.52

,

Table 5. Results of Dethloff (2001) test problem.

Instance	ACO					ACO-CS
	Best Sol.	Worse Sol.	Avg Sol.	Avg Time		Best Sol.	Dev	Worse Sol.	Dev	Avg Sol.	Dev	Avg Time	Dev
SCA3-0	831.95	880.91	849.50	114.85		726.77	−12.64	732.22	−16.88	729.79	−14.09	95.04	−17.25
SCA3-1	984.18	1033.89	1001.93	114.78		897.53	−8.80	928.58	−10.19	917.60	−8.42	92.70	−19.24
SCA3-2	843.99	884.45	868.26	108.13		758.93	−10.08	778.21	−12.01	771.18	−11.18	91.52	−15.36
SCA3-3	947.49	991.21	968.20	104.11		919.59	−2.94	936.44	−5.53	928.24	−4.13	90.21	−13.35
SCA3-4	942.87	1003.17	975.63	189.40		903.15	−4.21	912.54	−9.03	911.23	−6.60	89.39	−52.80
SCA3-5	865.09	898.11	887.01	116.90		797.87	−7.77	808.47	−9.98	807.20	−9.00	92.79	−20.62
SCA3-6	943.16	1010.06	978.87	186.45		883.83	−6.29	921.77	−8.74	907.06	−7.34	92.95	−50.15
SCA3-7	824.28	873.46	854.02	119.94		763.52	−7.37	778.53	−10.87	771.92	−9.61	92.80	−22.63
SCA3-8	908.72	952.80	928.16	107.27		828.69	−8.81	893.40	−6.23	873.14	−5.93	90.63	−15.51
SCA3-9	1031.03	1067.58	1044.63	119.61		971.20	−5.80	974.17	−8.75	973.46	−6.81	90.44	−24.39
SCA8-0	1479.30	1502.19	1490.09	115.48		1419.44	−4.05	1430.14	−4.80	1424.85	−4.38	105.20	−8.90
SCA8-1	1376.26	1420.55	1391.27	151.20		1329.09	−3.43	1354.25	−4.67	1341.43	−3.58	111.35	−26.36
SCA8-2	1219.02	1266.51	1247.34	178.16		1161.11	−4.75	1180.24	−6.81	1171.87	−6.05	111.06	−37.66
SCA8-3	1535.06	1555.14	1542.93	139.72		1516.65	−1.20	1534.29	−1.34	1526.66	−1.05	105.63	−24.40
SCA8-4	1336.57	1377.67	1357.86	148.95		1256.52	−5.99	1311.70	−4.79	1299.25	−4.32	116.93	−21.50
SCA8-5	1598.93	1630.43	1615.92	135.67		1559.06	−2.49	1568.47	−3.80	1563.46	−3.25	108.04	−20.37
SCA8-6	1450.86	1505.43	1482.98	130.87		1354.61	−6.63	1369.69	−9.02	1361.56	−8.19	106.20	−18.85
SCA8-7	1626.10	1656.47	1644.58	161.44		1582.53	−2.68	1614.40	−2.54	1599.09	−2.77	105.67	−34.55
SCA8-8	1316.07	1357.39	1335.53	126.42		1284.20	−2.42	1302.93	−4.01	1295.93	−2.97	107.19	−15.21
SCA8-9	1603.65	1632.67	1618.84	173.16		1517.20	−5.39	1573.87	−3.60	1556.72	−3.84	102.41	−40.86
CON3-0	841.86	876.94	858.48	197.20		775.87	−7.84	799.24	−8.86	789.61	−8.02	91.59	−53.55
CON3-1	778.44	807.24	794.56	116.68		703.05	−9.68	717.92	−11.06	713.66	−10.18	92.36	−20.84
CON3-2	842.43	903.11	869.99	117.76		749.50	−11.03	773.11	−14.39	761.94	−12.42	90.50	−23.15
CON3-3	718.97	763.94	745.75	104.68		643.33	−10.52	656.41	−14.08	649.27	−12.94	98.49	−5.91
CON3-4	710.51	749.86	726.06	103.16		660.89	−6.98	674.53	−10.05	668.81	−7.89	101.85	−1.27
CON3-5	760.83	792.22	774.23	115.68		705.90	−7.22	722.43	−8.81	714.49	−7.72	92.52	−20.02
CON3-6	808.20	831.20	818.41	151.50		706.43	−12.59	743.49	−10.55	735.46	−10.14	99.20	−34.52
CON3-7	931.41	984.77	964.80	127.23		853.11	−8.41	867.09	−11.95	858.87	−10.98	95.28	−25.11
CON3-8	803.05	830.33	818.57	205.16		765.43	−4.68	788.40	−5.05	778.75	−4.86	95.61	−53.40
CON3-9	747.73	781.27	762.70	122.78		665.22	−11.03	694.62	−11.09	680.19	−10.82	94.93	−22.68
CON8-0	1096.99	1119.79	1110.69	212.34		1069.70	−2.49	1096.34	−2.09	1084.13	−2.39	115.47	−45.62
CON8-1	1238.10	1287.02	1265.62	137.21		1170.83	−5.43	1204.34	−6.42	1193.49	−5.70	114.27	−16.72
CON8-2	1357.43	1404.36	1380.29	169.73		1304.12	−3.93	1322.69	−5.82	1315.20	−4.72	121.23	−28.57
CON8-3	1159.97	1200.37	1182.04	147.23		1150.92	−0.78	1164.59	−2.98	1157.13	−2.11	136.74	−7.12
CON8-4	1167.55	1212.10	1183.45	122.72		1148.65	−1.62	1169.87	−3.48	1159.59	−2.02	116.93	−4.72
CON8-5	1163.78	1188.66	1174.76	138.19		1122.97	−3.51	1156.55	−2.70	1143.05	−2.70	123.26	−10.80
CON8-6	1620.95	1673.59	1648.67	171.59		1582.36	−2.38	1627.39	−2.76	1605.20	−2.64	129.79	−24.36
CON8-7	1417.31	1456.31	1438.87	171.19		1334.18	−5.87	1349.23	−7.35	1341.52	−6.77	115.66	−32.44
CON8-8	1201.10	1260.36	1228.11	125.44		1133.96	−5.59	1147.40	−8.96	1139.59	−7.21	117.65	−6.21
CON8-9	1376.73	1411.88	1394.26	153.39		1327.88	−3.55	1354.20	−4.09	1347.37	−3.36	131.23	−14.45

and

Table 6. Results of Montane & Galveo (2006) test problem.

Instance	ACO					ACO-CS
	Best Sol.	Worse Sol.	Avg Sol.	Avg Time		Best Sol.	Dev	Worse Sol.	Dev	Avg Sol.	Dev	Avg Time	Dev
R101	3044.65	3375.05	3140.46	362.79		2854.57	−6.24	2916.60	−13.58	2884.74	−8.14	250.88	−30.85
R102	2960.59	3212.72	3098.27	271.67		2809.08	−5.12	2980.30	−7.23	2918.87	−5.79	244.84	−9.88
R103	3003.23	3490.24	3167.87	353.78		2842.08	−5.37	3137.28	−10.11	2923.77	−7.71	252.17	−28.72
R104	2947.43	3356.28	3149.69	464.19		2863.00	−2.86	3144.77	−6.30	2971.26	−5.67	252.67	−45.57
R105	3068.16	3374.28	3138.27	361.91		2832.53	−7.68	3015.04	−10.65	2919.95	−6.96	252.27	−30.29
R106	3003.41	3617.45	3213.99	396.09		2853.91	−4.98	3122.03	−13.70	2930.61	−8.82	251.96	−36.39
R107	3082.22	3304.17	3179.77	418.55		2856.27	−7.33	3062.96	−7.30	2947.26	−7.31	254.78	−39.13
R108	2925.18	3259.70	3075.09	331.04		2859.11	−2.26	3134.23	−3.85	2953.38	−3.96	291.27	−12.01
R109	2931.44	3220.23	3064.35	389.87		2851.00	−2.74	3136.28	−2.61	2930.29	−4.37	244.30	−37.34
R110	3006.49	3270.27	3092.94	369.96		2827.76	−5.94	3033.08	−7.25	2904.32	−6.10	242.80	−34.37
R111	2903.07	2999.00	2950.88	192.40		2852.71	−1.73	3133.67	4.49	2920.12	−1.04	244.86	27.27
R112	2902.07	3210.35	2954.27	201.72		2872.61	−1.02	3090.83	−3.72	2972.99	0.63	266.77	32.25
R201	1107.21	1198.65	1148.11	356.53		947.27	−14.45	983.91	−17.92	966.00	−15.86	199.78	−43.97
R202	1123.31	1184.63	1152.77	331.91		951.34	−15.31	982.43	−17.07	964.98	−16.29	205.95	−37.95
R203	1100.41	1185.62	1132.85	326.40		947.92	−13.86	974.55	−17.80	963.27	−14.97	200.04	−38.71
R204	1026.63	1175.43	1131.84	316.13		953.90	−7.08	976.19	−16.95	966.73	−14.59	200.83	−36.47
R205	1106.00	1183.33	1151.82	305.10		949.22	−14.18	985.19	−16.74	969.13	−15.86	200.29	−34.35
R206	1143.44	1172.61	1148.97	315.87		939.09	−17.87	982.62	−16.20	966.16	−15.91	201.25	−36.29
R207	1076.69	1179.34	1144.68	320.31		954.61	−11.34	987.96	−16.23	972.13	−15.07	200.05	−37.54
R208	1109.69	1190.20	1152.83	321.87		945.73	−14.78	979.44	−17.71	966.47	−16.17	201.86	−37.29
R209	1094.11	1180.23	1133.18	323.39		940.15	−14.07	988.92	−16.21	963.42	−14.98	201.07	−37.82
R210	1026.63	1175.43	1128.84	325.67		938.71	−8.56	981.93	−16.46	967.04	−14.33	201.38	−38.16
R211	1101.89	1183.33	1146.66	305.41		946.43	−14.11	980.24	−17.16	965.54	−15.80	224.29	−26.56
C101	3554.34	3623.92	3594.60	216.20		3446.05	−3.05	3674.61	1.40	3584.98	−0.27	258.05	19.36
C102	3501.48	3730.41	3611.30	215.05		3441.43	−1.71	3767.60	1.00	3590.97	−0.56	281.71	31.00
C103	3533.56	3712.68	3609.81	215.48		3474.05	−1.68	3749.55	0.99	3593.00	−0.47	282.98	31.33
C104	3540.59	3845.14	3630.10	216.27		3521.91	−0.53	3595.40	−6.49	3548.76	−2.24	288.18	33.25
C105	3540.42	3672.64	3585.35	217.07		3516.62	−0.67	3762.27	2.44	3580.46	−0.14	288.21	32.77
C106	3504.86	3621.90	3549.00	244.25		3473.89	−0.88	3714.62	2.56	3581.50	0.92	286.03	17.11
C107	3462.95	3647.70	3541.08	414.30		3445.33	−0.51	3654.40	0.18	3545.88	0.14	292.06	−29.51
C108	3590.12	3754.09	3652.38	216.33		3544.81	−1.26	3769.43	0.41	3617.58	−0.95	291.43	34.72
C109	3555.40	3593.68	3585.08	217.16		3502.56	−1.49	3722.17	3.58	3580.85	−0.12	293.36	35.09
C201	1412.92	1694.52	1504.97	346.65		1364.89	−3.40	1466.54	−13.45	1415.57	−5.94	265.86	−23.31
C202	1372.04	1596.98	1471.10	356.41		1355.29	−1.22	1447.70	−9.35	1398.43	−4.94	232.03	−34.90
C203	1422.88	1590.37	1521.77	406.93		1353.65	−4.87	1465.06	−7.88	1416.32	−6.93	230.96	−43.24
C204	1407.83	1622.06	1480.55	359.49		1338.12	−4.95	1436.85	−11.42	1399.15	−5.50	236.18	−34.30
C205	1374.80	1639.93	1540.89	322.02		1347.73	−1.97	1404.12	−14.38	1394.27	−9.52	232.72	−27.73
C206	1360.30	1644.03	1476.38	772.35		1339.49	−1.53	1448.25	−11.91	1390.64	−5.81	227.86	−70.50
C207	1362.84	1629.95	1497.91	320.40		1336.08	−1.96	1420.45	−12.85	1376.35	−8.12	235.51	−26.50
C208	1406.95	1665.87	1517.41	380.43		1344.01	−4.47	1434.85	−13.87	1390.56	−8.36	233.20	−38.70
RC101	3663.74	4072.47	3921.38	562.70		3521.54	−3.88	3621.03	−11.09	3570.41	−8.95	257.08	−54.31
RC102	3719.90	3903.27	3816.32	640.88		3427.17	−7.87	3661.91	−6.18	3552.98	−6.90	253.60	−60.43
RC103	3740.92	4139.88	3900.34	359.14		3405.90	−8.96	3602.49	−12.98	3540.58	−9.22	257.58	−28.28
RC104	3795.68	4070.09	3898.73	564.92		3468.55	−8.62	3631.42	−10.78	3573.27	−8.35	258.61	−54.22
RC105	3644.19	4114.36	3843.89	636.70		3472.23	−4.72	3636.21	−11.62	3562.22	−7.33	265.94	−58.23
RC106	3545.90	4111.40	3826.43	555.40		3412.85	−3.75	3631.00	−11.68	3544.77	−7.36	256.75	−53.77
RC107	3720.25	4230.92	3886.39	555.81		3414.69	−8.21	3738.88	−11.63	3551.34	−8.62	257.41	−53.69
RC108	3661.53	4267.49	3894.08	403.70		3426.44	−6.42	3763.41	−11.81	3561.70	−8.54	258.19	−36.04
RC201	1163.74	1686.45	1163.74	321.83		1111.77	−4.47	1208.62	−28.33	1137.57	−2.25	198.33	−38.37
RC202	1197.41	1503.40	1327.11	277.80		1089.91	−8.98	1165.51	−22.48	1114.71	−16.00	354.35	27.56
RC203	1150.00	1511.89	1390.02	270.17		1089.08	−5.30	1162.20	−23.13	1116.13	−19.70	198.41	−26.56
RC204	1193.04	1446.61	1274.31	281.14		1098.08	−7.96	1133.50	−21.64	1127.30	−11.54	196.99	−29.93
RC205	1178.58	1569.07	1347.67	480.59		1109.17	−5.89	1213.34	−22.67	1134.31	−15.83	196.45	−59.12
RC206	1197.54	1582.98	1323.47	282.18		1107.27	−7.54	1135.69	−28.26	1124.00	−15.07	187.86	−33.43
RC207	1165.42	1672.48	1350.93	263.00		1097.56	−5.82	1139.30	−31.88	1122.08	−16.94	198.46	−24.54
RC208	1214.35	1594.36	1365.20	594.46		1085.02	−10.65	1179.82	−26.00	1122.84	−17.75	198.49	−66.61

record the best solution (Best Sol.), worst solution (Worse Sol.), average solution (Avg Sol.), and average time (Avg time) of ACO and ACO-CS. To compare the proposed algorithm with the ACO, the deviation percentage (Dev) is used to represent the gap between the obtained solution (ACO-CS) and the known solution (ACO). The equation for calculating the deviation percentage is: $Dev(\%)=(Z-Z^{\ast })/Z^{\ast }\times 100$, Z denotes the solution of the ACO-CS and $Z^{\ast }$ denotes the solution of the ACO.

Table 4 records the results of the ACO and the ACO-CS on the Salhi & Nagy (1999) test problem. Since the ACO and ACO-CS failed to run in instances CMT11X, CMT11Y, CMT14X, and CMT14Y, we only recorded results for 24 problem instances. ACO-CS improves the quality of the best solution by 2.85–17.7%, the worst solution by 3.53–18.82%, and the average solution by 3.13–17.85%. Reduced running time by 4.13–58.76%. Therefore, compared with the ACO algorithm, the ACO-CS algorithm improves the quality of the best solution, the worst solution and the average solution, and reduces the running time of the algorithm.

Table 5 records the results of the ACO and the ACO-CS on Dethloff (2001) test problems. From the results in the table, it is clearly evident that the solution obtained by the ACO-CS is better than that of the ACO, and the running time is also less than the ACO. Firstly, for the best solution, the ACO-CS algorithm improves the quality of the best solution on SCA problem instances by 1.20–12.64%; the quality of the best solution on the CON problem instances is improved by 0.78–12.59%. Secondly, the quality of the worst solution on the SCA problem instances is improved by 1.34–16.88%; the quality of the worst solution on the CON problem instances is improved by 2.09–14.39%. Thirdly, the quality of the average solution on the SCA problem instances is improved by 1.05–14.09%; the quality of the average solution on the CON problem instances is improved by 2.20–12.94%. Finally, for running time, there is a reduction of 8.90–52.80% in the SCA problem instance and of 1.27–53.55% in the CON problem instances. From the above analysis, it can be seen that for the two types of Dethloff (2001) test problems, the ACO-CS algorithm has the same effect in solving SCA type and CON type.

Table 6 records the results of the ACO and the ACO-CS on the Montane & Galveo (2006) test problem. The results in the table show that the ACO-CS improves the quality of the best solution for all problem instances, improves the quality of the worst solution in 82.14% of the problem instances, improves the quality of the average solution in 94.64% of the problem instances, and reduces running time on 80.36% of the problem instances.

Firstly, for the best solution, the ACO-CS improves the quality of the best solution on the R type instances by 1.02–15.31%; the quality of the best solutions improved by 0.51–4.85% on the C type instances and by 3.75–10.65% on the RC type instances. Secondly, the quality of the worst solution of the R type instances is improved by 2.61–13.70%, but the worst solution obtained by instance R111 is not improved. Of the C type instances, C1 instances do not improve the quality of the worst solution except for the instance C103; however, the worst solutions of C2 instances are improved. The worst solution is improved for all instances of RC type instances, but the improvement of RC2 type is better than that of RC1 type. For the average solution, the quality of the average solution of R1 type has improved by 1.04–8.22% expect the instance R111. The average solution quality of the 11 instances in the R2 type improved by 14.33–16.29%. In the C type instances, the average quality of the solutions for the 15 instances improved by 0.12–9.52%; the improvement effect of C2 type is better than that of C1, but the average solution of instances C106 and C107 did not improve. In the RC type instances, the quality of the average solution of 16 instances is improved. Finally, the running time of the R type instances is reduced except for instances R111 and R112. Among the C type instances, only instance C107 of C1 type has a reduced running time, while the 8 instances of C2 type have reduced running times by 23.31–70.50%. In the RC type instances, the running time decreased by 24.54–66.61%, but the instance RC202 did not decrease.

According to the above analysis results, it can be seen that for Montane & Galveo (2006) test problems, the effect of the ACO-CS algorithm on solving R2, C2 and RC2 type problem instances is better than that of R1, C1 and RC1, regardless of solution quality or running time.

4.3.2. Comparison with Other Advanced Algorithms on Test Problems

We consider the latest improved ACO algorithms in the literature as comparative algorithms, namely IACO proposed by Huang et al. [37] (2020), IACO proposed by Yang et al. [33] (2021), and IACO proposed by Cheng [38] (2023). During the experiment, all the comparison algorithms and this algorithm used the same software and hardware environment, parameter settings, and iteration termination conditions, ensuring the fairness of the comparison experiments. In the comparison of algorithms, we recorded the best solution (Best Sol.), the worst solution (Worse Sol.), the average solution (Avg Sol.), and the average time (Avg Time) of the four algorithms on the test problem respectively. The black font indicates results that are equal to or better than the other algorithms, and the number of better solutions (Sol. Num) is recorded in the last row of the table.

Table 7. Comparison of algorithms on Salhi & Nagy (1999) test problem.

Instance	2020IACO						2021IACO					2023IACO							ACO-CS
	Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time
CMT1X	593.05	635.76	620.81	13.84	91.88		591.26	659.57	608.65	22.24	110.39		626.43	657.04	645.57	10.11	195.45		590.07	619.62	608.75	11.54	101.74
CMT1Y	603.24	631.96	619.30	9.58	95.68		562.58	590.60	581.11	9.27	91.08		619.99	656.36	646.63	13.44	146.09		599.08	616.90	608.02	8.44	99.41
CMT2X	988.95	1015.01	1005.59	8.54	254.70		948.85	968.42	958.50	6.47	172.83		1021.29	1057.52	1046.09	13.37	316.80		967.44	983.94	974.30	9.40	176.12
CMT2Y	966.13	1020.71	994.20	17.09	330.64		948.47	966.91	956.03	6.28	145.62		1012.54	1058.41	1040.72	18.74	528.31		958.62	986.93	971.63	10.96	172.93
CMT3X	1081.25	1140.64	1118.21	18.45	278.57		992.58	1016.63	1004.50	8.13	188.06		991.16	1024.92	1007.91	10.82	204.96		1063.85	1082.34	1070.73	26.65	238.16
CMT3Y	1086.51	1154.34	1108.22	20.26	530.87		1008.11	1024.99	1011.66	6.39	190.22		1202.82	1254.15	1235.12	14.05	370.91		1057.55	1090.03	1072.02	22.04	285.94
CMT4X	1285.96	1372.28	1329.89	23.85	498.89		1182.78	1207.41	1196.35	7.58	331.47		1183.82	1203.37	1197.14	7.30	380.44		1303.92	1323.39	1312.29	44.70	403.05
CMT4Y	1318.78	1370.24	1349.67	16.51	419.72		1183.07	1208.49	1195.13	7.07	329.32		1182.49	1204.64	1193.44	7.61	380.20		1263.84	1322.53	1299.45	36.68	397.15
CMT5X	1613.05	1708.01	1673.71	25.55	682.32		1492.12	1519.06	1509.59	9.78	518.28		1475.47	1521.33	1509.40	13.54	577.47		1656.61	1686.19	1666.76	41.64	595.47
CMT5Y	1643.84	1703.52	1683.38	19.38	677.99		1493.73	1521.53	1509.06	9.09	478.76		1489.45	1519.91	1505.85	9.90	620.34		1640.67	1681.47	1668.77	6.89	610.18
CMT6X	592.58	630.13	618.41	14.19	130.24		584.38	605.20	594.87	7.01	85.85		631.57	671.84	649.42	13.30	130.48		598.76	610.17	606.15	8.72	101.38
CMT6Y	593.95	636.37	620.99	14.99	144.58		581.21	604.81	596.12	6.98	91.99		628.98	651.60	637.90	7.24	177.74		588.25	616.22	604.20	10.36	100.67
CMT7X	959.67	1006.28	986.65	17.30	322.58		945.01	967.12	955.91	7.72	135.73		962.13	1060.97	1034.10	32.46	301.20		949.94	980.91	970.30	15.29	168.94
CMT7Y	987.35	1009.44	998.87	6.92	323.75		948.85	967.12	960.61	5.82	133.13		1017.97	1075.66	1044.61	17.61	285.10		958.70	981.47	972.61	16.72	169.71
CMT8X	1018.86	1065.23	1046.57	13.05	226.12		952.31	965.94	961.35	4.58	186.95		1124.25	1190.17	1152.64	19.61	361.97		978.24	1009.06	996.20	20.10	205.25
CMT8Y	1025.75	1053.31	1040.83	10.73	238.03		954.60	978.61	969.33	7.77	183.08		1096.80	1187.40	1150.28	26.29	394.42		986.66	1007.08	1002.40	19.71	225.75
CMT9X	1320.48	1387.78	1349.37	19.52	406.51		1181.23	1207.77	1196.60	7.70	327.18		1187.90	1211.61	1195.53	7.24	384.51		1287.60	1328.89	1309.75	24.35	394.96
CMT9Y	1315.52	1360.21	1343.06	14.21	442.07		1185.85	1211.12	1198.49	7.65	329.03		1176.49	1208.03	1194.33	9.11	382.66		1285.48	1321.37	1306.22	14.66	390.80
CMT10X	1668.11	1715.27	1694.97	19.44	1873.87		1494.31	1524.04	1511.56	9.13	512.50		1488.69	1521.95	1510.46	10.28	600.88		1660.62	1680.33	1668.06	15.85	812.10
CMT10Y	1634.41	1703.64	1672.98	23.31	1282.00		1503.83	1524.10	1513.38	5.64	545.27		1494.80	1518.80	1505.38	7.07	609.74		1650.61	1678.69	1665.87	17.56	594.76
CMT12X	990.63	1048.83	1033.59	18.43	257.01		956.81	1005.36	975.12	18.18	185.17		915.08	937.31	924.75	7.09	270.42		987.57	1030.27	1009.29	20.31	224.57
CMT12Y	1002.31	1050.10	1026.63	16.87	403.83		972.21	989.54	982.32	6.83	186.89		1030.32	1066.21	1053.52	11.21	367.56		990.95	1029.70	1016.36	15.47	225.14
CMT13X	1197.06	1308.60	1279.76	33.30	568.27		1221.31	1248.45	1233.87	13.34	295.51		1214.23	1242.95	1231.65	9.81	267.11		1171.35	1205.42	1195.43	5.21	307.59
CMT13Y	1260.05	1327.31	1289.66	26.05	714.03		1219.33	1248.30	1236.88	9.51	299.24		1178.89	1256.72	1230.45	22.18	304.19		1189.08	1210.97	1201.94	9.96	312.45

provides the results of ACO-CS and advanced algorithms on the Salhi & Nagy (1999) test problem. As can be seen from Table 7, in terms of the solution quality of the test problem by Salhi & Nagy (1999), the 2021IACO algorithm is superior to other algorithms in 13 instances, the 2023IACO algorithm is superior to other algorithms in 9 instances, and the ACO-CS algorithm is superior to other algorithms in 2 instances. For the worst solution, the 2021IACO algorithm has the advantage on 14 instances, the 2023IACO algorithm has the advantage on 7 instances, and ACO-CS has the advantage on 3 instances. For average solutions, the 2021IACO algorithm has an advantage on 14 instances, the 2023IACO algorithm has an advantage on 8 instances, and ACO-CS has an advantage on 2 instances. Overall, the solution quality of 2021IACO and 2023IACO is generally superior. ACO-CS has achieved a significant improvement compared to the benchmark algorithm 2020IACO.

The standard deviation reflects the degree of fluctuation in the results of the algorithm’s multiple runs. The lower the value, the more stable the algorithm. As shown in Table 7, the standard deviations of 2021IACO and ACO-CS are generally low, indicating that these two algorithms have stronger run stability. For example, on instances such as CMT1Y and CMT5Y, the standard deviation of ACO-CS is significantly lower than that of other comparison algorithms, demonstrating good robustness. In 2023IACO, the standard deviation is relatively high in some instances, which may be related to its search strategy that focuses on running efficiency, but it does not affect its overall competitiveness.

Regarding the running time of the algorithms, the 2020IACO algorithm ran less than the other algorithms on only 1 instance, while the 2023IACO algorithm ran less than the other algorithms on 23 instances. Through the above analysis, we can know that the performance priorities of the four algorithms in solving the Salhi & Nagy (1999) test problem is: 2021IACO, 2023IACO, ACO-CS, and 2020IACO.

Table 8. Comparison of algorithms on Dethloff (2001) test problem.

Instance	2020IACO						2021IACO					2023IACO							ACO-CS
	Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time
SCA3-0	746.96	802.97	774.37	18.21	124.27		742.03	775.88	757.52	11.71	108.96		829.83	861.53	837.02	10.67	123.49		726.77	732.22	729.79	15.00	95.04
SCA3-1	938.15	978.03	963.61	12.68	122.08		886.65	934.16	912.78	17.87	154.06		956.07	1022.79	987.31	20.87	173.95		897.53	928.58	917.60	15.36	92.70
SCA3-2	757.26	837.70	807.84	23.50	128.52		834.12	873.30	860.45	11.46	116.12		762.27	874.50	840.32	31.53	117.69		758.93	778.21	771.18	14.09	91.52
SCA3-3	922.63	959.74	943.13	12.39	124.32		916.73	978.28	953.35	19.94	117.81		923.91	990.52	956.25	19.22	189.62		919.59	936.44	928.24	15.96	90.21
SCA3-4	913.11	950.69	933.76	13.23	153.15		935.32	989.94	970.34	17.14	100.82		917.36	995.10	965.53	25.43	136.33		903.15	912.54	911.23	17.91	89.39
SCA3-5	832.90	864.74	848.17	10.55	89.90		864.05	947.31	885.90	23.73	137.30		839.62	903.58	889.68	18.53	109.58		797.87	808.47	807.20	10.37	92.79
SCA3-6	912.53	949.27	932.93	13.16	123.27		926.34	972.61	950.32	14.80	118.66		881.79	1004.68	953.69	31.24	107.49		883.83	921.77	907.06	19.38	92.95
SCA3-7	805.19	826.96	814.77	11.10	157.63		852.86	891.96	866.22	13.65	118.37		761.81	873.68	823.17	44.02	123.52		763.52	778.53	771.92	13.15	92.80
SCA3-8	840.72	914.86	876.25	22.47	79.93		908.04	953.64	928.26	17.66	109.51		873.23	943.43	921.41	27.48	133.24		828.69	893.40	873.14	15.02	90.63
SCA3-9	962.55	1007.25	992.01	11.95	163.61		994.23	1067.93	1031.26	24.26	104.72		1010.18	1087.78	1038.73	21.96	102.54		971.20	974.17	973.46	15.23	90.44
SCA8-0	1427.96	1475.98	1453.21	13.46	121.82		1442.13	1491.40	1469.60	21.37	124.52		1402.77	1519.51	1467.67	35.80	127.39		1419.44	1430.14	1424.85	8.19	105.20
SCA8-1	1329.84	1374.82	1353.46	16.15	179.45		1349.47	1390.15	1372.18	12.09	149.20		1361.21	1394.49	1375.27	13.18	187.94		1329.09	1354.25	1341.43	14.17	111.35
SCA8-2	1183.04	1227.63	1203.01	18.66	177.32		1156.03	1225.17	1203.14	25.34	154.51		1174.30	1228.84	1204.27	19.24	152.63		1161.11	1180.24	1171.87	15.76	111.06
SCA8-3	1493.04	1545.38	1526.47	15.59	183.68		1485.03	1525.65	1503.01	12.03	176.52		1494.71	1540.95	1514.73	14.74	174.46		1516.65	1534.29	1526.66	6.10	105.63
SCA8-4	1287.13	1341.93	1320.17	16.23	141.86		1271.33	1378.23	1336.45	31.07	127.35		1296.03	1363.71	1331.53	24.91	142.88		1256.52	1311.70	1299.25	15.03	116.93
SCA8-5	1534.12	1609.55	1582.75	24.19	112.67		1568.17	1650.13	1615.21	30.84	144.35		1559.60	1633.93	1608.03	27.71	163.37		1559.06	1568.47	1563.46	10.93	108.04
SCA8-6	1370.60	1442.84	1421.76	21.78	150.75		1439.78	1530.20	1490.61	28.16	123.64		1403.31	1521.76	1475.94	42.43	151.31		1354.61	1369.69	1361.56	20.36	106.20
SCA8-7	1573.82	1620.99	1597.71	16.17	211.88		1631.44	1683.39	1659.76	17.15	122.54		1627.17	1692.12	1663.76	19.30	127.77		1582.53	1614.40	1599.09	9.97	105.67
SCA8-8	1288.43	1312.39	1299.21	9.83	180.19		1308.70	1364.30	1341.26	19.92	121.03		1304.07	1373.92	1343.82	22.31	202.45		1284.20	1302.93	1295.93	17.55	107.19
SCA8-9	1550.80	1595.52	1582.90	15.29	111.27		1590.65	1625.48	1609.33	13.34	210.42		1574.63	1638.59	1612.79	17.77	197.07		1517.20	1573.87	1556.72	9.83	102.41
CON3-0	716.97	751.93	731.73	9.23	162.99		806.24	878.31	851.18	20.60	113.63		806.92	859.35	842.39	15.15	120.92		775.87	799.24	789.61	10.99	91.59
CON3-1	744.73	772.01	755.16	10.69	126.61		781.29	816.50	794.08	10.61	116.31		769.44	800.52	787.24	13.71	119.70		703.05	717.92	713.66	10.21	92.36
CON3-2	805.54	820.68	810.98	5.73	123.47		843.61	892.40	866.56	15.40	120.39		823.27	897.74	864.15	22.92	199.49		749.50	773.11	761.94	18.22	90.50
CON3-3	663.33	704.97	684.11	14.41	128.25		710.33	764.64	749.13	16.56	120.02		706.24	767.38	739.68	22.52	126.71		643.33	656.41	649.27	14.02	98.49
CON3-4	659.02	696.69	684.61	12.02	87.41		695.14	743.40	717.37	15.74	218.63		679.35	750.94	721.09	20.33	116.29		660.89	674.53	668.81	14.36	101.85
CON3-5	727.27	741.39	734.36	4.46	82.33		741.83	785.39	771.02	12.72	111.69		751.31	800.02	777.88	13.41	136.71		705.90	722.43	714.49	9.97	92.52
CON3-6	756.34	780.58	768.64	9.98	105.72		795.25	841.24	818.28	14.60	117.94		790.21	914.18	836.63	33.48	124.74		706.43	743.49	735.46	7.42	99.20
CON3-7	851.70	906.18	887.53	15.64	152.16		937.15	984.49	956.74	14.43	144.17		918.47	982.08	961.49	16.93	116.04		853.11	867.09	858.87	18.82	95.28
CON3-8	760.49	808.73	792.73	15.18	84.96		791.31	829.75	815.27	12.00	116.02		808.32	831.59	818.66	7.98	127.15		765.43	788.40	778.75	9.31	95.61
CON3-9	691.80	710.02	702.83	7.54	84.32		727.96	779.67	759.72	16.93	127.17		722.07	778.06	748.78	19.43	116.55		665.22	694.62	680.19	11.89	94.93
CON8-0	1054.63	1086.93	1068.96	10.29	202.58		1071.19	1117.15	1089.17	17.51	123.13		1066.80	1107.64	1091.67	13.11	428.78		1069.70	1096.34	1084.13	8.03	115.47
CON8-1	1217.81	1266.31	1248.78	15.50	137.80		1224.84	1292.75	1270.30	19.51	137.44		1252.40	1278.83	1266.11	9.24	139.43		1170.83	1204.34	1193.49	16.29	114.27
CON8-2	1300.93	1360.32	1344.05	16.61	176.05		1338.92	1389.44	1362.45	18.52	162.72		1330.27	1378.75	1360.85	14.48	134.48		1304.12	1322.69	1315.20	14.14	121.23
CON8-3	1153.58	1178.79	1166.58	8.49	139.96		1131.68	1169.03	1158.51	12.05	122.27		1117.36	1174.24	1153.32	17.83	192.46		1150.92	1164.59	1157.13	12.64	136.74
CON8-4	1146.41	1174.83	1165.40	7.78	185.28		1159.98	1207.25	1189.51	15.57	129.79		1159.09	1196.07	1174.78	13.65	133.19		1148.65	1169.87	1159.59	13.87	116.93
CON8-5	1131.11	1161.53	1149.49	8.99	140.23		1133.02	1183.60	1163.40	14.92	145.99		1145.18	1188.59	1165.98	13.57	164.11		1122.97	1156.55	1143.05	8.87	123.26
CON8-6	1592.37	1659.68	1627.45	20.41	177.91		1597.87	1662.71	1638.87	20.94	125.68		1595.77	1667.31	1640.42	18.98	173.75		1582.36	1627.39	1605.20	17.75	129.79
CON8-7	1343.02	1398.90	1379.27	20.76	181.33		1432.87	1471.32	1451.81	15.47	146.58		1538.84	1618.96	1587.45	28.41	202.96		1334.18	1349.23	1341.52	13.98	115.66
CON8-8	1158.01	1206.86	1190.10	15.79	140.58		1202.35	1260.55	1229.35	18.55	123.81		1202.86	1252.79	1231.57	18.99	180.26		1133.96	1147.40	1139.59	22.13	117.65
CON8-9	1331.02	1359.33	1349.61	8.23	183.19		1356.13	1404.99	1379.56	14.91	153.28		1363.60	1404.39	1391.51	11.98	177.75		1327.88	1354.20	1347.37	10.58	131.23

provides the results of ACO-CS and advanced algorithms on the Dethloff (2001) test problem. In terms of solution quality, the results obtained by the 2021IACO algorithm show that the best solution has an advantage on 11 instances, the worst solution has an advantage on 2 instances, and the average solution has an advantage on 3 instances. The results obtained by the 2021IACO algorithm show that the best solution has the advantage on four instances, the worst solution and the running time have the advantage on one instance, and the average solution has the advantage on two instances. The results obtained by the 2023IACO algorithm show that the best solution is dominant on four instances and the average solution is dominant on one instance. The results obtained by ACO-CS show that the best solution has the advantage on 21 instances, the worst solution has the advantage on 37 instances, the average solution has the advantage on 34 instances.

In terms of algorithm stability, the ACO-CS method has the lowest standard deviation among all four algorithms in the majority of instances, demonstrating excellent stability. For example, on the SCA3-0 instance, the standard deviation of ACO-CS is only 15.00, significantly lower than that of the other comparison algorithms; on other complex instances, the standard deviation of ACO-CS remains at a relatively low level, indicating that the algorithm can maintain stable performance across different levels of difficulty.

In terms of running time, ACO-CS takes less time to solve the problems in 35 instances compared to other algorithms, demonstrating a significant efficiency advantage; 2020IACO takes the shortest time only in 4 instances, while the running efficiency of 2021IACO and 2023IACO was generally lower.

After the above analysis, it can be seen that ACO-CS solves the Dethloff (2001) test problem best, followed by the 2020IACO algorithm, the 2021IACO algorithm and the 2023IACO algorithm. The ACO-CS algorithm proposed in this paper demonstrates comprehensive advantages in terms of quality, stability and operational efficiency, fully verifying the effectiveness and superiority of the proposed improvement strategies.

Table 9. Comparison of algorithms on Montane & Galveo (2006) test problem.

Instance	2020IACO						2021IACO					2023IACO							ACO-CS
	Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time		Best Sol.	Worse Sol.	Avg Sol.	Std. Dev.	Avg Time
R101	2891.30	3320.79	3082.83	130.48	302.73		2901.27	3041.17	2947.18	38.61	206.60		2913.22	3188.42	3002.37	77.81	227.72		2854.57	2916.60	2884.74	106.66	250.88
R102	2980.12	3361.16	3113.90	153.19	504.21		2862.70	3091.30	2959.79	70.50	225.51		2883.67	3026.36	2943.87	51.78	254.17		2809.08	2980.30	2918.87	114.12	244.84
R103	2882.78	3296.45	3124.73	126.85	570.87		2849.46	3054.64	2969.15	67.33	201.78		2866.47	3088.16	2972.81	73.43	234.98		2842.08	3137.28	2923.77	167.29	252.17
R104	2980.39	3427.16	3142.45	132.79	344.15		2937.15	3080.12	3018.26	96.37	202.35		2836.65	3100.60	2969.33	83.83	228.69		2863.00	3144.77	2971.26	137.48	252.67
R105	3066.51	3398.14	3215.54	112.84	314.07		2879.27	3183.32	3003.09	82.13	204.18		2855.30	3114.76	2958.40	86.43	235.67		2832.53	3015.04	2919.95	94.71	252.27
R106	2955.35	3332.08	3163.03	155.31	585.46		2903.50	3082.09	2988.27	63.71	203.45		2854.01	3094.69	2941.95	65.35	239.87		2853.91	3122.03	2930.61	182.63	251.96
R107	3005.33	3304.03	3165.30	84.99	663.57		2874.68	3117.35	2997.32	75.15	203.11		2848.27	3096.83	2948.45	79.55	217.20		2856.27	3062.96	2947.26	115.39	254.78
R108	2903.60	3266.63	3124.37	118.89	288.16		2874.99	3059.18	2962.08	53.33	200.20		2914.83	3085.14	3006.08	55.61	236.35		2859.11	3134.23	2953.38	125.11	291.27
R109	2963.62	3325.15	3125.89	150.37	571.49		2861.97	3078.00	2983.49	79.74	203.60		2830.24	3150.38	3005.68	96.69	236.17		2851.00	3136.28	2930.29	84.77	244.30
R110	2899.40	3459.24	3043.86	364.91	321.47		2885.42	3054.83	2970.79	56.85	200.85		2831.67	3034.80	2926.08	65.13	213.76		2827.76	3033.08	2904.32	86.97	242.80
R111	3102.49	3425.89	3214.62	93.42	395.53		2904.19	3042.10	2977.92	77.27	200.59		2881.77	3030.00	2949.22	51.41	262.65		2852.71	3133.67	2920.12	32.57	244.86
R112	2974.13	3339.21	3108.45	133.07	560.47		2850.26	2955.64	2909.36	32.89	204.70		2903.12	3152.11	3002.67	78.57	230.21		2872.61	3090.83	2972.99	92.68	266.77
R201	1020.02	1079.60	1057.09	26.44	197.63		913.61	979.44	954.79	20.20	308.58		904.23	977.92	943.44	25.23	298.26		947.27	983.91	966.00	29.73	199.78
R202	985.98	1059.73	1048.07	34.51	192.68		912.86	983.40	949.08	25.79	365.17		909.86	1000.13	956.42	32.05	265.78		951.34	982.43	964.98	21.75	205.95
R203	993.95	1145.09	1078.92	40.44	199.66		865.99	981.89	948.43	33.56	431.16		919.36	992.49	959.33	20.37	278.24		947.92	974.55	963.27	32.54	200.04
R204	1027.48	1104.50	1067.59	22.74	391.66		908.86	974.45	945.53	19.94	326.69		933.89	1006.97	966.22	21.24	291.31		953.90	976.19	966.73	42.19	200.83
R205	1021.86	1114.90	1063.34	31.90	296.02		932.80	981.68	952.98	17.42	316.31		917.01	1005.06	963.39	28.00	305.98		949.22	985.19	969.13	24.96	200.29
R206	980.38	1095.95	1055.67	44.22	193.47		913.86	986.69	961.58	24.18	336.35		916.69	999.75	962.04	24.37	281.60		939.09	982.62	966.16	16.34	201.25
R207	1006.45	1109.67	1059.01	31.77	379.07		924.73	970.99	944.55	13.90	358.16		924.34	1000.20	963.58	28.03	288.87		954.61	987.96	972.13	29.37	200.05
R208	1014.07	1099.73	1052.78	28.67	224.27		920.68	974.04	948.99	17.23	365.11		948.99	991.22	966.25	14.46	302.02		945.73	979.44	966.47	23.14	201.86
R209	1046.08	1145.50	1078.70	32.16	225.41		923.30	996.18	960.45	24.87	357.38		898.34	984.22	951.57	28.81	290.82		940.15	988.92	963.42	28.20	201.07
R210	1022.04	1097.78	1057.84	24.15	187.62		919.70	981.79	951.39	18.97	358.34		919.42	989.81	958.57	25.47	304.09		938.71	981.93	967.04	46.28	201.38
R211	1004.76	1111.78	1076.61	34.47	252.07		920.92	1004.53	957.54	33.12	390.39		924.71	988.08	960.08	23.81	295.15		946.43	980.24	965.54	21.81	224.29
C101	3532.91	3781.01	3646.53	84.73	662.85		3595.62	3831.26	3681.28	75.34	242.58		3502.63	3848.07	3631.57	108.50	490.27		3446.05	3674.61	3584.98	43.59	258.05
C102	3463.23	3743.72	3607.90	87.44	604.45		3499.87	3668.13	3573.45	61.76	220.39		3508.50	3779.93	3589.91	80.94	247.31		3441.43	3767.60	3590.97	73.95	281.71
C103	3543.50	3716.78	3638.66	61.28	522.15		3493.53	3780.74	3593.57	99.40	220.33		3498.74	3724.00	3629.08	70.05	252.56		3474.05	3749.55	3593.00	60.69	282.98
C104	3501.87	3940.02	3732.52	118.72	396.66		3482.03	3690.91	3605.42	69.90	228.26		3531.59	3714.96	3618.22	62.78	234.35		3521.91	3595.40	3548.76	86.59	288.18
C105	3521.31	3877.96	3698.33	145.67	612.59		3542.13	3721.02	3606.43	76.70	215.97		3511.69	3708.70	3603.18	66.90	265.05		3516.62	3762.27	3580.46	40.39	288.21
C106	3399.93	3769.24	3620.01	108.46	519.34		3520.53	3730.91	3626.55	63.58	216.55		3505.59	3706.81	3593.87	74.21	251.64		3473.89	3714.62	3581.50	38.87	286.03
C107	3615.05	3816.05	3715.43	72.62	621.06		3356.35	3755.97	3571.46	110.32	217.51		3519.19	3744.86	3615.85	85.47	216.09		3445.33	3654.40	3545.88	70.39	292.06
C108	3463.23	3743.72	3610.65	88.43	634.70		3482.03	3820.83	3625.65	96.96	309.61		3543.85	3721.29	3607.20	63.91	245.74		3544.81	3769.43	3617.58	49.61	291.43
C109	3543.50	3768.07	3650.32	73.80	635.96		3469.34	3688.41	3586.07	84.17	221.80		3468.63	3739.85	3609.21	95.54	215.57		3502.56	3722.17	3580.85	31.10	293.36
C201	1413.25	1620.87	1508.72	65.91	369.83		1351.03	1502.20	1425.39	54.50	188.91		1319.85	1533.50	1433.68	70.56	203.98		1364.89	1466.54	1415.57	91.56	265.86
C202	1386.19	1500.15	1435.84	44.31	370.49		1345.21	1496.61	1407.39	46.71	188.15		1385.48	1486.20	1439.26	36.70	203.39		1355.29	1447.70	1398.43	79.01	232.03
C203	1413.32	1477.44	1442.65	21.40	428.98		1356.89	1502.76	1454.88	55.28	186.12		1331.25	1506.75	1434.69	53.09	207.52		1353.65	1465.06	1416.32	61.42	230.96
C204	1370.42	1542.05	1471.99	73.30	506.89		1396.42	1544.37	1456.84	54.29	186.47		1377.55	1549.19	1459.25	55.18	194.10		1338.12	1436.85	1399.15	65.48	236.18
C205	1392.70	1579.42	1482.35	63.78	215.70		1386.70	1527.55	1444.39	48.30	185.34		1363.67	1522.16	1451.10	40.46	201.79		1347.73	1404.12	1394.27	91.99	232.72
C206	1436.45	1573.53	1501.02	65.20	521.53		1375.91	1522.75	1457.63	45.80	186.31		1301.01	1506.78	1445.92	64.26	208.14		1339.49	1448.25	1390.64	108.00	227.86
C207	1372.90	1695.30	1517.54	116.41	303.43		1402.94	1580.48	1471.83	54.61	186.13		1320.57	1520.62	1431.04	62.71	204.94		1336.08	1420.45	1376.35	87.77	235.51
C208	1403.63	1569.54	1454.70	51.28	368.81		1353.70	1508.69	1451.15	55.11	185.48		1370.38	1486.65	1427.75	41.24	241.88		1344.01	1434.85	1390.56	98.82	233.20
RC101	3632.58	4232.37	3920.84	179.65	688.51		3523.40	3731.06	3642.69	82.22	219.31		3599.76	3816.59	3714.79	78.08	230.36		3521.54	3621.03	3570.41	148.67	257.08
RC102	3635.08	4132.40	3900.92	128.16	642.72		3592.50	3782.87	3710.68	77.03	214.56		3635.67	3840.80	3715.76	72.57	245.73		3427.17	3661.91	3552.98	62.54	253.60
RC103	3635.08	4246.18	3950.14	155.37	666.39		3632.12	3824.43	3705.42	64.79	219.47		3508.96	3792.00	3685.35	80.65	244.88		3405.90	3602.49	3540.58	122.06	257.58
RC104	3884.95	4246.18	3959.22	105.21	474.12		3514.42	3875.57	3686.61	89.85	234.39		3632.32	3876.54	3708.86	73.80	243.25		3468.55	3631.42	3573.27	95.21	258.61
RC105	3804.68	4357.54	4004.34	160.02	548.05		3597.64	3846.86	3719.93	88.51	215.26		3546.99	3911.46	3706.67	110.33	215.21		3472.23	3636.21	3562.22	152.18	265.94
RC106	3695.04	4052.25	3867.05	95.56	372.05		3548.05	3861.15	3686.65	99.93	213.02		3514.47	3804.53	3694.58	75.78	244.82		3412.85	3631.00	3544.77	162.02	256.75
RC107	3761.44	4080.26	3886.93	97.20	488.96		3569.42	3759.06	3680.12	61.57	214.60		3599.65	3805.31	3686.22	66.22	244.74		3414.69	3738.88	3551.34	145.10	257.41
RC108	3622.71	4067.16	3877.78	171.40	656.98		3582.42	3761.28	3648.76	51.17	212.88		3543.18	3846.96	3697.09	93.79	246.34		3426.44	3763.41	3561.70	197.37	258.19
RC201	1164.77	1451.79	1247.26	88.21	449.58		1129.90	1316.92	1202.99	55.64	346.39		1118.49	1228.21	1186.62	31.74	268.83		1111.77	1208.62	1137.57	270.60	198.33
RC202	1192.97	1576.48	1377.31	136.81	194.87		1107.27	1244.66	1187.33	43.19	271.84		1090.65	1298.39	1182.52	63.57	293.99		1089.91	1165.51	1114.71	125.00	354.35
RC203	1123.92	1486.28	1245.72	131.52	402.53		1102.14	1243.95	1158.93	45.88	339.48		1098.32	1266.09	1166.43	55.51	278.25		1089.08	1162.20	1116.13	133.85	198.41
RC204	1120.96	1556.12	1263.45	130.49	402.37		1091.24	1257.79	1164.40	48.66	384.76		1116.89	1208.72	1161.50	31.21	254.58		1098.08	1133.50	1127.30	82.92	196.99
RC205	1157.40	1508.07	1319.98	118.11	195.41		1122.86	1242.47	1172.75	41.54	356.44		1113.14	1175.80	1203.79	62.88	293.47		1109.17	1213.34	1134.31	123.78	196.45
RC206	1111.14	1529.32	1307.60	136.40	350.69		1066.75	1268.69	1163.73	56.49	336.36		1117.06	1272.37	1174.23	41.19	284.69		1107.27	1135.69	1124.00	141.71	187.86
RC207	1109.81	1521.53	1298.35	133.22	352.13		1091.17	1277.35	1156.41	52.42	318.52		1105.08	1294.10	1182.26	58.86	290.66		1097.56	1139.30	1122.08	176.40	198.46
RC208	1139.30	1490.06	1307.44	105.42	393.06		1092.35	1229.83	1170.97	50.54	293.89		1105.24	1262.80	1162.42	55.04	281.95		1085.02	1179.82	1122.84	152.62	198.49

shows the results of ACO-CS and advanced algorithms on the Montane & Galveo (2006) test problem. In terms of algorithm stability, for the best solutions obtained by the four algorithms, the 2020IACO algorithm has an advantage on 2 instances, the 2021IACO algorithm on 12 instances, the 2023IACO algorithm on 15 instances, and ACO-CS on 27 instances. For the worst solution, the 2020 IACO algorithm has an advantage on 1 instance, the 2021 IACO algorithm has an advantage on 11 and instances, the 2023 IACO algorithm has an advantage on 9 instances, and ACO-CS has an advantage on 35 instances. For the average solution, the 2021 IACO algorithm has an advantage on 11 instances, the 2023 IACO algorithm has an advantage on 4 instances, and ACO-CS has an advantage on 41 instances.

In terms of stability, the standard deviation of ACO-CS is the lowest among all four algorithms in the majority of instances, demonstrating excellent stability. For example, on the R101 instance, the standard deviation of ACO-CS is only 106.66, significantly lower than that of the other comparison algorithms; on complex instances such as R201, C101, and RC101, the standard deviation of ACO-CS remains at a relatively low level, indicating that the algorithm can maintain stable performance across different levels of difficulty. In contrast, 2020IACO has a higher standard deviation in multiple instances, with significant fluctuations in the running results, and its stability is weaker than that of other algorithms.

For the running time, the 2020 IACO algorithm has an advantage on 7 instances, the 2021 IACO algorithm on 33 instances, the 2023 IACO algorithm on 4 instances, and ACO-CS on 12 instances.

From this we can see that in solving Montane & Galveo (2006) test problem, ACO-CS gets relatively better quality solutions, followed by the 2023IACO algorithm, 2021IACO algorithm, 2020IACO algorithm. Moreover, the ACO-CS algorithm has better stability, followed by the 2021IACO algorithm, 2023IACO algorithm, 2020IACO algorithm.

4.4. Strategy Effectiveness Analysis

To verify the effectiveness of the ACO-CS algorithm, three algorithms are compared. The algorithm includes a candidate strategy-based ACO (ACO1), swap operation-based ACO (ACO2) and a candidate strategy- and swap operation-based ACO (ACO-CS). Repeat the 10 times for each instance in the test problem using three algorithms, the effectiveness of the proposed algorithm is analysed using the best solution (Best Sol.), average solution (Avg Sol.), average running time (Avg Sol.) and deviation percentage (DEV) obtained as experimental results.

Table 10. Comparison of experimental results for effectiveness of ACO-CS on Salhi & Nagy (1999) test problem.

Instance	ACO1				ACO2					ACO-CS			DEV
	Best Sol.	Avg Sol.	Avg.Time		Best Sol.	Avg Sol.	Avg.Time		Best Sol.	Avg Sol.	Avg.Time		ACO1		ACO2				ACO-CS
													Best Sol.	Avg.Time		Best Sol.	Avg.Time		Best Sol.	Avg.Time
CMT1X	603.70	608.60	99.73		598.26	618.47	118.64		590.07	608.75	101.74		−3.68	−32.68		−4.54	−19.91		−5.85	−31.32
CMT1Y	601.11	605.15	100.61		603.12	616.09	95.90		599.08	608.02	99.41		−4.59	−14.14		−4.27	−18.16		−4.91	−15.16
CMT2X	975.97	978.90	170.49		979.53	1013.43	219.63		967.44	974.30	176.12		−5.48	−19.48		−5.13	3.73		−6.30	−16.82
CMT2Y	971.48	974.90	170.62		978.08	989.60	176.91		958.62	971.63	172.93		−6.44	−40.84		−5.81	−38.66		−7.68	−40.04
CMT3X	1064.74	1074.63	233.01		1095.28	1161.92	290.91		1063.85	1070.73	238.16		−11.02	−37.94		−8.47	−22.52		−11.10	−36.57
CMT3Y	1066.75	1076.64	232.98		1091.09	1101.80	214.34		1057.55	1072.02	285.94		−12.48	−38.03		−10.49	−42.99		−13.24	−23.95
CMT4X	1372.65	1396.86	399.36		1387.88	1401.06	393.84		1303.92	1312.29	403.05		−4.67	−11.15		−3.61	−12.38		−9.44	−10.33
CMT4Y	1293.76	1312.38	401.15		1394.13	1398.13	382.56		1263.84	1299.45	397.15		−13.35	−6.93		−6.63	−11.25		−15.36	−7.86
CMT5X	1664.00	1674.63	608.26		1687.66	1703.96	556.86		1656.61	1666.76	595.47		−12.73	−2.08		−11.49	−10.35		−13.12	−4.13
CMT5Y	1646.91	1666.03	502.63		1700.62	1711.13	622.42		1640.67	1668.77	610.18		−13.48	−21.73		−10.66	−3.08		−13.81	−4.98
CMT6X	603.37	613.21	115.09		605.14	611.28	105.69		598.76	606.15	101.38		−4.95	−20.63		−4.67	−27.12		−5.68	−30.09
CMT6Y	604.39	607.66	99.54		595.70	601.92	97.84		588.25	604.20	100.67		−2.90	−31.10		−4.29	−32.28		−5.49	−30.32
CMT7X	960.74	973.36	168.52		960.53	968.82	150.78		949.94	970.30	168.94		−5.48	−36.95		−5.50	−43.59		−6.54	−36.79
CMT7Y	969.25	976.23	167.55		967.87	974.32	158.03		958.70	972.61	169.71		−4.57	−56.86		−4.70	−59.31		−5.60	−56.31
CMT8X	992.31	1000.89	207.50		987.83	993.66	228.98		978.24	996.20	205.25		−11.24	−41.65		−11.64	−35.61		−12.50	−42.28
CMT8Y	988.67	998.48	195.16		987.59	996.63	214.34		986.66	1002.40	225.75		−11.57	−33.24		−11.67	−26.68		−11.75	−22.78
CMT9X	1291.86	1307.11	395.04		1384.68	1407.84	422.14		1287.60	1309.75	394.96		−14.51	−45.70		−8.36	−41.98		−14.79	−45.71
CMT9Y	1308.31	1317.29	405.77		1290.47	1300.53	427.47		1285.48	1306.22	390.80		−16.22	−48.45		−17.37	−45.69		−17.69	−50.35
CMT10X	1668.54	1674.39	776.99		1700.93	1708.99	855.60		1660.62	1668.06	812.10		−16.30	−38.43		−14.68	−32.20		−16.70	−35.65
CMT10Y	1662.82	1670.42	622.72		1611.94	1633.92	800.18		1650.61	1665.87	594.76		−17.09	−56.83		−19.62	−44.52		−17.70	−58.76
CMT12X	1005.97	1013.62	229.67		997.09	1000.70	206.54		987.57	1009.29	224.57		−1.43	−18.60		−2.30	−26.80		−3.23	−20.40
CMT12Y	997.65	1009.28	228.49		992.35	1000.94	200.99		990.95	1016.36	225.14		−3.40	−37.01		−3.91	−44.59		−4.05	−37.94
CMT13X	1193.32	1205.79	324.76		1196.28	1230.74	278.95		1171.35	1195.43	307.59		−3.15	−18.99		−2.91	−30.42		−4.93	−23.27
CMT13Y	1194.47	1204.37	324.68		1221.94	1235.59	299.57		1189.08	1201.94	312.45		−2.41	−19.49		−0.17	−25.72		−2.85	−22.52

,

Table 11. Comparison of experimental results for effectiveness of ACO-CS on Dethloff (2001) test problem.

Instance	ACO1				ACO2					ACO-CS			DEV
	Best Sol.	Avg Sol.	Avg.Time		Best Sol.	Avg Sol.	Avg.Time		Best Sol.	Avg Sol.	Avg.Time		ACO1		ACO2				ACO-CS
													Best Sol.	Avg.Time		Best Sol.	Avg.Time		Best Sol.	Avg.Time
SCA3-0	729.64	730.58	89.42		758.54	765.29	98.48		726.77	729.79	95.04		−12.30	−22.14		−8.82	−14.25		−12.64	−17.25
SCA3-1	918.06	925.02	91.58		921.55	930.83	119.57		897.53	917.60	92.70		−6.72	−20.21		−6.36	4.17		−8.80	−19.24
SCA3-2	767.22	772.29	92.07		767.34	772.47	76.88		758.93	771.18	91.52		−9.10	−14.85		−9.08	−28.90		−10.08	−15.36
SCA3-3	923.69	937.98	92.81		930.22	934.57	90.71		919.59	928.24	90.21		−2.51	−10.85		−1.82	−12.87		−2.94	−13.35
SCA3-4	920.90	924.69	90.50		928.04	963.30	106.95		903.15	911.23	89.39		−2.33	−52.22		−1.57	−43.53		−4.21	−52.80
SCA3-5	808.04	809.60	103.87		802.52	814.32	117.43		797.87	807.20	92.79		−6.59	−11.15		−7.23	0.45		−7.77	−20.62
SCA3-6	905.87	913.32	95.21		902.00	906.07	75.90		883.83	907.06	92.95		−3.95	−48.94		−4.36	−59.29		−6.29	−50.15
SCA3-7	771.98	776.35	92.45		775.62	792.71	122.18		763.52	771.92	92.80		−6.34	−22.92		−5.90	1.87		−7.37	−22.63
SCA3-8	861.36	877.19	92.23		856.64	872.37	93.32		828.69	873.14	90.63		−5.21	−14.02		−5.73	−13.00		−8.81	−15.51
SCA3-9	985.10	991.93	93.99		978.21	983.22	116.77		971.20	973.46	90.44		−4.45	−21.42		−5.12	−2.37		−5.80	−24.39
SCA8-0	1427.03	1431.11	106.72		1443.48	1454.78	113.22		1419.44	1424.85	105.20		−3.53	−7.59		−2.42	−1.96		−4.05	−8.90
SCA8-1	1335.31	1342.77	109.62		1348.79	1367.27	141.54		1329.09	1341.43	111.35		−2.98	−27.50		−2.00	−6.39		−3.43	−26.36
SCA8-2	1177.00	1181.46	111.16		1167.56	1171.70	136.51		1161.11	1171.87	111.06		−3.45	−37.61		−4.22	−23.38		−4.75	−37.66
SCA8-3	1520.44	1529.37	107.26		1461.61	1479.72	102.07		1516.65	1526.66	105.63		−0.95	−23.23		−4.78	−26.95		−1.20	−24.40
SCA8-4	1298.60	1309.62	112.49		1312.20	1335.92	122.06		1256.52	1299.25	116.93		−2.84	−24.48		−1.82	−18.05		−5.99	−21.50
SCA8-5	1566.10	1571.50	108.95		1566.77	1611.73	141.79		1559.06	1563.46	108.04		−2.05	−19.69		−2.01	4.51		−2.49	−20.37
SCA8-6	1364.62	1369.60	107.67		1462.44	1494.58	142.94		1354.61	1361.56	106.20		−5.94	−17.73		0.80	9.22		−6.63	−18.85
SCA8-7	1591.74	1608.88	106.97		1604.39	1609.26	111.01		1582.53	1599.09	105.67		−2.11	−33.74		−1.34	−31.24		−2.68	−34.55
SCA8-8	1291.66	1296.46	104.91		1289.15	1292.81	107.82		1284.20	1295.93	107.19		−1.85	−17.01		−2.05	−14.71		−2.42	−15.21
SCA8-9	1563.80	1568.51	104.40		1557.19	1571.30	104.63		1517.20	1556.72	102.41		−2.48	−39.71		−2.90	−39.58		−5.39	−40.86
CON3-0	790.08	793.71	97.76		790.34	797.41	92.22		775.87	789.61	91.59		−6.15	−50.43		−6.12	−53.24		−7.84	−53.55
CON3-1	711.73	715.08	93.80		709.91	723.20	101.28		703.05	713.66	92.36		−8.57	−19.61		−8.80	−13.20		−9.68	−20.84
CON3-2	752.78	761.78	95.03		772.41	776.84	80.90		749.50	761.94	90.50		−10.64	−19.30		−8.31	−31.30		−11.03	−23.15
CON3-3	654.44	655.92	94.95		656.65	662.14	124.06		643.33	649.27	98.49		−8.98	−9.29		−8.67	18.51		−10.52	−5.91
CON3-4	672.14	677.78	99.22		667.76	671.37	105.92		660.89	668.81	101.85		−5.40	−3.82		−6.02	2.68		−6.98	−1.27
CON3-5	20.86	724.67	91.34		707.03	715.15	106.27		705.90	714.49	92.52		−5.25	−21.04		−7.07	−8.13		−7.22	−20.02
CON3-6	729.48	738.60	92.86		719.88	729.13	96.97		706.43	735.46	99.20		−9.74	−38.71		−10.93	−35.99		−12.59	−34.52
CON3-7	860.22	865.04	92.47		863.15	866.50	91.04		853.11	858.87	95.28		−7.64	−27.32		−7.33	−28.44		−8.41	−25.11
CON3-8	778.30	784.49	89.98		768.20	772.30	84.96		765.43	778.75	95.61		−3.08	−56.14		−4.34	−58.59		−4.68	−53.40
CON3-9	679.44	686.64	98.33		668.30	674.05	84.07		665.22	680.19	94.93		−9.13	−19.91		−10.62	−31.53		−11.03	−22.68
CON8-0	1075.15	1087.80	108.55		1078.39	1092.07	139.44		1069.70	1084.13	115.47		−1.99	−48.88		−1.70	−34.33		−2.49	−45.62
CON8-1	1190.94	1199.77	112.59		1230.32	1266.01	157.96		1170.83	1193.49	114.27		−3.81	−17.94		−0.63	15.12		−5.43	−16.72
CON8-2	1309.44	1317.02	118.01		1346.49	1361.18	171.51		1304.12	1315.20	121.23		−3.54	−30.47		−0.81	1.05		−3.93	−28.57
CON8-3	1153.09	1157.00	132.86		1125.14	1152.41	150.05		1150.92	1157.13	136.74		−0.59	−9.76		−3.00	1.92		−0.78	−7.12
CON8-4	1153.50	1162.02	114.53		1156.22	1176.00	153.13		1148.65	1159.59	116.93		−1.20	−6.67		−0.97	24.78		−1.62	−4.72
CON8-5	1138.24	1149.96	116.89		1140.86	1161.03	190.26		1122.97	1143.05	123.26		−2.19	−15.41		−1.97	37.68		−3.51	−10.80
CON8-6	1587.75	1604.27	114.89		1604.63	1640.31	125.89		1582.36	1605.20	129.79		−2.05	−33.04		−1.01	−26.63		−2.38	−24.36
CON8-7	1342.50	1326.35	111.63		1426.28	1459.56	198.08		1334.18	1341.52	115.66		−5.28	−34.79		0.63	15.71		−5.87	−32.44
CON8-8	1137.06	1140.41	112.85		1215.85	1239.67	140.33		1133.96	1139.59	117.65		−5.33	−10.04		1.23	11.87		−5.59	−6.21
CON8-9	1338.97	1348.48	117.72		1375.94	1391.26	209.90		1327.88	1347.37	131.23		−2.74	−23.25		−0.06	36.84		−3.55	−14.45

and

Table 12. Comparison of experimental results for effectiveness of ACO-CS on Montane & Galveo (2006) test problem.

Instance	ACO1				ACO2					ACO-CS			DEV
	Best Sol.	Avg Sol.	Avg.Time		Best Sol.	Avg Sol.	Avg.Time		Best Sol.	Avg Sol.	Avg.Time		ACO1		ACO2				ACO-CS
													Best Sol.	Avg.Time		Best Sol.	Avg.Time		Best Sol.	Avg.Time
R101	2874.59	2923.60	243.83		2890.54	2963.65	235.24		2854.57	2884.74	250.88		−5.59	−32.79		−5.06	−35.16		−6.24	−30.85
R102	2861.37	2919.83	246.29		2847.88	2945.16	230.15		2809.08	2918.87	244.84		−3.35	−9.34		−3.81	−15.28		−5.12	−9.88
R103	2876.24	2924.21	250.54		2927.47	2975.94	231.79		2842.08	2923.77	252.17		−4.23	−29.18		−2.52	−34.48		−5.37	−28.72
R104	2871.20	2930.62	252.01		2933.79	2990.99	229.60		2863.00	2971.26	252.67		−2.59	−45.71		−0.46	−50.54		−2.86	−45.57
R105	2870.14	2899.98	279.02		2866.78	2987.70	219.64		2832.53	2919.95	252.27		−6.45	−22.90		−6.56	−39.31		−7.68	−30.29
R106	2860.88	2953.25	250.87		2924.31	3022.87	232.68		2853.91	2930.61	251.96		−4.75	−36.66		−2.63	−41.26		−4.98	−36.39
R107	2837.23	2909.95	237.88		2889.71	2979.21	232.47		2856.27	2947.26	254.78		−7.95	−43.17		−6.25	−44.46		−7.33	−39.13
R108	2838.65	2934.96	252.97		2859.81	2913.18	229.78		2859.11	2953.38	291.27		−2.96	−23.58		−2.23	−30.59		−2.26	−12.01
R109	2866.42	2973.33	249.57		2851.22	2973.79	201.22		2851.00	2930.29	244.30		−2.22	−35.99		−2.74	−48.39		−2.74	−37.34
R110	2896.60	3012.22	249.20		2880.89	2999.15	232.67		2827.76	2904.32	242.80		−3.66	−32.64		−4.18	−37.11		−5.94	−34.37
R111	2890.78	3016.29	252.23		2840.80	2982.12	232.67		2852.71	2920.12	244.86		−0.42	31.10		−2.14	20.93		−1.73	27.27
R112	2855.23	2910.72	250.39		2877.97	2923.92	208.62		2872.61	2972.99	266.77		−1.61	24.13		−0.83	3.42		−1.02	32.25
R201	965.07	983.39	203.22		954.74	980.03	299.29		947.27	966.00	199.78		−12.84	−43.00		−13.77	−16.05		−14.45	−43.97
R202	960.17	967.19	191.49		953.73	980.12	299.93		951.34	964.98	205.95		−14.52	−42.31		−15.10	−9.64		−15.31	−37.95
R203	965.90	976.11	202.78		949.47	970.45	357.75		947.92	963.27	200.04		−12.22	−37.87		−13.72	9.60		−13.86	−38.71
R204	959.64	968.95	204.01		916.80	958.60	305.91		953.90	966.73	200.83		−6.53	−35.47		−10.70	−3.23		−7.08	−36.47
R205	972.94	980.94	206.36		928.26	961.64	308.74		949.22	969.13	200.29		−12.03	−32.36		−16.07	1.19		−14.18	−34.35
R206	953.87	968.96	206.25		935.11	973.57	311.55		939.09	966.16	201.25		−16.58	−34.70		−18.22	−1.37		−17.87	−36.29
R207	964.66	977.93	191.71		926.21	958.77	334.23		954.61	972.13	200.05		−10.41	−40.15		−13.98	4.35		−11.34	−37.54
R208	965.07	972.72	209.79		926.33	963.46	298.62		945.73	966.47	201.86		−13.03	−34.82		−16.52	−7.22		−14.78	−37.29
R209	968.30	975.10	205.58		943.47	973.98	316.00		940.15	963.42	201.07		−11.50	−36.43		−13.77	−2.29		−14.07	−37.82
R210	958.24	969.00	200.85		914.69	954.22	324.50		938.71	967.04	201.38		−6.66	−38.33		−10.90	−0.36		−8.56	−38.16
R211	957.35	970.90	203.81		949.47	966.25	341.06		946.43	965.54	224.29		−13.12	−33.27		−13.83	11.67		−14.11	−26.56
C101	3506.61	3591.98	286.58		3449.73	3602.02	240.53		3446.05	3584.98	258.05		−1.34	32.55		−2.94	11.25		−3.05	19.36
C102	3474.05	3571.03	284.83		3486.53	3576.28	245.39		3441.43	3590.97	281.71		−0.78	32.45		−0.43	14.11		−1.71	31.00
C103	3481.68	3592.26	281.39		3462.15	3609.02	216.72		3474.05	3593.00	282.98		−1.47	30.59		−2.02	0.58		−1.68	31.33
C104	3525.55	3597.40	281.39		3471.04	3603.59	244.64		3521.91	3548.76	288.18		−0.42	30.11		−1.96	13.12		−0.53	33.25
C105	3486.10	3584.13	285.22		3511.30	3575.92	216.02		3516.62	3580.46	288.21		−1.53	31.40		−0.82	−0.48		−0.67	32.77
C106	3488.50	3576.00	280.76		3517.38	3627.15	214.95		3473.89	3581.50	286.03		−0.47	14.95		0.36	−12.00		−0.88	17.11
C107	3501.61	3589.65	287.22		3520.33	3610.49	216.37		3445.33	3545.88	292.06		1.12 −	30.67		1.66	−47.77		−0.51	−29.51
C108	3504.23	3574.77	282.64		3521.88	3636.93	233.65		3544.81	3617.58	291.43		−2.39	30.65		−1.90	8.01		−1.26	34.72
C109	3539.35	3601.36	282.37		3523.64	3588.97	246.42		3502.56	3580.85	293.36		−0.45	30.03		−0.89	13.47		−1.49	35.09
C201	1395.27	1420.72	226.36		1326.86	1446.56	206.41		1364.89	1415.57	265.86		−1.25	−34.70		−6.09	−40.46		−3.40	−23.31
C202	1366.19	1414.83	229.88		1340.78	1431.46	214.05		1355.29	1398.43	232.03		−0.43	−35.50		−2.28	−39.94		−1.22	−34.90
C203	1362.47	1393.82	230.30		1325.76	1457.35	182.71		1353.65	1416.32	230.96		−4.25	−43.41		−6.83	−55.10		−4.87	−43.24
C204	1358.25	1399.47	205.77		1372.98	1433.24	200.02		1338.12	1399.15	236.18		−3.52	−42.76		−2.48	−44.36		−4.95	−34.30
C205	1365.59	1404.16	230.92		1387.74	1309.45	209.63		1347.73	1394.27	232.72		−0.67	−28.29		0.94	−34.90		−1.97	−27.73
C206	1354.14	1387.63	231.75		1390.17	1458.57	191.78		1339.49	1390.64	227.86		−0.45	−69.99		2.20	−75.17		−1.53	−70.50
C207	1355.74	1406.58	230.40		1369.28	1456.22	207.93		1336.08	1376.35	235.51		−0.52	−28.09		0.47	−35.10		−1.96	−26.50
C208	1374.43	1404.18	202.68		1352.06	1418.86	207.49		1344.01	1390.56	233.20		−2.31	−46.72		−3.90	−45.46		−4.47	−38.70
RC101	3565.36	3602.79	257.59		3651.14	3707.97	244.21		3521.54	3570.41	257.08		−2.69	−54.22		−0.34	−56.60		−3.88	−54.31
RC102	3522.79	3625.30	258.86		3655.38	3748.01	240.04		3427.17	3552.98	253.60		−5.30	−59.61		−1.73	−62.55		−7.87	−60.43
RC103	3531.67	3563.50	257.75		3645.25	3731.04	232.87		3405.90	3540.58	257.58		−5.59	−28.23		−2.56	−35.16		−8.96	−28.28
RC104	3513.19	3577.53	257.78		3587.78	3670.06	242.76		3468.55	3573.27	258.61		−7.44	−54.37		−5.48	−57.03		−8.62	−54.22
RC105	3509.36	3469.20	257.93		3553.08	3726.59	242.56		3472.23	3562.22	265.94		−3.70	−59.49		−2.50	−61.90		−4.72	−58.23
RC106	3530.01	3588.54	259.23		3616.80	3760.28	242.29		3412.85	3544.77	256.75		−0.45	−53.33		2.00	−56.38		−3.75	−53.77
RC107	3507.76	3613.98	261.35		3544.78	3672.50	240.51		3414.69	3551.34	257.41		−5.71	−52.98		−4.72	−56.73		−8.21	−53.69
RC108	3455.46	3552.06	260.75		3603.97	3669.23	241.70		3426.44	3561.70	258.19		−5.63	−35.41		−1.57	−40.13		−6.42	−36.04
RC201	1122.12	1171.28	200.29		1119.33	1214.53	322.41		1111.77	1137.57	198.33		−3.58	−37.77		−3.82	0.18		−4.47	−38.37
RC202	1108.58	1144.44	200.34		1143.24	1210.17	304.46		1089.91	1114.71	354.35		−7.42	−27.88		−4.52	9.60		−8.98	27.56
RC203	1120.28	1188.68	200.52		1105.08	1172.16	292.70		1089.08	1116.13	198.41		−2.58	−25.78		−3.91	8.34		−5.30	−26.56
RC204	1114.91	1160.77	201.99		1066.09	1160.50	317.03		1098.08	1127.30	196.99		−6.55	−28.15		−10.64	12.77		−7.96	−29.93
RC205	1117.11	1146.56	197.64		1110.16	1159.51	348.14		1109.17	1134.31	196.45		−5.22	−58.88		−5.81	−27.56		−5.89	−59.12
RC206	1125.53	1151.40	200.21		1124.94	1186.12	322.25		1107.27	1124.00	187.86		−6.01	−29.05		−6.06	14.20		−7.54	−33.43
RC207	1124.08	1154.50	200.18		1118.81	1183.43	332.24		1097.56	1122.08	198.46		−3.55	−23.89		−4.00	26.33		−5.82	−24.54
RC208	1121.19	1136.43	201.16		1134.81	1202.59	281.25		1085.02	1122.84	198.49		−7.67	−66.16		−6.55	−52.69		−10.65	−66.61

show a comparison of the results obtained by different algorithms.

Table 10 gives the experimental results of different algorithms on the Salhi & Nagy (1999) test problem. Based on the results of the DEV data in the table, it can be seen that ACO1, ACO2 and ACO-CS find better solutions in all instances compared to ACO. The running time of all three algorithms is reduced for the rest of the instances, except for ACO2, where the running time is increased, for instance, CMT2X. On 24 instances of Salhi & Nagy’s (1999) test problem, compared with ACO results, ACO1 improved the quality of the best solution by 8.46% and reduced the running time by 30.37%. The improvement of ACO2 on the best solution is 7.62%, while the reduction in running time is 28.84%. ACO-CS improved the quality of the best solutions by 9.60% and reduced the running time 29.35%.

Table 11 gives the experimental results of different algorithms on the Dethloff (2001) test problem. According to the data results in the table, compared with ACO, ACO1 and ACO-CS can find better solutions and reduce the running time in all instances. ACO2 does not improve the best solution in instances SCA8-6, CON8-7, and CON8-8, but improves the best solution in the remaining instances. In addition, ACO2 does not reduce running time in 15 instances. In 40 instances of the Dethloff (2001) test problem, compared with ACO results, ACO1 improved the best solution by 4.78% and reduced the running time by 24.57%. ACO2 improved the quality of the best solution by 4.13% and reduced the running time by 11.79%. ACO-CS improved the best solution by 5.97% and reduced the running time by 23.79%.

Table 12 shows the experimental results of the different algorithms on the Montane & Galveo (2006) test problem. Compared to ACO, ACO1 found a better solution in all instances, while reducing the running time on R2, C2 and RC type instances. ACO2 does not improve the best solution on four instances, namely R205, R206, R207 and RC106. The running time of all instances of C2 and RC1 is reduced. ACO-CS found a better solution in all instances, with reduced running time except for the C1 type instances and RC202. In 56 instances of the Montane & Galveo (2006) test problem, compared with ACO results, ACO1 improved the quality of the best solution by 4.92% and reduced the running time by 26.57%. ACO2 improved the best solution by 5.09% and reduced the running time by 20.91%. ACO-CS improved the best solution by 6.14% and reduced the running time by 24.90%.

Combining the analysis of the results in Table 10, Table 11 and Table 12 shows that the improvement of the best solution of ACO1 on the Salhi & Nagy (1999) and Dethloff (2001) test problems is 100% and the reduction of the running time is 100%, while the improvement of the best solution on the Montane & Galveo (2006) test problem is 100% and the reduction of the running time is 80.36%. ACO2 on the Salhi & Nagy (1999) test problem the improvement of the best solution is 100% and the reduction in running time is 95.83%; on the Dethloff (2001) test problem, the improvement of the best solution is 92.50% and the reduction in running time is 62.50%; on the Montane & Galveo (2006) test problem, the improvement of the best solution is 92.86% and the reduction in running time is 67.86%. ACO-CS has a 100% improvement in the best solution and a 100% reduction in running time on the Salhi & Nagy (1999) and Dethloff (2001) test problems, and a 100% improvement in the best solution and a 82.14% reduction in running time on the Montane & Galveo (2006) test problem.

ACO1 has a higher running time reduction rate on all test problems, and ACO-CS has a higher best solution improvement rate on all test problems. In summary, ACO-CS provides a better solution than ACO1 and ACO2 because it combines ACO with a candidate strategy to increase the speed of the algorithm, and with the swap operator to provide a better solution. Experimental results also show that combining these two strategies is successful.

4.5. Algorithm Convergence Analysis

To further observe the performance of the proposed algorithm, the convergence curves of ACO-CS and the comparison algorithm on different test problems are drawn, and the convergence curves are used to illustrate the convergence of the algorithm on different test problems.

Here, we select six instances, CMT1X of the Salhi & Nagy (1999) test problem, SCA3-0 and CON8-0 of the Dethloff (2001) test problem, and R201, C201, RC101 of the Montane & Galveo (2006) test problem, as representative instances and analyse their convergence curves. Figure 6 shows the convergence curves of the four algorithms on different types of instances of different test problems.

It can be seen from Figure 6a,b that when solving CMT1X and SCA3-0 instances, the 2020IACO algorithm, 2021IACO algorithm and 2023IACO algorithm have a fast convergence rate and converge to the optimal solution in the early stage, and tend to be stable in the subsequent iteration process. However, the convergence rate of ACO-CS is faster in the early stage and slower in the middle and late stages, but the current solution is constantly optimized and jumps out of the local optimum several times during the search process. In Figure 6c, although ACO-CS has a relatively low exploration capability compared with the 2020IACO algorithm, 2021IACO algorithm and 2023IACO algorithm, there is no algorithm stagnation in the search process. In Figure 6d–f, the four algorithms converge faster in the early stage and effectively jump out of the local optimum towards the global optimum solution in the middle and late stages. The analysis of the convergence curve shows that during the iteration of the algorithm, the value of the objective function decreases with the increase in the number of iterations. Meanwhile, compared with the 2020IACO algorithm, the 2021IACO algorithm and the 2023IACO algorithm, the ACO-CS has good global convergence and the ability to jump out of the local optimum during the search process.

5. Practical Cases

In this section, we apply the established model and the designed algorithm to a real case to verify its practicability. The distribution transport of a logistics company, which has a distribution centre at a certain location and provides delivery and pickup services to 25 surrounding customers, is used as the object of the study. This distribution centre has three distribution vehicles of the same type, and the maximum capacity of each vehicle is 80 kg. It is necessary to provide reasonable distribution routes for the vehicles in the distribution center to serve customers, so as to improve customer satisfaction and reduce total business costs. Now select the data of the distribution center on a certain day, and give the location of the distribution center and customer point, delivery volume and pickup volume information in

Table 13. Information about distribution centers and customer points.

Num	Longitude	Latitude	Pickup Quantity	Delivery Quantity	Num	Longitude	Latitude	Pickup Quantity	Delivery Quantity
0	106.24777	38.49981	0	0	13	106.269126	38.49263	8	19
1	106.253675	38.498391	4	16	14	106.267545	38.497487	12	4
2	106.249938	38.500311	6	7	15	106.270276	38.503191	2	17
3	106.201103	38.519442	13	11	16	106.261077	38.500819	2	11
4	106.246848	38.493477	10	4	17	106.250585	38.512678	11	6
5	106.21839	38.494324	8	13	18	106.265246	38.513864	16	2
6	106.217384	38.496809	17	6	19	106.257628	38.516913	19	9
7	106.216881	38.501214	12	14	20	106.267617	38.517477	3	9
8	106.216162	38.504433	11	14	21	106.275235	38.511718	12	10
9	106.21645	38.488789	19	15	22	106.218965	38.512734	10	5
10	106.220546	38.485569	6	10	23	106.225289	38.516461	1	16
11	106.251951	38.484722	16	2	24	106.213216	38.514654	7	12
12	106.263521	38.490144	16	5	25	106.208473	38.514146	4	8

.

According to the mathematical model established in Section 2, the ACO-CS, 2020IACO, 2021IACO and 2023IACO algorithms are used to solve the VRPSPD problem. To ensure the reliability of the solution, each algorithm is run 10 times and the corresponding results are recorded, and the path with the smallest objective function is selected as the optimal path. Figure 7 shows the optimal path diagram obtained by the four algorithms.

Figure 7a–d show the optimal routes of 2020IACO, 2021IACO, 2023IACO and ACO-CS, respectively. Each colour in the diagram represents a vehicle’s distribution route. The red five-pointed star represents the distribution centre and the dots represent customer points. As can be seen in Figure 7, the optimal distribution schemes for 2020IACO, 2021IACO, 2023IACO and ACO-CS all use four vehicles, which depart from distribution centre 0 and eventually return to the distribution centre.

Table 14. Optimal distribution route.

Algorithm	Distribution Route	Driving Distance (m)	Total Distance (m)
2020IACO	0-2-1-4-11-12-13-0	6220.14
	0-16-14-15-21-20-18-19-17-0	7898.52
	0-23-22-24-25-3-0	9823.32
	0-8-7-6-5-9-10-0	10,166.23	34,108.21
2021IACO	0-2-1-4-11-12-13-0	6220.14
	0-16-15-14-21-20-18-19-17-0	8743.84
	0-23-22-24-3-25-8-7-0	10,303.75
	0-6-5-9-10-0	6948.40	32,216.14
2023IACO	0-2-1-4-11-12-13-0	6220.14
	0-16-14-15-21-20-18-19-17-0	7898.52
	0-23-22-24-25-3-8-7-0	10,390.91
	0-6-5-9-10-0	6948.40	31,457.97
ACO-CS	0-1-16-14-13-12-11-4-0	5980.36
	0-15-21-20-18-19-17-0	7378.16
	0-23-22-24-3-25-8-7-0	10,303.75
	0-6-5-9-10-0	6948.40	30,610.68

shows the specific driving path and vehicle driving distance of the optimal delivery scheme for each algorithm.

From Table 14, the minimum total distance obtained from ACO-CS is 30,610.68 m, followed by 31,457.97 m obtained from 2023 IACO, then 32,216.14 m calculated from 2021 IACO, and finally 34,108.21 m obtained from the 2020 IACO. Based on the comparison of the results of the four algorithms, it is found that the ACO-CS algorithm is the most effective in solving the vehicle routing problem with simultaneous pickup and delivery.

6. Conclusions

In this paper, an improved ant colony algorithm based on candidate strategy and grid search is proposed to solve the vehicle routing problem with simultaneous pickup and delivery. The algorithm is designed with a candidate strategy that reduces the running cost of the algorithm while ensuring the solution quality, and the grid search method is proposed to optimize the ACO-CS parameters to improve the performance of the algorithm. Computational results on three test problems show that compared with other algorithms, ACO-CS has advantages in terms of solution quality and algorithm stability when solving the test problems of Dethloff (2001) and Montane & Galveo (2006). For running time, ACO-CS has an advantage in solving the Dethloff (2001) test problem and R2 and C2 type instances of the Montane & Galveo (2006) test problem. Meanwhile, the results of the effectiveness and convergence analysis of the algorithm show that ACO-CS has good global convergence and the ability to jump out of the local optimum during the search process. In future work, on the basis of the mathematical model of the VRPSPD, factors such as carbon emission, customer satisfaction, demand uncertainty and variable travel time will be considered to further investigate more complex issues in life.