IMPROVED ANT COLONY OPTIMIZATION FOR SEAFOOD PRODUCT DELIVERY ROUTING PROBLEM

This paper deals with a real-life vehicle delivery routing problem, which is a seafood product delivery routing problem. Considering the features of the seafood product delivery routing problem, this paper formulated this problem as a multi-depot open vehicle routing problem. Since the multi-depot open vehicle routing problem is a very complex problem, a method is used to reduce the complexity of the problem by changing the multi-depot open vehicle routing problem into an open vehicle routing problem with a dummy central depot in this paper. Then, ant colony optimization is used to solve the problem. To improve the performance of the algorithm, crossover operation and some adaptive strategies are used. Finally, the computational results for the benchmark problems of the multi-depot vehicle routing problem indicate that the proposed ant colony optimization is an effective method to solve the multi-depot vehicle routing problem. Furthermore, the computation results of the seafood product delivery problem from Dalian, China also suggest that the proposed ant colony optimization is feasible to solve the seafood product delivery routing problem.


INTRODUCTION
Seafood processing, distribution and selling have been the main roles for a seafood company.Some literature concerned with these problems provides the awareness of the necessity of solving the problem concerned with seafood [1,2].The "Seafood", from the point of its definition, should include finfish, shellfish and crustaceans in fresh, frozen and processed product forms.In this paper the seafood in processed product forms is considered, noted as "seafood product".With the increasing demands of seafood product, the seafood product delivery cost constitutes a significant part of the operational costs of a seafood product company.
For a seafood company to ensure the quality of seafood product, the locations of the factories are mostly near the ports.In the factories, seafood will be processed according to the packaging standards.Then, there are many seafood product factories whose seafood products should be delivered from these factories to a set of customers (for example, hypermarkets).The management of the seafood company estimated that the fixed cost of the vehicles involved in the delivery process of seafood product is high enough.Therefore, the company decided to hire vehicles to complete the delivery tasks which cover only the distances from seafood product factories to the customers.Since the delivery cost paid by the seafood company is proportional to the total travel cost, the paper attempts to optimize the delivery routes for vehicles that minimize the total travelling and vehicle operating costs of the delivery process.The fact that the hired vehicles and the depot of the seafood product factories is not simple, led the authors to formulate this real-life problem as a combination of an open VRP(OVRP) [3,4] and a multi-depot vehicle routing problem (MDVRP) [5,6].However, these vehicle delivery routing problems have the following common characteristics: all the vehicles have the same capacity; the number of vehicles is unrestricted; each customer is visited once and only once by one of the vehicles and its demand must be completely fulfilled; the total demand of all the customers on a route must not exceed the capacity of the vehicle; each vehicle makes one trip only; the objective is to find the set of routes, and minimize the total distance travelled by the vehicles.
From the view of the model, both the OVRP and MDVRP do not completely fit our real-life problem.However, the concept of OVRP is involved in the formulation of the problem since it allows the hired vehicles to be assigned to routes in which they do not have to return to the seafood product factory after serving the customers, providing the company with the set of routes that minimizes the total distance travelled.Furthermore, the delivery process starts at more than one depot.Therefore, it will turn out that the model of the multi-depot seafood product delivery problem belongs to the open MDVRP (MDOVRP) based on the idea of OVRP and MDVRP.
Since both OVRP and MDVRP are considered as NP-hard problems, it is difficult to solve them by traditional methods.Recently, many studies have proved that heuristic techniques are suitable for solving this kind of complicated problems [6,[7][8][9][10][11][12][13][14][15].Brandao [4] presented a tabu search algorithm to solve the open vehicle routing problem, which uses an innovative concept that consists of integrating the lower bounds of a problem.Repoussis et al. [16] introduced a hybrid evolution strategy to solve the open vehicle routing problem and the results on well-known benchmark data sets suggested the effectiveness of the algorithm for this problem.Salari et al. [3] presented a heuristic improvement procedure based on integer linear programming techniques for solving the open vehicle routing problem.Mirabi et al. [17] presented a hybrid heuristics for solving the multi-depot vehicle routing problem.In the three hybrid heuristics, a constructive heuristic search and improvement techniques have been used to improve the performance of the algorithm in each hybrid heuristics.Ho et al. [18] presented a hybrid genetic algorithm to solve the multidepot routing problem.In the hybrid genetic algorithm, the initial solutions are generated randomly in one hybrid genetic algorithm and the initialization procedure is performed in other hybrid genetic algorithms.Yu et al. [6] attempted to solve the multi-depot vehicle routing problem with an improved ant colony optimization.In the algorithm, a coarse-grain parallel strategy, ant weight strategy and mutation operation have been used to improve the performance of ant colony optimization.
Ant colony optimization (ACO), an artificial intelligence procedure inspired by food-seeking behaviours of ant colonies in nature, was first proposed by Dorigo et al. [19].It has been successfully applied to solve these kind of complicated optimization problems, e.g.travelling salesman [19] quadratic assignment [20], job-shop scheduling [21], telecommunication routing [22], vehicle delivery routing problem [23], etc.Therefore, this paper also adopts ACO to solve the seafood product delivery routing problem.
The remainder of this paper is organized as follows.Section 2 describes the seafood product delivery routing problem as a multi-depot open delivery routing problem.Section 3 presents ACO and introduces some improving strategies to enhance the performance of ACO.Some computational results are discussed in Section 4 and finally, the conclusions are provided in Section 5.

PROBLEM FORMULATION 2.1 Seafood product delivery Routing Problem
The seafood product delivery Routing Problem can be described as follows: There are m seafood product depots, each of which owns Ki vehicles with the same capacity.All the vehicles are responsible for the service of n customers, the demand of customer i are di , , i n 1 f = ^h , and d q i 1 (q is the limit of vehicle load).The distance between two customers or between customers and the delivery points is described as the vertex set C which is partitioned into two subsets: , is the set of automobile part depots and , , , is the set of customers, respectively.The distance matrix is a real symmetric one satisfying the triangle inequality principle, that is, c c c ik ij jk # + .Each customer can be served by only one vehicle from any depot.Each vehicle can serve several customers whose demands must not exceed the transportation capacity of the vehicle.The objective of the seafood product delivery routing problem is to minimize the number of vehicles and, for a given number of vehicles, to minimize the total distance travelled by the vehicles.In the seafood product delivery routing problem, each customer can be served by only one vehicle from any depot.Each vehicle can serve several customers whose demands must not exceed the transportation capacity of this vehicle.And each route is a sequence of customers which should start from one seafood product depot and finish at one of the customers.In this paper, the number of vehicles is given and the objective of the seafood product delivery routing problem is to minimize the total delivery distance in serving all customers while it meets the following constraints.Thus, the model of the seafood product delivery routing problem is described as follows: x Constraint (a) service constraints: to assume that every customer node can be visited exactly once by one vehicle.Constraint (b) the capacity constraint: to assume the load of a vehicle cannot exceed its capacity.Constraint (c) the maximum number of vehicles constraint: to assure the number of vehicles used is less than the maximum.Constraint (d) ensures a vehicle serves a customer and leaves it.
The seafood product delivery routing problem can be described as in Figure 1.
sumed as 0 and ∞, respectively, is added.Each vehicle starts its route at the added dummy depot through one actual depot to a sequence of customers without exceeding the capacity constraints of each vehicle.Thus, the seafood delivery problem can be changed into a seafood delivery problem with a dummy depot, which is similar to an OVRP with the dummy depot as the origin.Figure 2 is an example of the seafood delivery problem with a dummy depot transferred from the same problem of Figure 1 by using a dummy depot.

Seafood product delivery problem with a dummy depot
Since the delivery routes consist of a combination of whichever depot and customers, the seafood product delivery problem is NP-hard and very difficult to solve even for relatively small size instances.It is necessary to reduce the complexity of the problem.In this paper, a method [6] is adopted to simplify the seafood delivery problem.First, a dummy depot c0, whose distances to H actual depots and N customers are as-

IMPROVED ACO FOR THE SEAFOOD PRODUCTION DELIVERY PROBLEM WITH A DUMMY DEPOT
The solution of the seafood delivery problem with a dummy depot is to find a set of minimum cost routes in order to facilitate the delivery from the dummy depot through m actual depot to a number of customer locations.This is very similar to food-seeking behaviour of ant colonies in nature.If we take the dummy depot as the nest, the actual depots as the entries of the nest and the customers as the food, the problem can be described as a process by which the ant colony searches for food starting from the nest through an entry.The general description of ACO can be seen in the literature [12,23].The specific steps of the improved ACO are as follows.

Generation of solutions
Using ACO whose colony scale is P, an individual ant simulates a vehicle, and its route is constructed by incrementally selecting customers until all customers have been visited.Initially, each ant starts at the central depot and the set of customers included in its route is empty.The ant selects the next customer to visit from the list of feasible locations and the storage capacity of the vehicle is updated before another customer is selected.The ant returns to the central depot when the capacity constraint of the vehicle is met or when all customers are visited.The total cost is computed as the objective function value for the complete route of the artificial ant.The ACO algorithm constructs a complete route for the first ant prior to the second ant starting its route.This continues until the last ant constructs a feasible route.In the iterative phase, the customers or partial routes are combined by sequentially choosing feasible entries from the actual depots and the customers.A combination is infeasible if it violates the capacity constraints or the customer has already been visited.
The decision making about combining customers is based on a probabilistic rule taking into account both the visibility and the pheromone information.The pheromone can tell us how good the combination of these two customers i and j was in previous iterations.The visibility is constructed based on the well-known savings Algorithm due to [24] and it contains the savings of combining two customers i and j on one route as opposed to serving them on two different routes.Thus, to select the next customer j for the kth ant at the ith node, the ant uses the following probabilistic formula.
where p k ij ^h = the probability of choosing to combine customers i and j on route; ij x = the pheromone density of edge , i j ^h; ij h = the visibility of edge , i j ^h; a and b = the relative influence of the pheromone trails and the visibility values, respectively; tabu = the set of infeasible nodes for the kth ant.

Adaptive strategy for parameters a and b
It can be attained from Formula (2) that two parameters a and b have important roles in ACO.Parameter a is a factor of the information heuristic to reflect the effect of pheromone.Parameter b is a factor of the expectation heuristic to express the relative importance of the visibility.Thus, the two parameters can greatly affect the randomness and determination of the algorithm.In general, before launching the algorithm the value of the two parameters generally are determined as a constant according to the user's experience, or by an external tuning method.The main drawback of the determination for the two parameters is that the values are constants during the searching process of the algorithm.However, the functions of the two parameters have in fact different effects during the searching process of the algorithm.
Parameter a determines the extent of randomization during the evolution process.The larger the effect of randomization the larger is the diversity while small value of parameter a can ensure a reasonable convergence velocity.Similarly, parameter b determines the extent of determination during the evolution process.The larger the effect of determination the larger is the convergence velocity while a small value of parameter b can ensure the diversity.Therefore, in the early stage of the searching process, parameter a with a small value and parameter b with a lager value can ensure a reasonable convergence velocity while parameter α with a larger value and parameter b with a small value can help the algorithm to search a wider space.More diversity leads to the algorithm escaping from the local optima during later stages.Therefore, a self-adaptive strategy for parameters a and b is introduced to improve the searching ability of the algorithm.
where, t is the current iteration and max T is the maximum number of iterations.Equations ( 3) and ( 4) provide a self-adaptive, non-linear model for optimizing the value of parameters a and b which is used to balance the global and local searching ability of the algorithm at a reasonable convergence velocity.

The 2-opt exchange
In the searching process of the algorithm, 2-opt is a simple local search algorithm [25] which is also applied to ensure local optimality.The main idea of 2-opt is to improve the routing by deleting two edges and by replacing them with the only possible couple of new edges in the route constraint.In the 2-opt exchange, all the possible pairwise exchanges of customer locations visited by individual vehicles are examined to see if an overall improvement in the objective function can be attained.If all pairs of the edges were considered and no further solution is found, a pareto local optimum solution is reached.The method has been used in several ACOs [23,[26][27][28][29] for VRP.

Crossover operation
Crossover is a genetic operator which exchanges genetic information between two parent chromosomes to form two new children at a predefined probability pc .The operators can help the algorithm to search a larger space.The idea of crossover operation is to randomly change two tours and produce two new solutions.In this paper, the crossover operator is designed to conduct customer exchanges in a random fashion.Firstly, two routes in the solution are selected B. Yao, P. Hu, M. Zhang, X. Tian: Improved Ant Colony Optimization for Seafood Product Delivery Routing Problem randomly and one customer in each selected route is selected randomly.Then, the two selected customers are exchanged and a new solution is created.Figure 3 shows the selection operation for the crossover operation and Figure 4 shows the exchange process for the crossover operation.
To achieve a better performance, the crossover rate pc setting should be done while solving the problem.It is important to control the amount of time invested for the crossover operation.While the crossover rate pc with a large value will increase the computation time, the crossover probability pc with a large value will decrease the efficiency of the algorithm.Therefore, a balance between a reasonable computation time and efficiency is necessary.Usually, the diversity of the solution is large at the beginning of a run and decreases with time.We adopt the crossover rate during a run to promote fast convergence to good solutions during the first generations and to introduce more diversity for escaping from local optima during later stages.The crossover rate pc at generation t is: where p min c and p max c denote the lower and upper crossover rates for the beginning and ending, respectively.Tmax and t are the maximum number of generations and the current generation, respectively.
According to preliminary tests, the lower crossover rate is set to = and nc is the number of the customers.The upper crossover rate is set to and nv is the amounts of the routes in the solution.

Update of Pheromone information
In order to improve future solutions, the pheromone trails of the ants must be updated to reflect the ant's performance and the quality of the solutions found.This updating is the key element to the adaptive learning technique of ACO and helps ensure the improvement of subsequent solutions.First, Pheromone updating is conducted by reducing the amount of Pheromone on all visited arcs in order to simulate the natural evaporation of Pheromone and to ensure that no one path becomes too dominant.This is done with the following Pheromone updating equation, where tij k D = the Pheromone increments of the kth route in the best solution on edge , i j ^h; t = the parameter that controls the speed of evaporation; k = the No. of the route.
From Formula ( 8) it can also be found that parameter t controls the quality of the Pheromone of the routes by a predefined value between 0 and 1.In the early stage, a larger value of parameter t is sufficient to simulate the natural evaporation.However, in the later stage, it is necessary to decrease the value of parameter t to improve the searching ability when the optimum routes have not improved after consecutive iterations.If the value of parameter t is too small, it will greatly affect the convergence velocity.Thus, an adaptive strategy for parameter t is used in this paper.
where min t is the lower value for parameter t.There are three popular Pheromone increment updating strategies: a) Ant-density; (b) Ant-quantity; and (c) Ant-cycle.Some of them omit global information, while others omit local information.This may result in false directive information and a large amount of invalid searches.Therefore, we have adopted a new Pheromone increment updating strategy: Ant-weight, which takes into account both global and local information.Specifically, the Pheromone increment update strategy [23] is written as: where Q = a constant; f k = the object value of the solution; fij k = the object value of the kth route on edge , i j ^h in the solution; n = the number of routes and n 2 2 .This increment update strategy consists of two components: the first one is global Pheromone increment / Q f k that is related to the quality of the kth op- that is related to the contribution of edge , i j ^h to the optimization route.In this way, valid information obtained from the previous search can be retained for further and more careful search in a more favourable area, which helps speed up the convergence of the algorithm.At the same time, the effectiveness of a largescale search can be ensured to facilitate the algorithm to find the overall optimal path.Additionally, to prevent local optimization and to enlarge the probability of gaining a higher-quality solution, an upper limit and a lower limit , min max x x 6 @ are introduced to the Pheromone on each link [30].
where d i 0 = the distance from the central depot to the ith customer.

NUMERICAL ANALYSIS
This paper attempts to use an improved ACO to solve a real-life delivery route problem which can be considered as MDOVRP.Since there is little mention in literatures of heuristics for solving MDOVRP, it is difficult for us to compare the adaptive algorithm with the previous methods.For comparison, some well-known benchmark problems of MDVRP were applied to examine the performance of the improved ACO.Then, the improved ACO was used to solve the seafood product delivery routing problem.The following will describe the two examples respectively.

Classical MDVRP
To examine the performance of the improved ACO in this paper, a set of instances derived from the MDVRP instances from literature [31][32][33] were used.These instances and the best known solutions are available at http://neo.lcc.uma.es/radi-aeb/We-bVRP//index.html?/ProblemInstances/MDVRPInstances.html.The main characteristics of these test problems are summarized in Table 1.The improved ACOs were coded in Visual studio.NET 2003: C++ and executed on a PC equipped with 3.25 GB of RAM and a Pentium processor running at 2.93 GHz.A simulation was performed to determine the parameters in the improved ACO.The parameter Q is set at 1,000 and the maximum number of generations is set at 1,000.In order to examine the performance of the algorithm, the results of the improved ACO algorithm were compared with FIND algorithm [5,34,6].Table 2 shows the results from these algorithms for solving MDVRP.From Table 2 it can be found that the efficiency and effectiveness of the improved ACO compared these well-known algorithms.Most of the results produced from the improved ACO have been close to the best solutions among these algorithms.It suggests that the improved ACO is suitable for MDVRP.Furthermore, some solutions are better than the given best solutions; thus, the improved ACO is fairly effective to solve MDVRP.To further test the performance of the adaptive strategy for ACO in this paper, a standard ACO and the improved ACO are used to solve the MDVRP simultaneously.The results are shown in Table 3.
It can be observed that the computation times and the solutions from the two algorithms are almost equal to solve some simple problems such as problems 1, 2 and 3.It is also found that when the problems are more complex, the computation times of the adaptive algorithm are more than the ones of the standard ACO.However, the solutions from the improved ACO are obviously better than the ones from the standard ACO.It can be explained that the adaptive strategy can explore larger solution space and prevent the algorithm from being trapped in the local optimum.However, the adaptive strategy also needs more time to enlarge the search space in the later iteration.Furthermore, we can find that the computation times are almost equal when solving larger problems such as problems 21, 22 and 23.It is due to the fact that a dummy depot is added to greatly improve the searching velocity when solving larger problems.

Seafood product delivery routing problem
The performance of the proposed algorithm is examined on the benchmark problems of MDVRP B. Yao, P. Hu, M. Zhang, X. Tian: Improved Ant Colony Optimization for Seafood Product Delivery Routing Problem which is in complete Euclidean graphs with straightline distance.However, in the real-life seafood product delivery routing problems, the distance between two points is based on the length of the routes.Figure 4 shows the location information of the real-life seafood product delivery routing problems in Dalian.There are several seafood product depots and some hypermarkets in Dalian city which can be seen in Figure 5.The problem is how these seafood product depots send the seafood product to the hypermarkets with the least delivery cost (the least length of delivery routes).
As shown in Figure 5 the rhombi stand for four seafood product depots in Dalian, and the dots represent the geographical positions of several hypermarkets in Dalian.The vehicle capacity is 200 and the specific information of consumers and distribution centres is shown in Tables 4 and 5, respectively.
Figure 6 shows the computing results of the proposed ACO algorithm while it continues calculating the seafood product delivery routing problem ten times.It can be found that the results are stable and the difference between the optimum and the worst plan is less than 3%.Meanwhile, the computing time, from 280 to 460 seconds, is very short for the larger complicated problems.Therefore, the algorithm has an excellent convergence performance to solving the seafood product delivery routing problem.The length of the routes is 66,375 KM and the optimized routes of the seafood product delivery routing problem can be shown in

CONCLUSION
We have considered the seafood product delivery routing problem in Dalian city as MDOVRP.In this problem, the seafood product depots expand and integrate with each other to keep their competitiveness in the market by reducing the delivery cost.The presence of multi-depot adds considerable difficulties to the standard VRP.A method is used to reduce the complexity of the problem by changing the MDOVRP into an OVRP with a dummy central depot in this paper.Then an improved ACO is presented to solve this seafood product delivery routing problem.In the algorithm, adaptive strategies for the selection of parameters α, β, ρ and the crossover rate are proposed to improve the performance of ACO.The algorithm was tested on the benchmark problems of MDVRP and the real-life sea-

14 Figure 1 -
Figure 1 -An example of the seafood product delivery routing problem where circles represent customers, and stars represent depots.

14 Figure 2 -
Figure 2 -An example of the seafood delivery problem with a dummy depot

14 Figure 3 - 14 Figure 4 -
Figure 3 -Selecting the points of the crossover

Figure 6 -Figure 7 -
Figure 6 -Computing results of the improved ACO after running ten times . Yao, P. Hu, M. Zhang, X. Tian: Improved Ant Colony Optimization for Seafood Product Delivery Routing Problem B

Table 1 -
Information of the test problems

Table 2 -
Computational results of some well-known published results and the improved ACO

Table 3 -
Comparison results between ACO and the improved ACO

Table 4 -
The information of the hypermarkets in Dalian

Table 5 -
The information of seafood product manufacture depots