An Imperialist-Based Optimization Algorithm for the ...

OVRP_ICA: An Imperialist-Based Optimization Algorithm for the Open Vehicle Routing Problem Shahab Shamshirband1, Mohammad Shojafar2(&), Ali Asghar Rahmani Hosseinabadi3, and Ajith Abraham4,5 1 Department of Computer System and Technology, Faculty of Computer Science and Information Technology, University of Malaya, Kuala Lumpur, Malaysia [email protected] 2 Department of Information Engineering Electronics and Telecommunications (DIET), University Sapienza of Rome, Rome, Italy [email protected] 3 Young Research Club, Behshahr Branch, Islamic Azad University, Tehran, Iran [email protected] 4 Machine Intelligence Research Labs (MIR Labs), Scientific Network for Innovation, and Research Excellence, Auburn, AL, USA [email protected] 5 IT4Innovations - Center of Excellence, VSB - Technical University of Ostrava, Ostrava, Czech Republic

Abstract. Open vehicle routing problem (OVRP) is one of the most important problems in vehicle routing, which has attracted great interest in several recent applications in industries. The purpose in solving the OVRP is to decrease the number of vehicles and to reduce travel distance and time of the vehicles. In this article, a new meta-heuristic algorithm called OVRP_ICA is presented for the above-mentioned problem. This is a kind of combinatorial optimization problem that can use a homogeneous fleet of vehicles that do not necessarily return to the initial depot to solve the problem of offering services to a set of customers exploiting the imperialist competitive algorithm. OVRP_ICA is compared with some well-known state-of-the-art algorithms and the results confirmed that it has high efficiency in solving the above-mentioned problem. Keywords: Metaheuristic algorithms
Open vehicle routing problem (OVRP)
Imperialist competitive algorithm (ICA)
Combinatorial optimization problem

1 Introduction The OVRP involves finding one of the best routes for a set of vehicles that must offer services to a set of customers. Every route in OVRP includes a sequence of customers that starts at the initial depot and ends at one of the customers [1]. The usual limitations of the OVRP are that all vehicles must have the same capacity, and each customer must © Springer International Publishing Switzerland 2015 E. Onieva et al. (Eds.): HAIS 2015, LNAI 9121, pp. 221–233, 2015. DOI: 10.1007/978-3-319-19644-2_19

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be visited by only one vehicle to receive the required service. The total requests of all the customers in a route should not exceed the total capacity of the vehicles. In some problems, the time limit is considered together with the distance limit in covering the specific route: the distance traveled and the travel time spent of vehicle for the route must not exceed the permissible limits [2]. Lowering the number of vehicles, decreasing the distance traveled in the routes, and reducing travel time are among the main purposes of solving this problem. The OVRPs are different and their most important difference is that there are Hamiltonian routes and Hamiltonian cycles in them, respectively. Hamiltonian routes start from one point and end in another Hamiltonian cycles finally return to the initial point [2]. Therefore, one of the main features of OVRP is that vehicles do not necessarily return to the initial depot after servicing to the customers, and if they do return to the initial depot, they will visit the same customers [3]. The OVRP is an NP-hard problem and its solution is considered a scientific challenge. In traditional studies, the OVRP was solved by assuming definite answers to the requests of all customers on the routes. Moreover, researchers have introduced various methods based on innovative and meta-heuristic methods for solving the OVRP, a few of which will be described here. Brandao [3] used Tabu Search to solve the VRP with limitations on vehicle capacity and on maximum travel distance in routes. Authors [4] introduced an algorithm to solve the OVRP for managing open routes. Fleszar et al. [5] implemented the variable neighborhood search technique for the OVRP with the purpose of reducing the number of vehicles and of decreasing travel time, and travel distance in the routes. Erbao and Mingyong [6] dealt with the OVRP by considering the fuzzy demands and combined the improved differential evolution algorithm and random simulation to solve the problem. In [7], a combination in the form of (GA + TS), in which the parallel computational power, the global optimization GA, and the rapid local search TS were used with the purpose similar to that of the above mentioned methods, was used for solving the OVRP. Repoussis et al. [8] introduced a combinatorial evolution strategy for the OVRP with the purpose of reducing the number of vehicles in the fleet and of decreasing the travel distance in the routes. Furthermore, Tabu Algorithms [9] and improved Tabu Algorithms [10] were implemented by Huang and Liu with the purpose of reducing the number of vehicles and travel costs for OVRP. Based on results of simulations, the introduced algorithm can decrease the number of required vehicles and reduce travel costs. Considering the slow rate of convergence and the weak search ability of the traditional genetic algorithm, the combinatorial genetic algorithm (which has greater convergence rate and rapid search ability) can be used to simplify the problem and improve its search efficiency [11, 12]. Authors in [13] introduced an algorithm that was based on genetic rules in order to upgrade the optimal performance of particle swarm and of differential evolution for solving the OVRP. In their algorithm, all members had dominant and recessive characters, optimization of particle swarm took place by the dominant character and differential evolution by the recessive one, and if the proportionality of the dominant character was smaller than that of the recessive one, the recessive character replaced the dominant one. In [14], the concepts of variable neighborhood search and evolutionary algorithms were used for optimization of the OVRP. This method could offer solutions with acceptable quality. Zachariadis and Kiranoudis [15] introduced a new search method for solving the

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OVRP that could investigate an extensive solution space to reduce the number of routes and decrease routing costs. Marinakis and Marinaki have introduced a new version of the BBMO algorithm for solving the OVRP problem [16], among the main parts of which is the replacement of outward movement by a local search that makes the proposed algorithm more effective for solving combinatorial optimization problems. They presented a special decoding method for implementing the PSO in which a vector including customers’ positions was produced in a descending order and then each customer was assigned to a specific route based on his/her position and, finally, a oneunit mutation was applied on all produced routes. This was an effective method for solving the problem because it allowed for studying the feasibility of the routes and for investigating the quality of the answers. The problem, and the in/equalities are defined in Sect. 2. Section 3 introduces the proposed algorithm which is based on imperialist competitive algorithms, and Sect. 4 explains in detail the results of algorithm simulations. Finally, conclusions are presented in Sect. 5.

2 Problem Definition and Notation The OVRP is a kind of the classic vehicle routing problem (VRP) in which the vehicles do not necessarily return to the initial depot after servicing to the customers. The OVRP applies in cases where either the company does not have the required vehicles or there are not enough vehicles in the company to distribute the product among the customers. In both cases, the company will rent a number of vehicles, and when these vehicles carry out their tasks, they do not return to the initial depot. This is one of the class 3PL problems [17]. In this research, the following assumptions were made in solving the OVRP. n is taken to be the number of customers, N ¼ f1; 2; . . .; ng the set of n customers, and V ¼ f0; 1; . . .; ng the set of customers and the starting point, where 0 represents the starting point, qi the request of customer i ϵ V − {0}, and Q is the capacity of each vehicle. It is assumed that K is the maximum number of available vehicles, and cij the cost for the sequence of customers from i to j and a scale equivalent to the appropriate adjustments. Moreover, the cost wk is considered for each travel of the vehicle k. Therefore, the OVRP will involve finding the minimum number of required vehicles and determining a route for each vehicle in a way that all customer requests are satisfied and each customer is visited exactly by one vehicle and no vehicle exceeds its capacity [16, 18]. Mathematical modeling of the OVRP requires two groups of variables, the first group for modeling a sequence in which customers are visited by vehicles and is defined as follows [16]: ( xkij

¼

1 if customer i precedes customer j visited by vehicle k 0

otherwise

ð1Þ

The second group of variables (shown as zk) is a binary variable and defined in ( zk ¼

1 if vehicle k is active 0 otherwise

ð2Þ

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If the same vehicle services at least one customer, it will be considered active [16]. Using the considered parameters and variables, we can express the OVRP problem in this way [16]: min

Xn Xn

Xk k¼1

c xk þ j¼o ij ij

i¼0

Xk k¼1

ð3Þ

wk zk

Subject to k X n X

xkij ¼ 1; 8j ¼ 1; 2; . . .; n

ð4Þ

k¼1 j¼1

Xn

Xk

xk j¼1 ij

k¼1

xkij  zk; Xn

8k ¼ 1; . . .; K;

xk  i¼0 iu X

Xn

xk j¼1 uj

8i ¼ 1; 2; . . .; n;

 jSj  1;

Xn

Xn

xk j¼1 0j

Xn

8j ¼ 1; 2; . . .; n

8S  V : 1  jSj  n;

xk Þ  Q; i¼0 ij

Xn

ð5Þ

¼ 0; 8k ¼ 1; 2; . . .; K; 8u ¼ 1; 2; . . .; n

xk ði;jÞ2ss ij qð j¼1 j

¼ 1; 8i ¼ 1; 2; . . .; n

 1;

xk i¼1 i0

k

8k ¼ 1; 2; . . .; K

8k ¼ 1; 2; . . .; K

 0; 8k ¼ 1; 2; . . .; K

ð6Þ ð7Þ ð8Þ ð9Þ ð10Þ ð11Þ

xkij 2 f0; 1g; 8k ¼ 1; 2; . . .; K; 8i ¼ 1; . . .; n; 8j ¼ 1; . . .; n

ð12Þ

zk 2 f0; 1g; 8k ¼ 1; 2; . . .; K

ð13Þ

The objective function in (3) will create equilibrium between the travel cost and the vehicle. Equation (4) represents the connectivity of all routes taken by all vehicles after they move from the initial point and the cost of the first part of each route too. Equations (5) and (6) guarantee the arrival to and departure of exactly one car to/from each customer. Equation (6) is related to variables x and z and indicates that all customers receive services from active vehicles. Equation (7) exhibits the continuity of each route, and (8) the identifier of the sub-tours. Inequality in (9) shows the maximum vehicle capacity, and inequalities in (10) and (11) indicate that only one vehicle must start its tour from the starting point to give service to a sequence of customers and that none of the vehicles returns to the starting point. Equations (12) and (13) are definitions for the variables x and z for each vehicle k [16]. There are some limitations considered in the problem: (1) All vehicles must be of the same capacity; (2) Travel time for each vehicle must be less than the threshold time considered; (3) The sum of the requests of the customers in each route must be less than the capacity of the vehicle assigned to the

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route; (4) Each customer is visited only once and by only one of the vehicles for his/her request to be satisfied. Put it simply, the purpose of OVRP is to reduce the number of vehicles considered for giving service to customers, and to decrease travel distance and time of the vehicles in the routes [4, 19, 20]. We use the ICA algorithm [21] to improve the consumed resources of an OVRP. ICA considers the algorithm has little information such as search space and definition of feasible answers for the problems, it can move toward finding better answers and progress as well as possible in the search space by creating random answers.

3 OVRP_ICA Algorithm The following six basic steps are taken to apply the proposed algorithm to the OVRP (which is one of the most frequently met problems in transportation). In the first step, the feasible answers (the initial countries) are defined. It must be noted that this is a very important step because these countries must be defined in a way that they have the necessary coordination with the structure of the problem. Therefore, in this step a twopart array is used as shown in Fig. 1. This array is prepared in such a way that customers met are ordered in turn from left to right in the first part of the array, and the number of customers visited by each vehicle is shown in the second part. It must be kept in mind that the number of the elements in the second part is equal to the total number of the vehicles. Moreover, in this figure, the first vehicle visits five customers, the second four, and, finally, the third vehicle visits two customers. It must be noted that the number of the customers allocated to each vehicle must not exceed its capacity.

Fig. 1. One country in the OVRP_ICA algorithm

In the OVRP_ICA, a determined and variable number P of random initial answers are produced and the values of the fm objective function are calculated for each member in m = 1,…,P. These answers are then placed in the D matrix together with their values (every row of this matrix shows one initial feasible answer, or one initial country, together with the related objective function). The use of a random structure in this step causes the obtained answers to have various structures in the feasible region. In the second step, values of the objective function for each county are compared, and the I number of countries that have lower objective function values are considered as independent countries (because OVRP is a minimization problem). By replacing the countries in the matrix, rows from 1 to I of the initial population in the D matrix are given to the independent countries. To determine the sphere of influence of each independent country, the number of dependent countries is allocated to the jth independent country according to Eq. (14) so that I empires are created

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0

1

 1 fj

kj ¼ int@PI

i¼1

1= fi

:ðP  I ÞA

j ¼ 1; . . .; I

ð14Þ

Equation (14) shows the number of dependent countries allocated to the jth independent country. It must be added that, according to this relation, each independent country that has a lower objective function value will acquire a larger number of dependent countries and a larger empire will be formed. Moreover, the int(.) function in Eq. (14) is the integer part function that causes each empire to be allocated an integer number of dependent countries. On the other hand, because the int(.) function is used, there may be some countries left in the end that have not been assigned to any of the empires. In this case, these countries are allocated to the most powerful independent country (which has the lowest objective function value). It must be noted that Eq. (14) determines only the number of countries allocated randomly to each empire. In step three, after the empires have formed, the dependent countries in each empire increases their quality by using the independent countries that have the role of local optimums. The noteworthy point is that, since a large number of dependent countries are combined with one independent country in each empire, some kind of an assimilation function having the random concept must be used so that similar answers are not produced. To achieve this, a modified nearest neighbor method is used in this step. For example, consider the two feasible answers [5 2 4 3 1 6 8 9 7 11 10| 3 3 5] as the independent country and the [2 6 5 3 1 11 9 6 8 10 4|2 4 5] as the dependent country. Now, start to move from 2, that is the first customer of the dependent country, and consider the two countries that are neighbors of 2 (that is, 4, 5, and 6) and have not been visited so far and put them in the S set. According to the modified nearest neighbor method, the probability of those customers being visited that belong to S is obtained from Eq. (15), in which Cij is the Euclidean distance between the customer I and the jth customer belonging to S. Let us suppose that 4 is selected. Therefore, so far, there are [2 4 - - - - - - - -|2 4 5] in the initial population. Moreover, for 4, the three customers 2, 3, and 10 are considered neighbors of the two countries, but since 2 has already been visited, the nearest possible neighbor is selected from 3 and 10 that form the new S set. This process continues until the rest of the customers in [2 4 - - - - - - - - - |2 4 5] are found, and it continues for all dependent countries. It must be noted that, in this method, all customers are selected only once, so that the obtained answer will be a feasible answer in this respect. The only thing that must be investigated is whether the capacity of each vehicle is less than the total capacities of the customers allocated to the vehicle which puts in s according to Eq. (15). If the answer is affirmative (i.e., accepted), the new answer replaces the old one; otherwise, the previous answer remains unchanged.  1 cij s¼P  ð15Þ R 1 j S cij Moreover, in this step, with the changed status of q percent of the dependent countries, the empires will have sufficient diversity. As in the previous step the first part of the answers were examined and upgraded according to the possible nearest neighbor

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algorithm, the second part in the vector of feasible answers will be studied in this step. For this purpose, one of the members is randomly selected and one unit is added to it. For example, let us consider the feasible answer [2 11 8 9 10 6 5 3 7 1 4|2 4 5]. Suppose 4 is selected from part 2 and one unit is added to it. Now, consider the neighbors of 4 (that are 2 and 5). As one unit was added to 4, therefore, one unit is deducted from one of the neighbors at random. For example, one unit is deducted from 5 and the new answer will be [2 11 8 9 10 6 5 3 7 1 4|2 4 5]. Here, again, the condition of the feasibility of the answer and its application within the limitations of the problem are examined. After calculating the answer and the new q value (q is the percentage of dependent countries), these countries may have greater power compared to the independent country in the empire. Therefore, a dependent country with the best value in each empire is selected and replaces the independent country. Care must be taken so that, if several dependent countries have the same objective function, to randomly select one of them, to compare it with the independent country in the empire and, if needed, to replace the independent country with this dependent one. In step four, since until the end of an iteration of the algorithm no other changes take place in the objective function, it is necessary to store the best answer and the value of the objective function. Therefore, in all iterations, after the replacement of independent countries takes place, the best answer among the independent countries is selected as the best current answer and, if it is better compared to previous answers, a local search is conducted on it and the new answer replaces the previous answer and value. The local search used in this step is based on omission of customer from the problem and on entering it in the appropriate position. This change is accepted if the new answer, besides applying within the limitations of the problem, is of higher quality compared to the previous answer. It must be noted that the customer is randomly selected, is tested for all entrance sites, and the best state is selected. Up to this step of the algorithm, the purpose is to conduct a global search to find good answers for important regions. Now, in the next step, these regions must be identified, and countries must converge towards them, so that important regions are more extensively examined, and the algorithm can find better answers. Therefore, in step five, the powers of the empires are compared by using Eq. (16). hj , fi , Si , and k represent the total power (the value of the objective function) of each empire, the power of each independent country, the mean power of the dependent countries in the jth empire, and the influence coefficient, respectively. The influence coefficient has values from zero to 1 and determines the importance of the mean power of the dependent countries compared to the power of the independent country in an empire. Now, the weaker empire gives its weakest dependent country to one of the empires. It must be kept in mind that if a weaker empire does not have a dependent country, it will be destroyed and its independent country will be allocated to the most powerful empire.   hj ¼ f i þ k Sj j ¼ 1; . . .; I ð16Þ In step six, the final condition will be examined, and if it is satisfied, the algorithm ends; otherwise, the algorithm will return to step three and repeat its operation. To end the algorithm, the two conditions of iterating the algorithm a determined T times or

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having only one empire left are used, and these two conditions are simultaneously examined at the end of each algorithm iteration. Either of the two conditions that are satisfied sooner will end the algorithm, and the best answer and value obtained until that time will be introduced as the final answer for the problem. Finally, the main steps in the OVRP_ICA are summarized in the pseudocode shown in Algorithm 1. Algorithm 1. OVRP_ICA Input: N, V, Q, ICA parameters 1: Output: minimum distance traveled N on V 2: 3: Begin: Randomly generate initial population of countries and initial empires 6: 7: 8:

Move colonies toward to the relevant empires(Assimilation) Randomly select some colonies and change the value of colonies( Revolution)

9:

If position a is better than the imperialist, exchanges position of colony and imperialist Compute total cost of each empire Peak the weakest colony from the weakest empire. Other imperialists compete to take possession of colonies. If an empire loses all colonies, collapse it If the stop condition is satisfied, stop, if not go to line 5 End Algorithm

10: 11 12: 13: 15:

4 Performance Evaluation In this section, we describe the test beds dataset and the show simulation results in small-scale and large-scale test problems.

4.1

Setup for Benchmark Data Sets

The proposed method was tested on the famous data set of Christofide et al. [22], Fisher [23], and Li et al. [1]. In Fisher’s model, there are C1 to C14 problems [23]. The numbers of customers in both sets were in the interval of 50 to 199, with Cartesian coordinates and Euclidean distance [23]. Among them, the limitations of route length (L) and service time (e) were considered for all customers. We Use problems which are specified as 8 problems (C1 to C8) introduced by Christofide et al. [22] and Fisher [23]. Among these problems, namely C6–C8, consider route length restrictions (180, 144, 207) and a uniform service time for all customers (10 for all), simultaneously. The maximum route length, compared to the VRP model, was a factor of 0.9 because the lengths considered in the VRP model are not suitable. The last column, in which Kmin (the total requests of the customers divided by the capacity of each vehicle), represents the lower limit. Moreover, the set of large-scale data of Li et al. [1] was also considered. The second case includes 8 problems in the form of O1–O8 with 200–480 customers without any limitations regarding route length. Each problem has a geometric symmetry and the customers are in a circle around the depot.

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4.2

229

Computational Results

Results of simulations conducted to solve the OVRP are presented in this section. To test the efficiency of the proposed algorithm on OVRP, and to compare it with other meta-heuristic algorithms, the proposed algorithm was tested on two classes of standard examples taken from [24]. Implementation took place in the C#.NET programming language and the programs were executed on a computer with 2.06 GHz Pentium-IV processor and 6 GB RAM. Results of simulations of the proposed algorithm (OVRP_ICA) were compared with several samples of various algorithms including HTS [9], ITS [10], DHPD [13] and Hybrid (1 + 1) ES [14] for small-scale and TR [1], HES [9], BLSA [15] and BBMOOVRP [17] for large-scale of problems which will be helpful in industries. Considering the steps mentioned in the previous section, results of outputs are presented in the following tables. In these tables, the k_min column shows the number of vehicles, the Fitness column the travel distance of the vehicle, and the Time column the time taken to solve the problem and to find the best answer. Results of the comparison indicate that the proposed algorithm is superior to the algorithms it was compared with respect to the time taken to solve the problem and the minimum number of vehicles, and with respect to the minimum travel distance of each vehicle. Small-Scale Test Problems: In Table 1, while the number of vehicles increase, the result of of fitness function which is calculated according to the aforementioned algorithm provide better (i.e., lower fitness values) results compared to HTS [9] and ITS [10]. In detail, it is better than HTS and ITS due to convergent and finding appropriate colony in each step and it search the problem space globally rather than Tabu searching which search the system locally. Also, OVRP_ICA total travel distacne is about 50 % less than other methods in medium and smooth restriction as are shown in rows C6 and C8 but it has approcimetely 1 % higher travel distance compared to the least one in C7. Table 1. Results of comparisons among OVRP_ICA, HTS [9] and ITS [10] methods for minimizing the total travel distance and time of the vehicle and the number of vehicles (small-scle) OVRP_ICA Time (s) Fitness 63.429 409.68 131.758 557.14 254.385 629.74 350.132 716.73 563.876 902.45 85.12 411.87 91.67 587.76 576.23 634.52

ITS [10] Time Fitness 14.9 461.37 26.7 663.28 218.4 954.34 531.5 1247.52 253.1 1059.47 61.4 890.63 17.9 602.71 553.1 1261.39

HTS [9] Time Fitness 16. 1 451.59 28.3 649.38 2 12.5 928.41 542.7 1201.95 268.6 1018.56 68.3 870.06 18.6 586.64 562.8 1254.28

The input Kmin Q 5 160 10 140 8 200 12 200 16 200 5 160 10 140 8 200

Problem N 50 75 100 150 199 50 75 100

C1 C2 C3 C4 C5 C6 C7 C8

In next Table, we compare our approach with ES [14] and DHPD [13] methods in the same problems to evaluate our approach with PSO-based and integrated

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neighborhood based methods for small-scale problems to minimize the total travel distance and time of the vehicle and the number of vehicles. As we seen in Table 2, our approach is able to minimize the total travel distance while the number of the customers and vehicles capacity increase well compared with the ES and DHPD [13] even with rising the restrictions (i.e., C6 an C8), but when the restriction is too tight DHPD [13] provide better result rather than others.

Table 2. Results of comparisons among OVRP_ICA, ES [14] and DHPD [13] methods (small-scale) OVRP_ICA

Time 63.429 131.758 254.385 350.132 563.876 85.12 91.67 576.23

Fitness 409.68 557.14 629.74 716.73 902.45 411.87 587.76 634.52

Hybrid (1 + 1) ES [14] Time Fitness – 412.95 – 564.06 – 639.25 – 733.13 – 868.44 – 412.95 – 566.93 – 640.89

DHPD [13]

The input

Time – – – – – – – –

Kmin 5 10 8 12 16 5 10 8

Fitness 408.5 567.14 617.0 733.13 897.93 412.96 552.87 644.63

Q 160 140 200 200 200 160 140 200

Problem

N 50 75 100 150 199 50 75 100

C1 C2 C3 C4 C5 C6 C7 C8

Large-Scale Test Problems: The OVRP contains eight large-scale problems that have no limitations regarding travel distance. Therefore, solving them is difficult and timeconsuming. In this research, the vehicle routing problem with large-scale as solved by Li et al. [1] was considered. Eight problems with 200–480 customers and no limitations regarding route distance were selected. Each of the problems had a geometrical symmetry and the customers were on a circular orbit around the depot. Each problem showed a geometrical symmetry that made it possible to estimate an answer. Tables 3 and 4 list the obtained results, the time spent in OVRP_ICA runs and the values related to answers estimated by the other algorithms. As can be seen in the tables, standard data and much more complex problems were used to compare the three methods of solving the OVRP. Results obtained showed the proposed algorithm could find considerably better answers compared to the other algorithms. Table 4 shows that the proposed method provides better results compared with two other methods in all huge problems except O7 even with lower execution time. This happens because while we fix the maximum capacity to 900 and increase slightly the active vehicles and customers the colonies increases and produce some new regin/ colony and the ICA unable to find precise imperialist somehow and the final result increases. On the other hand, ICA feels it finds proper imperlaist sooner than expected and the run time decreases with faster rate.

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Table 3. Comparison among ORTR [1] and HES[8] methods with the proposed method in 8 problems wirh huge customers (large-scale problems). OVRP_ICA Time Fitness

ORTR [1] Time Fitness

HES [8] Time Fitness

The input Kmin Q

N

331.14 438.35 492.76 573.98 765.43 971.29 933.74 1125.21

365 439 492 573 766 977 935 1126

452 613 736 833 1365 1213 1547 1653

5 9 7 10 8 9 10 10

200 240 280 320 360 400 440 480

6016.98 4572.40 7730.95 7158.36 9112.67 9743.72 10484.49 12412.56

6018.52 4584.55 7732.85 7291.89 9197.61 9803.80 10374.97 12429.56

6018.52 4583.70 7733.77 7271.24 9254.15 9821.09 10363.40 12428.20

Problem

900 550 900 700 900 900 900 1000

O1 O2 O3 O4 O5 O6 O7 O8

Table 4. Comparison among refrences [15, 17] methods with the proposed method for huge customers. OVRP_ICA Time Fitness 331.14 6016.98 438.35 4572.40 492.76 7730.95 573.98 7158.36 765.43 9112.67 971.29 9743.72 933.74 10484.49 1125.21 12412.56

Reference [17] Time Fitness 2.51 6021.11 3.05 4557.38 3.17 7735.14 3.27 7267.18 3.42 9198.25 4.06 9798.19 4.21 10351.18 4.41 12418.57

BLSA [15] Time Fitness 612 6018.52 774 4557.38 681 7731.00 957 7253.20 1491 9193.15 1070 9793.72 1257 10347.70 1512 12415.36

The input Kmin Q 5 900 9 550 7 900 10 700 8 900 9 900 10 900 10 1000

Problem N 200 240 280 320 360 400 440 480

O1 O2 O3 O4 O5 O6 O7 O8

5 Conclusion In this research, a meta-heuristic algorithm called OVRP_ICA is presented for solving the OVRP. The advantages of this algorithm include its minimum traveling distance for the vehicles, the time needed for its execution, the low number of required evaluators for small-scale and large-scale problems, simultaneously. In detail, OVRP_ICA reduces the execution time, to find the shortest route between the customers and reduce the number of vehicles. Its efficiency was compared with those of eight different algorithms in small and large scale benchmark problems. This algorithm has a competitive structure compared to other meta-heuristic algorithms and could obtain reasonable and suitable answers for standard examples of this problem especially for small-scale problem with approximately 50 % lower traveling distances. It seems use of other meta-heuristic algorithms such as Tabu Search or ant colony optimization in combination with this algorithm can result in finding better answers for OVRP, especially in large and very large examples due to covering locally the search space. On the other hand, this algorithm can be used for other OVRP expansions such as the vehicle routing problem together with receiving and delivering goods. The use of such combinations of algorithms and work on these suggestions will be studied in future articles.

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Acknowledgement. This work was also supported in the framework of the IT4 Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 by operational programme ‘Research and Development for Innovations’ funded by the Structural Funds of the European Union and state budget of the Czech Republic, EU.

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