An ant colony optimization technique for solving min–max Multi-Depot Vehicle Routing Problem

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An Ant Colony Optimization Technique for Solving Min-Max Multi-Depot Vehicle Routing Problem Koushik Narasimha, Elad Kivelevitch, Balaji Sharma, Manish Kumar

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S2210-6502(13)00043-6 http://dx.doi.org/10.1016/j.swevo.2013.05.005 SWEVO82

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Swarm and Evolutionary Computation

Received date: 24 February 2012 Revised date: 23 April 2013 Accepted date: 6 May 2013 Cite this article as: Koushik Narasimha, Elad Kivelevitch, Balaji Sharma, Manish Kumar, An Ant Colony Optimization Technique for Solving Min-Max Multi-Depot Vehicle Routing Problem, Swarm and Evolutionary Computation, http: //dx.doi.org/10.1016/j.swevo.2013.05.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

An Ant Colony Optimization Technique for Solving Min-Max Multi-Depot Vehicle Routing Problem Koushik Narasimha*, Elad Kivelevitch**, Balaji Sharma*, and Manish Kumar***c *School of Dynamic Systems, University of Cincinnati, OH, USA **School of Aerospace Systems, University of Cincinnati, OH, USA *** Department of Mechanical, Industrial, and Manufacturing Engineering, The University of Toledo, OH, USA Tel*c: 419-530-8227 [email protected] Abstract: The Multi-Depot Vehicle Routing Problem (MDVRP) involves minimizing the total distance travelled by vehicles originating from multiple depots so that the vehicles together visit the specified customer locations (or cities) exactly once. This problem belongs to a class of Nondeterministic Polynomial Hard (NP- Hard) problems and has been used in literature as a benchmark for development of optimization schemes. This article deals with a variant of MDVRP, called min-max MDVRP, where the objective is to minimize the tour-length of the vehicle travelling the longest distance in MDVRP. Markedly different from the traditional MDVRP, minmax MDVRP is of specific significance for time-critical applications such as emergency response, where one wants to minimize the time taken to attend any customer. This article presents an extension of an existing antcolony technique for solving the Single Depot Vehicle Routing Problem (SDVRP) to solve the multiple depots and min-max variants of the problem. First, the article presents the algorithm that solves the min-max version of SDVRP. Then, the article extends the algorithm for min-max MDVRP using an equitable region partitioning approach aimed at assigning customer locations to depots so that MDVRP is reduced to multiple SDVRPs. The proposed method has been implemented in MATLAB for obtaining the solution for the min-max MDVRP with any number of vehicles and customer locations. A comparative study is carried out to evaluate the proposed algorithm’s performance with respect to a currently available Linear Programming (LP) based algorithm in literature in terms of the optimality of solution. Based on simulation studies and statistical evaluations, it has been demonstrated that the ant colony optimization technique proposed in this article leads to more optimal results as compared to the existing LP based method.

Keywords: Multi-Depot Vehicle Routing Problem, Ant Colony Optimization, Combinatorial Optimization Problems, Min-max Objective

1 Introduction Transportation of goods at various levels including within a city, region, nation, or around the globe is an essential part of modern supply chains. The efficient transportation of goods holds immense value due to its high impact on cost and customer satisfaction by reducing energy consumption and speedy delivery. In the last decade, research [1] suggested that 10% to 15% of the traded goods corresponded to the transportation costs. Also, U.S. Bureau of 1

Labor Statistics estimates that transportation-related fields are growing by nearly 56,000 jobs a year, thus showing an increase in trade and logistic businesses [2]. Realizing the importance of this factor, researchers have devoted a lot of effort in finding novel and optimal ways for efficient transportation. According to Toth and Vigo [1], utilization of computational tools for transportation route optimization has a potential to result in significant cost savings ranging from 5% - 20%. A well-known problem in this field, which has emerged as a benchmark optimization problem during the past few decades, is the Single Depot Vehicle Routing Problem (SDVRP It is an extension of the classical Travelling Salesman Problem [3] in which one vehicle originates from a common depot to visit a set of customer locations with the objective of minimizing the total tour length. The goal in the SDVRP, on the other hand, is to minimize the total distance travelled by all the vehicles while meeting customer demands and vehicle constraints, e.g., maximum travel distance or vehicle capacity. SDVRP, or simply, VRP as commonly referred, was first proposed by Dantzig in 1959 [4]. Since then, it has been studied extensively and serves as one of the benchmark problems in the field of optimization. Just like TSP, VRP is known to be a computationally Nondeterministic Polynomial Hard (NP Hard) Problem [5-6]. Multi-Depot Vehicle Routing Problem (MDVRP) extends the SDVRP by having multiple depots where multiple vehicles can originate from. MDVRP can be traced back to 1976 when Gillet and Johnson published a paper on Multi Terminal Vehicle-Dispatch Algorithm [7]. In this paper, a heuristic algorithm was developed to obtain an approximate solution. Their objective was to determine a set of vehicle routes that originate in two or more depots, visit the collection of demand points and return to the depots, such that the total distance travelled was minimized. They employed a sweep algorithm which was based on a strategy to break the problem to single-terminal problem in order to significantly reduce the computational time. The solution was also extended to satisfy some of the constraints such as the vehicle capacity and the length of each route. After this paper, much effort has been dedicated by researchers around the globe and many have come up with different methods to solve this problem [8-11]. In 2005, Lim and Wang [12] proposed a more practical variant of this problem and it was named MDVRP with Fixed Distribution of vehicles (MDVRPFD). They proposed this problem with bounds on the number of vehicles in a depot unlike the traditional MDVRP where the limit was unrealizable infinite number of vehicles. With an assumption of 2

exactly one vehicle in each depot, they developed a binary programming technique to obtain the solution and generalized the solution for any number of vehicles in a depot. More importantly, they proposed a new one-stage approach where the assignment of customers to the depots and the route calculations were carried out in a single stage. This paper focuses on an interesting variant of the MDVRP called the min-max multi depot vehicle routing problem (min-max MDVRP). The objective of this problem is to minimize the maximum distance travelled by any vehicle instead of the total distance travelled which is the case in the conventional MDVRP. Clearly, similar to the manner in which the min-max SDVRP is different from the traditional SDVRP [13], the min-max MDVRP is fundamentally different from the traditional MDVRP. In the min-max MDVRP, an optimal solution makes use of all available vehicles in an attempt to reduce the distance travelled by those vehicles with the largest tours, and this leads to more equitable sharing of loads between the vehicles. This problem is often of interest when minimization of time taken to visit all points is more important than the total distance travelled. The application includes emergency management situations where the objective is to use all available vehicles to minimize the time taken to attend to all points needing emergency resources. The optimization of vehicle routes for emergency management and relief efforts [14] has been a topic of much interest recently, and different versions of vehicle routing problems have been formulated in literature motivated from issues in emergency management including the min-max VRP [15], average cost VRP [15], and last mile distribution [16]. Other applications of this problem are in defense and computer networking. For example, assigning tours to a group of Unmanned Aerial Vehicles (UAVs) engaged in large scale surveillance operation by solving min-max problem will minimize the maximum time of travel of UAVs, and hence help achieve desired objectives in time-critical scenarios. In computer networking, depots represent servers, vehicles represent data packets, and customers represent clients. In this problem, a network routing topology generated by solving the min-max problem would result in minimizing the maximum latency between any pair of server and a client. One of the first attempts to solve the min-max class of problems was by Gold et al in [13] where they proposed a method based on Tabu search and adaptive memory heuristic. Using several test cases, they showed that their method provided good quality solutions within reasonable computational time. However, they considered just the single depot problem. The min-max problem for the multiple depots case was first formulated by Carlsson et al. in 2007 3

[17]. Carlsson’s work, being one of the most widely accepted works in literature, is used in this paper for comparison purposes. In their paper, Carlsson et al. [17] performed a theoretical analysis by developing an asymptotic bound for longest tour length L and concluded that the optimal solution to min-max MDVRP with uniformly distributed n points would numerically approach a value proportional to

n /k

, which is the value of optimal TSP

tour of all customers split by number of vehicles ‘k’, under the constraint. Additionally, they developed two different heuristics to solve the min-max MDVRP. The first heuristic was a load-balancing technique based on linear programming, while the second heuristic is the region partition based method. In the second technique, noticing that a convex equitable partitioning of the region yields an even division of points, they divided the service region into a set of sub-regions with equal area and generated good initial solutions by assigning the customer points in the depot region to the respective depot. Other recent works on min-max VRP include [15] that uses insertion heuristics, [18] that uses hybrid Genetic Algorithm and Tabu search heuristic, and [19] that uses branch-and-bound method. However, all of these works pertain to the single depot version of min-max VRP. For min-max MDVRP, Carlsson’s method remains to be the most widely accepted method in literature to the best of the authors’ knowledge and hence has been used in this paper as a benchmark method for comparison purposes. The swarm intelligence technique, called Ant Colony Optimization (ACO), based on the foraging strategies of ants, was first applied to TSP in [19-22]. The basic idea underlying this ant based algorithm is to use a positive feedback mechanism, based on an analogy with the pheromone-laying, pheromone -following behavior of some species of ants and some other social insects, and to reinforce those portions of good solutions that contribute to the quality of these solutions. The initial algorithm developed by Dorigo and Colorni [20], called ant system, suffered from problems including non-convergence and local minima. Subsequent versions of the algorithm [21-22] introduced several mechanisms such as modified transition rules that promoted directed random search, use of candidate list, new pheromone update rule that promoted exploration of solution space, and local random searches such as 1-opt and 2opt techniques. There are several features of ant based algorithms that make them ideal for the combinatorial optimization problems such as the one considered in this paper. The unique mechanism of

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laying pheromones provides a positive feedback that exploits the global knowledge by reinforcing the good solutions and directs the search to solutions of good potential. Furthermore, mechanisms, such as the transition rule, introduce stochastic components that allow the algorithm to explore the environment and escape local minima situations. Since the algorithm involves simple rules of each ant with decentralized control where each ant makes its own decision, this technique is computationally efficient and fairly easy to implement. More importantly, the agent-based framework allows parallelization or distribution of a lot of computations, and hence provides a scalable mechanism to solve NP Hard problems such as the TSP and MDVRP. Motivated by the above features, ant based algorithms have been applied to solve a number of combinatorial optimization problems including Job Scheduling Problem [23], Graph Coloring Problem [24], Quadratic Assignment Problem [25], School Bus Routing Problem [26], and SDVRP [27]. Of particular interest to the problem considered in this paper is the application of ant based algorithms to SDVRP. For example, in [27], a hybrid ant system algorithm was presented that utilized information such as savings and capacity utilization to obtain the solution. In 2004, Bell et al [28] developed multiple ant colony methods to solve the vehicle routing problem. This method uses separate specialized ant groups with unique pheromone depositions for each vehicle to solve the VRP. This separation is intended to differentiate paths typically used in the route of the first vehicle from those used by subsequent vehicles. This technique is found to be more useful when the size of the problem, i.e., the number of cities and/or the number of vehicles, is large. Also, their paper reinstates the route improvement strategies such as 2-opt heuristic and candidate list technique which were initially proposed by Bullnheimer et al [27]. In [29], Rizzoli et al. studied ACO application to variants of VRP such as Capacitated VRP, VRP with Time Windows, Time Dependent VRP and Dynamic VRP. They showed the application of ACO to industry size problems for a super market chain in Switzerland and Pick-up and Delivery problem for an industry in Italy. In [30], the authors provide a number of applications of the ACO algorithm for practical business problems including call routing in telecommunication network, minimizing delays in internet traffic, routing of fleet of oil trucks, factory efficiency, and business idea management. In [31], Reimann et al. proposed D-Ants algorithm that improved the ant systems algorithm for VRP via decomposition of the problem into several disjoint subproblems and showed its applications to variants such as VRP with Time Window, VRP with Backhauls, and VRP with Backhauls and Time Windows. Ant based algorithms [32] have 5

also been applied to an interesting variant of VRP called Capacitated Arc Routing Problem [33] (CARP) where the objective is to determine minimum cost routes of a vehicle that is required to service demands located along routes (edges of the networks). The ant based methods [32] have been demonstrated to perform extremely well as compared to the state-ofthe-art meta-heuristics for the CARP. This article, based on the M.Sc. Thesis work by the first author [34], proposes an ant colony based algorithm to solve the min-max MDVRP. To the best of the authors’ knowledge, this is the first application of ant colony optimization method to solve this important class of problem. The algorithm is inspired by the algorithm developed in [27] to solve the traditional SDVRP. This article extends the algorithm developed in [27] in two ways. Firstly, it argues that minimization of the “distance constraint” used in [27] will lead to the solution of the min-max variant of SDVRP, and provides an algorithm to carry out the minimization. Secondly, it uses region partitioning technique to extend the min-max SDVRP to solve the min-max MDVRP. The paper is organized as follows. First, a mathematical formulation of the min-max MDVRP is presented. Then, the proposed ant colony approach to solve the min-max SDVRP is presented. This is followed by the extension of that approach to solve the min-max MDVRP using region partitioning technique. Next, the results from simulation studies are presented and compared with respect to the best known methods in literature to solve this class of problems. The paper ends with some conclusions and suggestions for future work.

2 Problem Formulation The paper focuses on the min-max MDVRP in which the objective is to minimize the maximal length of a tour by any vehicle in a MDVRP. To mathematically formulate this problem [17], consider ‘n’ customer/delivery points, ‘m’ depot points and a total number of ‘k’ vehicles at the depots. All the vehicles are initially located at their respective depots. The vehicles are required to visit all customer points and return to the same depot from where they started their journey. The problem as stated earlier is to decide the tours of each vehicle so that the maximum distance travelled by any vehicle is minimized. Mathematically, the aim is to

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Minimize λ subject to TSP ( S ) ≤ λ , ∀i i ∪S = N i

(1)

where N is the set of all customer locations, |N| = n, Si (⊂ N ) is the subset of customers assigned to vehicle ‘i’ and TSP(Si) is the length of the optimal tour for a single Traveling Salesman Problem applied to vehicle i and the customer set Si. The approach used in this article is based on ant colony algorithm used by Bullnheimer et al. [27] to solve the traditional SDVRP. We first present our algorithm to solve the min-max SDVRP and then extend that to solve the min-max MDVRP.

3 Approach: Min-Max SDVRP The traditional SDVRP can be represented by a complete weighted digraph G = (V, A, d ) where V = {v0 , v1 ,… v n } is a set of vertices and A = {(vi , v j ) : i ≠ j} is a set of arcs. The vertex v0 denotes the depot, and the other vertices of V represent cities or customers. A cost variable, dij, is associated with each arc and may represent either distance or travel time between city i and city j. The aim of the SDVRP is to find routes for all vehicles so that every city is visited and the total sum of costs associated with all vehicles is the minimum. In the ant colony based method to solve the SDVRP, the artificial ants searching the solution space simulate real ants searching the environment. In this method, the objective corresponds to the proximity of food sources to the nest and an adaptive memory corresponds to the pheromone trails. To aid the search procedure through a set of possible solutions, the artificial ants are provided with local heuristic functions. Ant colony algorithms typically use some parameters that include heuristic desirability, pheromone updating rule, and probabilistic transition rule as given by Eqs. (2)-(5) below. These parameters are defined in a problem-specific manner, and are critical to solving the optimization problem under consideration. For the min-max SDVRP, the definitions of these parameters are similar to the TSP problem. The heuristic desirability of visiting city j after city i (or the visibility) is represented by ηij and is equal to the reciprocal of dij as shown in Eq. (2):

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η ij =

1 d ij

(2)

Another heuristic desirability function [27] that can also be implemented to solve this problem is given by Eq. (3):

η ij = d i 0 + d 0 j − g ⋅ d ij + f ⋅ d i 0 − d 0 j

(3)

where di0 is the distance between the single depot and city i. g and f are optimization parameters. The probabilistic transition rule is indicated by pij which represents the probability of choosing to move to city j from city i, and is given by Eq. (4) ⎧⎪ [τ ij ]α [η ij ] β p ij = ⎨ if v j ∈ Ω α β ⎪⎩ ∑ h∈Ω [τ ih ] [η ih ] else p ij = 0

(4)

Here, τ ij is the pheromone concentration on the path from i to j, Ω ⊆ (V \ v0 ) is the set of feasible cities, i.e., cities that have not been visited, α and β represent the biases for pheromone trail and visibility respectively, and they represent parameters to weigh pheromone concentration (which represents learnt global knowledge) with respect to heuristic desirability (which represents local knowledge) in the transition rule. The pheromone update rule is given by Eq. (5): m

τ ijnew = ρτ ijold + ∑ Δτ ijk + σΔτ ij* k =1

(5)

The details of individual terms of Eq. (5) are explained in the algorithm in Table 1. At the beginning of the algorithm, a distance constraint L is defined that represents the maximum distance any vehicle is allowed to travel. The artificial ants construct vehicle routes by successively choosing the next cities using the probabilistic transition rule shown in Eq. (4). During every transition, the tour length is calculated and whenever the choice of next city would lead to an infeasible solution, i.e., if the tour length exceeds the vehicle distance constraint (the maximum distance that a vehicle can travel) L, then the depot is chosen as the next city (so that the tour of that vehicle is finished) and a new tour is started for a new vehicle. The vehicle capacity L is a critical parameter in finding out the min-max solution to

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the proposed problem. This parameter, as proven by Carlsson et al. in [17], is bounded between two values as per the following lemma: Lemma 1: For a general planar graph representing the VRP,

TSP( N ) TSP( D ∪ N ) − TSP( D) ≤L≤ + 2 ∗ d ( D, N ) k k

(6)

In Eq.(6), N denotes the set of customer points, D denotes set of depots, k denotes number of vehicles in a depot, d(A,B) denotes the largest distance between an arbitrary pair of points in two different sets A and B as shown in Eq.(7), i.e.,

d ( A, B) =

|| x − y || max x ∈ A, y ∈ B

(7)

A proof of the above lemma is provided in reference [17]. Since L represents the distance constraint, it is the maximum distance travelled by a vehicle. The approach here is based upon finding the least value of L, called L*, that will still lead to a valid solution of the traditional SDVRP. This solution of traditional SDVRP will also be the solution of the min-max SDVRP. Thus this constraint on L acts as a critical parameter in finding out the min-max solution. A high-level description of the algorithm used to solve the min-max SDVRP is shown in Table 1. It may be mentioned that the text between /* and */ represent the comments explaining the steps.

Table 1: SDVRP

(V , v0 ) : Ant Colony Optimization Algorithm to Solve the Min-Max SDVRP

Ant Colony Optimization Algorithm to Solve the Min-Max SDVRP /*Initialization*/

1. V = {v1 ,… v n } is the set of n cities, v0 is the depot TSP (V ) 2. L_max = + 2 ∗ d (v 0 , V ) k TSP (V ∪ v 0 ) 3. L_min = k 4. While (L>L_min) a. For every edge (i,j) do

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τij(0) = τ0 End For

/*Main Loop*/ b. For t = 1 to t_max do Initialize Length_tour and tour Tk For k = 1 to m do

Build tour Tk (t) by applying n - 1 times the following step [τ ij ]α [ηij ]β pij = if v j ∈ Ω Choose the next city j with probability, ∑h∈Ω [τ ih ]α [ηih ]β

else

pij = 0

Calculate length of the tour Length_tour If Length_tour + d0(j) > L

Choose depot as the next city Calculate the Length_tour accordingly End If End For

Find minimum of Length _tour and tour T If an improved tour is found then

Update it as T_opt and L_opt End If

Count the number of vehic, no_of_vehic based on number of times the ant has visited depot in the tour T_opt and also find tour of each vehicle, L_opt_vehic Using 2-opt Heuristic, if possible Obtain a better T_opt and L_opt For every edge (i,j) do m Update pheromone trails by applying the rule: τ new = ρτ old + ∑ Δτ k + σΔτ * ij ij ij ij k =1 Q ⎧ ⎧ Q if (i, j ) ∈ T _ opt if (i, j ) ∈ T ⎪ ⎪ k ∗ Where Δτ ij = ⎨ Length _ tour and Δτ ij = ⎨ L _ opt ⎪⎩ ⎪⎩ 0 otherwise 0 otherwise

End For

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End For

c. If no_of_vehic ≤ k, where k is the number of vehicles in the depot L = L – δL and go to ‘b’ until no_of_vehic = k+1 and the respective L is L* Else L = L + δL End If End While

5. Print T_opt, L_opt, L* 6 .Stop

4 Approach: Min-max MDVRP The approach used in this paper to solve the min-max MDVRP is to partition the problem into multiple SDVRPs and use the method proposed in Section 3 to solve those SDVRPs. The decomposition is carried out by dividing the region into as many numbers of equitable convex partitions as the number of depots. Once equitable partitions are found, solving minmax MDVRP reduces to solving min-max SDVRP for each partitioned region. However, it is necessary to achieve the partitioning in a way such that solving the min-max SDVRP for partitions would correspond to solving the complete min-max MDVRP. In order to show that, we first present below a result proven by Beardwood et al. in [35] as a lemma which can be used for arguing that, under certain assumptions, partitioning the polygon (convex polygon hull of the entire domain of customer points) equally would lead to an optimal partitioning for our problem.

Lemma 2: If Xi , 1≤ i ≤ ∞ are independently and identically distributed (i.i.d) random variables with bounded support in d-dimensional domain Rd, then the length Ln, under the usual Euclidean metric, representing the shortest path (or the TSP distance) through the points {X1, X2,….Xn} satisfies ( d −1) ⎛ ⎞ Ln ⎜ ⎟ d lim dx ( d −1) → β TSPd ∫ f ( x) ⎟ n→∞⎜ d d R ⎝ n ⎠

(8)

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In Eq. (8), f(x) is the probability density function representing the distribution of points Xi, and βTSP,d is a positive constant that depends on d but not on the distribution of the Xi. Based on the above lemma, we can deduce a proof for our 2-dimensional problem [36-39]. −1 Assuming the cities are i.i.d. with uniform distribution, i.e., f ( x ) = ( A) , where A = ∫ dx is

R2

the area A of the region, it is apparent with a few substitutions (d=2, n = no. of cities) in Eq. (8), that route length of each vehicle is asymptotically (i.e., when n is infinitely large) proportional to the square root of the area of the region, i.e., 1

Ln n

1 2

→ βTSPd A2

(9)

Thus establishing an asymptotic relation between tour length and area, it is appropriate to divide the region into partitions of equal areas since the lengths of optimal tours in each partition would approximately be the same. This holds importance for the min-max problem under consideration here because, for example, if an area of subregion (A1) is larger than subregion (A2), then route length (L1) of that area A1 becomes longer and hence the time taken to cater to that area increases. Hence, the most optimal heuristic is to divide the area into equal partitions. It may be noted that the above heuristic is optimal only when cities are large in number, cities and depots are i.i.d. with uniform distribution in the domain, and the number of vehicles in each depot is approximately the same. More discussion on this aspect is provided in the later sections. 4.1 Region Partitioning

Region partitioning is an important aspect of the solution because it is used to decompose the min-max MDVRP into min-max SDVRPs. There are a number of region partitioning techniques in literature [40] including: a) Triangulation method; b) Hertel-Mehlorn (HM algorithm); c) Chazelle’s complex cubic algorithm; d) Centroidal Voronoi Tessellation. The objective of the region partitioning algorithm for the problem in consideration is the determination of sub-regions that are equal in area. Although some of the above methods are relevant, the most relevant method to solve the problem considered here is Carlsson’s region

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partitioning method to find equitable convex partitions in a polygon [36]. The algorithm is based on generalized Ham Sandwich Theorem (HST) [37-39] which, for a planar case of finite sets of red and blue points, states that there exists a line that divides both red and blue points into sets of equal size. The technique used by the Carlsson in [36] assumes that the customer points in the 2-d plane are uniformly distributed. This method uses an approximation algorithm to determine the partitions by performing binary searches over the given set of depot points and customer points. Details of this method can be found in [17, 36]. Based on this method any given polygon can be partitioned so that a) All the partitions are convex polygons. b) All the partitions contain exactly one depot c) All the partitions have equal area A sample figure representing a partitioned region that was obtained using the above method is shown in Fig. 1.

Figure 1: A convex equitable partition for a case of 10 depots

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4.2 Min-Max MDVRP Algorithm

Explained in Table 2 is the proposed algorithm to obtain an approximate solution to the minmax MDVRP. Note that the subroutine SDVRP (V , v0 ) is the subroutine representing the min-max SDVRP solution shown in Table 1.

Table 2: Ant Colony Optimization Algorithm to Solve the Min-Max MDVRP

Ant Colony Optimization Algorithm to Solve the Min-Max MDVRP /*Inputs*/

1. cust_points = get_custpoints() /*set of x,y coordinates of the cities */ 2. depots = get_depotpoints() /*set of x,y coordinates of the depots*/ /*Generate convex hull*/

3. poly_points = [cust_points; depots] /*set of combination of depot and cust_points*/ 4. poly_vertices = convhull(poly_points) /*get the vertices of the convex hull using convhull function*/ /*Region Partitioning using Carlsson algorithm*/

5. subregions = region_partition(poly_vertices,depots) /*get the vertices of the partitioned polygon and also get the depot points corresponding to these set of subregion vertices*/ /*Main Loop*/

no_of_depots = size(depots) 6. For i = 1 to no_of_depots do IN = inpolygon() /*assign the cust_points that are in partition i to depot i, basically check the points if it inside or outside the polygon which is in focus*/

V = cust_points(IN) /*V is the set of customer points that are inside region ‘i’*/

v0 = depots(i) /*depot(i) corresponds to the x,y coordinate of the ith depot*/ /*Generate the min-max SDVRP tour and calculate optimum tour length and optimum vehicle tour for each vehicle*/

[opt_vehic_tour opt_tour_length] = SDVRP (V , v0 ) End For

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/*Plot the Vehicle Tours*/

7. For i = 1 to no_of_depots do For k=1 to no_of_vehic do

plot() /*Plot the vehicle tour*/ End For End For

4.3 Tour Improvement Strategies 4.3.1 Candidate List

Bullnheimer et al. [27] suggested using a candidate list. In their technique, a list of cities in the increasing order of distance from the current city will be assigned to each city. Only the closest cities will be included in the candidate list and are made available for selection of the next city to be visited in the route.

Figure 2: Candidate list technique: Ant in City ‘A’ can choose City ‘C’ which is in City ‘A’s candidate list, while ‘NC’ is not in candidate list

An example is provided in Fig. 2, where the candidate cities are marked with red circles for a current city A. Hence an ant in the city ‘A’ can’t choose city ‘NC’ which is far away from the

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city ‘A’ unlike u city ‘C C’. This is an a intuitive assumption a a it is not likkely that an optimal as o solution will w consist of routes con nnecting citiies that are farther f away from each other o while ignoring cities that arre closer. Ussually, the caandidate list size will be set to [n/4] where ‘n’ is the total number n of customers. Thhis applicatiion of candiddate list routte improvem ment techniquue has two advantages. a F First, the callculation tim me is reducedd because thee probability y calculationn is necessary only for much reducced number oof cities. Seccond, the sollution qualityy is improveed because selection s proobabilities arre no longer underminedd by highly uunlikely soluutions. In [28], Belll et al. experrimented witth different ccandidate lisst sizes and sstudied the im mpact of thee selection n size with reespect to the problem sizze.

4.3.2 r-o opt heuristic c This tourr improvemeent strategy was w used by Lin et al. in n [41]. They implementedd a local exchangee procedure after a locallly optimum tour was fou und. This tecchnique is coommonly called r-o opt heuristic where ‘r’ caan be 2, 3 orr more. In thiis paper, 2-oopt heuristic is used which ev valuates all possible p pairw wise exchannges in orderr to improve the solution n.

Figure 3: Vehicle V routes in a typical SD DVRP solution and sequentiall local exchangge procedure

For exam mple, consideer Fig. 3 andd say for 1st vvehicle the optimal o tour obtained usiing ant colony op ptimization technique t is D-1-2-3-4-55-D. The 2-oopt heuristic technique performs p a

1 16

sequential exchange of the cities and calculate the distances for all the pairwise permutations like for the set D-1-2-3, distances for D-1-3-2, D-2-3-1, D-2-1-3, D-3-1-2 and D-3-2-1 are calculated and if an overall improvement of tour length is obtained, the respective tour is used as an optimal tour. This adds greatly to the number of combinations that are explored by ant colony search technique and can be used in finding the best solution before applying the pheromone update rule.

5 Results and Discussions A MATLAB based program was developed to implement the proposed solution for min-max variants of SDVRP and MDVRP. The proposed algorithm was verified using a number of simulated scenarios and compared to the method proposed by Carlsson et al [17], which is considered to be the most widely accepted method in literature to solve the min-max MDVRP. A MATLAB program that implements the Carlsson’s Linear Programming based algorithm is freely available over the internet [42]. Going forward, this Carlsson’s Linear Programming based method is acronymically mentioned as LP based method. The simulations were performed on Dell PC with Intel Core i7 processor having 3.47 GHz clockspeed and 16GB RAM. Throughout the simulations, the values of various ant colony optimization parameters were assumed to be: α = 1, β = 5, ρ = 0.25, Q = 100, τ0 =10-6, σ = 5. 5.1 Min-Max SDVRP Results

To demonstrate the performance of our algorithm, we present results from five simulated scenarios: i) Scenario 1 with 25 cities and 3 vehicles (25x3); ii) Scenario 2 with 25 cities and 4 vehicles (25x4); iii) Scenario 3 with 25 cities and 5 vehicles (25x5); (iv) Scenario 4 with 30 cities and 4 vehicles (30x4); and (v) Scenario 5 with 65 cities and 6 vehicles (65x6). Each of the scenarios is obtained by distributing cities randomly via uniform distribution on space of size 100 by 100, and depot located randomly on the same space. For each of the scenario, we obtained solutions via 25 independent runs using the proposed ant colony based method as well as LP based method. It may be noted that the city and depot layouts remain unchanged 17

for each scenario for different runs. Tables 3 and 4 provide the detailed statistical analysis of the results obtained using the LP based method and the proposed ant colony method, respectively. Table 3: Summary of Solutions Obtained by Carlsson’s LP Based Method

Scenario

No. of Cities x

Best

Worst

Mean

Standard

Average

Number

No. of Vehicles

solution

Solution

Solution

Deviation of

Computational

Solution

Time (Sec)

1

25x3

255.7926

283.1943

263.3036

5.8583

0.6457

2

25x4

194.8816

267.9915

224.4329

13.9709

0.6396

3

25x5

190.6740

247.1371

195.0388

11.9106

0.8205

4

30x4

238.5980

281.0305

248.8421

8.9604

0.6418

5

65x6

211.3289

272.3252

232.1992

16.4602

0.9520

Table 4: Summary of Solution Obtained by the Proposed Ant Colony Based Method

Scenario

No. of Cities x

Best

Worst

Mean

Standard

Average

Number

No. of Vehicles

solution

Solution

Solution

Deviation of

Computational

Solution

Time (Sec)

1

25x3

224.5475

236.8301

227.5559

4.4003

17.9324

2

25x4

194.8816

196.8277

195.2202

0.4528

18.5404

3

25x5

186.1605

188.5581

186.6036

0.9104

18.2887

4

30x4

205.1138

214.7503

213.1760

2.6504

35.7768

5

65x6

213.6715

219.8546

217.5595

1.5731

328.7139

Figs. 4 show an instance of typical solutions obtained for Scenario 5 using both methods. Table 5 shows the respective distances travelled by the vehicles for the solutions obtained using both methods. The ant colony solution corresponds to distance travelled by vehicle 4 18

(219.69) while solution obtained by the LP based method corresponds to the distance travelled by vehicle 6 (238.58). SDVRP 100

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

Depot

0 0

20

40

60

80

100

LP Based Method

0 0

20

40

60

80

100

Ant Colony Based Method

Figure 4: Comparison of results obtained from the ant colony based and LP Based techniques for Scenario 5 (6 Vehicles – 65 Cities)

Table 5: Comparison of Vehicle Distances Obtained Using the Ant Colony Based and LP Based techniques for Scenario 5

Distance travelled by each vehicle (Ant Colony Based)

Distance travelled by each vehicle (LP Based)

194.80

216.91

162.63

206.19

195.06

183.47

219.69*

183.17

219.17

214.79

219.63

238.58*

* denotes the maximum distance travelled in the corresponding sets of vehicle routes obtained by both methods

5.2 Min-Max MDVRP Results

Before discussing the results for the min-max MDVRP case, let us capture some of the assumptions that are made: a) cities or customer points and depot points are uniformly distributed over 2-d space; b) the vehicle capacities remain the same throughout all depots. This paper uses six scenarios to present the results: i) Scenario 6 (3x80x6): 3 depots, 80

19

cities, and 6 vehicle (2 vehicles per depot); ii) Scenario 7 (3x140x9) : 3 depots, 140 cities, 9 vehicles (3 vehicles per depot) ; iii) Scenario 8 (4x80x8): 4 depots, 80 cities, 8 vehicles (2 vehicles per depot); iv) Scenario 9 (4x140x12): 4 depots, 140 cities, 12 vehicles (3 vehicles per depot); v) Scenario 10 (5x140x13): 5 depots, 140 cities, 12 vehicles (depots 1-3 have 2 vehicles each, and 4-5 have 3 vehicles each); and vi) Scenario 11 (5x140x15): 5 depots, 140 cities, 15 vehicles (3 vehicles per depot). For each scenario, we obtain solutions via 25 independent runs using the proposed ant colony based method as well as LP based method. It may be noted that the city and depot layouts remain unchanged for each scenario for different runs. Tables 6 and 7 provide the detailed statistical analysis of the results obtained using the LP based method and the proposed ant colony method, respectively.

Table 6: Summary of Solution Obtained by the Carlsson’s LP Based Method

Scenario

No. of Depot x

Best

Worst

Mean

Standard

Average

Number

No. of Cities x

solution

Solution

Solution

Deviation of

Computational

Solution

Time (Sec)

No. of Vehicles 6

3x80x6

193.7000

264.5546

224.8572

23.3321

1.7836

7

3x140x9

172.6441

202.5625

190.5611

7.3586

2.0648

8

4x80x8

207.6464

260.8396

213.9139

11.0426

1.6202

9

4x140x12

149.4203

223.2752

181.6720

20.6708

2.4278

10

5x140x12

129.0139

176.6352

150.7001

12.6358

2.5308

11

5x140x15

129.7286

200.5404

167.2335

20.7763

5.5811

20

Table 7: Summary of Solution Obtained by the Ant Colony Based Method

Scenario

No. of Depot x

Best

Worst

Mean

Standard

Average

Number

No. of Cities x

solution

Solution

Solution

Deviation of

Computational

Solution

Time (Sec)

No. of Vehicles 6

3x80x6

186.5168

199.7390

190.9416

4.2773

241.4765

7

3x140x9

184.5539

190.4588

188.4617

1.4237

602.4891

8

4x80x8

205.1906

205.1906

205.1906

0.0000

225.7526

9

4x140x12

146.8195

150.9697

149.8148

1.1232

484.5277

10

5x140x12

125.0439

125.8830

125.4495

0.2485

432.5970

11

5x140x15

110.2564

200.5404

112.0314

0.5435

443.8635

Figs. 5 show an instance of typical solutions obtained for Scenario 10 using both the methods. Table 8 shows the respective distances travelled by the vehicles for the solutions obtained using both the methods. The ant colony solution corresponds to the distance travelled by vehicle 1 of depot 5 (125.32) while solution obtained by the LP based method corresponds to the distance travelled by vehicle 2 of depot 5 (166.14). MDVRP 100

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

Depot Depot Depot

0 0

20

40

60

80

100

0 0

Depot

20

(A)

40

Depot

60

80

100

(B)

Figure 5: For the case of Scenario 10 (140 cities, 5 depots, and 12 vehicles): (A) Results obtained using the LP based approach (B) Results obtained using the ant colony based approach

21

Table 8: Comparison of the Ant Colony Based and LP Based Techniques to Solve Min-Max MDVRP for Scenario 10

Depot No.

Vehic No.

1 2 3

Ant Colony based method Distance Travelled by Each Vehicle

LP based method Distance Travelled by Each Vehicle

Vehic No.

1

114.17

1

103.41

2

124.30

2

106.21

1

113.97

1

119.79

2

110.15

2

148.57

1

46.04

1

121.10

2

116.90

2

105.07

1

114.80

1

164.27

2

111.89

2

126.71

3

83.69

3

129.92

1

125.32*

1

141.94

2

106.71

2

166.14*

3 83.92 * denotes the maximum distance travelled in a set of vehicle routes

3

86.63

4

5

Finally, we carried out an extensive Monte Carlo simulation study using 300 runs where we used Scenario 12 (5x140x16) that has 5 depots, 140 cities, and 16 vehicles. However, in this study, for each Monte-Carlo simulation, the locations of the cities and depots were randomly generated using a uniform distribution in the space of size 100 by 100. The mean solution obtained from LP based method was 186.1105, and that obtained from ant colony based method was 173.0535.

22

Normalized ACO cost / LP cost vs. run 1

ACO cost / LP cost

0.95

0.9

0.85

0.8

0.75

0

50

100

150 Run #

200

250

300

Figure 6: Monte-Carlo comparison of maximum tour lengths (min-max cost) of vehicle for 5 depots-140 cities16 vehicles problem: The Y-axis represent the ratio of cost obtained by ant colony (ACO) method to the cost obtained by LP based method

The plot summarizing the results obtained by each method obtained in each run is shown in Fig. 6. As depicted in the figure, the ant colony cost is less than the cost obtained by the LP solution, for every run. The ratio of costs ranges from 76.1% to 96.6%, with a mean of 92.9% and a standard deviation of 1.95%. In order to carry out a more thorough statistical characterization of the performance of the proposed ant colony method as compared to the Carlsson’s LP method, based on the experiments carried out as reported in Tables 3-4, 6-7, two pairwise comparison tests, the Sign test and the Wilcoxon signed rank test, were carried out as suggested in [43]. The results obtained from these tests are presented in Tables 9 and 10. It may be noted that Scenarios 1-5 correspond to the SDVRP and Scenarios 6-11 correspond to the MDVRP. In Table 9, number of wins for each scenario corresponds to the number of times the ant colony based method outperforms the LP based method. In Table 10, R+ is the sum of ranks for the 23

cases in which the ant colony based method outperforms the LP based method, and R- is the sum of ranks for the opposite. More details about the calculations of metrics reported in Tables 9 and 10 can be found in [43]. It may be noted that the results are based on a level of significance, α = 0.1. The sole case of tie (of single run in Scenario 2) is ignored, as suggested in [43]. As can be seen from the last columns of these two tables that provide the final outcomes of these tests for different scenarios, the results overwhelmingly suggest the superior performance of the proposed ant based method as compared to that of Carlsson’s LP based method in terms of the optimality of the results obtained. It may be noted that there is one negative result for Scenario 7 where the proposed ant based method’s performance cannot be claimed to be better than the LP method based on level of significance α = 0.1. However, it may also be noted that the proposed method does outperform the LP based method by winning 16 times and losing only 9 times. Overall, out of 275 total runs, the proposed method wins 257 times, loses 17 times, and ties 1 time. Table 9. Sign Test for Pairwise Performance Comparisons Between the Ant Colony Based Method (ACO) and LP Based Method Scenario

Wins

Losses

Ties

Critical Value (α = 0.1)

ACO better than LP?

1

25

0

0

17

Yes

2

24

0

1

16

Yes

3

25

0

0

17

Yes

4

25

0

0

17

Yes

5

17

8

0

17

Yes

6

25

0

0

17

Yes

7

16

9

0

17

No

8

25

0

0

17

Yes

9

25

0

0

17

Yes

10

25

0

0

17

Yes

11

25

0

0

17

Yes

24

Table 10. The Wilcoxon Signed Ranks Test for Pairwise Performance Comparisons Between the Ant Colony Based Method (ACO) and LP Based Method

Scenario

R+

R-

p-value

ACO better than LP?

1

325

0

2.9802 E-08

Yes

2

300

0

5.9605 E-08

Yes

3

325

0

2.9802 E-08

Yes

4

325

0

2.9802 E-08

Yes

5

279

46

5.1355 E-04

Yes

6

325

0

2.9802 E-08

Yes

7

222

103

5.6746 E-02

Yes

8

325

0

2.9802 E-08

Yes

9

325

0

2.9802 E-08

Yes

10

325

0

2.9802 E-08

Yes

11

325

0

2.9802 E-08

Yes

Some of the observations that can be made from the results obtained for both min-max SDVRP and MDVRP cases are as follows. First, the ant colony based method provides more optimal solutions on average as compared to the LP based method for both min-max SDVRP and MDVRP cases. This can be seen from the results presented in Tables 3-4, 6-7, 9-10 and Fig. 6 which demonstrate that the proposed ant colony based method performs better than the LP based method in terms of i) the mean value of the solution for every scenario, ii) best and worst case solutions for almost all scenarios, and iii) the two sign tests for pairwise statistical comparisons for almost all scenarios. It may be noted that, for min-max MDVRP, the optimality of the solutions obtained by the proposed ant based method becomes better than the LP based method as the scenarios become larger (i.e., the number of cities increases). This can be explained by the fact that, for the ant based method, we use region portioning method that assumes uniform distribution of cities. This assumption becomes more and more valid as the number of cities increase. Second, the solutions obtained by the proposed ant colony approach are more consistent than those obtained by the LP-based method. This is evident from the lower values of the 25

standard deviation in solutions obtained for different scenarios from the ant colony based approach. A lower value of standard deviation for optimization algorithms based on stochastic components is a desirable property which demonstrates that the solutions are more consistent due to the convergence of the solutions to a close neighborhood of each other. Finally, it may be noted that the computational time required for the proposed method is more than that of the LP based method as can be seen from Tables 3-4, 6-7. This does show that Carlsson’s approach is computationally efficient. This is partly due to the utilization of the well-optimized Concorde TSP solver [44] by the Carlsson’s approach which can get to nearexact TSP solution in a very computationally efficient manner. However, it is to be noted that the major advantage of the proposed ant colony based method is that the method lends an inherent mechanism for distributed computation, which enables exploitation of parallel processing to improve computation time. Essentially, computations carried out by each ant/agent in the algorithm can be done pretty much independently with minimal communication with centralized unit that stores the global information. Furthermore, execution of each SDVRP after the partitioning on a separate CPU would decrease the computation time by a factor that is close to the number of depots, and would close much of the gap in computation time when compared with Carlsson’s LP based approach. Apart from the above points, there are some more observations to be noted. For example, getting a guaranteed optimal solution to the min-max MDVRP is known to be NP hard. However, approximate methods, such as proposed here, can reduce the computational burden and are scalable to larger problems. It may be noted that the proposed approach makes an assumption regarding the uniform distribution of cities, depots and vehicles in a depot. In practice, the cities may not be distributed uniformly or may be very few in number. In these cases, the quality of the results obtained from this approach may be lower. However, these sub-optimal solutions can be used with other optimization methods to obtain better solutions. For example, the methods such as 2-opt or 3-opt can be extended to exchange cities between tours after this initial partitioning is done. In fact, a more detailed and comparative study on the use of local search algorithms to augment the proposed ant based method would be very interesting future work. There are several local search algorithms in literature [45-46] 26

including those based on iterative improvement, simulated annealing, genetic algorithms, tabu search, large neighborhood search, and variable neighborhood search. These local search algorithms have the potential to not only reduce the computational time requirements of the proposed method, but also improve the optimality of results. Also, although this paper carries out comparison of the proposed method with respect to the Carlsson’s method which is the most widely accepted method in literature for solving the min-max class of VRP, it will be an interesting future work to carry out comparisons of the proposed method with respect to other heuristic methods available in literature such as those reported in references [15], [18], and [19].

6 Conclusions The paper presents an approximate solution to the min-max MDVRP based on an ant colony optimization. Unlike the traditional MDVRP, which minimizes the total distance travelled, the min-max MDVRP minimizes the maximal distance travelled by a vehicle. This variant of the problem holds special relevance for time-critical problems. The approach is based on decomposing the overall problem into several min-max SDVRPs via equitable partitioning of the region consisting of depots and cities. The min-max SDVRP is then solved using the ant colony method that finds out the minimum value of distance constraint that yields a solution to the traditional SDVRP. This optimal distance constraint, when used in a traditional SDVRP, minimizes the maximal distance travelled by a vehicle. Simulation studies and statistical tests validate the effectiveness of the proposed method and demonstrate that the proposed approach provides improvements in terms of optimality of the solution as compared to one of the best known methods in literature. Furthermore, the proposed ant colony based method provides much consistent solution over different runs showing convergence of the solution to a small neighborhood. Future work includes developing region-partitioning techniques for cases where we have non-uniform distribution or smaller number of cities. Further attempt should be made to implement the inherently distributed ant colony based method in a parallel computing framework, thus fully utilizing its potential. Additionally, an evaluation of the performance of various local search algorithms to complement the proposed ant based method for the minmax MDVRP would be an interesting direction for future work. 27

References [1] Toth,P., Vigo, D., "The Vehicle Routing Problem", Monographs on Discrete Mathematics and Applications, SIAM, Philadelphia, 2001. [2] US Department of Labor, “Occupational Outlook Handbook”, Bureau of Labor Statistics, 2010-11. [3] Lawler, E. L., Lenstra, J. K., Rinnooy-Kan, A. H. G., and Shmoys, D. B., eds., The Travelling Salesman Problem, Wiley & Sons, New York, NY, 1985 [4] Dantzig, G.B., Ramser J.H., “The truck dispatching problem”, Management Science, Vol. 6, No. 1, pp. 8091, 1959. [5] Karp, R.M., “Reducibility among combinatorial problems”, Complexity of Computer Computations, Advances in Computing research, pp. 85-103, 1972. [6] Garey, M. R. and Johnson, D. S., Computers ans Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman & Co., San Francisco, CA, 1979. [7] Gillett, B.E., Johnson, J.G., “Multi-Terminal Vehicle-Dispatch Algorithm”, Omega, Vol.4, pp. 771-718, 1976. [8] Tillman, F., “The multiple terminal delivery problem with probabilistic demands”, Transportation Science, Vol. 3, pp. 192-204, 1969. [9] Chao, I.M., Golden, B., Wasil, E., “A new heuristic for the multidepot vehicle routing problem that improves upon best-known solutions”, American Journal of Mathematical and Management Sciences, Vol. 13, No. 3, pp. 371-406, 1993. [10] Laporte, G., Nobert, Y., Taillefer, S., “Solving a family of multi-depot vehicle routing and location-routing problems”, Transportation Science, Vol. 22, No. 3, pp. 161-172, 1988. [11] Cordeau, J.F., Gendreau, M., Laporte, G., “A Tabu search heuristic for periodic and multi-depot vehicle routing problems”, Networks, No. 30, pp. 105-119, 1997. [12] Lim, A., Wang., “Multiple-Depot Vehicle Routing Problem: A One-Stage Approach”, IEEE Transactions on Automation Science and Engineering, Vol. 2, No. 4, pp. 397- 402, 2005. [13] Golden, B. L., Laporte, G., Taillard, E. D., “An adaptive memory heuristic for a class of Vehicle Routing Problems with minmax objective”, Computers & Operations Research, Vol. 24, No. 5, pp. 445-452, 1997. [14] De Angelis, V., Mecoli, M., Nikoi, C. and Storchi, G., “Multiperiod integrated routing and scheduling of World Food Programme cargo planes in Angola”, Computers & Operations Research, Vol. 34, pp. 16011615., 2007 [15] Campbell, A., Vandenbussche, D., and Hermann, W., 2008, “Routing for relief efforts”, Transportation Science, Vol. 42, No. 2, pp. 127-145, 2008. [16] Balcik, B., Beamon, B. ,and Smilowitz, K., “Last mile distribution in humanitarian relief”, Journal of Intelligent Transportation Systems, Vol. 12, No. 2, pp. 51-63, 2008. [17] Carlsson, J., Ge, D., Subramaniam, A., Wu, A., Ye, Y., “Solving Min-Max Multi-Depot Vehicle Routing Problem”, Lectures on Global Optimization, American Mathematical Society, pp. 31-46, 2007. [18] Ren, C., “Solving min-max Vehicle Routing Problem”, Journal of Software, Vol. 6, No. 9, pp. 1851-1856, 2011.

28

[19] Applegate D., Cook W., Dash S., “Solution of a min-max vehicle routing problem,” Informs Journal on Computing, Vol. 14, No. 2, pp.132-143, 2002. [20] Dorigo, M., Colorni, A., “The Ant System: Optimization by a colony of cooperating agents”, IEEE Transactions on Systems, Man and Cybernetics –Part B, Vol. 26, No. 1, pp. 1-13, 1996. [21] Dorigo, M., Gambardella, L.M., “Ant Colonies for the Travelling Salesman Problem”, BioSystems, Vol. 43, pp. 73-81, 1997. [22] Dorigo, M., Gambardella, L.M., “Ant Colony System: A Cooperative Learning Approach to the Travelling Salesman Problem”, IEEE Transactions on Evolutionary Computation, Vol.1, No.1, pp. 53-66, 1997. [23] Colorni, A., Dorigo, M., Maniezzo, V., and Trubian, M., “Ant System for Job Shop Scheduling”, Belgian Journal of Operations Research, Statistics, and Computer Science, Vol. 34, No. 1, pp. 39, 1994. [24] Costa, D. and Hertz, A., “Ants Can Colour Graphs”, Journal of Operations Research Society, Vol. 48, No. 3, pp. 295-305, 1997. [25] Maniezzo, V. and Colorni, A.; , "The Ant System Applied to the Quadratic Assignment Problem," IEEE Transactions on Knowledge and Data Engineering, Vol.11, No.5, pp.769-778, 1999. [26] Euchi, J., and Mraihi, R., “The Urban Bus Routing Problem in the Tunisian Case by the Hybrid Ant Colony Algorithm”, Swarm and Evolutionary Computation, Vol. 2, pp. 15 – 24, 2012. [27] Bullnheimer, B., Hartl, R.F., Strauss, C., “An improved ant system algorithm for vehicle routing problem”, Anals of Operations Research, Vol. 89, pp. 319-328, 1999. [28] Bell, John E., McMullen, Patrick R., “Ant Colony Optimization Techniques for the Vehicle Routing Problem”, Advanced Engineering Informatics 18, pp. 41 – 48, 2004. [29] Rizzoli, A.E., Oliverio F., Montemanni R., Gambardella L.M., “Ant Colony Optimization for vehicle routing problems: from theory to applications”, Technical Report IDSIA-15-04, Istituto Dalle Molle di Studi sull’Intelligenza Artificiale, September 2004. [30] Bonabeau, E., Meyer, C., “Swarm Intelligence: A whole new way to think about business”, Harvard Business Review, R0105G, pp. 105-114, May 2001. [31] Reimann, M., Doerner, K., and Hartl, R. “D-Ants: Savings based ants divide and conquer the Vehicle Routing Problem”, Computers Operations Research, Vol. 31, No. 4, pp. 563—591, 2004. [32] Satos, L., Coutinho-Rodrigues, J., and Current, J. R., “An improved ant colony optimization based algorithm for the capacitated arc routing problem”, Transportation Research Part B: Methodological, Vol. 44, No. 2, pp. 246 – 266, Feb. 2010. [33] Tang, K. , Mei, Y., and Yao, X., “Memetic algorithm with extended neighborhood search for

Capacitated Arc Routing Problems”, IEEE Transactions on Evolutionary Computation, Vol. 13, No. 5, pp. 1151-1166, October 2009. [34] Narasimha, K., Ant Colony Optimization Technique to Solve Min-Max Multi Depot Vehicle Routing Problem, M.S. Thesis, University of Cincinnati, Cincinnati, OH, 2011. [35] Beardwood, J., Halton, J.H., Hammersley, J.M., “The shortest path through many points”, Proceedings of Cambridge Philosophical Society, Vol. 55, pp. 299-327, 1959. [36] Carlsson, J., Armbruster, B., Ye, Y., “Finding equitable convex partitions of points in a polygon efficiently”, ACM Transactions on Algorithms, Vol. 6, No. 4, article no. 72, 2010.

29

[37] Stone, A.H., Tukey, J.W., “Generalized sandwich theorems”, Duke Mathematical Journal, Vol. 9, pp. 356359, 1942. [38] Zivaljevic, R.T., “Topological Methods”, Handbook of Discrete and Computational Geometry, chapter 11, pp. 209-224, 1997. [39] Bespamyatnikh, S., Kirkpatrick, D.G., Snoeyink, J., “Generalizing Ham Sandwich Cuts to Equitable Subdivisions”, Symposium on Computational Geometry (SoCG), pp. 49-58, 1999. [40] O’Rourke, J., “Computational Geometry in C (Second Edition)”, Chapter 2, 1998. [41] Lin, S., Kernighan, B.W., “An effective heuristic algorithm for the TSP”, Operations Research, vol. 21, pp. 498-516, 1973. [42] MATLAB Code for Carlsson Method for Min-Max MDVRP, http://www.stanford.edu/class/msande310/handout.shtml [43] Derrac, J., García, S., Molina, D., and Herrera, F., "A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms", Swarm and Evolutionary Computation, vol. 1, no. 1, pp. 3-18, Mar 2011. [44] CONCORDE TSP Solver, http://www.tsp.gatech.edu/concorde.html [45] Braysy, O. and Gendreau, M., “Vehicle Routing Problem with Time Windows, Part I: Route Construction and Local Search Algorithms, Transportation Science, vol. 39, No. 1, pp. 104-118, 2005. [46] Aarts, E. H. L., van Laarhoven, P. J. M., Lenstra, J. K., and Ulder, N. L. J., “A Computational Study of Local Search Algorithms for Job Shop Scheduling”, Informs Journal on Computing, vol. 6, no. 2, pp. 118125, 1994.

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