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A swift heuristic algorithm based on capacitated clustering for the Open Periodic Vehicle Routing Problem ANNA DANANDEHa, MEHDI GHAZANFARIb, REZA TAVAKOLI-MOGHADDAMc, MEHDI ALINAGHIANb1 a

Strategic Planning & Control Vice Presidency, Iran Power Plant Project Management Company (MAPNA) b

c

Department of Industrial Engineering, Iran University of Science & Technology

Department of Industrial Engineering, Faculty of Engineering, University of Tehran Tehran, Iran 1

[email protected]

Abstract: Distribution problem is one of the important subsidiaries of Supply Chain Management (SCM), which is in relation to finding the best schedule and routs to any material transfer within the system components. Open Periodic vehicle routing problem (OPVRP) is a novel generated problem, in this group, in which, like a PVRP, the vehicles pay visits to clients over a given time horizon so as to satisfy some service level. however, they do not come back to depot at the end of each day. This paper introduces OPVRP and then presents a mathematical model and a new, simple and effective heuristic algorithm based on clustering for it. For evaluating the algorithm, it is compared with results from lingo software. The algorithm compares good in terms of accuracy with optimal solutions while benefiting from short solving time.

Keywords: Supply Chain Management, Distribution problems, Open Periodic Vehicle Routing Problem (OPVRP), Heuristic Algorithm, Clustering, Davis Criterion,

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distance traveled by the vehicles in a way that only one vehicle handles the deliveries for a given customer and the total quantity of goods that a single vehicle delivers is not larger than Q [1].

Introduction

The classical vehicle routing problem (VRP) is defined as follows: vehicles with a fixed capacity Q must deliver order quantities qi (i=1,..,n) of goods to n customers from a single depot (i=0). Knowing the distance dij between customers i and j (i,j=1,..,n), the objective of the problem is to minimize the total

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In PVRP, Each customer i∈I={1,2,..,i,..,n} specifies a set k(i) of combinations, and the visit days are assigned to the customer by selecting one of these

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algorithm with four kinds of move. Pisinger and Ropke [17] developed an adaptive large neighborhood search in which a feasible solution is constructed and then modified in each iteration. Finally, F. Li et al [18] provided a literature review on the OVRP and then proposed a variant of recordto-record algorithm.

combinations. Thus, the vehicles must visit the customer i on the days belonging to the selected combination. Early formulations of the PVRP were developed by Beltrami and Bodin [2] and by Russell and Igo [3] who proposed heuristics applied to waste collection problems. Tan and Beasley [4] use the idea of the generalized assignment method proposed by Fisher and Jaikumar [5] and assign a visiting schedule to each vertex. Eventually a heuristic for the VRP is applied to each day. Russell and Gribbin [6] developed a heuristic organized in four phases. Solution methods in these papers have focused on two-stage (construction and improvement) heuristics. Cordeau et al. [7] present another algorithm: The solution algorithm is a TS heuristic which, differently from the above heuristics, may allow infeasible solutions during the search process. Similarly, good results were obtained in the more recent work of Ha. Dji constantinuo and Baldacci [8], Angelelli and Speranza [9], And Blakeley et al. [10] who provide specific practical applications of the PRVP.

The other problem which is the focus of this paper is open periodic vehicle routing problem (OPVRP). In the best of our knowledge, it is the first time that this problem is introduced. This problem has a new characteristic rather than those in PVRP: a vehicle is not obliged to return to depot. The objective of the problem is to find the minimum paths for each vehicle needed to service the customers. Like OVRP, OPVRP can be easily applied to any couriers to reduce costs. Many of these companies (like FedEx) work with contractors which do the service (whether pickup or delivery) and are not obliged to come back to the depot. In this paper, a new heuristic algorithm based on clustering has been developed and applied on the basic model of OPVRP with an additional constraint; the number of vehicles is limited. The algorithm is also compared with lingo software so that we can have an evaluation of the accuracy of it. It has been proved that it has a significant ability in solving PVRPs and its noticeable point is that it can reach to the answer in quite small amount of time.

The open vehicle routing problem (OVRP), is a variant of the VRP in that vehicles are not obliged to return to the depot. So that they do not form a rout, instead the problem is to find the best Hamiltonian paths. The first notion of the OVRP goes to 20 years ago; however, only over the last decade some serious works have been done in this area. Sariklis and Powell [11] propose a two phase algorithm which firsts clusters and then builds the routs by solving a minimum spanning tree. Brandão [12] presents a tabu search algorithm with just two types of move: an insert move and a swap move. Tarantilis et al. [13] developed a two phase adaptive memory-based tabu search. They create the routs over a sequence of points called bone. Tarantilis et al. [14],[15] developed two variants of simulated annealing algorithms: Backtracking adaptive threshold accepting (BATA) and List-based threshold accepting (LBTA). Fu et al. [16] proposed an algorithm which creates initial solution with a farthest first heuristic and it improve it by a TS

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This paper is organized as follows: first in section 2 the considered heuristic algorithm is described, in Section 3 the computational results are presented. The Paper is followed by the conclusion in Section 4.

2

Mathematical Model of the OPVRP

The Proposed OPVRP model is developed from an integer programming formulation by Christofides and Beasley which uses two types of decision variables: one for the assignment of customers to schedules,

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Si

and another for the routing of a given vehicle on a given day (20).

� zis = 1 𝑠𝑠=1

In order to develop OPVRP model (from PVRP model), the constraint that necessitates vehicles return to depot should be dismissed. To do that, an artificial node with zero distance with any node (including depot) is considered. Also some other constraints are added to the model. The variables of the model are introduced here:

𝑣𝑣𝑖𝑖𝑑𝑑 𝐾𝐾

N+1

𝑑𝑑 ∑𝑗𝑗N+1 =1 𝑥𝑥0𝑗𝑗𝑗𝑗 ≤ 1

𝑘𝑘 ∈ 𝐾𝐾, 𝑑𝑑 ∈ 𝐷𝐷

𝑖𝑖,𝑗𝑗 ∈𝑄𝑄

wi = total demand of node 𝑖𝑖 over the planning period

𝑁𝑁

N+1

∀d∈D

∀d∈D

(6) (7)

(8)

(9)

(10)

𝑖𝑖=1

𝑗𝑗 =1

𝑑𝑑 � 𝑤𝑤𝑖𝑖 � 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 ≤C

∀ 𝑘𝑘 ∈ 𝐾𝐾, 𝑑𝑑 ∈ 𝐷𝐷

(12)

𝑑𝑑 ≤L � � 𝑐𝑐𝑖𝑖𝑖𝑖 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖

∀ 𝑘𝑘 ∈ 𝐾𝐾, 𝑑𝑑 ∈ 𝐷𝐷

(13)

𝑑𝑑 =0 � � 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖

∀𝑑𝑑 ∈ 𝐷𝐷

(14)

N N+1

1 if day 𝑑𝑑 is in combination 𝑠𝑠 =� 0 otherwise

i=0 𝑗𝑗 =1

K N+1

k=1 𝑖𝑖=0

The formulation of the OPVRP is:

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𝑘𝑘 ∈ 𝐾𝐾, 𝑑𝑑

(5)

𝑑𝑑 � 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 ≤ |𝑄𝑄| − 1 ∀ 𝑄𝑄 ∈ 𝑁𝑁, 𝑘𝑘 ∈ 𝐾𝐾, 𝑑𝑑 ∈ 𝐷𝐷 (11)

cij = distance between nodes 𝑖𝑖 and 𝑗𝑗 , cost of arc (𝑖𝑖, 𝑗𝑗)

𝑑𝑑 =1 𝑖𝑖=0 𝑗𝑗 =1 𝑘𝑘=1

∀ j ∈ N ,d ∈ D

𝑑𝑑 ≤1 � 𝑥𝑥𝑖𝑖(𝑁𝑁+1)𝑘𝑘

𝑑𝑑 ∑Kk=1 ∑𝑁𝑁+1 𝑖𝑖=1 𝑥𝑥𝑖𝑖0𝑘𝑘 = 0

L: maximum daily length for each vehicle

𝑑𝑑 𝑚𝑚𝑚𝑚𝑚𝑚 � � � � 𝑐𝑐𝑖𝑖𝑖𝑖 . 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖

j=0

∀ k ∈ K , i ∈ 𝑁𝑁, 𝑑𝑑 ∈ 𝐷𝐷

𝑑𝑑 ∑Kk=1 ∑𝑁𝑁 𝑖𝑖=0 𝑥𝑥(𝑁𝑁+1)𝑖𝑖𝑖𝑖 = 0

1 if customer 𝑖𝑖 is visited in day 𝑑𝑑 0 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

𝑁𝑁 𝑁𝑁+1 𝐾𝐾

𝑑𝑑 = � 𝑥𝑥𝑗𝑗𝑗𝑗𝑗𝑗

𝑖𝑖=0

C: vehicle capacity

𝐷𝐷

𝑁𝑁

∀ 𝑑𝑑 ∈ 𝐷𝐷 , 𝑖𝑖, 𝑗𝑗 ∈ 𝑁𝑁 , (𝑖𝑖 ≠ 𝑗𝑗) (4)

𝑁𝑁

1 if combination 𝑠𝑠 is selected for servicing customer 𝑖𝑖 =� 0 otherwise

𝑎𝑎𝑠𝑠𝑠𝑠

K

N

2

k=1 𝑖𝑖=0

1 if vehicle k passes rout 𝑖𝑖𝑖𝑖 in day 𝑑𝑑 =� 0 otherwise

𝑣𝑣id = �

�

vid + vjd

𝑑𝑑 � � 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑣𝑣jd

K: The number of vehicles

𝑧𝑧𝑖𝑖𝑠𝑠

𝑠𝑠=1

𝑑𝑑 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖

j=1

(2)

= � 𝑧𝑧𝑖𝑖𝑠𝑠 . 𝑎𝑎𝑠𝑠𝑠𝑠 ∀ 𝑑𝑑 ∈ 𝐷𝐷, 𝑖𝑖 ∈ 𝑁𝑁 ∪ 0 ∪ 𝑁𝑁 + 1 (3)

k=1

Nodes have i and j indices, Depot is indexed i=0, and the artificial node is specified by N+1 index

𝑑𝑑 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖

𝑆𝑆𝑖𝑖

𝑑𝑑 � 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 ≤

N: The number of customers: nodes excluding depot and the artificial node

∀ 𝑖𝑖 ∈ 𝑁𝑁 ∪ 0 ∪ 𝑁𝑁 + 1

𝑧𝑧is ∈ {0,1}

(1)

∀ i ∈ N ∪ 0 ∪ N + 1, s ∈ Si

(15)

d 𝑥𝑥ijk ∈ {0,1} ∀ i, j ∈ N ∪ 0 ∪ N + 1, k ∈ K, d ∈ D(16)

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The objective function (1) minimizes the travel distance. Constraint (2) ensures that from several proposed combinations, one and only one combination is chosen for each node. Constraint (3) defines service days for each customer. It is supposed that depot and the artificial node have all the days in their day combinations. Constraint (4) allows arcs only between customers assigned for delivery on day d. Constraint (5) is to make sure vehicles visit and leave a node in the same day. This constraint is not considered for depot and the artificial day. Constraint (6) is about servicing customers in selected days. Constraint (7) ensures that routs end to the artificial node, while constraint (8) doesn’t let routs begin from the artificial node. Constraint (9) ensures that a vehicle is used no more than once a day. Constraint (10) makes it impossible to return to depot. Constraint (11) is the sub tour elimination constraint. Constraint (12) is the physical capacity constraint of a vehicle, while Constraints (13) confines route lengths. Constraint (14) is to omit loops. And Finally, constraints (15, 16) define the sets of variables.

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a constant value rather they oscillate between some points. Before exploring the algorithm, the two definitions and procedures used in the algorithm are explained. Definition 1. The tour core is determined by the average of radian value of customer’s polar coordinate. The initial value of the core is considered to be zero. Definition 2. The Davis criterion is a measure for cluster desirability. Sum of the Davis criteria in a cluster shows its desirability. This criterion is calculated by the equation 1. In the equation 1, x and c respectively represent the density of the cluster and the cluster core. The density is the average of absolute node distances from the core.

D=

3.1

The Heuristic Algorithm

c j − ci

(17)

Assign procedure

This procedure assigns the customers to a path without considering capacity limitation in following steps:

This algorithm is trying to simultaneously construct m×v tours in which m and v are respectively representing the number of days and the number of vehicles. To form the paths, the algorithm puts the adjacent customers in the same path. In the first step the coordinate of the customers is changed from Cartesian to polar. In the next step which is improvised to let the procedure catch to final cores faster, an initial core is assigned to each tour, without considering the capacity limitation, using an assign procedure. The algorithm then improves the initial cores by applying k-means clustering algorithm, again without considering capacity limitation. After attaining the improved cores, the algorithm considers capacity limitation and assigns customers to the path. In practice, we face and solve two problems; first, there is the possibility of facing infeasible solutions which will be amended by applying feasibility procedure. Second, by adding the customers to the path after capacity limitation, the cores do not tend to

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xi + x j

1. Calculate the cost of all combinations of a customer. This cost is the sum of combination’s daily costs which is the minimum distance (according to their radian value) between path cores of the day and the customer. Figure 1 illustrates the procedure of finding the cost of allocating customer to its combinations. 2. For each customer, Choose the combination with the least cost and assign it to the path with the least distance in related day. 3. Update the path cores which new customer has been assigned to. Use the average value of assigned customer’s radian value for each path.

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3.3 Table 1. The procedure of finding the cost of allocating a customer with two combinations ([1,2] and [2,3]) Vehicle 1

Vehicle 2

Day1

C11

C21

Day2

C12

C22

Day3

C13

C23

Choosing best vehicle in each day Daycost1=Min{ c11,c12}

Following, the steps of proposed algorithm is presented:

Cost of combinations

1. First Step 1.1. Change the coordinate of depot to the zero point of coordinate system by subtracting its coordinate from all the nodes. 1.2. Change the coordinate of the nodes from Cartesian to polar It is obvious that Radian coordinate of the path can vary from 0 to π and from -π to 0.

Combination one=

Daycost2=Min{ c21,c22} Sum{Daycost1, Daycost2}

Daycost3=Min{ c31,c32}

Combination two= Sum{Daycost2, Daycost3}

3.2

2. Second Step In this phase the initial cores are considered for the paths and then are improved by the k-means clustering algorithm. The generated cores in this phase are used to build feasible paths regarding to capacity limitation.

Feasibility procedure

The Feasibility procedure is applied when there is no feasible combination for a customer. In order to solve this problem these steps are taken.

2.1. Core formation 2.1.1. Sort the nodes in a non-decreasing order according to their radian value. 2.1.2. Let the initial value of the cores be equal to zero. 2.1.3. Pick one of the possible combinations of the first customer and assign it to a random path in a random day combination. 2.1.4. Update the core of the paths which nodes have been assigned to by using average value of customer’s radian value with the zero angle of central depot. 2.1.5. Apply the assigning procedure to the all other customers.

1. Consider all combinations of a given customer and define the infeasible days of them. 2. Sort combinations of a given customer non_decreasingly according to number of infeasible days. 3. Select sorted combinations from the top, in each infeasible day of selected combination, sort the paths non-increasingly according to their remained capacity. 4. In each path from the top, check whether it is possible to move some already assigned customers, preferably customers with the maximum demand which theirs removal frees the most space, to other paths of the same day or, if not, to their other combination to let the infeasible customer meet the capacity limitation. 5. If yes for at least one path of each infeasible day of selected combination, do the movements and exit the procedure otherwise check the next combination. 6. Announce the problem infeasible if there is no left combination.

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Main Algorithm

2.2. Core improvement using k-means clustering algorithm 2.2.1. Remove all the customers from path. 2.2.2. Appoint the created core as the initial core. 2.2.3. Sort the customers non-decreasingly according to their radian part. 2.2.4. Assign the customers to the paths using assignment algorithm. 2.2.5. Compare the new core with the initial one. If the difference between sum of the new cores and the initial ones is less than ε =0.01, go to next step, otherwise make new cores the initials and repeat the subroutines

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RAM-memory. In order to avoid the error caused by randomly generated problems, 5 problems were generated in each class and the average solution and average error were reported. Computational results in small classes showed that the computational error of the algorithm is about 6%. The results are reported in table 1.

of core improvement. 3. Third Step Up to now, the best tours have been created but the capacity limitation has not been satisfied. In this phase the customers cluster according to the improved cores gained from the later phase. The problem in this part, as mentioned before, is that the cores do not tend to a constant value that is solved by applying Davis criterion.

Table 2. Error estimation from small instances

# of rows 1 2 3 4 5 6 7 8 9

3.1. Path creation considering capacity limitation 3.1.1. Sort the customers non-increasingly according to their demand. 3.1.2. Pick the customers from the top and assign them using assign procedure and considering feasible combinations. The feasible combination is the one in each days of it there is at least one path with sufficient capacity for the customer. 3.1.3. Apply feasibility procedure if no feasible combination is found for a customer. 3.1.4. Save the created core and calculate the Davis criterion of created paths. 3.1.5. Appoint the core from former step to the initial core and repeat the sub routines 10 times. Let’s mention that the number 10 is chosen by the experiences. 3.1.6. Select the paths with the least value of Davis criterion from all created paths.

average

5

# of nodes 6 7 8 6 7 8 6 7 8

%Error 3.2 8.8 6.3 11 6.7 6.1 0 7.3 5.6 6.11

Conclusion

This paper was intended to present a solution for one of the most important problems in Supply Chain Management, Distribution problems. The aim of this paper was to develop a heuristic algorithm for the novel model of PVRP in condition that vehicles are hired and each vehicle does not return to depot, called OPVRP. This problem has additional constraint which is the limitation in the number of vehicles. The proposed algorithm employs k-means clustering algorithm, and nearest neighbor heuristic algorithm for OPVRP. This algorithm offers appropriate solutions in a very small amount of time. This especial attribute makes it suitable to be used as an upper bound for OPVRPs. Furthermore, the

Computational Results

The aim of this section is to test the algorithm’s result via the small instances. However, in the best of our knowledge OPVRP is not suggested with other researchers so there is not any test problem for this special problem. In order to test the proposed algorithm some samples were generated and solved both by Lingo software and the heuristic algorithm. Then, the results were compared .The computer code of the heuristic algorithm was written using Matlab 7 on Intel dual core, 2.33 GHz processor and 2 GB of

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# of days 2 2 2 3 3 3 2 2 2

As it is apparent from the results, the algorithm has the ability to find satisfactory answers in a quite short time. This especial attribute makes it appropriate to be used as an upper bound for OPVRPs or PVRPs. Furthermore, the answers can be used as initial answers for improvement algorithms like metaheuristics.

4. Fourth Step The aim of this phase is to arrange the customers of each path. To achieve good paths in this step, the nearest neighbor heuristic algorithm is applied.

4

# of vehicle 2 2 2 2 2 2 3 3 3

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Journal of the Operational Research Society; 51:564–73. [12] Brando. J. (2004). A tabu search algorithm for the open vehicle routing problem. European Journal of Operational Research; 157:552–64. [13] Tarantilis. C., Diakoulaki. D., & Kiranoudis C. (2004). Combination of geographical information system and efficient routing algorithms for real life distribution operations. European Journal of Operational Research; 152:437–53. [14] Tarantilis. C., Ioannou. G., Kiranoudis. C., & Prastacos. G. (2004). A threshold accepting approach to the open vehicle routing problem. RAIRO Operations Research; 38:345–60. [15] Tarantilis. C., Ioannou. G., Kiranoudis. C., & Prastacos. G. (2005). Solving the open vehicle routing problem via a single parameter metaheuristic algorithm. Journal of the Operational Research Society; 56:588–96. [16] Fu. Z., Eglese. R., & Li. L. (2005). A new tabu search heuristic for the open vehicle routing problem. Journal of the Operational Research Society; 56:267–74. [17] Pisinger. D., & Ropke. S. (2005). A general heuristic for vehicle routing problems. Working paper, Department of Computer Science, University of Copenhagen, Copenhagen, Denmark,. [18] Feiyue. Li., Bruce Golden., & Edward Wasil. (2007). The open vehicle routing problem: Algorithms, large-scale test problems, and computational results. Computers & Operations Research 34 2918 – 2930. [19] Christofides. N., & Beasley, J.E. (1984). The period routing problem. Networks, 14, 237±256.

answers can be used as initial solutions for improvement algorithms like meta-heuristics which can be considered in the next studies.

Acknowledgement Our sincere gratitude is towards Iran Power Plant Project Management Company (MAPNA) due to the generous support of this research.

References: [1] Alegre. J., Laguna. M., & Pacheco. J. (2005). Optimizing the periodic pick-up of raw materials for a manufacturer of auto parts. European Journal of Operational Research. [2] Beltrami, E.J., & Bodin, L.D. (1974). Networks and vehicle routing for municipal waste collection. Networks 4, 932. [3] Russell. R.A. & Igo. W. (1979). An assignment routing problem. Networks, 9, 1±17. [4] Tan. C.C.R., & Beasley. J.E. (1984). A heuristic algorithm for the period vehicle routing problem. OMEGA, International Journal of Management Science, 12, 5, 497±504. [5] Fisher. M.L., & Jaikumar. R. (1981). A generalized assignment heuristic for vehicle routing. Networks,11, 109±124. [6] Russell. R.A., & Gribbin. D. (1991). A multiphase approach to the period routing problem. Networks, 21, 747±765. [7] Cordeau. J.F., Gendreau. M.G., & Laporte. (1997). A tabu search heuristic for periodic and multi-depot vehicle routing problems. Networks, 30, 105±119. [8] Dji constantinuo. Ha, & Baldacci. E. (1998). A multi-depot period vehicle routing problem arising in the utilities sector. Journal of Operations Research Society, 49, 12, 1239–1248. [9] Angelelli. E., & Speranza. M.G. (2002). The periodic vehicle routing problem with intermediate facilities. European Journal of Operational Research 137, 2, 233–247. [10] Blakeley. F., Bozkaya. B., Cao. B., Hall. W., & Knolmajer. J., (2003). Optimizing periodic maintenance operations for Schindler Elevator Corporation. Interfaces, 33, 1, 67– 79. [11] Sariklis. D., Powell. S., (2000). A heuristic method for the open vehicle routing problem.

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