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A new intuitional algorithm for solving heterogeneous fixed fleet routing problems: Passenger pickup algorithm
Applied Mathematics and Computation 181 (2006) 1552–1567 www.elsevier.com/locate/amc
A new intuitional algorithm for solving heterogeneous fixed fleet routing problems: Passenger pickup algorithm Cevriye Gencer a
a,*
_ , Ismail Top b, Emel Kizilkaya Aydogan
c
Department of Industrial Engineering, Faculty of Engineering and Architecture, Gazi University, Ankara, Turkey Department of Operational Research, Defence Sciences Institute, Turkish Military Academy, Ankara, Turkey c Department of Industrial Engineering, Faculty of Engineering, Erciyes University, Kayseri, Turkey
b
Abstract Fixed-fleet heterogeneous vehicle routing is a type of vehicle routing problem that aims to provide service to a specific customer group with minimum cost, with a limited number of vehicles with different capacities. In this study, a new intuitional algorithm, which can divide the demands at the stops for fixed heterogeneous vehicle routing, is developed and tested on tests samples. The algorithm is compared to the BATA Algorithm available in the literature in relation to the number of vehicles, fixed cost, variable cost and total cost. 2006 Elsevier Inc. All rights reserved. Keywords: Routing; Algorithm; Heterogeneous fleet
1. Introduction Vehicle routing problems are divided into various areas. In most of the studies, the vehicle fleets that are studied consist of homogenous vehicles. Problems related to providing service through heterogeneous vehicles/ vehicle fleets are more comprehensive and more complicated in comparison to homogenous fleet vehicle routing problems. Therefore, the number of studies on heterogeneous fleet vehicle routing has been rather limited. Heterogeneous fleet vehicle routing problems (HFVRP) are various. One of these problems is the heterogeneous fleet mixture-vehicle routing problem (HFMVRP) which aims to find out how many of each vehicle with different capacities must be purchased or leased in order to provide a minimum-cost service to an existing customer group. HFMVRP is widely studied in the literature mostly on heuristics by researchers like Salhi et al. (1992) [1], Salhi and Rand (1993) [2], Osman and Salhi (1996) [3]. The other type of problem is the limited number heterogeneous vehicle fleet routing. Problems that involve a limited number of vehicles, constitute the most complicated and most difficult vehicle routing problems. In these problems, the aim is to provide service to the customer group with minimum cost and with the available vehicles if problem is feasible. The first study
*
Corresponding author. E-mail address: [email protected] (C. Gencer).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.02.043
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on fixed-fleet heterogeneous vehicle routing problems (FFHVRP) was carried out by Taillard (1999) [4], and the recent study was conducted by Tarantilis et al. (2004) [5]. This study also suggests a new intuitional algorithm—passenger pickup algorithm—for FFHVRP, tests it using the test samples available in the literature, and compares it to the BATA Algorithm solutions of Tarantilis et al. (2004) [5]. 2. Passenger pickup algorithm (PPA) In the literature, it is seen that studies related to vehicle routing do not divide the node demands, and node locations are designated taking into account the grid (x–y coordinates), and the distance between nodes is mostly assumed as the Euclidian distance. This assuming, eliminates the possibility of passing by another node on the way to any one from the main depot as can be seen in Fig. 1(a). However, in urban transportation, as seen in Fig. 1(b), there is the necessity for vehicles to follow specific paths to reach any node, and this path may include node(s) that are not included in the specified route. The distances between stops as used in the test samples and in the algorithms in the literature are symmetric. However, it is not always the case in real life. In urban transportation, the symmetricity can disappear. In addition, on the results of algorithms in the literature on FFHVRP, the sample node clusters does not divide the node demands in a way to maximize the utilization rate of vehicle(s). However, in daily life, the route followed may include other nodes with unsatisfied demands that are not included in current route. In such a case, vehicle has to pass from the node with unsatisfied demands, and then vehicle can fulfil its capacity by maximizing the number of passengers in the vehicle. As a result, both cost and the total number of vehicles serving is reduced. The passenger pickup algorithm (PPA) developed in this study takes into account the possibility of vehicle lease when available vehicles fall short and the divisible character of the demands, both of which are ignored in other algorithms available in the literature for FFHVRP. Moreover, it can use both symmetrical and asymmetrical inter-stop distances, in line with the daily life conditions. 2.1. Phases of PPA The algorithm functions according to the principle of first clustering-then routing. According to this principle, sub-clusters are created from the cluster of all stops, and these sub-clusters are solved as travelling salesman problems. An overview of how the algorithm works is shown in Fig. 2. It will be explained fully on further steps. 2.1.1. Step 1: the starting phase Data related to nodes is required for solution, such as number of stops, distance between stops; data related to vehicles such as vehicle types available at hand, their numbers, capacities, fixed costs and fuel rates. Distances between nodes are also acquired.
Depot
(a) According to Euclidian distance
Depot
(b) According to Urban Transportation
Fig. 1. Routes according to Euclidian and urban transportation.
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-Step 1: Enter data about nodes, P’, TD’,PP and vehicles. -Step 2: Outbound loops Do the following while (P) is not equal P’ Continue while (TD)is not equal TD’ -Step 3: While (iterate) : inbound iterate loop -Produce solution from a different starting point.S’ ∈ N(s) Accept if c(s’)
The algorithm also takes into account the cases where the vehicles available at hand are not sufficient. In other words, in this problem it also includes the leasing of additional vehicles to meet the whole node demands. Hence, the rental vehicle types, their variable and fixed leasing costs must be provided for the algorithm at the starting phase. 2.1.2. Step 2: loops This step consists of two main inter-loops. The first while loop is used for P, which is called the maximum deviation value (P) and taken as a basis in establishing the routes. P represents the acceptable changes that may occur in the route due to additional nodes inserted, if possible, when creating a route. Further details are given in Step 3. The second while loop is created for the deviation penalty coefficient (TD), which penalties the deviation and enables the creation of a route by grouping different stops as a result of an increase or decrease in its value. 2.1.3. Step 3: routing The third while (iterate) loop shows the maximum number of different stops from which the algorithm can start according to the current TD, since any different node is selected as the starting node for the first vehicle leaving the main depot. This step consists of the stages of identifying the starting node, selecting the vehicle, routing, finding the shortest route length and vehicle changes according to the number of passengers on the route. • The phase of identifying the starting node: Starts to create the first route by selecting the station farthest from the main depot among the nodes that were not selected as starting nodes before and if the demands are not met yet. If the starting station is not changed the algorithm will select the same node in each iteration. Selection of the same station prevents the occurrence of different routes and different station groups. Hence, the starting station must be changed for each iteration. According to the algorithm, after selecting the nonneighbouring starting nodes in the first two iterations, the third starting node that will be selected must be the one farthest to the last selected two starting ones. With each change of the deviation penalty coefficient (TD) within the algorithm, the iteration number is reset to null, and hence all stops are given the opportunity to be re-selected as starting nodes in later iterations. • Vehicle selection phase: Among the available unused vehicles, the one with the largest capacity is selected. If the vehicle’s capacity is not fully used in the generated route, it is changed to a vehicle of lesser capacity. It is also noted that the vehicle with lower capacity spends less fuel than the vehicle with higher capacity. In case of running out of vehicles that will be used during route generation, the algorithm chooses to rent vehicles, and rents the highest capacity vehicle available. • Routing phase: Routes are generated according to the gain formula where minimum cost algorithms developed by Clarke and Wright (1964) [6] and Rosenkrantz et al. (1977) [7], which are tour-constructing algo-
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rithms from demand-not-met nodes; maximum-gain station(s) are selected, and the routes are created through insertion. The node that is the farthest to the main depot among nodes whose demands are not met according to the algorithm is selected as the starting point (Fig. 3). Depending on the fact that the demand in some nodes are larger than some of the vehicle capacities, the station which has the highest number of passengers whose demands are not met may be selected instead of the farthest station at the beginning of the algorithm. Thus, at the beginning the route 1–2–1 is created. According to the Clarke and Wright (1964) [6] algorithm, the savings of each station with another neighbouring station is calculated as Eq. (1). S ij ¼ 2c1i þ 2c1j ðc1i þ c1j þ cij Þ ¼ c1i þ c1j cij .
ð1Þ
According to the method of Rosenkrantz et al. (1977) [7], the insertion costs for insertions at any section of the route are calculated as per Eq. (2). This cost corresponds to the deviation value. The deviation value for insertion of node j between nodes i and k is expressed as in Eq. (2) according to Fig. 4. SAijk ¼ cij þ cjk cik .
ð2Þ
In PPA, the highest-gain station (j) among those whose demands are not met is found using the gain formula (3). If SAijk 6 P ;
then Gain ¼ S ij þ S jk TD SAijk .
ð3Þ
6
2 5
7 4 3 8 9
Depot(1) Fig. 3. Selecting the route starting point.
Fig. 4. Deviation value.
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If the deviation of candidate nodes with uncovered demands that will be inserted between any two nodes on route is smaller than P, the gain is calculated as per Eq. (3), and the node with the highest gain is included in the route (e.g. 1–2–5–1). This operation is continued until the vehicle chosen for the route reaches its full capacity. If the vehicle is not completely full, the algorithm initiates the passenger pickup (capacity integration) part, where a portion of the node demand that maximizes the vehicle capacity is picked up. According to this, the node for partial passenger pickup is not as per maximum gain but as per capacity integration constraint (PP). To determine the candidate node(s) for partial passenger pickup on the generated route, the deviation values of stops whose demands are not supplied are calculated. All stops with deviations smaller than PP value are included in the candidate nodes group. In case of more than one node in the candidate station group, then the one farthest to the main depot is selected. A portion of the passengers sufficient to maximize the vehicle capacity is picked up from the selected node, and the new passenger number of the station is equalized with the number of remaining passengers. This node is considered as an unsatisfied one, until all its passengers are picked up. There is no requirement for the vehicle to always have maximum capacity. It may not be possible to find a candidate node as per PP value. In that case, the algorithm ends the routing process, and returns to the vehicle selection and starting point selection phase in order to draw a new route. These operations continue until the demands of all nodes are satisfied. The variation in the probable nodes group to be selected as per P and TD values of the gain formula is shown in Fig. 5. In Fig. 5, as the P value increases, the route will include stops with a long distance to the main depot and with high deviation values. The TD value penalizes the deviation as described in gain formula (3); therefore, nodes 5, 6 and 7 cannot be included in the route-generating nodes group after node 2 selected as the first one; and the nodes close to the 1–2–1 route generated initially, that is towards the main depot, are included in the route. Considering that the nodes are dispersed on a plane, the gain value for the selected route initial node (like node no. 2) changes by decreasing P and TD values. Hence, creating all probable groups can assess the stops. At the end of the iterations, the TD value is subtracted and reset to null again, and when the decrease in TD value finishes, the P value is subtracted, and therefore TD regains its maximum value. • Finding the shortest path for the route: The nodes constituting the vehicle route are sub-cluster of the all nodes cluster. All stops in this sub-cluster must be travelled from the shortest path. At this stage, the problem is reduced to a travelling salesman problem (TSP). Here, also the least costly insertion method of Rosenkrantz et al. (1977) [7] is used. Among the nodes creating the route, the one farthest to the main depot is selected as the starting point. After drawing the starting route (such as 1–2–1) the node (j) with the least deviation (SAijk) is inserted between nodes i and k (1, . . . , i j k, . . . , 1). The insertion process is continued until the routing is complete for all nodes in the sub-cluster.
6
2
5
7
6
2
5
7 4 8
4 3
9
Depot(1)
(a) The effect of change of P to the stops group
8
3 9
Depot(1)
(b) Effect of TD change to stops group
Fig. 5. The effect of P and TD variation to routing.
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• Vehicle changing phase: After the route is generated, the vacancy status of the vehicle assigned to this route is checked. If there is an unused vehicle that has the capacity larger than the number of passengers on board, it is changed with the vehicle on this route. The purpose of this changing procedure is to serve the given route with minimum cost. Using the inert-capacity vehicle on the said route means ignoring situations of possible economy in the subsequent route selection. The vehicle allocated to any route must have rate of maximum utilization when considering all the other vehicles. For example, considering a route with 25 passengers, assuming that the available vehicles have a passenger capacity of 29 and 44, assigning the vehicle with 29-passenger capacity to this route would be economical. Therefore, at the vehiclechanging phase, the vehicle that will minimize inert capacity and route cost should be allocated to the route.
2.1.4. Step 4: termination After working within the framework of the constraints in the first three steps, the algorithm compares the value acquired in each iteration to the best value; if this value is smaller than the minimum value in the memory, this value is taken as the minimum value. Following the loops, the termination phase starts. The minimum value found, the route number, together with the route length, route cost, whether vehicle is rented or not, number of passengers on the route, the capacity of the vehicle on the route, how many additional passengers must be picked up from which stops to maximize the capacity, and the routes to be followed starting from the main depot selected as the first depot and then again returning to the main depot as the finish line, are printed out.
3. Applying the algorithm to test samples In the literature, Golden et al. (1984) [8] suggests test samples for different types of vehicle routing problems. Those samples between 13 and 20 are rearranged by Taillard (1999) [4] for FFHVRP. The relevant samples are also used for PPA testing in the study. In PPA, first of all the P value which determines the maximum deviation must be identified. The tests carried out indicate that relating P value to the distance of the stop farthest to the main depot (xu) gives the best result. Therefore, P = xu/4 was used in the algorithm test. Hence, the algorithm starts working by determining the P value, and continues to work while the value of P decreases. During the tests, to enable the algorithm to reach the result in a short time, P was reduced as P = P 5. TD must be changed according to the distribution of nodes. The tests revealed that, in cases where the main depot as location in the samples is away from the group of nodes, a good result is acquired when TD value is 5 or lower; however, in cases where the main depot is in the centre and where the stops are not collected at specific locations, it is observed that a good result is acquired when TD assumes a value between 5 and 10. Building on these tests, TD value is assumed as (0 6 TD 6 10) and reduced as (TD = TD 1), and applied to the problems. The algorithm can produce n solutions with different starting points when the number of nodes excluding the main depot is n. Since selecting the nodes in the vicinity of main depot as starting solution does not contribute to generating a different route, these nodes must not be selected for starting solutions. Assuming half of the nodes are in the vicinity of main depot, to shorten the run-time of the algorithm and to reduce the number of iterations it was anticipated that an iteration number of n/2 would be useful during the initial test phase of PPA. Generating the nodes cluster by distribution on a random plane and using Euclidean distances between nodes is not compatible with the development purpose and working principles of the PPA. Therefore, applying PPA, which enables additional passenger pickup via dividing the node demands and which aims to maximize the vehicle capacity, will result in increased route distances, and therefore increased costs. For that reason, first the passenger pickup (capacity integration) part of the algorithm was eliminated in the PPA test by assuming the capacity integrator as (PP < 0), and the algorithm was run and tested without dividing the demands of the stops. The best result was gathered in this way in the 16th and 19th test
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Problem no.
Tarantilis (BATA) Type of vehicle used and default
Total # of vehicles used
Fixed cost
Variable cost
Overall cost
Type of vehicle used and default
Total # of vehicles used
Fixed cost
Variable cost
Overall cost
13 14 15 16 17 18 19 20
4A, 3A, 4A, 2A, 4A, 3A, 3A, 6A,
17 6 9 9 11 13 8 13
1685 6800 2050 2200 1040 1930 9300 3300
1519.96 611.39 1015.29 1145.52 1071.01 1846.35 1123.83 1556.35
3204.96 7411.39 3065.29 3345.52 2111.01 3776.35 10,423.83 4856.35
2A, 4A, 4A, 2A, 3A, 3A, 3A, 6A,
15 6 9 9 10 13 8 13
1640 6800 2050 2200 1015 1930 10,200 3300
1602.862 693.8265 1069.991 1192.126 1151.405 1982.482 1280.747 1791.045
3242.8622 7493.8265 3119.9912 3392.1263 2166.4053 3912.4821 11,480.747 5091.0454
2B, 2B, 3B, 4B, 4B, 4B, 3B, 4B,
4C, 4D, 2E, 1F 1C SC 3C 2C, 1D 2C, 2D, 1E, 1F 2C 3C
Passenger pickup algorithm (PPA)
2B, 2B, 3B, 4B, 4B, 4B, 2B, 4B,
4C, 4D, 2E, 1F 1C 2C 3C 2C, 1D 2C, 2D, 1E, 1F 3C 3C
PPA BATA derivation (%)
1.183 1.112 1.785 1.393 2.624 3.605 10.139 4.833
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Table 1 A comparison of BATA and PPA according to overall costs
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problems compared to PPA. Then, tests were conducted with a capacity integration coefficient of PP = P/5 and PP = P/10. In the literature, only the variable costs are taken into account when testing FFHVRP operations. However, in real life, the costs of vehicles do not only consist of variable costs. Particularly when considering that the vehicles are used for profit purposes, it is clear that fixed costs also have a great significance. Not including the fixed costs in the assessments would be a great mistake for enterprises that aim to maximize their profits. Therefore, it would be more appropriate to test the algorithms by taking into account the fixed costs, variable costs and overall costs. There are no studies on FFHVRP in the literature that makes an assessment that takes into account the overall and fixed costs. PPA was compared to Tarantilis et al. (2004) [5] BATA algorithm. Although PPA test was conducted on the stops cluster generated contrary to its development purposes, good results were obtained. PPA and BATA Algorithms were subjected to eight test problems; tests and comparisons were made for four different aspects, namely the number of vehicles, fixed costs, variable costs and overall costs. Tarantilis’ study includes only the variable costs. To enable a comparison with the PPA, overall costs were calculated by adding the fixed costs of Taillard’s (1999) [4] study to the BATA algorithm results. The results acquired according to the test problems are shown in Table 1. Detailed results as per variable costs are given in Appendix. When Table 1 is examined in terms of total number of vehicles used, it is seen that PPA and BATA give the same results for problems 14, 15, 16, 18, 19, 20; whereas PPA gives better results for problems 13 and 17. PPA divides the node demands and maximizes the vehicle utilization; hence reveals the distinction of the algorithm as it aims for minimum cost with fewer vehicles. When Table 1 is examined in terms of fixed costs, it is seen that PPA gives better results for problems 13 and 17, and both PPA and BATA give the same results for problems 14, 15, 16, 18, 20; whereas BATA algorithm gives a better result for problem 19. The results found based on fixed costs also reveal a parallelism with total numbers of the used vehicles. Since fixed costs are the total fixed costs of vehicles used, PPA gives a better result than BATA for test problems 13 and 17. When Table 1 is examined in terms of variable costs, it is seen that PPA fails to reach a better result than BATA. However, this is due to the fact that the test problems do not correspond to the real life problems for which the algorithm was developed. When Table 1 is examined in terms of overall costs; PPA fails to get a better result than BATA. The reason for it is because test problems are very specific problems. In the problems, the total passenger capacity of the fleets is bigger than the total demand at the stops, but only at the rate of approximately the smallest vehicle capacity. Therefore, PPA fails to reduce the vehicle numbers in many of the problems. For example in problem 15, there is the obligation to use all vehicles. If the stop demands and vehicle numbers increase, PPA will achieve a better result due to its capacity integration ability, in which case the fixed costs and overall costs would be reduced. However, the studies available in the literature do not have the capacity integration ability, therefore when the stop demands and the vehicle numbers increase, the costs will also increase. In addition, when the number of nodes is decreased without changing the demand or the number of available vehicles, the demand per station would rise, and therefore, according to the algorithms in the literature, the vehicle number would become insufficient and the problem will remain unsolved or will not be feasible. Yet, PPA will be able to solve the problem thanks to its capacity integration feature. In all problems, when the deviation values are taken into consideration, it is worth noting that PPA’s results are rather close to BATA results. In eight test problems, the mean deviation of PPA is 3.334%.
4. Conclusion In the study, an algorithm oriented to full-use of the existing vehicle capacity was developed. The developed algorithm was compared to the BATA Algorithm in the literature, using the test problems.
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Table 2 Capacity utilization of BATA and PPA in problem 13 Vehicle no.
Tarantilis (BATA) # of passengers
PPA Vehicle Capacity
# of passenger
Vehicle Capacity
Problem 13 [4] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 20 19 19 30 27 33 40 37 39 67 67 62 68 118 117 192
20 20 20 20 30 30 40 40 40 40 70 70 70 70 120 120 200
19 17 27 30 40 40 40 40 70 70 70 70 120 120 200
20 20 30 30 40 40 40 40 70 70 70 70 120 120 200
Total
973
1020
973
980
A comparison of the number of passengers at the nodes and the vehicle capacities of the test samples reveals very small values. This makes it harder to understand the differences between PPA and other algorithms. The results of the 13th test sample partially reveal the difference of capacity-integrating PPA. The number of vehicles used by PPA and BATA and the full-capacity utilization status of these vehicles for Problem 13 is shown in Table 2. When Table 2 is examined, it is seen that BATA uses only three vehicles with full capacity; however PPA uses only three vehicles without full capacity. Additionally, for 973 passengers BATA uses 17 vehicles whereas PPA uses 15 vehicles. In other words, although the number of transported passengers is the same, the number of vehicles used is less. Considering that each vehicle used would increase the fixed cost, it is possible to say that in real life PPA is more effective than BATA in terms of the number of vehicles used and fixed costs. If the passenger demands at the stops exceed half of the vehicle capacities (e.g. demand = 18, capacity 24 or demand = 21, capacity = 30), according to the results of the algorithms in the literature, the inert capacities of the vehicles used would be quite high, which implies more vehicles and more costs. Under such circumstances, the purpose and significance of PPA, which reduces excessive vehicle use and cost, would be better understood. When the algorithms in the literature developed to solve the limited number heterogeneous vehicle fleet routing problems are examined, it is seen that when solution is sought for the same problem at different times, the same solution is not achievable due to the rational functioning of the algorithms. However, the developed PPA always leads to the same result since the algorithm does not work randomly. In other words, in PPA, there is always one intuitional best solution for the same problem. Moreover, PPA can be used both in symmetrical and asymmetrical stop distances, and can also be applied to homogenous-fixed fleet vehicle routing problem. The studies in the literature use fixed fleets and identify the problem as fixed fleet heterogeneous vehicle routing problems. In the suggested algorithm, fixed fleet can be used, but in addition to the fixed fleet there is the possibility to rent vehicles when the existing vehicles fall short. This situation can be identified as fixed-rented fleet heterogeneous vehicle routing problem. Hence, a new perspective can be introduced into the literature.
Appendix
Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
14.1421 25.9761 22.3607 38.3164 31.3242 34.9312 35.6264 44.4785 72.8962 65.4524 69.9531 83.9783 91.1478 107.6892 101.9995
14.1421 25.9761 24.5967 42.148 37.589 41.9174 42.7516 53.3742 123.9236 111.2691 118.9202 142.7631 227.8695 269.223 326.3983
No No No No No No No No No No No No No No No
19 17 27 30 40 40 40 40 70 70 70 70 120 120 200
Total length of routes: 840.2721. Total cost of routes: 1602.8622. Total # of vehicles used: 15. Demands of 50 nodes are met.
# of nodes on the route
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
20 20 30 30 40 40 40 40 70 70 70 70 120 120 200
– – – 7 from node 5 7 from node 13 4 from node 5 13 from node 7 – – 3 from node 7 8 from node 18 2 from node 7 4 from node 46 – 10 from node 27
1 2 1 3 2 3 2 2 5 4 5 6 7 6 10
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51 27 13 1 47 1 7 31 5 1 13 41 1 35 28 5 1 7 34 1 18 33 1 46 30 49 16 14 1 3 29 23 7 1 17 50 25 4 18 1 24 42 43 44 2 7 1 22 48 37 21 38 6 46 1 10 32 26 19 51 45 1 27 40 11 39 12 15 20 9 36 8 1
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Problem 13–50 nodes, fleet capacity = 1020, total demand = 973 [4]
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Problem 14–50 nodes, fleet capacity = 1100, total demand = 973 [4] Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6
61.5075 70.7044 66.948 103.9303 111.8433 183.7969
61.5075 70.7044 66.948 114.3233 123.0277 257.3156
No No No No No No
113 120 120 160 160 300
# of nodes on the route
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
120 120 120 160 160 300
– 12 from node 7 6 from node 5 7 from node 5 3 from node 3 –
6 7 7 9 10 15
1 7 17 18 41 13 27 1 1 5 35 14 28 46 31 7 1 1 5 47 9 20 15 36 8 1 1 34 29 23 43 42 44 2 3 5 1 1 30 16 21 38 6 37 48 22 49 3 1 1 24 50 25 4 45 33 51 19 26 10 40 32 11 39 12 1
Total length of routes: 598.7304. Total cost of routes: 693.8265. Total # of vehicles used: 6. Demands of 50 nodes are met.
Problem 15–50 nodes, fleet capacity = 820, total demand = 777 [4] Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9
47.6007 29.5296 33.6006 23.4962 103.1214 81.6457 64.8963 143.8849 124.2663
47.6007 29.5296 33.6006 23.4962 164.9942 130.6331 103.8341 287.7699 248.5327
No No No No No No No No No
49 41 26 42 100 100 100 159 160
Total length of routes: 652.0419. Total cost of routes: 1069.9912. Total # of vehicles used: 9. Demands of 50 nodes are met.
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
50 50 50 50 100 100 100 160 160
– – – – 12 from node 48 12 from node 28 1 from node 47 – –
1 1 1 1 1 1 1 1 1
15 19 47 13 48 28 47 18 23
26 1 1 33 2 28 1 48 1 5 43 20 41 42 14 1 49 24 8 44 25 7 1 6 50 10 17 39 12 1 38 45 16 46 34 11 40 31 35 22 30 51 1 9 27 32 29 4 37 36 21 3 1
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Route number
Problem 16–50 nodes, fleet capacity = 820, total demand = 777 [4] Route length (km)
Route variable cost
1 2 3 4 5 6 7 8 9
36.1574 16.1245 91.926 48.8894 55.1652 39.6248 118.5173 132.0092 112.7476
36.1574 16.1245 147.0816 78.2231 88.2644 63.3997 248.8863 277.2193 236.77
Vehicle Rented or not
Passengers on board
Capacity
Passengers for capacity utilization
No No No No No No No No No
32 29 79 78 72 75 137 138 137
40 40 80 80 80 80 140 140 140
– – – – – – – – –
Route (sequence of nodes)
1 1 1 1 1 1 1 1 1
7 49 1 13 1 9 8 44 25 24 1 12 39 50 6 47 1 28 2 23 3 33 1 19 5 48 1 18 45 16 46 34 11 40 31 35 10 1 27 32 29 4 37 36 21 30 22 51 17 1 38 43 20 41 42 14 26 15 1
Total length of routes: 651.1615. Total cost of routes: 1192.1263. Total # of vehicles used: 9. Demands of 50 nodes are met.
Problem 17–75 nodes, fleet capacity = 1430, total demand = 1364 [4] Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10
39.1473 35.962 15.4562 99.0592 79.2657 80.3436 42.0552 119.5908 109.3356 198.1009
39.1473 35.962 15.4562 118.8711 95.1188 96.4124 50.4663 179.3862 164.0034 356.5817
No No No No No No No No No No
50 50 46 120 117 120 116 200 200 345
50 50 50 120 120 120 120 200 200 350
2 2 – 8 – – – – 1 –
1 76 7 34 69 1 1 76 31 3 1 1 76 5 1 1 7 74 64 24 57 25 50 17 52 1 1 27 59 32 41 18 1 1 46 30 16 58 55 14 28 53 1 1 35 47 9 36 8 68 1 1 13 73 11 40 10 26 56 19 51 33 45 4 1 1 9 20 15 60 67 66 39 12 54 1 1 6 38 21 71 61 72 70 37 48 49 22 62 29 23 65 43 42 44 2 63 75 1
from node 76 from node 76 from node 7
from node 9
1563
Total length of routes: 818.3166. Total cost of routes: 1151.4053. Total # of vehicles used: 10. Demands of 75 nodes are met.
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
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Route number
1564
Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10 11 12
12.1655 14.5602 31.9296 14.1421 37.7511 46.1115 69.1204 80.114 98.6204 104.6512 102.4158 131.6086
12.1655 14.5602 31.9296 18.3848 49.0764 59.9449 89.8566 152.2165 187.3788 251.1628 245.7978 381.665
No No No No No No No No No No No No
13
152.6073
488.3432
No
Total length of routes: 895.7976. Total cost of routes: 1982.4821. Total # of vehicles used: 13. Demands of 75 nodes are met.
# of nodes on the route
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
9 6 20 29 50 50 50 100 100 150 150 250
20 20 20 50 50 50 50 100 100 150 150 250
– – 9 from node 27 – 4 from node 69 22 from node 41 – 12 from node 18 – 3 from node 4 1 from node 3 1 from node 5
1 1 2 1 3 4 3 5 7 10 9 17
400
400
–
19
1 27 1 1 69 1 1 27 41 1 151 1 7 34 69 1 1 52 4 41 18 1 1 75 62 3 1 1 59 11 32 13 18 1 1 17 50 25 57 24 64 74 1 1 73 40 10 26 56 19 51 33 45 4 1 1 3 63 29 23 65 43 42 44 2 1 1 5 46 30 6 16 58 38 21 71 61 72 70 37 48 22 49 31 1 1 76 68 35 47 53 28 14 55 9 36 20 15 60 67 66 39 12 54 8 1
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Problem 18–75 nodes, fleet capacity = 1430, total demand = 1364 [4]
Route number
Route length (km)
Route variable cost
1 2 3 4 5 6
88.5505 73.1181 67.6165 104.1546 85.8002 168.7501
7 8
Vehicle
Passengers for capacity utilization
Route (sequence of nodes)
1 19 9 47 46 85 6 84 61 90 1 1 14 88 43 44 16 58 3 1 1 29 69 30 25 81 13 1 1 93 99 101 15 39 45 17 62 92 86 38 94 100 7 1 1 95 96 98 60 87 18 97 1 1 51 77 78 4 80 79 35 10 82 34 52 21 67 66 36 72 31 71 2 70 28 1 1 32 63 11 91 33 64 65 12 20 50 37 48 49 83 8 89 53 1 1 27 55 56 26 5 40 68 24 57 76 42 23 75 73 74 22 41 59 54 1
Rented or not
Passengers Capacity on board
88.5505 73.1181 67.6165 145.8165 120.1203 286.8752
No No No No No No
100 83 89 199 135 291
100 100 100 200 200 300
– – – – – –
159.5547
271.243
No
273
300
–
133.7688
227.407
No
288
300
–
Total length of routes: 881.3136. Total cost of routes: 1280.7471. Total # of vehicles used: 8. Demands of 100 nodes are met.
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Problem 19–100 nodes, fleet capacity = 1900, total demand = 1458 [4]
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1566
Route number
Route length (km)
Route Variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10 11 12 13
62.9381 56.3679 35.1496 35.0283 24.8577 26.8343 99.7775 112.7228 93.9717 107.0747 107.1597 157.9954 158.2649
62.9381 56.3679 35.1496 35.0283 24.8577 26.8343 169.6217 191.6288 159.7519 182.027 214.3194 315.9909 316.5299
No No No No No No No No No No No No No
60 60 60 59 23 47 140 140 140 139 194 198 198
Total length of routes: 1078.1426. Total cost of routes: 1791.0454. Total # of vehicles used: 13. Demands of 100 nodes are met.
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
60 60 60 60 60 60 140 140 140 140 200 200 200
3 2 8 – – – 1 – – – – – –
1 1 1 1 1 1 1 1 1 1 1 1 1
from node 90 from node 28 from node 54
from node 7
90 19 49 89 1 28 2 31 32 1 54 27 13 29 1 90 95 96 1 28 53 1 54 59 14 1 7 62 17 87 39 15 45 92 101 1 9 47 46 84 61 6 85 18 86 94 93 1 88 3 58 16 44 43 98 38 99 60 100 97 51 77 78 69 81 30 25 55 56 26 5 1 22 73 57 40 68 24 76 42 23 75 74 41 71 21 52 36 66 67 72 10 82 34 35 79 70 11 63 91 33 64 65 12 20 50 37 48
71 1 80 4 1 83 8 1
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Problem 20–100 nodes, fleet capacity = 1520, total demand = 1458 [4]
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References [1] S. Salhi, M. Sari, D. Saidi, N.A.C. Touati, Adaptation of some vehicle fleet mix heuristics, OMEGA International Journal of Management Science 20 (1992) 653–660. [2] S. Salhi, G.K. Rand, Incorporating vehicle routing into the vehicle fleet composition problem, European Journal of Operational Research 66 (1993) 313–330. [3] I. Osman, S. Salhi, Local search strategies for the vehicle fleet mix problem, in: V.J. Rayward-Smith, I.H. Osman, C.R. Reeves, G.D. Smith (Eds.), Modern Heuristic Search Methods, Wiley, 1996, pp. 131–153. [4] E.D. Taillard, Heuristic column generation method for the heterogeneous fleet VRP, Recherche-Operationnelle 33 (1999) 1–14. [5] C.D. Tarantilis, C.T. Kiranoudis, V.S. Vassilidis, A threshold accepting metaheuristic for the heterogeneous fixed fleet vehicle routing problem, European Journal of Operational Research 152 (2004) 148–158. [6] G. Clarke, J. Wright, Scheduling of vehicles from a central depot to a number of delivery points, Operations Research 12 (1964) 568– 581. [7] D. Rosenkrantz, R. Stearns, P. Levis, An analysis of several heuristics for the traveling salesman problem, SIAM Journal on Computing 6 (1977) 563–581. [8] B. Golden, A. Assad, L. Levy, F. Gheysen, The fleet size and mix vehicle routing problem, Computers and Operations Research 1 (1) (1984) 49–66.
A new intuitional algorithm for solving heterogeneous fixed fleet routing problems: Passenger pickup algorithm Cevriye Gencer a
a,*
_ , Ismail Top b, Emel Kizilkaya Aydogan
c
Department of Industrial Engineering, Faculty of Engineering and Architecture, Gazi University, Ankara, Turkey Department of Operational Research, Defence Sciences Institute, Turkish Military Academy, Ankara, Turkey c Department of Industrial Engineering, Faculty of Engineering, Erciyes University, Kayseri, Turkey
b
Abstract Fixed-fleet heterogeneous vehicle routing is a type of vehicle routing problem that aims to provide service to a specific customer group with minimum cost, with a limited number of vehicles with different capacities. In this study, a new intuitional algorithm, which can divide the demands at the stops for fixed heterogeneous vehicle routing, is developed and tested on tests samples. The algorithm is compared to the BATA Algorithm available in the literature in relation to the number of vehicles, fixed cost, variable cost and total cost. 2006 Elsevier Inc. All rights reserved. Keywords: Routing; Algorithm; Heterogeneous fleet
1. Introduction Vehicle routing problems are divided into various areas. In most of the studies, the vehicle fleets that are studied consist of homogenous vehicles. Problems related to providing service through heterogeneous vehicles/ vehicle fleets are more comprehensive and more complicated in comparison to homogenous fleet vehicle routing problems. Therefore, the number of studies on heterogeneous fleet vehicle routing has been rather limited. Heterogeneous fleet vehicle routing problems (HFVRP) are various. One of these problems is the heterogeneous fleet mixture-vehicle routing problem (HFMVRP) which aims to find out how many of each vehicle with different capacities must be purchased or leased in order to provide a minimum-cost service to an existing customer group. HFMVRP is widely studied in the literature mostly on heuristics by researchers like Salhi et al. (1992) [1], Salhi and Rand (1993) [2], Osman and Salhi (1996) [3]. The other type of problem is the limited number heterogeneous vehicle fleet routing. Problems that involve a limited number of vehicles, constitute the most complicated and most difficult vehicle routing problems. In these problems, the aim is to provide service to the customer group with minimum cost and with the available vehicles if problem is feasible. The first study
*
Corresponding author. E-mail address: [email protected] (C. Gencer).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.02.043
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on fixed-fleet heterogeneous vehicle routing problems (FFHVRP) was carried out by Taillard (1999) [4], and the recent study was conducted by Tarantilis et al. (2004) [5]. This study also suggests a new intuitional algorithm—passenger pickup algorithm—for FFHVRP, tests it using the test samples available in the literature, and compares it to the BATA Algorithm solutions of Tarantilis et al. (2004) [5]. 2. Passenger pickup algorithm (PPA) In the literature, it is seen that studies related to vehicle routing do not divide the node demands, and node locations are designated taking into account the grid (x–y coordinates), and the distance between nodes is mostly assumed as the Euclidian distance. This assuming, eliminates the possibility of passing by another node on the way to any one from the main depot as can be seen in Fig. 1(a). However, in urban transportation, as seen in Fig. 1(b), there is the necessity for vehicles to follow specific paths to reach any node, and this path may include node(s) that are not included in the specified route. The distances between stops as used in the test samples and in the algorithms in the literature are symmetric. However, it is not always the case in real life. In urban transportation, the symmetricity can disappear. In addition, on the results of algorithms in the literature on FFHVRP, the sample node clusters does not divide the node demands in a way to maximize the utilization rate of vehicle(s). However, in daily life, the route followed may include other nodes with unsatisfied demands that are not included in current route. In such a case, vehicle has to pass from the node with unsatisfied demands, and then vehicle can fulfil its capacity by maximizing the number of passengers in the vehicle. As a result, both cost and the total number of vehicles serving is reduced. The passenger pickup algorithm (PPA) developed in this study takes into account the possibility of vehicle lease when available vehicles fall short and the divisible character of the demands, both of which are ignored in other algorithms available in the literature for FFHVRP. Moreover, it can use both symmetrical and asymmetrical inter-stop distances, in line with the daily life conditions. 2.1. Phases of PPA The algorithm functions according to the principle of first clustering-then routing. According to this principle, sub-clusters are created from the cluster of all stops, and these sub-clusters are solved as travelling salesman problems. An overview of how the algorithm works is shown in Fig. 2. It will be explained fully on further steps. 2.1.1. Step 1: the starting phase Data related to nodes is required for solution, such as number of stops, distance between stops; data related to vehicles such as vehicle types available at hand, their numbers, capacities, fixed costs and fuel rates. Distances between nodes are also acquired.
Depot
(a) According to Euclidian distance
Depot
(b) According to Urban Transportation
Fig. 1. Routes according to Euclidian and urban transportation.
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-Step 1: Enter data about nodes, P’, TD’,PP and vehicles. -Step 2: Outbound loops Do the following while (P) is not equal P’ Continue while (TD)is not equal TD’ -Step 3: While (iterate) : inbound iterate loop -Produce solution from a different starting point.S’ ∈ N(s) Accept if c(s’)
The algorithm also takes into account the cases where the vehicles available at hand are not sufficient. In other words, in this problem it also includes the leasing of additional vehicles to meet the whole node demands. Hence, the rental vehicle types, their variable and fixed leasing costs must be provided for the algorithm at the starting phase. 2.1.2. Step 2: loops This step consists of two main inter-loops. The first while loop is used for P, which is called the maximum deviation value (P) and taken as a basis in establishing the routes. P represents the acceptable changes that may occur in the route due to additional nodes inserted, if possible, when creating a route. Further details are given in Step 3. The second while loop is created for the deviation penalty coefficient (TD), which penalties the deviation and enables the creation of a route by grouping different stops as a result of an increase or decrease in its value. 2.1.3. Step 3: routing The third while (iterate) loop shows the maximum number of different stops from which the algorithm can start according to the current TD, since any different node is selected as the starting node for the first vehicle leaving the main depot. This step consists of the stages of identifying the starting node, selecting the vehicle, routing, finding the shortest route length and vehicle changes according to the number of passengers on the route. • The phase of identifying the starting node: Starts to create the first route by selecting the station farthest from the main depot among the nodes that were not selected as starting nodes before and if the demands are not met yet. If the starting station is not changed the algorithm will select the same node in each iteration. Selection of the same station prevents the occurrence of different routes and different station groups. Hence, the starting station must be changed for each iteration. According to the algorithm, after selecting the nonneighbouring starting nodes in the first two iterations, the third starting node that will be selected must be the one farthest to the last selected two starting ones. With each change of the deviation penalty coefficient (TD) within the algorithm, the iteration number is reset to null, and hence all stops are given the opportunity to be re-selected as starting nodes in later iterations. • Vehicle selection phase: Among the available unused vehicles, the one with the largest capacity is selected. If the vehicle’s capacity is not fully used in the generated route, it is changed to a vehicle of lesser capacity. It is also noted that the vehicle with lower capacity spends less fuel than the vehicle with higher capacity. In case of running out of vehicles that will be used during route generation, the algorithm chooses to rent vehicles, and rents the highest capacity vehicle available. • Routing phase: Routes are generated according to the gain formula where minimum cost algorithms developed by Clarke and Wright (1964) [6] and Rosenkrantz et al. (1977) [7], which are tour-constructing algo-
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rithms from demand-not-met nodes; maximum-gain station(s) are selected, and the routes are created through insertion. The node that is the farthest to the main depot among nodes whose demands are not met according to the algorithm is selected as the starting point (Fig. 3). Depending on the fact that the demand in some nodes are larger than some of the vehicle capacities, the station which has the highest number of passengers whose demands are not met may be selected instead of the farthest station at the beginning of the algorithm. Thus, at the beginning the route 1–2–1 is created. According to the Clarke and Wright (1964) [6] algorithm, the savings of each station with another neighbouring station is calculated as Eq. (1). S ij ¼ 2c1i þ 2c1j ðc1i þ c1j þ cij Þ ¼ c1i þ c1j cij .
ð1Þ
According to the method of Rosenkrantz et al. (1977) [7], the insertion costs for insertions at any section of the route are calculated as per Eq. (2). This cost corresponds to the deviation value. The deviation value for insertion of node j between nodes i and k is expressed as in Eq. (2) according to Fig. 4. SAijk ¼ cij þ cjk cik .
ð2Þ
In PPA, the highest-gain station (j) among those whose demands are not met is found using the gain formula (3). If SAijk 6 P ;
then Gain ¼ S ij þ S jk TD SAijk .
ð3Þ
6
2 5
7 4 3 8 9
Depot(1) Fig. 3. Selecting the route starting point.
Fig. 4. Deviation value.
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If the deviation of candidate nodes with uncovered demands that will be inserted between any two nodes on route is smaller than P, the gain is calculated as per Eq. (3), and the node with the highest gain is included in the route (e.g. 1–2–5–1). This operation is continued until the vehicle chosen for the route reaches its full capacity. If the vehicle is not completely full, the algorithm initiates the passenger pickup (capacity integration) part, where a portion of the node demand that maximizes the vehicle capacity is picked up. According to this, the node for partial passenger pickup is not as per maximum gain but as per capacity integration constraint (PP). To determine the candidate node(s) for partial passenger pickup on the generated route, the deviation values of stops whose demands are not supplied are calculated. All stops with deviations smaller than PP value are included in the candidate nodes group. In case of more than one node in the candidate station group, then the one farthest to the main depot is selected. A portion of the passengers sufficient to maximize the vehicle capacity is picked up from the selected node, and the new passenger number of the station is equalized with the number of remaining passengers. This node is considered as an unsatisfied one, until all its passengers are picked up. There is no requirement for the vehicle to always have maximum capacity. It may not be possible to find a candidate node as per PP value. In that case, the algorithm ends the routing process, and returns to the vehicle selection and starting point selection phase in order to draw a new route. These operations continue until the demands of all nodes are satisfied. The variation in the probable nodes group to be selected as per P and TD values of the gain formula is shown in Fig. 5. In Fig. 5, as the P value increases, the route will include stops with a long distance to the main depot and with high deviation values. The TD value penalizes the deviation as described in gain formula (3); therefore, nodes 5, 6 and 7 cannot be included in the route-generating nodes group after node 2 selected as the first one; and the nodes close to the 1–2–1 route generated initially, that is towards the main depot, are included in the route. Considering that the nodes are dispersed on a plane, the gain value for the selected route initial node (like node no. 2) changes by decreasing P and TD values. Hence, creating all probable groups can assess the stops. At the end of the iterations, the TD value is subtracted and reset to null again, and when the decrease in TD value finishes, the P value is subtracted, and therefore TD regains its maximum value. • Finding the shortest path for the route: The nodes constituting the vehicle route are sub-cluster of the all nodes cluster. All stops in this sub-cluster must be travelled from the shortest path. At this stage, the problem is reduced to a travelling salesman problem (TSP). Here, also the least costly insertion method of Rosenkrantz et al. (1977) [7] is used. Among the nodes creating the route, the one farthest to the main depot is selected as the starting point. After drawing the starting route (such as 1–2–1) the node (j) with the least deviation (SAijk) is inserted between nodes i and k (1, . . . , i j k, . . . , 1). The insertion process is continued until the routing is complete for all nodes in the sub-cluster.
6
2
5
7
6
2
5
7 4 8
4 3
9
Depot(1)
(a) The effect of change of P to the stops group
8
3 9
Depot(1)
(b) Effect of TD change to stops group
Fig. 5. The effect of P and TD variation to routing.
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• Vehicle changing phase: After the route is generated, the vacancy status of the vehicle assigned to this route is checked. If there is an unused vehicle that has the capacity larger than the number of passengers on board, it is changed with the vehicle on this route. The purpose of this changing procedure is to serve the given route with minimum cost. Using the inert-capacity vehicle on the said route means ignoring situations of possible economy in the subsequent route selection. The vehicle allocated to any route must have rate of maximum utilization when considering all the other vehicles. For example, considering a route with 25 passengers, assuming that the available vehicles have a passenger capacity of 29 and 44, assigning the vehicle with 29-passenger capacity to this route would be economical. Therefore, at the vehiclechanging phase, the vehicle that will minimize inert capacity and route cost should be allocated to the route.
2.1.4. Step 4: termination After working within the framework of the constraints in the first three steps, the algorithm compares the value acquired in each iteration to the best value; if this value is smaller than the minimum value in the memory, this value is taken as the minimum value. Following the loops, the termination phase starts. The minimum value found, the route number, together with the route length, route cost, whether vehicle is rented or not, number of passengers on the route, the capacity of the vehicle on the route, how many additional passengers must be picked up from which stops to maximize the capacity, and the routes to be followed starting from the main depot selected as the first depot and then again returning to the main depot as the finish line, are printed out.
3. Applying the algorithm to test samples In the literature, Golden et al. (1984) [8] suggests test samples for different types of vehicle routing problems. Those samples between 13 and 20 are rearranged by Taillard (1999) [4] for FFHVRP. The relevant samples are also used for PPA testing in the study. In PPA, first of all the P value which determines the maximum deviation must be identified. The tests carried out indicate that relating P value to the distance of the stop farthest to the main depot (xu) gives the best result. Therefore, P = xu/4 was used in the algorithm test. Hence, the algorithm starts working by determining the P value, and continues to work while the value of P decreases. During the tests, to enable the algorithm to reach the result in a short time, P was reduced as P = P 5. TD must be changed according to the distribution of nodes. The tests revealed that, in cases where the main depot as location in the samples is away from the group of nodes, a good result is acquired when TD value is 5 or lower; however, in cases where the main depot is in the centre and where the stops are not collected at specific locations, it is observed that a good result is acquired when TD assumes a value between 5 and 10. Building on these tests, TD value is assumed as (0 6 TD 6 10) and reduced as (TD = TD 1), and applied to the problems. The algorithm can produce n solutions with different starting points when the number of nodes excluding the main depot is n. Since selecting the nodes in the vicinity of main depot as starting solution does not contribute to generating a different route, these nodes must not be selected for starting solutions. Assuming half of the nodes are in the vicinity of main depot, to shorten the run-time of the algorithm and to reduce the number of iterations it was anticipated that an iteration number of n/2 would be useful during the initial test phase of PPA. Generating the nodes cluster by distribution on a random plane and using Euclidean distances between nodes is not compatible with the development purpose and working principles of the PPA. Therefore, applying PPA, which enables additional passenger pickup via dividing the node demands and which aims to maximize the vehicle capacity, will result in increased route distances, and therefore increased costs. For that reason, first the passenger pickup (capacity integration) part of the algorithm was eliminated in the PPA test by assuming the capacity integrator as (PP < 0), and the algorithm was run and tested without dividing the demands of the stops. The best result was gathered in this way in the 16th and 19th test
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Problem no.
Tarantilis (BATA) Type of vehicle used and default
Total # of vehicles used
Fixed cost
Variable cost
Overall cost
Type of vehicle used and default
Total # of vehicles used
Fixed cost
Variable cost
Overall cost
13 14 15 16 17 18 19 20
4A, 3A, 4A, 2A, 4A, 3A, 3A, 6A,
17 6 9 9 11 13 8 13
1685 6800 2050 2200 1040 1930 9300 3300
1519.96 611.39 1015.29 1145.52 1071.01 1846.35 1123.83 1556.35
3204.96 7411.39 3065.29 3345.52 2111.01 3776.35 10,423.83 4856.35
2A, 4A, 4A, 2A, 3A, 3A, 3A, 6A,
15 6 9 9 10 13 8 13
1640 6800 2050 2200 1015 1930 10,200 3300
1602.862 693.8265 1069.991 1192.126 1151.405 1982.482 1280.747 1791.045
3242.8622 7493.8265 3119.9912 3392.1263 2166.4053 3912.4821 11,480.747 5091.0454
2B, 2B, 3B, 4B, 4B, 4B, 3B, 4B,
4C, 4D, 2E, 1F 1C SC 3C 2C, 1D 2C, 2D, 1E, 1F 2C 3C
Passenger pickup algorithm (PPA)
2B, 2B, 3B, 4B, 4B, 4B, 2B, 4B,
4C, 4D, 2E, 1F 1C 2C 3C 2C, 1D 2C, 2D, 1E, 1F 3C 3C
PPA BATA derivation (%)
1.183 1.112 1.785 1.393 2.624 3.605 10.139 4.833
C. Gencer et al. / Applied Mathematics and Computation 181 (2006) 1552–1567
Table 1 A comparison of BATA and PPA according to overall costs
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problems compared to PPA. Then, tests were conducted with a capacity integration coefficient of PP = P/5 and PP = P/10. In the literature, only the variable costs are taken into account when testing FFHVRP operations. However, in real life, the costs of vehicles do not only consist of variable costs. Particularly when considering that the vehicles are used for profit purposes, it is clear that fixed costs also have a great significance. Not including the fixed costs in the assessments would be a great mistake for enterprises that aim to maximize their profits. Therefore, it would be more appropriate to test the algorithms by taking into account the fixed costs, variable costs and overall costs. There are no studies on FFHVRP in the literature that makes an assessment that takes into account the overall and fixed costs. PPA was compared to Tarantilis et al. (2004) [5] BATA algorithm. Although PPA test was conducted on the stops cluster generated contrary to its development purposes, good results were obtained. PPA and BATA Algorithms were subjected to eight test problems; tests and comparisons were made for four different aspects, namely the number of vehicles, fixed costs, variable costs and overall costs. Tarantilis’ study includes only the variable costs. To enable a comparison with the PPA, overall costs were calculated by adding the fixed costs of Taillard’s (1999) [4] study to the BATA algorithm results. The results acquired according to the test problems are shown in Table 1. Detailed results as per variable costs are given in Appendix. When Table 1 is examined in terms of total number of vehicles used, it is seen that PPA and BATA give the same results for problems 14, 15, 16, 18, 19, 20; whereas PPA gives better results for problems 13 and 17. PPA divides the node demands and maximizes the vehicle utilization; hence reveals the distinction of the algorithm as it aims for minimum cost with fewer vehicles. When Table 1 is examined in terms of fixed costs, it is seen that PPA gives better results for problems 13 and 17, and both PPA and BATA give the same results for problems 14, 15, 16, 18, 20; whereas BATA algorithm gives a better result for problem 19. The results found based on fixed costs also reveal a parallelism with total numbers of the used vehicles. Since fixed costs are the total fixed costs of vehicles used, PPA gives a better result than BATA for test problems 13 and 17. When Table 1 is examined in terms of variable costs, it is seen that PPA fails to reach a better result than BATA. However, this is due to the fact that the test problems do not correspond to the real life problems for which the algorithm was developed. When Table 1 is examined in terms of overall costs; PPA fails to get a better result than BATA. The reason for it is because test problems are very specific problems. In the problems, the total passenger capacity of the fleets is bigger than the total demand at the stops, but only at the rate of approximately the smallest vehicle capacity. Therefore, PPA fails to reduce the vehicle numbers in many of the problems. For example in problem 15, there is the obligation to use all vehicles. If the stop demands and vehicle numbers increase, PPA will achieve a better result due to its capacity integration ability, in which case the fixed costs and overall costs would be reduced. However, the studies available in the literature do not have the capacity integration ability, therefore when the stop demands and the vehicle numbers increase, the costs will also increase. In addition, when the number of nodes is decreased without changing the demand or the number of available vehicles, the demand per station would rise, and therefore, according to the algorithms in the literature, the vehicle number would become insufficient and the problem will remain unsolved or will not be feasible. Yet, PPA will be able to solve the problem thanks to its capacity integration feature. In all problems, when the deviation values are taken into consideration, it is worth noting that PPA’s results are rather close to BATA results. In eight test problems, the mean deviation of PPA is 3.334%.
4. Conclusion In the study, an algorithm oriented to full-use of the existing vehicle capacity was developed. The developed algorithm was compared to the BATA Algorithm in the literature, using the test problems.
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Table 2 Capacity utilization of BATA and PPA in problem 13 Vehicle no.
Tarantilis (BATA) # of passengers
PPA Vehicle Capacity
# of passenger
Vehicle Capacity
Problem 13 [4] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 20 19 19 30 27 33 40 37 39 67 67 62 68 118 117 192
20 20 20 20 30 30 40 40 40 40 70 70 70 70 120 120 200
19 17 27 30 40 40 40 40 70 70 70 70 120 120 200
20 20 30 30 40 40 40 40 70 70 70 70 120 120 200
Total
973
1020
973
980
A comparison of the number of passengers at the nodes and the vehicle capacities of the test samples reveals very small values. This makes it harder to understand the differences between PPA and other algorithms. The results of the 13th test sample partially reveal the difference of capacity-integrating PPA. The number of vehicles used by PPA and BATA and the full-capacity utilization status of these vehicles for Problem 13 is shown in Table 2. When Table 2 is examined, it is seen that BATA uses only three vehicles with full capacity; however PPA uses only three vehicles without full capacity. Additionally, for 973 passengers BATA uses 17 vehicles whereas PPA uses 15 vehicles. In other words, although the number of transported passengers is the same, the number of vehicles used is less. Considering that each vehicle used would increase the fixed cost, it is possible to say that in real life PPA is more effective than BATA in terms of the number of vehicles used and fixed costs. If the passenger demands at the stops exceed half of the vehicle capacities (e.g. demand = 18, capacity 24 or demand = 21, capacity = 30), according to the results of the algorithms in the literature, the inert capacities of the vehicles used would be quite high, which implies more vehicles and more costs. Under such circumstances, the purpose and significance of PPA, which reduces excessive vehicle use and cost, would be better understood. When the algorithms in the literature developed to solve the limited number heterogeneous vehicle fleet routing problems are examined, it is seen that when solution is sought for the same problem at different times, the same solution is not achievable due to the rational functioning of the algorithms. However, the developed PPA always leads to the same result since the algorithm does not work randomly. In other words, in PPA, there is always one intuitional best solution for the same problem. Moreover, PPA can be used both in symmetrical and asymmetrical stop distances, and can also be applied to homogenous-fixed fleet vehicle routing problem. The studies in the literature use fixed fleets and identify the problem as fixed fleet heterogeneous vehicle routing problems. In the suggested algorithm, fixed fleet can be used, but in addition to the fixed fleet there is the possibility to rent vehicles when the existing vehicles fall short. This situation can be identified as fixed-rented fleet heterogeneous vehicle routing problem. Hence, a new perspective can be introduced into the literature.
Appendix
Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
14.1421 25.9761 22.3607 38.3164 31.3242 34.9312 35.6264 44.4785 72.8962 65.4524 69.9531 83.9783 91.1478 107.6892 101.9995
14.1421 25.9761 24.5967 42.148 37.589 41.9174 42.7516 53.3742 123.9236 111.2691 118.9202 142.7631 227.8695 269.223 326.3983
No No No No No No No No No No No No No No No
19 17 27 30 40 40 40 40 70 70 70 70 120 120 200
Total length of routes: 840.2721. Total cost of routes: 1602.8622. Total # of vehicles used: 15. Demands of 50 nodes are met.
# of nodes on the route
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
20 20 30 30 40 40 40 40 70 70 70 70 120 120 200
– – – 7 from node 5 7 from node 13 4 from node 5 13 from node 7 – – 3 from node 7 8 from node 18 2 from node 7 4 from node 46 – 10 from node 27
1 2 1 3 2 3 2 2 5 4 5 6 7 6 10
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
51 27 13 1 47 1 7 31 5 1 13 41 1 35 28 5 1 7 34 1 18 33 1 46 30 49 16 14 1 3 29 23 7 1 17 50 25 4 18 1 24 42 43 44 2 7 1 22 48 37 21 38 6 46 1 10 32 26 19 51 45 1 27 40 11 39 12 15 20 9 36 8 1
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Problem 13–50 nodes, fleet capacity = 1020, total demand = 973 [4]
1561
1562
Problem 14–50 nodes, fleet capacity = 1100, total demand = 973 [4] Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6
61.5075 70.7044 66.948 103.9303 111.8433 183.7969
61.5075 70.7044 66.948 114.3233 123.0277 257.3156
No No No No No No
113 120 120 160 160 300
# of nodes on the route
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
120 120 120 160 160 300
– 12 from node 7 6 from node 5 7 from node 5 3 from node 3 –
6 7 7 9 10 15
1 7 17 18 41 13 27 1 1 5 35 14 28 46 31 7 1 1 5 47 9 20 15 36 8 1 1 34 29 23 43 42 44 2 3 5 1 1 30 16 21 38 6 37 48 22 49 3 1 1 24 50 25 4 45 33 51 19 26 10 40 32 11 39 12 1
Total length of routes: 598.7304. Total cost of routes: 693.8265. Total # of vehicles used: 6. Demands of 50 nodes are met.
Problem 15–50 nodes, fleet capacity = 820, total demand = 777 [4] Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9
47.6007 29.5296 33.6006 23.4962 103.1214 81.6457 64.8963 143.8849 124.2663
47.6007 29.5296 33.6006 23.4962 164.9942 130.6331 103.8341 287.7699 248.5327
No No No No No No No No No
49 41 26 42 100 100 100 159 160
Total length of routes: 652.0419. Total cost of routes: 1069.9912. Total # of vehicles used: 9. Demands of 50 nodes are met.
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
50 50 50 50 100 100 100 160 160
– – – – 12 from node 48 12 from node 28 1 from node 47 – –
1 1 1 1 1 1 1 1 1
15 19 47 13 48 28 47 18 23
26 1 1 33 2 28 1 48 1 5 43 20 41 42 14 1 49 24 8 44 25 7 1 6 50 10 17 39 12 1 38 45 16 46 34 11 40 31 35 22 30 51 1 9 27 32 29 4 37 36 21 3 1
C. Gencer et al. / Applied Mathematics and Computation 181 (2006) 1552–1567
Route number
Problem 16–50 nodes, fleet capacity = 820, total demand = 777 [4] Route length (km)
Route variable cost
1 2 3 4 5 6 7 8 9
36.1574 16.1245 91.926 48.8894 55.1652 39.6248 118.5173 132.0092 112.7476
36.1574 16.1245 147.0816 78.2231 88.2644 63.3997 248.8863 277.2193 236.77
Vehicle Rented or not
Passengers on board
Capacity
Passengers for capacity utilization
No No No No No No No No No
32 29 79 78 72 75 137 138 137
40 40 80 80 80 80 140 140 140
– – – – – – – – –
Route (sequence of nodes)
1 1 1 1 1 1 1 1 1
7 49 1 13 1 9 8 44 25 24 1 12 39 50 6 47 1 28 2 23 3 33 1 19 5 48 1 18 45 16 46 34 11 40 31 35 10 1 27 32 29 4 37 36 21 30 22 51 17 1 38 43 20 41 42 14 26 15 1
Total length of routes: 651.1615. Total cost of routes: 1192.1263. Total # of vehicles used: 9. Demands of 50 nodes are met.
Problem 17–75 nodes, fleet capacity = 1430, total demand = 1364 [4] Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10
39.1473 35.962 15.4562 99.0592 79.2657 80.3436 42.0552 119.5908 109.3356 198.1009
39.1473 35.962 15.4562 118.8711 95.1188 96.4124 50.4663 179.3862 164.0034 356.5817
No No No No No No No No No No
50 50 46 120 117 120 116 200 200 345
50 50 50 120 120 120 120 200 200 350
2 2 – 8 – – – – 1 –
1 76 7 34 69 1 1 76 31 3 1 1 76 5 1 1 7 74 64 24 57 25 50 17 52 1 1 27 59 32 41 18 1 1 46 30 16 58 55 14 28 53 1 1 35 47 9 36 8 68 1 1 13 73 11 40 10 26 56 19 51 33 45 4 1 1 9 20 15 60 67 66 39 12 54 1 1 6 38 21 71 61 72 70 37 48 49 22 62 29 23 65 43 42 44 2 63 75 1
from node 76 from node 76 from node 7
from node 9
1563
Total length of routes: 818.3166. Total cost of routes: 1151.4053. Total # of vehicles used: 10. Demands of 75 nodes are met.
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
C. Gencer et al. / Applied Mathematics and Computation 181 (2006) 1552–1567
Route number
1564
Route number
Route length (km)
Route variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10 11 12
12.1655 14.5602 31.9296 14.1421 37.7511 46.1115 69.1204 80.114 98.6204 104.6512 102.4158 131.6086
12.1655 14.5602 31.9296 18.3848 49.0764 59.9449 89.8566 152.2165 187.3788 251.1628 245.7978 381.665
No No No No No No No No No No No No
13
152.6073
488.3432
No
Total length of routes: 895.7976. Total cost of routes: 1982.4821. Total # of vehicles used: 13. Demands of 75 nodes are met.
# of nodes on the route
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
9 6 20 29 50 50 50 100 100 150 150 250
20 20 20 50 50 50 50 100 100 150 150 250
– – 9 from node 27 – 4 from node 69 22 from node 41 – 12 from node 18 – 3 from node 4 1 from node 3 1 from node 5
1 1 2 1 3 4 3 5 7 10 9 17
400
400
–
19
1 27 1 1 69 1 1 27 41 1 151 1 7 34 69 1 1 52 4 41 18 1 1 75 62 3 1 1 59 11 32 13 18 1 1 17 50 25 57 24 64 74 1 1 73 40 10 26 56 19 51 33 45 4 1 1 3 63 29 23 65 43 42 44 2 1 1 5 46 30 6 16 58 38 21 71 61 72 70 37 48 22 49 31 1 1 76 68 35 47 53 28 14 55 9 36 20 15 60 67 66 39 12 54 8 1
C. Gencer et al. / Applied Mathematics and Computation 181 (2006) 1552–1567
Problem 18–75 nodes, fleet capacity = 1430, total demand = 1364 [4]
Route number
Route length (km)
Route variable cost
1 2 3 4 5 6
88.5505 73.1181 67.6165 104.1546 85.8002 168.7501
7 8
Vehicle
Passengers for capacity utilization
Route (sequence of nodes)
1 19 9 47 46 85 6 84 61 90 1 1 14 88 43 44 16 58 3 1 1 29 69 30 25 81 13 1 1 93 99 101 15 39 45 17 62 92 86 38 94 100 7 1 1 95 96 98 60 87 18 97 1 1 51 77 78 4 80 79 35 10 82 34 52 21 67 66 36 72 31 71 2 70 28 1 1 32 63 11 91 33 64 65 12 20 50 37 48 49 83 8 89 53 1 1 27 55 56 26 5 40 68 24 57 76 42 23 75 73 74 22 41 59 54 1
Rented or not
Passengers Capacity on board
88.5505 73.1181 67.6165 145.8165 120.1203 286.8752
No No No No No No
100 83 89 199 135 291
100 100 100 200 200 300
– – – – – –
159.5547
271.243
No
273
300
–
133.7688
227.407
No
288
300
–
Total length of routes: 881.3136. Total cost of routes: 1280.7471. Total # of vehicles used: 8. Demands of 100 nodes are met.
C. Gencer et al. / Applied Mathematics and Computation 181 (2006) 1552–1567
Problem 19–100 nodes, fleet capacity = 1900, total demand = 1458 [4]
1565
1566
Route number
Route length (km)
Route Variable cost
Vehicle Rented or not
Passengers on board
1 2 3 4 5 6 7 8 9 10 11 12 13
62.9381 56.3679 35.1496 35.0283 24.8577 26.8343 99.7775 112.7228 93.9717 107.0747 107.1597 157.9954 158.2649
62.9381 56.3679 35.1496 35.0283 24.8577 26.8343 169.6217 191.6288 159.7519 182.027 214.3194 315.9909 316.5299
No No No No No No No No No No No No No
60 60 60 59 23 47 140 140 140 139 194 198 198
Total length of routes: 1078.1426. Total cost of routes: 1791.0454. Total # of vehicles used: 13. Demands of 100 nodes are met.
Route (sequence of nodes)
Capacity
Passengers for capacity utilization
60 60 60 60 60 60 140 140 140 140 200 200 200
3 2 8 – – – 1 – – – – – –
1 1 1 1 1 1 1 1 1 1 1 1 1
from node 90 from node 28 from node 54
from node 7
90 19 49 89 1 28 2 31 32 1 54 27 13 29 1 90 95 96 1 28 53 1 54 59 14 1 7 62 17 87 39 15 45 92 101 1 9 47 46 84 61 6 85 18 86 94 93 1 88 3 58 16 44 43 98 38 99 60 100 97 51 77 78 69 81 30 25 55 56 26 5 1 22 73 57 40 68 24 76 42 23 75 74 41 71 21 52 36 66 67 72 10 82 34 35 79 70 11 63 91 33 64 65 12 20 50 37 48
71 1 80 4 1 83 8 1
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Problem 20–100 nodes, fleet capacity = 1520, total demand = 1458 [4]
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1567
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