ATTRIBUTE EXPLAINATION ON CUSTOMER SATISFACTION
4.4 TWO MODELS
In this section, we formulate two models that are both based on mathematical formulations in Appendix C. Connexxion wants to see the effects when using fixed detour time. This means no matter the duration of the ordered ride, the detour time remains the same. The second model uses a factor, in this case de detour time is the shortest ride time times the factor. This means short rides have a relative short detour time, and longer rides a relative longer detour time. In Section 4.4.1 the model with a fixed detour time is explained, while in Section 4.4.2 the second model with detour factors is explained.
4.4.1 EXPLANATION OF MODEL 1
Times of a request
Model 1: Here we use the concept of a fixed detour time. The latest pickup time is equal to the earliest pickup time plus flexibility time (TW size). The latest arrival time is equal to the latest possible pickup time plus the direct ride time. Hence the planning flexibility might be ‘considered in two ways, either by waiting at the pickup location or by the detour time.
Example
The basic idea of the multi vehicle DARP, is serving all the customer requests. The routes are not fixed but are created based on the customer requests. Below a simplified example is shown of Model 1. The example has the following parameters: a maximum detour of 10 minutes, a maximum call time of 5 minutes, and a flexibility of 15 minutes.
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[8:30 ; 8:45] [9:15]
Legend
Route of Vehicle 1 [U:MM ; U:MM] Time Window [E,L]
[U:MM] Latest Arrival Time
50 Connexxion Figure 4.2 shows the route of vehicle 1 servicing only one customer request, since no more requests are
known. Vehicle 1 starts the route to service customer 1. At 8:29 another customer sends a request, see Figure 4.3.
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[8:30 ; 8:45] [8:33 ; 8:48] [9:15] [9:24]Infeasible route of Vehicle 1 Route of Vehicle 2 Legend 15 min26 min 13 min Route of Vehicle 1 [U:MM ; U:MM] Time Window [E,L]
[U:MM] Latest Arrival Time
Figure 4.3: Model 1 with two customers
Figure 4.3 shows that a second vehicle is needed to serve the second request, since the agreed latest arrival time, and the maximum detour time of customer 1 are violated, if vehicle 1 arrives at 8:30 at the pickup location of customer 1, then vehicle 1 can pickup customer 2 at 8:45, so the time windows of both customers
are feasible. To check if customer 2 can be inserted in the current route we check the following sequence options before visiting the drop-off location 1, the sequences and the time of arrival in the drop-off location is calculated (assuming pickup location (1+) is visited at 8:30): 1+, 1−= 9: 00; 1+, 2+, 1−= 9: 11; 1+, 2+, 2−, 1−= 9: 34; 1+, 2+, 1−, 2−= 9: 24. The sequence 1+, 2+, 2−, 1− is infeasible due to the violation
of the latest arrival time of customer 1. The sequence 1+, 2+, 1−, 2− is not violating the pickup TW or the
latest arrival times of both request, but this sequences is infeasible due to violating the maximum detour time. The detour for customer 1 is 11 minutes, while the maximum detour time is 10 minutes. So a second vehicle is needed to accept the request of customer 2. In Figure 4.4, the model is extended with a third customer.
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Infeasible route of Vehicle 1
10 min 23 min 18 min 2 min 19 min [U:MM ; U:MM]
Time Window [E,L]
[U:MM]
Latest Arrival Time
S o l u t i o n D e s i g n
M.H. Matena 51
Figure 4.4 shows that the third customer sends in a request at 8:33, the current situation consists of two routes 1+, 1− and 2+, 2−, driven by two vehicles. If we add customer 3 to the route of vehicle 1. The pickup
time of customer 3 is violated, so an insertion in route 1 is infeasible. When we add customer 3 in the route
of vehicle 2, the pickup TW, and the maximum detour time are not violated. The following route sequence is
possible 2+, 3+, 2−, 3− without violating any restriction. Customer 3 is inserted in vehicle 2. It is also possible to pickup customer 3 before customer 2, the route then becomes 3+, 2+, 2−, 3−, but the request of customer
3 is done at 8:33, exactly the moment that vehicle 2 arrives at the pickup location of customer 2. So this sequence is rejected, since customer 2 is already in the vehicle. The option of using a new vehicle, if available, is also considered, but using a new vehicle comes with high start costs, so this option is only considered when a request cannot be assigned to one of the current routes.
4.4.2 EXPLANATION OF MODEL 2
Times of a request
Model 2: Here we use the concept of using a factor. The maximum detour is determined by multiplying the direct driving time with the detour factor. All the other parameters determined in the same manner as model 1.
Example
Model 2 uses another TW strategy, using a detour factor. The maximum detour time is the shortest path time times the detour factor. The model is explained with the use of a simplified example given below. The insertion of the first request is the same as shown in Figure 4.2, the insertion of the second customer is different. The example has the following parameters: a detour factor of 1.5, a maximum call time of 5 minutes, and a TW size of 15 minutes. Legend
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[8:30 ; 8:45] [8:33 ; 8:48] [9:15] [9:24]Infeasible route of Vehicle 1 15 min 26 min
13 min
Route of Vehicle 1 [U:MM ; U:MM] Time Window [E,L]
[U:MM] Latest Arrival Time
Figure 4.5: Model 2 with two requests
Figure 4.5 shows a second customer has send a request. Compared to the Figure 4.3 it is possible to service both requests with one vehicle. Since the detour time of customer 1 is 15 minutes, and the latest arrival times
52 Connexxion are not violated. The feasible route shown in Figure 4.5, shows that the arrival time at the drop-off location of the second request is 9:24.
Legend
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Infeasible route of Vehicle 1 Route of Vehicle 2 Route of Vehicle 1
Infeasible route of Vehicle 1
[U:MM ; U:MM]
Time Window [E,L]
[U:MM]
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In Figure 4.6 a third request is made by customer 3. Since the actual pickup time is already communicated to customer 2, it is not allowed to insert customer 3 before customer 2. If we insert customer 3 after the pickup location of customer 2, the pickup TW of customer 3 is violated. If an extra vehicle is available the request is assigned to that vehicle, else if no more vehicles are available the request is outsourced to a third party.
4.4.3 CONCLUSION
From the examples we see both models handle the requests differently. In the first model, customer 2 is assigned to a new vehicle, and customer 3 inserted in the route of the second vehicle. The second model shows that customer 2 is inserted in the route of the first vehicle while the customer 3 is assigned to a new vehicle.
4.5
CONCLUSION
This chapter introduced an insertion method that incorporates all restrictions mentioned in the mathematical model. The insertion method assigns the requests that are known to a vehicle with the lowest costs. The narrow time windows reduces the number of possible insertions drastically.
In this chapter we formulated two construction models, the first model makes use of a fixed maximum ride time. The second model uses a maximum ride time that is based on a factor times the direct driving time. The next chapter simulates both models and report the results.
M.H. Matena 53
5
PERFORMANCE EVALUATION
“An ounce of performance is worth pounds of promises.” - Mae West
In this chapter, we test our solution models, and present the results. We start by describing the experiments in Section 5.1, followed by the results of the experiments in Section 5.2. Section 5.3 provides an overall conclusion.