The aim of this model is to find the optimal headways respectively for the two systems. The principal trade-off is between the passengers waiting time and operating costs. As the bus headway becomes shorter, passengers waiting time decreases and the service level to passenger increases.
However, simultaneously the operating cost for bus operators increases too, because by which it means dispatching a larger number of buses for the same period of time. In contrast, if the headway becomes longer, the number of buss dispatched will be smaller. The passenger waiting time will then be longer and the service level lower. Such relationship can be illustrated in Figure 3 below.
Figure 3 Cost function of the system
Basic assumptions of the model are as follows:
1. Arrival of passenger at a bus stop is assumed to be random
2. Passengers are assumed to board the first bus arrived at the bus stop.
3. For the system with the information, perceived waiting time is assumed to be the same as the actual one.
4. Every bus will experience the same condition during the trip, the running time is stochastic; and as soon as the bus arrives at the terminal station, it will return to the starting station along the same route.
5. The number of buses is sufficient to cover the capacity requirements of different headway.
The objective function is the minimization of the system’s total cost ( ), which consists of passenger cost ( ) and operator cost (Co), that is,
(1) The passenger cost per day is equal to the passenger waiting cost q multiplied by the perceived waiting time, total ridership per unit time R and total service hours per day T, which gives:
(2) The passenger waiting cost can be estimated by wage rate per unit time. Based on the utility function theory, we can build the waiting cost function as:
(3)
where g is the wage rate, t is the waiting time, and k is the growth constant.
The value of the growth constant k can be derived from the passenger survey, and it indicates the frequency per unit time for the cost increasing by a factor e. The value of the growth constant related with the time threshold the passengers have for waiting. It is different for each person, depends on socio-economy aspects, such as jobs, social status, and how much their concern about time.
The visibility provided by RFID will remove passengers’ uncertainty before or while waiting.
If the passengers can get the information before their arrival to the bus station, their arrival is less random, because they will be able to coordinate their arrival with the bus arrival to minimize the
waiting time.
Figure 4 Different passengers behavior for system with vs. without RFID
Figure 4 describes the difference will experienced by passengers when the RFID is used in the bus route. When the passengers provided with the information on bus location, they can time their arrival, so the arrival will be less random (although it will be still random), and in the bus station they also will get the information about the recent location of the bus, which will remove the uncertainty while waiting. In this case, the passengers expected waiting time will be shorter.
Based on previous researches on waiting time, the derivation of Ew[t] is as follows. For the passengers arriving randomly at a stop, the probability density function (p.d.f.) of the waiting time
can be estimated by the p.d.f. of the headway on a specific bus route ; and the relationship is given as below (Larson et al., 1981):
= λ H(t) (4)
where is cumulative distribution function of the headway, H(t) =1- , E [h]is the mean
headway, and is the bus frequency where λ= .
Furthermore, the expected value or the mean of the waiting time can be calculated as:
Ew[t] =
(5)Marguier et al. (1984) give the direct result of the mean waiting time as:
Ew[t] =
[1+ ] (6)where varH is the variance of the buses headway.
Instead of the difference on expected waiting time, there also will be difference in the
passengers’ perceived waiting time. Understanding the relationship between perceived waiting time
and actual waiting time is important. In this model, we use perceived waiting time instead of actual waiting time as the variable of the passengers cost, as it is related to the service level for passengers.
Using the same headway value, the passengers perceived waiting time for the system without RFID will be higher, since the passengers do not have information. We use perceived waiting time
coefficient (α) to distinguish the perceived waiting time from the actual one, thus:
= α Ew[t] (7)
where: 1 ≤ α ≤ 2In the non-RFID system, the perceived waiting time is higher than the actual one. Based on the study by Mishalani et al. (2006), the perceived waiting time is assumed to have a linear correlation with the expected waiting time Ew[t], with:
= 1.33+0.92 Ew[t] +ε (8)
Another study by Hess et al. (2004) proved the previous studies conclusion that bus riders perceived their wait time to be almost twice what it actually is (α = 2), so:= 2 Ew[t] (9)
In the system with RFID, the perceived waiting time is assumed to be the same with the actual waiting time (α = 1). It is because the customers have the real-time location information of the bus.It will give:
= Ew[t] (10)
The operator cost per day is equal to the n routes sum of bus operating cost per hour (p)multiplied by the vehicle service hours per day. The bus operating cost can be estimated from fuel cost, wage rate, maintenance expenses, and the depreciation of the bus. Vehicle service hours per day is the ratio of the average round trip running time E [t] to the average headway h multiplied by total service hours per day T, that is,
=p (11)
Each trip is a stochastic process with the p.d.f. of the bus service time in one route being r (t).
Therefore, the mean running time per trip will be given as:
E[t]=
(12)The system total cost can then be obtained by putting together operator costs and passenger costs, and with h as a decision variable. The optimal headway must meet the capacity constraint of the bus c, and it should meet the passengers demand (ridership), which gives:
(13)
Then the system total cost can be formulated as follows.
min E[ ]= q(t)
+p T
= q(t) p (14)
Where 0 < h
Solving the objective function, we can get the system total cost and optimal headway for the system with RFID and the one without. The difference between the two costs can then be derived as the benefit of using RFID in the system.
Furthermore, the verification process should be followed by Cost-Benefit Analysis which is the process of measuring the trade-off between the total cost required for RFID investment and the benefits derived from the project. This is a fundamental and indispensable step, because it is the main analysis behind a “go” or “no-go” decision. This analysis can be formulated as a return on investment value, as follow:
ROI =
(15)
By comparing the minimum-cost headway for the system using RFID technology with the one without, the amount in cost reduction can be obtained and will be considered as the benefit of RFID application. The RFID implementation cost consist of RFID tag for each bus, RFID reader installed in every bus station, bus stops, traffic lights and lamp posts, the display infrastructure at every bus stops, and the RFID middleware infrastructure – software, servers, and database.
Table 1 Cost and Benefit Analysis for RFID application in public transportation
Additionally, the payback period also can be used as the analysis tools, and when the analysis resulted in the feasible payback period and return on investment, the operator can be confidently implement RFID in their system.