## 3 Multi-Type Bus Operating Organization Energy Consumption Optimization Model
### Model Establishment
#### 3.2.2 Objective Function  
In the process of optimizing the multi-type bus operating plan while controlling the total energy consumption, we consider the service level of the bus, which includes the comfort indicators (waiting time, full load rate), convenience indicators (transfer distance, station coverage rate) and economic indicators (fare). However, for fixed bus lines, the transfer distance and station coverage indicators have been determined, and the ticket fare has no relation to the departure frequency, so the service level indicators considered in the paper mainly contains the waiting time and full load rate. The time cost of passengers waiting for a bus is mainly related to the time of waiting, and the full load rate is associated with the time cost of the passengers' in-bus time, because passengers have different perceptions of the travel time for the same travel distance under different full load rates. Therefore, we establish a multi-objective bus operating organization energy consumption optimization model. The objectives include the total energy consumption, passengers waiting time and passenger's in-bus time. By transforming different objectives into cost, multiple objectives can be converted into a single optimization objective with a unified unit of measurement, which can reduce the complexity and difficulty of solving the model. The minimum total energy consumption \(E\) can be calculated as the sum of the product of vehicle per unit energy consumption and driving mileage for each task, which is shown as Formula (1). \[\text{min}E=\sum_{p\in\Omega}\sum_{k\in K}\sum_{m\in M}e_{k}L_{p}\beta_{mp}^{k }\theta_{p} \tag{1}\]  
\(\Omega\) is the set of all the feasible bus order chain solutions. \(P\) represents the bus order chain. \(M\) is the set of buses. \(m\) represents the bus. \(K\) represents the vehicle type set. \(e_{k}\) is the unit (per kilometer) energy consumption of type k vehicles. \(L_{P}\) is the mileage of the bus order chain \(P\). \(\beta_{mp}^{k}\) is a variable ranging from zero to one, representing whether the bus order chain \(P\) is executed by the bus \(m\) of the vehicle type \(k\). If it is, \(\beta_{mp}^{k}\) is one, otherwise, \(\beta_{mp}^{k}\) is zero. \(\theta_{p}\) is a variable ranging from zero to one, indicating whether the bus order chain \(P\) is in the feasible solution. If it is, then \(\theta_{p}\) is one, otherwise \(\theta_{p}\) is zero. \(P=\{1,2,3,\ldots,i,n\}\) is a set representing the sequence of the exit arc, task arc, waiting arc, empty-driving arc, and entry are executed by a bus departing from a certain parking yard. \(C_{1}\) is the minimum passengers' waiting time. Since the passengers' waiting time cost is mainly decided by the waiting time, and the waiting time is associated with the departure frequency of the bus, the relationship between the departure frequency and the passengers' waiting time can be described as follows. Assuming that the law of the buses and the passenger arriving at the bus stations are subject to the Poisson Distribution and uniform distribution, respectively. The passengers' waiting time on average is the half of the departure interval and the bus lines' departure interval in unit time is equal to the reciprocal of departure frequency. The passengers' waiting time cost is inversely proportional to the departure frequency, and directly proportional to the number of people boarding the bus in the stations, so the calculation of the waiting time is shown in Formula (2). \[\text{min}C_{1}=\frac{1}{2f}\sum_{n}\chi^{n} \tag{2}\]  
\(f\) is the departure frequency of the research bus line in the unit time period, with the unit of times/h. \(\chi^{n}\) is the number of passengers boarding the bus at station \(n\) in a unit time period, with the unit of persons. \(C_{2}\) is the minimum passengers in-bus time. The passengers' perception of the in-bus time cost is mainly influenced by the degree of the in-bus crowding (full load rate), which is determined by the departure frequency to some extent.