برنامه ریزی اتوبوس بین شهری تحت سهم بازار متغیر و خواسته های بازار نامشخص

Bus scheduling is essential to a carrier's profitability, its level of service and its competitiveness in the market. In past research most inter-city bus scheduling models have used only the projected (or average) market share and market demand, meaning that the variations in daily passenger demand that occur in actual operations are neglected. In this research, however, we do not utilize a fixed market share and market demand. Instead, passenger choice behaviors and uncertain market demands are considered. Stochastic and robust optimizations and a passenger choice model are used to develop the models. These models are formulated as a nonlinear integer program that is characterized as NP-hard. We also develop a solution algorithm to efficiently solve the models. They are tested using data from a major Taiwan inter-city bus operation. The results show the good performance of the models and the solution algorithm.

Bus scheduling is a critical activity in inter-city bus operations and is essential to a carrier's profitability, its level of service and its competitiveness in the market. There has already been some research devoted to inter-city bus scheduling problems; for example, see [1], [2], [3] and [4]. However, in these researches fixed parameters, such as the projected market share and market demand, have usually been used for bus scheduling. The resultant bus schedules are produced neglecting variations in daily passenger demand that occur in actual operations. Actually, market share may vary with passenger choice behaviors, especially in competitive markets. The actual market share may also decrease with respect to the projected market share if a bus schedule is inferior, and vice versa. Moreover, market demand usually varies on a daily basis due to disturbances that may occur in actual operations. Passenger fluctuations arising from variable market share and uncertain market demands can affect the “actual” performance of the planned schedules. These planned schedules are the basis for real operations, while, on the other hand, real operations must fulfill the planning objectives by implementing the planned schedules. This interrelationship between the planned schedules and real operations cannot be neglected in the bus scheduling process. However, the traditional models, based on the projected (or average) market share or market demand, do not consider passenger fluctuations, so produce overly optimistic optimal schedules where resources may be used too tightly. Unfortunately, this may produce large variations in “actual” performance when applied in real operations, where passenger numbers often fluctuate. Therefore, to set a good schedule, not only does related bus supply have to be considered, but also variations in the market share and uncertain market demands also have to be taken into account. Yan and Chen [3] developed an inter-city bus scheduling model, based on fixed projected market share and market demand. Their model would be a useful tool for inter-city bus scheduling provided that the market share projection was accurate. However, in inter-city bus scheduling problems, the estimation of the target carrier's market share, for each OD pair from the origin–destination table (known as the OD table), and for each time interval, is complicated. The market share for all OD pairs, and for all time intervals, will vary simultaneously with changes in the supply (for example, the bus schedule). In practice, market share is correlated with passenger choice behaviors. It is not easy for a carrier to make an accurate projection of the competitive market share without taking into consideration of passenger choice behaviors. To improve Yan and Chen [3]'s model, Yan et al. [4] employed a time–space network technique and a passenger choice model so as to incorporate variable market share into a bus scheduling model. It was expected that with Yan et al.'s [4] model, bus scheduling and passenger choice behaviors in a competitive market could be more efficiently integrated. However, as was the case in Yan and Chen [3]'s model, market demand was also assumed to be fixed. In other words, daily variations of the market demand in actual operations were neglected. In practice, the market demand for each OD pair, for each time interval, usually follows a specific distribution. If the uncertainties of market demand do not conform to the bus schedule, then, to solve Yan et al.'s [4] model, the carrier needs to make repeated adjustments to the market demand in relation to the distribution, until a satisfactory bus schedule is acquired. Such a trial-and-error process is less systematic and less efficient. As a result, it is very difficult for a carrier to incorporate both variations in the market share and market demand, to perform a sensitivity analysis, using this model. Nevertheless, Yan et al.'s [4] model could serve as a basis for the development of an inter-city bus scheduling model under variable market share and uncertain market demands. Therefore, to improve on Yan et al.'s work [4], both variable market share and uncertain market demands, such as that occur in actual operations, are considered here. Our research differs from Yan et al.'s [4] on several other points: 1. Yan et al. [4] utilized a fixed market demand in their model; we not only consider variable market share but also take uncertain market demands into account. Such an approach should more accurately reflect variations in market share and market demand, and thus be more practical in actual operations. 2. To consider uncertain market demands, two stochastic and robust optimization concepts (which will be introduced later) are employed in our modeling. These were not considered in Yan et al.'s [4] research. 3. Yan et al. [4] considered the scheduling of non-stop and one-stop bus trip operations. However in Taiwan, inter-city bus trips are typically short-haul so that in actuality, Taiwan inter-city bus carriers mainly provide non-stop bus trips. For example, Yan et al. [4] found that about 93% of all passengers were transported on non-stop bus trips (the remainder were served by one-stop bus trips). We consider both variations in the market share and market demand in the modeling, a more complicated procedure (details will be discussed in Section 2), to reduce the problem complexity, so we only consider non-stop bus trips. 4. In Yan et al.'s [4] model, a draft timetable was not utilized. In theory, their model had the capability of directly and systematically managing the interrelation between supply and demand. However, such an approach, without the prior consideration of service frequency and other supply constraints in the draft timetable, would be inclined to result in an impractical bus schedule that needs to be adjusted by a post-optimization process. In our research, the draft timetable is used as a medium for the scheduling process, meaning the obtained bus schedule will be more practical and suitable for Taiwan carrier operations. Since we consider passenger choice behaviors and uncertain market demands during the bus scheduling process, we draw on past research in both these fields for reference purposes. For the former, there has been research applying multinomial or nested logit models to formulate passenger choice behaviors in competitive market situations. For examples, see [5], [6], [7], [8] and [9]. For the latter, there has been research considering uncertain disturbances in other fields. For examples, see [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33] and [34]. In general, stochastic and robust optimization concepts have recently been employed to deal with these types of planning problems under uncertain disturbances. The stochastic optimization models have usually been designed to optimize the expected value of all possible scenarios, while the robust optimization models are sensitive to variations in different scenarios. The objective functions of therobust optimization models have usually been designed to include the expected value of all possible scenarios, plus the expected variations in all possible scenarios, multiplied by a weighting value. In addition, Yan et al. [35] utilized passenger choice behaviors to develop a short-term flight scheduling model that could evaluate variable market share. However, in their model, the daily market demand is assumed to be fixed, and any stochastic variations in the market demand are neglected. Moreover, their problem is not the same as the inter-city bus scheduling problem addressed in this paper. To the best of the authors’ knowledge, there has been no research on inter-city bus scheduling problems that includes both stochastic and robust optimization models. In addition, none has considered passenger choice behaviors in stochastic and robust optimal models that are capable of reflecting the variable market share and uncertain market demands. To remedy this, in this research, we first utilize the stochastic and robust optimization concepts to develop a stochastic optimization bus scheduling model (SOBSM) and a robust optimization bus scheduling model (ROBSM). The passenger choice model developed by Wen et al. [9] is incorporated into both the SOBSM and the ROBSM. To evaluate the performance of the SOBSM and the ROBSM, we also develop a deterministic-demand bus scheduling model (DDBSM), given a fixed market share and market demand, and a fix market demand bus scheduling model (FMDBSM), given a fixed market demand but variable market share. In addition, the SOBSM and the ROBSM are formulated as a nonlinear integer program that can be characterized as NP-hard [36]. A solution algorithm to effectively solve the SOBSM and the ROBSM is formulated. Finally, to compare the performance of the models, we also develop a simulation-based evaluation method. For simplicity, a single type of bus is considered in the models; this, however, is extendable to multiple types of buses if needed. In this research, we only address a variable market share and uncertain market demand, such as those that occur in regular operations, rather than other larger types of fluctuations, such as increased passenger demand that might occur during major festivals. We should note that the Taiwan inter-city bus carrier studied performs bus maintenance and crew scheduling activities after the bus schedule has been completed. This is because these activities are rather flexible, due to the use of stand-by crews and spare buses. In only a few cases do bus schedules have to be slightly modified to meet maintenance or crew scheduling constraints. Thus, to reduce problem complexity, these constraints are excluded in this research. The incorporation of larger fluctuations as well as the bus maintenance and crew scheduling can be a direction of future research. The remainder of this paper is organized as follows: in Section 2, we discuss the SOBSM and ROBSM. In Section 3, a solution algorithm and an evaluation method are introduced. In Section 4, numerical tests are performed. Finally, in Section 5, we conclude.

In this research we use stochastic and robust optimization concepts to develop the SOBSM and the ROBSM. To consider passenger choice behaviors in actual operations, a passenger choice model is incor- porated into both the SOBSM and the ROBSM. This approach is expected to more accurately reflect any variations in the market share and market demand. Numerical tests, including three case studies as well as sensitivity analyses of the value and the number ofscenarios, are performed to evaluate the performance of the models. It is found that, although the DDBSM and the FMDBSM produce a better objective value than the SOBSM and the ROBSM in the planning stage, their optimal planned schedules lose optimality when applied in actual operations. It is therefore important that the actual performance of different planning models should be evaluated, after schedules are applied in actual oper- ations, instead of only evaluating their planning results. Such findings and other test results could serve as use- ful references for transportation planners and operators. Finally, aside from passenger demand, uncertain bus delays may occur in actual operations. This could also be incorporated into future models. In addition, we have only addressed the planning stage of the bus schedul- ing problem; the interrelationship between the planning and real-time stages, as affected by such uncertain fluc- tuations, is not considered. If the planned bus routes and real-time schedule adjustments can be integrated into the same model or framework, then the two stages could be systematically analyzed together, which might help to produce a better bus schedule. This is a future research topic.