Determining optimal frequency and vehicle capacity for public transit routes: A generalized newsvendor model

Transportation Research Part B 71 (2015) 85–99

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Determining optimal frequency and vehicle capacity for public transit routes: A generalized newsvendor model Avi Herbon, Yuval Hadas ⇑ Department of Management, Bar-Ilan University, Israel

a r t i c l e

i n f o

Article history: Received 7 April 2014 Received in revised form 21 October 2014 Accepted 22 October 2014

Keywords: Public transit Frequency Vehicle size Non-linear optimization Newsvendor model

a b s t r a c t The level of service on public transit routes is very much affected by the frequency and vehicle capacity. The combined values of these variables contribute to the costs associated with route operations as well as the costs associated with passenger comfort, such as waiting and overcrowding. The new approach to the problem that we introduce combines both passenger and operator costs within a generalized newsvendor model. From the passenger perspective, waiting and overcrowding costs are used; from the operator’s perspective, the costs are related to vehicle size, empty seats, and lost sales. Maximal passenger average waiting time as well as maximal vehicle capacity are considered as constraints that are imposed by the regulator to assure a minimal public transit service level or in order to comply with other regulatory considerations. The advantages of the newsvendor model are that (a) costs are treated as shortages (overcrowding) and surpluses (empty seats); (b) the model presents simultaneous optimal results for both frequency and vehicle size; (c) an efficient and fast algorithm is developed; and (d) the model assumes stochastic demand, and is not restricted to a specific distribution. We demonstrate the usefulness of the model through a case study and sensitivity analysis.  2014 Elsevier Ltd. All rights reserved.

1. Introduction Public transit is a capital-intensive system that operates within a dynamic environment. One of the key factors in this environment is passenger demand that is basically uncertain and depends on spatial and temporal attributes such as the location of the stations along the route and service frequency during the day. The awareness of public transit passengers for higher service standards are increasing. Characterized by adequate timetables, high frequencies and available seats, this awareness make the problem of determining route frequencies and vehicle capacity an appealing problem both to researchers and practitioners. Analyzing the monetary properties (resulting from frequencies and vehicle capacity) of the demand side (Jansson, 1980;Mohring, 1972) is an important issue that provides the authorities and operators with the means to better design competitive fare structures. A common practice in public transit planning is to determine the frequency of service based on accumulated hourly passenger counts; average travel time; vehicle capacity; desired occupancy (load standard); and the minimum frequency permitted according to the time of day (Ceder, 2007). Criteria for quality of public transport can also include vehicle and transfer comfort at the terminal; regularity of service; service coverage; frequency level; and crowding (Pel et al., 2014;Tirachini et al., 2013; Wardman and Whelan, 2011). From the operator’s perspective, on the other hand, the objective is to make as ⇑ Corresponding author. E-mail addresses: [email protected] (A. Herbon), [email protected] (Y. Hadas). http://dx.doi.org/10.1016/j.trb.2014.10.007 0191-2615/ 2014 Elsevier Ltd. All rights reserved.

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much profit as possible (Guihaire and Hao, 2008). While the perspectives may seem to differ, decisions taken by the operator that do not consider consumer criteria will deteriorate profits due to competition and possible decline of passenger demand. In this paper, we consider the operator’s monetary perspective as the focal objective. In order to realize his monetary perspective, the operator must achieve appropriate service levels and frequencies. A suitable frequency setting should provide sufficiently regular service to satisfy the users and sufficiently sparse service to reduce the required departures and thereby the operator’s costs (Guihaire and Hao, 2008). The emergence of automated identification technology can assist in coping with the above problem. Automatic vehicle location (AVL) and automatic passenger counting (APC) are well-known technologies that can track the location of vehicles en route and collect the number of alighting\boarding passengers at each stop (Tétreault and El-Geneidy, 2010). According to Hadas and Shnaiderman (2012), the data acquired can be used to analyze as well as to enhance the performance of public transit systems and to support the development of advanced models. Using AVL technology, it is possible to accurately forecast the estimated arrival times of buses and to implement bus-holding strategies to coordinate transfers (Dessouky et al., 2003). Strathman et al. (2002) analyzed the operating performance for Tri-Met, the transit provider for the metropolitan area of Portland, Oregon. They demonstrated how AVL–APC data can be utilized to monitor and evaluate service performance in relation to adopted standards. Tétreault and El-Geneidy (2010) used AVL and APC data to select stops and estimate run times for a new service that will run parallel to a heavily used bus route in Montreal, Canada. According to Desaulniers and Hickman (2007) summary of PT planning optimization models, the problem of setting frequencies can be approached both at the network level and for a single route. At the route level, the primary goal is to select frequencies that maximize passenger service subject to various constraints, such as overall fleet size, route capacity in relation to demand and policies regarding minimum desirable frequencies. The transit network frequency setting problem (TNFSP) was mathematically addressed by Salzborn (1972). His work was based on determining frequencies given passenger arrival rates with the objective of decreasing fleet size (peak period) and passenger waiting time (off-peak period). The model assumes deterministic travel time and demand. Furthermore, bus capacity (overcrowding) is not a constraint. Schéele (1980) proposed a nonlinear model (solved by a non-optimal algorithm) for bus network vehicle allocation and frequency setting by minimizing the total travel time given passenger trip assignment. The model assumes deterministic demand and overcrowding is not allowed. Another mathematical approach is suggested by Furth and Wilson (1982). They used a relaxation of a non-linear model for maximizing ridership benefit and waiting time saving benefit under bounded fleet size, headway and budget. Gao et al. (2004) proposed a heuristic solution based on sensitivity analysis designed to optimize frequencies settings. The algorithm is designed to help transit planners to adjust an existing transit network to evolutions in demand and various other parameters. Constantin and Florian (1995) developed a non-linear, non-convex mixed integer programming model for the TNFSP with the goal of minimizing the total expected travel and waiting time of passengers under fleet size constraints. The TNFSP was also addressed heuristically. Han and Wilson (1982) considered the problem as one of allocating vehicles among the routes of the network. They proposed a two-stage heuristic to reach their objective, minimizing the maximum ‘‘occupancy level’’ at the maximum load point for each route. Chowdhury and I-Jy Chien (2002) considered transfer coordination for intermodal transit networks by optimizing both headways and slack times. A mathematical programming model is first developed and then a procedure is presented that first optimizes headways without taking coordination into account. It then optimizes slack times in the context of intermodal transit. All the above mentioned TNFSP models share several properties. First, the objective functions are time base. Secondly, stochastic demand is not considered, hence overcrowding (if addressed) does not lead to fail to board passengers. Finally, operators under-utilizing their vehicle is not considered. Another approach for setting frequencies stems from microeconomics. This is a broader approach since it considers in detail the costs of both passengers and operators. Mohring (1972) developed a model that considers optimal frequency and stop spacing as a means of minimizing bus company operating costs and user costs (walking, waiting, and riding). The model assumes deterministic demand and does not takes into account overcrowding (bus capacity is realized from frequency and demand). Jansson (1980) further investigated the above mentioned model (the square root formula) for peak and off-peak periods, each having different deterministic demand patterns. An optimal frequency and bus size model was introduced as well. More recently, Jara-Dfaz and Geschwender (2003) extensively reviewed the evolvement over the years of the models that Mohring and Jansson developed. Specifically, they investigated the effect of vehicle size on operating costs and passenger crowding that resulted from delayed boarding. The model assumes deterministic demand that is uniformly distributed. Jara-Díaz et al. (2008) address the goal of maximizing social benefits (considering both users and operators) in the case of inelastic demand and where the cost function includes waiting time and travel time. They found the optimal levels of frequency and vehicle size on a public transport corridor. This model, like that of Jara-Dfaz and Geschwender, does not treat overcrowding. A variation of the model developed by Delle Site and Filippi (1998) adds operational tactics in the form of short-turns in addition to setting vehicle size and frequencies. This model also assumes that all passengers are able to board. A better realization of overcrowding was introduced by Oldfield and Bly (1988) in the form of an infinite waiting time component. Tirachini and Hensher (2011) increased bus capacity (spare capacity) to absorb random variation in demand. From the above it is clear that a more detailed analysis of the effect of passenger demand on capacity should be sought. When demand is lower than capacity (crowding), there is a more positive effect on passengers since boarding time decreases (Jara-Dfaz and Geschwender, 2003) and perceived time decreases (Wardman and Whelan, 2011), (Pel et al., 2014). Furthermore, crowdedness has an effect on wellbeing,

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route choice, and frequency (Tirachini et al., 2013). On the other hand, these models neglect to take into account the case of overcrowding, specifically the case of passengers who fail to board. Hadas and Shnaiderman (2012) were among the first to consider extra waiting time when passengers fail to board. More recently, Wang et al. (2014) considered the effect of queues on passenger patience. Another interesting model considers limiting boarding (Delgado et al., 2012) in order to regulate the service and prevent bunching. Finally, crowdedness should be evaluated from the operator’s perspective while empty seats, considered as unused resources (Hadas and Shnaiderman, 2012), should be included within the analysis. In a recent work, Hadas and Shnaiderman (2012) addressed the problem of setting the optimal frequency under uncertain demand and travel time. They construct a monetary objective function that includes two cost elements. The first component is empty-seat and the other is un-served demand. The goal is to minimize expected total costs with decision variables, either frequency or vehicle capacity. Their proposed formulation is described in a discrete framework that acquires data about alighting passengers at each stop and about newly arriving passengers after the previous vehicle has left. They assumed that the number of alighting passengers at each stop is binomial distributed and that the probability distribution of the number of newly arriving passengers at each stop is a distributed Poisson with a mean that depends on headway. A promising approach to simultaneously deal with costs and service levels under stochastic demand is associated with the newsvendor model. The classical single-period newsvendor problem is to find for a product the order quantity that maximizes the expected profit under probabilistic demand. Interest in the newsboy or newsvendor problem remains unabated and many extensions to it have been proposed (Khouja, 1999). Real-life applications for utilizing this model for decisionmaking have been developed for the fashion and sporting industries both at the manufacturing and retail levels (Gallego and Moon, 1993;Kogan and Herbon, 2008). The single-period newsvendor problem has also been studied in regard to managing capacity and evaluating advanced booking of orders in service industries such as airlines and hotels (Weatherford and Pfeifer, 1994). Khouja (1999) offers a review of literature on the single-period newsvendor problem and its extensions. This paper generalizes the newsvendor classical approach in inventory for modeling the problem of setting vehicle frequency and capacity for a given fixed route and under stochastic demand. The main enhancements with respect to Hadas and Shnaiderman (2012) are: (1) surplus costs also depend on vehicle capacity; (2) shortage costs are associated with lost sales instead of running time; (3) fleet operational costs are considered; (4) the model is simplified by omitting detailed data about alighting and boarding passengers; (5) optimality conditions are derived simultaneously for decision variables (frequency and vehicle capacity); and (6) the analysis of the model is not restricted to a specific distribution. We start with the formulation of the model and an optimal algorithm, followed by a case study, a sensitivity analysis and conclusions. 2. Modeling the problem 2.1. Model description The aim of the model is to find the optimal vehicle frequency (or headway) and capacity for serving a route for a specified planning period. It is assumed that demand and travel time retain their characteristics throughout that period. Consider a route that consists of K stops, k = 1, 2, . . . , K. Identical vehicles with capacity V serve the route for N trips to attain a desired headway, i = 1, 2, . . . , N, starting from the originating stop (k = 1) and ending at the final stop (k = K). The entire travel time of each one of the vehicles for completing the route is T, where the travel time (including dwell time) between successive stops P k  1 and k is denoted by tk such that T ¼ Kk¼2 t k . In order to model demand rate, we define an aggregate demand. Aggregate demand in stop k is obtained by summing up all passengers in a single cycle wishing to pass through stop k and to reach one of the subsequent stops. According to this definition, a passenger who is included within the aggregate demand in stop k be is also included within the aggregate demand in stop k + 1 in case a passenger wishes to pass through stop k + 1. This definition differs slightly from the more common term departing volume by including also passengers in stop k who failed to P board. The aggregate demand in stop k is distributed along all vehicles such that dk ¼ Ni¼1 dik , where dik denotes the total number of passengers wishing to depart from stop k, k = 1, 2, . . . , K, in vehicle i in a given cycle. Specifically, the demand includes passengers who are currently in the vehicle before arrival at stop k and passengers waiting at the stop less the alighting passengers at stop k. According to this definition, demand at the first stop includes solely passengers waiting at the stop while demand at the last stop is zero. Since demand is stochastic in nature, we describe dik as a random variable P with distribution fik(x,N) and expectation is denoted by lik, where lk ¼ Ni¼1 lik . It is assumed that capacity is enforced. Since demand in this context includes passengers wishing to board, not all demand can be satisfied when demand is higher than capacity. As introduced by Hadas and Shnaiderman (2012), the total cost is composed of a shortage component (when demand is higher than capacity) and a surplus component (when demand is lower than capacity). We now model each component as follows. 2.1.1. Shortage component Whenever a vehicle is overcrowded, a passenger will be unable to board and will have the choice of waiting for the subsequent vehicle or of leaving the station to seek another mode of transportation (Yulin et al., 2010). We model the passenger’s choice as a two-component formula. In the first component, the minimal proportion of passengers leaving the stop from

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among the passengers who failed to board is denoted by r0k ; k ¼ 1; 2; . . . ; K  1. This proportion depends on the stop index since the availability of transportation alternatives, a key consideration in the decision to leave or wait, might vary with location. In the second component, an additional proportion of passengers leaving the stop from among the passengers who failed to board due to the waiting time (or headway), T/N (Ceder, 2007), might also lead passengers to leave the station. Following these notations, the total proportion of passengers leaving the stop from among the passengers who failed to board is denoted by rk, where

rk ¼ r 0k þ r

 b1 T ; N

8k ¼ 1; 2; . . . K  1

ð1Þ

Expression (1) reflects passenger reaction to boarding failure or a long waiting time, (r > 0). The proportion of passengers who failed to board and choose to leave the stop grows with increasing waiting time. The proportion rk is assumed to be polynomial increasing with the headway and b1 a parameter presenting the power, b1 > 0. The proportion rk increases with waiting time since a passenger who failed to board will have to wait further for boarding while the waiting time is a key factor in the decision to remain or leave. The selection of coefficient r should not cause rk to exceed 1. We model the shortage  cost coupled with a failed to board passenger by c k . Following (1), ck can be computed as

ck ¼

r0k þ r

 b1 ! T pk ; N

8k ¼ 1; 2; . . . K  1

ð2Þ

where

    1þb 1 b þp ðk  1Þ; pk ¼ p  K 1 K  1 ðK  1ÞðK  2Þ

8k ¼ 1; 2; . . . ; K  1

ð3Þ

p is a parameter that represents a possible penalty caused by a passenger who left due to failed boarding; it is at least the average ticket price, where 0 6 b 6 K  2. Expression (3) approximates the penalty costs due to unsatisfied passengers who   1þb   1 b leave the stop. The factor K1 ðk  1Þ makes it possible to control the penalty cost distribution among þ K1  ðK1ÞðK2Þ stops by manipulating b. With respect to K  2, it is possible to define increasing (b < K  2), decreasing (b > K  2), or constant (b = K  2) cost distribution. In our case study, an increasing factor was used in order to reflect that passengers failing to board and leaving the stop earlier increases the probability of passengers waiting at successive stops to board and thus prevents additional lost sales. 2.1.2. Surplus component A vehicle departing a given stop with unoccupied seats indicates, from the operator’s viewpoint, the non-utilization of resources which imposes additional monetary costs. We model surplus costs associated with every empty seat in a vehicle þ departing from stop k by cþ k . The cost of surplus c k is associated with the travel time from stop k to stop k + 1 and the capacity of the vehicle. Examples of such costs are additional operational costs that are related to the vehicle size, such as additional energy consumption due to greater weight, additional parking areas, etc. For simplicity, we compute surplus as

cþk ¼ ðc  aV b2 Þtkþ1 ;

8k ¼ 1; 2; . . . K  1

ð4Þ

which reflects a generalized polynomial decreasing function of the capacity V, where a and b2 are parameters (a > 0, b2 P 0) and c is the maximal cost associated with empty seat per unit of time. The polynomial function allows flexible modeling (both concave and convex dependency on capacity) of the negative effect of increasing the capacity on the operational costs associated with the economy of size. 2.1.3. Operational component A standard vehicle (as set by the operator) is regarded as the base-line for measuring both shortage and surplus components. The third component of the objective is associated with the direct cost of operating a standard vehicle; the monetary and financial costs of acquisition; the costs of employing logistical personnel and of acquiring equipment needed to support management and maintain the fleet over a given cycle. The essence of our model directly refers to these three components. Passengers who leave the stop due to a failure to board, excessive available capacity, or large operational costs are undesirable from the operator’s viewpoint and should be eliminated as possible. 2.2. Modeling assumptions and formulation We develop a generalized newsvendor model and formulate an optimization model where the decision-maker (the transit authority or operator) wishes to determine both headway (T/N) and capacity (V) under stochastic demand, while minimizing expected surplus, shortage and operation costs.

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2.2.1. Assumptions (A1) All route cycles are repeated with identical characteristics. (A2) Failed to board passengers continue to use the same transportation mode for future trips. (A3) Distribution of demand functions fik(x,N) are known for any vehicle i, "i = 1, 2, . . . , N, and for any stop k, "k = 1, 2, . . . , K. (A4) Passengers only leave the stop due to a failure to board the vehicle. (A5) An available seat (or maximal capacity) is a condition for passenger boarding. 2.2.2. Formulation The classical newsvendor model seeks to determine the optimal initial inventory level for reducing expected surplus and shortage costs when a single source of demand is given as well as both surplus and shortage costs. We model the transportation problem above by developing a generalized newsvendor model to include multiple sources of demand for cases where more than one decision variable is required and where both surplus and shortage costs are functions of the decision variables. The following notations are used throughout the paper. Decision variables: N V

the number of trips per planning period the vehicle’s capacity

Parameters:

a

K Vregul

the marginal cost reduction with capacity the wait time coefficient of passengers leaving a stop the power in the expression of shortage cost the power in the expression of surplus cost the maximal cost associated with empty seat per unit of time the maximal average passenger waiting time the costs over a route cycle associated with operating a standard vehicle the price of a single ticket (or larger parameter) the fractional penalty price assigned to rejected or failed to board passenger at stop k which leaves that stop the minimal proportion of passengers leaving stop k from among the failed to board passengers the route’s travel time the travel time between successive stops k  1 and k the expectation of demand at vehicle i leaving stop k number of stops along the route the maximal capacity of a vehicle

Distributions: fik(x,N)

the demand probability density function of vehicle i leaving stop k given N

r b1 b2 c w g p pk r 0k T tk

lik

Thus, the optimization problem is

(

K1 X N h   i X  b  Min E c  aV b2 t kþ1 maxðV  dik ; 0Þ þ pk r 0k þ r NT 1 maxðdik  V; 0Þ þ gN N;V

)

k¼1 i¼1

s:t ð5:1Þr 0k þ r ð5:2ÞaV

b2

 T b1 N

6 1; 8k ¼ 1; 2 . . . ; K  1

ð5Þ

6 c

ð5:3ÞV 6 V regul T ð5:4Þ 2N 6 w

The objective includes three components. The first refers to the expectation of the sum of surplus costs. The second refers to the expectation of the sum of shortage costs. The third component is associated with the direct cost of operating a standard vehicle. The first constraint ensures that the proportion of passengers leaving stop k out of the passengers who failed to board, rk never exceeds 1. The second constraint does not allow surplus cost to be negative. The third constraint limits the vehicle size according to regulations or design limitations. The fourth constraint limits the average waiting time of passengers according to regulations that require a minimal service level.

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3. Analysis of the model In this section, we present the necessary optimality condition of optimization problem (5). Since the aggregate demand of passengers at each stop k is distributed over all vehicles i, i = 1, 2, . . . , N the distribution of passenger demand fik(x) in general depends on decision variable N. To simplify the analysis, we assume that distributions of demand as well as the decision variables are continuous. The objective is extracted to be

"Z K1 X N X

V



Z  c  aV t kþ1 ðV  xÞf ik ðxÞdx þ

0

k¼1 i¼1

1

b2

pk

V

r 0k

#  b1 ! T þr ðx  VÞf ik ðxÞdx þ gN: N

ð6Þ

We use a sensible simplifying assumption in which passenger demand at a given stop is equally distributed among all vehicles, that is lik = lk/N, "i,k and hence modify the notation of the distributions fik(x) to fk(x,N) along the paper. Due to the assumption of continuity, objective (6) is reduced to

" Z Z K1 V X N ðc  aV b2 Þt kþ1 ðV  xÞf ik ðx; NÞdx þ N 0

k¼1

1

pk V

r 0k

#  b1 ! T ðx  VÞf k ðx; NÞdx þ gN þr N

ð7Þ

The first component in the objective penalizes the decision-maker for utilizing larger vehicles whose capacity exceeds demand. Although we expect a decrease in the marginal cost of each empty seat with an increase in vehicle capacity, we also expect that the surplus cost for each stop k increases with vehicle capacity. Therefore, the condition to be satisfied is

Z

V

@=@V 0

  c  aV b2 ðV  xÞf k ðx; NÞdx P 0;

8k ¼ 1; 2; . . . ; K  1

In order to satisfy the last condition, it is sufficient that problem (5) includes the following bonding constraint:

aðb2 þ 1ÞV b2 6 c

ð8Þ

Since constraint (8) makes constraint (5.2) redundant, we substitute them in problem (5). In order that the surplus costs is a concave function of capacity it is that b2 P 1.  sufficient b The K  1 constraints, r 0k þ r NT 1 6 1; 8k ¼ 1; 2; . . . ; K  1 in problem (5) can be replaced by a single constraint that mostly restricts the number of vehicles from below among all stops. Denoting r0max ¼ maxk2f1;2;...;K1g r0k the above constraints are replaced by

r0max þ r

 b1 T 6 1 N

Let us define, N min

N P Nmin Let us define, V max

ð9Þ

 h ib1 1 T ¼ max T 1rr0 ; 2w . Constraints (5.1) and (5.4) are replaced by the constraint max

  1=b2 ¼ min V regul ; aðb2cþ1Þ . Constraints (5.2) and (5.3) are replaced by the constraint

V 6 V max

ð9:1Þ

ð10Þ

The Lagrangian of the modified problem is now

LðN; V; kÞ ¼

K1 h K1 h i X i X  b  RV R1  N 0 ðc  aV b2 Þtkþ1 ðV  xÞf k ðx; NÞdx þ N V pk r 0k þ r NT 1 ðx  VÞf k ðx; NÞdx k¼1

k¼1

þ gN þ k1 ðNmin  NÞ þ k2 ðV  V max Þ According the signs of k1, k2 the necessary optimality condition is separated into four cases. A detailed analysis is given in Appendix B. (a) k1 = k2 = 0 This case addresses the situation when constraints of both frequency and vehicle size are non-binding. Mathematically, it is equivalent to solving the reduced problem. (b) k1 > 0, k2 = 0 This case addresses the situation where only the constraint on the vehicle size is non-binding. This sub problem is an optimization problem of a single variable V given frequency N. (c) k1 = 0, k2 > 0 This case addresses the situation where only the constraint on the frequency is non-binding. This sub problem is an optimization problem of a single variable N given vehicle size V. (d) k1 > 0, k2 > 0 The last case addresses the situation where both constraints are binding.

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4. Optimal search algorithm (1) Solve set of Eqs. (13) and (14) for variables V,N. If the solution is feasible, then an optimal solution is derived, stop.  h Otherwise go to (2). ib1 1 T (2) If infeasibility is in variable N, set N = Nmin, where N min ¼ max T 1rr0 ; 2w , and solve the reduced problem by (19) max

for variable V. If the solution is feasible, stop. Otherwise go to  (3).  1=b2 and solve the reduced problem (24) (3) If infeasibility is in variable V, set V = Vmax, where V max ¼ min V regul ; aðb2cþ1Þ for variable N. If the solution is feasible, stop. Otherwise go to (4). (4) The optimal solution is N = Nmin and V = Vmax.

5. Numerical example and sensitivity analysis In this section, we present a numerical example that illustrates the applicability of the suggested solution process to the problem. Additionally, a sensitivity analysis of the solution for changing the set of parameters is presented. A real-world bus route from Auckland, New-Zealand was selected. Due to data sensitivity, the raw data (travel time and load profiles) were manipulated and the route’s characteristics are undisclosed.

Table 1 Travel times (s) and demand during morning and afternoon peaks. Stop k

AM peak (7–9) Average demand lk

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

19.5 22.0 11.8 22.5 36.0 32.2 38.3 47.8 37.8 103.5 25.0 84.8 135.7 93.5 99.0 33.3 48.4 92.7 43.6 194.5 174.3 176.5 181.5 191.7 173.2 249.2 209.7 93.6 142.2 141.7 87.5 276.0 235.0 311.5 323.5 318.7 213.8 105.5 200.7 420.7 318.2 166.4

Minimal proportion leaving r 0k

PM peak (16–18) Travel time tk

Average demand lk

Travel time tk

86.36 31.40 30.39 45.13 53.66 32.91 41.89 26.98 51.85 23.82 36.65 103.58 26.19 33.13 27.78 42.30 67.32 24.16 121.33 45.65 50.18 35.74 44.47 36.47 44.81 90.29 29.93 33.34 58.49 25.92 166.10 301.35 51.43 41.07 81.58 38.99 45.14 115.35 139.77 88.84 54.93 211.50

34.2 35.5 8.5 13.2 37.7 15.3 18.8 46.2 15.7 50.0 50.0 70.7 79.3 31.3 102.2 46.7 21.3 83.7 30.0 211.7 107.3 126.5 120.6 92.0 68.5 218.7 166.0 93.5 118.2 145.5 102.8 306.8 254.7 175.7 100.7 223.5 215.7 61.1 249.0 343.3 301.5 90.2 0.0

94.03 25.42 17.96 27.14 36.52 27.64 30.13 17.43 30.79 19.93 36.65 91.25 20.78 28.70 25.87 42.12 46.10 20.02 106.44 40.36 44.69 28.89 31.37 31.76 61.09 115.04 31.43 26.82 62.35 37.61 196.90 357.24 33.54 29.15 67.91 34.22 46.02 133.45 137.27 87.23 40.49 277.99

0.1 0.1

0.1 0.2 0.2 0.2 0.2

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A. Herbon, Y. Hadas / Transportation Research Part B 71 (2015) 85–99 Table 2 Optimal results compared to current practices. Optimal

Current

AM peak (7–9)

V N Cost Cost reduction (%)

23 7 23,170 12.3

37 5 26,438

PM peak (16–18)

V N Cost Cost reduction (%)

23 6 19,250 19.5

37 5 23,920

For the sake of comparison, the current policy entails five departures during morning peak (7–9), and five departures during afternoon peak (16–18). The average ticket price is 4.40NZD, and vehicle capacity is 37 seats. Travel time was calculated based on automatic vehicle location (AVL) data. Demand was estimated based on smart card transactions. The minimal proportion of passengers leaving was set according to the layout of the road network and alternative transportation modes. 5.1. Numerical example Consider an operator of a bus line on a given route of 43 stops, k = 1, 2, . . . , 43. The entire route’s travel time for each time period is approximately 45 min. Travel time between successive stations is presented in Table 1. The aggregate passenger demand dk in a single cycle is randomly distributed with expectation as presented in Table 1. The operator has to find the optimal values of V and N, where the number of passengers dik wishing to pass through stop k in bus i, i = 1, 2, . . . , N h i in a given cycle is uniformlydistributed with f k ðx; NÞ  U 0; 2Nlk . To complete the set of data: p ¼ 30$; r0max ¼ 0:2; r ¼ 0:1; b1 ¼ 0:1; c ¼ 0:08$; b2 ¼ 2:0; a ¼ 0:00005; b ¼ 25; g ¼ 35; V regul ¼ 500; w ¼ 6 min. Table 2 summarizes the optimal solutions.

Table 3 Optimal solution with percent of deviation from original parameter (Uniform distribution). Parameter

Change in % in parameter value 50

25

10

10

25

50

Influence on variable

b1

6 23 20,219 33 3 25,151 5 32 23,592 4 23 14,659 7 23 23,047 5 23 17,052 15 16 17,136 7 23 22,371 7 23 21,480 6 23 18,102 8 23 23,416

6 23 21,701 33 3 25,143 6 26 23,461 6 23 19,373 7 23 23,108 6 23 20,386 9 20 20,528 7 23 22,770 7 23 22,325 7 23 21,987 7 23 23,170

7 23 22,573 5 34 23,125 7 24 23,321 7 23 21,777 7 23 23,145 7 23 22,140 8 21 22,482 7 23 23,010 7 23 22,832 7 23 22,723 7 23 23,170

7 23 23,804 9 16 23,573 7 22 23,129 7 23 24,562 7 23 23,194 7 23 24,200 6 24 24,142 7 23 23,330 7 23 23,508 7 23 23,622 7 23 23,170

8 23 24,787 13 11 23,614 8 20 23,304 8 23 26,406 7 23 23,231 8 23 25,599 6 25 25,579 7 23 23,569 7 23 24,015 7 23 24,307 7 23 23,170

8 23 26,374 19 7 23,746 9 18 23,356 9 23 29,198 7 23 23,292 9 23 27,765 4 28 27,162 7 23 23,969 7 23 24,859 7 23 25,461 7 23 23,170

N⁄ V⁄

b2

a p

g

r

c

b

r 0max

maxk(lk)

w

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄

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A. Herbon, Y. Hadas / Transportation Research Part B 71 (2015) 85–99 Table 4 Optimal solution with percent of deviation from original parameter (Erlang distribution). Parameter

b1

b2

a p

g

r

b

c

r 0max

maxk(lk)

w

Change in % in parameter value 50

25

10

10

25

50

Influence on variable

5 23 20,154 32 4 24,791 5 32 23,891 4 23 14,423 6 23 23,349 5 23 16,883 6 23 22,428 13 16 17,908 6 23 21,608 6 23 18,346 8 23 23,965

6 23 21,723 2 74 23,860 6 26 23,710 5 23 19,333 6 23 23,402 6 23 20,405 6 23 22,941 9 20 20,974 6 23 22,531 6 23 22,254 6 23 23,454

6 23 22,730 5 34 23,530 6 24 23,524 6 23 21,835 6 23 23,433 6 23 22,235 6 23 23,249 7 21 22,789 6 23 23,085 6 23 23,000 6 23 23,454

7 23 24,117 9 16 23,676 7 22 23,348 7 23 24,880 6 23 23,475 7 23 24,521 7 23 23,643 6 24 24,249 6 23 23,504 6 23 23,913 6 23 23,454

7 23 25,160 12 11 23,634 7 20 23,495 7 23 26,991 6 23 23,507 7 23 26,093 7 23 23,899 5 25 25,472 7 23 23,578 7 23 24,600 6 23 23,454

8 23 27,027 18 7 23,666 8 18 23,490 8 23 30,098 6 23 23,559 8 23 28,484 7 23 24,325 4 28 26,881 7 23 23,685 7 23 25,749 6 23 23,454

N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄

An additional set of results is presented in Appendix A for a different setting (a = 0.00001) in which the cost of an empty seat is significantly increased.

5.2. Sensitivity analysis We analyzed the sensitivity of the optimal fleet size, N⁄; the optimal capacity of each bus V⁄; and the minimal cost, p⁄, to changes in the values of the model parameters (AM peak). Table 3 presents results for the uniform distribution. In order to further validate our conclusions in Table 4 we present the results for the Erlang distribution (with two degrees of freedom). From the results in Tables 3 and 4, it is evident, based on the case study, that, increasing each one of parameters g; b; r0max ; b1 ; r; p; c and maxk(lk) increases the optimal cost. Furthermore, the optimal cost is extremely sensitive to changes in parameters b; r0max ; b1 ; r; p; c and maxk(lk). On the other hand, the optimal cost is relatively insensitive to changes in parameters b2, a, g and w. It is worth noticing that for the values of the parameters of the case study (and also of those presented in Appendix A), increasing the surplus cost parameter c increases vehicle size and decreases optimal fre the optimal  quency. In order to explain both results, a closer look of the expression c  aV b2 ðV  xÞ in the objective is required. To avoid high penalty costs due to empty seats, two options are at hand. The first is to choose a very small vehicle V (where demand

Table 5 Optimal results. Optimal AM peak (7–9)

V N Cost

51 4 24,527

PM peak (16–18)

V N Cost

3 27 20,551

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A. Herbon, Y. Hadas / Transportation Research Part B 71 (2015) 85–99

always exceeds capacity). The second is to increase V, but faster than a linear function, so the penalty cost will decrease faster than the increase of empty seats. By setting b2 = 2, we enable this scenario. Increasing the optimal vehicle size subsequently decreases optimal frequency N. It is worth noticing that Jara-Díaz and Gschwender (2003) obtained relatively contradicting conclusions regarding optimal vehicle size and optimal frequency. This may be linked to the assumption that the operator cost is a linear function of the number of seats. As a result, in order to avoid high penalty costs due to empty seats, a small vehicle size V should be chosen. Another important property of the model is that the optimal decision variables V⁄ and N⁄ are barely altered for changes in parameter maxk(lk). This indicates that the model is insensitive to demand fluctuations. Another interesting behavior is linked to parameter b2. With this parameter, optimal frequency introduces a convex curve while optimal vehicle size introduces a concave curve. In general, for both tested distributions, all the parameters consistently affect the optimal decision variables. One of several other parameters sets that were used in order to analyze and verify the algorithm is presented in Appendix A.

6. Conclusions We introduce an analytical model for simultaneously setting frequency and vehicle size in a public transit system as well as the necessary optimality conditions and an efficient algorithm. In contrast to previous models, the model offers the following features: (a) (b) (c) (d) (e)

Flexible cost functions provide easy calibration for different demand characteristics and public transit operations. The spatial characteristics of the route, as well as competitive modes are taken into account within the model. Passenger waiting time is directly realized in the model as a constraint on the vehicle size. By omitting detailed data about alighting and boarding, the model is simplified. Obtaining optimal operator costs and decreasing waiting time are not necessarily contradicting goals and may simultaneously be achieved.

The sensitivity analysis of the costs introduced portrays various ways the authorities and operators can improve the service and efficiently use resources. With regard to crowding and overcrowding, the model directly realizes overcrowding and waiting time. On the other hand, in order to decrease the empty seat cost as capacity increases, crowding can be indirectly addressed by the surplus component. c  aV b2 by setting larger a and setting b P 1. Detailed modeling, which addresses crowding as one of the objectives without neglecting surplus costs, should be considered as a future research direction. Other extensions to our study could address stochastic travel times. An additional line of research might also consider moving from a route level model to a system-wide analysis of multiple routes and transfers.

Appendix A A.1. Additional analysis A second set of parameters was used to further present the algorithm’s performance and to further strengthen the validity of the conclusions. p ¼ 30$; r0max ¼ 0:2; r ¼ 0:1; b1 ¼ 0:1; c ¼ 0:08$; b2 ¼ 2:0; a ¼ 0:00001; b ¼ 25; g ¼ 35; V regul ¼ 500; w ¼ 6 min. Table 5 summarizes the optimal solutions. Tables 6 and 7 present sensitivity analysis for uniform and for Erlang distributions, respectively. When 45 min are required to travel the entire route, the implication is that every 11.25 min a bus departs on with an average waiting time of 5.63 min. Though the average waiting time slightly increases for the AM peak, it does not exceed external regulations (an average wait of 6 min). A.2. Sensitivity analysis See Tables 6 and 7.

Appendix B. The Lagrange multipliers method According the signs of k1, k2 the necessary optimality condition is separated into four cases. (a) k1 = k2 = 0

95

A. Herbon, Y. Hadas / Transportation Research Part B 71 (2015) 85–99 Table 6 Optimal solution with percent of deviation from original parameter (Uniform distribution). Parameter

Change in % in parameter value

b1

b2

a p

g

r

c

b

r 0max

maxk(lk)

w

@=@V

50

25

10

10

25

50

Influence on variable

4 30 22,188 33 3 25,160 33 3 25,159 26 2 14,733 88 1 24,285 21 3 17,675 7 36 17,259 42 2 23,937 43 2 22,861 4 51 19,831 33 3 25,155

4 51 23,294 33 3 25,159 33 3 25,157 36 2 20,305 4 51 24,492 27 3 21,660 4 44 21,263 4 51 24,237 4 51 23,841 4 51 23,365 33 3 25,155

4 51 24,010 33 3 25,157 4 54 24,897 30 3 23,308 4 51 24,513 4 51 23,709 4 48 23,054 4 51 24,411 4 51 24,253 4 51 24,085 33 3 25,155

4 51 25,080 5 35 23,863 4 49 24,192 4 51 25,637 4 51 24,541 4 51 25,346 43 2 25,683 4 51 24,643 4 51 24,802 4 51 24,978 4 51 24,527

4 51 25,978 7 22 23,741 4 46 23,903 4 51 27,302 4 51 24,562 4 51 26,574 39 2 26,330 4 51 24,817 4 51 25,213 4 51 25,665 4 51 24,527

4 51 27,687 12 12 24,061 4 42 23,685 4 51 30,076 4 51 24,597 4 51 28,620 34 2 27,210 4 51 25,107 4 51 25,899 4 51 26,830 3 51 24,045

N⁄ V⁄

" Z K1 X N

V



Z  c  aV b2 t kþ1 ðV  xÞf k ðx; NÞdx þ N

0

k¼1

" Z K1 X N @=@N

pk r0k þ r

V V

Z   c  aV b2 t kþ1 ðV  xÞf k ðx; NÞdx þ N

0

k¼1

1

1

r 0k

pk

V

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄

#  b1 ! T ðx  V Þf k ðx; NÞdx ¼ 0 N

#  b1 ! T þr ðx  V Þf k ðx; NÞdx þ gN ¼ 0 N

ð11Þ

ð12Þ

In order to take the derivatives in Eqs. (11) and (12), we apply Leibniz’s rule. For Eq. (11),

" K1 Z X k¼1

V

0

Z   @=@V c  aV b2 tkþ1 ðV  xÞf k ðx; NÞdx þ

V

1

@=@V pk r 0k þ r

! #  b1 ! T ðx  V Þf k ðx; NÞ dx ¼ 0 N

or,

" K 1 Z X k¼1

V

0

Z   c  aðb2 þ 1ÞV b2 þ axb2 V b2 1 t kþ1 f k ðx; NÞdx 

1

V

pk r 0k þ r

#  b1 ! T f k ðx; NÞdx ¼ 0 N

For Eq. (12),

Z K1

X t kþ1 k¼1

þ

Z   c  aV b2 ðV  xÞf k ðx; NÞdx þ Ntkþ1 @=@N

0

" K 1 Z X k¼1

or,

V

V

0

1

pk

V

    c  aV b2 ðV  xÞf k ðx; NÞ dx þ g

! !#  b1 !  b1 ! Z 1 T T ¼0 r 0k þ r ðx  V Þf k ðx; NÞdx þ N@=@N pk r0k þ r ðx  V Þf k ðx; NÞ dx N N V

ð13Þ

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A. Herbon, Y. Hadas / Transportation Research Part B 71 (2015) 85–99

Table 7 Optimal solution with percent of deviation from original parameter (Erlang distribution). Parameter

Change in % in parameter value

b1

b2

a p

g

r

b

c

r 0max

maxk(lk)

w

Z K1

X t kþ1

V

50

25

10

10

25

50

Influence on variable

17 5 21,855 32 3 24,799 32 3 24,798 26 2 14,467 33 3 24,225 21 3 17,362 30 3 23,548 6 36 18,191 42 2 22,502 29 3 19,824 32 3 24,795

29 3 23,343 32 3 24,798 32 3 24,797 36 2 19,963 33 3 24,514 27 3 21,296 31 3 24,181 4 44 21,696 44 2 23,659 32 3 23,649 32 3 24,795

4 51 24,775 32 3 24,796 32 3 24,795 42 2 22,935 32 3 24,683 30 3 24,440 32 3 24,553 4 48 23,687 32 3 24,351 32 3 24,364 32 3 24,795

4 51 25,902 4 35 24,081 32 3 24,794 4 51 26,477 32 3 24,907 4 51 26,183 32 3 25,037 42 2 25,272 4 51 25,596 32 3 25,289 32 3 24,795

4 51 26,849 6 22 23,949 4 46 24,474 4 51 28,211 31 3 25,073 4 51 27,476 33 3 25,392 39 2 25,885 4 51 26,009 32 3 25,882 32 3 24,795

4 51 28,649 11 12 24,042 4 42 24,080 4 51 31,102 4 51 25,390 4 51 29,631 4 51 25,978 35 2 26,730 4 51 26,698 32 3 26,975 3 51 24,265

N⁄ V⁄

Z   c  aV b2 ðV  xÞf k ðx; NÞdx þ Ntkþ1

0

k¼1

þ

1

pk

V

k¼1

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄ N⁄ V⁄

p⁄

    c  aV b2 ðV  xÞ@f k ðx; NÞ=@N dx þ g

0

" K1 Z X

þ

V

p⁄ N⁄ V⁄

" Z K1 X pk N

1 V

k¼1

#  b1 ! Z 1    T b1 ðb1 þ1Þ 0 rk þ r ðx  V Þf k ðx; NÞdx  pk N b1 rT N ðx  V Þf k ðx; NÞ dx N V r0k þ r

#  b1 ! T ðx  V Þ@f k ðx; NÞ=@Ndx ¼ 0 N

or,

Z K 1

X t kþ1 0

k¼1

þ

V

"Z

K1 X

V

k¼1

Z   c  aV b2 ðV  xÞf k ðx; NÞdx þ Ntkþ1

0

1

pk

V

    c  aV b2 ðV  xÞ@f k ðx; NÞ=@N dx þ g

# " #  b1 !  b1 ! Z 1 K X T T 0 0 r k þ ð1  b1 Þr ðx  V Þf k ðx; NÞdx þ pk N rk þ r ðx  V Þ@f k ðx; NÞ=@Ndx N N V k¼1

¼0

ð14Þ

The set of Eqs. (13) and (14) is a non-linear system with two variables V,N. (b) k1 > 0, k2 = 0 The necessary optimality condition for the case is

" K 1 Z X k¼1

0

V



Z  c  aðb2 þ 1ÞV b2 þ axb2 V b2 1 t kþ1 f k ðx; NÞdx 

1 V

pk r 0k þ r

#  b1 ! T f k ðx; NÞdx ¼ 0 N

ð15Þ

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A. Herbon, Y. Hadas / Transportation Research Part B 71 (2015) 85–99

Z K1

X t kþ1

Z   c  aV b2 ðV  xÞf k ðx; NÞdx þ Ntkþ1

V 0

k¼1

þ



0

"Z

K1 X

   c  aV b2 ðV  xÞ@f k ðx; NÞ=@N dx

# " #  b1 !  b1 ! Z 1 K X T T 0 0 r k þ ð1  b1 Þr ðx  V Þf k ðx; NÞdx þ pk N rk þ r ðx  V Þ@f k ðx; NÞ=@Ndx N N V k¼1

1

pk

V

k¼1

V

 k1 þ g ¼ 0

ð16Þ

k1 ðNmin  NÞ ¼ 0

ð17Þ

The set of Eqs. (15) and (17) is a non-linear system with three variables V,N and k1. By extracting N from (17) we obtain

N ¼ Nmin

ð18Þ

By substituting (18) in (15) and (16), we again obtain a non-linear system of two equations with two variables V, k1. After the substitution, Eq. (15) depends only on a single variable V. Therefore, Eq. (15) is reduced to

" K1 Z X

V

0

k¼1

b1 !

b1 ! # Z 1     1 1 r r 0 0 dx  dx c  aðb2 þ 1ÞV b2 þ axb2 V b2 1 t kþ1 f k x; T p 1 þ r  r x; T f k k max k 1  r 0max 1  r 0max V

¼0 ð19Þ (c) k1 = 0, k2 > 0 The necessary optimality conditions for the case are:

" K1 Z X

V

0

k¼1

Z   c  aðb2 þ 1ÞV b2 þ axb2 V b2 1 t kþ1 f k ðx; NÞdx  Z   c  aV b2 ðV  xÞf k ðx; NÞdx þ Ntkþ1

V 0

þ

pk

V

Z K1

X t kþ1 k¼1

1



#  b1 ! T f k ðx; NÞdx þ k2 ¼ 0 þr N

ð20Þ

   c  aV b2 ðV  xÞ@f k ðx; NÞ=@N dx þ g

0

" K1 Z X

# " #  b1 !  b1 ! Z 1 K X T T 0 0 r k þ ð1  b1 Þr ðx  V Þf k ðx; NÞdx þ pk N rk þ r ðx  V Þ@f k ðx; NÞ=@Ndx N N V k¼1

1

pk

V

k¼1

V

r 0k

¼0

ð21Þ

k2 ðV  V max Þ ¼ 0

ð22Þ

The set of Eqs. (20) and (22) is a non-linear system with three variables V,N and k2. By extracting V from (22) we obtain

V ¼ V max

ð23Þ

By substituting (23) in (20) and (21), we again obtain a non-linear system of two equations with two variables N, k2. After the substitution, Eq. (21) depends only on a single variable N. Therefore, Eq. (21) is reduced to K 1

X

tkþ1

k¼1

þ

Z

V max

0

" K 1 Z X

 2 c  aV bmax ðV max  xÞf k ðx; NÞdx þ Ntkþ1

1

V max

k¼1



pk

Z

V max

0



   2 c  aV bmax ðV max  xÞ@f k ðx; NÞ=@N dx þ g

# "  b1 ! Z 1 K X T r 0k þ ð1  b1 Þr ðx  V max Þf k ðx; NÞdx þ pk N N V max k¼1

r 0k þ r

#  b 1 ! T ðx  V max Þ@f k ðx; NÞ=@Ndx ¼ 0 N

ð24Þ

(d) k1 > 0, k2 > 0 The necessary optimality condition for the case is

" K1 Z X k¼1

V

Z   c  aðb2 þ 1ÞV b2 þ axb2 V b2 1 t kþ1 f k ðx; NÞdx 

0

V

Z K1

X t kþ1 k¼1

þ

V

Z   c  aV b2 ðV  xÞf k ðx; NÞdx þ Ntkþ1

0

" K1 Z X k¼1

pk

r 0k

#  b1 ! T f k ðx; NÞdx þ k2 ¼ 0 þr N

ð25Þ

    c  aV b2 ðV  xÞ@f k ðx; NÞ=@N dx þ g

0

1

pk V

V

1

# " #  b1 !  b1 ! Z 1 K X T T r 0k þ ð1  b1 Þr ðx  V Þf k ðx; NÞdx þ pk N r 0k þ r ðx  V Þ@f k ðx; NÞ=@Ndx  k1 ¼ 0 N N V k¼1 ð26Þ

N ¼ Nmin

ð27Þ

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A. Herbon, Y. Hadas / Transportation Research Part B 71 (2015) 85–99

V ¼ V max

ð28Þ

The set of Eqs. (25)–(28) is a non-linear system with four variables V, N, k1, k2. By substituting (27) and (28) into (25) and (26), we again obtain a non-linear system of two equations with two variables k1, k2. Therefore, Eqs. (25) and (26) are reduced to:

 h   ib1  3 b2 1 1 2 dx 7 c  aðb2 þ 1ÞV bmax þ axb2 V max tkþ1 f k x; T 1rr0 max 7 6   7 þ k2 ¼ 0 6 1 h i   R 5 4 b1 1 k¼1  dx p 1 þ r0k  r0max f k x; T 1rr0 V max k 2

R V max K1 6 0 X

ð29Þ

max

3

2

  h ib1  R V max  1 2 dx c  aV bmax ðV max  xÞf k x; T 1rr0 6 tkþ1 0 max K 1 6 X 0 6 6 h ib1  6 R V B 1 r 2 k¼1 6 þ T t kþ1 0 max @ c  aV bmax ðV max  xÞ@f k ðx; NÞ=@Nj h 4 1r 0max T

 h K 1

 ib1   X  T b1  R1 1 0 r x  V dx p r þ ð 1  b Þr ð Þf x; T þ max 0 1 k k k V max N 1r

r 1r0 max

1

ð30Þ

max

k¼1

2 K X  b  R1  6 þ 4pk N V max r0k þ r NT 1 ðx  V max Þ@f k ðx; NÞ=@Nj h k¼1

7 7 7 7þg 7 C 7 ib1 Adx 5 1

T

3 r 1r0 max

7 ib1 dx5  k1 ¼ 0 1

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