Transportation Research Record 1733 ■ Paper No. 00-0738

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Optimization Model for Large-Scale Bus Transit Scheduling Problems Mohamadreza Banihashemi and Ali Haghani A procedure is presented for solving real-world large-scale multiple depot vehicle scheduling (MDVS) problems considering the route time constraints (RTCs). The procedure is applied to some test problems and then to a real-world problem. The real-world problem is the transit bus scheduling problem of the mass transit administration (MTA) in Baltimore, Maryland. The RTCs are added to the MDVS problem to account for real-world operational restrictions such as fuel consumption. Formulation of the MDVS problem, the set of constraints for considering the time restriction, and a heuristic procedure for solving the MDVS problems with RTCs are discussed. Application of the proposed procedures in solving bus scheduling problems in large cities requires a reduction in size of those problems in terms of number of variables and constraints. Two techniques are proposed to decrease the size of the real-world problems. Combining these techniques results in a strategy to reduce the MTA problem size into a manageable and solvable size. The solutions to the reduced size problems are further improved by solving a series of single depot vehicle scheduling problems for each of the MTA depots. The final results from the proposed model are compared with the MTA’s January 1998 schedule. The comparison indicates that the proposed model improves on the MTA schedules in all aspects. The improvements are 7.90 percent in the number of vehicles, 4.66 percent in the operational time, and 5.77 percent in the total cost.

The transit planning process usually has four major steps: designing routes, setting frequencies and building timetables, scheduling vehicles, and scheduling crew. All steps except the first are usually revisited three or four times a year because of major changes in demand. This process is usually called a major schedule change. Upon completion of the second step of the planning process, a number of trips are identified that are to be served by the available buses. All these trips have a fixed starting and ending location and time. Vehicle scheduling, or blocking, consists of sequencing a series of trips into blocks of trips, where each block is the duty for a single bus for a day. The lengths of the blocks are restricted by the fuel capacity of the bus and other operational constraints set forth by the agency. Designing these blocks usually accounts for about 30 percent of the total schedule makers’ time during the process of a schedule change. In most agencies the problem is split into several smaller scheduling problems for individual depots, and each of these problems is treated as a separate problem. The blocking problem can be formulated and solved as a multiple depot vehicle scheduling (MDVS) problem. Bodin et al. (1) and Ball and Bodin (2) provide a clear definition of this problem. Adding the route time constraints (RTCs) restricts the blocks from being longer than a specified time. Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742.

MDVS problems with RTCs (MDVSRTC) can be formulated as an integer-programming problem. Because of their large-scale nature, real-world MDVSRTC problems cannot be solved to optimality. Heuristic procedures are needed to find acceptable solutions to these problems. The objective function of a MDVSRTC problem is usually minimizing a combination of the capital or fixed cost (as determined by the number of vehicles used) and the total deadhead cost. The basic constraints of the problem are as follows: 1. 2. 3. 4. 5.

All the trips should be serviced. Each trip is run by just one vehicle. Each depot has a limited number of vehicles. Each vehicle returns to the same depot it started from. The total time a vehicle is away from its depot is limited.

This paper focuses on proposing a new approach for formulating and solving these problems. In the following sections, a brief review of the existing models, the proposed formulation, the solution strategy used, the results of its implementation in some test problems and a case study of the mass transit administration (MTA) operational problem, and, finally, the results of a sensitivity analysis with respect to the main parameters of the model are presented.

PREVIOUS FORMULATIONS AND SOLUTION PROCEDURES FOR MDVS PROBLEMS Traditionally, transit agencies solve different problems involved in the transit planning process based on some guidelines. NCHRP Synthesis of Highway Practice 69 (3) presents a practical set of guidelines for different steps of the process. Different approaches have been proposed for solving the MDVS problem. These approaches find either an exact optimal solution or an acceptable one. Two of these approaches that use a single depot vehicle scheduling (SDVS) problem formulation as their basis are called cluster first–schedule second and schedule first–cluster second. These approaches are used more than others in real-world large-scale scheduling. These approaches are summarized in the next section.

Cluster First–Schedule Second This method relies on formulating the SDVS problem. Carraresi and Gallo described this method as follows (4): The set of trips is partitioned into l subsets, one for each depot. Then for each subset a single-depot problem is solved. If the capacity constraint for some depot is not satisfied, the partition is modified and the

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new single-depot problems are solved. The process continues until a satisfactory solution is obtained. This approach corresponds to the way in which most companies operate in practice, since quite often the trips are assigned to the depots a priori.

Transportation Research Record 1733

The proposed formulation is as follows: min Σ d , i ad , i Ad , i + Σ d , i ed , i Ed , i + Σ i , j , d ci , j , d Xi , j , d + Σ d , i bi , d Bi , d + Σ d , i fi , d Fi , d (1)

Schedule First–Cluster Second

subject to

In this approach, first an SDVS problem on the entire set of trips is solved, yielding a set of blocks. A major assumption in this step of the solution is that the set of best blocks constructed from trips is independent of the location of the depots. This is not a completely accurate assumption. Then each block is assigned to one of the depots in such a way that the total cost of deadhead trips is minimized and the capacity constraints are satisfied. Gavish and Shlifer proposed such an approach for scheduling vehicles in a large transit agency (5). The MDVS problem is formulated as a mixed integer programming problem in different ways. Bertossi et al. formulated the MDVS problem as a multicommodity matching problem and proposed a heuristic procedure for solving this problem (6 ). There is another multicommodity formulation proposed by Lamatsch based on a timespace network (7 ). Forbes et al. presented a shorter form of the problem formulated as a multicommodity problem that appears to be more practical (8). To account for RTCs, to the best of the authors’ knowledge, the schedule makers generally use manual methods. However, there are some considerations in the literature. Bodin et al. (1) and Branco (9) [described by Freling and Paixao (10)] have taken these constraints into account in the SDVS problem context. In both approaches, they looked at only the first and the last trips of the blocks and did not pay attention to the trips in the middle that might be connected to the depot in the middle of the day. This makes the models unsuitable for consideration of fuel consumption. Based on the review of the literature and the authors’ experience with the scheduling procedures implemented at the MTA in Baltimore, Maryland and some other agencies, it has been concluded that most transit agencies still perform a sizable portion of the scheduling by conventional manual approaches despite using different software. Existing software is used to streamline the schedule makers’ duties by performing some minor optimizations and reducing the time for some repetitive calculations. In this paper a computational procedure is proposed for solving the large-scale MDVSRTC problems and the procedure is applied to the MTA scheduling problem in January 1998.

Σ i Ad , i ≤ rd

PROPOSED FORMULATION AND SOLUTION OF THE MDVSRTC PROBLEM Formulation This formulation consists of a new formulation for the MDVS problem and a new set of constraints considering the time restrictions. Two trips are deemed compatible when the same bus can run one after the other. Two trips are depot compatible if the same vehicle can run the trips consecutively but it is less costly for the vehicle to return to the depot instead of waiting on the street after it serves the first trip. For presenting this formulation, the original trip set is partitioned into three sets: morning trips, midday trips, and afternoon trips. This partitioning is based on the compatibilities of the morning trips and the afternoon trips. All the morning trips are depot compatible with all the afternoon trips.

∀d

(2)

Σ d Bi , d + Σ j , d Xi , j , d + Σ d Fi , d = 1 Σ i Ed , i − Σ i Fi , d = 0

∀i

(3)

∀d

Ad , i + Ed , i + Σ j X j , i , d − Bi , d − Fi , d − Σ j Xi , j , d = 0

( 4) ∀i, d

(5)

Ad , t1 + Xt1 , t2 , d + Xt2 , t3 , d + L + Xt( p −1) , t p , d + Bt p , d ≤ p ∀d and ∀ blocks with block time greater than Tmax (6) All variables integer

( 7)

where 1 if Trip i is the first trip run by a vehicle from Ad , i =  Depot d 0 otherwise 1 if Trip i is in the afternoon trip set and is the first  trip run by a vehicle from Depot d returning to Ed , i =  the street  0 otherwise 1 if compatible Trips i and j are run consecutively  by a vehicle from Depot d (not for depot com Xi , j , d =  patible Trips i and j, with i in the first and j in the  last trip sets) 0 otherwise 1 if Trip i is the last trip run by a vehicle from  Bi , d =  Depot d 0 otherwise 1 if Trip i is in the morning trip set and is the last  trip run by a vehicle from Depot d returning to Fi , d =  the depot  0 otherwise ad,i = travel cost between Depot d and the start point of Trip i plus (fixed cost)/2, fixed cost = fixed cost of one vehicle converted into an equivalent operational cost, ed,i = travel cost between Depot d the start point of Trip i, travel cost of Trip i plus the time between start  time of Trip j and end time of Trip i (if travel to   the Depot d is not feasible in the time between   2 trips) Ci , j , d =  min of the above and the total travel cost from   Trip i to Depot d and from Depot d to Trip j if  travel to the Depot d is feasible in the time   between 2 trips) bi,d = travel cost from the ending point of Trip i to the Depot d plus travel time of Trip i plus (fixed cost)/2, fi,d = travel cost from the ending point of Trip i to the Depot d plus travel time of Trip i, rd = number of vehicles at Depot d,

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P = number of trips of the block, ti = ith trip of the block, and Tmax = maximum allowed block time. The objective Function 1 minimizes the total cost including the capital cost and the operational cost. The function’s Constraint Set 2 ensures that the total number of the blocks started by vehicles from each depot is limited to the number of available vehicles in that depot. Constraint Set 3 ensures that all the trips are serviced. Constraint Set 4 ensures that the number of vehicles that return to the depots in the middle of the blocks is equal to the number of vehicles that return to the street in the middle of the blocks. Constraint Set 5 ensures that vehicles start from and return to the same depot. Constraint Set 6 ensures that the construction of the blocks with the route times more than the maximum allowed block time is prevented. Finally, Constraint Set 7 is the set of integrality constraints. An exact solution for this problem can be obtained by relaxing Constraint Set 6, solving a regular MDVS problem, adding the RTCs 6 as required to prevent creation of any blocks that violate the time restrictions, and solving the resulting problem. An optimal solution is obtained when there are no blocks that violate the RTCs. This procedure may have to be repeated several times before a final solution is obtained. The exact solution for test problems with 300 and 400 trips is presented in Table 1. Seven classes of test problems used in this research are generated based on the operational characteristics of the MTA in Baltimore. The test problems used in this study are problems with 300, 400, 500, and 600 trips (five problems in each group) and problems with 700, 800, and 900 trips (one problem in each group).

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Step 1. Solve the MDVS problem optimally. Step 2. Construct the established blocks and recognize the violated blocks. If there is no such block, the solution is the optimal solution for the MDVSRTC problem as well. Stop. Step 3. Corresponding to each violated block with p trips, there are p + 1 variables equal to 1 in the solution. For each violated block containing p trips ( p + 1 variables) decrease the number of trips one by one until the block time becomes legal. Build a constraint set containing the equality constraints (variable = 1) for all the variables associated with all the legal blocks. Step 4. Add the constraints generated in Step 3 to the problem at hand to create a new MDVS problem. Solve this problem optimally and go to Step 2. The results of applying this heuristic solution procedure on several test problems are also presented in Table 1. As the table indicates the procedure provides excellent solutions that are either optimal or very close to the optimal solutions in all these test problems. As the results indicate, the MDVS and the MDVSRTC solutions are practically the same. Because the optimal solution of the MDVSRTC problem cannot be obtained for all test problems, for consistency in solution evaluation the calculated gaps are the solution gaps with the MDVS solutions. The problems that can be solved by the preceding heuristic procedure are much smaller than real-world problems. The number of trips in a large city is usually more than 5,000. The MTA problem has 5,650 trips. This problem may have more than 20,000,000 variables. The following two techniques are proposed to reduce the size of the problem.

Heuristic Procedure for Solving MDVSRTC Problem Decreasing the Number of Trips Large problems cannot be solved by using the exact solution procedure outlined above. The details of the heuristic procedure used to solve the MDVSRTC problem are presented by Banihashemi (11). The major steps of this heuristic are as follows: TABLE 1

Join the trips and make one trip from two or more trips and consider it as a single trip a priori. All pairs of compatible trips, where the ending point of the first one is the same as the starting point of the sec-

Solution Results of the Heuristic Solution Procedure

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TABLE 2

Maximum Layover Time of 15 min for Joining the Trips

ond one, are studied. The time difference between the ending time of the first trip and the starting time of the second trip is called the layover time. A minimum layover time is selected. All the pairwise compatible trips with layover time equal to or less than the minimum layover time are joined and treated as a single trip with the starting time and starting point of the first trip and the ending time and ending point of the second trip; the trip time equals the time difference between the ending time of the second trip and the starting time of the first trip. A minimum layover time of 15 min was chosen in this solution by testing different minimum layover times. This method was applied to some test problems. Table 2 presents the results of solving the reduced size test problems and the gaps of these solutions with the MDVS solutions for the same problems.

Decreasing the Number of Variables The number of variables can be reduced in a preprocessing step. The variables corresponding to the compatible trips (Xi,j,d variables) are chosen for this variable reduction. The cost associated with each Xi,j,d variable is the summation of the cost of the deadhead between the two compatible trips and the cost of the first trip. The 20th percentile of these deadhead costs is calculated. Only the Xi,j,d variables with deadhead cost equal to or less than this percentile are kept. The new problem has almost 20 percent of the variables of the original problem. Then, as the last step for improving the solution, the problem

TABLE 3

is split into subproblems, one for each depot. The subproblems are created without any variable reduction and they are solved with the proposed heuristic. The solution results for some test problems are presented in Table 3. In general, in the variable reduction step, whatever percentile is desired can be chosen. The lower the percentile, the smaller the size of the problem compared with the original problem size and the larger the solution gap compared with the optimal solution. MTA PROBLEM CHARACTERISTICS AND TEST PROBLEM GENERATION The MTA bus transit system has three schedule changes a year. In each schedule change the number of trips and some other information may change. After the schedule change in January 1998 this system had 5,650 trips for weekdays run on 53 routes (lines). The basic area of the operation is almost a square of 25 × 25 mi (40.2 × 40.2 km). Close to 94 percent of the trips are run in this area. This area is referred to as the regular operational area. It is assumed that the other 6 percent of trips that belong to the “long lines” are running in a square of 50 × 50 mi (80.5 × 80.5 km) that is referred to as the wide operational area. There is an area almost in the middle of the operational area with a size of 4 × 4 mi (6.4 × 6.4 km) where 78 percent of trips start or end. This area is a little larger than the real central business district (CBD) of Baltimore and is referred to as the CBD. The system is operated

Decreasing the Number of Variables

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24 h a day. Based on the starting times of trips, the operational time is split into five segments as follows: 1. 2. 3. 4. 5.

Morning off-peak period (from 3:00 a.m. to 6:00 a.m.); Morning peak period (from 6:00 a.m. to 10:00 a.m.); Midday off-peak period (from 10:00 a.m. to 1:30 p.m.); Afternoon peak period (from 1:30 p.m. to 7:00 p.m.); and Evening off-peak period (from 7:00 p.m. to 3:00 a.m.).

The percentages of trips, starting in each of the five time intervals, are 3, 32, 13, 36, and 16 percent, respectively. For the preceding time periods and for the whole day, the number of starting times of trips, the number of trips running, and the travel speed are studied. Figure 1a presents the number of starting times and the number of trips running in 1-h intervals during a day. In the MTA system each line has different patterns whose starting and ending points might be different from the original starting and ending points of the line. In each schedule change, some of the patterns of the lines are chosen for trips of that line to be able to better satisfy the demand. The average deadhead speed for deadhead travels in the MTA solution is 15.5 mph (25 km/h). To provide a safety margin in the

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calculations, the MTA average deadhead speed was decreased by 10 percent and 14 mph (22.5 km/h) was used as average deadhead speed in this model. There are 321 starting points of trips and 310 ending points of trips in the MTA problem. For establishing the problem it was necessary to estimate all the distances between all the ending points and all the starting points, between all the depots and all the starting points, and between all the ending points and all the depots. The network data of Baltimore in MINUTP software format was used to estimate these distances. The morning peak period was partitioned into two periods and the afternoon peak period was partitioned into three periods. This resulted in eight time periods with linearly distributed starting times. The preceding information and some other statistics were used to develop a code to generate the test problems. The input file to this code takes the following elements into account: 1. The total number of trips, the non-CBD number of trips, the number of trips in the long lines, the number of depots, the number of available vehicles in each depot, the capital cost of one vehicle, the maximum allowed block time, and the average deadhead speed;

FIGURE 1 Comparing the starting times and trips running at 1-h intervals for 900-trip problem to MTA problem: (a) MTA problem; (b) 900-trip problem (MPM  minutes past midnight).

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2. The total number of lines, the non-CBD number of lines, the number of long lines, the number of extra starting and ending points of each line, and the percentage of trips starting and ending at extra starting and ending points; 3. The starting and ending of the different operational periods in minutes past midnight; 4. The minimum, the average, the maximum, the standard deviation, and the correlation factor for the trip speeds in the morning peak, the afternoon peak, and the off-peak periods; 5. The dimensions of all operational areas; 6. The minimum and maximum lengths of routes; 7. The percentages of the trip starting times in different operational periods; 8. The tangents and offsets of the lines fitted to the starting times distributions; and 9. A seed number. The flow of the simulation code for generating the test problems is as follows: 1. Randomly determine the location of the depots in the regular area of the operation. 2. Randomly choose the original starting points and the original ending points of the lines in the regular operational area out of the CBD, in the CBD, and in the wide operational area out of the regular operational area. 3. Randomly choose three extra starting and three extra ending points for each line in the first and the last quarter segments of the line. It was assumed that 40 percent of trips are running starting or ending at these extra points. 4. Distribute the trips among the lines evenly between two directions. Distribute the trips of a line among different eight time intervals of the day based on the percentages of the trips of the MTA problem for these times (3, 16, 16, 13, 8.9, 12.2, 14.9, and 16 percent). In each time interval the starting times of the trips are randomly chosen based on a linear distribution. 5. For each trip the speed is randomly generated. The operation time is split into three different periods for generating the speed, offpeak period, morning peak period, and afternoon peak period. It was assumed that the speeds and the distances are normally distributed in each of these periods correlated with each other. For each trip a normally distributed random variable Z2 was generated on (0,1). Then the trip speed X2 was calculated from the following equation:  X − µ1  X2 = µ 2 + σ 2  ρ 1 + 1 − ρ 2 Z2    σ1 where Z2 = generated normally distributed random variable, X1 = trip distance, X2 = generated trip speed, µ1 = average distances of trips of the MTA, µ2 = average trip speed of the MTA, σ1 = standard deviation of the trip distances of the MTA, σ2 = standard deviation of the speeds in the MTA, and ρ = correlation factor between speed and distance in MTA. 6. Calculate the ending times from the starting times, speed, and trip distances. For validating the generated test problems the variation of the starting times and trips running was drawn in 1-h intervals during a whole

Transportation Research Record 1733

day for the first test problem of each class. All the drawings are similar. The one corresponding to the 900-trip problem is presented in Figure 1b. Comparing Figure 1b with Figure 1a indicates that the distribution of the trips in a day is quite similar to the distribution of the trips for the MTA problem.

PROPOSED SOLUTION STRATEGY AND ITS APPLICATION TO TEST PROBLEMS The strategy has the following steps: Step 1. Choose a minimum layover time and apply the method for joining the trips. Name the newly created problem Revision1 problem. Step 2. Choose an appropriate percentile and apply the method for decreasing the number of the variables. Name the newly created MDVS problem Revision2 problem. Step 3. Solve the MDVS Revision2 problem to optimality. Step 4. Based on the solution found in Step 3, partition the trip set of the Revision2 problem into separate sets for each depot. Expand each new trip set to the trips in the original trip set and build singledepot subproblems, one for each depot. Apply the heuristic approach to these subproblems. Step 5. Combine the solutions found in Step 4. The procedure was applied to the 700-trip, 800-trip, and 900-trip test problems using 15 min of layover time in the first step and the 7th percentile in the second step. The results are presented in Table 4 with an average gap of 5.2 percent with the MDVS solution.

SOLVING MTA SCHEDULING PROBLEM The proposed procedure is applied to the January 1998 MTA scheduling problem. The results of applying the above procedure to the MTA problem are as follows: 1. A minimum layover time of 15 min was chosen and Step 1 was applied. From the original 5,650-trip problem the Revision1 problem with 2,218 trips is created. This problem has 5,168,108 variables and 11,098 constraints. The required times for the preparing process and the joining process are about 720 min (12 h) and 5 min, respectively. 2. Step 2 is applied to the Revision1 problem with the 7th percentile chosen as the criterion for eliminating variables. The Revision2 problem has 364,925 variables and 11,098 constraints. This step takes about 90 min. 3. The integer solution of the Revision2 problem is equal to 518,092 min of operation that includes the fixed cost of 590 buses. The total solution time for this step is about 190 min.

TABLE 4

Applying the Solution Procedure to Some Test Problems

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4. Partitioning the Revision2 problem into four parts results in construction of four subproblems with 1,541 trips, 1,018 trips, 1,693 trips, and 1,398 trips related to Depots 1, 2, 3, and 4, respectively. The heuristic procedure for solving the MDVSRTC problem is applied to all four problems. The final solution, which is the sum of the four solutions, is 510,939 min of operation that uses 571 buses. The total preparation and solution time for this step is about 900 min. In the January 1998 schedule of the MTA, 620 vehicles were used. The fixed cost of each vehicle is assumed to be equivalent to the cost of about 300 min of operation. This value is based on some assumptions discussed by Banihashemi (11). A summary of this solution and comparison of this solution with the actual schedule implemented by the MTA are presented in Table 5. The total time required for applying this procedure to the MTA problem is 1,905 min. CPLEX software (version 6.1) on a 600-MHz Pentium PC with 256 megabytes of random-access memory was used to solve the test problems and the MTA problem.

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the deadhead speed. As the results indicate the model is very sensitive to this parameter. Proper allocation of the vehicles to different depots affects the total cost of the MDVS problem significantly, but it does not change the total number of vehicles used in the solution. The change in the total cost is due to the change in the operational cost. As experiments on the test problems show, on the average, there is a 12.6 percent difference between the best- and the worst-case costs. The basic parameter that differentiates the MDVSRTC and the MDVS problems is the maximum allowed block time—namely, Tmax. The Tmax used in the test problems was 1,200 min or 20 h based on the estimation provided by MTA personnel. The problems in the 300-trip problem group are tested and the heuristic solution procedure is used to get the MDVSRTC problem solutions. The problems were tested for Tmax equal to 960, 1,020, 1,080, 1,200, 1,320, and 1,440 min. As expected, as the Tmax increases fewer violated blocks are formed in the MDVS solution and, as a result, fewer iterations are needed for obtaining the MDVSRTC solution. The results also show that as Tmax increases the total cost decreases. Figure 2d and 2e shows the changes in the total cost and in the number of iterations versus the changes in Tmax.

SENSITIVITY ANALYSIS The most important parameters affecting the solutions of the MDVS and MDVSRTC problems are the fixed cost of one vehicle, the deadhead speed, the number of vehicles located in each depot, and the maximum allowed route time. The effects of changes in those parameters on the solutions were studied using the 300-trip test problems. For the first three parameters the revised MDVS problems were solved and for the last parameter the MDVSRTC problems were solved. The costs associated with the first and the last trips of a block include the fixed cost of a vehicle. This cost is translated into minutes of operation and the value of 300 min of operation is used in the test problems. To observe the effect of this parameter on the solution and starting with the fixed cost equal to zero the MDVS problems were solved to optimality. The experiments were continued by increasing the fixed cost. Increasing the fixed cost results in dramatic reduction in the number of vehicles up to the point where the fixed cost is equal to 50, at which point the number of vehicles reaches the minimum number needed for servicing the trips. Figure 2a presents the variation of total cost, vehicle cost, and operational cost versus fixed cost and Figure 2b presents the variation of the number of vehicles versus fixed cost. In this research an average deadhead speed of 14 mph was used. Variation in the deadhead speed would affect not only all the objective function’s coefficients but also the number of variables in the problem. Figure 2c presents the effect on the total cost of increasing

TABLE 5

CONCLUSIONS AND FUTURE RESEARCH The results of this research indicate that the proposed model works well in solving real-world bus transit scheduling problems. Based on the size of the problem and based on the computer capabilities we can choose different values for the minimum layover time for joining the trips and different percentiles for eliminating the variables to reduce the size of the problem. Any heuristic solution of the MDVS or MDVSRTC problems may be improved if the problems are partitioned, if some SDVS problems are created, and if these problems are solved separately. In general, multiple depot scheduling is more efficient than single depot scheduling but it may result in more complicated blocks that in turn may increase the costs of driver training. The ultimate decision on single-depot or multiple-depot scheduling should be made by considering the trade-offs between the savings in operating costs and the complexities that are introduced by multiple-depot scheduling. In this research the construction of the MDVS problem is based on a fixed timetable of trips. If there is some flexibility in modifying the starting and ending times of the trips based on the results of the blocking, more cost savings could be realized. This consideration makes the problem much more complicated but also builds significant extra compatibilities among the trips, which may lead to better solutions.

Summary and Comparison of Solutions

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FIGURE 2 Sensitivity analysis: (a) costs versus fixed cost; (b) number of vehicles versus fixed cost; (c) cost versus deadhead speed; and (d) cost versus T max .

Besides the RTCs, considering distance restriction constraints or a combination of the two is more realistic. This becomes especially important for long trips with high average speeds. Finally, a simultaneous consideration of vehicle and crew scheduling for multipledepot problems is another fruitful area for future work. REFERENCES 1. Bodin, L., B. Golden, A. Assad, and M. Ball. Routing and Scheduling of Vehicles and Crews: The State of the Art. Computer & Operations Research, Vol. 10, 1983, pp. 63–211. 2. Ball, M., and L. Bodin. A Matching Based Heuristic for Scheduling Mass Transit Crews and Vehicles. Transportation Science, Vol. 17, No. 1, Feb. 1983, pp. 4 –31. 3. NCHRP Synthesis of Highway Practice 69. TRB, National Research Council, Washington, D.C., 1980. 4. Carraresi, P., and G. Gallo. Network Models for Vehicle and Crew Scheduling. European Journal of Operations Research, Vol. 16, 1984, pp. 139–151.

5. Gavish, B., and E. Shlifer. An Approach for Solving a Class of Transportation Scheduling Problems. European Journal of Operations Research, Vol. 3, 1978, pp. 12–134. 6. Bertossi, A. A., P. Carraresi, and G. Gallo. On Some Matching Problems Arising in Vehicle Scheduling Models. Networks, Vol. 17, 1987, pp. 271–281. 7. Lamatsch, A. An Approach to Vehicle Scheduling with Depot Capacity Constraints. Proc., Fifth International Workshop on Computer-Aided Scheduling of Public Transit, Montreal, Canada, 1990. 8. Forbes, M. A., J. N. Hotts, and A. M. Watts. An Exact Algorithm for Multiple Depot Vehicle Scheduling. European Journal of Operational Research, Vol. 72, 1994, pp. 115–124. 9. Branco, I. M. Algorithms para modelos matemátição de quasi-afectacao e extensões. Ph.D. thesis, DEIOX-FCUL, Universidade de Lisboa, Portugal, 1989. 10. Freling, R., and J. Paixao. Vehicle Scheduling with Time Constraints. Proc., Sixth International Workshop on Computer-Aided Scheduling of Public Transit, Lisbon, Portugal, 1993. 11. Banihashemi, M. Multiple Depot Transit Scheduling Problem Considering Time Restriction Constraints. Ph.D. dissertation. University of Maryland, Oct. 1998.