Bus garage location planning with dynamic vehicle assignments: A methodology

Tmnsppn.Res.~B Vol 19B, No. I. pp. I-13, Pnntcd in the U.S A.

0191-2615/X5 $3 CO+ 00 0 1985 Pergamon Press Ltd

1985

BUS GARAGE VEHICLE

LOCATION PLANNING WITH DYNAMIC ASSIGNMENTS: A METHODOLOGY T. H. MAZE

The Oklahoma

Highway

and Transportation Engineering Center, The University Norman, OK 73019, U.S.A.

of Oklahoma.

and

Department

SNEHAMAY KHASNABIS of Civil Engineering, Wayne State University, (Received

15 March

1983; in revised form

Detroit, MI 49202, U.S.A

15 January

1984)

Abstract-A simultaneous vehicle scheduling and bus garage location and sizing optimization is described. The methodology’s importance lies in its treating garage locations and sizes and vehicle schedules as dynamic. In other bus garage planning methodologies, vehicle schedules are assumed fixed.

1. INTRODUCTION

In the last two years there have been two significant advances in the state of the art of analytical methodologies used in bus garage planning. In a recent paper we reported the application of our bus garage location and sizing optimization method (Maze et al., 1983a). At approximately the same time but in a completely independent effort, Ball et al. (198 1) developed a heuristic technique to size and locate bus garages. Although the two techniques approach the problem differently, they both assume that the structure of bus assignments to routes is independent of location and sizes of new or existing garages. In practice, the structure of vehicle schedules is not independent of the location and size of garages, nor would it be efficient for them to be treated independently. Both techniques assume that the basic structure of the way in which buses are assigned to one or a series of journeys (this assignment is termed a block) will remain unchanged when new garages are located. In practice blocks are rescheduled several times per year, and with each rescheduling minor modifications to the block designs are made to accommodate the most recent service modifications. When there is a change in garage facilities, major modifications in block designs are typically made. For example, when a new garage is located, blocks will be redesigned such that blocks originating from the new site start at the termini of routes closest to the new site or route termini may even be moved closer to the new site to minimize deadhead travel to the new location. Therefore, the assumption that blocks remain unchanged with new garage locations and capacities does not realistically characterize the problem. In this paper, we develop a technique which simultaneously locates and sizes garages and redesigns blocks. Simultaneously preforming these activites increases the realism of the characterization of the problem in two respects. First, it minimizes the bias toward existing garages that is included in previous block designs, and, second, it permits a realistic testing of the potential cost implications of locating at new sites by redesigning blocks with respect to these new sites. The development of the methodology is divided into three portions. First, an existing block design optimization is explored. This optimization considers garage location and size as being fixed. Next, our garage location and sizing optimization is briefly covered. This optimization, in contrast to block design optimizations, considers block designs as being fixed. Lastly, the two optimizations are coupled into a single optimization which simultaneously designs blocks (assigns vehicles to journeys) and locates and sizes garages. The simultaneous optimization is demonstrated with a small-scale hypothetical example problem. 2.

VEHICLE

ASSIGNMENT

OPTIMIZATION

Vehicle assignment optimizations are included as a module of most automatic vehicle scheduling and driver assignment computer packages. For example, RUCUS, an early automatic

T. H. MAZE and S. KHASXABIS

2

vehicle scheduling and driver assignment package that has been installed at approximately 40 North American transit properties, starts with such an optimization module (Luedtke 1983). The purpose of this module is to take a first cut at grouping journeys into feasible and efficient assignments for both vehicles and drivers (Roberts 1973). However, the constraints considered in the vehicle assignment optimization are only related to forcing a feasible assignment for buses. The resulting block designs may or may not be feasible (and possibly inefficient) with respect to labor constraints. A later module in the flow of the RUCUS package generally goes through the result of the vehicle assignment optimization and matches blocks to labor, and may even reconstruct some blocks so that they are feasible and efficient labor assignments. After iterating through the matching routine a feasible, and hopefully efficient, vehicle and driver assignment schedule is produced. The mathematical formulation of the vehicle assignment optimization of a widely used interactive scheduling package (MUNI-SCHEDULER) is utilized as an example (Keaveny 1981). The right to distribute this bus scheduling package has recently been purchased by the Urban Mass Transportation Administration (UMTA) and will be distributed under the name RUCUS II. Unlike previous versions of RUCUS, which are primarily executed in batch mode. RUCUS II is an interactive package. The vehicle assignment optimization model is separate from the driver assignment module (the run cutter) and the scheduler is permitted the capability of iterating with the vehicle assignment optimization until a desirable block design is reached. The desirable block design is then input to the run cutter. The vehicle assignment optimization is solved as a single commodity, linear network flow problem. The objective of the model is to determine the minimum cost groupings of journeys into blocks. There costs are considered in the model. They include: (1) the costs of pulling a bus out of garage to a journey beginning, (2) the travel costs of buses hooking between journeys, and (3) the costs of pulling the bus back into the garage after the block is completed. The objective function is shown in eqn (1). In eqn (1), zeroone integer variables X,, Z;, and Y, represent the assignment of a bus pullout to a journey, the assignment of a hook between journeys i andj, and a pullin from journey j, respectively. The constraints in eqns (2) and (3) force all journeys to end either with the bus pulling in or hooking to another journey. Note that because resources (vehicles) are being allocated to activities (journeys on a oneto-one basis, this is a classical “assignment problem.” At the same time, this is a simple network problem where the links are the deadhead travel and layover between journeys or between garages and journeys. The nodes in the network that correspond to journey ends are transshipment nodes because, at the end of a journey, it is uncertain whether the vehicle will return to its ultimate destination (the garage) or travel on another intermediate link (a hook). Because of the network structure, the assignment variables are naturally occurring integers (0 or 1) in the optimal solution

Minimize

Z = c

C,X, + 2

2

D:Z; + 2

,=I

,=I

,=-I

,=I

Subject to X, + 2

Z; = 1 for all j

E,Y,

(1)

, n)

(j = 1, 2,

(2)

,=I

“‘,

Y, + C Z:, = 1 for all j p= 1

(j = 1, 2, .

, n)

(3)

where j is the journey number (j = 1, 2, . , n), i is the journey number of all feasible hooks to journey j (i = 1, 2, , m,) p is the journey number of all feasible hooks from journey j (k = 1, 2, . , oI) C, is the cost of pulling out a bus to journey j, 0; is the cost of hooking from journey i to journey j, E, is the cost of pulling in a bus from journey j, X, is the assignment of a pullout to journey j (X, = 0 or l), Z; is the assignment of a hook from journey i to journey j (Z; = 0 or l), Y, is the assignment of a pullin from journey j (Y, = 0 or 1).

Location planning

with dynamic

vehicle assignments

3

The vehicle assignment optimization will later be coupled with the garage location and sizing optimization. However, before the two can be coupled, the vehicle assignment model must be revised. To understand why revisions in the vehicle assignment optimization must be made, note that none of the assignment variables are associated with a garage. Given the existing structure of the optimization, it is impossible to force a block which begins with a pullout from one garage to return the bus to the same garage. Because the vehicle assignment optimization is insensitive to the garage location, pullout and pullin costs can not be associated with the travel costs from and to a particular garage. Therefore, all pullout and pullin costs are in practice generally assigned the same very large, arbitrary cost. The impact of assigning a large value to pullout and pullin costs is to cause the optimization to choose the block design which minimizes the number of pullouts and pullins. Since pullouts and pullins are the beginning and end of blocks, the resulting block design is the one which minimizes the total number of blocks. Whether minimizing the number of blocks is the appropriate objective is debatable, but certainly insensitivity to garage location is an infeasible circumstance for a model employed in a garage location and sizing analysis. A feasible approach has been found to make the vehicle assignment optimization sensitive to garage location. This approach will be described in a later section of this paper.

3.

GARAGE

LOCATION

AND

SIZING

OPTIMIZATION

The model formulation for the bus garage location and sizing optimization is one developed by Maze et al. (1982). This optimization starts with a fixed set of existing sites and must be given a fixed block design. Only discrete locations are chosen for potential garage locations, because in most urban areas there are only a few locations (usually less than twenty even in very large urban areas) which are both politically feasible sites for a garage and which have adequate access to the transit and highway network. The mathematical model is only briefly described in this paper. The objective of the mathematical program is to minimize the sum of three costs related to the cost of locating and sizing garages. These costs are: (1) Nonrevenue transportation costs, which are the time and distance costs of travel between block assignments and the garage. These costs are estimated for each potential assignment of a block to a garage. (2) Garage operating costs, which are the costs of maintaining the buses housed at a garage and operating the garage. Engineering cost studies of bus garage operation have shown that there is a fixed charge for the first increment of capacity assigned to a garage and that after the first bus, total costs increase linearly with garage capacity. (3) Garage construction costs, which are the costs of the garage building, land, and equipment. Engineering cost studies of bus garage construction have shown that there is a fixed charge for the first increment of capacity assigned to a new garage and that after the first bus, total costs increase linearly with garage capacity. The nonrevenue transportation costs are expressed in terms of blocks, while the other costs are expressed in terms of bus capacity. To convert blocks assigned to a garage to an equivalent garage capacity, capacity assigned to a garage is set equal to the maximum number of blocks assigned to a garage at any one time. We divide the blocks into four types with respect to the time period they cover. These block types are: (1) A.M. blocks, which are made during the morning peak periods, (2) midday blocks, which cover neither the A.M. nor P.M. peak periods, (3) P.M. blocks, which cover the P.M. peak period, and (4) all-day blocks, which cover or at least partially cover both peaks. The capacity of a garage must be greater than or equal to the number of all-day plus A.M. blocks assigned to it, since both types of blocks cover the A.M. peaks. Similarly, the garage must have capacity greater than or equal to the number of midday plus all-day blocks and the number of P.M. plus all-day blocks. These block types will occur on weekday Saturday and Sunday schedules. The objective of the mathematical program is to minimize the sum of all costs subject to garage capacity constraints and to meet all scheduled service. The objective function for the model is shown in eqn (4). The first term sums the nonrevenue transportation cost (Tkiy) for all blocks assigned to each garage (zeroone assignment variable X,&. The second and third terms sum the variable portion of the garage operating cost and garage construction cost,

T. H. MAZE and S. KHASNABIS

4

respectively. The fixed charges are not included because their inclusion would make the problem an integer program. Specifically, the fixed charge would have to be multiplied by a zeroone switch variable. But the very large number of potential assignment variables in actual system application would make such an integer program beyond the normal limits of feasible computing times. Instead, the problem is solved as linear program and garages are switched off and on by setting their capacity to zero or to their maximum capacity. When a garage is switched on, the appropriate fixed charge is added to the solution of the linear program. All combinations are evaluated through a set of efficient implicit enumeration rules which will be described later. The constraint in eqn (5) forces all blocks to be assigned a bus. Equations (6)-(S) are capacity constraints allowing no more blocks to be assigned to a garage at any period of the day or day of the week than there are available spaces for buses. Even though weekend service will generally not create a capacity problem for garages, Saturday and Sunday service must also be accounted in the assignment problem. Weekend service must deadhead to and from garages and these nonrevenue costs must be accounted for when locating a garage.

min Z = c

2

2

TLII,X,,, +

2

O(N;) +

2

CN;)

(4)

subject to

2

Xk,q =

1

for all i

(i =

, tz) and for all k

1, 2,

(k = a, b, c,

. , s)

(5)

y=l

‘i,,

c XL,, + ,=I

i

,=I

x,,,-

ti,. c Xk,, + c

Nz5-O

X,,, s 0

forallq

(q = 1,2,.

,m)andfork

for all y

(q = 1, 2,

, m) and for k = e, f and h

(q = 1, 2, . .

for all q Ng -

N; -

K; 5 0

N” 5 K”y Y

for all y

forallq

, m) and for k

(q = 1, 2, .

(4=

= a,bandd

1,2,...,m)

, m)

q

1, o and s

(6)

(7)

(8) (9) (IO)

Restrictions N;; I 0 for all q N; 2 for all q

(q = I, 2, (9 = 1, 2,

Xk,, = 0 or I

, mj . , m)

(II) (12) (13)

where u is the weekday A.M. blocks, b is the weekday P.M. blocks, c is the weekday all-day blocks, d is the weekday midday blocks, e is the Saturday A.M. blocks, f is the Saturday P.M. blocks, g is the Saturday all-day blocks, h is the Saturday midday blocks, 1 is the Sunday A.M. blocks, o is the Sunday P.M. blocks, p is the Sunday all-day blocks, s is the Sunday midday blocks, k = a, 6, c, , s, i is the block number (i = 1, 2, . , n), q is the garage site number (4 = 1, 2, , m), X is block assignment (X = 0 or l), N” is the number of active buses assigned to a garage site, N’ is the number of active buses assigned to a garage site that

Location

planning

with dynamic

vehicle assignments

5

require the construction of new spaces, KY is the maximum (m) capacity of garage q, which may be greater than the capacity of an existing garage if the facility can be expanded, 4 is the existing capacity (e) of garage site q (for candidate sites kz = 0). This problem is also an “assignment problem.” Resources (garage spaces at particular times of the day and week) are allocated to activities (blocks) on a one-to-one basis. This also is a linear network problem. In this case, the nodes are blocks and garages, and the links are the possible assignments of blocks to garages. However, each block time distinction represents another type of flow through the network and all flow types have links between their individual sink capacity (garage capacity) constraints. These types of problems are known as multicommodity network flow problems. Unlike a single commodity network flow, there is no assurance that a multicommodity network flow problem is unimodular (Evans and Jarvis 1978). However, the model in eqns (4)-( 13) has been solved with a- standard linear programming package numerous times, with different sets of data and several perturbations of constraining values and never was a noninteger values included in the solution (Maze et al., 1982). The model repeatedly arriving at all integer solutions strongly suggests unimodularity, although these integer solution do not necessarily constitute proof of unimodularity (proof is, for practical purposes, impossible to show). Assuming the model is unimodular, the integer restriction of the zero-one assignment variables is preserved. To implicitly enumerate all combinations of sets of garages, flow problems are solved with garages switched on and off according to the decision rules developed by Khumawala (1972). In the decision rules, the candidate sites are divided into three sets: set {K,} where all members are fixed closed, set {K,} where all members are fixed opened, and a set {K2} where the sites have not been fixed open or closed. At the beginning of the process the sets {K,} and {K,} will be empty and the object is to place as many members of {Kz} into either {K,} or {K,} as possible through the two decisions rules (the delta and omega rules). The first step is to apply the delta rule, which evaluates members of {Kl} as candidates for {K,}. The second step is to apply the omega rule which evaluates members of {KJ as candidates for {K,,}. The delta and omega decision rules are stated below. (1) The delta rule creates the lower bound for optimal solution. The linear program is first run with all sites switch open. The solution will correspond to the least possible nonrevenue transportation cost solution since more sites will be closer, in cost, to more block ends that in any other solution. Next, one site is switched off and the linear program is run again. The difference between the second and the first solution represents the least possible nonrevenue cost savings of opening that particular site. If the least possible cost savings is greater than the fixed charge, then the site will remain open in the optimal solution and a member of {K,} and, if not, then the site remains a member of {K2}. All sites are similarly tested. (2) The omega rule creates an upper bound for the optimal solution. Only the sites moved to {K,} are switch open and the linear program is run again. Since the least number of sites of any other solution are switched open, the solution to the linear program is the greatest nonrevenue transportation cost solution. Next, one member of {K2} is switched open and the linear program is run again. The difference between the two solutions represents the greatest possible cost savings due to opening that particular site. If the greatest possible cost saving of opening the site is less than the site’s fixed charge then the site will be closed in optimal solution and a member of {K,} and, if not, then the site remains a member of {K,}. All sites are similarly tested. The two decision rules can be run through until the sites remaining in {K2} cannot be opened or closed by the delta or omega decision rules. At this point, the remaining sites can be evaluated through an explicit enumeration. However, the number of nodes remaining to be evaluated (members of {K2}) are typically few. will

4.

SIMULTANEOUS

GARAGE

LOCATION

ASSIGNMENT

AND

SIZING

AND

VEHICLE

OPTIMIZATION

The problem now is one of merging the two independent optimizations discussed in the previous sections. However, merging presents two problems. (1) The vehicle assignment optimization cannot treat multiple garages. Specifically, given its existing structure, the optimization cannot force buses originating from one garage to return

T. H. MAZE and S. KHASNABIS

6

to the same garage. However, by definition the simultaneous optimization must be able to treat multiple garage sites. (2) The garage location and sizing optimization relates blocks to garage capacity through the time period the block covers. However, in the merged model, blocks must be designed within the optimization. As a result, the time period that blocks will span is not known until after a solution to the optimization is found. Therefore, the resulting model must be able to assign capacity to garages without prior knowledge of the block design. The second of the above problems can be dealt with by recognizing that no more blocks (or buses) can leave the garage during the morning than the capacity allocated to the garage, and, in the afternoon and evening, no more blocks can return to the garage than the capacity allocated to the garage. Therefore, the capacity must greater than or equal to the number of morning pullouts from the garage and greater than or equal to the number of afternoon and evening pullins assigned to the garage. Capacity constraints formatted in this manner are shown in eqns (18) and (19) in the simultaneous garage location and sizing and vehicle assignment optimization is shown in eqns (14)-(27).

Min Z = 2

i

all k y=l

+

2

C,,,X,,,

+ i

,=I

q=I

c i f:

E,,,Y,,,

ill1Ay=l /=I

2

2

,=I

,=I

D_i,Z$

(k = a, b,

.

.

.

, f)

+ i ON; + i GN; + i 2 OS,, C/=1 q=l y=l ,=I (k = a, 6,

. .

3

f)

(14)

Subject to

X,qk+z Z;,-S,,=O i=l

for allj

(j= 1, 2,

.

, n)andforallq(q=1,2,.

.0

(15)

Y,,,+ i Zyq-S,,=O “)=I

for allj

(j= 1, 2,

. . , n)andforallq.(q=1,2,.

>0

(16)

. . , 12)

(17)

1,2,...,1)andfork=a,b,andc

(18)

i S,, = 1 for all j (j = 1, 2, 0=1

i:

‘id

-N;5Oforallq(q=

i=l

2 Yjyk - N$ 5 0 for all q (q = 1, 2, . . ,=I N; - N;; - K;5Oforallq(q

= 1,2,.

N; 5 KY for all q (q = 1, 2, .

, 1) and for k = d, e, and f

I

4 3

(19)

(20)

0

(21)

X,,, = 0 or 1

(22)

Z;, = 0 or 1

(23)

Y wk =Oorl

(24)

S,, = 0 or 1 N:; 2 0 for all q (q = 1, 2,

(25) . 0

(26)

Location N;

2

planning

with dynamic

0 for all 4 (q = 1, 2,

vehicle assignments

.

, I)

I

(27)

where j is the journey number (j = 1, 2, . , n), q is the garage site number (q = 1, 2, . , I), a is the weekday morning pullout and pullin activites, b is the Saturday morning pullout and pullin activities, c is the Sunday morning pullout and pullin activities, d is weekday afternoon and evening pullout and pullin activities, e is the Saturday afternoon and evening pullout and pullin activities, f is the Sunday afternoon and evening pullout and pullin activities, k = u, b, . . , f, i is the journey number of all feasible hooks to journey j (i = 1, 2, . , m,), P is the journey number of all feasible hooks from journey j (P = 1, 2, . . , o,), C,+ is the cost of pulling out a bus to journey j from garage site q during time period k, Q, is the cost of hooking between i and j for a block originating from garage site y, E,,, is the cost of pulling in a bus from journey j to garage site 9 during time period k, 0 is the variable garage operating cost per bus, G is the variable garage construction cost per bus, X,,, is the assignment of a pullout to journey j from garage q and at time period k (x,<,~= 0 or l), Z;, is the assignment of a hook from journey i to journey j (Z;,/ = 0 or l), Y,,r is the assignment of a pullin from journey j to garage q and at time period k (Y,,k = 0 or l), N” is the number of active buses assigned to a garage site, N’ is the number of active buses assigned to a garage site that will require the construction of new spaces, S,, is the auxiliary variable (S,, = 0 or l), K; is the maximum (m) capacity of garage q which may be greater than the existing capacity if the garage can be expanded, KG is the existing (e) of garage site y (for candidate sites K; = 0). The first problem of forcing a block to return to the same garage it originates from is solved by creating an auxiliary variable (S,,). The auxiliary variable tracks the block through its journeys and forces the block to originate and return to the same garage. Equation (17) forces S,, to equal one if journey j begins with a pullout from garage y(X,, = 1) or if journey j begins with a hook which initially began with a pullout from garage q(Z;, = 1). Equation 15 forces the block to end with either a pullin or a hook with the same garage indication as the block began with. Equation 16 forces the journey assignment to be associated with only one garage. The objective function, in eqn (14), minimizes the sum of six terms: (1) pullout costs, (2) the layover and deadheading costs of hooks, (3) pullin costs, (4) variable garage operating costs, (5) variable garage construction costs, and (6) the auxiliary variable, which has zero cost. The model is, again, an assignment problem and provides a one-to-one match between journeys, blocks and garage spaces. At the same time, it is also a network problem with the nodes at the end of journeys representing transshipment nodes.? The return of the flow of the vehicle to the same garage is governed by the auxiliary variables. Again, the problem is a multicommodity network flow problem because the flow from individual sites at the various times represent different types of flows. When the simultaneous garage location and sizing and vehicle assignment optimization was repeatedly solved with various data sets and the constraining values were perturbed, all integer solutions resulted in every case (Maze et al., 1983b). Although these trials do not prove unimodularity, they strongly suggest the functional constraints are unimodular. Assuming that the constraints of the model are unimodular, the integer restriction of the model is observed when it is solved as a linear program. The optimal combination of sites is determined by utilizing the delta and omega implicit enumeration rules. Once the optimal combination of sites is located, capacities are allocated to garages, blocks are designed with respect to the sites and capacities selected, and the appropriate fixed charges are added to the solution.

5.

SAMPLE

PROBLEM

To demonstrate the simultaneous run assignment and garage location and sizing model, a small-scale hypothetical example is developed. A map of the example system is drawn in Fig. 1, showing the relative position of four routes, one existing garage (garage A) and three candidate sites (site B, C and D). Table 1 shows the example journey schedule. Only a partial weekday morning schedule is used to keep the example short and small in scale. To demonstrate how to read the journey schedule, under the inbound half of the table and under the left column for fThe link-node description

of the network is similar to the description

given for the vehicle assignment

model

T. H. MAZE and S. KHASNABIS

Fig.

1. Sample problem map

route 1, at 6:30 A.M. a bus pulls out to point 1 on route 1 and starts journey 1. Going to the next column to the right, journey 1 of route 1 ends at 7:15 A.M. and at that time the bus is free to return to the garage or hook to another journey. The schedule has been divided into 15 min increments for the purposes of this demonstration, but when working with an actual problem the schedule could be divided into even finer time increments (for example, one-minute increments). Table 2 shows the deadhead travel costs for the example problem between the garages and the pullout and pullin points. Table 3 shows the deadhead travel costs between potential hooks. Listed in Table 4 are the times of travel between all combinations of pullout and pullin points. The times are used to determine which journeys are potential hooks. Specifically, the journeys being hooked into must start later in time than the sum of the time of the end of the journey the bus is hooking from, plus the travel times between journeys plus a minimum layover time. The number of potential hooks examined in one interation of the model can be limited by a maximum cost per hook, a maximum layover to be included in a hook, or a maximum number of hooks to be considered for the end of each journey. However, a bus would never be assigned a hook which implies a larger cost than the cost of pulling the bus into the garage at the end of the first journey and pulling another bus out to the second journey. As a practical matter, experience with vehicle assignment optimization in RUCUS II has shown that vehicle assignments are rarely effected by the consideration of more than 5 or 6 hooks (Campbell, 1982). In the construction of the example for input into the linear programming package, a maximum of three potential hooks is considered. The three considered are those three with the

Location planning

with dynamic

schedule

Inbound

1

2

Route

1

3 4 Route 2

9

vehicle assignments

Table 1. Example journey

Outbound

5 6 Route 3

1 2 Route 1

7 8 Route 4

3 4 Route 2

5 6 Route 3

7 8 Route 4 __-

Time I

6.30

1

6~45

2

7:oo

3

1:15

4

1

4

: 30 7:45

5

2

5

1

6

3

b

2

6.00

7

4

7

3

8: 15

8

5

8

4

H:,O

9

b

9

5

a:45

10

1

10

6

Y:OO

11

8

11

7

Y: 15

12

9

12

R

Y.30

12

10

13

9

9-45

14

11

14

lo

IO:00

15

13

15

11

7

16

1

11

2

18

9

9

10

1

14

10

16

3

11

18

2

20

12

19

3

21

5 4

12

20 22

b

13

21

5

23

13

22 24

7

14

23 25

13

15

16

11.30

18

22

19

23

20

24

21

25

22

26

23

27

24

28

25

29

26

3P

27

31

28

32

8 9

9 10

10

31 32

25

lb

21

30 26

15

17

29

14

24

7

8

20 19

4

14

19

11

17

12

11:OO 11-15

17 18

3

15

IO.41

1

2

13

1”: 15 Ill30

1

15 26

1,145 12~00

least cost. To provide an example of how the hooks and hooking costs are determined, refer back to Table 1 and note journey 2 on route 1. Journey 2 starts during the 15minute interval starting at 6:45 and finishes during the time interval which starts at 7:30. The most obvious (but not necessarily optimal) hook for journey 2 would be journey 19 of route 1. Journey 19 starts at the point where Journey 1 ends and during the same time interval. Because the bus does not have to travel to hook to journey 19, there is no transportation cost for the hook and only a $408 cost per year for a 5 min layover. The next best hookmate is journey 22 of route 3. The cost of hooking to journey 22 includes a transportation cost for travel point 2 to point 6 of $300 per year plus approximately $1000 per year for a 12.5 min layover once the bus reaches point 6. The three least-cost hooks (if three feasible hooks exist) are determined by

Table 2. Example problem deadhead

travel costs between garage and pullout and pullin points COST

Pullout-Pullin Point

(per year) Garage

A

B

C

D

1

$3,500

$2,400

$6,100

$6,800

2

$3,400

$4,600

$2,300

$2,100 $3,000

3

$5,000

$3,000

$1,400

4

$3,500

$4,600

$2,200

$2,000

5

$6,400

$5,000

$1,500

$2,500

6

$3,550

$4,650

$2,200

$1,900

7

$9,000

$8,000

$4,000

$2,500

8

$3,600

$4,700

$2,300

$1,900

10

T. H. MAZE and S. KHASNABIS Table 3. Example problem deadhead

mileage costs between potential hook points

COST

Point 1

1

2

$0

2

(per year)

3

6

same

in both

7

directions

8

$9700

$4600

$4550

$ 300

$5000

$ 300

$4700

$ 320

$0

$3250

$1500

$3300

$4500

$3350

SO

$5000

$ 320

$4600

$ 340

$0

$4950

$3300

$5000

$0

$5000

$ 200

$0

$4900

4 5

7

$46GO

$0

are

6

$6000

$3500

Cost

Point 5

$4550

$4500

3

4

the

8

$0

hand calculation and coded for the linear program. If the generation of hook costs were determined by machine rather than by hand, it would be feasible to code many more than three possible hooks. In the example problem, the variable garage operating cost and variable construction cost are $2000 and $1000 per bus per year, respectively. The fixed garage operating charge and fixed garage construction charge are $15,000 and $5000 per bus per year, respectively. Existing garage A is given a capacity of eight buses. IBM’s mathematical programming package, MPSX-MIPI370, is used for solving all linear programs. The solution to the problem with all sites switched on results in a total cost (not including the fixed charges) of S255,l 10. The vehicle assignments are listed in Table 5. In this table, the vehicle assignments are listed for each bus assigned to each garage. The numbers defining the block’s vehicle assignment list the route number, followed by a common and then the journey number. Deadhead legs, either from or to the garage or hooking between journeys, are identified with a dash. For example, bus 1 in garage A starts by pulling out to journey 1 on route 1, after finishing journey 1 it hooks to journey 21 on route 3, then it hooks to journey 9 on route 3, then it hooks to journey 29 on route 3, and then it hooks to journey 15 on route 3, and then it pulls back into the garage. The next step in performing the delta decision rule is to switch off one garage at a time and resolve the linear program. The difference between solution with one garage switched off and the solution with all garages minus the fixed charge is equal to the delta value. If the delta value is greater than or equal to zero, then the site becomes a member of set {K,} if not, the site remains a member of the set {K,}. Listed in Table 6 are the values for the delta decision rules. Only site C has a positive value, and therefore site C is moved to set {K,} and sites A, B and D remain in set {K,}. Table 4. Example

problem deadhead MINUTES

Point

1

2

1

0

30 0

2

5 6 7 8

Travel are

Point 5

30

30

45

30

60

30

30

15

30

15

30

15

0

30

15

30

30

30

0

30

15

30

15

0

30

30

30

0

30

both

15

0

30

times

the same

in

directions

(per trip)

4

3.

3 4

travel times between potential hook points

---v-7-

8

0

Location planning Table 5. Vehicle assignment Existing BUS

with dynamic

with all garages switched on the example problem

Garage

Vehicle

A

Assignment

-1,I - 3,21 - 3,9 - 3,29 - 3,15 - 1.3 - 1,s - 1,7 - 1,9 - 1,ll - 1.13 -

1

2 3 4 5 6 7

Site

1

11

vehicle assignments

-

B

- 1,2 - 1,19 - 1,8 - 1,22 - 1,14 - 1,26

-

Site C

-

1

2 3 4 5 6 I 8 9 10 11 12 13

2,l - 1,18 2,3 2,4 - 2,12 2,5 2,7 - 2,13 2.8 2,9 - 2,14 3,l 3,2 - 3,22 3,3 - 3.23 3,4 - 3.24 3,17 - 3,5 3,19 - 3,7 Site

-

1

2 3 4 5 6 7 8

- 1.6

1.21 - 1,12 - 1,24 -

- 2,6.- 2,lO - 2,15 -

3,lO 3.11 3,12 3,25 3,27

- 3.30 - 3,31 - 3,13 - 3,32 -

D

4,l - 1,20 - 1.10 - 1,23 - 1,16 4,2 4,3 - 4.9 - 4.8 - 4,4 1,17 - 4,l - 4,8 - 4,4 4,5 - 4,lO 2,ll - 2,2 3,18 - 3,6 - 3,26 - 3,14 3,20 - 3.8 - 3,28 - 3,16 -

The next stage is to apply the omega decision rule. The first step in the omega rule is to solve a linear program with only members of set {K,} switched on. In the case of the example problem this means switching only site C and running the linear program. The next step in the omega decision rule is to switch on one additional garage at a time and resolve the linear program. The difference between the solution with only site C and the solution with one additional site minus the fixed charge is equal to the omega value. If the omega value is less than or equal to zero, then the site becomes a member of {K,,}; if not, the site remains a member of the set {K,}. The results of the application of the omega decision rule to the sample problem are shown in Table 7. Only site D can be moved to set {&} and A and B remain in set {K2}. As a result of the decision rules, site C is fixed open and site D is fixed closed. Sites A and B are still undecided. To determine which combination of open and closed sites is optimal, a short enumeration process is utilized. The enumeration involves switching open sites A, B and C and solving the linear program. The optimal combination includes opening sites A and C and fixing closed sites B and D. The optimal solution has a total costs of $305,689 per year. Table 6. Application Delta

Value

of the delta rule to the example problem -LActive

BUS Assignments

AA = -12,183

NA=

0, NB=

9, NC=

13, ND=

8

AB = -17,800

NA=

9, NB=

0, NC=

13, ND=

8

AC =

NA=

8, NB=

4, NC=

20, ND=

0

8, NB=

2, NC=

22, ND=

8

a, =

1,200 -5,389

NA=

T. H. MAZE and S. KHASNABIS

12

Table 7. Application omega

of the omega rule to the example problem

Value

Active

BUS Assignments

NA= 9,

NB= 0,

NC= 21,

ND= 0

402

NA= 0,

NB= 9,

NC= 21,

ND= 0

Qc= -4,980

NA= 0,

NB= 0,

NC= 22,

ND= 8

dA=

OB’

6,989

Table 8 shows the vehicle assignments location and size combination.

for the optimal vehicle assignment

6.

schedule and garage

CONCLUSION

In this paper the mathematical formulation of two existing models, a vehicle assignment optimization and a garage location and sizing model, is introduced. The two models are then merged to create an optimization which simultaneously assigns vehicles (schedules blocks) and locates and sizes garages. The difficulty in structuring the model is in creating a mechanism which forces a vehicle which leaves a garage to be scheduled through its assigned journeys and returned to the same garage. This problem is overcome by building auxiliary variables into the mathematical program which keeps track of the flow of each bus. Using the resulting model, a short, hypothetical sample problem is solved. The new model has advantages over both an existing vehicle assignment model (one which is widely used in North America) and the existing garage location and sizing model. The existing vehicle assignment model cannot track a bus from a particular garage back to the same garage. Thus, when it groups journeys into blocks it does not have the capacity of including in the design of vehicle assignments any consideration for the cost trade-offs of deadheading

Table 8. Vehicle assignments

with garages opened at optimal locations in the example problem

Existing Vehicle

BUS

1

2 3 4 5 6 7 8 9

-

Garage

A

Assignment

1,l - 3,21 - 3,9 - 3,27 1,2 - 1,19 - 1,8 - 1,22 1.3 1,5 1,7 1,9 1.11 1,13 1,15 - 1,27 -

- 3,15 - 1,14 - 1,26 -

Site C

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 21

-

2.1 - 1,18 - 1,6 - 1,21 - 1,12 - 1.24 2,3 2,4 - 2,12 - 2,6 2.5 2;7 - 2,13 - 2,lO 2.8 2,9 3,l 3,2 - 3,22 - 3,lO - 3,30 3,3 3,4 - 3,24 - 3.12 4.2 4;l - 4,8 - 4,4 4.3 - 4,9 - 4,6 - 2,14 4,5 - 4,lO 1,17 - 1,4 - 1,20 - 1,lO - 1,23 - 1,16 2,ll - 2,2 - 3,24 - 3,ll - 3,31 3,17 - 3.5 - 3,25 - 3,13 - 3,32 3.18 - 3.6 - 3,26 - 3,14 - 2,15 3,19 - 3.7 - 3,27 3,20 - 3,8 - 3,28 - 3.16 -

Location planning

with dynamic

vehicle assignments

I3

and layover costs between journeys (hooking costs) and garage operational costs and deadheading costs from and to the garage. The new model considers all of these costs when it designs a vehicle assignment. The existing garage location and sizing model considers the blocking of journeys to be fixed. A major consideration in the design of blocks and assignment beginning and ends is the location of the garage from which buses are to originate. The existing schedule of journeys into blocks will have been designed with the location of the existing garages in mind. Hence, if the existing assignment is used, existing garages appear to be well located whether or not they actually are. Because the new model simultaneously assigns vehicles and locates and sizes garages, it is not biased by the goals of prior block designs. A major drawback of the proposed model is that when a problem of practical dimensions (hundreds of journeys) is solved using a standard linear program package, lengthy computing times are required (Maze et al., 1983b). Computing times are basically a function of the number of constraints and is roughly proportional to the cube of this number (Hillier and Lieberman, 1980). In our model there are 2 constraints for each journey-garage combination, 1 for each journey and 6 for each garage. For a problem of any size, the number of constraints add up very quickly and generally makes the solution impractical. Note, however, that our model is a multicommodity network flow problem and such problems can be solved with efficient labeling algorithms (Evans and Jarvis, 1978). For example, Ali et al. (1980), have reported solving a problem equivalent to a linear program with 3705 rows and 6405 columns in approximately 2 min on a CDC Cyber 72 using a multicommodity flow code. This scale of problem and computing times are both within the realm of what would be practical to employ the model in garage planning and vehicle assignment situations. Further, computing time should be much shorter on larger and faster machines or more recent vintage.? AcknoM,lcdgem~nts-We would like to thank the Urban Mass Transportation Administration of the U.S. Department of Transportation for the funds that were made available to us through a grant to Wayne State University (MI-I I0005). We would also like to thank the anonymous referee for his or her comments.

REFERENCES Ali, A., Helgason R., Kennington J. and Hall H. (1980) Computational comparison among three multicommodity network flow algorithms, Oper. Rcs. 28, 995-1000. Ball M., Assad A., Bodin L. and Golden B. (1981) Garage location for an urban mass transit system, College of Business and Management Working Paper Series MS/S 81-039, Univ. Maryland, College Park, MD. Campbell R. (1982) Head of Scheduling for the Southeastern Michigan Transportation Authority, private communication. Evans J. R. and Jarvis J. J. (1978) Network topology and integral multicommunity flows Networks 8, 107-I 19. Hillier F. S. and Lieberman G. J. (1980) Introduction ro Operurions Research. p. 194. Holden-Day, San Francisco, CA. Keaveny, 1. T. (1981), Sage Managment Consultants (Developers of RUCUS II), private communication. Khumawala B. M. (1972) An efficient branch-bond algorithm for plant location, Managemenr Sci. 18, 7 18-73 1, Luedtke L. K. (1973) RUCUS II: A review of System Capabilities, Presented to 3rd fnr. Workshop on Transit Vehicle and Crew Scheduling, Montreal, Canada. Maze T. H.. Khasnabis S., Kapur K. and Kutsal M. D. (1982). A Methodology for Locating and Sizing Transit Fixed Facilities and the Detroit Case Study, prepared by Wayne State Univ. for the Urban Mass Transportation Administration, U.S. Dep. Transportation, Washington, DC. Maze T. H., Khasnabis S. and Kutsal M. D. (1983a) Application of a bus garage location and sizing optimization. Transpn. Res. 17A, 65-72. Maze T. H.. Khasnabis, S., Kutsal, M. D. and Dutta U. (1983b). An Analysis of Total System Costs Related to Bus Garages and Network Configurations, prepared by Wayne State Univ. for the Urban Mass Transportation Admin. istration, U.S. Dep. Transportation. Washington. DC. Roberts K. R. (1973) Vehicle Scheduling and Driver Run Cutting: RUCUS Package Overview, prepared by the MITRE Corporation for the Urban Mass Transportation Administration. U.S. Dep. Transportation, Washington, DC.

tResearch

on this topic is currently

in progress

in The University

of Oklahoma.