Abstract: We compare the linear programming relaxation of different mathematical formulations for the multi-depot vehicle scheduling problem. As a result of this ...
Exact Algorithms for the Multi-Depot Vehicle Scheduling Problem Based on Multicommodity Network Flow Type Formulations Marta Mesquita! aud Jose Paixao 2
2
Instituto Superior de Agronomia, Dep. de Matematica, Tapada da Ajuda, 1300 Lisboa, Portugal DElO, Faculdade de Ciencias de Lisboa, Bloco C2-Campo Grande, 1700 Lisboa, Portugal
Abstract: We compare the linear programming relaxation of different mathematical formulations for the multi-depot vehicle scheduling problem. As a result of this theoretical analysis, we select for development a tree search procedure based on a multicommodity network flow formulation that involves two different types of decision variables: one type is used to describe the connections between trips, in order to obtain the vehicle blocks, while the other type is related to the assignment of trips to depots. We also develop a branch and bound algorithm based on the linear relaxation of a more compact multicommodity network flow formulation, in the sense that it contains just one type of variable and fewer constraints than the previous model. Computational experience is presented to compare the two algorithms.
1 Introduction This paper is concerned with the multi-depot vehicle scheduling problem (MDVSP) which consists of grouping a set of timetabled trips into vehicle blocks and, at the same time, assigning these vehicle blocks to depots. Each vehicle block represents a set of trips that can be carried out by one vehicle. A set of n timetabled trips T], ... ,Tn has to be operated by vehicles housed at k different depots, D], ... ,Db at the £ th of which d f! vehicles are stationed (£ = 1, ... ,k). The vehicles are treated as identical. Each trip Tj is characterized by its starting and ending times and locations. An ordered pair of trips (Tj , ~) is said to be
compatible if the bus released after the completion of trip Tj can be assigned to trip
N. H. M. Wilson (ed.), Computer-Aided Transit Scheduling © Springer-Verlag Berlin Heidelberg 1999
222 ~. The objective is to group compatible trips into vehicle blocks and at the same time assign these vehicle blocks to depots, in order to minimize the cost associated with the schedule. The MDVSP has been shown to be NP-Hard (Bertossi/CarraresilGallo (1987». Several mathematical formulations and different solution approaches, both heuristics (see Dell' AmicolFischettilToth (1993), MesquitalPaixao (1992), Lamatsch (1992), Bertossi/Carraresi/Gallo (1987), EI-Azm (1985), SmithlWren (1981), Ceder/Stern (1981» and exact methods (see ForbeslHoltlWatts (1994), Ribeiro/Soumis (1994), Carpaneto/Dell' AmicolFischettilToth (1989» have been proposed for the MDVSP. The last survey on the subject can be found in BalllMagnantilMonmalNemhauser (1995). In this paper we focus on exact methods for solving the MDVSP. In particular, we discuss different branching strategies for a tree-search scheme based on a multicommodity network flow formulation that involves two different types of decision variables: one is associated with the connections between trips while the other is related to the assignment of trips to depots. We also consider a more compact multicommodity network flow formulation, in the sense that it contains just one type of variable and less constraints than the previous model. Both formulations are used to compare different branch and bound procedures. The paper is organized as follows. First, in Sect. 2 we review the mathematical formulations proposed for the MDVSP. In Sect. 3 we compare the linear programming relaxation for the different mathematical formulations mentioned above. As a result of this theoretical analysis, in Sect. 4, two branch and bound procedures based on different multicommodity network flow type formulations are presented. Computational experience with real data from a bus operator, CarrisLisbon, is shown in Sect. 5 and is analyzed in Sect. 6. Finally some conclusions are drawn in Sect. 7.
2 Integer Programming Models of the MDVSP The integer programming models presented in this section can be grouped into three basic approaches: 1) multicommodity flow formulations; 2) singlecommodity flow formulations, namely, an assignment and a quasi-assignment formulation; 3) set partitioning formulation.
2.1 Multicommodity Flow Formulations BertossilCarraresi/Gallo (1987) formulated the MDVSP as a multicommodity network flow problem in a complete bipartite graph. They consider the complete
223 E 5, i = 1, ... ,n correspond to the ending = 1,... ,n correspond to the starting locations of
bipartite graph (5, T,A) where nodes i locations of trips, and nodes JET, j
trips. The arc set, A is partitioned into two subsets AI and A 2 . An arc
(i, j) E
Al
represents the deadhead trip from the end of trip Ti to the start of trip ~, while an arc (i, j) E A2 denotes a sequence of two deadhead trips, from the end of trip Ti to the depot and from the depot to the start of trip ~. If costs
c6
MDVSP can be defined as the problem of finding a partition of
are given, the
{I, ... , n} , Il> ... ,h
and k arc sets MI> ... ,Mk such that: (i) M f, (ii)
e= l, ... ,k is a perfect matching for (5, T, A);
I. I. c6 fEK
is minimum.
(i,j)EM,
Each set If, e= 1, ... ,k corresponds to the set of trips assigned to each depot whereas M f defines the schedule of such trips.
e
Next; we define the mathematical formulation presented by Bertossil Carraresi/Gallo (1987), upon a sparse graph, such as the graph used by RibeirolSoumis (1994). Let N = {I, ... , n} represent the set of trips, I ~ N x N denote the set of compatible pairs of trips and K
= {n + 1 ,... , n + k}
represent the set of depots.
With no loss of generality, we assume that trips are ordered by increasing value of their starting times. A graph C f and
= (V f , A f)
is associated with depot
e,
where V f
= N u {n + e}
Af =Iu({n+e}xN)u(Nx{n+e}), e=l, ... ,k.
The following example will be used to illustrate the different network flow approaches. Consider a problem with four trips and two depots, each one having a single vehicle. The starting and ending time of each trip is: Trip start time end time
9:00 10:30
2 9:30 12:00
3 11:00 13:30
4 12:30 14:00
Figure 2.1 shows graph C 1 , associated with depot D I , represented by full lines and graph C 2 , associated with depot D2 , represented by dotted lines.
224
(depot D2 )
Fig. 2.1 Example of a graph for the multicommodity flow models.
Costs while
C ij'
(i, j) E A e are and
cn+€.i
c i •n+€'
known and are independent of depot £ if
(i, j) E
I,
i EN, are usually dependent on depot £ . Note that, in
the multicommodity network flow model presented by Bertossi/Carraresi/Gallo (1987) costs
cO,
(i, j) E
I , always depend on depot £ that supplies the vehicle to
perforni the deadhead trip. Further, consider the binary variables: xo'
(i, j) E
I ,
£ E K , which indicate whether trips i and j are run in order by the same vehicle
housed at depot £ , xo
= 1 , or not,
xo
=0;
xi,n+€ (xn+€,i), i EN, £ E K, which
indicate whether the bus immediately returns to depot £ after trip i, xi,n+€ not, xi,n+€
= 0 ( whether depot
= 1, or
£ directly supplies a bus for trip i), and yf, which
indicate whether trip i is assigned to depot £,
yf
= 1 , or not,
yf = o. Then the
mathematical formulation presented in Bertossi/Carraresi/Gallo (1987), will take the form:
s. t.
L. Xij€ + xi,n+€ = Yif
~
'11£
E
K, 'IIi EN
(2.1)
j:(i ,j)EI
~ n· = )~ L. xt.IJ +x n+t,J J
(2.2)
i:(i,j )EI IXn+€,j JEN
~df
(2.3)
225
LA =1
'\Ii
E
N
x;] E{O,I}
'\1£
E
K, '\I(i,j)
'\1£
E
K, '\Ii
(2.4)
I!EK
Xi,n+l!, xn+I!,i'
yf E {o, I}
E
E
I
N
(2.5) (2.6)
The objective function for (MFx).) consists of two major components. The first is related to the linking of trips and may account for both the corresponding travel costs and eventual penalties imposed by the user, for instance on idle times. The second refers to the costs of linking trips and depots which, besides the travel costs, may include a penalty associated with the use of a vehicle. The constraint set of (MFx).) includes two types of decision variables: the assignment variables,
A, which determine the assignment of trips to the depots
and the scheduling variables, x;], Xi,n+l!, xn+l!,i which establish the assignment between trips. Constraints (2.1) and (2.2) relate for each trip the corresponding two types of assignment variables. Constraints (2.3) refer to the number of vehicles available at each depot. Constraints (2.4) guarantee that each trip will be assigned to only one depot. Since trips are ordered by increasing value of their starting times, set I contains only arcs (ij) with i < j . Therefore, no circuit containing only vertices i E N exists. Ribeiro/Soumis (1994) also presented an integer multicommodity network formulation which is more compact, in the sense that it uses only one of the sets of
xy)
variables involved in (MF
and contains less constraints than the integer
program described above.
(MFx)
mm S.t.
I
L LCijX;]+L I(Ci,n+I!Xi,n+I!+Cn+l!,iXn+l!,i) I!EK iEN I!EK(i,j)EI
LX;] + LXi,n+1! I!EK {j:(i,j)EI} CEK
=1
Lxt+xn+l!,j- LXA-Xj,n+I!=O i:(i,j )EI i:(j,i )EI LXn+l!,j jEN
~ dl!
'\Ii
E
N
(2.7)
'\I£EK,'\IjEN
(2.8)
'\1£
(2.9)
E
K
226
V£EK,V(i,j)EI (2.10) Xi,n+Ji, xn+Ji,i E
{o, I}
"1£ E K, Vi
E
N
(2.11)
Constraints (2.7) ensure that each vertex i E N is visited exactly once, while constraints (2.8) are flow conservation constraints, for each vertex j EN. For example, a feasible solution for the problem stated above is shown in Fig. 2.2.
XI X 5,1
_
1,3 -
=1
~
1 G) /'
(depot D2 )
__ ~
- 2- - - ~r\ X 4,6
X ,
2,4
=1
~
=
1 /
I
/
I
2
Y2
2
=Y4
=
1
/'
- - - - - - - -X6,2 = 1
Fig. 2.2 A feasible solution.
The remammg variables, not shown in Fig. 2.2, are equal to zero. Variables
e
xij ,xi,n+( ,xn+(,i
are shared by formulations (MFX).) and (MFx) while variables
e
Yi
only exist in (MFx)).
2.2 Single-Commodity Flow Formulations Next, we are going to describe two alternative single-commodity flow formulations. The first is based on the assignment problem while the second is based on the quasi-assignment problem. Consider a graph G = (V, A), in which the vertex set V = {I, ... , n + k}, is partitioned into N
= {l, ... ,n}
corresponding to trips and K={n+l, ... ,n+k}
referring to depots, and the arc set associated with each arc example.
A = r u (K x N) U (N x K).
(i, j) EA. Fig.
A cost cij is
2.3, below, shows graph G = (V, A) for the
227
Fig. 2.3 Example of a graph for the single-commodity flow models.
CarpanetolDell' AmicolFischettirroth (1989) split each depot vertex, n + f, into formulation based on the assignment
d f! vertices and present a mathematical
model with subtour breaking constraints. In order to facilitate comparisons with other formulations, we present this formulation in a slightly different form. Consider the binary variables xij = 1, (i, j) E I , if trip i is directly connected to trip j, xij
=0
otherwise, and xi,n+fi., xn+fi.,i,
yf,
i EN, f
E K ,
as defined for
models (MFx;.) and (MFx). Let II denote the family of all elementary paths P joining different vertices in K.
(MA)
min
'~>ijxij + (i,j)EI
S.t.
LXij j:(i,j)EI
L
L(Ci,n+fi.Xi,n+fI +Cn+fI,iXn+fI,i) f!EK iEN
+ LXi,n+fI = 1
N
(2.12)
VjE N
(2.13)
VfE K
(2.14)
PEll
(2.15)
V(i,j)EA
(2.16)
Vi
E
fEK
LXij + LXn+f!.,j flEK i:(i,j)EI LXn+f,j Ipi-i. Without loss of generality, let us
ends at different depots and
(i,j)EP
represent the above path P by n + 1 ---? 1 ---? 2 ---?
... ---?
r - 1 ---? r
---?
n+2
233
where n+ 1 and n+2 are vertices corresponding to two different depots, n + 1 :;t: n + 2, and = r + 1.
Ipl
Since variables xij, (i,
(Xij, l)
j) E
A, are the same in both models, we want to construct
feasible for constraints (2.20), (2.21), (2.22), (2.22a) and (2.23).
The ending depot of the above path P is represented by vertex n+2. For a trip i, included in path P, the minimum value 2
Yr = x r ,n+2'
2 Yr-l
2
yf can take is given by:
= Yr + xr-l,r
-1,
2
2 YI =Y2 +xl,2- 1
On the other hand, referring to the assignment of trip i, included in path P, to the starting depot, the minimum value
y}
can assume is the following: 1 1 Y r-l = Y r-2 + x r-2,r-1 -1,
1
Yl = xn+l,l ' 1 1 Yr = Yr-l +X r-l,r- 1
Specifically for vertex 1, we have
yi + yt From
yf
2>ij - (r -1) > Ipl-I-lpl + 2 = 1. (i,j)E P ~ 0, we have
I. yf > 1 . £EK
We proved that the set of feasible solutions defined by the linear relaxation of (MA) contains the projection into the xij, (i, j) E A, subspace of the set of feasible solutions defined by the linear relaxation of (QA). Hence, if solution feasible in the linear relaxation of (QA) then solution
(Xij)
(xij, l)
is
is feasible in the linear
relaxation of (MA) and, since the two formulations have the same objective function value, we conclude, v LP (QA) ~ v LP (MA) .
234
Proposition 3.4: There are instances with v LP (QA) > v LP (MA) .
proof Our aim is to present a solution (xij) feasible in the linear relaxation of (MA) for which there is no
(Y/)
such that (xij,
l) is a feasible solution for the
linear relaxation of (QA). Consider the example below, depicted in Fig. 3.1, where the value of each variable xij is near the corresponding arc.
~l
1.0
[;QJ
~n+2
0.5
Fig. 3.1 Example of a feasible solution for the linear relaxation of (MA) but not feasible for the linear relaxation of (QA) Note that the solution shown in Fig. 3.1 can be easily completed so that flow conservation constraints are valid for depots n+ 1 and n+2. Consider path P, drawn from the above solution ~81--_0_.7_5_-J..~8+-
__
0._5_--1.. ~~ _ _1._0--l"~~
Using (2.20), (2.21) and (2.23) we establish:
~01
23
~~ YI =l;Yl =YI =0;
) .3 2 > -
0) 5
1.0
r:::-::;l
0 .25', ),34 > - 025' . ,
- - - - - l....I~ ),1 _ ),3 _
O.
),2
-1
5-5-'5-
235
From (2.22) and (2.22a) we have ~f0
0.75
~ vf---....c..c:..:-.. •
L . :. .:..f
0)
0t::'\
0.5 ~.75 ...f------I..... 4 5
yi :2: 0.75;
----I....[;i] 1.0
n+
2
Y22> - 025 .
3
Concerning vertex 2 we have
Lyi :2: 1.25> 1 , which contradicts (2.23). [=]
Note that the solution presented in Fig. 3.1, which is not feasible in the linear relaxation of (QA), is feasible in the linear relaxation of (QA) without the constraint set (2.22a).
4 Branch and Bound In the previous section we saw that the bound obtained by the linear relaxation of (MFX)) is equal to the bound obtained by the linear relaxation of (MF
0::
i
i
0 C\I
c..
~
C\I
c..
i
C\I C\I
c..
C')
C\I
c..
Fig. 6.4 Linear relaxation. Results from table 5.7.
The linear relaxation of (MFx).) takes less time than the linear relaxation of (MFJ, which is a more compact model since it contains fewer variables and constraints than (MFX)')' A heuristic explanation for these results follows from the fact that the decision variables
yf,
which are not essential in a multicommodity flow type
formulation, help in the computation of the
x&
variable values, which are
fundamental in such mathematical approaches. In the integer formulation (MFX).) the number of variables whenever the
yf
x&
is greater than the number of variables
variable values are known then the
x&
yf,
but
variable values are easily
determined by solving k vehicle scheduling problems with one depot. If we compute the average time for the different branch and bound algorithms, we can see that the best time was attained for model (MFX).) with branching on variables
x&, yf . If instead of the average time we compute the median, which is
less sensitive to outliers, we conclude that the best strategy is to use model (MFX).) and to branch on variables
yf.
242
7 Conclusions The first exact method for the MDVSP was presented by CarpanetolDell' Amico/ FischettilToth (1989). The authors formulated the MDVSP as an assignment problem with side constraints and proposed a branch and bound algorithm based on the computation of the lower bounds through an additive lower bound procedure. Computational experience is presented for problems with up to 70 trips and three depots. Alternatively, Ribeiro/Soumis (1994) proposed an exact algorithm based on the continuous relaxation of (MP). The authors report the solution of test problems with up to 300 trips and six depots. At the same time, ForbeslHoltlWatts (1994) presented an exact algorithm based on the linear relaxation of the multicommodity network flow type formulation (MFx). They consider test problems with three depots and varying in size from 100 to 600 trips. Ribeiro/Soumis (1994) proved that the linear relaxation of (MFx) is equivalent to the linear relaxation of (MP). Now, we have compared the bounds given by linear programming relaxations of the different MDVSP formulations presented in Sect. 2 together with the bound given by the linear relaxation of the (QA) model. The main results obtained in this paper can be compiled in the following scheme.
Ribeiro/Soumis (1994)
Fig. 7.1 Classification of the MDVSP linear relaxations
As a result of this theoretical analysis we selected the multicommodity network flow formulation (MF.'C)') for use in a tree search procedure. Model (MFC'C).) allows different branching strategies according to the subset of variables on which branching is possible. We also develop a branch and bound algorithm based on the
243
linear relaxation of a more compact multicommodity network flow formulation, (MFx)·
Computational results show that, concerning the computational time required either for solving the linear relaxation or for performing a tree search procedure based on this linear relaxation, model (MFxy) outperforms model (MFx).
Acknowledgments: The authors would like to thank the referees for their helpful suggestions.
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