Stability and Performance of Scheduling Policies
in a Transportation Node
Mauro Boccadoro
1
, Francesco Martinelli
2
and Paolo Valigi
1
1
DIEI, University of Perugia, Italy
2
DISP, University of Rome ”Tor Vergata”, Italy
Abstract. In this paper we consider the dynamic model of a logistic node of a
transportation network and study dispatching feedback policies in terms of sta-
bility and optimality. A necessary and sufficient condition for the existence of a
stable feedback policy is given and a policy is presented which would be optimal
if the transportation resources were continuous.
1 Introduction
An intermodal logistic system can be modeled as a network comprising a set of nodes
(hubs and terminals) connected by the links established by the transport operations,
which, in general, take place under different modes. The management of logistic nodes
in this network is a complex problem where several factors have to be taken into ac-
count, from the availability of carriers and their assignment to particular tasks (in terms
of products to be shipped, destinations or routes), and fulfillment of various perfor-
mance criteria such as timely delivery, minimization of transportation and inventory
costs (possibly, both at the logistic nodes and at the destinations); see among others,
[1–4].
Many instances of decisional problems for these systems are presented and solved
in the literature; often transportation problems can be addressed in terms of linear pro-
gramming problems, see e.g., [5–8], and developing ad-hoc techniques to obtain the
solutions, such as dynamic programming with linear approximation of the (unknown)
value function. It is worth mentioning that by the approach of [5, 6] the framework of
the Logistic Queuing Networks is introduced.
A slightly different paradigm considers shipping policies for simplified models of a
logistic network (or a part of it) and addresses the minimization of transportation and
inventory costs, see for example [9,10] and [15] where a stochastic setting is adopted.
Still another example of management problems for a logistic node is represented by the
optimization of space allocated for containers in ports (e.g., [13]); or the optimization
of the operations of discharging containers from a ship, their location in the terminal
yard and the upload of new containers [14]. In these last two cases, the performances
considered can be also viewed in terms of the necessity to maintain low levelsof stocked
products in the logistic node (in this case, a port). The stability of the dynamics of the
stock at a logistic node is therefore a relevant issue to be taken into account.
Boccadoro M., Martinelli F. and Valigi P. (2009).
Stability and Performance of Scheduling Policies in a Transportation Node.
In Proceedings of the 3rd International Workshop on Intelligent Vehicle Controls & Intelligent Transportation Systems, pages 86-95
Copyright
c
SciTePress
In order to study the stability (and the performance of stabilizing policies) from
a dynamical point of view, in this paper a simplified model of a logistic node in a
transportation network is considered and feedback policies based on the current state
of the system are defined to control the node. The scenario is similar to those arising in
other applicative domains (like in manufacturing, communications, computer systems
or queuing networks in general), so the feedback policies considered in this paper have
been inspired by various well established techniques developed in those domains. The
stability of these policies will be investigated and a necessary and sufficient condition
for the possibility of stabilizing the system will be determined. The sufficiency will be
established in a constructive way by determining a class of policies which guarantee the
stability of the system. A comparison among the performances of different stabilizing
policies will be carried out through simulations, showing that a policy, inspired by a
control known as optimal for the fluid version of the problem, will provide the best
results among the policies considered in the paper.
2 Problem Formulation
Consider the discrete time model of a logistic node collecting Q different types of wares
which have to be shipped to P different locations, and let x
ij
(k) 0 be the quantity of
items of type j = 1, . . . , Q, with destination i = 1, . . . , P , stocked at the logistic node
at time t
k
, and collected in the buffer
3
B
ij
. In this model a destination could be more
in general considered as a route among different locations, established through some
routing algorithm.
The time evolution of each x
ij
is observed at various decisional time instants t
k
,
and as such characterized by a discrete time dynamics. Denoting d
ij
(k) the amount of
goods of type j to be sent to destination i arriving in the node in the interval (t
k
, t
k+1
)
and u
ij
(k) the amount of goods of type j shipped to destination i from the node in the
same interval (t
k
, t
k+1
), we have:
x
ij
(k + 1 ) = x
ij
(k) + d
ij
(k) u
ij
(k) (1)
In addition to this dynamics, we consider that of the vehicles executing the shipping
task. Let n
i
(k) be the number of vehicles assigned to destination i in the interval
(t
k
, t
k+1
); the total number N (k) of vehicles present in the node at time t
k
obeys to the
following equation:
N(k + 1) = N(k) + R(k)
P
X
i=1
n
i
(k) (2)
where R(k) is the number of vehicles arriving from outside in the interval (t
k
, t
k+1
).
Notice that, according to the above dynamics, the total number of vehicles available for
a shipping task at time t
k
is given by
3
These buffers could be considered as virtual, in the sense that in some cases we may have
items which are physically stocked in different places according to their type (in such a way
that the physical content of a buffer is given by
P
P
i=1
x
ij
) as it happens for the stocked finished
products in a factory.
N
a
(k) := N (k) + R(k) (3)
To model the inflow of vehicles R(k), first consider the simple scenario where there is
a fixed quantity N
c
of vehicles which could serve the logistic node (as if for example,
the logistic node is a shipper who owns a certain quantity N
c
of trucks). A traveling
time T
i
N is associated to each route (i.e. destination) i; in particular T
i
denotes the
round-trip time, i.e. the interval after which a vehicle is again available at the node after
completion of a shipping task to destination i. In this case R(k) =
P
P
i=1
n
i
(k T
i
)
is the number of vehicles coming back from their expedition, and therefore (2) reads as
follows:
N(k + 1) = N(k) +
P
X
i=1
n
i
(k T
i
)
P
X
i=1
n
i
(k); N(0) = N
c
(4)
Notice that the total sum of vehicles (those at the logistic node and those traveling)
equals N
c
at each time instant.
In other cases we can consider the logistic node and the shippers as separate entities,
so that the total number of vehicles which are going to access the logistic node varies
with time; in such situations N
c
can be obtained through a suitable average of the expe-
dition history in the node, and can be possibly perturbed when new vehicles are assigned
to (or removed from) the node. A possible way to model this situation is by perturbing
the signal R(k), i.e., R(k) =
P
P
i=1
n
i
(k T
i
) + (k), where is a disturbance signal
characterized by certain statistical properties (e.g. zero mean). Another interesting ex-
tension would be to add to the round trip time some noise (possibly asymmetric, in the
sense that positive perturbations, so that
˜
T
i
> T
i
, are more likely to occur than negative
ones). As a first approach to the problem, in the following we consider the simplified
model (4), i.e., assuming that the T
i
s are deterministic quantities and N
c
is fixed. In this
case the number of available vehicles at time t
k
is: N
a
(k) = N (k) +
P
P
i=1
n
i
(k T
i
).
Let’s now consider the interaction between the stock dynamics (1) and the vehicle
dynamics (4). To this end, assume that each vehicle has identical volume capacity and
that each item of type j = 1, . . . , Q has a relative volume with respect to vehicle
capacity v
j
1 (that is, a vehicle has unit capacity). Accordingly,we havethe following
constraint for any route i:q
Q
X
j=1
v
j
u
ij
(k) [0 , n
i
(k)]. (5)
Since n
i
(k) vehicles are used at time t
k
for route i, it is reasonable that the above
quantity is larger than n
i
(k) 1 (actually, by the policies that will be considered in this
paper, vehicles travel completely loaded).
The objectives of this work will be essentially two. First, derive conditions on the
stability of the system, that is conditions on the inflow process d(·), (relative) part vol-
umes v
j
, traveling times T
i
and number of vehicles N
c
such that there exists a policy of
selection of n
i
(·) and u
ij
(·) which maintains limited all the buffers x
ij
(·). Second, ana-
lyze the performance of some class of policies, trying to solve the optimization problem
consisting in the selection of the n
i
(k) and of the u
ij
(k) to minimize:
J =
K
X
k=1
g[x(k)]γ
k
(6)
where γ (0, 1] is a discount factor and K a planning horizon, possibly infinite. The
function g(·) penalizes waiting freights in the node, e.g., for a linear g(x),
g(x) =
P
X
i=1
Q
X
j=1
c
ij
x
ij
(7)
We now make a fluid approximation for the variables involved in (1), considering
x
ij
, d
ij
and u
ij
as continuous quantities. Accordingly, the information about the vol-
ume of each type j = 1, . . . , Q is now carried by the continuous variables (now the
relative volumes v
j
have no sense per se, hence they will be dropped in the follow-
ing) and each cost c
ij
, assuming a fixed i, now has the meaning of holding cost of part
j = 1, . . . , Q per unit volume
4
. Notice that also the variables N, N
c
, n
i
will represent
volumes (multiples of the unit volume).
We will deal with the two problems above by restricting the control policies to those
which make vehicles travel completely full (this is possible under the fluid approxima-
tion of the materials): this should represent, as remarked below, a correct choice under
heavy traffic conditions. Notice, also, that transportation costs have not been included
in the cost index: this depends on the fact that (i) transportation costs are considered
constant in time; (ii) we restrict the analysis to policies which make all vehicles travel
completely full. The assumptions above imply that the transportation cost is a fixed
component that does not influence the optimization problem. The choice of considering
the vehicles fully loaded is reasonable under heavy traffic conditions (where allowing
the possibility of sending partially full vehicles may even compromise the stability), but
may become significantly sub-optimal in the case of reduced inflow rates, large holding
costs c
ij
and small traveling costs.
3 Stability
As an introduction, consider a one part-type system (Q = 1) with constant inflow
processes d
i
and equal transportation times T
i
= T , i. The equations are then:
x
i
(k + 1) = x
i
(k) + d
i
u
i
(k), i = 1, . . . , P (8)
u
i
(k) [0, n
i
(k)] (9)
P
X
i=1
n
i
(k) N
a
(k) (10)
N(k + 1) = N(k) +
P
X
i=1
n
i
(k T )
P
X
i=1
n
i
(k) (11)
4
Formally, as if the system were described by new variables x
ij
= x
ij
v
j
(and similarly for d
ij
and u
ij
) and c
ij
= c
ij
/v
j
; dropping the “prime” and remaining with the same notation.
Based on the Little’s law, the necessary and sufficient condition of stability for this
system should be:
P
X
i=1
d
i
N
c
T
(12)
In fact, N
c
/T is actually the effective number of vehicles available at each time unit,
and hence also the volume of goods the node may handle in each unit of time. This must
be equal to the volume arriving from outside, i.e.
P
P
i=1
d
i
.
The stronger condition that there exists a static vehicle allocation such that:
T d
i
n
i
(13)
for all i, which implies condition (12), actually is not necessary (but clearly sufficient,
since if it holds, allows to apply a policy where vehicles are divided once for ever among
the tasks and each task is fulfilled, with no interaction among them), as shown in the
following simple example.
Example. Consider a system with d
1
= d
2
= .5, T = 1, N
c
= 1. Clearly it is
not possible to distribute vehicles once for ever (in fact for any static selection of n
i
,
condition (13) does not hold). However (12) holds and, in fact, the periodic allocation
n
1
(·) = {1, 0, 1, . . .} and n
2
(k) = 1 n
1
(k), maintains the buffers bounded.
Let us now return to the general case, but considering at first a constant inflow
process. Condition (12) should be substituted by:
P
X
i=1
Q
X
j=1
d
ij
T
i
N
c
(14)
which will be shown to be necessary and sufficient for the stability of the node. In this
case, in fact, the quantity d
ij
T
i
plays the role of a work inflow in the system per unit of
time (in the sense that for each item of type j to be sent to i, the system must allocate
a working capacity of T
i
, where the total working capacity is N
c
). In the case of time
varying inflow rates (but with the inflow rate oscillating in a bounded interval), the same
condition should hold with average inflow rates
¯
d
ij
.
Remark 1. Actually, while (14) is necessary for stability, the proof reported below only
holds if the inequality in (14) is strict. We believe however that also the equality ensures
the stability. Notice, in any case, that a strict inequality should be considered in practical
settings to guarantee a certain degree of robustness of the stability property.
The previous discussion can be formalized in the following theorem.
Theorem 1. Condition (14) is necessary and sufficient (if taken with strict inequality)
to maintain all the buffers in the node bounded at all times.
Proof. Necessity. The necessity of (14) can be shown by relaxing the integer constraint
on the n
i
(k). If the vehicle resource is not discrete, it is possible to maintain all the
buffers bounded only if there exists a static assignment of the vehicles (notice in fact
that in our model the inflow process is constant) which balances the freight inflow into
the system for all the routes i. The freight inflow into the system of parts to be sent
on the route i is given by D
i
:=
P
Q
j=1
d
ij
. If n
i
vehicles are assigned to this route,
since each transport requires T
i
time units, n
i
vehicles are available only every T
i
time
units. The amount of wares accumulated in such a period is given by D
i
T
i
. So it must
be D
i
T
i
n
i
. Summing over i, we get the condition (14). Since this condition is
necessary for the relaxed problem, it is necessary also for the original problem.
Sufficiency. The proof of sufficiency is constructive: we exhibit a class of policies
which, if (14) holds with strict inequality, ensures that all the buffers remain bounded.
The proof is very similar to the proof of Theorem 1 in [11]. The class of policies ensur-
ing stability is like the CAF policies in [11] where, however, the buffer x
ij
is processed
not until it is cleared (level zero) but until its level becomes lower than N
c
. That is: all
the vehicles are assigned to a single route by filling them with the products of a certain
buffer B
ij
(selected according to the CAF rule (15) reported below) only if this buffer
has sufficient stock to use all vehicles, and the buffer is changed when this is no more
possible. If no buffer can fill all the vehicles, the system remains idle until this becomes
possible. Let τ
n
denote the time a buffer has been finished to be processed. At each time
τ
n
the next buffer will be the one (denoted with a ) satisfying:
x
(τ
n
) ǫ
X
i,j
x
ij
(τ
n
) (15)
for some ǫ > 0 (e.g. the policy which selects the buffer with the largest content will
belong to this class, satisfying (15) with any ǫ (0, 1/P ), see [11]). Let
¯
T
i
:= T
i
/N
c
.
Performing a derivation similar to the one reported in [11], it is possible to show that:
τ
n+1
τ
n
¯
T
x
(τ
n
)
1 ρ
+
N
c
d
(16)
where the denotes the quantities corresponding to the buffer selected at time τ
n
and
ρ
:=
¯
T
d
. The terms in (16) have been obtained as follows: the first term
¯
T
x
(τ
n
)
1ρ
corresponds to the time to bring the buffer x
from its initial level x
(τ
n
) to a value
below N
c
and is derived from [11] setting the setup time δ to 0 and considering that we
only need to reach a value below N
c
and not 0; the second term
N
c
d
takes into account
that when a buffer is selected, perhaps its content is less than N
c
. We define, as in [11]:
w(k) =
X
i,j
¯
T
i
x
ij
(k)
Then we have:
w(τ
n+1
) =
X
i,j
¯
T
i
x
ij
(τ
n+1
) =
=
X
i,j6=
¯
T
i
[x
ij
(τ
n
) + d
ij
(τ
n+1
τ
n
)] +
¯
T
x
(τ
n+1
)
= w(τ
n
) +
X
i,j6=
¯
T
i
d
ij
(τ
n+1
τ
n
) +
¯
T
[x
(τ
n+1
) x
(τ
n
)]
w(τ
n
) +
X
i,j6=
¯
T
i
d
ij
(τ
n+1
τ
n
) +
¯
T
N
c
¯
T
x
(τ
n
)
where the last inequality is implied by the fact that x
(τ
n+1
) N
c
(we stop processing
x
at time τ
n+1
, when its content is below N
c
). Exploiting (16),
w(τ
n+1
) w(τ
n
) +
X
i,j6=
¯
T
i
d
ij
¯
T
x
(τ
n
)
1 ρ
+
N
c
d
+
¯
T
N
c
¯
T
x
(τ
n
)
Now, introducing the notation ρ :=
P
i,j
¯
T
i
d
ij
, we have that
P
i,j6=
¯
T
i
d
ij
= ρ ρ
.
Introducing this in the equation above and simplifying, we get:
w(τ
n+1
) w(τ
n
)
¯
T
x
(τ
n
)
1 ρ
1 ρ
+
ρ
d
N
c
Using (15), the previous becomes:
w(τ
n+1
) w(τ
n
)
¯
T
ǫ
X
i,j
x
ij
(τ
n
)
1 ρ
1 ρ
+
ρ
d
N
c
w(τ
n
)
¯
T
¯
T
M
ǫ
X
i,j
¯
T
i
x
ij
(τ
n
)
1 ρ
1 ρ
+
ρ
d
N
c
where
¯
T
M
= max
i
¯
T
i
. So,
w(τ
n+1
) w(τ
n
)
1 ǫ
¯
T
¯
T
M
1 ρ
1 ρ
+
ρ
d
N
c
Notice that condition (14) under strict inequality can be written as ρ < 1, which is
exactly the condition considered in [11]. The proof can be continued exactly as in [11]
where, however, for us α
ij
=
¯
T
i
1ρ
1ρ
ij
(the same as in [11]) and β
ij
=
ρ
d
ij
N
c
. So, as in
[11], it is possible to obtain:
sup
n
w(τ
n
)
¯
T
M
ǫ
max
ij
β
ij
α
ij
hence
w(t
k
)
¯
T
M
ǫ
max
ij
β
ij
α
ij
+ ρ
N
c
d
m
where d
m
= min
ij
d
ij
. This allows to obtain that
X
ij
x
ij
(t
k
)
1
¯
T
m
w(t
k
)
¯
T
M
¯
T
m
ǫ
max
ij
β
ij
α
ij
+ ρ
N
c
ρ
m
is bounded for all t
k
(where
¯
T
m
:= min
i
¯
T
i
).
4 Optimization
Consider for now Q = 1 (one part type system). Now, under condition (14), if every-
thing is approximated through continuous variables, the optimal policy is myopic [12],
that is, it is the rule if dealing with a linear cost function g(x) as the one consid-
ered in (7). The rule consists in processing the buffers B
ij
according to a priority
established by the product c times µ, where in the present problem, the cost coefficient
c associated to the buffer B
ij
is given by the coefficient c
ij
in (7) and the maximum
processing rate µ for this buffer is given by µ
i
=
N
c
T
i
: this is actually the maximum
processing capacity for goods with destination (route) i. There are however two major
differences:
vehicles are not continuous resources;
the capacity allocation has an influence also on the future (if we allocate all vehicles
to destination i we have to wait T
i
time units before we can change allocation)
while in the scheduling machine case, where the policy has been proved optimal,
capacity allocations can change instantaneously at each step.
4.1 A Possible Heuristic
According to the above observations, we propose here a policy that we believe repre-
sents a promising and simple real time rule. We do not give here a proof of optimality
for this policy and neither give a proof of stability: the performance of this policy will
be explored from a computational point of view. According to the simulations, the sta-
bility appears to hold whenever condition (14) holds: this is not surprising since the
policy reported below reduces the idle periods with respect to the one considered in the
proof of the sufficiency of Theorem 1. This depends on the fact that, even if also this
policy (as the one considered in the proof of Theorem 1) does not allow vehicles to
travel partially loaded, it is no more required here that all the vehicles travel together to
the same destination.
In particular, at each time step, the policy considered in this section allocates the ve-
hicles available at that moment to the buffer B
ij
which, among the ones with
P
j
x
ij
1 (that is, among the ones which allow to complete the load of a vehicle) has the largest
index (where, as mentioned above, for the buffer B
ij
, the index is given by
c
ij
N
c
/T
i
). To illustrate the policy more in detail, assume for simplicity Q = 1 and
let i
1
, . . . , i
P
be the priority established according to the rule (that is c
i
k
/T
i
k
c
i
k+1
/T
i
k+1
for all k). Then, the policy is given by:
n
i
1
(k) = min {N
a
(k), x
i
1
(k)⌋}
n
i
2
(k) = min {N
a
(k) n
i
1
(k), x
i
2
(k)⌋}
and so on, where N
a
(k), defined in (3), is the numberof available vehicles in the interval
(t
k
, t
k+1
). Then, to fill all the vehicles assigned to route i, we set:
u
i
(k) = n
i
(k).
4.2 Simulative Results
We tested the policies discussed above in a system with Q = 1 (a single product), P =
3, characterized by the following parameters: delays T
1
= 4, T
2
= 3 , T
3
= 5 ; arrivals,
constant in time, d
1
= d
2
= 7, d
3
= 5; with this choice the minimum N
c
guaranteeing
stability is 74,according to condition (14). In figure are shown the performances(6), (7),
with unit costs c
1
= c
3
= 1, c
2
= 2, and γ = 1, of three policies derived by simulating
the system, for a finite time horizon, for various values of the parameter N
c
. The dash
dotted line shows the performances of the stabilizing policy described in Theorem 1;
the dashed line the performances of the policy which allocates at each time instant all
the available vehicles prioritizing the buffers with higher content, and the continuous
line the performances of the policy” (those coefficients, by the parameters chosen,
make buffer 2 the one with higher priority followed by buffer 1 and 3).
It is possible to observe that for values of N
c
lower than the stabilizing value (74),
none of the policy described can achieve stability, consistently with Theorem 1 (for
N
c
< 74 the costs reported in Figure 1 result finite as a consequence of the finite
time horizon considered). For N
c
> 74 the policy performs better than the policy
prioritizing the higher buffers.
70 72 74 76 78 80 82 84 86 88 90
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
5
N
c
J
Fig.1. Performances of the policy (continuous line), the serve-largest-buffer policy (dashed),
and the basic stabilizing policy (dash-dotted), as a function of N
c
.
5 Conclusions
In this paper, a simplified model of a logistic node has been considered, where items
arrive from outside to the node and must be routed to different destinations. Waiting
items are stored in different buffers, according to their class and destination. At first,
a necessary and sufficient condition is given in the paper for the possibility of finding
dispatching dynamic policies that maintain all the buffers bounded. Subsequently an
optimization problem is considered and a simulative comparison of the performance of
different feedback policies is presented in the paper. The problem has been studied un-
der a fluid approximationof the items traveling in the node: this allows to completely fill
the vehicles (and to neglect complex combinatorial loading problems). This possibility
is actually used by the policies studied in this paper that do not allow the vehicles to
travel partially filled. This is actually a reasonable choice under heavy traffic conditions
where allowing the possibility of sending partially full vehicles may even compromise
the stability, but may become significantly sub-optimal in the case of reduced inflow
rates, large holding costs and small traveling costs.
References
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