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directional simple case, and that we are imposed some
time horizon [0, T
max
]. Then, we consider a vehi-
cle fleet, with n vehicles k = 1,...,n, whose activity
is scheduled between 0 and T
max
, according to rout-
ing strategies (Γ
k
,v
k
),k = 1,...,n, where Γ
k
is a path
which starts from some origin o
k
and arrives at some
target destination d
k
. b (Γ
k
,v
k
),k = 1,...,n, allow us
to derive, for any arc e a risk function t 7→ Π
e
(t) de-
fined with:
• A collection of auxiliary risk functions t 7→ Π
e
k
(t)
whose meaning is: Π
e
k
(t) is the risk function,
which we obtain while removing vehicle k and re-
lated routing strategy (Γ
k
,v
k
)
• A collection of marginal risk functions t 7→ Π
e
k
(t),
obtained from equation 2 and whose meaning is:
if we consider all vehicles but k, and if we make
vehicle k run along e between t and t +dt at speed,
v ≤ 1, then additional expected damage explicitly
involving for k is equal to H(v).Π
e
(t).dt
Then we may extend the well-known notion of
Wardrop Equilibrium by defining a Risk Wardrop
Equilibrium as follows: A collection of strategies
{(Γ
k
,v
k
),k = 1,...,n} define a Risk Wardrop Equi-
librium if, for any k = 1,...,n : (Γ
k
,v
k
) is an optimal
strategy for vehicle k in the sense of the SSPP instance
involving marginal risk functions t 7→ Π
e
k
(t) and time
horizon [0,T
max
]. We state:
Proposition 4. If T
max
≥ sup
k
{L
∗
(o
k
,d
k
)}, where L
∗
means the shortest path distance in the sense of the
length L, then there exists a Risk Wardrop Equilib-
rium.
Proof. It directly derives from the way we defined
t 7→ Π
e
k
(t) as related to the marginal expected dam-
age induced by the introduction of additional vehicle
k into a transit network where other vehicles 1,...,k −
1,k + 1, . . . , n, are supposed to be already evolving.
More precisely, function t 7→ Π
e
k
(t) is defined for
any k by equations 2 which links it to risk function
t 7→ Π
e
(t) and risk functions t 7→ Π
e
k
(t). Let us con-
sider an optimal solution {(Γ
opt
k
,u
opt
k
),k = 1,...,n} of
the following GRM: Global Risk Minimization prob-
lem:
Compute a strategy collection {(Γ
k
,v
k
),k = 1,...,n}
such that resulting global risk value
∑
e
R
[0,T
max
]
Π
e
(t)dt
be the smallest possible.
Let us denote by t 7→ R
opt
e
(t), t 7→ R
opt
e
k
(t) and
t 7→ Π
opt
e
k
(t),k = 1,...,n, related risk func-
tions and marginal risk functions. Then we see
that, for any k, (Γ
opt
k
,u
opt
k
), which meets time
horizon [0,T
max
], minimizes under this con-
straint
∑
e
R
[0,T
max
]
Π
e
(t)dt −
∑
e
R
[0,T
max
]
R
opt
e
k
(t)dt =
∑
e
R
[0,T
max
]
H(v(t)).Π
opt
e
k
(t)dt, and so is an optimal so-
lution of the SSPP instance induced by marginal risk
functions t 7→ Π
opt
e
k
(t). If T
max
≥ sup
k
{L
∗
(o
k
,d
k
)}
then GRM clearly admits a feasible solution. What
remains to be proved is that it admits an optimal
solution. We notice that no more than n vehicles
can be located at a given time t on a same arc
e, and that, once a vehicle leaves e, it does not
come back later. It comes that the number of
breakpoints of a function Π
e
k
(t) cannot exceed n.
Then a simple topological argument about com-
pactness allows us to check that from any sequence
{(Γ
p
k
,u
p
k
),k = 1,. . . , n, p = 1,··· + ∞} , we may
extract a convergent subsequence in the simple sense,
and that Lebesgue Theorem may be applied to this
convergence. We deduce that there must exist a
collection {(Γ
opt
k
,u
opt
k
),k = 1,...,n} which achieves
In f
(Γ
k
,v
k
),k=1,...,n
(
∑
e
R
[0,T
max
]
Π
e
(t)dt).
3.3 Risk Versus Distance SSPP
Reformulation
Remark 2 leads us to define the Risk versus Time
coefficient for arc e
i
as the value 2H
′
(v
q
)Π
e
i
q
in-
volved in Proposition 3. This proposition, combined
with Proposition 1, allows us to significantly simplify
SSPP: We define a risk versus distance strategy as a
pair (Γ,λ
RD
) where:
• Γ is a path, that means a sequence {e
1
,...,e
n
} of
arcs, which connects origin node o do destination
node d;
• λ
RD
e
associates, with any arc e in Γ, Risk ver-
sus Distance coefficient λ
RD
e
= 2H
′
(v)R
e
. In case
H(v) = v
2
, we notice that this coefficient means
the amount of risk per distance unit induced on
arc e at any time t such that v(t) < 1, by any tra-
jectory (Γ,v) which satisfies Proposition 3.
Let us suppose that we follow a trajectory (Γ,v) which
meets Proposition 3, and that we know value λ
RD
e
for
any arc e of Γ.Since H is supposed to be convex and
such that H(v) ≪ v, we may state that H
′
admits a
reciprocal function H
′−1
. Then, at any time t when
vehicle V is inside arc e, we are able to reconstruct
value
v(t) :
(
H
′−1
(
λ
RD
e
2R
e
), if H
′−1
(
λ
RD
e
2R
e
) < 1
1, otherwise
(3)
According to this and Proposition 3, SSPP may be
A Comparison of Several Speed Computation Methods for the Safe Shortest Path Problem
19