Minimizar la suma de las k-mayores funciones entre n y sus

Transcripción

Minimizar la suma de las k-mayores funciones entre n
Rectilinear k-centrum problem
Discrete k-centrum location problem
Discrete convex ordered median location problem
Locating subtrees
Minimizar la suma de las k-mayores funciones
entre n y sus aplicaciones a localización
J. Puerto† and A. Marı́n‡
† Dpto.
de Estadı́stica e Investigación Operativa. Univ. de Sevilla
‡ Dpto.
de Estadı́stica e Investigación Operativa. Univ. de Murcia
J.Puerto, A. Marı́n
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Locating subtrees
Notation and preliminares
1
2
3
4
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Locating subtrees
Tactical ordered median objective
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Locating subtrees
Minimizar la suma de las k-mayores funciones lineales
entre n
1
2
3
4
5
Locating subtrees
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Locating subtrees
Notation
For a real number z, z+ = max{0, z}.
Let be y = (y1 , . . . , yn ) ∈ Rn and y() the vector obtained by
sorting its elements in nondecreasing order, i.e.,
y(1) ≥ y(2) ≥ . . . ≥ y(n) .
P
For any k, let Sk (y ) = ki=1 y(i) .
The key function:
h(t) =
n
X
k(t − yi )+ + (n − k)(yi − t)+ .
i=1
Piecewise linear and convex with slopes from −n(n − k) < 0 to
nk > 0.
The minimum is reached when t = y(k) since the one side
derivatives are of opposite sign.
Óptimos 2007
Locating subtrees
t
t < y(n)
y(n) ≤ t < y(n−1)
y(n−1) ≤ t < y(n−2)
..
.
h(t)
n(k − n)t + αn
n(k − n + 1)t + αn−1
n(k − n + 2)t + αn−2
..
.
Slope sign
<0
<0
<0
..
.
y(k+1) ≤ t < y(k)
y(k) ≤ t < y(k+1)
..
.
n(k − n + n − k)t + αn−k
n(k − n + n − k + 1)t + αn−k−1
..
.
=0
>0
..
.
y(1) < t
nkt + α1
>0
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Locating subtrees
The expression for the k-centrum
h(y(k) ) = k
n
X
(y(k) − y(i) ) − (n − k)
i=1
i=k+1
= n
k
X
k
X
y(i) − k
i=1
= nSk (y ) − k
n
X
y(i)
i=1
n
X
y(i) .
i=1
Isolating, one has:
Sk (y ) =
n
1 X
k
yi + min h(t)
t∈R
n
i=1
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(y(k) − y(i) )
Locating subtrees
The expression for the k-centrum
h(y(k) ) = k
n
X
(y(k) − y(i) ) − (n − k)
i=1
i=k+1
= n
k
X
k
X
y(i) − k
i=1
= nSk (y ) − k
n
X
y(i)
i=1
n
X
y(i) .
i=1
Isolating, one has:
Sk (y ) =
n
1 X
k
yi + min h(t)
t∈R
n
i=1
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(y(k) − y(i) )
Locating subtrees
The linear programming representation
min
1
n
−
i=1 (kyi + kdi + (n −
di+ − di− = yi − t,
di+ , di− ≥ 0.
Pn
s.t.
k)di+ )
Making the change di− = di+ − yi + t
min f (t) := (kt +
s.t.
Pn
+
i=1 di )
di+ ≥ yi − t,
di+ ≥ 0.
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Locating subtrees
The linear programming representation
min
1
n
−
i=1 (kyi + kdi + (n −
di+ − di− = yi − t,
di+ , di− ≥ 0.
Pn
s.t.
k)di+ )
Making the change di− = di+ − yi + t
min f (t) := (kt +
s.t.
Pn
+
i=1 di )
di+ ≥ yi − t,
di+ ≥ 0.
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Locating subtrees
Example
Consider an example with n = 6, k = 2, (y(i) ) = (5, 3, 2, 1, 1, 0).
f (t) = 2t + max{0, 0 − t} + 2 max{0, 1 − t} + max{0, 2 − t}+
max{0, 3 − t} + max{0, 5 − t}.
f (t) is a convex piecewise linear function with slopes moving from
k − n + 1 < 0 to k > 0 in integer steps, whose minimum is reached
when t equals y(k) .
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Locating subtrees
Example
Consider an example with n = 6, k = 2, (y(i) ) = (5, 3, 2, 1, 1, 0).
f (t) = 2t + max{0, 0 − t} + 2 max{0, 1 − t} + max{0, 2 − t}+
max{0, 3 − t} + max{0, 5 − t}.
f (t) is a convex piecewise linear function with slopes moving from
k − n + 1 < 0 to k > 0 in integer steps, whose minimum is reached
when t equals y(k) .
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Locating subtrees
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Locating subtrees
The rectilinear k-Centrum Problem
Given is a set {v 1 , . . . , v n } of n points in Rd . Suppose that v i ,
i = 1, . . . , n, is associated with a nonnegative real weight wi . For
each point x ∈ Rd define the vector D(x) ∈ Rd by
D(x) = (w1 d1 (x), . . . , wn dn (x)). For a given k = 1, . . . , n, the
single facility rectilinear k−centrum problem in Rd is:
min Sk (x) :=
k
X
x∈Rd
D(i) (x)
i=1
Using the above results,
P
min
kt + ni=1 d̃i
s.t.
(1)
d̃i ≥ wi di (x) − t,
x = (x1 , . . . , xd ) ∈
d̃i ≥ 0,
Rd .
Óptimos 2007
i = 1, . . . , n,
Locating subtrees
The rectilinear k-Centrum Problem
Given is a set {v 1 , . . . , v n } of n points in Rd . Suppose that v i ,
i = 1, . . . , n, is associated with a nonnegative real weight wi . For
each point x ∈ Rd define the vector D(x) ∈ Rd by
D(x) = (w1 d1 (x), . . . , wn dn (x)). For a given k = 1, . . . , n, the
single facility rectilinear k−centrum problem in Rd is:
min Sk (x) :=
k
X
x∈Rd
D(i) (x)
i=1
Using the above results,
P
min
kt + ni=1 d̃i
s.t.
(1)
d̃i ≥ wi di (x) − t,
x = (x1 , . . . , xd ) ∈
d̃i ≥ 0,
Rd .
Óptimos 2007
i = 1, . . . , n,
Locating subtrees
To obtain a linear program we replace each one of the n nonlinear
constraints d̃i ≥ wi di (x) − t by a set of 2d linear constraints. For
i = 1, . . . , n, let ∆i = (δ1d , . . . , δdi ) be a vector all of whose
components are equal to +1 or -1.
P
min
kt + ni=1 d̃i
P
s.t. d̃i + t ≥ dj=1 δij wi (xj − vji ), i = 1 . . . , n,
δji ∈ {−1, 1},
d̃i ≥ 0,
j = 1, . . . , d,
i = 1 . . . , n,
i = 1, . . . , n.
Note that the linear program has n + d + 1 variables, d̃1 , . . . , d̃n ,
x1 , . . . , xd , t and 2d n + n constraints. This formulation is the dual
of linear multiple-choice knapsack problems. Therefore, using the
results of Zemel, when d is fixed, an optimal solution can be found
in O(n) time. If d = 1 the problem reduces to the k-centrum
problem on the line.
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Locating subtrees
To obtain a linear program we replace each one of the n nonlinear
constraints d̃i ≥ wi di (x) − t by a set of 2d linear constraints. For
i = 1, . . . , n, let ∆i = (δ1d , . . . , δdi ) be a vector all of whose
components are equal to +1 or -1.
P
min
kt + ni=1 d̃i
P
s.t. d̃i + t ≥ dj=1 δij wi (xj − vji ), i = 1 . . . , n,
δji ∈ {−1, 1},
d̃i ≥ 0,
j = 1, . . . , d,
i = 1 . . . , n,
i = 1, . . . , n.
Note that the linear program has n + d + 1 variables, d̃1 , . . . , d̃n ,
x1 , . . . , xd , t and 2d n + n constraints. This formulation is the dual
of linear multiple-choice knapsack problems. Therefore, using the
results of Zemel, when d is fixed, an optimal solution can be found
in O(n) time. If d = 1 the problem reduces to the k-centrum
problem on the line.
Óptimos 2007
Locating subtrees
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Locating subtrees
Óptimos 2007
Locating subtrees
The k-Centrum Problem
In this case, it is the sum of the k maximum costs which has to be
minimized.
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Locating subtrees
The k-Centrum Problem
Consider the first formulation for the p-median problem. After
getting the x-values, let us sort the values of
X
Ci :=
cij xij
∀i ∈ I .
j∈J
The objective value we want to minimize is the sum of the k
largest C -values. We represent with C(i) the values Ci sorted in
non increasing order:
C(1) ≥ C(2) ≥ . . . ≥ C(n) .
Consider the following function defined in [0, +∞):
X
f (t) = kt +
max{0, Ci − t}.
i∈I
Óptimos 2007
Locating subtrees
Then the minimum value of f is
n
X
f (C(k) ) = kC(k) +
max{0, Ci − C(k) }
i=1
k
X
= kC(k) +
(C(i) − C(k) ) =
i=1
k
X
C(i) ,
i=1
i.e., the sum of the k maximum values of C .
Therefore, the k-centrum problem can be formulated as
P
min
kt + i∈I di
P
s.t.
j∈J yj = p
xij ≤ yj
P
di ≥
P
j∈J
(3)
∀i ∈ I , j ∈ J
(4)
∀i ∈ I
(5)
xij − t ∀i ∈ I
(6)
xij = 1
j∈J
(2)
di ≥ 0
∀i ∈ I
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(7)
Locating subtrees
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Locating subtrees
We are given λ = (λ1 , . . . , λn ) such that
λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 0
ci (X ) := mink∈X cik
σX is a permutation of {1, . . . , n}
cσX (1) (X ) ≥ cσX (2) (X ) ≥ · · · ≥ cσX (n) (X )
Discrete ordered median problem (DOMP)
min
X ⊆A , |X |=N
n
X
λi cσX (i) (X ) .
i=1
Óptimos 2007
Locating subtrees
Recall Sk (X ) =
Pk
i=1 c(i) (X ),
the k-largest values of c(X ).
λ1 S1 (X )− λ2 S1 (X )
λ2 S2 (X )− λ3 S2 (X )
λ3 S3 (X )− λ4 S3 (X )
...
λ1 c(1) (X )
λ2 c(2) (X )
λ3 c(3) (X )
λn Sn (X )
. . . λn c(n) (X )
Then, taking λn+1 = 0:
n
X
i=1
λi c(i) (X ) =
n
X
(λk − λk+1 )Sk (X ).
k=1
Óptimos 2007
Locating subtrees
Recall Sk (X ) =
Pk
i=1 c(i) (X ),
the k-largest values of c(X ).
λ1 S1 (X )− λ2 S1 (X )
λ2 S2 (X )− λ3 S2 (X )
λ3 S3 (X )− λ4 S3 (X )
...
λ1 c(1) (X )
λ2 c(2) (X )
λ3 c(3) (X )
λn Sn (X )
. . . λn c(n) (X )
Then, taking λn+1 = 0:
n
X
i=1
λi c(i) (X ) =
n
X
(λk − λk+1 )Sk (X ).
k=1
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Locating subtrees
Therefore, the convex discrete ordered p-median problem can be
formulated as
Pn
P
(9)
min
k=1 (λk − λk+1 )(ktk +
i∈I dik )
P
s.t.
(10)
j∈J yj = p
xij ≤ yj
P
j∈J
dik ≥
P
∀i ∈ I , j ∈ J (11)
xij = 1
j∈J
xij − tk
dik ≥ 0
xij , yj ∈ {0, 1}
Óptimos 2007
∀i ∈ I
(12)
∀i ∈ I
(13)
∀i ∈ I
(14)
∀i ∈ I , j ∈ J. (15)
Locating subtrees
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2
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Locating subtrees
Óptimos 2007
Locating subtrees
Notation
T = (V , E ) undirected tree network with node set V = {v1 , ..., vn }
and edge set E = {e2 , ..., en }. Each edge ej , j = 2, 3, ..., n, has a
positive length lj , and is assumed to be rectifiable. A(T ) denote
the continuum set of points on the edges of T .
Let P[vi , vj ] denote the unique simple path in A(T ) connecting vi
and vj .
Suppose that the tree T is rooted at some distinguished node, say
v1 . For each node vj , j = 2, 3, ..., n, let p(vj ), the parent of vj , be
the node v ∈ V , closest to vj , v 6= vj on P[v1 , vj ]. vj is a child of
p(vj ). ej is the edge connecting vj with its parent p(vj ). Sj will
denote the set of all children of vj . A node vi is a descendant of vj
if vj is on P[vi , v1 ]. Vj will denote the set of all descendants of vj .
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Locating subtrees
Notation II
d(x, y ) denote the length of P[x, y ], the unique simple path in
A(T ) connecting x and y . A(T ) is a metric space with
respect to the above distance function.
We call a subtree discrete if all its (relative) boundary points
are nodes of T . For each i = 1, . . . , n, we denote by Ti the
subtree induced by Vi . If Y is a subtree we define the length
or size of Y , L(Y ), to be the sum of the lengths of its partial
edges.
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Locating subtrees
The model
In our location model the nodes of the tree are viewed as demand
points (customers), and each node vi ∈ V is associated with a
nonnegative weight wi . The set of potential servers consists of
subtrees.
The OM function for parameters Λ = (λ1 , ..., λn ) at the subtree S:
X (S) = {w1 d(v1 , S), ..., wn d(vn , S)} be the set of weighted
distances of the n demand points from S and X() (S) be the
sequence obtained by sorting the elements in X (S) in
nonincreasing order. The value of OM function is then the scalar
product of X() (S) with the sequence Λ.
Given a real L, the tactical problem consists of finding a subtree
(facility) Y of length smaller than or equal to L, minimizing the
ordered median objective.
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Locating subtrees
Optimal tactical continuous rooted subtree
The discrete model is NP-hard even for the median function, since
the Knapsack Problem is a special case.
Optimal tactical continuous subtree containing a given node
For each edge ej of the rooted tree, connecting vj with its parent,
p(vj ), assign a variable xj , 0 ≤ xj ≤ lj .
We also need the following condition to be satisfied:
xj (li − xi ) = 0, if vi = p(vj ), vi 6= v1 , and j = 2, ..., n.
(16)
Equation (16) ensures the connectivity.
For each node vt , the weighted distance of vt from a subtree is
X
yt = w t
(lk − xk ).
vk ∈P[vt ,v1 )
Óptimos 2007
Locating subtrees
We call a set X = {x2 , ..., xn } admissible if x2 , ..., xn , satisfy
0 ≤ xj ≤ lj , j = 2, ..., n, and
n
X
xj ≤ L.
j=2
Theorem
Let X =X
{x2 , ..., xn } be admissible. For each vt ∈ V , define
yt = w t
(lk − xk ). Then, there exists an admissible set
X0
vk ∈P[vt ,v1 )
0 , ..., x 0 },
= {x2P
n
yt0 = wt
satisfying (16), such that for each vt ∈ V
0
vk ∈P[vt ,v1 ) (lk − xk ) ≤ yt .
Óptimos 2007
(17)
Locating subtrees
Optimal tactical continuous rooted subtree II
The above LP formulations for the convex ordered median
objectives will provide a compact LP formulation for finding a
continuous subtree rooted at a distinguished point, whose length is
at most L.
min
n
X
(λk − λk+1 )(ktk +
+
di,k
)
i=1
k=1
s.t.
n
X
+
+
di,k
≥ yi − tk , di,k
≥ 0,
X
yt = w t
(lk − xk ),
i = 1, ..., n, k = 1, ..., n,
for each vt ∈ V ,
vk ∈P[vt ,v1 )
0P≤ xj ≤ lj ,
n
j=2 xj ≤ L.
j = 2, ..., n,
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Locating subtrees
Optimal tactical continuous rooted subtree III
The above formulation uses p = O(n2 ) variables and q = O(n2 )
constraints. Assuming integer data, let I denote the total number
of bits needed to represent the input. Then, by Vaidya[?], the
above LP can be solved by using only O(n6 + n5 I ) arithmetic
operations. We also note that in the unweighted model, where the
distances of all demand points are equally weighted, all the entries
of the constraint matrix can be assumed to be 0, 1 or −1.
Therefore, by Tardos[?], the number of arithmetic operations
needed to solve the unweighted version is strongly polynomial, i.e.,
it is bounded by a polynomial in n, and is independent of the input
size I .
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Locating subtrees
Where To Find This Formulation
S. Nickel, J. Puerto. “Location Theory: A unified approach”.
Springer Series in Operations Research and Decision Theory.
ISBN: 3-540-24321-6. (2005).
J. Puerto, A. Tamir, “Locating tree-shaped facilities using the
ordered median objective”. Mathematical Programming
Volume 102:2, 313 - 338, 2005.
Ogryczak, W. and Tamir, A. Minimizing the sum of the k
largest functions in linear time. Information Processing Letters
85 (2003) 117-122.
See also
Tamir. The k-centrum multi-facility location problem.
Discrete Applied Mathematics, 2000.
Óptimos 2007

Minimizar la suma de las k-mayores funciones entre n y sus

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