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Um problema de múltiplas instalações com distâncias euclideanas pode ser formulado da seguinte forma (Francis et al., 1974, p. 227 ):
W
M
(
X
)
=
∑
i
=
1
m
∑
j
=
1
n
w
1
i
j
[
(
x
i
1
−
a
j
1
)
2
+
(
x
i
2
−
a
j
2
)
2
]
1
/
2
+
∑
i
=
1
m
−
1
∑
r
=
i
+
1
m
w
2
i
r
[
(
x
i
1
−
x
r
1
)
2
+
(
x
i
2
−
x
r
2
)
2
]
1
/
2
{\displaystyle WM(X)=\sum _{i=1}^{m}\sum _{j=1}^{n}w_{1ij}[(x_{i1}-a_{j1})^{2}+(x_{i2}-a_{j2})^{2}]^{1/2}+\sum _{i=1}^{m-1}\sum _{r={i+1}}^{m}w_{2ir}[(x_{i1}-x_{r1})^{2}+(x_{i2}-x_{r2})^{2}]^{1/2}}
m é o número de novas instalações;
n é o número de instalações já existentes;
w
1
i
j
{\displaystyle \ w_{1ij}}
distância entre uma nova instalação
i
{\displaystyle \ i}
j
{\displaystyle \ j}
w
1
i
j
≥
0
{\displaystyle w_{1ij}\geq 0}
w
2
i
r
{\displaystyle \ w_{2ir}}
i
{\displaystyle \ i}
r
{\displaystyle \ r}
w
2
i
r
≥
0
{\displaystyle w_{2ir}\geq 0}
(
x
i
1
,
x
i
2
)
{\displaystyle \ (x_{i1},x_{i2})}
localização da nova instalação
i
{\displaystyle \ i}
(
a
j
1
,
a
j
2
)
{\displaystyle \ (a_{j1},a_{j2})}
j
{\displaystyle \ j}
∂
f
∂
x
f
{\displaystyle \ \partial f \over \partial x_{f}}
∂
f
∂
y
j
{\displaystyle \ \partial f \over \partial y_{j}}
∂
f
∂
x
f
{\displaystyle \ \partial f \over \partial x_{f}}
∑
i
=
1
m
w
1
i
j
(
x
i
1
−
a
j
1
)
E
i
j
+
∑
k
=
1
n
w
2
i
r
(
x
i
1
−
x
r
1
)
D
i
r
,
j
=
1
,
.
.
.
,
n
{\displaystyle \ \sum _{i=1}^{m}{{w_{1ij}(x_{i1}-a_{j1})} \over E_{ij}}+\sum _{k=1}^{n}{w_{2ir}(x_{i1}-x_{r1}) \over D_{ir}},j=1,...,n}
e
∂
f
∂
y
j
{\displaystyle \ \partial f \over \partial y_{j}}
∑
i
=
1
m
w
1
i
j
(
x
i
2
−
a
j
2
)
E
i
j
+
∑
k
=
1
n
w
2
i
r
(
y
i
2
−
y
r
2
)
D
i
r
,
j
=
1
,
.
.
.
,
n
{\displaystyle \ \sum _{i=1}^{m}{{w_{1ij}(x_{i2}-a_{j2})} \over E_{ij}}+\sum _{k=1}^{n}{w_{2ir}(y_{i2}-y_{r2}) \over D_{ir}},j=1,...,n}
E
i
j
=
[
(
x
i
1
−
a
j
1
)
2
+
(
x
i
2
−
a
j
2
)
2
]
1
/
2
{\displaystyle \ E_{ij}=[(x_{i1}-a_{j1})^{2}+(x_{i2}-a_{j2})^{2}]^{1/2}}
e
D
i
r
=
[
(
x
i
1
−
x
r
1
)
2
+
(
x
i
2
−
x
r
2
)
2
]
1
/
2
{\displaystyle \ D_{ir}=[(x_{i1}-x_{r1})^{2}+(x_{i2}-x_{r2})^{2}]^{1/2}}
x
{\displaystyle \ x}
(
a
1
,
0
)
,
.
.
.
,
(
a
m
,
0
)
{\displaystyle \ (a_{1},0),...,(a_{m},0)}
a
1
{\displaystyle \ a_{1}}
a
m
{\displaystyle \ a_{m}}
Francis et al., 1992, p. 368 ).
![{\displaystyle WM(X)=\sum _{i=1}^{m}\sum _{j=1}^{n}w_{1ij}[(x_{i1}-a_{j1})^{2}+(x_{i2}-a_{j2})^{2}]^{1/2}+\sum _{i=1}^{m-1}\sum _{r={i+1}}^{m}w_{2ir}[(x_{i1}-x_{r1})^{2}+(x_{i2}-x_{r2})^{2}]^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6809851160687e35f7b97d425b19c6e22bee5bb5)
![{\displaystyle \ E_{ij}=[(x_{i1}-a_{j1})^{2}+(x_{i2}-a_{j2})^{2}]^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad95fff752f4947e55b09bef62ea94e3c13186b)
![{\displaystyle \ D_{ir}=[(x_{i1}-x_{r1})^{2}+(x_{i2}-x_{r2})^{2}]^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1325ec2d0c2fa22d087e37a9ee0870c7499b05)
