Mixing
Dualization of a Linear Program for City Logistics
$begingroup$
I am trying to use Bender's decomposition for a Location-Allocation Problem. The aim is to dualize the slave problem to generate the cuts for the Restricted Master Problem. Therefore, I have identified a slave problem and am trying to dualize it.
The problem consists of a two-stage distribution network. There are external zones E outside of a city. Satelites S are potential locations for opening a satelite within the city. Between external zones and satelites only trucks T are operating. Delivery from satelites to customer zones Z within the city is done using eco-friendly city-freighters V.
To give a bit of background I'll introduce the variables:
We have two decision variables in the Master problem:
begin{align*}f_{esz}^{pvt} &= text{flow of product } p in P text{ from external site } e in E text{ to satelite } s in S text{ using truck } t in T $ \
&text{ and delivering to customer zone } z in Z text{ with city freighter } v in V \
end{align*}
begin{align*}
y_s &= begin{cases} 1 & text{if satelite } s in S text{ is opened}, \
0 & text{otherwise.} end{cases} \
end{align*}
In the slave problem y is given so y becomes a normal variable.
$k_p t_s^p$ are the handling cost of product p per time unit multiplied times the handling time of product p at satelite s.
$t_{es}$ are the travel times from external site e to satelite s and $t_{sz}$ the travel times from s to customer zone z respectively.
$k_t$ are the operations cost for a truck t and $k_v$ the operations cost for a city freighter.
$u_t$ is the capacity of a truck and $u_v$ the cap. of a city freighter.
$g_{ez}^p$ is the demand of customer z for product p from external e
At satelite s there is a capacity of $u_s^T$ trucks and $u_s^V$ city freighters. If a satelite s is opened then at least $l_s^T$ trucks have to be used at the satelite.
The slave problem in the primal looks as follows:
begin{align*}
min Z left( f right) = &sum_{s in S} sum_{p in P} k_pt_s^p sum_{e in E}sum_{z in Z}sum_{t in T}sum_{vin V} f_{esz}^{pvt} + \
& sum_{p in P} left( sum_{e in E}sum_{s in S} t_{es}sum_{t in T} k_t sum_{z in Z} sum_{v in V} frac{f_{esz}^{pvt}}{u_t} + sum_{s in S}sum_{z in Z} t_{sz}sum_{v in V} k_v sum_{e in E}sum_{t in T} frac{f_{esz}^{pvt}}{u_v}right)
end{align*}
begin{align}
text{s.t.} quad &sum_{s in S}sum_{t in T}sum_{v in V} f_{esz}^{pvt} = g_{ez}^p quad forall e in E, z in Z, p in P \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} leq u_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} geq l_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_v} leq u_s^V y_s quad forall sin S \
& y in lbrace 0,1 rbrace quad forall s in S \
& f_{esz}^{pvt} geq 0 quad forall e in E, z in Z, s in S, p in P ,v in V, t in T
end{align}
My current version of the dualization looks like this:
begin{align*}
max Z(f) sum_{s in S} u_s^T y_s alpha_s + sum_{s in S} l_s^T y_s gamma_s + sum_{s in S} u_s^V y_s beta_s + sum_{e in E}sum_{z in Z}sum_{p in P} g_{ez}^p delta_{ez}^p
end{align*}
begin{align}
text{s.t.} quad delta_{ez}^p &in mathbb{R} quad forall ein E, z in Z, p in P \
alpha_s &leq 0 quad forall s in S \
gamma_s &geq 0 quad forall s in S \
beta_s &leq 0 quad forall s in S \
frac{alpha_s+gamma_s}{u_t} + frac{beta_s}{u_v} &leq frac{t_{es}k_t}{u_t} + frac{t_{sz} * k_v}{u_v} quad forall e in E, s in S, t in T, z in Z, v in V \
delta_s &leq k_p t_s^p quad forall s in S, p in P, z in Z, e in E
end{align}
However, I am really unsure about the constraints. How do I know which variables should appear in which constraint and generally, what would be the correct formulation?
optimization linear-programming duality-theorems
asked Jan 13 at 14:11
user3578476user3578476
12
$endgroup$
add a comment |
$begingroup$
I am trying to use Bender's decomposition for a Location-Allocation Problem. The aim is to dualize the slave problem to generate the cuts for the Restricted Master Problem. Therefore, I have identified a slave problem and am trying to dualize it.
The problem consists of a two-stage distribution network. There are external zones E outside of a city. Satelites S are potential locations for opening a satelite within the city. Between external zones and satelites only trucks T are operating. Delivery from satelites to customer zones Z within the city is done using eco-friendly city-freighters V.
To give a bit of background I'll introduce the variables:
We have two decision variables in the Master problem:
begin{align*}f_{esz}^{pvt} &= text{flow of product } p in P text{ from external site } e in E text{ to satelite } s in S text{ using truck } t in T $ \
&text{ and delivering to customer zone } z in Z text{ with city freighter } v in V \
end{align*}
begin{align*}
y_s &= begin{cases} 1 & text{if satelite } s in S text{ is opened}, \
0 & text{otherwise.} end{cases} \
end{align*}
In the slave problem y is given so y becomes a normal variable.
$k_p t_s^p$ are the handling cost of product p per time unit multiplied times the handling time of product p at satelite s.
$t_{es}$ are the travel times from external site e to satelite s and $t_{sz}$ the travel times from s to customer zone z respectively.
$k_t$ are the operations cost for a truck t and $k_v$ the operations cost for a city freighter.
$u_t$ is the capacity of a truck and $u_v$ the cap. of a city freighter.
$g_{ez}^p$ is the demand of customer z for product p from external e
At satelite s there is a capacity of $u_s^T$ trucks and $u_s^V$ city freighters. If a satelite s is opened then at least $l_s^T$ trucks have to be used at the satelite.
The slave problem in the primal looks as follows:
begin{align*}
min Z left( f right) = &sum_{s in S} sum_{p in P} k_pt_s^p sum_{e in E}sum_{z in Z}sum_{t in T}sum_{vin V} f_{esz}^{pvt} + \
& sum_{p in P} left( sum_{e in E}sum_{s in S} t_{es}sum_{t in T} k_t sum_{z in Z} sum_{v in V} frac{f_{esz}^{pvt}}{u_t} + sum_{s in S}sum_{z in Z} t_{sz}sum_{v in V} k_v sum_{e in E}sum_{t in T} frac{f_{esz}^{pvt}}{u_v}right)
end{align*}
begin{align}
text{s.t.} quad &sum_{s in S}sum_{t in T}sum_{v in V} f_{esz}^{pvt} = g_{ez}^p quad forall e in E, z in Z, p in P \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} leq u_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} geq l_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_v} leq u_s^V y_s quad forall sin S \
& y in lbrace 0,1 rbrace quad forall s in S \
& f_{esz}^{pvt} geq 0 quad forall e in E, z in Z, s in S, p in P ,v in V, t in T
end{align}
My current version of the dualization looks like this:
begin{align*}
max Z(f) sum_{s in S} u_s^T y_s alpha_s + sum_{s in S} l_s^T y_s gamma_s + sum_{s in S} u_s^V y_s beta_s + sum_{e in E}sum_{z in Z}sum_{p in P} g_{ez}^p delta_{ez}^p
end{align*}
begin{align}
text{s.t.} quad delta_{ez}^p &in mathbb{R} quad forall ein E, z in Z, p in P \
alpha_s &leq 0 quad forall s in S \
gamma_s &geq 0 quad forall s in S \
beta_s &leq 0 quad forall s in S \
frac{alpha_s+gamma_s}{u_t} + frac{beta_s}{u_v} &leq frac{t_{es}k_t}{u_t} + frac{t_{sz} * k_v}{u_v} quad forall e in E, s in S, t in T, z in Z, v in V \
delta_s &leq k_p t_s^p quad forall s in S, p in P, z in Z, e in E
end{align}
However, I am really unsure about the constraints. How do I know which variables should appear in which constraint and generally, what would be the correct formulation?
optimization linear-programming duality-theorems
asked Jan 13 at 14:11
user3578476user3578476
12
$endgroup$
add a comment |
$begingroup$
I am trying to use Bender's decomposition for a Location-Allocation Problem. The aim is to dualize the slave problem to generate the cuts for the Restricted Master Problem. Therefore, I have identified a slave problem and am trying to dualize it.
The problem consists of a two-stage distribution network. There are external zones E outside of a city. Satelites S are potential locations for opening a satelite within the city. Between external zones and satelites only trucks T are operating. Delivery from satelites to customer zones Z within the city is done using eco-friendly city-freighters V.
To give a bit of background I'll introduce the variables:
We have two decision variables in the Master problem:
begin{align*}f_{esz}^{pvt} &= text{flow of product } p in P text{ from external site } e in E text{ to satelite } s in S text{ using truck } t in T $ \
&text{ and delivering to customer zone } z in Z text{ with city freighter } v in V \
end{align*}
begin{align*}
y_s &= begin{cases} 1 & text{if satelite } s in S text{ is opened}, \
0 & text{otherwise.} end{cases} \
end{align*}
In the slave problem y is given so y becomes a normal variable.
$k_p t_s^p$ are the handling cost of product p per time unit multiplied times the handling time of product p at satelite s.
$t_{es}$ are the travel times from external site e to satelite s and $t_{sz}$ the travel times from s to customer zone z respectively.
$k_t$ are the operations cost for a truck t and $k_v$ the operations cost for a city freighter.
$u_t$ is the capacity of a truck and $u_v$ the cap. of a city freighter.
$g_{ez}^p$ is the demand of customer z for product p from external e
At satelite s there is a capacity of $u_s^T$ trucks and $u_s^V$ city freighters. If a satelite s is opened then at least $l_s^T$ trucks have to be used at the satelite.
The slave problem in the primal looks as follows:
begin{align*}
min Z left( f right) = &sum_{s in S} sum_{p in P} k_pt_s^p sum_{e in E}sum_{z in Z}sum_{t in T}sum_{vin V} f_{esz}^{pvt} + \
& sum_{p in P} left( sum_{e in E}sum_{s in S} t_{es}sum_{t in T} k_t sum_{z in Z} sum_{v in V} frac{f_{esz}^{pvt}}{u_t} + sum_{s in S}sum_{z in Z} t_{sz}sum_{v in V} k_v sum_{e in E}sum_{t in T} frac{f_{esz}^{pvt}}{u_v}right)
end{align*}
begin{align}
text{s.t.} quad &sum_{s in S}sum_{t in T}sum_{v in V} f_{esz}^{pvt} = g_{ez}^p quad forall e in E, z in Z, p in P \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} leq u_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} geq l_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_v} leq u_s^V y_s quad forall sin S \
& y in lbrace 0,1 rbrace quad forall s in S \
& f_{esz}^{pvt} geq 0 quad forall e in E, z in Z, s in S, p in P ,v in V, t in T
end{align}
My current version of the dualization looks like this:
begin{align*}
max Z(f) sum_{s in S} u_s^T y_s alpha_s + sum_{s in S} l_s^T y_s gamma_s + sum_{s in S} u_s^V y_s beta_s + sum_{e in E}sum_{z in Z}sum_{p in P} g_{ez}^p delta_{ez}^p
end{align*}
begin{align}
text{s.t.} quad delta_{ez}^p &in mathbb{R} quad forall ein E, z in Z, p in P \
alpha_s &leq 0 quad forall s in S \
gamma_s &geq 0 quad forall s in S \
beta_s &leq 0 quad forall s in S \
frac{alpha_s+gamma_s}{u_t} + frac{beta_s}{u_v} &leq frac{t_{es}k_t}{u_t} + frac{t_{sz} * k_v}{u_v} quad forall e in E, s in S, t in T, z in Z, v in V \
delta_s &leq k_p t_s^p quad forall s in S, p in P, z in Z, e in E
end{align}
However, I am really unsure about the constraints. How do I know which variables should appear in which constraint and generally, what would be the correct formulation?
optimization linear-programming duality-theorems
asked Jan 13 at 14:11
user3578476user3578476
12
$endgroup$
I am trying to use Bender's decomposition for a Location-Allocation Problem. The aim is to dualize the slave problem to generate the cuts for the Restricted Master Problem. Therefore, I have identified a slave problem and am trying to dualize it.
The problem consists of a two-stage distribution network. There are external zones E outside of a city. Satelites S are potential locations for opening a satelite within the city. Between external zones and satelites only trucks T are operating. Delivery from satelites to customer zones Z within the city is done using eco-friendly city-freighters V.
To give a bit of background I'll introduce the variables:
We have two decision variables in the Master problem:
begin{align*}f_{esz}^{pvt} &= text{flow of product } p in P text{ from external site } e in E text{ to satelite } s in S text{ using truck } t in T $ \
&text{ and delivering to customer zone } z in Z text{ with city freighter } v in V \
end{align*}
begin{align*}
y_s &= begin{cases} 1 & text{if satelite } s in S text{ is opened}, \
0 & text{otherwise.} end{cases} \
end{align*}
In the slave problem y is given so y becomes a normal variable.
$k_p t_s^p$ are the handling cost of product p per time unit multiplied times the handling time of product p at satelite s.
$t_{es}$ are the travel times from external site e to satelite s and $t_{sz}$ the travel times from s to customer zone z respectively.
$k_t$ are the operations cost for a truck t and $k_v$ the operations cost for a city freighter.
$u_t$ is the capacity of a truck and $u_v$ the cap. of a city freighter.
$g_{ez}^p$ is the demand of customer z for product p from external e
At satelite s there is a capacity of $u_s^T$ trucks and $u_s^V$ city freighters. If a satelite s is opened then at least $l_s^T$ trucks have to be used at the satelite.
The slave problem in the primal looks as follows:
begin{align*}
min Z left( f right) = &sum_{s in S} sum_{p in P} k_pt_s^p sum_{e in E}sum_{z in Z}sum_{t in T}sum_{vin V} f_{esz}^{pvt} + \
& sum_{p in P} left( sum_{e in E}sum_{s in S} t_{es}sum_{t in T} k_t sum_{z in Z} sum_{v in V} frac{f_{esz}^{pvt}}{u_t} + sum_{s in S}sum_{z in Z} t_{sz}sum_{v in V} k_v sum_{e in E}sum_{t in T} frac{f_{esz}^{pvt}}{u_v}right)
end{align*}
begin{align}
text{s.t.} quad &sum_{s in S}sum_{t in T}sum_{v in V} f_{esz}^{pvt} = g_{ez}^p quad forall e in E, z in Z, p in P \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} leq u_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_t} geq l_s^T y_s quad forall sin S \
& sum_{e in E}sum_{z in Z}sum_{p in P}sum_{t in T}sum_{v in V} frac{f_{esz}^{pvt}}{u_v} leq u_s^V y_s quad forall sin S \
& y in lbrace 0,1 rbrace quad forall s in S \
& f_{esz}^{pvt} geq 0 quad forall e in E, z in Z, s in S, p in P ,v in V, t in T
end{align}
My current version of the dualization looks like this:
begin{align*}
max Z(f) sum_{s in S} u_s^T y_s alpha_s + sum_{s in S} l_s^T y_s gamma_s + sum_{s in S} u_s^V y_s beta_s + sum_{e in E}sum_{z in Z}sum_{p in P} g_{ez}^p delta_{ez}^p
end{align*}
begin{align}
text{s.t.} quad delta_{ez}^p &in mathbb{R} quad forall ein E, z in Z, p in P \
alpha_s &leq 0 quad forall s in S \
gamma_s &geq 0 quad forall s in S \
beta_s &leq 0 quad forall s in S \
frac{alpha_s+gamma_s}{u_t} + frac{beta_s}{u_v} &leq frac{t_{es}k_t}{u_t} + frac{t_{sz} * k_v}{u_v} quad forall e in E, s in S, t in T, z in Z, v in V \
delta_s &leq k_p t_s^p quad forall s in S, p in P, z in Z, e in E
end{align}
However, I am really unsure about the constraints. How do I know which variables should appear in which constraint and generally, what would be the correct formulation?
optimization linear-programming duality-theorems
optimization linear-programming duality-theorems
asked Jan 13 at 14:11
user3578476user3578476
12
asked Jan 13 at 14:11
user3578476user3578476
12
edited Jan 13 at 14:17
user3578476
asked Jan 13 at 14:11
user3578476user3578476
12
asked Jan 13 at 14:11
user3578476user3578476
12
asked Jan 13 at 14:11
user3578476user3578476
12
12
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