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I have an element e $\in E$ with $E$ the set containing all elements e and $e \in Y_i$ with $Y_i \subseteq E$. Each set $Y_i$ has different attributes.

$G_j$ is a set of sets and the following holds: $ Y_i\in\ G_j $ and $\cup_j G_j = E$

Example:

$e=5$ and sets $Y_2$={5,2,7}; $Y_{40}$={5,100,7}; ...; $Y_t$={300,400,2,5} are in $G_1$ so:

$ G_1=\{Y_2, Y_{40},..., Y_t\}$

1. In the end, I want to write a sum over a decision variable, that sums all variables found in the different sets for a particular element e.

This should look likes this -> $\sum\limits^{G_1}_{Y_i \ni e}x_e(Y_i)$ or $\sum\limits^{G_j}_{Y_i}x_{e,Y_i} \; \forall e \in E: e \in Y_i:Y_i \in G_j$

2. I want a sum over all elements within a set $Y_i$

This should look like this -> $\sum \limits^{Y_i}_e x_{e,Y_i} \; \forall\ Y_i \in G_j: e\in Y_i $

x is a binary decision variable.

With my knowledge, I do not see how a solver could work like this.

I could have used indices for the attributes found in the sets $Y_i$. This way I would avoid using $Y_i$ as an index. That though would probably add too many unnecessary decision variables that would be set to zero, since not for all elements exists these combinations of attributes.

Is there a way to formulate such a thing?

I have an element e $\in E$ with $E$ the set containing all elements e and $e \in Y_i$ with $Y_i \subseteq E$. Each set $Y_i$ has different attributes.

$G_j$ is a set of sets and the following holds: $ Y_i\in\ G_j $ and $\cup_j G_j = E$

Example:

$e=5$ and sets $Y_2$={5,2,7}; $Y_{40}$={5,100,7}; ...; $Y_t$={300,400,2,5} are in $G_1$ so:

$ G_1=\{Y_2, Y_{40},..., Y_t\}$

1. In the end, I want to write a sum over a decision variable, that sums all variables found in the different sets for a particular element e.

   This should look likes this -> $\sum\limits^{G_1}_{Y_i \ni e}x_e(Y_i)$ or $\sum\limits^{G_j}_{Y_i}x_{e,Y_i} \; \forall e \in E: e \in Y_i:Y_i \in G_j$

2. I want a sum over all elements within a set $Y_i$

   This should look like this -> $\sum \limits^{Y_i}_e x_{e,Y_i} \; \forall\ Y_i \in G_j: e\in Y_i $

x is a binary decision variable.

With my knowledge, I do not see how a solver could work like this.

I could have used indices for the attributes found in the sets $Y_i$. This way I would avoid using $Y_i$ as an index. That though would probably add too many unnecessary decision variables that would be set to zero, since not for all elements exists these combinations of attributes.

Is there a way to formulate such a thing?

I have an element e $\in E$ with $E$ the set containing all elements e and $e \in Y_i$ with $Y_i \subseteq E$. Each set $Y_i$ has different attributes.

$G_j$ is a set of sets and the following holds: $ Y_i\in\ G_j $ and $\cup_j G_j = E$

Example:

$e=5$ and sets $Y_2$={5,2,7}; $Y_{40}$={5,100,7}; ...; $Y_t$={300,400,2,5} are in $G_1$ so:

$ G_1=\{Y_2, Y_{40},..., Y_t\}$

1. In the end, I want to write a sum over a decision variable, that sums all variables found in the different sets for a particular element e.

This should look likes this -> $\sum\limits^{G_1}_{Y_i \ni e}x_e(Y_i)$ or $\sum\limits^{G_j}_{Y_i}x_{e,Y_i} \; \forall e \in E: e \in Y_i:Y_i \in G_j$

2. I want a sum over all elements within a set $Y_i$

This should look like this -> $\sum \limits^{Y_i}_e x_{e,Y_i} \; \forall\ Y_i \in G_j: e\in Y_i $

x is a binary decision variable.

With my knowledge, I do not see how a solver could work like this.

Is there a way to formulate such a thing?

I have an element e $\in E$ with $E$ the set containing all elements e and $e \in Y_i$ with $Y_i \subseteq E$. Each set $Y_i$ has different attributes.

$G_j$ is a set of sets and the following holds: $ Y_i\in\ G_j $ and $\cup_j G_j = E$

Example:

$e=5$ and sets $Y_2$={5,2,7}; $Y_{40}$={5,100,7}; ...; $Y_t$={300,400,2,5} are in $G_1$ so:

$ G_1=\{Y_2, Y_{40},..., Y_t\}$

1. In the end, I want to write a sum over a decision variable, that sums all variables found in the different sets for a particular element e.

This should look likes this -> $\sum\limits^{G_1}_{Y_i \ni e}x_e(Y_i)$ or $\sum\limits^{G_j}_{Y_i}x_{e,Y_i} \; \forall e \in E: e \in Y_i:Y_i \in G_j$

2. I want a sum over all elements within a set $Y_i$

This should look like this -> $\sum \limits^{Y_i}_e x_{e,Y_i} \; \forall\ Y_i \in G_j: e\in Y_i $

x is a binary decision variable.

With my knowledge, I do not see how a solver could work like this.

I could have used indices for the attributes found in the sets $Y_i$. This way I would avoid using $Y_i$ as an index. That though would probably add too many unnecessary decision variables that would be set to zero, since not for all elements exists these combinations of attributes.

Is there a way to formulate such a thing?

I have an element e $\in Y_i$. Each set $Y_i$ has different attributes.

$G_1$ is a set of sets and the following holds: $ Y_i\in\ G_1 $

Example:

$e=5$ and sets $Y_2$={5,2,7}; $Y_{40}$={5,100,7}; ...; $Y_t$={300,400,2,5} are in $G_1$ so:

$ G_1=\{Y_2, Y_{40},..., Y_t\}$

In the end, I want to write a sum over a decision variable, that sums all variables found in the different sets for a particular element e.

This should look likes this -> $\sum\limits^{G_1}_{Y_i \ni e}x_e(Y_i)$

x is a binary decision variable.

With my knowledge, I do not see how a solver could work like this.

I am formulating the problem like this, and avoid using the sets as indices in order to minimize the decision variables that will be set to zero. Of course, in this case, an additional parameter would be needed that would state that a combination is impossible (0 value) and possible (1).

Is there a way to formulate such a thing?

I have an element e $\in E$ with $E$ the set containing all elements e and $e \in Y_i$ with $Y_i \subseteq E$. Each set $Y_i$ has different attributes.

$G_j$ is a set of sets and the following holds: $ Y_i\in\ G_j $ and $\cup_j G_j = E$

Example:

$e=5$ and sets $Y_2$={5,2,7}; $Y_{40}$={5,100,7}; ...; $Y_t$={300,400,2,5} are in $G_1$ so:

$ G_1=\{Y_2, Y_{40},..., Y_t\}$

1. In the end, I want to write a sum over a decision variable, that sums all variables found in the different sets for a particular element e.

This should look likes this -> $\sum\limits^{G_1}_{Y_i \ni e}x_e(Y_i)$ or $\sum\limits^{G_j}_{Y_i}x_{e,Y_i} \; \forall e \in E: e \in Y_i:Y_i \in G_j$

2. I want a sum over all elements within a set $Y_i$

This should look like this -> $\sum \limits^{Y_i}_e x_{e,Y_i} \; \forall\ Y_i \in G_j: e\in Y_i $

x is a binary decision variable.

With my knowledge, I do not see how a solver could work like this.

Is there a way to formulate such a thing?