Respuesta :
(a) **LP Formulation**
**Decision Variables:**
- \(x_i = 1\) if researcher \(i\) is selected to attend the meeting, 0 otherwise.
**Objective Function:**
- Minimize \(z = \sum_{i=1}^n c_ix_i\), where \(c_i\) is the cost of sending researcher \(i\) to the meeting.
**Constraints:**
- Each group must be represented by at least one member:
- \(x_i \ge 1\) for all \(i\) such that researcher \(i\) is in group \(j\).
- The attendance at the meeting must be as small as possible:
- \(\sum_{i=1}^n x_i \le k\), where \(k\) is the maximum number of attendees allowed.
(b) **Dual of the LP**
The dual of the LP is:
**Decision Variables:**
- \(y_j \ge 0\) for all groups \(j\).
**Objective Function:**
- Maximize \(w = \sum_{j=1}^m y_j\), where \(m\) is the number of groups.
**Constraints:**
- The cost of sending each researcher to the meeting must be at least as great as the sum of the benefits of representing each group that the researcher is a member of:
- \(c_i \ge \sum_{j=1}^m y_j\) for all researchers \(i\).
(c) **Meaning of the Dual**
The dual of the LP models the situation in which the department is trying to find the minimum cost of sending a set of researchers to the meeting such that each group is represented by at least one member. The decision variables \(y_j\) represent the benefits of representing group \(j\), and the objective function maximizes the total benefit of the meeting.
(d) **Complementary Slackness**
Complementary slackness states that for any optimal solution to the primal and dual problems, the following conditions hold:
- If \(x_i = 1\), then \(c_i = \sum_{j=1}^m y_j\).
- If \(y_j > 0\), then there exists at least one researcher \(i\) such that \(x_i = 1\) and researcher \(i\) is in group \(j\).
In this case,
**Decision Variables:**
- \(x_i = 1\) if researcher \(i\) is selected to attend the meeting, 0 otherwise.
**Objective Function:**
- Minimize \(z = \sum_{i=1}^n c_ix_i\), where \(c_i\) is the cost of sending researcher \(i\) to the meeting.
**Constraints:**
- Each group must be represented by at least one member:
- \(x_i \ge 1\) for all \(i\) such that researcher \(i\) is in group \(j\).
- The attendance at the meeting must be as small as possible:
- \(\sum_{i=1}^n x_i \le k\), where \(k\) is the maximum number of attendees allowed.
(b) **Dual of the LP**
The dual of the LP is:
**Decision Variables:**
- \(y_j \ge 0\) for all groups \(j\).
**Objective Function:**
- Maximize \(w = \sum_{j=1}^m y_j\), where \(m\) is the number of groups.
**Constraints:**
- The cost of sending each researcher to the meeting must be at least as great as the sum of the benefits of representing each group that the researcher is a member of:
- \(c_i \ge \sum_{j=1}^m y_j\) for all researchers \(i\).
(c) **Meaning of the Dual**
The dual of the LP models the situation in which the department is trying to find the minimum cost of sending a set of researchers to the meeting such that each group is represented by at least one member. The decision variables \(y_j\) represent the benefits of representing group \(j\), and the objective function maximizes the total benefit of the meeting.
(d) **Complementary Slackness**
Complementary slackness states that for any optimal solution to the primal and dual problems, the following conditions hold:
- If \(x_i = 1\), then \(c_i = \sum_{j=1}^m y_j\).
- If \(y_j > 0\), then there exists at least one researcher \(i\) such that \(x_i = 1\) and researcher \(i\) is in group \(j\).
In this case,
