You are considering forming a team with entrepreneurs for your startup company Driverless & Co. You plan to form a founding team of 5 members (except yourself). You have narrowed down a list of 10 candidates from your collaborators, friends, and colleagues. Table 1 lists, for each candidate, your subjective score to quantify their ability in "Technology", "Management", and "Team Spirit".

You aim to form the team that maximizes the average score of technology. But in addition, you need to comply with the following consideration:

• The average score of management across your team needs to be at least 6.0 and the average score of team spirit needs to be at least 7.0.
• You need at least 3 members who are in charge of technology development, and at least 1 member who are in charge of internal management. Candidates 1, 3, 4, 5, and 8 can take charge of the technology development. Candidates 2, 5, 7, and 10 can take charge of the internal management.
• You need to select at least 1 candidate from previous employees of Uber UTG. Candidates 3, 4, and 8 are previous employees of Uber.
• Due to earlier conflicts, Candidate 3 tell you that he would not join the team if Candidate 5 or Candidate 7 is in the team.
• Candidate 4 and 8 are very good collaborators and they tell you that they can only join the team together.

Table 1. Scores for each candidate

Candidate	Technology	Management	Team Spirit
1	8	6	9
2	7	8	9
3	9	7	6
4	9	5	8
5	8	9	6
6	7	6	9
7	7	8	6
8	8	5	8
9	7	7	7
10	6	9	8

1. Please formulate the problem of team formation as an integer programming model.
2. Make a spreadsheet model for this problem, and report the optimal solution. What is the selected team? What is the average score of technology across all selected candidates?
3. Even though there is no direct way of performing sensitivity analyses in integer programming, you still want to test the impact of the requirement of minimum values of the average scores of management and team spirit. Vary these numbers and re-run your model a few times. Report the average score of technology across all selected candidates as a function of these two values.

In many cases, a model is not only used to select the one absolute "best" solution, but to generate a small number of "good" solutions which can then be discussed among decision-makers. You now aim to generate three solutions instead of a single one even though they may not all maximize the average score of technology. You revert here to the original case by ensuring that the average scores of management and team spirit exceed 6.0 and 7.0, respectively.

1. Propose a methodology to generate the second-best solution and the third-best solution. Implement it on your spreadsheet. What is the average score of technology across all selected candidates across the three solutions?