Question

At its meeting next month, the Dean at the Telfer School of Management will be making plans to allocate funds for the coming school year. Nine projects are being considered for the coming school year. The estimated cost of each project and the estimated number of students the project would benefit are summarized in the table below. X1, X2, ..., X, are binary decision variables, where X; = 1 if project i is funded, and 0 if project i is not funded. Dec. Var. X X2 X X4 XS X6 X X8 X9 Project Hire seven new staff members Renovate the management library Fund five scholarships for the PhD program Renovate the student lounge Hire four new faculty positions for research Host an entrepreneur pitch competition Restore previous cuts to undergrad student clubs Restore previous cuts to graduate student clubs Buy new computers for the school Cost ($1000) 400 350 50 100 500 90 220 150 140 Students Impacted 350 1000 5 2500 900 400 5000 250 1800 Write down the corresponding constraints for the following conditions: (a) The budget remaining this year is $900,000. (2 points) (b) Recognizing that students will need space to study, the school has decided that only one of the renovation projects can be funded at this time. Thus, either the management library is renovated, or the student lounge is renovated but not both. (2 points) (c) In order to avoid conflict between the student clubs over who gets money and who does not, the Dean has decided that if the undergrad student clubs receive funding, then graduate student clubs must also be funded and vice versa. (3 points) (d) As a priority for the school, new computers and the five PhD scholarships must be funded before new staff members can be hired. This does not imply that by funding the new computers and PhD scholarships, that new staff members will be hired, only that they can be hired. (3 points)

Answer 1

(a)

400 X₁ +350 X₂ + 50 X₃ + 100 X₄ + 500 X₅ + 90 X₆ + 220 X₇ + 150 X₈ + 140 X₉ <= 900

(b)

X2 + X4 <= 1

(c)

Here, X₇ and X₈ will either both be zero or both be one.

X₇ - X₈ = 0

(d)

Truth table:

X3	X9	X1	Condition	X3 + X9 - 2 X1
1	1	1	Allowed	0
1	0	1	Barred	-1
0	1	1	Barred	-1
0	0	1	Barred	-2
1	1	0	Allowed	2
1	0	0	Allowed	1
0	1	0	Allowed	1
0	0	0	Allowed	0

We have to keep the function X₃ + X₉ - 2 X₁ above -1 in order to prevent the barred condition from happening.

So, the required constraint is:

X₃ + X₉ - 2 X₁ >= 0