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 Xi = 1 if project i is funded, and 0 if project i is not funded. Dec. Var. Project Cost ($1000) Students Impacted Xi Hire seven new staff members 400 350 X Renovate the management library 350 1000 Fund five scholarships for the PhD program 50 5 X4 Renovate the student lounge 100 2500 Xs Hire four new faculty positions for research 500 900 Host an entrepreneur pitch competition 90 400 X Restore previous cuts to undergrad student clubs 220 5000 X Restore previous cuts to graduate student clubs 150 250 X Buy new computers for the school 140 1800 X Xs 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) (©) 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 (all costs in thousands)

This inequality ensure that budget is not crossed for project selection.

b. X2 + X4 = 1

This equation ensures that only 1 of 2 projects are selected. If X2 is 1, X4 cannot be 1 and vice versa.

c. X7 = X8

This equation ensure that either both projects are funded or both aren't. As if X7 is 1, X8 has to be 2 and be X7 is 0, X8 has to be 0.

d. X9>= X1, X3>= X1

This inequality ensure that X1 cannot be selected when X9 or X3 is not selected. X1 cannot be 1, if X3/X9 is 0.

However, X1 can be 1 if X3/X9 is 1.