Exercises on 0-1 Variable Representations BSNS5230, J. Wang There are four projects. Four binary variables are defined as below: Yi = 1 if project i is selected, and Yi = 0 otherwise, for i = 1, 2, 3, 4. Represent the following conditions by using constraints with the binary variables: 1. “At least two of the four projects must be selected”. 2. “Among project 1, project 2 and project 4, exactly one is selected”. 3. “At most three projects of the four can be selected”. 4. “Project 2 and project 3 must go together. That is, it is not allowed to select one while deselect the other”. 5. “The four projects can not be all selected.”. 6. “If project 4 is selected, then project 2 must be selected; but if project 4 is not selected, then there is no restriction on project 2”. 7. “If project 1 is selected then project 4 must not be selected; and if project 1 is not selected, then project 4 must be selected; and if project 4 is selected then project 1 must not be selected, and if project 4 is not selected then project 1 must be selected. ” (Hint: Reword this first) 8. “If project 1 is not selected, then project 3 is not selected (no restriction on project 3 if project 1 is selected)”. (Can you reword it to facilitate formulation?) (Continued on the back) For Problem 9 and 10: Let X1 be an integer between 0 and 20 (not binary!). Let Y1 be a binary variable, i.e., Y1 can be 0 or 1 only. 9. Write a constraint to represent “If X1=0 then Y1=0, and if X1>0 then Y1 can be either 0 or 1”. 10. Write a constraint to represent “If X1>0 then Y1=1, and if X1=0 then Y1 can be either 0 or 1”. Note: The constraints in question 9 and 10 together indicate such a relationship between variables X1 and Y1: If X1 is zero, then Y1 must be zero; and if X1 is not zero, then Y1 must be 1. Such a relation is what we want to formulate in the “fixed charge problem”.