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. (A tip: Use Y, for "being selected"; use ( IY1) for "not being selected") (1) If project 1 is selected then project 4 must not be selected: (2) If project 1 is not selected, then project 4 must be selected: (3) If project 4 is selected then project 1 must not be selected: (4) If project 4 is not selected then project 1 must be selected: 8. "If project 1 is not selected, then project 3 is not selected (no restriction on project 3 if project 1 is selected)". 9. If either project 1 or 2 is selected, then project 4 must be selected. 10. If both project 1 and 2 are selected, then project 4 must be selected. 11. If project 1 is selected, then either project 3 or 4 must be selected. 12. If project 1 is selected, then both project 3 and 4 must be selected. For Problem 13 and 14 : Let X1 be an integral variable (not binary!). Let Y1 be a binary variable, i.e., Y1 can be 0 or 1 only. 13. Write a constraint to represent "If X1=0 then Y1=0, and if X1>0 then Y1 can be either 0 or 1".. 14. 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 13 and 14 together indicate such a relationship between variables X1 and Yt : 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