Linear Programming, logical constraints to linear contraint in a Linear Program

2. A factory produces x1 widgets, x2 gizmos, and x3 doodads in a way that maximizes profit according to some constraints that we don't need to know about for this problem. Let's just assume that from these constraints, it follows that at most 1000 total items can be produced. Now, we want to add some additional constraints that require integer variables to formulate. (a) Let y1,y2,y3 be binary variables: y1,y2,y3Z with 0y1,y2,y31. Formulate constraints that, for each i, force us to set yi=1 if xi>0. (b) Using the variables defined in part (a), formulate the constraint that widgets and gizmos are mutually exclusive: if the factory produces any gizmos, then it can't produce any widgets, and vice versa. (c) Using the variables defined in part (a), formulate the constraint that if the factory produces any doodads, then it can produce at most 100 widgets. 2. A factory produces x1 widgets, x2 gizmos, and x3 doodads in a way that maximizes profit according to some constraints that we don't need to know about for this problem. Let's just assume that from these constraints, it follows that at most 1000 total items can be produced. Now, we want to add some additional constraints that require integer variables to formulate. (a) Let y1,y2,y3 be binary variables: y1,y2,y3Z with 0y1,y2,y31. Formulate constraints that, for each i, force us to set yi=1 if xi>0. (b) Using the variables defined in part (a), formulate the constraint that widgets and gizmos are mutually exclusive: if the factory produces any gizmos, then it can't produce any widgets, and vice versa. (c) Using the variables defined in part (a), formulate the constraint that if the factory produces any doodads, then it can produce at most 100 widgets