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You have been hired by your old summer camp to help them assign kids to cabins and weeks. Let K be the set of kids, C the set of cabins, and W the set of weeks. Each kid has to be assigned to a cabin and a week. There can be at most 8 kids to a cabin. Each kid ranks the weeks that they want to go to, where 1 is the value of their least-preferred choice. (a) Formulate a mathematical model (possibly with integer and/or binary variables) to solve this problem, maximizing the total satisfaction across all kids. (b) Suppose that N groups of kids, each denoted by G1, G2, ..., GN, reach out and request you to schedule them in the same week and in the same cabin. (G; is a subset of K for all i = 1,..., N, and suppose that different Gi's do not overlap.) Knowing that you may not be able to fulfill all such requests, you associate each failure-to-fulfill with a penalty in the amount of p. For example, suppose that G1 = {1,2,3} and you schedule kids 1 and 2 to the same week, same cabin but kid 3 to a different week, then you incur a penalty of 2p, for not scheduling kids (1,3) and (2,3) to the same week, same cabin. Revise your model to maximize the total satisfaction across all kids, minus the total penalty. Please note that your model needs to use a linear objective function and linear constraints