C&O-Cafe is considering to open several restaurants in Waterloo. In particular, C&O-Cafe considers n possible locations. Due to the city regulations, a restaurant at location i, i = 1, . . . , n can be of capacity at most ui persons. The cost of opening a restaurant at the location i, i = 1, . . . , n equals fi + ci t dollars, where fi is a fixed part of the cost and ci t is the part of the cost proportional to the restaurants capacity. (Annual costs of just running a restaurant of capacity t at location i equals ci t dollars). C&O-Cafe specializes in lunches and is planning to have the whole profit generated from employees of t companies in Waterloo. There are ej employees working at the company j, j = 1, . . . , t and they will have lunches at the restaurant built at the location `j {1, 2, . . . , n} as long as there are free spots at this restaurant (at noon). Note, that we assume that C&O-Cafe cannot prescribe customers which of their restaurants to visit. If C&O-Cafe serves lunch (to one person) during one year at a restaurant at the location i, i = 1, . . . , n, then C&O-Cafe gets a profit of pi dollars this year. (a) Construct an IP model to decide where to open restaurants and how large should be the capacity of these restaurants in order to maximize the total profit of C&O-Cafe in the first y years. (b) Describe all the constraints for your IP and justify them (including the relationships between the constraints and the objective function). (c) Explain whether your IP can be generalized to the case, when employees working at the company j, j = 1, . . . , t are ready to have lunches only at restaurants built in the locations Lj {1, 2, . . . , n} as long as there are free spots at these restaurants (at noon). In the case of a positive answer provide a generalization, in the case of a negative answer explain the limitations of your IP.

C&O-Cafe is considering to open several restaurants in Waterloo. In particular, C&O-Cafe considers n possible locations. Due to the city regulations, a restaurant at location i, i = 1,...,n can be of capacity at most ui persons. The cost of opening a restaurant at the location i, i = 1,...,n equals fi + Cit dollars, where fi is a fixed part of the cost and cit is the part of the cost proportional to the restaurant's capacity. (Annual costs of just running a restaurant of capacity t at location i equals ci t dollars). C&O-Cafe specializes in lunches and is planning to have the whole profit generated from employees of t companies in Waterloo. There are e; employees working at the company j, j = 1, ...,t and they will have lunches at the restaurant built at the location l; {1, 2, ..., n} as long as there are free spots at this restaurant (at noon). Note, that we assume that C&o-Cafe cannot prescribe customers which of their restaurants to visit. If C&O-Cafe serves lunch (to one person) during one year at a restaurant at the location i, i = 1,...,n, then C&O-Cafe gets a profit of p; dollars this year. (a) Construct an IP model to decide where to open restaurants and how large should be the capacity of these restaurants in order to maximize the total profit of C&O-Cafe in the first y years. (b) Describe all the constraints for your IP and justify them (including the relationships between the con- straints and the objective function). (c) Explain whether your IP can be generalized to the case, when employees working at the company j, j = 1, ..., t are ready to have lunches only at restaurants built in the locations L;