An art gallery is considering installing a video camera security system to reduce its insurance premiums. A diagram of the seven display rooms with possible security camera locations is shown in the figure below: room 1 room 2 room 3 room 6 room 4 room 5 room 7 A security firm has proposed that two-way cameras be installed at some of the doorways of the rooms. Each camera has the ability to monitor the two rooms between which the camera is located. For example, if the camera in doorway is installed, rooms 1 and 2 are covered. The security firm objective is to provide coverage for all seven rooms. (a) Formulate a linear programming problem to achieve the firm's objective with the minimal number of security cameras. (10 marks) (b) Assume now that the costs of installing cameras at different doorways vary. Moreover, no more than one camera can be installed at doorways and and at least two cameras to be installed at doorways 0.8 and 9. Reformulate your model for the new settings of the problem Specify which extra data you would need to solve the problem (10 marks) (c) A night security guard whose main position is in room 1. wants to know whether it is possible to make a tour visiting all rooms without being in the same room twice. Reformulate this "night security guard problem into an instance of the well known problem in the graph theory, and advise on a solution [14 marks)