You are hired by the City of Montreal to optimize the performance of two signalized intersections...

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You are hired by the City of Montreal to optimize the performance of two signalized intersections in the city. You have determined the traffic flows (passenger cars-pc per hour) in different directions, as shown the following figure: 3000 3000 2000 2000 2000 2000 The assumptions you made are as follows: (i) drivers start passing as soon as the signal becomes green (i.e., no reaction time gap); (ii) each cycle (pre-timed) has only two lights: green and red (i.e., no yellow); the green time for the north/south bound equals to the red time for the west/east bound and vice versa; and (iii) the red time for each direction should be shorter than 2 minutes and longer than 0.5 minutes; and (iv) the green light for the straight, right and left turns are the same, i.e., each traffic signal has only two phases, e.g., for Intersection 1, when the light is green for the west/east bound, both vehicles going east (2000 passenger cars per hour) and vehicles going west (1000 1000 1000 passenger cars per hour) could pass the intersection, no matter if they turn left or right, or continue straight. In addition, you know the average delay function for each direction i (Webster's formula for a uniform delay) as follows-the function is the same for both intersections: di = c²-g₁² 2cx(1-4000) where, x₁ = Traffic volume entering the intersection from direction i (west, east, north, or south bound), in vehicle (passenger car) per hour. g₁= Green time of the traffic light-direction i, in minutes. c = Cycle length (both green and red) of the traffic light, in minutes. di-Average uniform delay at the intersection-direction i, in minutes (for each vehicle). (a) Formulate the problem as an LP: What can be a proper objective function, based on the problem description? How many decision variables do you need (at most) and what could those be? (b) If you would like to modify/improve the formulation and use more realistic assumptions, what could you do differently? With your formulation in (a), could you analyze/optimize the intersections together (think about how they are connected to each other)? If not, how could you link them? You are hired by the City of Montreal to optimize the performance of two signalized intersections in the city. You have determined the traffic flows (passenger cars-pc per hour) in different directions, as shown the following figure: 3000 3000 2000 2000 2000 2000 The assumptions you made are as follows: (i) drivers start passing as soon as the signal becomes green (i.e., no reaction time gap); (ii) each cycle (pre-timed) has only two lights: green and red (i.e., no yellow); the green time for the north/south bound equals to the red time for the west/east bound and vice versa; and (iii) the red time for each direction should be shorter than 2 minutes and longer than 0.5 minutes; and (iv) the green light for the straight, right and left turns are the same, i.e., each traffic signal has only two phases, e.g., for Intersection 1, when the light is green for the west/east bound, both vehicles going east (2000 passenger cars per hour) and vehicles going west (1000 1000 1000 passenger cars per hour) could pass the intersection, no matter if they turn left or right, or continue straight. In addition, you know the average delay function for each direction i (Webster's formula for a uniform delay) as follows-the function is the same for both intersections: di = c²-g₁² 2cx(1-4000) where, x₁ = Traffic volume entering the intersection from direction i (west, east, north, or south bound), in vehicle (passenger car) per hour. g₁= Green time of the traffic light-direction i, in minutes. c = Cycle length (both green and red) of the traffic light, in minutes. di-Average uniform delay at the intersection-direction i, in minutes (for each vehicle). (a) Formulate the problem as an LP: What can be a proper objective function, based on the problem description? How many decision variables do you need (at most) and what could those be? (b) If you would like to modify/improve the formulation and use more realistic assumptions, what could you do differently? With your formulation in (a), could you analyze/optimize the intersections together (think about how they are connected to each other)? If not, how could you link them?