Set up a linear programming model for the following problem. Clearly define the decision variables and their units, and the mathematical expressions for the objective function and all the constraints. Clarity of presentation is important.

Streets A$,$ B and C of the Figure below all dump traffic onto Boulevard D$.$ There are three types of vehicles that use the roadways, and each type requires a different amount of time traversing the intersection. The objective is to time the light cycle so that the maximum number of vehicles per hour can enter Boulevard D$.$ Assume that Avenue D can accommodate as many vehicles as can be routed onto it$,$ but that only one of the streets A$,$ B or C can be flowing traffic at one time. Excess traffic on the converging streets will seek alternate routes. A traffic light cycle should not exceed $3$min, and each street should have a green light for at least $40\frac{s}{c}$ycle. Develop a mathematical model that could be used to determine the optimum timing for the light. Use the data reported.