Wayne the rancher needs to move 100 cattle from the winter pasture to the summer pasture. He would like to drive them along the most direct route possible, but the most direct route runs straight through the town of Tumbleweed. Fortunately, the Tumbleweed town council has offered Wayne a solution. They will allow him to drive cattle along two roads that traverse the town. In order for cattle to be driven along a road, that road must be temporarily closed. Wayne must reimburse the town for the costs of setting up roadblocks and safety measures along any roads he uses. He must pay $700 to close Road 1 and $500 to close Road 2. Additionally, if Wayne chooses to use both roads instead of just one, he must compensate the citizens of Tumbleweed an additional $400 for their inconvenience. Wayne must also hire cowboys to drive the cattle. Each cowboy costs $200. Wayne can hire up to five cowboys (total), and each cowboy can be assigned to drive cattle along one closed road. The marginal increase in the number of cattle that can be supervised per additional cowboy hired decreases with the number of cowboys already assigned to a particular road, as shown in Table 1. Table 1: Marginal increase in cattle supervised per cowboy added, by road. 1st Cowboy 2nd Cowboy 3rd Cowboy 4th Cowboy 5th Cowboy 30 25 20 15 100 Road 1 Road 2 26 19 14 12 9 For example, if Wayne closes both Road 1 and Road 2, and he assigns two cowboys to Road 1 and one cowboy to Road 2, he will be able to drive a total of 30+25+26-81 cattle. (Assume that all cattle must be moved in a single drive. For example, Wayne cannot hire a single cowboy on Road 1 and have that cowboy execute multiple drives of 30 cattle each.) a) Formulate an integer linear program that minimizes Wayne's total cost while ensuring that all cattle reach the summer pasture. Clearly indicate your decision variables, objective function and constraints. Hint: You will need to define a number of binary decision variables. One of the binary decision variables you will need is xr.c, which is equal to 1 if at least c cowboys are assigned to Road r, and 0 if strictly less than c cowboys are assigned to Road r. For our example above, we would have .x1.1= x1,2x2,1-1 and x13x1,4-X1.5X2.2-x23x2.4=X2,5-0. Wayne the rancher needs to move 100 cattle from the winter pasture to the summer pasture. He would like to drive them along the most direct route possible, but the most direct route runs straight through the town of Tumbleweed. Fortunately, the Tumbleweed town council has offered Wayne a solution. They will allow him to drive cattle along two roads that traverse the town. In order for cattle to be driven along a road, that road must be temporarily closed. Wayne must reimburse the town for the costs of setting up roadblocks and safety measures along any roads he uses. He must pay $700 to close Road 1 and $500 to close Road 2. Additionally, if Wayne chooses to use both roads instead of just one, he must compensate the citizens of Tumbleweed an additional $400 for their inconvenience. Wayne must also hire cowboys to drive the cattle. Each cowboy costs $200. Wayne can hire up to five cowboys (total), and each cowboy can be assigned to drive cattle along one closed road. The marginal increase in the number of cattle that can be supervised per additional cowboy hired decreases with the number of cowboys already assigned to a particular road, as shown in Table 1. Table 1: Marginal increase in cattle supervised per cowboy added, by road. 1st Cowboy 2nd Cowboy 3rd Cowboy 4th Cowboy 5th Cowboy 30 25 20 15 100 Road 1 Road 2 26 19 14 12 9 For example, if Wayne closes both Road 1 and Road 2, and he assigns two cowboys to Road 1 and one cowboy to Road 2, he will be able to drive a total of 30+25+26-81 cattle. (Assume that all cattle must be moved in a single drive. For example, Wayne cannot hire a single cowboy on Road 1 and have that cowboy execute multiple drives of 30 cattle each.) a) Formulate an integer linear program that minimizes Wayne's total cost while ensuring that all cattle reach the summer pasture. Clearly indicate your decision variables, objective function and constraints. Hint: You will need to define a number of binary decision variables. One of the binary decision variables you will need is xr.c, which is equal to 1 if at least c cowboys are assigned to Road r, and 0 if strictly less than c cowboys are assigned to Road r. For our example above, we would have .x1.1= x1,2x2,1-1 and x13x1,4-X1.5X2.2-x23x2.4=X2,5-0