8. (One dimensional Crabs Business) Two people have decided to go into business together selling crabs that they catch along the Baltimore coast line After they catch the crabs, they bring them back to their processing plant where they are cleaned, clawed, and wrapped to go. Following this process, they ship the crabs to n different seafood restaurants located in a 10 mile line segment. Each restaurant i has two coordinates a; and demands d; crabs. We will measure the distance between the processing plant and the restaurant using the abso- lute value. The unit transportation cost of shipping a crab is $c per mile (so it costs $mc per mile to ship $m crabs). Their goal is to locate the processing plant at a point x inside the line segment to minimize the total transportation costs. (a) Formulate the problem of determining where the processing plant should be located as a mathematical program to meet all demands while minimizing the transportation costs. (Your answer to this part does not need to be a linear program.) (Hint: use absolute value.) (b) Convert the mathematical program of part (a) to a linear programming problem. (c) It is claimed that there is always an optimal solution to the linear programming problem in which the processing plant is located at one of the restaurants. Prove or disprove this claim. 8. (One dimensional Crabs Business) Two people have decided to go into business together selling crabs that they catch along the Baltimore coast line After they catch the crabs, they bring them back to their processing plant where they are cleaned, clawed, and wrapped to go. Following this process, they ship the crabs to n different seafood restaurants located in a 10 mile line segment. Each restaurant i has two coordinates a; and demands d; crabs. We will measure the distance between the processing plant and the restaurant using the abso- lute value. The unit transportation cost of shipping a crab is $c per mile (so it costs $mc per mile to ship $m crabs). Their goal is to locate the processing plant at a point x inside the line segment to minimize the total transportation costs. (a) Formulate the problem of determining where the processing plant should be located as a mathematical program to meet all demands while minimizing the transportation costs. (Your answer to this part does not need to be a linear program.) (Hint: use absolute value.) (b) Convert the mathematical program of part (a) to a linear programming problem. (c) It is claimed that there is always an optimal solution to the linear programming problem in which the processing plant is located at one of the restaurants. Prove or disprove this claim