Plant	District			Supply
	1	2	3
A	5	8	9	100
B	4	4	7	300
C	3	3	5	300
Demand	300	200	200

Problem 1: Logistics

Consider the following transportation problem. A single product is produced in three plants and is sold in 3 different districts. The following table summarizes the data for the problem:

The LP formulation to minimize the shipping costs can be written as follows:

Let A1, A2, A3, B1, B2, B3, C1, C2, C3 be number of units shipped from plants (A, B, C) to districts (1, 2, 3)

Minimize Total Costs = 5A1+ 8A2+ 9A3+ 4B1+ 4B2+ 7B3+ 3C1+ 3C2+ 5C3

Constraints:

A1 + A2+ A3<= 100

B1 + B2+ B3<= 300

C1 + C2+ C3<= 300

A1 + B1+ C1>= 300

A2 + B2+ C2>= 200

A3 + B3+ C3>= 200

A1, A2, A3, B1, B2, B3, C1, C2, C3 >= 0

(a) Assume that the fixed costs of $500000, $700000, and $450000 are associated with the use of plants 1, 2, and 3, respectively. For example, if any amount of material is shipped out of plant 1, then a fixed charge of $500000 is incurred. If nothing is shipped from plant 1, then there is no associated cost. How would the problem formulation change to account for this?

(b) Suppose that we wish to impose the constraint that at least two plant locations must be used. How would the problem formulation change to account for this?

(c) Assume there is a fixed overhead cost of $150 associated with the maintenance of each shipping route that arrive to district 3. For example, if any amount of material is shipped from plant 1 to district 3, the overhead cost is $150. How would the problem formulation change to account for this?

(d) Suppose that the route from plant 1 to district 3 can be used only if plant 2 is also used. What additional variables and/or constraints do we need? How would the problem formulation change to account for this?

Note: you do not need to solve the problem above just add the required variables and/or change the objective function/constraints as required.