A construction company has undertaken three projects, one each in City A, City B and City C. Each project requires a specific supply of raw material which the firm sources from the Cities D, E, F, and G. Shipping costs (in rupees) differ from location to location. The following table summarizes the problem the manager of the construction firm faces:

The contractor learns that due to some difficulties, currently no shipments are possible between the cities D and A, and between F and C.

a) Obtain the optimal transportation plan for the contractor. Calculate the total cost. At what location is some supply going to be short? By how much?

b) Is the solution in (a) is unique. If not, identify another optimal solution.

c) Write the dual of this problem and use the values of the dual variables to check the optimal solution.

a) Obtaining the optimal transportation plan and calculating the total cost:

The transportation problem can be solved using the transportation method in operations research. The objective is to minimize the total transportation cost.

Let's set up the initial transportation table:

```
   From\To     A    B    C    Supply
   D          0    0    0      750
   E          0    0    0     1200
   F          0    0    0      900
   G          0    0    0       
   Demand   1800 1500 1650
```

Now, allocate shipments in the cells while respecting the supply and demand constraints and considering the restrictions between D and A, and between F and C.

1. For D:
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