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1. The following table gives the cost of transporting one tonne of goods from the origins A, B, C to the destinations F, G, H. Also shown are the availabilities of the goods at the origins and the requirements at the destinations.

The transportation problem implied by this table can also be written in the form

$$\text{minimize} \: \: \underline{c} \: ^T \: \underline{x}$$

$$\text{subject to :} \: \: Ax= \underline{b}$$

$$\underline{x} \geq 0$$

Display $\underline{c} \: ^T , A$ and $\underline{b}$ if $\underline{x}$ is the vector

(XAF, XAG, XAH, XAH, XBF, XBG, XBH, XCF, XCG, XCH)

Where $x_{ij}$ represents the shipment from $i$ to $j$.
2. Given that XAG, XBH, XCF, XC are the variable in the basis, solve for the values of these variables in the above question(i).

3. For the solution of (ii) above, calculate the values of the duals and determine whether this is an optimal solution.

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