2. A company has three factories, \(\mathrm { A } , \mathrm { B }\) and C . It supplies mattresses to three shops, \(\mathrm { D } , \mathrm { E }\) and F . The table shows the transportation cost, in pounds, of moving one mattress from each factory to each shop. It also shows the number of mattresses available at each factory and the number of mattresses required at each shop. A minimum cost solution is required.

1. Use the north-west corner method to obtain an initial solution.
2. Show how the transportation algorithm is used to solve this problem. You must state, at each appropriate step, the
   • shadow costs,
   • improvement indices,
   • route,
   • entering cell and exiting cell,
     and explain clearly how you know that your final solution is optimal.