7. A distributor has six warehouses. At one point the distributor needs to move 25 lorries from warehouses \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\) to warehouses \(W _ { \mathrm { A } } , W _ { \mathrm { B } }\) and \(W _ { \mathrm { C } }\) for the minimum possible cost. The transportation tableau below shows the unit cost, in tens of pounds, of moving a lorry between two warehouses, and the relevant figures regarding the number of lorries available or required at each warehouse.

1. Write down the initial solution given by the north-west corner rule.
2. Obtain improvement indices for the unused routes.
3. Use the stepping-stone method to find an improved solution and state why it is degenerate.
4. Placing a zero in cell \(( 2,2 )\), show that the improved solution is optimal and state the transportation pattern.
5. Find the total cost of the optimal solution. \section*{Please hand this sheet in for marking}

   Stage	State	Destination	Cost	Total cost
   \multirow[t]{3}{*}{1}	Marquee	Deluxe Cuisine
   	Castle	Deluxe Castle Cuisine
   	Hotel	Deluxe Cuisine Hotel
   \multirow[t]{3}{*}{2}	Church	Marquee Castle Hotel
   	Castle	Marquee Castle
   	Registry Office	Marquee Castle Hotel
   3	Home	Castle Church Registry

   \section*{Please hand this sheet in for marking}

   1. 	A	B	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
      A	-	85	59	31	47	52	74	41
      B	85	-	104	73	51	68	43	55
      C	59	104	-	54	62	88	61	45
      D	31	73	54	-	40	59	65	78
      E	47	51	62	40	-	56	71	68
      \(F\)	52	68	88	59	56	-	53	49
      \(G\)	74	43	61	65	71	53	-	63
      \(H\)	41	55	45	78	68	49	63	-

   2. 	A	\(B\)	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
      A	-	85	59	31	47	52	74	41
      B	85	-	104	73	51	68	43	55
      C	59	104	-	54	62	88	61	45
      D	31	73	54	-	40	59	65	78
      E	47	51	62	40	-	56	71	68
      \(F\)	52	68	88	59	56	-	53	49
      G	74	43	61	65	71	53	-	63
      \(H\)	41	55	45	78	68	49	63	-