Transportation problem: stepping-stone method

3. The table below shows the cost of transporting one block of staging from each of two supply points, X and Y , to each of four concert venues, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . It also shows the number of blocks held at each supply point and the number of blocks required at each concert venue. A minimal cost solution is required.

1. Use the north-west corner method to obtain a possible solution.
   (1)
2. Taking the most negative improvement index to indicate the entering square, use the stepping stone method twice to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, routes, entering cells and exiting cells.
3. Is your current solution optimal? Give a reason for your answer.
   (1)

2. The table below shows the cost of transporting one unit of stock from each of four supply points, 1 , 2, 3 and 4, to each of three demand points, \(\mathrm { A } , \mathrm { B }\) and C . It also shows the stock held at each supply point and the stock required at each demand point. A minimal cost solution is required.

1. Add a dummy demand point and appropriate values to Table 1 in the answer book. Table 2 shows an initial solution given by the north-west corner method.
   Table 3 shows some of the improvement indices for this solution. \begin{table}[h]

   	A	B	C	D
   1	20
   2	15	7
   3		18	2
   4			28	10

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table} \begin{table}[h]

   	A	B	C	D
   1		- 13		- 9
   2			- 11
   3
   4	1	- 7

   \captionsetup{labelformat=empty} \caption{Table 3}
   \end{table}
2. Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in the answer book.
3. Taking the most negative improvement index to indicate the entering square, use the steppingstone method once to obtain an improved solution. You must make your route clear and state your entering cell and exiting cell.

3. The table below shows the cost, in pounds, of transporting one tonne of concrete from each of three supply depots, \(\mathrm { A } , \mathrm { B }\) and C , to each of four building sites, \(\mathrm { D } , \mathrm { E } , \mathrm { F }\) and G . It also shows the number of tonnes that can be supplied from each depot and the number of tonnes required at each building site. A minimum cost solution is required. The north-west corner method gives the following possible solution. Taking AG as the first entering cell,

1. use the stepping stone method twice to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, routes, entering cells and exiting cells.
2. Determine whether your current solution is optimal. Justify your answer.

7. Mrs. Hartley organises the tennis fixtures for her school. On one day she has to send a team of 10 players to a match against school \(A\) and a team of 6 players to a match against school \(B\). She has to select the two teams from a squad that includes 7 players who live in village \(C\), 5 players who live in village \(D\) and 8 players who live in village \(E\). Having a small budget, Mrs. Hartley wishes to minimise the total amount spent on travel. The table below shows the cost, in pounds, for one player to travel from each village to each of the schools they are competing against.

1. Use the north-west corner rule to find an initial solution to this problem.
2. Obtain improvement indices for this initial solution.
3. Use the stepping-stone method to obtain an optimal solution and state the pattern of transportation that this represents. \section*{Please hand this sheet in for marking}

   Stage	State	Action
   \multirow[t]{2}{*}{1}	G	GI
   	H	HI
   \multirow[t]{3}{*}{2}	D	DG DH
   	E	EG \(E H\)
   	F	FG FH
   \multirow[t]{3}{*}{3}	A	AD \(A E\) \(A F\)
   	B	BD BE \(B F\)
   	C	CD CE CF
   4	Home	Home-A Home-B Home-C

   \section*{Please hand this sheet in for marking}
   1. \includegraphics[max width=\textwidth, alt={}, center]{4e50371b-0c1c-4b4e-b21d-60858ae160df-8_662_1025_529_440}
   2. Sheet for answering question 6 (cont.)

4. A furniture manufacturer has three workshops, \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\). Orders for rolls of fabric are to be placed with three suppliers, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). The supply, demand and cost per roll in pounds, according to which supplier each workshop uses, are given in the table below. Starting with the north-west corner method of finding an initial solution, find an optimal transportation pattern which minimises the total cost. State the final solution and its total cost.
(11 marks)

5. A car-hire firm has six branches in a region. Three of the branches, \(A , B\) and \(C\), have spare cars, whereas the other three, \(D , E\) and \(F\), require cars. The total number of cars required is equal to the number of cars available. The table below shows the cost in pounds of sending one car from each branch with spares to each branch needing more cars and the number of cars available or required by each branch.

1. Use the north-west corner method to obtain a possible pattern of moving cars and find its cost. The firm wishes to minimise the cost of redistributing the cars.
2. Calculate shadow costs for the pattern found in part (a) and improvement indices for each unoccupied cell.
3. State, with a reason, whether or not the pattern found in part (a) is optimal.

5. A carpet manufacturer has two warehouses, \(W _ { 1 }\) and \(W _ { 2 }\), which supply carpets for three sales outlets, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). At one point \(S _ { 1 }\) requires 40 rolls of carpet, \(S _ { 2 }\) requires 23 rolls of carpet and \(S _ { 3 }\) requires 37 rolls of carpet. At this point \(W _ { 1 }\) has 45 rolls in stock and \(W _ { 2 }\) has 40 rolls in stock. The following table shows the cost, in pounds, of transporting one roll from each warehouse to each sales outlet: The company's manager wishes to supply the 85 rolls that are in stock such that transportation costs are kept to a minimum.

1. Use the north-west corner rule to obtain an initial solution to the problem.
2. Calculate improvement indices for the unused routes.
3. Use the stepping-stone method to obtain an optimal solution.

3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H , to three sales points, \(\mathrm { A } , \mathrm { B }\) and C . It also shows the stock held at each supply point and the amount required at each sales point.
A minimum cost solution is required.

1. Explain why it is necessary to add a dummy demand point.
2. On Table 1 in the answer book, insert appropriate values in the dummy demand column, D. After finding an initial feasible solution and applying one iteration of the stepping-stone method, the table becomes

   	\(A\)	\(B\)	\(C\)	\(D\)
   \(E\)	21
   \(F\)	19	13
   \(G\)		6	23
   \(H\)	5			18

3. Starting with GD as the next entering cell, perform two further iterations of the stepping-stone method to obtain an improved solution. You must make your method clear by showing your routes and stating the

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