D2 June 2013 (R) Q3

EdexcelOld spec8 marksTransportation Problems

3. Table 1 below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to four demand points 1, 2, 3 and 4. It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.

	1	2	3	4	Supply
A	22	36	19	37	35
B	29	35	30	36	15
C	24	32	25	41	20
D	23	30	23	38	30
Demand	30	20	30	20

Table 1

Table 2 shows an initial solution given by the north-west corner method.

Table 3 shows some of the improvement indices for this solution.

	1	2	3	4
A	30	5
B		15	0
C			20
D			10	20

Table 2

	1	2	3	4
A	x	x
B		x	x
C	8	2	x	1
D	9	2	x	x

Table 3

(a) Explain why a zero has been placed in cell B3 in Table 2. (1)

(b) Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in your answer book. (4)

(c) Taking the most negative improvement index to indicate the entering cell, state the stepping-stone route that should be used to obtain the next solution. You must state your entering cell and exiting cell. (3)

Scheme	Marks
The solution would otherwise be degenerate	B1
	(1)

Mark scheme (b)

Scheme

Marks

		22	36	31	46
		1	2	3	4
0	A	x	x	−12	−9
−1	B	8	x	x	−9
−6	C	(8)	(2)	x	(1)
−8	D	(9)	(2)	x	x

M1 A1
M1 A1

(4)

Notes

b1M1: 8 shadow costs stated.

b1A1: CAO

b2M1: Remaining 4 IIs stated.

b2A1: CAO

(b) Alternative shadow costs:

1(0) 2(14) 3(9) 4(24)
A(22) B(21) C(16) D(14)

Mark scheme (c)

Scheme	Marks
Route is e.g. A3 – B3 – B2 – A2	M1 A1
entering cell A3, Exiting cell B3	A1
	(3)
	8 marks

Notes

c1M1: A valid route (possibly drawn), their most negative II chosen, only one empty square used, \(\theta\)’s balance.

c1A1: CAO – stepping stone route stated or clearly shown on separate diagrams

c2A1: CAO for entering and exiting cells.