Week 7
Exercise 22 Solve the transportation problem having the following cost,
requirement, availability table. Start with your first solution coming from
the matrix method.
80
100
20
55 70 35 40
13 11 18 17
2 14 10 1
5 8 18 11
Exercise 23 It is required

Week 2
Exercise 5 Consider a linear programming problem given in standard form
(S). That is to say,
maximise z = c x
subject to:
Ax b
x 0n
where A Rmn , x Rn and b Rm . Write down the associated problem
in canonical form (C). Prove that if (C) has an opti

Week 9
Exercise 28 Find a maximal flow and a minimal cut for the following network. Show explicitly the flows in the individual arcs.
0
1
0 9 6 2 0 0 0
B 0 0 0 0 6 0 0 C
B
C
B 0 0 0 4 2 5 0 C
B
C
B 0 0 3 0 0 5 0 C
B
C
B 0 6 3 0 0 0 5 C
B
C
@ 0 0 5 5 0 0 1

Week 3
Exercise 8 A mining company produces 100 tons of red ore and 80 tons of
black ore each week. These can be treated in dierent ways to produce three
dierent alloys, Soft, Hard or Strong. To produce 1 ton of Soft alloy requires
5 tons of red ore and 3

Week 6
Exercise 19 Go back and do part (iii) of Exercise 17.
Exercise 20 Show without using the Simplex Method that
xT = (
5 5 27
, , )
26 2 26
is an optimal solution to the following linear programming problem:
maximise z = 9x1 + 14x2 + 7x3
subject to:
2

Week 8
Exercise 25 Suppose now in question 23 there is a reduction in transport
cost of 10 per machine taken from A for any machines above six. (This
is separate from and not in addition to the change described question 3.)
Use the same device as in quest

Week 4
Exercise 11 Solve the following problem by the two phase method.
maximise z = x1 + x2 2x3 + 2x4
subject to:
x1 x2 x3 2x4 2
x1 + x2 + x4 8
x1 + 2x2 x3 = 4
x1 , ., x4 0.
Exercise 12 A health food shop packages three types of snack foods; chewy,
crunc

Linear Programming Exercises
Week 1
Exercise 1 Consider the case of the Betta Machine Products Company described in the lecture notes.
(a) Use a graphical method to obtain the new optimal solution when the
selling price of product 2 changes to (i) 55 poun

Week 5
Exercise 15 Suppose that we consider the asymmetric dual to the primal
canonical linear programming problem
maximise z = c x
subject to:
Ax = b
x 0n
where A Rmn , b Rm and c Rn are given. Prove that the conclusion
of the Weak Duality Theorem is sti