1) A company makes two products (X and Y) using two machines (A and B). Each unit
of X that is produced requires 50 minutes processing time on machine A and 30
minutes processing time on machine B. Each unit of Y that is produced requires 24
minutes processing time on machine A and 33 minutes processing time on machine B.
At the start of the current week there are 30 units of X and 90 units of Y in stock. Available
processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35
hours.
The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95
units. Company policy is to maximise the combined sum of the units of X and the units of Y
in stock at the end of the week.
2) A company is involved in the production of two items (X and Y). The resources need
to produce X and Y are twofold, namely machine time for automatic processing and
craftsman time for hand finishing. The table below gives the number of minutes
required for each item:
Item X
Y
Machine time Craftsman time
13
20
19
29
The company has 40 hours of machine time available in the next working week but only 35
hours of craftsman time. Machine time is costed at £10 per hour worked and craftsman time is
costed at £2 per hour worked. Both machine and craftsman idle times incur no costs. The
revenue received for each item produced (all production is sold) is £20 for X and £30 for Y.
The company has a specific contract to produce 10 items of X per week for a particular
customer.
3) Blending problem
Consider the example of a manufacturer of animal feed who is producing feed mix for dairy
cattle. In our simple example the feed mix contains two active ingredients and a filler to
provide bulk. One kg of feed mix must contain a minimum quantity of each of four nutrients
as below:
Nutrient
gram
A
90
B
50
C
20
D
2
The ingredients have the following nutrient values and cost
Ingredient 1 (gram/kg)
Ingredient 2 (gram/kg)
A
100
200
B
80
150
C
40
20
D
10
-
Cost/kg
40
60
What should be the amounts of active ingredients and filler in one kg of feed mix?
1) Capital budgeting
There are four possible projects, which each run for 3 years and have the following
characteristics.
Project
1
2
3
4
Available
Return (£m)
0.2
0.3
0.5
0.1
capital (£m)
Capital requirements (£m)
Year
1
2
3
0.5
0.3
0.2
1.0
0.8
0.2
1.5
1.5
0.3
0.1
0.4
0.1
3.1
2.5
0.4
We have a decision problem here: Which projects would you choose in order to
maximise the total return?
What if at least, exactly, one of the projects 1 or 2 has to be selected?
2) Blending problem extension:
if we use any of ingredient 2 we incur a fixed cost of 15
we need not satisfy all four nutrient constraints but need only satisfy three of them (i.e.
whereas before the optimal solution required all four nutrient constraints to be satisfied
now the optimal solution could (if it is worthwhile to do so) only have three (any three) of
these nutrient constraints satisfied and the fourth violated.
3) The assignment problem
The matrix below shows the time required (in minutes) for each combination of a worker
and a job. The jobs are denoted by J1, J2, J3, and J4, the workers by W1, W2, W3, and
W4.
W1
W2
W3
W4
J1
82
77
11
8
J2
83
37
69
9
J3
69
49
5
98
J4
92
92
86
23
Each worker should perform exactly one job and the objective is to minimize the total
time required to perform all jobs.