Programmed to pass
Ian Janes, CIMA course leader at Newport Business School, supplies and
explains the answers to the supplementary question asked in March 2011
Financial Management’s graphical linear programming feature.
Before attempting this question, please read the aforementioned article, and
download and read question 6 (RT) from the May 2010 P2 exam paper.
To answer the question posed, we should first isolate the effect of having
more direct labour hours available, as this is the binding resource constraint.
We see from graph 7 (below) that an expansion of hours will mean an
outward movement of that line. Hence point C will move. This will bring about
increased production of units of T, with units of R remaining the same. This
will mean greater total contribution.
With this in mind, I asked you to consider these questions as an alternative
application of the linear programming model in the original examination
answer:
•
calculate the shadow price of direct labour
•
assume that it is just more direct labour hours which become
available and there’s no further availability of the other three
scarce resources. State the upper limit of units and hours over
which the shadow price calculated above remains the same.
State and explain the shadow price of labour beyond that limit.
To review the main points of May 2010’s Q6:
RT produces two products from different quantities of the same resources
using a just-in-time (JIT) production system. The selling price and resource
requirements of each of the products are shown below:
Product
Unit selling price (USD)
Resources per unit:
Direct labour (USD8/hr)
Material A (USD3/kg)
Material B (USD7/litre)
Machine hours (USD10/hr)
R
130
T
160
3 hours
5 kg
2 litres
3 hours
5 hours
4 kg
1 litre
4 hours
Market research shows that the maximum demand for products R and T
during June 2010 is 500 units and 800 units respectively. This does not
include an order that RT has agreed with a commercial customer for the
supply of 250 units of R and 350 units of T.
At a recent meeting of the purchasing and production managers to discuss
the production plans of RT for June, the following resource restrictions for
June were identified:
Direct labour hours
Material A
Material B
Machine hours
7,500 hours
8,500 kg
3,000 litres
7,500 hours
It becomes apparent that the predicted resource restrictions were rather
optimistic and that their availability could be as much as 10% lower than their
original predictions.
It is not possible for RT to meet customer demand: there are three scarce
resources (direct material A, direct material B and machine hours), as shown
below:
Resource
Initial
10%
quantity reduction
Needed
for
contract
Direct
labour
Material A
Material B
Machine
hours
7,500
750
2,500
Resources Resources
available
needed for
maximum
demand
4,250
5,500
8,500
3,000
7,500
850
300
750
2,650
850
2,150
5,000
1,850
4,600
5,700
1,800
4,700
To illustrate how the numbers in the table have been calculated, look again at
the direct labour hours. Originally it was thought that 7,500 labour hours would
be available but the initial forecast has now been reduced by 10%.
The contract requires the following hours:
• R
250 units x 3 hours = 750 hours
• T
350 units x 5 hours = 1,750 hours
Total = 2,500 hours
This leaves 7,500 - 750 - 2,500 = 4,250 hours available. Maximum demand
requires (500 x 3) + (800 x 5) = 5,500 hours, and hence the direct labour is a
scarce resource.
Similar calculations show that material A and machine hours are also scarce.
If there is more than one scarce resource we cannot solve the problem using
‘contribution per unit of scarce resource’ and then ‘ranking’ the products as we
could for parts (a) and (b) of the exam question.
The examiner effectively told you this in the requirement for part (c).
‘Assuming that RT completes the order with the commercial customer, and
using graphical linear programming, prepare a graph to show the optimum
production plan for RT for June 2010 on the basis that the availability of all
resources is 10% lower than originally predicted.’
Full details of the answer to part (c) are available in the Financial
Management article, but in summary we formulated a model which is
reproduced below.
Let R be the number of units of product R to be produced in June
Let T be the number of units of product T to be produced in June
RT wishes to maximise total contribution (TC) where TC = 47R + 61T, subject
to the following (numbered) constraints:
3R + 5T <= 4,250
5R + 4T <= 5,000
1 (direct labour)
2 (material A)
2R + 1T <= 1,850
3R + 4T <= 4,600
0 <= R <= 500
0 <= T <= 800
3 (material B)
4 (machine hours)
5 (demand limit)
6 (demand limit)
The constraints were then taken as equations and put on a graph, to which
was added the total contribution function:
Interpreting the graph shows that point C was the furthest point we could
reach within the feasible region of OABCD given the relative contributions of
R and T (which determine the slope of the iso profit line). So point C is the
optimum point. We can see from the graph that point C is at the intersection of
lines 1 and 5 and that the production quantities are:
R = 500 and T = 550
Given that point C is at the intersection of lines 1 and 5 we can prove these
values mathematically (instead of reading them from the graph). At any point
on line five, R = 500. This must hold true for the point at which it intersects
with line 1, i.e. 3R+5T = 4,250.
Substituting R = 500 into that equation:
(3 x 500) + 5T = 4,250
5T = 2,750
T = 550
This production plan will yield a total contribution of (47 x 500) + (61 x 550) =
57,050. This was not specifically asked for in the original examination
question but will help us in the calculation of shadow price later.
The answer is that the graph shows – in addition to meeting the contract – the
best production plan is to produce 500R and 550T.
In part (d) of the examination question, candidates were asked the following.
‘Discuss how the graph in your solution to (c) above can be used to help to
determine the optimum production plan for June 2010 if the actual resource
availability lies somewhere between the managers’ optimistic and pessimistic
predictions.’
With this in mind, but not as an answer to the examiner’s part (d), I asked you
to consider the effect of a more optimistic view, which is having more direct
labour hours available, chosen since they are the binding resource constraint.
We can see from graph 7 that an expansion of hours will mean an outward
movement of that line and hence point C will also move. This will bring about
increased production of units of T, with units of R remaining the same. This
will mean more total contribution.
Therefore, I asked you to consider how much more RT would pay for an extra
hour of direct labour, i.e. calculate the shadow price of direct labour.
Answer
The shadow price of labour can be calculated by measuring the effect of
making available one more labour hour.
Equation 1 now becomes
3R + 5T = 4,251
which together with equation 5
R = 500
Gives a new optimum solution of
(3 x 500) +5T = 4,251
1,500 + 5T = 4,251
5T = 2,751
T = 550.2
This new optimum solution gives a total contribution of
Total contribution = (47 x 500) + (61 x 550.2)
= 23,500 + 33,562.20
= 57,062.20
This is a rise in total contribution of USD12.20 compared to the original
solution, and is hence the shadow price of labour, the maximum premium
(amount over and above the normal price) which RT would pay for an extra
hour of labour.
I also asked, ‘Assuming that it is just more direct labour hours which become
available, and there’s no further availability of the other three scarce
resources, state the upper limit of units and hours over which the shadow
price calculated above remains the same, and state (and explain) the shadow
price of labour beyond that limit’
Answer
The shadow price of USD12.20 per labour hour will hold as long as labour
remains a binding constraint. This will be the case up until another scarce
resource ‘takes over’.
The extra labour hours purchased will move line 1 out in a parallel fashion. It
remains a binding constraint up until it cuts the intersection of line 2 and line
5, at which point line 2 becomes the binding constraint.
This is at the intersection of:
5R + 4T = 5,000 (line 2)
R = 500 (line 5)
i.e. where
(5 x 500) + 4T = 5,000
2,500 + 4T = 5,000
4T = 2,500
T = 625
In other words, it is worth purchasing extra labour hours (at a premium up to
USD12.20 per hour) for another 75 (625 – 550) units of T, or for another 375
labour hours (5 hours x 75 units).
Beyond this, extra labour hours have no value as they cannot expand
production, due to the new material A constraint (2). Hence, extra labour
hours beyond this point have a zero shadow price.