2 Higgins Co (HC) manufactures and sells pool cues and snooker cues. The cues both use the same type of good quality
wood (ash) which can be difficult to source in sufficient quantity. The supply of ash is restricted to 5,400 kg per
period. Ash costs $40 per kg.
The cues are made by skilled craftsmen (highly skilled labour) who are well known for their workmanship. The skilled
craftsmen take years to train and are difficult to recruit. HC’s craftsmen are generally only able to work for 12,000
hours in a period. The craftsmen are paid $18 per hour.
HC sells the cues to a large market. Demand for the cues is strong, and in any period, up to 15,000 pool cues and
12,000 snooker cues could be sold. The selling price for pool cues is $41 and the selling price for snooker cues is
$69.
Manufacturing details for the two products are as follows:
Pool cues Snooker cues
Craftsmen time per cue 0·5 hours 0·75 hours
Ash per cue 270 g 270 g
Other variable costs per cue $1·20 $4·70
HC does not keep inventory.
Required:
(a) Calculate the contribution earned from each cue. (2 marks)
(b) Determine the optimal production plan for a typical period assuming that HC is seeking to maximise the
contribution earned. You should use a linear programming graph (using the graph paper provided), identify
the feasible region and the optimal point and accurately calculate the maximum contribution that could be
earned using whichever equations you need. (12 marks)
Some of the craftsmen have offered to work overtime, provided that they are paid double time for the extra hours over
the contracted 12,000 hours. HC has estimated that up to 1,200 hours per period could be gained in this way.
Required:
(c) Explain the meaning of a shadow price (dual price) and calculate the shadow price of both the labour
(craftsmen) and the materials (ash). (5 marks)
(d) Advise HC whether to accept the craftsmens’ initial offer of working overtime, discussing the rate of pay
requested, the quantity of hours and one other factor that HC should consider. (6 marks)
(25 marks) 2 Higgins Co
(a) Contribution per cue
Pool cue Snooker cue
$ $
Selling price 41·00 69·00
Material cost at $40/kg (10·80) (10·80)
Craftsmen cost at $18/hr (9·00) (13·50)
Other Variable cost (1·20) (4·70)
–––––– –––––––
Contribution per cue 20·00 40·00
–––––– –––––––
(b) Formulation of the linear programming problem
Variables
Let P and S be the number of pool and snooker cues made and sold in any three month period.
Let C represent the contribution earned in any three month period
Constraints:
Craftsmen: 0·5P + 0·75S ≤ 12,000
Ash: 0·27P + 0·27S ≤ 5,400
Demand levels – Pool cues P ≤ 15,000
– Snooker cues S ≤ 12,000
Non negativity: P, S ≥ 0
Objective: Higgins seeks to maximise contribution in a three month period, subject to:
20P + 40S = C
See diagram on next page
The feasible region is identified as the area inside OABCDE.
The contribution line is identified as the dotted line. Pushing the contribution line outward increases the contribution gained
(theory of iso-contribution). The contribution line last leaves the feasible region at point D which is the intersect of the skilled
labour line and the maximum demand line for S.
Solving at point D:
Maximum demand S = 12,000 (1)
Craftsmen 0·5P + 0·75S = 12,000 (2)
Substituting S = 12,000 in equation (2)
0·5P + (0·75 x 12,000) = 12,000
0·5P + 9,000 = 12,000
0·5P = 12,000 – 9,000
0·5P = 3,000
P = 6,000
Therefore the maximum contribution is earned when 6,000 pool cues and 12,000 snooker cues are made and sold in a three
month period.
The contribution earned is
C = (20 x 6,000) + (40 x 12,000)
C = 120,000 + 480,000
C = $600,000 480 000
15 000
30 15 , 000 30 000
,
– , ,
⎛
⎝ ⎜
⎞
⎠ ⎟
= Adv
Production schedule
16
E
0
2
4
6
8
10
12
14
16
18
20
22
24
2 4 6 8 10 12 14 16 18 20
S
contribution
A
Ash
Craftsmen
Max S
Max P
Max contribution
F
D
C
B
P
Feasible region = OABCDE
Optimal point at point D
(c) Shadow prices
A shadow price is the value assigned to changes in the quantity of a scarce resource available, normally measured in terms
of contribution. If more critical scarce resource becomes available then the feasible region would tend to expand and this
means that the optimal point would tend to move outward away from the origin thus earning more contribution. It is this
increase in the contribution that is the shadow price measured on a per unit of scarce resource basis.
Management can use the shadow price as a measure of how much they would be willing to pay to gain more of a scarce
resource. It represents the maximum they should be willing to pay for more scarce resource over and above the normal price
subject to any non-financial issues that may be present.
If the availability of a non-critical scarce resource increased then the feasible region would not tend to expand and therefore
no more contribution could be earned. In this case extra non-critical scarce resource has no value and a nil shadow price.
Calculation of shadow prices:
Ash: This is a non-critical scarce resource and as such it has a shadow price of nil. Put simply we have slack (spare material)
of ash and therefore have no desire to pay more to get more of it.
Craftsmen: This is a critical scarce resource and if more became available then the feasible region would expand and the
optimal point would move outward thus earning more contribution. Assuming that just one more hour becomes available it
is necessary to find the new optimal point and measure the increase in contribution earned.
At point D, we re-solve based on the available craftsmen hours being one more than previously.
S = 12,000 (3)
0·5P + 0·75S = 12,001 (4)
Substituting S = 12,000 in equation (4)
0·5P + 0·75(12,000) = 12,001
0·5P + 9,000 = 12,001
0·5P = 3,001
P = 6,002
The new optimal solution would be where 12,000 snooker cues and 6,002 pool cues are made. This would earn an extra
$40 (2 x $20) in contribution.
The shadow price is therefore $40 per extra hour of craftsmen time.
(d) Acceptability of the craftmens’ offer.
Rate of pay
The rate of pay requested (double time) is on the face of it less than the shadow price and is therefore affordable by Higgins
Co. The business would be better off by accepting the offer.
However, it is common for overtime to be paid at time and a half ($27 per hour) and Higgins would be well advised to
negotiate on this point. Higgins takes the commercial risks in this business and would therefore be justified in keeping the
majority of the rewards that come with it. Equally it is a dangerous precedent to accept the first offer and pay such a high
rate for overtime, Higgins would have to ask itself what would happen next time an overtime situation arose. It is also possible
that double time, being so generous, encourages slow working in normal time so as to gain the offer of overtime.
How many hours to buy?
The problem here is that as Higgins buys more craftsmen time, the craftsmen constraint line will move outward, changing
the shape of the feasible region. Once the craftsmen line reaches point F (see diagram) then there would be little point buying
any more hours since Higgins would then not have the materials (ash) to make more cues.
We need therefore to calculate the number of hours needed at point F.
At F
Maximum demand for S S= 12,000 (5)
Ash 0·27P + 0·27S = 5,400 (6)
Substituting S = 12,000 in equation (6)
0·27P + 0·27(12,000) = 5,400
0·27P + 3,240 = 5,400
0·27P = 2,160
P = 8,000
Point F falls where S = 12,000 and P = 8,000
The craftsmen hours needed at this point would be given by putting the above P and S values in the craftsmen constraint
formula.
Craftsmen hours = (0·5 x 8,000) + (0·75 x 12,000)
Craftsmen hours =13,000 hours Therefore Higgins should only buy 1,000 hours (13,000 – 12,000).
In general terms Higgins need only buy the number of hours that the business can use to make and sell more product. If
more ash can also be bought then more labour hours may be desirable.
Quality of work
Higgins should consider the quality of work. Overtime hours can force tiredness on craftsmen that have already worked a full
day. Tired people often produce sub-standard work. If quality is important then this could damage the reputation of the
business.
Any other feasible points would be accepted |
发表于 2008-10-11 14:00
| 