- This topic has 3 replies, 3 voices, and was last updated 7 years ago by
.
- Posts
Hi,
I have a revision question from a kit, and I am unsure about the approach taken. Any help would be gladly appreciated!
Q) A company makes two products, X and Y, on the same type of direct labour, and production capacity per period is restricted to 60,000 direct labour hours. The contribution per unit is $8 for Product X and $6 for Product Y. The following constraints apply to production and sales:
x < 10,000 (Sales demand for Product X)
y < 12,000 (Sales demand for Product Y)
5x + 4y < 60,000 (Direct labour hours)The contribution-maximising output is to produce and sell 10,000 units of Product X and 2,500 units of Product Y.
What is the shadow price per direct labour hour and for how many additional hours of labour does this shadow price per hour apply?
A $1.50 per hour for the next 38,000 direct labour hours
B $1.50 per hour for the next 47,500 direct labour hours
C $1.60 per hour for the next 38,000 direct labour hours
D $1.60 per hour for the next 47,500 direct labour hoursAnswer: A
The correct answer is: $1.50 per hour for the next 38,000 direct labour hours.
If one extra direct labour hour is available, the optimal solution will change to the point where:
(1): sales demand for X: x = 10,000
(2): direct labour 5x + 4y = 60,001Multiply (1) by 5
(3) 5x = 50,000Subtract (3) from (2) 4y = 10,001
y = 2,500.25Now Total contribution = $(10,000 × $8) + $(2,500.25 × $6) = $80,000 + $15,001.15 = $95,001.50
Total contribution in original solution = $(10,000 × $8) + $(2,500 × $6) = $95,000The shadow price per direct labour hour is therefore $1.50.
The solution is changing because each additional labour hour allows the company to produce an additional 0.25 units of Product Y, to increase total contribution by $1.50.
This shadow price will cease to apply when the direct labour hours constraint is replaced in the optimal solution by the sales demand for Product Y constraint. At this level of output, total labour hours would be (10,000 units of X at 5 hours) + (12,000 units of Y at 4 hours) = 98,000 hours.The shadow price of $1.50 per hour therefore applies for an additional 38,000 hours above the current limit.
My Question
Why has x = 10,000 been taken as the starting point for the optimal solution, as opposed to y = 12,000. There is no explanation given, but it is crucial in getting the correct answer, as to choose y = 12,000 gives the apparently incorrect answer of $1.60 per hour.
Is x chosen as the sales demand for it in the optimal solution gives more contribution? Or is it because the sales demand constraint for x matches the demand in the optimal solution, so you need to go with it instead of y?
The choice seems arbitrary – and is not explained.
Can anyone guide me here?
Many thanks!
When calculating the shadow cost you are calculating the contribution generated of one extra hour of labour. So you must start from the point of maximum contribution, which was given to you as x = 10,000 and y = 2,500. You are then saying, what if we had one extra hour of labour, what could we do with it and what contribution can we make with it? This then determines the shadow price, which is the maximum premium you would pay for this extra labour e.g. by offering overtime.
Starting with y = 12,000 in this case wouldn’t make sense because that’s not the optimal production plan. If you are making 12,000 y then either you are not making enough x and therefore have a sub-optimal production plan (in terms of contribution generated), or you are making the maximum x and y and therefore there is no demand for you to make any more products, so the shadow price would be $0.
You must start from the point of maximum contribution within the constraints, which will either be given to you or you may have to calculate yourself.
Hi Chris,
Many thanks for your reply – it is much appreciated. What I had meant to say though in the initial question, was why not choose the optimal solution constraint of demand for y = 2,500 – particularly given a linear relationship exists with the demand for x for Direct Labour Hours?
Yet, when I choose y = 2,500, and therefore for 1 extra unit of direct labour hours:
5x + 4y = 60,001, I get x = 10,000.20I then get a total value of contribution in this scenario of $95,001.60, which when set against the initial scenario, gives a shadow price of $1.60 – not $1.50.
Given a linear relationship exists between x & y for direct labour hours, I would have thought it was irrelevant whether x or y’s optimal demand amount is chosen for maximum contribution, as you would arrive at the same value for the shadow price of $1.50. But this is clearly not the case.
That is why I am confused. There seems to be some reasoning to choose the higher demand figure, that gives the higher contribution on it’s own – separate to y?
I might be wrong here, but I think the issue lies in that we can’t sell any more of product X as the demand is only 10,000, so an extra .2 of a unit of X wouldn’t generate any extra contribution as we couldn’t sell it.
Whereas Y has a demand of 12,000 units and we are only producing 2,500 units, so extra units of Y would generate extra contribution and therefore a shadow price.Hope this helps.
- You must be logged in to reply to this topic.