Limiting factors

Dear Mr Moffat,

I did the following question which was fine until the end where I have a doubt for which I would like to ask you question. I will be writing the whole question, apologies in advance if it will be a lot of reading.

A manufactures and sells three products X-Y-Z for which budgeted sale demand, unit selling price and unit variable costs are are follows:

Product X Y Z

Budget sale demand 550 500 400

Unit sale price 16 18 14
Variable cost
Material 8 6 2
Labour 4 6 9

Contribution 4 6 3

The organization has existing inventory of 250 units of X and 200 units of Z which is quite willing to use up to meet the sale demand. All three products use the same direct materials and the same type of direct labour. In the next year the available supply of materials will be restricted to $4800 (at cost) and the available supply of labour $ 6600 (at cost)

Determine which product mix and sale mix would maximise the organization profit in the next year.

Solution:
Define the which is the scars resource

Product X Y Z
Sale 550 500 400
Inventory 250 – 200
Production 300 500 200

Product Qty to produce Material Labour

X 300 2400 1200
Y 500 3000 3000
Z 200 400 1800

Material supplied 4800
Labour supplied 6600

Shortfall (1000) 600

Material is the limiting factor.

Rank the product in order of contribution earns of direct material

Product X Y Z

Contribution 4 6 3
material 8 6 2
0,5 1 1,5

Profit maximizing budget

Product Unit Material/Cost Contribution T.C
z 400 2 =800 1,5 1200
Y 500 6 =3000 1 3000
x 425 8=3400 0,5 1700

Why have total units been multiplied by contribution earn of direct material and not by the contribution per unit (4-6-3)?

Thanks in advance

Gabriella