Linear programming

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 hours

Answer: 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 = 60,001

Multiply (1) by 5
(3) 5x = 50,000

Subtract (3) from (2) 4y = 10,001
y = 2,500.25

Now 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) = $95,000

The 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.

i dont understand why has x = 10,000 been taken as the starting point for the optimal solution, as opposed to y = 12,000. if i choose y = 12,000 gives the apparently incorrect answer of $1.60 per hour.