Sleak Teak builds yard furniture using hardwoods and (in a smaller shop) handcra

Question

Sleak Teak builds yard furniture using hardwoods and (in a smaller shop) handcrafted knick-knacks from the same sort of wood. The hardwood usage in the two lines of product are
Yard Furniture: Y = 2 Ty - .001Ty2
Knick-knacks: Z = 20Tz - .01 Tz2
where Y and Z are units of furniture and knick-knacks respectively. Ty and Tz are the amounts of hardwood used in Y and Z production respectively. Yard furniture can be sold at a profit (i.e., net of costs) of $100 per unit and knick-knacks can be sold at a profit of $25 per unit. Sleak Teak has 1300 units of hardwood available that can be allocated between these two lines of production.
(a) Using the Lagrangean Multiplier method, determine how should the hardwood be allocated between the two lines of product so that total profit can be maximized. Also calculate the optimal amounts of Y and Z and total profit from each product line.
(32 points)
(b) Using Excel Solver verify your answers to (a) above. (Show your work.)

Explanation / Answer

Production should be allocated so the marginal profitability of yard furniture equals that of knick knacks. Why? Because if the profitability of one is greater than the other more of the former and less of the latter should be produced.

y' (derivative)= 2-.002Ty marginal profiatbility is 100(2-.002Ty)
K'(derivative)= 20-.02Tk marginal profitability is 25(20-.02Tk)

So we have 100(2-.002Ty)= 25(20-.02Tk)
200-.2Ty= 500-.5Tk
.5Tk-.2Ty= 300
Tk+Ty= 1300 (constraint)
5Tk -2Ty= 3000
2Tk +2Ty= 2600
7Tk= 5600
Tk= 800
Ty= 1300-800=500
Production should be allocated to 500 for Ty and 800 for Tk

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