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A cabinetmaker uses mahogany wood to produce five desks each day
A cabinetmaker uses mahogany wood to produce five desks each day. Each delivery of one container of wood is $5000, whereas the storage of that material is $10 per day per unit stored, where a unit is the amount of material needed by her to produce one desk.
How much material should be ordered each time, and how often should the material be delivered, to minimize her average daily cost in the production cycle between deliveries?
Expert Solution
According to question,
For delivery every xx days then it must order 5x5x unit
The average amount in storage is approximately one half of the delivery amount that is 5x25x2
Cost per cycle == delivery cost ++ storage cost
As we know,
Delivery cost =5000=5000 dollar (given)
Storage cost =5x2×10×x=25x2=5x2×10×x=25x2
So, cost per cycle =5000+25x2=5000+25x2
Average daily cost =5000+25x2x=5000+25x2x
For minima, differentiate above expression and put the value equal to zero
ddx(5000+25x2x)=0=ddx(5000x+25x)=0⇒−5000x2+25=0⇒5000x2=25⇒5000=25x2⇒x2=200⇒x=±√200ddx(5000+25x2x)=0=ddx(5000x+25x)=0⇒−5000x2+25=0⇒5000x2=25⇒5000=25x2⇒x2=200⇒x=±200
Neglecting the negative part
We have
x=√200x=200
For x=√200x=200
Average daily cost =707.10678=707.10678
Hence, the cabinet maker should schedule delivery of 5×14=705×14=70 unit of mahogany every 1414 day.
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