A widget distributor wants to find the minimum cost necessary to keep widgets in

Question

A widget distributor wants to find the minimum cost necessary to keep widgets in inventory. The distributor expects to sell 1600 widgets at a steady rate during the next year. Orders of the same number of widgets will be evenly spaced throughout the year. The shipping cost for each order is $40. Carrying costs, based on the average number of widgets in inventory, amount to $5 per widget. Let x be the number of widgets per order and r be the number of orders placed during the year.

Write an equation to express annual ordering costs as a function of r:
ordering cost =

Write an equation to express average number of widgets in inventory as a function of x:
average number of widgets in inventory =

Write an equation to express carrying costs as a function of x:
carrying cost =

Write an equation to express total cost, using both x and r:
This is the objective equation.
total cost =

Write the constraint equation, based on the expected widget sales, using both x and r: 1600 =

Use the constraint equation to rewrite the total cost equation as a function of only x:
C(x) =


Determine the number of orders per year and the number of widgets per order that minimizes cost: orders per year and widgets.
The minimal inventory cost (i.e. total cost) is $ .

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Explanation / Answer

Expected number of sale : 1600
That means r*x= 1600

r=1600/x    --- (1)

Shipping cost = 40*r =40*1600/x

Average inventory = (x) /2 = x/2

Inventory carrying cost = 5*x/2

Total cost = Shipping cost + inventory carrying cost

C = 64000/x+ 5*x/2

To calculate optimum :

dC/dx = -64000/x^2 + 5/2 =0

x^2 = 25600

x= 160

Then r=1600/160 = 10

Minimum cost : 64000/160+5*160/2 = $800

All the answers in the sequence below :
1) Annual ordering cost in the function of r = Shipping cost(40*r) + = 40*r + 4000/r

2) average number of widgets in inventory= (x)/2 [ because Order is evenly placed ]

3) Carrying cost = ( (x/2)* 5) ) = 5x/2
4) Total Cost = 40*r + 5*x/2

5) Constraint = r*x = 1600
6) C = 64000/x+ 5*x/2

7) Number of order (r) =10

8) Number of widget (x) = 160

9) Minimum cost = $800

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