suppose it cost $1000 to prepare a factory to produce a certain item,after the preparation is complete, it cost $40 to produce each item. After the items are produced, it cost an average of $10 per year for each item held in inventory. if the company needs 5000 units per year. how many unit should be run off in each batch in order to minimize cost. in managerial economics

To find the number of units that should be run off in each batch in order to minimize cost, we need to consider the setup cost and the holding cost.

Let's define the variables:
C = total cost
N = number of units in each batch
S = setup cost
P = production cost per item
H = holding cost per item per year
D = demand per year

The total cost is given by the sum of setup cost and production cost for each batch, plus the holding cost for each item in inventory:

C = (S + (P * N)) + (H * (N/2))

Given the following information:
S = $1000 (setup cost)
P = $40 (production cost per item)
H = $10 (holding cost per item per year)
D = 5000 (demand per year)

To minimize the cost, we need to differentiate the total cost equation with respect to N and find its minimum:

dC/dN = 0

Differentiating the equation:
dC/dN = P - (H * N/2) = 0

Solving for N:
P = (H * N/2)

N = (2P) / H

Substituting the given values:
N = (2 * $40) / $10
N = 8

Therefore, the company should produce 8 units in each batch in order to minimize cost.

To find the number of units that should be run off in each batch in order to minimize cost, we need to consider the fixed costs, variable costs, and holding costs associated with the production of each item.

Let's break down the costs involved:

1. Fixed Cost (Preparation Cost): $1000

2. Variable Cost (Cost of producing each item): $40

3. Holding Cost (Cost per year for each item held in inventory): $10

Now, we can calculate the total cost per batch based on the number of units produced:

Total Cost = Fixed Cost + (Variable Cost per unit × Number of units) + (Holding Cost per unit per year × Number of units ÷ Production per year)

To minimize cost, we can take the derivative of the total cost with respect to the number of units and set it equal to zero.

d(Total Cost) / d(Number of units) = 0

Let's calculate the derivative:

d(Total Cost) / d(Number of units) = 40 - (10 × 5000 / Number of units^2)

Setting this derivative equal to zero:

40 - (10 × 5000 / Number of units^2) = 0

Simplifying the equation:

40 = 10 × 5000 / Number of units^2

Multiplying both sides by Number of units^2:

40 × Number of units^2 = 10 × 5000

Dividing by 10:

4 × Number of units^2 = 5000

Dividing by 4:

Number of units^2 = 5000 / 4

Taking the square root:

Number of units = √(5000 / 4)

Number of units ≈ √1250

Number of units ≈ 35.35

Therefore, in order to minimize costs, the company should run off approximately 35 units in each batch.