Mason’s Meows is a company that makes cat toys. The company sells 1200 toys per year. The firm incurs a fixed cost of $150 in labor each time it starts up the manufacturing process to begin a new batch of toys. Each toy costs Mason’s Meows $9 to produce. The company's accountant recommends using a holding cost equal to 20% of the cost of the toy, per year. a) What is the optimal batch size, Q*? If the company uses batches of size Q*, how many times per year, on average, will it start up the manufacturing process?
b) After careful analysis, the inventory team at Mason's Meows realized that the per-unit production cost is smaller if the batch size is larger. In particular, the production cost is $9 per unit for batches of fewer than 400 units and $7.50 per unit for batches of 400 or more units. Now what is the optimal batch size?

a. The optimal batch size, Q * is Economic order quantity (EOQ)

Annual demand = 1,200 toys  

Cost of each toy =$9

Fixed cost starts up the manufacturing process (S) = $150/batch

Inventory carrying cost (H) = 20% of the cost of the toy, per year

Inventory carrying cost (H) = 20% *$9 per unit

Inventory carrying cost (H) = $1.80 per Toy per year

EOQ = Q* = √ (2 * Annual Demand *fixed processing cost/ Inventory carrying cost)

EOQ = √ (2 * 1,200 *$150 / $1.80)

EOQ = Q* = 447.21

EOQ = Q* = 447 toys

The optimal batch size is 447 toys

Number of orders per year, on average, if it will start up the manufacturing process

= Annual demand / EOQ

= 1,200 / 447

= 2.68 times per year

b. Annual demand = 1200 toys

Cost of each toy =$7.5 (assume that batch size is more than 400)

Fixed cost starts up the manufacturing process (S) = $150/batch

Inventory carrying cost (H) = 20% of the cost of the toy, per year

Inventory carrying cost (H) = 20% *$7.5 per unit

Inventory carrying cost (H) = $1.50 per Toy per year

EOQ = Q* = √ (2 * Annual Demand *fixed processing cost/ Inventory carrying cost)  

EOQ = √ (2 * 1,200 *$150 / $1.50)

EOQ = Q* = 489.90

EOQ = Q* = 490 toys

The optimal batch size is 490 toys.

you have to put the parentheses like this:

in fact, you have to perform the operation inside parentheses first, and then proceed in order: