# Lecture Video 2

## Lecture 2: Optimum Batch Size for a Production Run

The economic order quantity is a calculation we do to determine the optimal size of an order in a retail company. When we place frequent small orders, our ordering costs will be high. And if it’s infrequent large orders, the average inventory will be higher, which will increase the holding cost. And the economic order quantity was to determine the perfect balance between the ordering cost and the holding cost.

In this lecture, we will look at how to use the economic order quantity to determine the optimum batch size for a production or manufacturing company. If you think of a manufacturing company, if we produce small batches, say one unit at a time; every time we produce a new unit, we have to incur setup costs. Setup costs are like cleaning the machines before we put in new raw materials. And programming the machines, handling the materials etc. So that’s setup costs. So small batches or small production runs would increase our total setup costs. That is why we would want to increase the batch sizes; to limit the setup costs. But too large batch sizes will mean higher average inventory levels, which will mean higher holding costs again.

So you can see the problem is similar to a retail company, but some of the costs are a little bit different.

So this is just a recap of economic order quantity.

Economic Order Quantity = 2 x U x C
H + (P x I)

U = annual demand
S = variable setup cost per batch
H = variable carrying cost per unit
P = manufacturing cost per unit
I = interest rate

We have our annual demand, multiplied by our ordering cost per order, time two. And then we divide it by the direct holding cost, plus the opportunity cost that’s tied up in the inventory.

So, let’s see how this needs to change then in order to determine the optimum batch size for a production run. So we’ve got the same calculation; we still have the annual demand, so that doesn’t change. We don’t have order costs now; we replace our order costs with setup costs, so every time we run a production run or produce a batch, we incur setup costs. We still have holding costs, so once we have the inventory, we still have to store it somewhere. And then we don’t have a purchase price anymore; now it’s manufacturing costs per unit. And we still have the interest rate, so our opportunity cost is made up of the costs tied up in the inventory, times the interest we could’ve earned on it.

So, the calculation is exactly the same; the main thing is that we substitute ordering costs with the setup costs per batch. So let’s look at an example to illustrate this.

U = annual demand 18 000 units
S = variable setup cost per batch 45 cost per batch
H = variable carrying cost per unit 1.50 per unit
P = manufacturing cost per unit 5 per unit
I – interest rate 10%

So let’s do this calculation:
2 x 18 000 x 45
1.50 + (5x10%=0.50 opportunity cost)
= 900 units

We’ve got 2, times the demand, times the setup costs. Usually, we would’ve had the order costs there. Divide it by the holding cost, that’s the direct holding cost, and then our opportunity cost. So we’ve got 5 times 10%, so that would be 0,50, that’s our opportunity cost.

So if you do this calculation on your calculator, you’ll get 900 units. That’s our optimum batch size. So whenever we produce a batch, we would want to produce 900 units. So if we produce more than 900 units per batch, that will increase our holding cost, because we’ll have more inventory to hold. If we produce less than 900 units, we will increase our setup costs, because we will have more production runs per year.

Let’s quickly calculate the total setup costs for the year.

Annual demand
We’ve got an annual demand of 18 000, divided by 900 units per batch. So that means we’ll have 20 production runs or 20 batches per year.
And then the setup cost per batch is 45, so our total cost for setup in currency units would be 900 per year.

Holding costs
We need our average inventory levels. So average inventory would be the 900 units we produce, divided by 2. Why do we divide it by 2? Because we assume we use the inventory evenly throughout the period.

So we start off with a value or an inventory level of 900 after we produced the batch. And then we use it up until we don’t have anything left. So at any point in time, it’s going to be decreasing at a constant rate, so the average inventory we’ll keep, will be halfway between the maximum and the minimum. And that’s why we divide it by 2 to get 450.

We multiply that by holding costs, so that’s 2. That’s the 1,50 plus the 50 cents, and that’ll give us the total holding cost of 900 per annum.

As you can see, the setup costs and holding costs are equal. And that’s when we have our optimal batch size. This example is similar to the economic order quantity for a retail company, where our ordering costs were equal to the holding costs. That is where we have our optimum batch size or optimal order quantity in that case.

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