In this video, I'll talk about the economic order quantity (EOQ) inventory model with
zero lead time and deterministic demand.
Inventory theory studies how to minimize the costs associated with maintaining inventory
to meet customer demand.
It usually involves answering questions such as when should an order be placed for products?
How large should each order be?
These decisions will directly affect the inventory level and the total cost.
Inventory models may involve the following costs.
There may be an ordering or setup cost each time you place an order.
This cost does not depend on the size of the order.
They include things like paperwork, billing, or machine setup time.
The unit purchasing or producing cost is simply the variable cost associated with purchasing
or producing a single unit.
It may include the variable labor cost, variable overhead cost, and raw material cost for producing
one unit of product.
The holding cost is the cost of carrying one unit of inventory for one time period.
It usually includes the storage cost, the insurance cost, and the opportunity cost.
The shortage cost.
A shortage occurs when a customer's demand is not met on time.
The shortage cost is related to the lost sales and low customer satisfaction.
The basic EOQ model is the simplest inventory model.
It assumes that the demand is deterministic and occurs at a constant rate.
The ordering actions are repetitive.
Orders are placed in a regular fashion.
The basic EOQ model is a continuous review model.
The inventory is monitored all the time and an order can be placed at any time when the
inventory is low.
The lead time, which is the time between the instant when an order is placed and the instant
when the order arrives, is zero.
That means, the products arrive immediately after each order is placed.
No shortage is allowed.
That means, all the demand is met on time.
Finally, the ordering cost, the unit purchasing cost, and the unit holding cost are all constant.
Although some assumptions are unrealistic, the basic EOQ model is still useful and some
other more complex inventory models are derived from the basic EOQ model.
In the basic EOQ model, we know the following parameters.
They are represented by capital letters because they are all constant.
D is the total demand.
It is measured in units of products per year.
K is the unit ordering cost.
It is measured in $ per order.
P is the unit purchasing cost.
It is measured in $ per unit of products.
H is the unit holding cost.
It is measured in $ per unit of products per year.
The basic EOQ model determines the most economic order quantity (x) that minimizes the yearly
sum of ordering cost, purchasing cost, and holding cost.
The order quantity is measured by units per order.
The total cost is denoted by TC, which is a function of x.
Before we calculate the total cost, let's see how the inventory level will change over
time.
Assume that we place an order of size x at time 0.
Because the lead time is zero.
The order arrives immediately and the inventory level is x.
Since the total demand D will be consumed in one year, we would like to know how long
it will take to consume x units.
It will take x/D years for the inventory to reach zero again.
The inventory level will decline along a straight line of slope -D.
When it reaches zero, an order of size x is placed again and arrives instantaneously,
raising the inventory level back to x.
This is called a cycle.
It begins with the arrival of an order and ends immediately before the next order is
received.
The length of the cycle is x/D.
We should never place an order when the inventory level is greater than 0; Otherwise we need
to store the extra order in the warehouse, which will incur unnecessary inventory cost.
If the inventory level is 0, we must immediately place an order to prevent a shortage.
Each order should be placed in the same quantity x.
Now let's compare two different strategies.
The first one is a single large order.
It means we place an order of D units of products at the beginning of the year and everything
will be stored in the warehouse.
The inventory over the year is the area below this line, which is equal to the area of the
triangle: the length of the base times the height, divided by 2.
The second one is multiple small orders.
It means we first place a small order that is less than D, then wait for the inventory
to be reduced to 0 and place another order and so on.
Let's just use x=D/2 as an example.
Then we need a total of two orders in a year.
The inventory over the year is the sum of the areas of these small triangles, which
is less than the area of the big triangle.
That means the corresponding inventory cost will be less in the case of multiple small
orders than in the case of a single large order.
However, the total ordering cost will be greater in the case of multiple small orders than
in the case of a single large order because the total ordering cost is related to how
many orders are placed and irrelevant to the quantity of each order.
That's why we need to find the trade-off between the holding cost and the ordering cost.
Now let's calculate the annual ordering cost and the annual purchasing cost.
The annual ordering cost is equal to the unit ordering cost times the number of orders.
The unit ordering cost is $K per order, and the number of orders is D/x orders per year.
The annual purchasing cost is equal to the unit purchasing cost times the demand in the
year.
The unit purchasing cost is $P per unit, and the demand is D units per year.
So the annual purchasing cost is $PD per year.
Now let's calculate the holding cost per cycle and the annual holding cost.
The holding cost per cycle is equal to the unit holding cost times the inventory in a
cycle.
The unit holding cost is $H per unit per year, and the inventory in a cycle is the area of
the triangle, which is the length of the base, x/D, times the height x, divided by 2.
The annual holding cost is equal to the holding cost per cycle times the number of cycles
in a year.
The holding cost per cycle is just this, and the number of cycles is D/x.
So the annual holding cost is $Hx/2 per year.
The annual total cost is equal to the sum of the annual ordering cost, the annual purchasing
cost, and the annual holding cost.
This is a nonlinear objective function of x because x is in the denominator in the first
item.
Let's take the second order derivative.
We know that the unit ordering cost K is greater than 0, the demand is greater than 0, and
the order quantity x is also greater than 0.
So the second order derivative must be greater then 0.
This objective function is a convex function.
This is a minimization problem.
In order to find the optimal solution, we will let the first order derivative equal
to 0 and solve for x.
The optimal solution is x*, which is the economic order quantity, or EOQ.
Let's plug this value back into the objective function.
It is equal to the square root of 2KDH plus PD.
We can also move this item to the rhs and get this equation.
Let's multiply both sides by x*.
We have this new equation.
The lhs is the annual ordering cost and the rhs is the annual holding cost.
It means, at the optimal solution, the annual ordering cost must be equal to the annual
holding cost so the total cost is minimized.
Let's see an example.
A car dealer sells 300 cars per year.
This is D.
The dealer needs to decide the EOQ so he can place the order with a local car manufacturer
to meet the customer demand.
The lead time is zero.
The ordering cost is $400 per order.
This is K.
The purchasing cost (or price) is $20,000/car.
This is P.
The holding cost is $500/car/year.
This is H. Assume that demand occurs at a constant rate
and shortages are not allowed.
What is the EOQ?
What is the cycle time?
How many orders will be placed each year?
Now, let's just use the formula to calculate the EOQ, which is 21.9 cars per order.
The cycle time is 0.073 years, or 26.7 days.
There will be a total of 13.7 orders during the year.
The annual ordering cost is equal to the annual holding cost, and they are both equal to $5477.
Okay, that is about the EOQ inventory model with zero lead time and deterministic demand.
Thanks for watching.