Deterministic models: independent demand
Deterministic models are those that assume an approximately constant demand that is known with certainty. This model answers the question about when to release the order. Since the demand is known and does not vary over time, it will be enough to know the waiting time to know the exact moment in which the order must be released.
Therefore, the question we have to answer in deterministic models is the one regarding the order quantity. To answer it we determine the so-called economic order lot.
To develop a simple model we will consider a series of hypotheses:
• - Demand is continuous at a constant rate.
• - This situation continues indefinitely.
• - There are no restrictions regarding order quantity, storage capacity and available capital.
• - The delivery time is zero.
• - Stockouts are not allowed.
• - Discounts are not considered on the purchase price of the item.
• - Costs are considered invariable over time.
Although these assumptions are very restrictive, and it is unlikely that there will be stock situations that conform to them simultaneously, their consideration simplifies the analysis. The greatest usefulness of this simple model is that, based on the results obtained, it allows the development of decision rules for more complex systems adapted to real situations.
In this simple model, due to the accepted hypotheses, the costs that will be taken into account are those of issuing orders and the cost of storage.
Variable and symbols used:
When a purchase order of a size of Q units is launched, as the supply is assumed to be instantaneous, the stock level immediately reaches said value Q. From that moment, as demand is continuous and at a constant rate, there is a decrease in the stock level until it reaches zero, at which time a new order is launched, repeating the process on a recurring basis. The evolution over time of the stock level will have the characteristic sawtooth shape.
The annual cost of issuing orders will be:.
Therefore, the larger the batch, the fewer the number of orders that will have to be placed and consequently the annual costs of issuing orders will be lower.
Keeping one unit of the item in stock for one year costs r x c (monetary units). The order is placed in fixed quantities and dates, so the average stock is Q/2 units, and the annual storage cost is:.
As the order lot size increases, the storage cost increases.
The total annual cost of inventories will be the sum of these two previous concepts:
Wilson's formula:
Named after R. H. Wilson"), this formula allows us to obtain the value of the lot Q that minimizes the total cost. It is represented by Q* and is obtained by differentiating the expression for CT with respect to Q and setting it equal to zero.
The minimum total cost will be obtained by substituting the value obtained from Q* in the corresponding expression:.
Usually, a purchase order is followed by a production order for the ordered item, so a certain period of time is necessary to complete said production order. During this time the item is being produced and in demand. For this case to make sense, the production rate has to be greater than the demand rate, since if this were not the case there would be no stock at any time.
The production rate, P, is defined as the number of units produced in a period of time, generally one year.
When stocks are depleted, point A, production of the order for lot Q begins. A production time Q/P is required. During this time, stocks accumulate at a P-D rate, so when the production of the lot of size Q runs out, the maximum inventory level I (point B) will be reached, which is:
From this point, the level of stocks decreases, as a consequence of a uniform and constant demand, when stocks are depleted the cycle begins again.
Annual issue cost:.
Average stocks:
So the annual storage cost is:.
The total annual cost:.
It is possible to obtain the value of the optimal lot that minimizes costs in the same way as in the case of the simple model:
As expected, for instantaneous provisioning, P = ∞, we obtain the Wilson formula.
• - Formula based on EOQ Economic Order Quantity.
• - Takes into account volume discounts.
• - The new total cost formula includes the unit price:.
• - Calculate for each price using the EOQ formula Adjusted Economic Order Quantity:.
• - If it is less than the minimum amount for the discount, adjust it up to this.
• - Calculate CT for each.
• - Choose the one with which the lowest Total Costs can be obtained.
The company you work for wants to know what would be the most efficient procedure for purchasing tubes of specific lubricant essential for the operation of its machines. The price per tube depends on the quantity to be purchased: up to 99 tubes the price is €50, between 100 and 149 tubes the price is €44 and for an order greater than 150 tubes the price drops to €42. Shipping costs amount to €55 regardless of the quantity ordered. Since the lubricant loses its properties over time, storage costs account for 40% of the price per unit. Calculate the order quantity that minimizes inventory costs for an annual demand of 1,500 lubricant tubes..
In commercial practice, the existence of discounts based on the quantities purchased is common, so they must be considered when deciding on the quantity of items to order.
There are usually two types of discounts: global or total discounts and incremental discounts.