Economic order quantity

The classical EOQ formula or Losformel (English Economic Order Quantity, EOQ formula ) is a German-speaking countries in 1929: made ​​by Kurt Andler method to determine the optimal lot size in the context of a single-stage, unkapazitierter industrial production. The approach has already been developed by Ford W. Harris in 1913.

In a release in 2005 trial demonstrated Georg war on the one hand important differences between the works of K. Andler and F. W. Harris back as well as on the resulting divergence in the storage costs. Second, it makes the use of the term EOQ formula on the Harris formula in question because K. Andler the Harris formula is not derived, but developed their own, more detailed Losformel that would actually carry the designation Andlersche lot size formula right.

In Anglo-Saxon literature, the term Economic order quantity dominates ( EOQ formula ), the problem is investigated for optimal order quantity. On the similarities between order and production quantity determination of optimal order quantity is discussed in the section.

• 2.3.1 infinite production rate
• 2.3.2 Open Manufacturing
• 2.3.3 Closed production

Basic assumptions and definitions

Approach

The classic Losformel is designed for companies with a batch production where a lot caused when placed in storage and set-up costs on the way to the customer storage costs. Because a lot by running as a (closed ) post production stages, increases with its size and inventory costs. The set-up costs, however, decline because fewer lots placed and thus less setup procedures must be performed to produce the same amount. The sum of the two types of costs therefore depends on the lot size. They can be represented as a function of the lot size and find its minimum with the EOQ formula.

The procedure can also be applied to open and closed production, where arising only different storage costs. Although the approach is carried out to the optimum in the form of a cost minimum of the cost side, the profit maximization is ( in linear sloping price-demand function) to the same result.

Assumptions of the classical lot size model:

• Production: single-stage production with spare capacity without intermediate storage or multi-level production without waste, interruptions and identical speeds.
• Realistic, finite production rate (equivalent to the storage access rate )
• Arbitrary divisibility of lot size
• Existing capacity to produce the determined optimal lot size

• Constant storage cost rate
• Storage with unlimited storage capacity
• Exactly one product in exactly one warehouse

• No shortages
• Infinite planning horizon
• Constant period requirements (according to the stock issue rate )

• The production of the determined optimal lot size is possible and was not compromised by the time between production and sales

• Static approach with the assumption that the data remain constant over time and costing takes place continuously.

Symbols:

• Variables:   - the batch size
• - Period length

• - Costing rate or paragraph speed
• - Maximum sales volume
• - Variable costs
• - Finite production rate with
• - Losfixe costs (eg, setup costs )
• - Losvariable costs (eg storage costs set)

• - Time Period

The optimal batch size is now available where the sum of all controllable costs, that is, from set-up costs and storage costs, reaches a minimum.

Theoretical concept

In the first step, the two types of costs are considered storage costs and setup costs and subsequently presented optimization approaches in terms of cost minimization and profit maximization.

Set-up costs

The number of set-up processes is directly related to the quantity of production: it decreases with increasing lot size, drop the set-up costs (based on the total amount ), and the setup time is now available for production.

For production of different varieties of variable setup times and setup costs so that may be present, so that in the optimization process, the relative contribution margin should be used as a decision criterion. In the following, however, we refrain from interdependencies between different varieties and going of isolation, of the varieties of.

The multiplication of the number of required set-up procedures with cost-per- conversion gives the relationship between setup costs and lot size, which is summarized in the following table:

As can be seen from Figure 1, the curve of the set-up costs, depending on the batch size is progressively reduced. Thus we obtain the first component of the optimization process.

Storage costs

For storage costs are in the short run mainly by cost of capital employed. In the longer term, the storage and capacity costs must be considered additionally. When determining the storage costs but further assumptions about the production technology are to be made, which in turn affects the amount of storage. A distinction is made between open and closed production, because the two types of production lead to different maximum and average inventory levels.

First, it is assumed that the classical assumption that the average stock of half the batch size corresponds. In his work, K. Andler mentioned, however, that even the average stock issue must also be taken into account in determining the inventory. In this ( important ) detail but initially omitted below.

Rate of production

Assuming an infinite production rate, so is the whole produced Los available immediately, so that the pure production time is close to zero and the production rate tends to infinity: leaves as the last piece of the previous lot, the warehouse, the next Los reached the warehouses and stands for sales purposes to the full extent available. The storage costs are then obtained as follows:

The stock over time at infinite speed of production is illustrated in Figure 2. At this point is to refer to the first -mentioned work of G. War, comes to a different average stock in their results, than previously thought.

Open manufacturing

If open production before production at finite speed, then leave individual products which last significant manufacturing stage before the entire lot has been produced, so that the products can be delivered earlier and the average inventory decreases.

The rate of production is higher than the sales rate, so not the entire lot must be stored, but only the resulting difference. As already can be delivered to production time, also eliminates the need to keep a minimum amount in stock, if no production interruptions are expected.

Figure 3 makes the history of the stock significantly in this type of production.

Closed production

In this case, the production reached the warehouses until the production of a batch has been completed, what (if they are collectively transported to the warehouse ) may be conditionally technically (if jointly receive an imprint in the final stage all parts) or logistically.

The average inventory is divided into a storage building and a destocking phase, but both have the same average storage costs. In the build phase is positive in inventories produced with speed and deposited with the speed. In the decomposition phase, only the sales will be held and the available capacity can be used to manufacture other products.

In continuous costing the production begins much earlier by the length of the production period of a lot, so that when the last piece of the stock is delivered, the new lot will be incorporated in full. Thus, the average inventory increases compared to open production:

The situation described at the beginning with an infinite production rate is thus a special case of production with a closed production due.

Zeitvariate period demand

Contrary to the assumption of classical Losformel the required quantities are usually not constant over time. In this case we speak of dynamic demand. However, this fact can first be integrated into the model by the zeitinvariaten period needs replacing, for example, by the average of periods occurring in the planning horizon or with the help of statistical tools such as regression analysis estimates.

This policy, however, leads either to increased storage costs, or is associated with the risk of shortages. But even with the changes the classical model is not suitable to determine the optimal lot size and Losauflagezeitpunkte because its determination does not take place simultaneously.

Minimum lot size costs

The last section clearly showed that the average inventory depends on the type of production. The following are the respective minimum cost batch sizes are derived based on this knowledge by means of the differential calculus.

Rate of production

Respectively

Because of

Open manufacturing

Respectively

Because of

Closed production

Respectively

Because of

Thus, the minimum cost lot size can now be determined if the terms of the underlying model are met.

Profit Maximum batch size

A profit maximizing firm is faced with usually an elastic demand, which decreases with rising prices. This relationship is described in business administration in general by means of the price-demand function. The relationship between the price-demand function and the revenue function is shown in Figure 5.

The maximum profit is achieved in this case under Cournot quantity, that quantity, then, in which the difference between revenue and cost of producing this quantity is maximal ( monopoly case). The cost step on the variable costs of production, storage and conversion costs. The optimization problem for the case of open production can be formulated as follows:

The profit-maximizing quantity is then given by:

The profit Maximum batch size corresponds then:

The procedure can also be applied to the situation with free competition and constant selling prices. Thus we obtain another important result that the profit maximum lot size corresponds to when capacity of kostenminmalen lot size.

However, the output situation changes radically when additional, more realistic constraints such as capacity shortages and competing objectives such as full paragraph satisfaction must be considered, so that the relative contribution margin per unit of time comes as a decision criterion in question.

Determination of optimal order quantity

The classic Losformel can be applied to other problems that are based on the same scenario. So one among other determining optimal order quantities of the tasks of procurement logistics, which also make up the total costs depending on the amount of linear storage costs and volume independent, degressive order costs. Thus, the ordering costs of procurement logistics and set-up costs in batches, describe exactly the same problem. Based on the symbolism of the classical lot size formula, the variables used in the fitted model can be described as follows:

• Variables:   - Quantity ordered
• - Period length

• - Costing rate or paragraph speed
• - Maximum sales volume
• - Variable costs
• - Finite speed of delivery
• - Quantity-independent ordering costs
• - Dependent on the amount of storage costs

The optimal order quantity is present at infinite speed of delivery

A situation in which a minimum amount of stock to be kept on hand, corresponds to the production with a closed production and corresponds to optimum

The premises of the application of classical lot size must be respected accordingly in determining optimal order quantities, which inevitably brings advantages and disadvantages.

Evaluation and limits

The criticism of the classical Losformel depends primarily on the underlying assumptions. In particular, the restriction to single-stage or multi-stage construction is criticized severely restricted: a broadcasting on multi-stage production processes is only possible if no committee and disruption of production at identical speeds of the stages are present, which is also not realistic.

On the other hand, further restrictions may exist that are not included in the methodology. So it may happen that the production of an optimal lot size is not possible and at scarce capacity to develop large batches to save set-up time; lack of storage capacity can force to suboptimal smaller lots about it. The permissible duration of storage of products (eg food production ) also sets the limits of the classical lot-size optimization. An immediate production of an optimal batch might not be financially viable, because the time was between production and sales can lead to liquidity problems.

Another basic premise - a constant, continuous costing - is not, or only very rarely encountered in reality, because only in this case can the storage costs accurately determine and avoid shortages. Also, the isolated consideration of any sort due to spare capacity is unrealistic, because they compete for bearing and machine capacities. In optimum all varieties must be equal often established to solve the problem of sequence planning for interdependencies between the varieties. With limited capacity, taking into account full demand satisfaction, the model does not necessarily lead to an optimal solution, so that may compromise solutions need to be considered, the approach does not provide this assistance. This means that the model in its application by the strict and impractical assumptions is very limited and the problem of scheduling is not solved with this method. Thus, the classical Losformel has more textbook character as a practical use.

How could an example be shown by the open and closed production, the classic Losformel has been adapted in various ways to more realistic basic requirements and expanded. Among other things, was replaced by a time-delayed, for example, the immediate order fulfillment, added backlogs in the calculation, etc. incorporated variable set-up processes in the formula. The fundamental problem of the lack of adaptation to the needs met these modifications not sufficient. Significant progress has been made in the area of ​​optimizing the lot size only with the dynamic batch sizes investigations that allow a more complex problem detection.

Swell

• Procurement Logistics
• Production Management