Classic ticket formula

The classic batch formula or Andler formula (English Economic Order Quantity , EOQ formula ) is a method made known by Kurt Andler in the German-speaking area in 1929 for determining the optimal batch size in the context of single-stage, uncapacitated industrial production . However, the approach was already developed by Ford W. Harris in 1913.

In a study published in 2005, Georg Krieg points out, on the one hand, important differences between the work of K. Andler and F. W. Harris and the resulting divergences in the area of ​​storage costs. On the other hand, he questions the application of the name Andler's formula to the Harris formula, because K. Andler does not derive the Harris formula, but rather developed his own, more precise lot formula, which would actually rightly be called Andler's lot size formula .

In Anglo-Saxon literature, the term economic order quantity (EOQ formula) dominates, whereby the problem is examined with regard to the optimal order quantity . The similarities between the order and production quantity are discussed in the section Determining the optimal order quantity .

Basic assumptions and definitions

approach

The classic Losformel was for companies with a lot size developed where a lot when placing set-up costs and the camps on the way to the customer storage costs caused. Because a lot goes through the production stages as a (closed) item , the storage costs increase with its size. The set-up costs, on the other hand, decrease because fewer batches have to be placed and therefore fewer set-up processes have to be carried out to produce the same quantity. The sum of the two types of costs therefore depends on the lot size. It can be represented as a function of the lot size and its minimum can be found with the Andler formula.

The procedure can also be used with open and closed production, with only different storage costs resulting. Even if the approach to the optimum takes place in the form of a cost minimum from the cost side, profit maximization (with a linearly inclined price-sales function) comes to the same result.

Premises of the classic lot size model:

• Production :
  • single-stage production with free capacities without intermediate storage or multi-stage production without rejects, interruptions and identical speeds.
  • realistic, finite production speed (corresponds to the warehouse access rate)
  • any divisibility of the lot size
  • existing capacity for the production of the determined optimal lot size
• camp
  • constant storage cost rate
  • Warehouse with unlimited storage capacity
  • exactly one product in exactly one warehouse
• paragraph
  • no shortages
  • infinite planning horizon
  • Constant period requirement (corresponds to the warehouse issue rate)
• financing
  • the production of the determined optimal lot size is possible and not endangered by the time lag between production and sales
• Time component
  • Static approach with the assumption that the data will remain constant over time and that inventory will be continuously displaced.

Symbols :

• Variables:
  • ${\displaystyle y}$ - lot size
  • ${\displaystyle T}$ - Period length
• Parameter:
  • ${\displaystyle V}$ - Storage rate or sales speed
  • ${\displaystyle R}$ - maximum sales volume
  • ${\displaystyle k}$ - variable costs
  • ${\displaystyle M}$ - finite production speed with ${\displaystyle M>V}$
  • ${\displaystyle C_{\mathrm {R} }}$- Lot-fixed costs (e.g. set-up cost rate )
  • ${\displaystyle C_{\mathrm {L} }}$- losvariable costs (eg. Storage costs sentence )
• Clues:
  • ${\displaystyle t}$ - time period

The optimum lot size is now available where the sum of all controllable costs, i.e. set-up costs and storage costs, reaches a minimum.

Theoretical concept

In the first step, the two types of costs storage costs and setup costs are considered and then optimization approaches with regard to cost minimization and profit maximization are presented.

Setup costs

Figure 1 Cost history for batch production

The number of setup processes is directly related to the production quantity: it decreases with increasing lot size, the setup costs (in relation to the total quantity) decrease, and the setup time is now available for production.

When producing several types, there may be variable setup times and thus setup costs, so that the relative contribution margin must be used as a decision criterion in the optimization process. In the following, however, we will refrain from interdependencies between individual varieties and assume that the varieties are considered in isolation.

The multiplication of the number of necessary setup processes by the costs per conversion results in the relationship between setup costs and lot size, which is summarized in the following table:

Setup costs

Expression	interpretation
${\displaystyle {\frac {R}{y}}}$	Changeover frequency
${\displaystyle {\frac {R}{y}}C_{R}}$	Changeover costs per period
${\displaystyle {\frac {R}{y}}C_{R}/R={}^{C_{R}}\!\!\diagup \!\!{}_{y}\;}$	Changeover costs per unit of measure

As can be seen in Figure 1, the setup costs are degressive as a function of the lot size. This gives us the first component of the optimization process.

storage costs

In the short term, storage costs are mainly the costs of tying up capital . In the longer term, storage and capacity costs must also be taken into account. When determining the storage costs, however, further assumptions must be made regarding the production technology, which in turn influences the storage quantity. A distinction is made between open and closed production because the two types of production lead to different maximum and average storage quantities.

First of all, it is based on the classic assumption that the average inventory is half the lot size. In his work, however, K. Andler mentions that when determining the stock level, the average stock issue must also be taken into account. However, this (important) detail will initially not be used in the following.

Infinite production speed

Figure 2 Inventory with immediately available delivery (M → ∞)

Assuming infinite production speed, the entire lot produced is immediately available, so that the pure production time runs towards zero or the production speed towards infinity: when the last piece of the previous lot leaves the warehouse, the next lot reaches the warehouse and is fully available for marketing purposes. The storage costs are then as follows: ${\displaystyle M}$

storage costs

Expression	interpretation
${\displaystyle {\frac {y}{2}}}$	average stock
${\displaystyle {\frac {y}{2}}C_{L}}$	average storage costs
${\displaystyle {\frac {y}{2}}C_{L}T}$	total storage costs

Inventory over time at infinite production speed is illustrated in Figure 2. At this point, reference should be made to the work by G. Krieg mentioned at the beginning, the result of which is a different average inventory than previously assumed.

Open manufacturing

Figure 3 Inventory with open manufacturing

If there is open production at a finite production speed, then individual products leave the last production stage before the entire batch has been produced, so that the products can be delivered earlier and the average inventory is reduced.

The speed of production is higher than the sales speed , so that the entire lot does not have to be stored, but only the resulting difference. Because deliveries can be made during production time, there is no need to keep a minimum amount in stock if no interruptions in production are expected. ${\displaystyle M}$ ${\displaystyle V}$

storage costs

Expression	interpretation
${\displaystyle {\frac {y}{M}}}$	Production time of a lot
${\displaystyle V{\frac {y}{M}}}$	Sales volume during production
${\displaystyle y-y{\frac {V}{M}}}$	maximum storage quantity
${\displaystyle y\left(1-{\frac {V}{M}}\right)}$	maximum storage quantity (modified)
${\displaystyle y\left(1-{\frac {V}{M}}\right)/2}$	average storage quantity
${\displaystyle {\frac {y}{2}}\left(1-{\frac {V}{M}}\right)C_{L}T}$	Storage costs per period
${\displaystyle {\frac {y}{2}}\left(1-{\frac {V}{M}}\right)C_{L}T/R}$	Storage costs per unit of measure
${\displaystyle {\frac {y}{2V}}\left(1-{\frac {V}{M}}\right)C_{L}}$	Storage costs per unit of measure because of ${\displaystyle R=VT}$

Figure 3 shows the course of the inventory in this type of production.

Closed manufacturing

Figure 4 Inventory with closed production

In this case, production only reaches the warehouse when the production of a batch has been completely completed, which can be technically (if all parts are embossed together in the final stage) or logistically (if they are transported to the warehouse in one block).

The average inventory is split into a build-up and a tear-down phase, but both have the same average inventory costs. In the build-up phase, positive stocks are produced at high speed and sold at high speed . In the dismantling phase, only sales take place and the free capacity can be used to manufacture other products. ${\displaystyle M}$ ${\displaystyle V}$

In the case of continuous stock outflows, production starts earlier by the length of the production time of a lot, so that when the last piece of stock is delivered, the new lot is stored in full. This increases the average inventory compared to open production:

storage costs

Expression	interpretation
${\displaystyle {\frac {y}{M}}V}$	Storage quantity at the start of production
${\displaystyle \underbrace {{}^{{\frac {y}{M}}V}\!\!\diagup \!\!{}_{2}\;} _{\text{Anfangsbestand}}+\underbrace {{\text{ }}{}^{y}\!\!\diagup \!\!{}_{2}\;{\text{ }}} _{\text{Endbestand}}}$	Ø storage quantity in the construction phase
${\displaystyle {\frac {y}{2}}\left(1+{\frac {V}{M}}\right)}$	Modified equation
${\displaystyle \underbrace {{}^{y-{\frac {y}{M}}V}\!\!\diagup \!\!{}_{2}\;} _{\text{variierender Bestand}}+\underbrace {{}^{{\frac {y}{M}}2V}\!\!\diagup \!\!{}_{2}\;} _{\text{Mindestbestand}}}$	Ø storage quantity in the dismantling phase
${\displaystyle {\frac {y}{2}}\left(1+{\frac {V}{M}}\right)}$	Modified equation
${\displaystyle {\frac {y}{2}}\left(1+{\frac {V}{M}}\right)C_{L}T}$	Storage costs per period
${\displaystyle {\frac {y}{2V}}\left(1+{\frac {V}{M}}\right)C_{L}}$	Storage costs per unit of measure

The situation described at the beginning with infinite production speed is therefore a special case of production with closed manufacturing . ${\displaystyle \lim _{M\to \infty }{\frac {V}{M}}=0}$

Time variant period demand

Contrary to the assumption of the classic batch formula, 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 replacing the time-invariant period requirement, for example, with the mean value of the periods occurring in the planning horizon, or by estimating it with the help of statistical instruments such as regression analysis . ${\displaystyle R}$ ${\displaystyle {\bar {R}}}$

However, this policy either leads to increased storage costs or is associated with the risk of shortages. But even with the adjustments, the classic model is not suitable for optimally determining the lot size and the time of lot production, because they are not determined simultaneously.

Minimum cost lot size

The last section clearly showed that the average stock level depends on the type of production. In the following, the respective minimum cost lot sizes are derived based on this knowledge with the help of the differential calculation.

Infinite production speed

${\displaystyle K(y)=\underbrace {{\frac {R}{y}}C_{R}} _{{\text{Umr }}\!\!{\ddot {\mathrm {u} }}\!\!{\text{ stkosten}}}+\underbrace {{\frac {y}{2}}C_{L}T} _{\text{Lagerkosten}}\rightarrow \min !}$

${\displaystyle {K}'(y)=-{\frac {R}{y^{2}}}C_{R}+{\frac {1}{2}}C_{L}T{\overset {!}{\mathop {=} }}\,0}$ ${\displaystyle \Rightarrow 2RC_{R}=y^{2}C_{L}T}$

${\displaystyle y_{k\min }={\sqrt {\frac {2RC_{R}}{C_{L}T}}}}$ respectively

${\displaystyle y_{k\min }={\sqrt {\frac {2VC_{R}}{C_{L}}}}}$ because of ${\displaystyle R=VT}$

Open manufacturing

${\displaystyle K(y)=\underbrace {{\frac {R}{y}}C_{R}} _{{\text{Umr }}\!\!{\ddot {\mathrm {u} }}\!\!{\text{ stkosten}}}+\underbrace {{\frac {y}{2}}\left(1-{\frac {V}{M}}\right)C_{L}T} _{\text{Lagerkosten / ME ZE}}\rightarrow \min !}$

${\displaystyle {K}'(y)=-{\frac {R}{y^{2}}}C_{R}+{\frac {1}{2}}\left(1-{\frac {V}{M}}\right)C_{L}T{\overset {!}{\mathop {=} }}\,0}$

${\displaystyle y_{k\min }={\sqrt {\frac {2RC_{R}}{\left(1-{\frac {V}{M}}\right)C_{L}T}}}}$ respectively

${\displaystyle y_{k\min }={\sqrt {\frac {2VC_{R}}{(1-{\frac {V}{M}})C_{L}}}}}$ because of ${\displaystyle R=VT}$

Closed manufacturing

${\displaystyle K(y)=\underbrace {{\frac {R}{y}}C_{R}} _{{\text{Umr }}\!\!{\ddot {\mathrm {u} }}\!\!{\text{ stkosten}}}+\underbrace {{\frac {y}{2}}\left(1+{\frac {V}{M}}\right)C_{L}T} _{\text{Lagerkosten / ME ZE}}\rightarrow \min !}$

${\displaystyle {K}'(y)=-{\frac {R}{y^{2}}}C_{R}+{\frac {1}{2}}\left(1+{\frac {V}{M}}\right)C_{L}T{\overset {!}{\mathop {=} }}\,0}$

${\displaystyle y_{k\min }={\sqrt {\frac {2RC_{R}}{\left(1+{\frac {V}{M}}\right)C_{L}T}}}}$ respectively

${\displaystyle y_{k\min }={\sqrt {\frac {2VC_{R}}{\left(1+{\frac {V}{M}}\right)C_{L}}}}}$ because of ${\displaystyle R=VT}$

This means that the minimum cost lot sizes can now be determined if the conditions of the underlying model are met.

Profit maximum lot size

Figure 5 Profit maximization problem

A profit-maximizing company is usually faced with elastic demand that decreases as prices rise. This relationship is generally described in business administration using the price-sales function . The relationship between the price-sales function and the revenue function is shown as in Figure 5. ${\displaystyle p(R)=a-bR}$

In this case, the profit maximum is reached with Cournot quantities , i.e. the quantity at which the difference between the proceeds and costs for the production of this quantity is at its maximum ( monopoly case ). The variable costs of production, storage and retrofitting costs occur as costs. The optimization problem for the case of open production can be formulated as follows:

${\displaystyle G(R,y)=aR-bR^{2}-kR-{\frac {R}{y}}C_{R}-{\frac {y}{2}}\left(1-{\frac {R}{MT}}\right)C_{L}T\rightarrow \max !}$

${\displaystyle {\frac {\partial G}{\partial R}}=a-2bR-k-{\frac {C_{R}}{y}}+{\frac {y}{2}}{\frac {1}{MT}}C_{L}T{\overset {!}{\mathop {=} }}\,0}$

The profit maximum amount is then given by:

${\displaystyle R(y)={\frac {a-k-{\frac {C_{R}}{y}}+{\frac {y}{2}}{\frac {1}{M}}C_{R}}{2b}}}$

The winning maximum lot size then corresponds to:

${\displaystyle {\frac {\partial G}{\partial y}}={\frac {C_{R}}{y^{2}}}-{\frac {1}{2}}\left(1-{\frac {R}{MT}}\right)C_{L}T{\overset {!}{\mathop {=} }}\,0}$

${\displaystyle y(R)={\sqrt {\frac {2C_{R}R}{(1-{\frac {R}{MT}})C_{L}T}}}}$

The procedure can also be transferred to the situation with free competition and constant sales prices. This gives us another, important result that the maximum profitable lot size with free capacities corresponds to the minimum cost lot size.

However, the initial situation changes fundamentally if additional, more realistic restrictions such as scarce capacities and competing goals such as complete sales satisfaction have to be taken into account, so that the relative contribution margin per unit of time comes into question as a decision criterion.

Determination of the optimal order quantity

The classic lottery formula can be transferred to other problems that are based on the same scenario. Thus, among other things, the determination of optimal order quantities is one of the tasks of procurement logistics , with the total costs also being made up of linear storage costs, depending on the quantity, and quantity-independent, degressive order costs. The order costs for procurement logistics and the set-up costs for batch production thus describe exactly the same problem. Based on the symbolism of the classic lot size formula, the variables used in the adapted model can be described as follows:

• Variables:
  • ${\displaystyle y}$ - order quantity
  • ${\displaystyle T}$ - Period length
• Parameter:
  • ${\displaystyle V}$ - Storage rate or sales speed
  • ${\displaystyle R}$ - maximum sales volume
  • ${\displaystyle k}$ - variable costs
  • ${\displaystyle M}$ - finite speed of delivery ${\displaystyle M>V}$
  • ${\displaystyle C_{\mathrm {R} }}$ - Quantity-independent order costs
  • ${\displaystyle C_{\mathrm {L} }}$ - storage costs depending on the quantity

The optimal order quantity is with infinite speed of delivery

${\displaystyle y_{k\min }={\sqrt {\frac {2RC_{R}}{C_{L}T}}}}$

A situation in which a minimum amount must be kept in stock corresponds to production with closed production and corresponds to the optimum

${\displaystyle y_{k\min }={\sqrt {\frac {2RC_{R}}{\left(1+{\frac {V}{M}}\right)C_{L}T}}}}$

The premises of the use of classic lot sizes must be taken into account when determining optimal order quantities, which inevitably has advantages and disadvantages.

Evaluation and Limits

The criticism of the classic lottery formula is primarily aimed at the assumptions on which it is based. Above all, the restriction to single-stage or highly restricted multi-stage production is criticized: a transfer to multi-stage production processes is only possible if there are no rejects and interruptions in production at identical speeds of the stages, which is also hardly realistic.

On the other hand, there may be other restrictions that are not taken into account in the methodology. So it can happen that the production of an optimal lot size is not possible and when capacity is limited, lots that are too large are created in order to save set-up times; A lack of storage capacity, on the other hand, can force suboptimal smaller lots. The permissible duration of the storage of products (e.g. in food production) also sets limits to classic lot size optimization. Immediate production of an optimal lot could also not be financially viable because the time lag between production and sales can lead to liquidity problems.

Another basic premise - constant, continuous stock removal - is not or only very rarely found in reality, because only in this case can the storage costs be precisely determined and shortages avoided. The isolated consideration of each variety based on free capacities is also unrealistic because they compete for storage and machine capacities. In the optimum, all varieties have to be placed equally often in order to solve the problem of sequence planning in the event of interdependencies between the varieties. In the case of scarce capacities, taking full demand satisfaction into account, the model does not necessarily lead to an optimal solution, so that compromise solutions may have to be considered, whereby the solution approach does not provide any assistance. As a result, the model is severely limited in its application by the strict and unrealistic assumptions and the problem of process planning is not solved with this method. The classic lottery formula has more of a textbook character than a practical use.

As could be shown using the example of open and closed production, the classic lot formula was adapted and expanded in various ways to more realistic basic requirements. Among other things, the immediate order fulfillment has been replaced by a time-delayed one, order backlogs have been included in the calculation, variable set-up processes etc. have been incorporated into the formula. These modifications did not adequately address the fundamental problem of the lack of adaptation to requirements. Significant progress in the area of lot size optimization was only made with the dynamic lot size determinations, which allow a more complex problem recording.

swell

1. ↑ Kurt Andler: Rationalization of the production and optimal batch size . Munich: R. Oldenbourg, 1929
2. ↑ Andler refers in his dissertation to an unspecified Hrs. in the journal Technik und Betrieb , Volume 1 (1924), pp. 81–83; Rights held by Orell Füssli Verlag, Zurich
3. ^ FW Harris (1913) How Many Parts to Make at Once Factory: The Magazine of Management 10 (2): 135-136,152. Also reprinted in Operations Research 38 (6): 947-950, 1990; Online (PDF; 292 kB)
4. ↑ Georg Krieg (2005) New findings on Andler's lot size formula . Working paper, Catholic University of Eichstätt-Ingolstadt
5. ^ Wallace J. Hopp and Mark L. Spearman (2000) Factory Physics: foundations of manufacturing management ; 2nd ed. McGraw-Hill Higher Education; ISBN 0-256-24795-1