Lot size

In the context of industrial management and production management, the batch size is the amount of products in a production order in the case of batch production , which go through the stages of the production process as closed items.

Classification features of the models to determine the optimal lot size

Several parameters relevant to practice can be incorporated into the modeling of corresponding lot size problems:

Level of information

Quality of the data used in the modeling: in contrast to deterministic information, stochastic data, i.e. data subject to fluctuations, are expressed in higher stocks or production or set-up times.

Development over time

with static and dynamic demand.

Choice of planning period

the width of the planning horizon to be taken into account when planning . In particular, with rolling planning, one assumes finite planning periods. However, dynamic models assume infinite planning horizons.

Number of Products

Scope of the production program as part of a one-product or multi-product company .

Number of planning levels

Depth of the production structure with single or multi-level manufacturing.

Consideration of capacities

capacitive or non-capacitive consideration with the inclusion of available resources or financial means.

Costs to Consider

Consideration of set-up , storage and production costs .

Type of product transfer

Type of exchange of lots between individual stages or intermediate storage. With closed production , the complete lot leaves the last production stage, while with open production the first completed piece of a lot can be passed on.

Product structure

Type of manufacturing with serial, converging, diverging and general manufacturing.

Consideration of shortages

If shortfalls are allowed, a distinction is made between delayed amounts that are subsequently delivered and orders that are completely lost.

Manufacturing speed

simple models assume an infinitely high production speed, more complex models a finite one, which is usually assumed to be constant. If necessary, sequence-dependent set-up times can also be taken into account.

aims

Most models try to minimize the total cost . However, some models relate to maximizing the level of service ( readiness for delivery ) or to the most even possible utilization of capacity .

Types of lot sizes

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A distinction is made between the following lot sizes:

Technical lot size

This is understood to mean the net requirement of the lot at a specific point in time in production.

Capacitive lot size

This lot size is used to optimum utilization of the capacity to achieve.

Economic lot size

It is chosen so that the costs for the requirement are kept as low as possible.

Logistic lot size

Different loading space capacities of means of transport and transport quantities are the reasons for this size.

Bottleneck-oriented lot size

This lot size results from the trade-off that a customer has an urgent need for some good, but the capacities are either very scarce or overloaded.

Lot size models

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The methods available for determining the lot size are divided into three groups:

• Static lot sizing procedures

With the static lot-sizing procedure, the lot size is formed exclusively on the basis of quantity specifications from the respective material master record. There are different criteria according to which the lot size can be calculated:

Exact lot size

Fixed lot size

Refill to the maximum stock

Order point disposition with or without consideration of external requirements

• Periodic lot-sizing procedure

With the period lot-sizing procedure, the required quantities of one or more periods are combined into one lot. The number of periods that are to be combined into an order proposal can be set as desired. One differentiates:

Day lot size

Week lot size

Monthly lot size

Lot sizes according to flexible period lengths, analogous to accounting periods (period lot sizes)

• Optimizing lot size procedures

With the optimizing lot-sizing method, required quantities of several periods are combined into one lot-size, whereby an optimum cost is determined between lot-size-fixed costs and storage costs. The different optimization methods only differ in the type of cost minimum.

Other models

Static-deterministic models (one product)

What these models have in common is that the demand is constant (static) and known in advance (deterministic). The basic model is the uncapacitated, single-stage, single-product model with an infinite production speed. For the details of the basic model, see the classic lot formula .

Finite production speed

The costs per unit of time result

${\displaystyle K(\tau )={\frac {f}{\tau }}+{\frac {1}{2}}l_{\mathrm {max} }c}$

With

• ${\displaystyle l_{\mathrm {max} }}$and are auxiliary variables: ${\displaystyle t^{p}}$
  • ${\displaystyle t^{p}=q/p}$
  • ${\displaystyle l_{\mathrm {max} }=q-t^{p}b=q(1-\rho )=b(1-\rho )\tau }$
• ${\displaystyle \tau }$ - cycle duration
• ${\displaystyle c}$ - Storage costs per unit of time
• ${\displaystyle f}$ - Fixed setup costs per setup process
• ${\displaystyle b}$ - Demand per time (sales speed)
• ${\displaystyle p}$ - production per time (production speed)
• ${\displaystyle q}$ - lot size
• ${\displaystyle \rho :=b/p}$ - Share of occupancy time on the machine

The costs are thus or . ${\displaystyle K(\tau )={\frac {f}{\tau }}+{\frac {1}{2}}b(1-\rho )\tau c}$ ${\displaystyle K(q)={\frac {fb}{q}}+{\frac {1}{2}}q(1-\rho )c}$

By deriving and setting to zero you get the optimal sizes or . ${\displaystyle q^{*}={\sqrt {\frac {2bf}{c(1-\rho )}}}}$ ${\displaystyle \tau ^{*}={\sqrt {\frac {2f}{c(1-\rho )b}}}}$

Further generalizations

• Discounts: A distinction is made between two different types:
  • The discount applies to the entire order quantity. Example: € 5 / piece - from a quantity of 1000 pieces: € 4 / piece for the entire amount
  • The discount applies to all units above a certain limit. E.g .: 5 € / piece - from a quantity of 1000 pieces: 4 € / piece for the quantity that exceeds 1000.

In both cases, the first step is to use the standard model for the individual discount scales with corresponding minimal costs. The global minimum is obtained by comparing the individual minima.

• Jump-fixed storage costs
• Variable cost prices

Static-deterministic models (several products)

In the event that there is no coupling between the products, the optimal order policy results from the separate application of the standard model to the individual products. Couplings result, for example, from lots from different products for which the fixed lot costs only arise once. The order quantity model with collective orders is analogous, in which the fixed order costs are incurred only once for the collective order. There are also models with storage capacity restrictions and models in which all products are to be manufactured on a bottleneck machine . The latter is known as the problem of optimal brand switching .

Sample model

A simple, capacity-constrained model looks like this:

• The individual products are stored in a common warehouse. The capacity is [m²] ${\displaystyle \kappa }$
• ${\displaystyle b_{j}}$be the demand of the product in units of measure per time unit [ME / ZE] ${\displaystyle j=1,2,\dots ,n}$
• ${\displaystyle f_{j}}$ Fixed lot production costs / (fixed order costs) for product j
• ${\displaystyle c_{j}}$ Storage costs for product j
• ${\displaystyle k_{j}}$ Capacity requirement of a ME of product j (e.g. m² of floor space)
• ${\displaystyle w:=c_{j}b_{j}/2}$ is used to simplify the representation

The objective function is

${\displaystyle \min K(q)=\sum _{j=1}^{n}(b_{j}/q_{j}f_{j}+0{,}5c_{j}q_{j})}$

Under the constraints

${\displaystyle \sum _{j=1}^{n}q_{j}k_{j}\leqq \kappa }$

In the worst case, all lots will be published at the same time. The sum of the lot sizes to be determined must therefore be less than or equal to the storage capacity . ${\displaystyle \kappa }$

${\displaystyle q_{j}>0}$

Solution of the model

If the storage capacity is neglected and the formula of the standard model is used for each product, the optimum batch size or cycle duration is obtained:

${\displaystyle q_{j}^{e}={\sqrt {\frac {2b_{j}f_{j}}{c_{j}}}}\quad {\text{und}}\quad \tau _{f}^{e}={\frac {q_{j}^{e}}{b_{j}}}={\sqrt {\frac {f_{j}}{w_{j}}}}}$

If the capacity restriction is met, the optimal solution has been found. If not so:

${\displaystyle \sum _{j=1}^{n}q_{j}k_{j}=\kappa }$

The storage capacity is fully used. The optimal lot size results from Lagrange multiplication with the capacity restriction:

${\displaystyle L(q,\lambda )=\sum _{j=1}^{n}(b_{j}/q_{j}f_{j}+0{,}5c_{j}q_{j})+\lambda \left(\sum _{j=1}^{n}q_{j}k_{j}-\kappa \right)}$

If you now divide it up and down, you get a system of equations with n + 1 equations and unknowns. ${\displaystyle q_{1},q_{2},\dots ,q_{n}}$ ${\displaystyle \lambda }$

You get

${\displaystyle q_{j}^{*}={\sqrt {\frac {2b_{j}f_{j}}{c_{j}+2\lambda k_{j}}}}}$

the optimal lot sizes depending on . ${\displaystyle \lambda }$

This can be inserted into the partial derivatives and receives a strictly monotonic function that takes on the value at one point : ${\displaystyle \kappa }$

${\displaystyle \sum _{j=1}^{n}k_{j}{\sqrt {\frac {2b_{j}f_{j}}{c_{j}+2\lambda k_{j}}}}=\kappa }$

This point can be determined using iterative methods such as Newton's method .

Static-deterministic models with multi-level production

In the case of multi-level production, batch sizes must be formed not only for each product, but also for each production level. The design of a specific problem depends heavily on the product structure. With a converging structure, several individual parts are put together to form an overall product. With a divergent structure, various end products are made from an intermediate product. Mixtures can also occur.

Dynamic-deterministic models

→ Main article : Dynamic lot size determination

With dynamic-deterministic models, the planning period is divided into a finite number of equally long periods, while the static-deterministic models usually assume an infinitely long planning period. The basic model by Wagner and Whitin, sometimes also called the Wagner-Whitin model , only deals with one product, with only one production stage and without capacity limits. It can be solved with dynamic optimization . It can be interpreted as a warehouse location problem : The opening of a location then corresponds to the issue of a lot and the customers correspond to the individual periods. The Wagner-Whitin algorithm delivers an optimal result. There are also some heuristics:

• sliding economic or dynamic lot size ( least-unit-cost method )
• Piece compensation period ( part-period method )
• Silver-meal heuristic
• Procedure by Groff

Stochastic models

Stochastic models are based on random requirements. The best known is the newspaper boy model .

literature

• Wolfgang Domschke , Armin Scholl , Stefan Voss : Production planning. Process organizational aspects. 2nd, revised and expanded edition. Springer, Berlin et al. 1997, ISBN 3-540-63560-2 .
• Horst Tempelmeier : Material logistics. Basics of requirement and lot size planning in PPS systems. 3rd, completely revised and expanded edition. Springer, Berlin et al. 1995, ISBN 3-540-58928-7 .
• Sigfrid Gahse: Warehouse disposition with electronic data processing systems, Neue Betriebswirtschaft, 18th year (1965), p. 4

Individual evidence

1. ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 69-75.
2. ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 79-81.
3. ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 84.
4. ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 90.
5. ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 87-89.
6. ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 103.
7. ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 115-128.