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 setup 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 oneproduct or multiproduct company .
 Number of planning levels
 Depth of the production structure with single or multilevel manufacturing.
 Consideration of capacities
 capacitive or noncapacitive consideration with the inclusion of available resources or financial means.
 Costs to Consider
 Consideration of setup , 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, sequencedependent setup 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
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.
 Bottleneckoriented lot size
 This lot size results from the tradeoff that a customer has an urgent need for some good, but the capacities are either very scarce or overloaded.
Lot size models
The methods available for determining the lot size are divided into three groups:
 Static lot sizing procedures
With the static lotsizing 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 lotsizing procedure
With the period lotsizing 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 lotsizing method, required quantities of several periods are combined into one lotsize, whereby an optimum cost is determined between lotsizefixed costs and storage costs. The different optimization methods only differ in the type of cost minimum.
Other models
Staticdeterministic 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, singlestage, singleproduct 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
With

and are auxiliary variables:
  cycle duration
  Storage costs per unit of time
  Fixed setup costs per setup process
  Demand per time (sales speed)
  production per time (production speed)
  lot size
  Share of occupancy time on the machine
The costs are thus or .
By deriving and setting to zero you get the optimal sizes or .
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.
 Jumpfixed storage costs
 Variable cost prices
Staticdeterministic 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, capacityconstrained model looks like this:
 The individual products are stored in a common warehouse. The capacity is [m²]
 be the demand of the product in units of measure per time unit [ME / ZE]
 Fixed lot production costs / (fixed order costs) for product j
 Storage costs for product j
 Capacity requirement of a ME of product j (e.g. m² of floor space)
 is used to simplify the representation
The objective function is
Under the constraints
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 .
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:
If the capacity restriction is met, the optimal solution has been found. If not so:
The storage capacity is fully used. The optimal lot size results from Lagrange multiplication with the capacity restriction:
If you now divide it up and down, you get a system of equations with n + 1 equations and unknowns.
You get
the optimal lot sizes depending on .
This can be inserted into the partial derivatives and receives a strictly monotonic function that takes on the value at one point :
This point can be determined using iterative methods such as Newton's method .
Staticdeterministic models with multilevel production
In the case of multilevel 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.
Dynamicdeterministic models
With dynamicdeterministic models, the planning period is divided into a finite number of equally long periods, while the staticdeterministic models usually assume an infinitely long planning period. The basic model by Wagner and Whitin, sometimes also called the WagnerWhitin 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 WagnerWhitin algorithm delivers an optimal result. There are also some heuristics:
 sliding economic or dynamic lot size ( leastunitcost method )
 Piece compensation period ( partperiod method )
 Silvermeal 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 3540635602 .
 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 3540589287 .
 Sigfrid Gahse: Warehouse disposition with electronic data processing systems, Neue Betriebswirtschaft, 18th year (1965), p. 4
Individual evidence
 ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 6975.
 ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 7981.
 ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 84.
 ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 90.
 ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 8789.
 ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 103.
 ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 115128.