Lot-sizing models differ in their underlying assumptions and in the details they incorporate. Figure 2.2 illustrates a classification of lot-sizing problems in which each characteristic strongly impacts the modeling and complexity of the problem.
Lot-sizing problems
Multi-facility Single facility
Single level Multi-level Multi-level
Single item Multi-item
Uncapacitated Capacitated Deterministic demand Stochastic demand Static demand Dynamic demand
Figure 2.2: A category of lot-sizing problems
Lot-sizing problems can be characterized by a variety of aspects and classification criteria, which are explained in the following subsections.
2.3.1 Planning Horizon
A planning horizon is the length of time into the future for which plans are made. The length of the horizon can be finite or infinite. A finite planning horizon is typically accompanied by a time-varying demand and an infinite planning horizon by a constant demand rate. Furthermore, the time horizon can be divided into discrete or continuous time periods.
As defined by Belvaux and Wolsey (2000), lot-sizing problems can be either small bucket or big bucket. Big bucket problems allow for the production of many items at the same time period without taking into account sequencing issues. Small bucket models
considers short time periods in order to be able to model start-ups, switch-offs and/or changeovers. The small bucket models are then split further into those in which only one item can be setup per period, and those with possibly two setups per period.
2.3.2 Number of Products
Single item models consider one type of product at a time. Multi-item models consider a number of products simultaneously. These products must have at least one interrelating or binding factor such as budget, capacity constraint, or a common setup.
2.3.3 Number of Levels
If multiple items are considered, they can either be from a single level of the product structure, i.e. multiple independent final products are considered, or they can be on different levels, i.e. parent-component relationships between the items are present. In such multi-level production systems, end products are assembled from intermediate products (sub-assemblies), which might in turn require raw materials or parts for production. The output of one stage is thus the input for the next stage.
2.3.4 Capacity Constraints
Resources in a manufacturing system contain manpower, equipment, machines, budget, and so forth. If the models assume unlimited capacities of resources, they are considered as uncapacitated problem. Capacitated models recognize that some resources are given in a limited number or amount so that planning and scheduling systems need to avoid over utilizing these resources.
In some cases, it is essential to consider capacity utilization more accurately in order to achieve a feasible production plan. For instance, the capacity utilized when a machine begins or finishes a production batch, or when a machine shifts from one product to
another, may need to be considered. In such cases, models deal with setup times, changeover times, or sequencing restrictions.
2.3.5 Setup Structure
A particular setup is often necessary to prepare a machine for the production of a specific product if this machine produces different types of products. Whenever this changeover causes setup times and/or cost, a lot-sizing problem arises.
Setup times implies the capacity consumed because of cleaning, warming, machine adjustments, calibration, inspection, test runs, tool changes, and so forth, when the production for a new product begins. Setup times can be included explicitly in a model. However, due to the complexity in such a case, they are often incorporated indirectly via the setup costs (Jans & Degraeve, 2008). Setup costs and setup times, are generally modeled by considering zero-one variables in the mathematical models and make the problem solving harder.
2.3.6 Demand
Another important characteristic of lot-sizing problem is the nature of demand. Static demand models assume that parameter’s value does not vary over the planning horizon, while dynamic demand models allow for variation over time. If the demand value is known in advance, the demand stream is considered as deterministic. If the demand is based on distribution or a measure of uncertainty, it is considered as stochastic.
Independent demand refers to the demand for a product which is independent of demand for other items. Independent demand for end products is triggered by the market. Dependent demand for components is triggered by scheduled production on superior levels.
2.3.7 Inventory Shortage
Shortage occurs when demand exceeds the available inventory for an item, and can be divided into two categories, namely backordering and lost sale. Backordering occurs when it is probable to fulfill demand of the current period in the next time period. If demand cannot be satisfied at all, lost sale can happen. The combination of backordering and lost sale is also plausible. However, both cases incur penalty cost as they have a negative impact on customer satisfaction.
2.3.8 Deterioration
Another aspect that affects the problem complexity is deterioration or shelf life constraint where items can be held only for a limited lifetime. Deterioration refers to a process in which inventories undergo a change in storage over time, such that they become partially or completely unsuited for consumption and therefore, may impose additional costs for inventory storage. Ignoring deterioration of the items may bring about misleading replenishment policies and shortage in demand which in turn incurs additional shortage cost.