Based on fuzzy sets, a fuzzy inference system can be built. In such a system, fuzzy rules are implemented for modeling, which in turn make it possible to carry out the process of fuzzy inference. Fuzzy inference is a multi-step process in which:

1. Quantitative variables are transformed into linguistic concepts,
2. Linguistic concepts are modeled on a rule base that reflects our knowledge of the problem,
3. The linguistic effects of inference back are converted into quantitative variables.

In practice, many decision-making processes are not formal and do not explicitly refer to the principles of classical logic. Fuzzy systems help mimic human reasoning and generally perform well in complex situations. A classical fuzzy system consists of four elements: a rule base, a blurring block, an inference block and a sharpening block (Figure 1).

Figure 1. Diagram of the fuzzy system

The input data (signals) are fed into the blurring block, where they are transformed from quantitative to qualitative form, expressed in linguistic form. The input data transformed in this way is represented by fuzzy sets, which boils down to determining the membership function. In the inference block, rules are invoked whose premises are satisfied. These rules lead to the determination of a fuzzy set that represents the resulting conclusion. Since the product of the inference block is a fuzzy set, it should be transformed in the sharpening block to a numerical form, which will be the output signal. Thus, the input variables as well as the output variables are real values, so in practice, the range of their variability is scaled to an interval, generally speaking [-1; 1]

The rule base represents qualitative knowledge, which can come from a variety of sources: expert knowledge, qualitative modeling, automatic knowledge extraction algorithms. The rules are defined in the form of IF-THEN expressions, so they refer to the implication known from classical logic. Premises (predecessors of implications) consist of linguistic expressions linked together by operators (conjunctions) of conjunction or alternative, while conclusions (successors of implications) are generally single expressions. An example of an inferred rule might be as follows:

Yes, as noted, the premises are linguistic expressions, in the example given they are the expressions “the stock of x1 is low”, “the demand for x1 in the last week was high”. The terms were connected by a conjunction of conjunction expressed in natural language by the word “AND”. Linguistic expressions contain imprecise terms “is low”, “was high”, which require the use of fuzzy sets, which are obtained in the process of blurring numerical data. In the example, the data would be on inventory and demand volumes over a specific time interval. The conclusion “the stock should be replenished to a high level” is also given in the form of a linguistic expression that must be converted to numerical form.

It is worth noting that the fuzzy equivalent of implication can be defined in an infinite number of ways. This means that many types of inference rules can be used in the inference process.

Fuzzy systems are used, among others, in electronic control systems, in medicine, in data mining tasks, in expert systems, etc. It is also possible to build a system to determine the size of an order based on current inventory, projected demand or inventory costs.

This article was written thanks to the funds from the European Union’s co-financing of the Operational Program Intelligent Development 2014-2020, a project implemented under the competition of the National Center for Research and Development: under the “Fast Track” competition for micro, small and medium-sized entrepreneurs – competition for projects from less developed regions under Measure 1.1: R&D projects of enterprises Sub-measure 1.1.1 Industrial research and development work carried out by enterprises. Project title: “Developing software to improve forecast accuracy and inventory optimization from the perspective of customer and supplier collaborating in the supply chain using fuzzy deep neural networks.