Fuzzy logic applied to inventory management

Introduction

You have certainly heard of Artificial Intelligence. Perhaps you remember the theme from fiction films, such as “Bicentennial Man” or “AI – Artificial Intelligence”. Or those confrontations created to compare performances between man and machine, like the chess challenge between the supercomputer Deep Blue of IBM and super champion Garry Kasparov, 20 years ago. Or, more recently, the developments of Google's AI division, Google Brain, and its research that managed to make two computers communicate with each other, in a completely confidential way.

Artificial Intelligence is part of a branch of study of computing that has been studied by researchers for decades and has varied applications in the most diverse areas of science: computing soft ou soft computing. To better understand what it is about, it is important to define the “hard computing” or conventional computing.

Computers have helped decision makers through their vast information processing potential, which far surpass human capacity with regard to volume, speed and accuracy. Through the concepts of logic, we program computational algorithms that perform ordered tasks and provide satisfactory answers to many of our problems. These existing algorithms today use, for the most part, Probability Theory, in which modeling, definition of variables and calculations use measures of position and dispersion, such as mean and standard deviation. The algorithms also use the theory of classical logic, represented, for example, by AND/OR operators. Conventional computing therefore requires exact mathematical modeling and usually requires a high computational cost.

Already in computing soft, Probability Theory does not necessarily form the conceptual basis for algorithm operations. Its different methodologies (genetic algorithms, neural networks, logic fuzzy, etc) aim to exploit the imprecision tolerance inherent in human thinking and the real world to deliver robust solutions at a low computational cost. This adaptive capacity to inexact conditions is an interesting attribute when we analyze the evolution of businesses throughout history. Over the years, the complexities that permeate business environments have grown a lot. At the same time, the pressure for faster decision-making has also increased. These two factors, combined, bring uncertainty, ambiguity and imprecision to the managerial environment. It is not by chance that the application of computational techniques soft in multiple areas, from industrial to financial and administrative.

In the case of logic fuzzy, researcher Lofti Zadeh, in the mid-60s, developed the concept of sets fuzzy, which opposes the Boolean logic of 0 or 1, adding the possibility of countless degrees of membership between these two values. If, according to classical set theory, an element can only belong or not belong to a set, in reality fuzzy an element can belong partially to that set. The answers do not necessarily need to be “yes” or “no”, to be able to assume values ​​such as “I believe so”, “I strongly believe not” or “I am almost absolutely sure that yes”. In this way, it was possible to transform these linguistic values, laden with inaccuracies, into values ​​that could be programmed. From there the logic fuzzy it became a research topic, and its practical applications did not take long to occur, as we can see in table 1. What is proposed in this article is to go into more detail on an application of logic fuzzy in the context of inventory management.

FUZZY LOGIC IN INVENTORY MANAGEMENT

In the context of inventory management, works using logic fuzzy have been developed since the 90s. Whether for determining the Economic Purchase Lot, for solving continuous and periodic review models, for determining the order size in the newsboy problem or even in the ABC classification of items, see a number of published academic papers.

In the context of stock allocation, there is a work that created a decision-making system for stock allocation (or in English DMS-SA), which used as inputs the demand and maintenance cost of any item to determine of the quantity of items allocated in each retailer. The work described in this article is a revision of the DMS-SA (therefore called DMS-SA Rev.), considering more cost items in the composition of the input variables.

In a distribution system consisting of a warehouse and a quantity N points of sale, treated as retailers, we need to distribute a stock E⁰, with the constraint that the sum of the quantities demanded Dⁿ by retailers n exceed the existing quantity in the warehouse, that is, a decision model is needed that defines, in view of the size of the orders oⁿ placed by retailers, how much stock qⁿ should be allocated to each of them, considering two performance parameters: the service level (represented by the allocated quantity divided by demand) and the total cost (represented by the sum of maintenance costs, order cost, transportation cost, lack). A system summary is shown in Figure 1.

Figure 1 – Scheme of the distribution system containing warehouse and N retail outlets.

The purpose of the DMS-SA Rev. is to allocate inventories in order to optimize the trade-offs level of service and total cost of the distribution system as a whole. The system's service level is calculated as the average of perceived service levels at each retailer, while the total cost is the sum of costs incurred at each POS.

A summary of model steps fuzzy is shown in Figure 2, which will be described below.

Figure 2 – Structure of the DMS-SA Rev.

1) Fuzzifying interface

The input variables of the model are the demand D, the maintenance cost cm, the shortage cost cf and the fixed cost cfi (represented by the sum of order and transport costs).

As mentioned earlier, the logic fuzzy allows variables to assume infinite values ​​between 0 and 1, with 0 being the minimum value and 1 the maximum value. It is necessary to define, with the help of a process specialist, which would be the maximum and minimum values ​​of that variable, according to the context of the operation. For example, for order cost, a storekeeper could determine that his costs are never less than 100 nor greater than 1000. In this case, the low function would take a maximum value of 1 in 100 and the high function would take a value of 1 in 1000 .

Still on the variables, we have to define how many membership functions will be established for each variable. In the example above, the low and high functions were mentioned and, therefore, there would be two membership functions for the variable. For the research in question, three membership functions were used: low, medium and high. The specialist also defines the type of function (triangular, trapezoidal, among others) and the most suitable format for the operation. For the cost of absence variable, the functions chosen are of the trapezoidal type, with the format shown in Figure 3. Each function is represented by a trapezoidal figure of a different color.

Figure 3 – Failure cost variable, represented by three trapezoidal membership functions.

The trapezoidal function allows a range of values ​​to assume the maximum degree of membership 1. For the average function in Figure 2, for example, we can see from the graph that shortage costs between 45 and 55 would have an average membership degree equal to 1.

With the functions defined, we will finally be able to transform the input variables of the model into variables fuzzy.

2) Base rule

After the panel of experts, when the membership functions and their values ​​are defined to represent the model variables, it is necessary to create the Base Rules: set of IF-THEN rules that relate the inputs and provide the output. In the proposed model, the output is called the Priority Index, or PI. We would have something like “IF the maintenance cost is low, the fixed cost is high, the outage cost is high and the demand is high, THEN the PI will be high”. For each possible combination of interaction between the variables, a rule similar to the one presented must be established. At the end of the simulation, each retailer will have its respective PI', which will serve as the basis for the proportional allocation of stock E⁰ throughout the system. The higher the PI, the more product the retailer will receive. It is important to emphasize that the output of this step is the PI', still in number format fuzzy.

3) Defuzzifying interface

With the value of PI' in format fuzzy, a method is chosen to turn the output into a real PI number, which can be used for subsequent calculations. In the research in question, the centroid method was used, which, in short, consists of obtaining the centroid of the figure demarcated by the output membership functions and the indices obtained from the base rule.

To simulate the model, four scenarios were created, varying the number of retailers in the system: 2, 3, 5 and 10. For each of these scenarios, 200 simulations were performed, in which demand and cost values ​​varied randomly within intervals pre-established. After carrying out each simulation, the service level and total cost values ​​were calculated, so that with the data from the 200 simulations, it was possible to observe the performance of the DMS-SA Rev model. against three other comparative models: the DMS-SA (model fuzzy original), the Even Allocation (division of the quantity in stock equally between the retailers) and the Fair Share Allocation (division proportional to the order size of each retailer).

The results for the scenario of 10 retailers, service level and total cost are presented in Figures 4, through graphs of the type boxplot.

Figure 4 – Level of service and Total cost of the distribution system for the scenario of 10 retailers.

Statistical tests were performed on these results in order to verify whether there was, in fact, a significant difference between the models. What was observed was a small loss in terms of service level, but a substantial gain in model costs. fuzzy proposed compared to the other three models. This gain increases as the number of retailers in the system increases, reaching 8% in scenarios with 15 or more retailers. In day-to-day business, the manager can conclude that a loss in the level of service is justified, if it is accompanied by a considerable reduction in costs, demonstrating that such a model has potential for a real application.

Conclusion

the logic fuzzy, as well as other computational logics soft, are increasingly present in day-to-day business. Proximity to human reasoning and the ability to deal with uncertainties and subjectivities form the main attributes that give the soft computing a great advantage for applications in the context of administration.

The positive result of the research presented in this article is, in a certain way, fragile, given the absence of a probabilistic model in the comparison of performance. Still, he brings out the ability of logic fuzzy to deal with inaccurate information and provide satisfactory results at low computational costs. What is expected is, increasingly, the application of such techniques by company managers in the search for robust solutions to the challenges faced.

References

• ALVARENGA, Henry. Review of Fuzzy Inference Systems Applied to Inventory Allocation Decisions. 2016. 53 f. Dissertation (Master in Administration) – Coppead Institute of Administration, Federal University of Rio de Janeiro, Rio de Janeiro.
• AZADEGAN, Arash et al. Fuzzy logic in manufacturing: A review of literature and a specialized application. International Journal of Production Economics, v. 132, no. 2, p. 258-270, 2011.
• KO, Mark; TIWARI, Ashutosh; MEHNEN, Jörn. A review of soft computing applications in supply chain management. Applied Soft Computing, v. 10, no. 3, p. 661-674, 2010.
• XIE, Ying; PETROVIC, Dobrilla. Fuzzy‐logic‐based decision‐making system for stock allocation in a distribution supply chain. Intelligent Systems in Accounting, Finance and Management, v. 14, no. 1-2, p. 27-42, 2006.