Forecast Method Based on the Time-Delay Mean Field Boltzmann Machine Oleg Grygor, Eugene Fedorov and Olga Nechyporenko Cherkasy State Technological University, Shevchenko blvd., 460, Cherkasy, 18006, Ukraine Abstract The problem of insufficient forecast efficiency for supply chain management is solved. A neural network forecast model based on the Time-Delay Mean Field Boltzmann Machine with time delays in the visible layer has been created. In the process of adjusting the structure of the developed model, the length of the hidden layer was determined, and the calculation of the model parameters was carried out on the basis of the parallel computing platform CUDA. Improving forecast accuracy and speed of calculations makes it possible to improve the quality of the forecast, resulting in increased supply flexibility and reduced logistics costs. A software toolkit based on the Matlab package has been developed, which makes it possible to implement the proposed method. The developed software tools are used to solve the problem of supply chains forecasting. Keywords 1 Forecast efficiency, supply chain management problem, neural network forecast model, Time-Delay Mean Field Boltzmann Machine, positive and negative learning phase 1. Introduction Supply chains are complex adaptive systems characterized by structural and dynamic complexity, operating under a large number of random factors. Supply chain management is based on forecasting the demand for the final product. This requires efficient and intelligent supply chain planning. Planning challenges include, but are not limited to, fragmented data across the organization and difficulty in forecasting deliveries. This leads to low accuracy of sales plans, a large volume of illiquid products and, as a result, to losses for the company. The tasks of reducing inventory and increasing turnover are directly related to the accuracy of forecasting sales. When calculating safety stock, the average deviation of sales from forecasts is one of the main components. Today, one of the main problems in the field of supply chain management is the lack of forecast efficiency [1-4]. Therefore, the decisions made may not be accurate and fast enough. Improving forecast accuracy can lead to an increase in inventory turnover, as well as increase sales due to a decrease in the number of out-of-stock. Thus, the creation of effective forecasting methods for supply chain management is an urgent task. There is a set of methods as a means for forecasting, among which are:  logical forecasting methods based on classification and regression trees [5];  forecasting methods based on exponential smoothing [6];  regression and autoregressive forecasting methods [7];  neural network forecasting methods [8, 9];  structural forecasting methods based on Markov chains [10]. Using artificial neural networks for forecasting provides the following advantages:  assumptions about the distribution of input features are not required;  analysis of systems with a high degree of nonlinearity is possible; Information Technology and Implementation (IT&I-2021), December 01-03, 2021, Kyiv, Ukraine EMAIL: chdtu-cherkassy@ukr.net (O. Grygor); fedorovee75@ukr.net (E. Fedorov); olne@ukr.net (O. Nechyporenko) ORCID: 0000-0002-5233-290X (O. Grygor); 0000-0003-3841-7373 (E. Fedorov); 0000-0002-3954-3796 (O. Nechyporenko) ©️ 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) 114  high adaptability;  rapid model development;  the relationships between the input features are investigated on ready-made models;  a priori information about the input features may be missing;  the original data may be incomplete or contain noise, as well as highly correlated;  analysis of systems with a large number of input features is possible;  analysis of systems with heterogeneous characteristics is possible;  a complete enumeration of all possible models is not required. Therefore, a neural network forecasting method will be used in the article. 2. Formal problem statement Let the training set S  {( x , d  )} ,  1, P be given for the forecast. Then the problem of improving the forecast accuracy for the Time-Delay Mean Field Boltzmann Machine (TDMFBM) model is g ( x,W ) , where x – is the input vector, W – is the vector of parameters, represented as the problem of finding such a vector of parameters W * for this model, that 1 P satisfies criterion F   P  1 ( g ( x ,W * )  d  ) 2  min . The aim of the work is to create an effective forecasting method for supply chain management. To achieve this goal, the following tasks were set and solved:  analyze existing neural network forecasting methods;  create a neural network forecast model based on the mean field Boltzmann machine;  choose a criterion for evaluating the effectiveness of a neural network forecast model based on the mean field Boltzmann machine;  develop a method for identifying the parameters values of the neural network forecast model based on the mean field Boltzmann machine;  perform numerical studies. 3. Literature review The number of publications demonstrates significant attention to the application of advanced analytics, methods and modern computer tools of artificial intelligence in the field of supply chain management, but also leaves unresolved and insufficiently studied a number of problems regarding the development and synthesis of methods and models of artificial intelligence. The most commonly used forecast neural networks are: 1. Gateway neural networks:  long short-term memory (LSTM) [11, 12];  bidirectional long short-term memory (BLSTM) [13, 14];  gateway recurrent unit (GRU) [15-17];  bidirectional gateway recurrent unit (BGRU) [18, 19]. 2. Reservoir neural networks:  echo state network (ESN) [20, 21];  liquid state machine (LSM) [22-24]. Table 1 shows the comparative characteristics of forecasting neural networks. The learning rate is directly proportional to the computational complexity. For LSTM computational complexity ~PN(1)(5M(0)+ 3M(0)S+24S+S2), for BLSTM computational complexity ~2PN(1)(5M(0)+ 3M(0)S+24S+S2)), for GRU computational complexity ~PN(1)6(M(0)+N(1)), for BGRU computational complexity ~PN(1)6(M(0)+N(1)), for ESN computational complexity ~PN (M +N )+(max{P,M +N }) , for LSM computational complexity ~PN(r)(N(r)M(0)+ N(1)), (1) (0) (1) (0) (1) 2 115 where M(0) – the number of unit delays for the input layer, S – the number of cell, N(1) – the number of neurons in the first layer, N(r) – the number of neurons in the reservoir layer, P – training set cardinality, N(1)<