One-third of the global plastics production is processed by injection molding [1], making injection molding machines (IMMs) widely used industrial assets. Figure 1 depicts the main components of a typical IMM. During the injection process, plastic pallets stored in a feed hopper are melted along the barrel by heating bands and injected via pressure into a mold cavity filled with plastic material. After solidifying, the material is ejected from the mold cavity, producing a three-dimensional part. The complexity of the injection molding process makes the maintenance of IMMs and their components a challenge [2], which has particular importance when it is acknowledged that expenses related to maintenance are typically a considerable part of an enterprises’ total expenses [3]. For this reason, adopting sensors to monitor IMMs and their components is very common, especially concerning injection molds ^[4][5][4,5], as they are a key component of the injection process [6].

Injection molds are subjected to degradation due to repeated use, high temperatures, pressure, and contact with melted materials, which may cause issues including such as surface degradation, dimensional changes, or misalignment. These issues may be time-consuming, leading to high periods of unavailability, or may be easily solved without a major intervention. Scheduling maintenance actions for these components is challenging, as the time between two failures does not follow a pattern, and requires balancing the mold’s performance with the expected maintenance costs. Furthermore, keeping detailed records of mold maintenance activities, including repairs, modifications, and maintenance history is essential for maintenance planning; however, this can be challenging in large-scale manufacturing.

IMMs are widely used assets for which maintenance is challenging due to the high number of parameters and components involved in the injection process. Research has been carried out on ways to improve maintenance actions for IMMs and their components, particularly by exploiting process parameters monitored by sensors. In ^[8][16], the authors used the maximum entropy principle ^[9][17] together with a fuzzy technique to deal with only censored data uncertainty while incorporating the knowledge of the maintenance staff to reduce annual maintenance costs of IMMs. On the other hand, ^[10][18] used a copulas model that considered the dependence of different failure modes to optimize maintenance actions on an IMM.

In ^[11][19], the authors presented an interesting work in which a statistical method based on kernel density was mixed with outlier detection for PdM in the case of the hydraulic pump of an IMM. Although an interesting path, the proposed approach is in an embryonic stage and lacks results. The process parameters of injection were analyzed by ^[12][20], who adopted the Nelson Rules ^[13][21] to detect abnormal patterns in the injection process. Other authors have exploited machine learning (ML) techniques to handle IMM maintenance. For instance, ^[14][22] employed several data mining techniques to forecast machine-related disruptions on IMMs, while ^[15][23] employed reinforcement learning for real-time production planning and scheduling in smart manufacturing of injection molding thermoplastics. The authors of [2] exploited ML classification algorithms to distinguish between optimal and borderline functioning of IMMs in order to trigger maintenance actions when borderline functioning is detected. Other research has focused on creating the necessary apparatus to monitor and collect relevant data for further analysis ^[16][17][24,25], which is the basis for applying maintenance actions based on the actual condition of the equipment.

One of the main fields within PdM is prognostics, in which the main focus is determining the number of working cycles a given industrial asset can perform before its failure ^[18][7]. One of the main challenges in such approaches is the fact that the sensor data often present stochastic patterns that do not allow for determination of the asset’s degradation ^[19][26]. As is apparent from this overview of maintenance solutions for IMMs and their components, the majority of research has focused on classifying the working state of the assets or detecting anomalous behaviors, as there are no monotonic sensor signals able to describe the degradation process. In the present work, new features are created through the combination of anomaly detection and GFTs, enabling degradation assessment of injection molds by determining their failure probability. This contributes to the field of prognostics more broadly, as the same approach may be employed in other industrial use cases where the degradation of assets follows a stochastic distribution of abnormal events.

Reliability analysis is concerned with the failure probability of a given system or part. It relies on statistical data from relevant parameters of a given process. One of the most widely used reliability methodologies is FTA ^[20][27], which is exploited by well-known entities such as NASA ^[21][28].

FTA graphically represents a failure event, designated as the top event (TE) using a fault tree (FT). An example of a generic FT is depicted in Figure 2. This structure has two main components, namely, the basic events (BEs) and the gates or operators. BEs are the root factors responsible for causing a given TE, while gates are logic operators (𝐴𝑁𝐷 and 𝑂𝑅) that compute the probability of the TE based on the probability of the BEs according to 𝐴𝑁𝐷(𝐹𝑋,𝐹𝑌)=𝐹𝑋𝐹𝑌 and 𝑂𝑅(𝐹𝑋,𝐹𝑌)=𝐹𝑋+𝐹𝑌−𝐹𝑋𝐹𝑌, where X and Y are two BEs and 𝐹𝑋 and 𝐹𝑌 are the respective cumulative density functions (CDF) of X and Y. Note that the 𝐴𝑁𝐷 gate triggers when both of the input BEs have occurred, while the 𝑂𝑅 gate triggers when at least one of the input BEs has occurred. In addition to quantitative analysis enabled by the calculation of TE probability, FTA allows for qualitative analysis of the minimal cut sets (MCS), which are the sets containing the minimum number of BEs to ensure the occurrence of a TE ^[22][29] (for example, the minimal cut sets for the FT shown in Figure 2 are 𝐶1={𝐵𝐸1} and 𝐶2={𝐵𝐸2,𝐵𝐸3}.

There have been several approaches for quantitative analysis of FT proposed in the literature, such as state-space models ^[23][24][25][30,31,32], simulation methods ^[22][26][27][29,33,34], algebraic approaches ^[28][29][35,36], and commercial software tools and model checkers such as Galileo ^[30][37], Coral ^[31][38], DFTCalc ^[32][39], and Storm ^[33][40], among others ^[34][41]. Although research effort that has been dedicated to FTA, the methodology does have a number of issues ^[35][36][42,43]. Typically, the TE represents the failure of a complex system, while BEs represent the failure of parts or components within the main system. This has two limitations: (1) expert knowledge is necessary to build the FT structure according to the possible failure modes of the components, which may be tedious and expensive; and (2) the probability of BEs, which is the failure probability of a given part or component, does not accord with the actual state of degradation of these components, as its calculation is only based on the hazard rate calculated from historical data on the malfunctioning of these components.

To overcome the aforementioned issues, the GFT methodology was proposed in ^[35][42]. Contrary to FTA, the GFT methodology does not rely on previous expert knowledge of the system, as the BEs are generated automatically from data; in addition, the tree structure that best describes a given TE is automatically obtained through a training procedure. The definition of BEs from continuous data can be achieved by discretizing the data, using cutting-edge parameters ^[35][36][42,43], or using an anomaly detection technique to detect abnormal events which are then used as BEs in the GFT analysis ^[37][13]. With respect to the training process, it can be used to either minimize the error between the CDF obtained by the GFT and the real CDF of the TE for a given dataset, or to optimize the maintenance costs.