# Selection of Production Reliability Indicators for Project Simulation Model
## 2 Literature Background  
### Modeling Project Manufacturing Processes  
Due to the challenges of globalization, customers require products that have a high degree of customization. This urges the companies to be flexible in terms of manufacturing, which induces numerous production routings, with multi-purpose tools and machinery and supports the emerge of industry 4.0 [1]. In this special situation, manufacturing companies deal with project manufacturing, because the main features of a product change within a short time, and there is a high chance that the same type of product will never be created anymore [2]. To successfully keep the project management's \"golden triangle\", 4 stages must be performed thoroughly, see [2]:  
1. Project initiation: The need for special services or products has arisen. The main specification is settled and discussed in financial and other terms, such as agreed time frames. 2. Project planning: A more detailed analysis is performed in terms of production: activities, dependencies are revealed, resources are estimated, initial risk assessment is performed. 3. Execution: Value creation occurs in this stage: the provision of the services or product manufacturing. In this phase, all the activities are monitored and compared to the plan, and where needs and workflows are carried out. 4. Closure: This stage is for the evaluation of positive and negative impacts of the project execution. Drawing conclusions are crucial and these can support later projects. One of the most critical stages is the project planning, since in this step, the quality, price, delivery time and other indicators are fixed and contracted, and other stages should resonate with these points. A widely known method for estimating the project lead time is called the Critical Path Method (hereinafter CPM) [3]. This method identifies what activity should be included in the process, what are the dependencies among them, and there should be estimations about the activity times as well. With the use of this method, bottleneck activities (critical activities) can be identified, total project lead time can be calculated, and time slacks can be assigned to those activities that are not considered constraints [4]. The result of this method in the early stage is a project network (G(\(N\),\(A\)) network) that represents the production process [5]. Transforming into an operations research model, the following equations are applied [4]:  
Objective:  
\[\min z=-x_{0}+x_{n} \tag{1}\]  
Subject to:  
\[-x_{i}+x_{j}\;\geq\;t_{ij} \tag{2}\]  
where: \(x_{i};0\leq i\leq n\) are the nodes, \(z=\) total project lead time, \(t_{ij}\) activity time between \(x_{i}\) and \(x_{j}\) nodes. Besides this method which contributes valuable information to the decision-makers, literature identifies some drawbacks, as per [4; 6]:  
* A small change in the activity times can induce a change in the list of bottleneck activities (critical activities). * It ignores the stochastic feature of the production; the model uses deterministic values for human resources and raw materials. * It does not provide the possibility of risk management. * It does not take the resource allocation into account. As the previously mentioned drawback list displays, CPM conducts production planning in a deterministic way. As a complement of the critical path method, Project Evaluation and Review Technique (hereinafter PERT) is created and used. The main idea behind thismethodology is to build a stochastic evaluation of the activity durations with the use of a beta distribution [3;4]:  
\[E\big{(}t_{ij}\big{)}=\frac{a+4b+c}{6} \tag{3}\]  
\[Var\big{(}t_{ij}\big{)}=\frac{\left(c-a\right)^{2}}{36} \tag{4}\]  
where: \(t_{ij}\), activity time between \(x_{i}\) and \(x_{j}\) nodes; \(a\): minimum time duration of an activity \(t_{ij}\); \(b\): most likely duration of an activity \(t_{ij}\); \(c\): maximum time duration of an activity \(t_{ij}\). Similarly to CPM method, PERT also has shortcomings [4], especially:  
* It only deals with time indicator, it excludes financial calculations.