MAKING ENGINEERING PROJECTS MORE THOUGHTFUL WITH THE USE OF FUZZY VALUE-BASED PROJECT PLANNING

: Generally, engineering projects are getting bigger and bigger and more complex to handle. Due to the developments of information technology, managers have the opportunity to plan the execution of their projects and calculate the critical paths of those projects precisely, regardless of the types and sizes of their projects. However, these indicators are not realistic; in many cases the predetermined partial-deadlines cannot be kept, therefore the end of the project must be postponed. The aim of the research is to modify the inputs of the critical path method with the application of fuzzy values. The use of fuzzy values in business planning can support project managers in building the uncertainty factor into their model.


Introduction
As a result of globalization -in almost every competitive market -companies have to put a lot of effort into making their production efficient and keeping it updated. There are a lot of existing tools, which play an important role in keeping companies efficient and productive. Mathematical methods (e.g. linear programming, simulation techniques, etc.) in the field of production management are widely applied, and due to Information Technology (IT) developments in recent decades these methods have become easier to use [1]. As the project management has a very good IT-supported background (e.g.
Project manufacturing has technological focus, which means that companies must build their machines in a way that makes it possible for them to respond flexibly to their customers' needs in project production systems [2]. According to product feature approach, the product must be unique or it must be produced in a small variety in project manufacturing systems [2].
Recently, project managers have to face more challenging and more complex tasks. Not only logical thinking and good soft skills, but mathematical tools are also required to handle these projects [4], [5]. Companies should monitor and control their projects in order to be able to accomplish sub-tasks on time and within planned costs [4]. What is more, scheduling is one of the most important segments in project planning, because it provides a rough estimation for total process time and total process cost [4].
Critical Path Method (CPM) was invented as a result of demands for chemical plants at DuPont in the late 1950s [4]. The essence of this technique is to determine critical and non-critical tasks in a project. Those tasks, which are on the critical path has a crucial effect on the project's lead time, because this path is the longest of all the paths in the project [5].
Operations research can contribute to the success of identifying the critical path in a given project with the help of a mathematical solution [6]. This type of technique is called network method and the equations can be seen below [6], [7]: Objective: Constraints: where x n is the closing event of the project (the latest event time is when the event must be finished without delay in the network (LT)); x 1 is the early event time of the project (the earliest event time is when the event can begin (ET)); x j is the next activity within the project; x i is the previous activity within the project; t ij is the process time of activity x i .

Fuzzy logic and fuzzy values
Modeling real life is always a complex and complicated task. In several fields of life there is only a fine line between yes and no statements. In management [8], [9] fields, managers have to face a lot of problems and their decisions can lead to several outcomes and consequences.
Uncertainty factor can make Boolean logic less precise, therefore Zadeh [10] created fuzzy logic. With the application of fuzzy logic and its values, real life can be modeled more precisely than by Boolean logic, because the validity of a statement is determined not only by using 0 or 1 values but by any values of the [0;1] interval [11]. And if the planning method is meticulous, it can result in savings in the company's budget, which can further lead to the company's survival and advantage in the market [12].

Combining critical path method with fuzzy logic
Critical path method is a very good tool in the planning phase of a project. But planning a project and real life is not always in accordance with each other: there can be a lot of unpredictable obstacles, e.g. uncertainty and constraints in the implementation stage of a project [13]. Therefore, there is a great potential in fuzzy logic to make project planning more thoughtful and realistic than conventional methods. Several papers were published in the topic of how fuzzy logic was implemented in many fields, like engineering, IT fields [13], thermal image processing [14], as well as articles were written about how fuzzy logic was added to CPM and led to useful results [15], [16], [17].

Decision making within a company
Regardless of a company's size or complexity, managers often have to face problems and decision-makings [18]. Some of these decisions seem trivial, but many of them require a great variety of information and a good decision-making tool to decide what alternative is worth choosing. For this purpose, a lot of multi-criteria decisionmaking techniques were invented, for instance Analytic Hierarchy Process (AHP) or Weighted Sum Model (WSM) [19]. In this model there are n decision criteria. Every single criterion has a weight, and it is assumed that the higher value a criterion gets, the more important the criteria is. Alternatives can be evaluated with the sum achieved by multiplying criteria values and their weights [19]: where A i is the total importance of the alternative; a ij is the performance of the i-th alterative by j criteria. W i is the weight assigned to i-th criteria. Then, with the help of this multi-criteria model, decision makers can compare each alternative by their weighted score, and they can decide which alternative gives the best solution to that problem [17].

Product presentation
The manufacturing process we analyzed is an assembly process of an agricultural cultivator trailer. This machine helps limbering up the soil with needles and makes football courts suitable for playing as well as it can enliven other green areas by providing the soil with a better water drainage. Besides, this equipment can help to develop green areas by making grass stronger and denser. There is a great variety of this machine, and the demand for it is quite hectic, therefore the number of different products produced in a given period is low. Similar tools and equipment can also be used for assembling other machines. Considering these two features it can be stated that this assembly line works well in project manufacturing.

Data collection and processing
During the research at the company, 10-15 measurements were taken depending on the process steps. The next scene of the research was the Gemba, where the assembly line is located, and the researchers were able to get acquainted with the process while the duration of every single activity was timed. These measurements provided the basis for further work. Data analysis was made by MATLAB and Microsoft Excel with its Solver plug-in.
Steps of the research 1) All the data necessary for the research work at the company were collected. The duration of the data collection was 20 days; 2) The project net was drawn up; 3) The descriptive statistics of the collected data was prepared and the indicators and definitionsthat helped to compare the results were selected

Research goal
During the research the following question was determined:

Can the projects be made more reliable with the use of fuzzy-values?
At the same time the goal of the research was established:

It is possible to get closer to the project's real time value through fuzzy-CPM.
The reason for choosing this research goal was to make production more efficient and more cost-effective through precise predictions, because a more realistic planning can assist the company's management to create a more accurate business plan. Furthermore, the assembly line can prove to be more effective if a company is operated by a well-considered plan.

Case study
The examined process is an assembly line. The main process can be divided into two main parts: the first part is a pre-assembly, where the raw materials to be built into the main product are assemblied (Nodes 1, 2, 3, 4). The second part is the main assembly process where raw materials and semi-processed products are built into one cultivator machine (From Node 5 to Node 15). The relationship between the activities can be seen in Fig. 1. The next step in the research was to make a brief descriptive statistics from the measured activity times. At the request of the company, times were distorted, therefore the results can be seen in Time Measurement Unit (hereinafter tmu).
First of all the critical path was calculated. One full assembly process was measured from the first step to the last, and the values were applied to the model. This served as a basis for the further research.
After using the values in the model, we calculated the following results: The company wanted to plan their production by using the mean values. Table I of the solver's sensitivity report revealed that the critical path was the following: 1-5-6-7-8-9-11-12-13-14-15.

CPM method complemented with fuzzy triangle numbers
The first method of creating fuzzy values was the Fuzzy Triangular Number (FTN) method of CPM. The following formula was applied to calculate FTN, which serves as an input for further calculation [16]: where a is the minimum time of a process step; b is the average time of a process step; c is the maximum time of a process step. The research group created the FTN-integrated model, and the following results were calculated: Critical path of the project was: 1-5- 6-7-8-9-11-12-13-14-15. It was the same as in the first model: Compared to the first values some differences in the slack times can be detected.

CPM method complemented with fuzzy sigmoid function values
Not only triangular funcitons exist to create fuzzy-values, but there are other opportunities as well e.g. sigmoid functions. Pokorádi and Menyhárt [20] examined the parameter tolerances of batteries in their study with the use of this fuzzy function. The applied function is the following: This formula was used to create both critical paths of the project and to diagnose the time slacks within the project. Two axes of the cumulative distribution functions are the following: y stands for the interval between [0;1], while x is the range of the activity's time. At the first attempt the y value equall to 0.2. This led to the results below: Critical path of the project was: 1-5-6-7-8-9-11-12-13-14-15 Same equation was applied to demonstrate the last examined model. In this case the y value was equall to 0.8. Results can be seen below: Critical path of the project was: 1-5-6-7-8-9-11-12-13-14-15 The next step of our research was to evaulate and compare these results: weighted sum model was applied to compare all the calculated models. the best, the importance of each criterion must be decided. There was a common agreement between the company's management and the research group in assigning weight to indicators, which is demonstrated in Table II.

Validation and decision making
Furthermore, the fit of each criterion was calculated by applying the following equation: The decision making matrix shows the final rank. The higher final value a method reaches; the better method it is. According to Table III considering the fit of predetermined indicators by real life value, the best method was the sigmoid method in which the y value is equal to 0.8. If the company applies the best-fit fuzzy-value, the production planning can become more accurate than the result of the traditional network model. By using the better method, a company's planning can get closer by 7% to the real project process time.
Taking real life circumstances into account, the company in focus would be able to produce about 10 machines a week, but in their plan this number was 12. The company would have not been able to reach the production aimed at in its business plan.

Discussion
If the SME in the assembly industry cannot keep its predetermined partial-deadlines due to supplier's mistakes or to other reasons, the managers can get in a situation in which they have to choose whether they change suppliers or standardize the assembly line. However, if it is not viable, the managers have to take the uncertainty factors into account and design their business plan in accordance with the available resources to make their production more efficient and profitable. In this paper, a case study is presented, where uncertainty factors were built into an operations research network model. Four fuzzy calculation types were compared to one another. On the basis of our case study, the results of the sigmoid fuzzy function proved to be the best, in which the following indicators were taken into account: total process time, free time slack, safety time slack and total time slack. As we assumed in the hypothesis, fuzzy-based production planning became more accurate and more realistic than the result of a traditional network model due to the factors of unpredictability of the human workforce and those of unforeseeable interruptions in the production.