Theoretical framework of Fuzzy-AI model in quantitative project management

Abstract

Management issue incorporates a huge amount of fuzzy information processing, that is exactly representing the essentials of human intelligence in reasoning for managing problems. This paper distinguishes the qualitative (empirical) management versus the quantitative (digital) management and focuses the studies of the latter one and verifies the possibility and necessity of adopting quantitative project management in many real world problems. The theoretical arguments of “Fuzzy-AI model” are presented for dealing with the quantization of uncertain management events, which simulates the intelligence of human being by AI (Artificial Intelligence) technology and quantizes the management solution by fuzzy approach. Several practical examples are presented in this paper as the evidence of the applicability of this model in the project management issues, which provides digital support to decision maker in the management. Finally, the theoretical framework of applying this model in different stages of project management is presented for the purpose of calibrating its application scope.

Keywords

Fuzzy-AI Model quantitative management project management theoretical framework performance matrix

1 Introduction

Management essentially is a series of decision making under uncertainties. Moreover, management process can be regarded as a highly intelligent activity conducted by human being. How can IT through AI (Artificial Intelligence) engage the simulation of human intelligence for support decision making in the management process? The fuzzy reasoning in AI could accomplish this process and more accurately assist the decision making.

“Fuzzy-AI Model” is an effective information treatment methodology integrated by fuzzy information processing and AI technology. Since AI can be simplified as the technique of simulating human intelligence by means of computer, yet the most powerful capability of human being is its ability of treating massive fuzzy information. Therefore, using fuzzy information processing and AI technology for simulating human intelligence and apply it in the management science will have broad perspectivefuture.

Uncertainties exist in almost every field no matter in natural, social, engineering and managerial problems. Nevertheless, quantitative assessment of those managerial factors of them needs in-deterministic mathematical modeling, in this regard; fuzzy set theory will work in the management issues. Zadeh [26] has defined fuzzy set theorem, it explored the possibility of providing quantized solutions for a family of uncertain problems. Moreover, the integration of fuzzy set to AI inference will great enhance its potentiality in application to problem solving. The “Analytic Hierarchy Process (AHP)” suggested by Saaty [17] has widely applied to solve managerial problems, which is so-called “Non-Structured” problems and cannot be easily expressed by analytic mathematical modeling. Introducing fuzzy set to AHP will extensively widening the range of its application, thus, the “Fuzzy-AI model” would explore a new approach of solving those “Non-Structured Problem”.

As regard to fuzzy approach for solving real world problems, it is much questionable that the arbitrarily selection of membership function without verification in mathematical strictness is unreasonable, therefore fuzzy approach will not in the same degree of rationality compared with probabilistic approach. However, it was pointed out by Lin Shaopei [8] that fuzzy membership function has the same degree of arguments compared with the probabilistic density function. It proved that through repeatedly sampling objective events for AI machine learning, which could derives reasonable parameters for constructing fuzzy membership function; just similar to collecting practical samples through observation for building the probabilistic density function. Thus, theoretically it is evident that the fuzzy membership function will have the same degree of reasonability with the probabilistic functions. This situation provides the theoretical basis of “Fuzzy-AI Model” and promotes the quantitative analysis in managerial science [6].

To simulate human experience and intelligence in decision making is the essentials of management decision processes, and the “Fuzzy-AI Model” will naturally reflect the logic sense of decision making for different project management events. It is reasonable that through retrieving past human experiences during decision making for constructing fuzzy membership function. Since management decision is base on a series of facts and events which are a kind of “Uncertain in degree” in nature and is not sensitive to whatever the original information is subjected to slightly varying, and it fits the fuzziness of the membership function in fuzzy set analysis.

The “Fuzzy-AI model” realizes quantitative management by fuzzy reasoning, through quantizing of uncertain attributes, optimizing the inter-relationship between these attributes based on minimum resource consumption or other principles of comparison for the scheduling, budgeting and quality control for the project, thus make sure how long does it take? How much does it cost? How to identify the best way for the project to determine what, why, who, when, where and how to do the jobs during project implementation in different stages, etc.

In order to rectify the scope of further application of this model, it seems necessary to develop the theoretical framework of “Fuzzy-AI model” for the purpose of calibrating and facilitating further applications of the model in different areas of project management according to the contents of knowledge category defined in PMI-PMBOK [16].

As a matter of fact, the “Fuzzy-AI Model” explores one of the approaches to quantitative management — a new area in management science. Lin Shaopei [15] proved that this Model can be one of the theoretical bases in “Quantitative management”.

Eventually, some researchers introduced another branch of AI application approaches which is using a series of modern soft computing in simulating human heuristic thinking for the optimization of decision making [18 –21]. Such intelligent algorithms can be applied to various problems in the fields of engineering, business and economics.

Besides, the fuzzy and AI methods have been effectively applied in the past 20 more years by the authors in different applications, including risk management, operation and maintenance, engineering planning, engineering design, transportation, real estate, investment, finance management problems [1–7 , 22–25]. This paper focuses on the possibility of exploring a new scenario of management science through the attempt of further extending the application of this new technique and establishing the theoretical framework of the Model.

2 Qualitative and quantitative management methodologies

Since the management means to make a series of decision making under uncertain environment, it needs high intelligent performance of human being. In the past, the decision maker solves management problems mostly by the synthesis of his/her personal experience and judgment, which can be concluded as qualitative (or empirical based) management. Nevertheless, under realistic project environment, the effectiveness and exactness of human experience are somewhat heuristic and vague; the philosophy of qualitative management may cause misleading since subjective judgment could be apart from the objective reality. The situation may even worse when the event consists of numerous factorial attributes and the measurement of which yet remains uncertain especially when the attributive interactions of the event still remain unknown.

For above reasons, in this paper the quantitative management is introduced, which makes problem solving more exactly coincide to its nature. It provides the possibility of solving managerial problems taking consideration of multi-attribute interaction of in a fairly reasonable approach. We recognize that the project management is aimed to achieve specific project objective(s) with limited resource (including time) limitation and satisfying specific quality requirements. The quantitative management method could derive a series of managerial decisions through quantizing all those uncertain attributive factors during project implementation. In this regard, “Fuzzy-AI model” provides an efficient approach of problem solving.

3 Mathematical argumentations

It is recognized that there are two different kinds of uncertainty: “Uncertain in occurrence” and “Uncertain in degree”. The mathematical modeling should be accommodating to the nature of uncertainty of the event. In other word, a correct mathematical formulation for modeling the uncertain event should be performed only if the nature of uncertain event has clarified. Probability model is appropriate to those uncertain events in which it is “uncertain in occurrence”; but it seems improper to use for modeling managerial event which is natured in “uncertain in degree”. In such case, one needs to simulate the event by means of another non-deterministic mathematical model other than probabilistic one, such as fuzzy logic, nearness theory and grey theory etc., for it represents the essential of the managerial uncertain event in degree uncertainty.

The arguments of why people accept probability theory are based on two aspects: Firstly, the probability function is analytic and mathematically derivative and secondly, the parameters of the probability function are based on statistical data from observation. However, the vulnerability of the fuzzy approach comes from arbitrarily of its fuzzy membership function building, which is originally determined only by heuristic knowledge of human experience; this attracts criticism from different aspects. In solving this dilemma problem, one may choose fuzzy membership function an analytic and mathematically derivative one, then to define its parameters through AI supervised machine learning from real world samples, replaced the parameters in probabilistic functions through the statistical data from objective world observations in the probability approach. When membership function is constructed by parameters obtained by AI machine learning from specimens, a highly coincidence of Fuzzy-AI approach in comparison with reality can be found [8]. Thus it was evident that “Fuzzy-AI model” possesses the same rationality with the probabilistic one in mathematical argumentation.

4 Performance matrix of managerial events

The events concerning to project management are very complicated. Firstly, during project life-cycle there are series of events in different stages, such as project initiation and verification, planning, scheduling, implementation control and completion etc. Secondly, project management covers a broad knowledge fields and even possesses competence, as stated in PMI-PMBOK, there will be different management events to be treated in different stage of operation. Since human intelligence is characterized by massive fuzzy information processing, it is a hereditary relationship between AI and fuzzy inference. However, the modeling of fuzzy reasoning under different situations for a variety of application events should be modeled by explicit mathematical forms. This strategy can be concluded as to identify the concrete modeling concept by the performance matrix, which is constructed by specific knowledge component as its row; and the specific management operation issue as its column. For the purpose of calibrating the application scope and the region of project management, a theoretical framework is built accordingly, which is in form of a two dimensional matrix with the knowledge component as its row and management event as its column.

The intersection space of each row and column represents the specific managerial event of the specific professional managerial issue (column) engaged in specific managerial aspect of PMBOK knowledge component (row). The particular quantitative project management event in each intersection of the performance matrix is presented in Table 1.

Table 1 shows the theoretical framework defining the project management event numbers to 10×8 = 80 though the managerial events in column are limited to eight and the managerial aspect (knowledge component) during operation in row are limited to ten.

It assumes that e_ij is the element of afore said performance matrix E, which represents a specific project management event: $E = \sum_{i = 1}^{m} \sum_{j = 1}^{k} {e_{ij}} (i = 1, 2 \dots, m; j = 1, 2, \dots, k)$ (1)

Where temporarily we have m = 10 and k = 8 as shown in Table 1. For each e_ij, there is a mathematical formulation described the implicit nature of the managerial event. Generally, it is a decision making model under uncertain environment and in various forms of problem orientation.

The element e_ij of performance matrix is the topic that we are interested in and being modeled, which is a specified managerial event both involving certain professional managerial event in certain stage of operation in project life-cycle and in certain professional managerial aspect. As a matter of fact, the management decision model e_ij can be formulated in different forms, such as:

Network Solution Model (NS): Optimization/Minimization in form of a non-linear network model, to represent the project management event by the network and pursuing its appropriate network system performance.

Mathematical Programming Model (MP): Optimization in form of a mathematical programming. If fuzzy set is involved, the optimum fuzzy variable vector will be the objective of the fuzzy MP model.

Nearness and Matching Model (NM): Optimization in form of matching certain managerial indicator(s) through certain defined benchmarks or assigned samples, while the model is represented in fuzzy variables.

Assessment and Evaluation Model (AE): Optimization in form of fuzzy assessment of uncertain managerial factor(s).

Max/Min indicator model (MM): Optimization in form of a maximizing or minimizing the system indicator(s), such as fuzzy state indicator of the managerial system, such as in project initiation stage, to maximize or minimize fuzzy internal rate of return IRR, fuzzy profitability or deficiency in economic feasibility study, etc.

The project management objective(s) in e_ij event can be achieved through above-mentioned models or through the solution of its corresponding mathematical formulation. Whiles, the uncertain management environment and its in-deterministic information can be quantized and inferred by “Fuzzy-AI Modeling”. Therefore, the quantitative management solution of e_ij can be achieved in fuzzy digital form.

Though this paper is not aimed to formulate every e_ij in performance matrix E, the expression of formulation for each e_ij could be different from each other; however, typical forms of modeling are presented hereby through case studies for reader’s reference.

5 Case study

5.1 Case study of managerial events by network solution

A construction project risk management problem is introduced as the example of “Fuzzy-AI Model” by network solution (NS).

5.1.1 Establish the Risk-Knowledge-Framework (RKF) network

If the project risk can be expressed as the node of a part of the network, and the countermeasure for constraining the risk as another part of the network, the linking of each corresponding nodes in these two parts of the network can form an overall network of project risk management. Since the physical meaning of the linking line represents the constraint or the effectiveness of individual measure to mitigate the corresponding risk, which is an uncertain connection with fuzzy nature. We try to introduce a fuzzy vector to represent the nonlinear fuzzy relation between the measure and risk. If we take account of linking line could be “one-to-one”, “one-to-multiple” and “multiple-to-multiple”; also the risk could be hierarchically arranged due to its nature, we would have a sophisticated multi-layer nonlinear network called Risk-Knowledge-Framework (RKF), which is the global expression of all measures against all the risks for the project and exactly the knowledge representation for the project risk management.

Even extremely complicated relationships in risk evaluation can be simplified to a series of choice among the fuzzy vectors reflecting the one-to-one links in “RKF” network. Nevertheless, the logical structure of RKF is a complex nonlinear network which composes the basic knowledge structure of project risk control, its logical structure RKF is shown in Fig. 1. The knowledge based fuzzy decision supporting system (KB-FDSS) software for project risk management is developed with the framework of RKF [24].

Since the experts’ knowledge and experience is reflected by the effectiveness of the countermeasures against risks, the risk evaluation process is just to assess every single risk constrained by the countermeasure through choice among 4 kinds of fuzzy vectors, which is composed by four areas of fuzzy membership grades of “effectiveness degree”:R = {very effective (VE) , rather effective (RE) , slightly effective (SL) , negligibly effective (NG)}. Define:

Classified fuzzy vector “1” (R₁ = {0.85, 0.1, 0.05, 0}) is very effective (VE) to risk control;

Classified fuzzy vector “2” (R₂ = {0, 0.85, 0.1, 0.05}) is rather effective (RE) to risk control;

Classified fuzzy vector “3” (R₃ = {0, 0.2, 0.7, 0.1}) is slightly effective (SL) to risk control;

Classified fuzzy vector “4” (R₄ = {0.05, 0.1, 0.85, 0}) is negligibly effective (NG) to risk control.

Through RKF the complicated relationships in project risk evaluation can be simplified as to make a choice of number among 1, 2, 3 and 4. On the basis of quantifying the risk by fuzzy inference, the quantitative evaluation of each risk of the project can be obtained.

5.1.2 Fuzzy inference assessment and quantification of risks

The information of fuzzy vectors is utilized to get the effectiveness of the measures (knowledge), so as to build the relation between the risk and knowledge. In project risk management, for making risk management decision, it is necessary to quantize the risk level by following fuzzy mathematical model.

Define: risk factor set U ={Political Risk, Economic Risk, Legal Risk, Social Environment Risk, Security Risk (assault, slaughter), Technical Risk, Management Risk, etc.}. Then define V is the effectiveness of knowledge in reducing risks, V can be expressed by four grades: Very Effective (VE), Rather Effective (RE), Slight Effective (SE) and Negligible (NG). Thus, V = {VE, RE, SE, NG}.

Introduce fuzzy sets to describe degrees of the problems, then the fuzzy relation R between U and V can be described as: $R_{i} = r_{ij} (i = 1, 2, 3, 4 \dots, 7), (j = 1, 2, 3, 4)$ (2)

Where r_ij is the membership of the i-th risk in U to the j-th domain of V. Fixing i, then we have: R_i = {r_i1, r_i2, r_i3, r_i4}. It is membership of the i-th risk in U to the four domains of V. When i = 1, 3, the risks can be effectively reduced, and the fuzzy risk matrix of the project R should be [12]: $R = [r_{ij}] = [\begin{matrix} 0.85 & 0.1 & 0.05 & 0 \\ 0 & 0.2 & 0.7 & 0.1 \\ 0.85 & 0.1 & 0.05 & 0 \\ 0 & 0.05 & 0.1 & 0.85 \\ 0 & 0.05 & 0.1 & 0.85 \\ 0 & 0.2 & 0.7 & 0.1 \\ 0 & 0.2 & 0.7 & 0.1 \end{matrix}]$ (3)

Introduce the Risk weight matrix of each functional system: $P = [P_{i}] (i = 1, 2, \dots, 7)$ (4)

Considering the political risk, economic risk and social security risk of the local region are most influential to the safety of the project, these risks should be considered in priority. Then, P = {0.25, 0.20, 0.10, 0.10, 0.25, 0.05, 0.05}. Where, P is a risk weight matrix revealing the influences of the risks, then the vector quantity E representing the comprehensive fuzzy risk assessmentshould be: $E = P \times R = {0.1275, 0.0745, 0.2525, 0.3275} .$

After nominalization, $E = {0.163, 0.095, 0.322, 0.419} .$

According to the principle of maximization the membership function, the domain where maximum membership of E is located reflects the effectiveness of risk reduction.

In this project, Max(E) = 0.419 is in the fourth NG region of V, which means the risks are barely reduced. For the countermeasures that we have taken are negligibly effective for both social environment risks and security risks, the project should be denied or reconsidered.

5.1.3 Example verification of risk evaluation

A railway project in Africa with the main track around 350 kilometers is studied as an example [24]. The project is tested and verified for the risks by means of the prototype software KB-FDSS. Sequentially input the rank of fuzzy membership vectors of the measures to be taken for constraining the sub-risks to the main risks (political, economic, policy & legislation, environmental, implementation and technical risks), which represent the effectiveness of control to these risks, the prototype software KB-FDSS could automatically calculate and output the fuzzy risk membership vector of each main risk, as well as the “zone” where the maximum membership of the vector is located. The state of this “zone” represents the risk state of the project after those measures have been taken place. The results of fuzzy risk assessment of the railway project by KB-FDSS are shown in Fig. 2.

Fig. 2 shows the distribution map of the fuzzy risk assessment data listed in Table 2. Among them, the grey one shows the maximum membership of the entire project is 0.365, which means the risk is “rather effectively” controlled, and thus the project is in the domain of “Slight Risk”. The distribution of project fuzzy risk membership vector of main risks and sub-risks of the project is shown in Table 2. The result of assessment is “Slightly Risk”.

5.2 Case study of managerial events by mathematical programming model

A maintenance management problem is introduced as the example of “Fuzzy-AI Model” by using mathematical programming.

The planning of Metro vehicle repair cost management can be solved by mathematical programming model [10, 11], which can be stated as:

${\begin{matrix} Maximize (or Minimize) the objective function \\ F_{i} (X) (i = 1, 2, \dots) \\ To find the variables X = {x_{j}} (j = 1, 2, \dots) \\ Subject to constraints C_{k} (k = 1, 2, \dots) \end{matrix}$ (5)

The solution of decision model (5) is the vector of variable X = {x_j} (j = 1, 2, …), which can be served as the argument to support the management decision making. It is noticed that the fuzzy inference for quantizing uncertainties for the decision problem has treated before the modeling. Therefore, decision model (5) can be dealt with as the conventional optimization procedures.

The optimization modeling of repair investment planning decision for vehicle system, related to each kind of vehicle and each distress attribute, can be expressed by mathematical programming with the fuzzy objective function VDSI (vehicle distress state indicator): ${\begin{matrix} Min (VDSI) = \sum_{j = 1}^{k} \sum_{i = 1}^{m} {w_{i} [(1 - u_{i} (t_{ij})) \\ - (1 - u_{i} (t_{ij} - Δ t_{ij}^{(g)}))]} \\ Subject to : \\ Cost constraint \sum_{j = 1}^{k} C_{j} \leq C_{0} \\ Manpower constraint \sum_{j = 1}^{k} [MP]_{j} \leq [MP]_{0} \\ Vehicle safety state constraint \\ \sum_{i = 1}^{m} {[\frac{w_{i} (t_{i}) \times u_{i} (t)}{w_{i} (t_{i}) \times u_{i} (t^{cr})}]}^{2} < 1 \end{matrix}$ (6)

The optimization model (6) is the formulation of minimizing an objective function VDSI, which is defined by fuzzy distance of distressed vehicle with the perfect vehicle; where the fuzzy concept is introduced. The more the vehicle is distressed, the larger the fuzzy expression VDSI will be.

The goal of optimization model (6) is to simulate human’s intelligence of finding system optimum solution —- the integer variable vector G_ijg. Where, G_ijg is a set of 3mk integer number of repair state g (1 - capital repair, 2 - moderate repair, or 3 - slightmaintenance) for each i-th distress attribute (i = 1, 2, …, m) and for each j-th vehicle (j = 1, 2, …, k). If G_ijg is found, the repair state of each distress attribute of each vehicle can be defined respectively.

In Cost constraint $\sum_{j = 1}^{k} C_{j} \leq C_{0}$ , C₀ and C_j represent annual cost budget limit and cost for j-th vehicle repair. In Manpower constraint $\sum_{j = 1}^{k} {[MP]}_{j} \leq {[MP]}_{0}$ , [MP] ₀ and [MP] _j represent annual manpower limit and manpower for j-th vehicle repair respectively.

The vehicle safety state constraint $\sum_{i = 1}^{m} {[\frac{w_{i} (t_{i}) \times u_{i} (t)}{w_{i} (t_{i}) \times u_{i} (t^{cr})}]}^{2} < 1$ is an expression of safe bund of distressed vehicle by means of a hyper-ellipsoid in m distress attribute dimensions. Where, the axis of the hyper-ellipsoid is expressed by the weighted relative distress rate of individual attributes. Either weighted distress rate of individual attributes approaches to 1.0 means damage of this attribute, it means the VDSI approaches to the surface of the hyper-ellipsoid and will bring the vehicle out of work. Usually, the vehicle safety state constraint — the hyper-ellipsoid, should not be happened for it will not be allowed the working state of any attribute to be approached or exceeded to its critical state.

It is obvious that this model combines the essentials of “Fuzzy-AI Model” and could be applied to a kind of investment and cost control problems in project management practice.

5.3 Case study of managerial events by nearness and matching model

A real estate management problem is introduced as the example of “Fuzzy-AI Model” by nearness and matching model (NM).

The NM model can be served for the problem of price determination for real estate properties [1]. Since the sale price of property $\underset{\sim}{B}$ is critical important for real estate developer. It is recognized that there are roughly 10 influence factors affected to property price as below:

Geographic location of the property;

Structural type and status of property;

Level of exterior and interior decoration of property;

Floor location of the flat property;

Real estate market demand and supply situation;

Potential of value adding of property;

Transportation convenience of property;

Auxiliary facilities status of property;

Surrounding and environmental conditions of property;

Political and economic situation of local region.

Denote the price alternative $\underset{\sim}{B}$ , samples in the sample base are $\underset{\sim}{A} = {\underset{\sim_{i}}{A}} (i = 1, 2, \dots n)$ . Considering the criterion of i-th sample and $\underset{\sim}{B}$ in j-th attribute based on membership difference is: $Min [| D_{j} (A_{\sim i}^{j}, B_{\sim}^{j}) | | {\underset{\sim}{A}}_{i} \in \underset{\sim}{A}, i = 1, 2, \cdot \cdot \cdot, n] \leq θ_{j}$ (7)

Where, $D_{j} (A_{\sim i}^{j}, B_{\sim}^{j})$ is the distance of j-th attribute of management decision $\underset{\sim}{B}$ and sample $\underset{\sim i}{A}$ ; θ_j is the threshold of classification.

The fuzzy classification based on nearness is:

$Max [| k_{ij} (A_{\sim i}^{j}, B_{\sim}^{j}) | | \underset{\sim i}{A} \in \underset{\sim}{A}, i = 1, 2, \cdot \cdot \cdot, n] ⩾ v_{j}$ (8)

Where, $k_{ij} (A_{\sim i}^{j}, B_{\sim}^{j})$ is the nearness of j-th attribute of decision solution $\underset{\sim}{B}$ and sample $\underset{\sim i}{A}$ , 0 < k_ij < 1.0, v_j represents the threshold value.

The samples satisfied Equations (7) or (8) are regarded as in the same rank with the management decision solution $\underset{\sim}{B}$ ; θ_j and v_j will be determined according to the requirements of the rank.

Since the assessment of influence factors t_i (i = 1, 2, 3, …, 10) is a matter of degree uncertainty, denote the fuzzy membership function is defined by μ (t_i) (i = 1, 2, 3, …, 10); applying 10 samples A_j (t_i) (i = 1, 2, 3, …, 10 ; j = 1, 2, …, 10) from sample base, which satisfy Equations (7) and (8) to infer the sale price of property $\underset{\sim}{B}$ . The processes of determining fuzzy distance are presented in Table 3, and fuzzy identification processes through Equation (9) are presented in Table 4 respectively.

In Tables 3 and 4, if we assume $\underset{\sim}{B}$ is also a sample, then we have the new sample set $\underset{\sim i}{A}$ ’ (i = 1, 2, … , n, n + 1). Through pair comparison, following distance matrix can be obtained:

$\underset{\sim}{D} (\underset{\sim p}{A}, \underset{\sim q}{A}) = \sum_{i = 1}^{m} w_{i} (d_{i}) \times d_{i} (\underset{\sim p}{A}, \underset{\sim q}{A})$ (9)

Where, p, q = 1, 2, …, n, n + 1; n is dimension of sub-space (the attribute number of the decision-making); m is number of attributes of management event state $\underset{\sim}{A}$ and $\underset{\sim}{B}$ in any level; w_i is the weight of i-th attribute.

Then the r-th attribute of management decision alternative $\underset{\sim}{B}$ in j-th attribute can be determined by Equation (10):

$\underset{\sim}{B_{r}} = \frac{\sum_{i = 1}^{n} {(1 - D_{i (n + 1)})}^{K} \times A_{ij}}{\sum_{i = 1}^{n} {(1 - D_{i (n + 1)})}^{K}} (j = 1, 2, 3, \dots, m)$ (10)

Where, K is weight indicator, empirically we take K = 2.

Selecting from Table 4, the three minimum values of D_i are from sample A₅, A₈ and A₉ respectively, thus, we obtain the sale price $\underset{\sim}{B}$ should match to the nearest values of D_i by Equation (10): $\begin{matrix} \underset{\sim r}{B} ’ & = (1 / 3) \times [(1 - D_{5}) \times A_{5} + (1 - D_{8}) \times A_{8} \\ + (1 - D_{9}) \times A_{9}] \\ = (1 / 3) \times [(1 - 0.0024) \times 3800 + (1 - 0.0071) \\ \times 3200 + (1 - 0.0008) \times 3700] = 3555.07 \end{matrix}$

Taking account of inflation and other un-foreseeing factors, a coefficient of 1.2 is adopted for determine the marketing price, therefore, the decision of sale price $\underset{\sim r}{B}$ ’ will be: $\underset{\sim r}{B} ’ = 1.2 \times 3555.07 = 4266 .$

In this example, the fuzzy distance is used for matching the appropriate sale price with the most closed past sale price (best practice) as the simulation process of human experience. It is another expression of “Fuzzy-AI” modeling.

5.4 Case study of managerial events by maximum / minimum model

Maximum/Minimum (MM) model is widely used for selecting management alternative decision, which is introduced as the example of expression in “Fuzzy-AI Model”.

For instance, to make a decision which is functioned by implicit interaction of attributes, can be served as an example of MM model. Obviously, certain unwritten rules and understanding between people are necessarily under consideration. In MM model, fuzzy set expression can be introduced for decision evaluation.

The management event is characterized of maximizing or minimizing the project objective parameter(s), which is a quantitative representation either tangible or intangible for the solution alternative $\underset{\sim}{B}$ . It is necessary

to assess those quantitative representations by further fuzzy synthetic assessment.

Assume that the evaluation attribute set is defined by:

$\underset{\sim}{U} = (u_{1}, u_{2}, \dots, u_{m})$ (11)

Where, $\underset{\sim}{U}$ is attributive set of the management event which comprised of u₁, u₂, …, u_m attributes.

The fuzzy evaluation subset of solution alternative $\underset{\sim}{B}$ with respect to each attribute can be expressed by membership function of satisfaction:

$μ (B) = \frac{μ_{B} (u_{1})}{u_{1}} + \frac{μ_{B} (u_{2})}{u_{2}} + \dots + \frac{μ_{B} (u_{m})}{u_{m}}$ (12)

The weight coefficient between u_i (i = 1, 2, …, m)are p_i (i = 1, 2, …, m), then the fuzzy synthetic evaluation indicator η of the solution alternative for selection are:

$Max / Min η = \sum_{i = 1}^{m} p_{i} \times μ_{B} (u_{i}) \geq I$ (13)

Where, I represents the satisfactory threshold of the solution. The solution can be accepted if Equation (13) is satisfied. Moreover, it will be more acceptable as the threshold I increases (or decreases).

This case implies the applying of fuzzy set expression for representing internal implicit relation between the attributes for achieving the objective goal of the project in comprehensive approach.

5.5 Case study of managerial events by assessment and evaluation model

Assessment and Evaluation (AE) Model is used for evaluating human factors influence to the alternative decision of a joint ventured project, which is introduced as the example of “Fuzzy-AI Model”.

Human factor and cultural risk will seriously affect to the success of joint-ventured project. If a joint-ventured project which is targeting in the right goal, taking the right form, at the right time and under right management etc., it will certainly be successful. Nevertheless, the human factors shown below could perform in serious influences:

Strategic thinking and management of two (or more) decision-maker of the project;

Personality of multiple administrators of the project;

Ideological believing or religion believing;

Personality of people who involve in the project;

Transferability of basic management works by the counter-partners;

Individual style in work and working qualification;

Interfaces and channel of communication between two parties for implementing strategic management etc.

Obviously, above-mentioned are crucial to the success of the project, the human factor is also dominant to all business with people having different cultural background on following aspects:

Human factor in the understanding of project contract;

Human factor in business psychology;

Human factor in strategic planning of marketing;

Human factor in utilizing techniques and resources;

Human factor in leading the organization;

Human factor in motivating and forming the enterprise culture.

Fuzzy assessment can be used for quantitative decision making with proper cooperative mode of joint ventured project. It is possible to apply fuzzy analytic hierarchy process (FAHP) to quantize the human factor by Fuzzy-AI Model or by heuristic knowledge. Therefore, fuzzy assessment and evaluation model can be established once the membership function μ is defined. μ can be constructed through different conditions with respect to different cooperative modes. The joint-ventured modes and the conditions are listed in Table 5.

For joint-ventured project in Table 5, there are two kinds of fuzzy membership function with respect to different conditions:

μ_a in Fig. 3(a) represents membership function which varies in terms of different cooperative modes with human factor involved. μ_a consists of following conditions:

Favorableness in Economic Policies,

Availability of Infrastructure,

Stability of Political Situation, and

Working Efficiency of Local Staff.

μ_b in Fig. 3(b) represents membership function for “Marketing and Profitability”, which maintains constant in terms of different cooperative modes disregard of human factor involved.

It is shown by Fig. 3 that the modes of the membership function are defined in four cases, namely:

Mode 1: Foreign ventured,

Mode 2: Joint ventured,

Mode 3: Local ventured with foreign loans, and

Mode 4: Local ventured with foreign loans and local invest.

It is obvious that μ_a is sensitive for cooperative modes 1 and 2; but almost negligible for modes 3 and 4. However, μ_b maintains unity while whatever modes are being adopted; it is less sensitive for all cooperative modes and implies that whilst the “condition” of “marketing and profitability” is optimistic, any cooperative mode will have positive response. It is clear that by using membership functions, one can quantitatively determine which joint-venture mode can be fairly accepted in the decision making of joint-ventured projects.

Combining Assessment and Evaluation (AE) model with Maximum/Minimum (MM) model, it is possible to evaluate the market and profitability of joint ventured project fuzzily and quantitatively based on different environmental conditions and with respect to different cooperation modes.

This example extends the area of how the “Fuzzy-AI Model” is being applied for different managerial events. By means of constructing fuzzy membership function and simulating of decision logic thinking of human being following AE and MM, the concerning point(s) of project objective under different operation mode can be quantized fuzzily.

6 Conclusions

Management process is tightly related to decision making, which represents the highest human intelligence, characterized of processing of massive fuzzy information heuristically. Through “Fuzzy-AI model”, it is possible to simulate these processes fuzzily in digital form. The theoretical framework of “Fuzzy-AI model” is presented in this paper, which laid out a global insight of the project management problems relating to so-called “Quantitative Management”, which is a new branch in modern managerial science.

It is clear in Table 1 that the elements in the performance matrix E have almost covered all the fields of management events; and the types of modeling presented in Section IV are sufficient enough for representing a majority of managerial events. However, further works in modeling of specific management events including fuzzy reasoning and different membership function configuration, are needed in the future.

It seems that one cannot expect to complete such a complicated theoretical framework for managerial science in one paper; nevertheless, it is expected that this paper could raise interests from our readers for further using different quantitative management model (such as Fuzzy-AI model and other deterministic and in-deterministic intelligent algorithms) for solving management problems in more extensive areas.

References

Chen

H.Y.

and Lin

S.P.

, Studies of intelligent decision supporting system for real estate, Proceedings of 6-th National Conference on Computer Applications in Civil Engineering, 1995, pp. 43–50. (in Chinese).

Lin

S.P.

and Wu

W.R.

, On human factors affected to cooperative project with people of different cultural background, Proceedings of the 3rd International Symposium on Human Factors in Organizational Design and Management, Kyoto, Japan, 1990.

Lin

S.P.

, et al., Fuzzy economic assessment for offshore oil fields, Journal of China Ocean Engineering 4(1) (1990), 97–108.

Lin

S.P.

, et al., A conceptual approach of fuzzy decision for systems by “deep knowledge” and “deep data”, Proceedings of 2nd International Conference on the Applications of AI Technology in Civil and Structural Engineering, UK, 1991.

Lin

S.P.

, et al., Fuzzy reasoning and machine learning of expert system for structural design, Computational Mechanics in Structural Engineering, Elsevier Applied Science, 1992, pp. 450–463.

Lin

S.P.

, Fuzzy-AI Model, Proceedings of 2nd International Conference on Artificial Intelligence for Engineering, Wuhan, China, 1998a, pp. 56–72.

Lin

S.P.

, Fuzzy modeling of investment decision in engineering project, Uncertainty Modeling and Analysis in Civil Engineering, CRC press, USA, 1998b, pp. 167–188.

Lin

S.P.

, On paradox of Fuzzy-AI modeling: Supervised learning for rectifying fuzzy membership functions, Journal of Artificial Intelligence Review 23 (2005), 395–405.

Lin

S.P.

, Fuzzy-AI model for managerial science, Keynote Plenary Speech of the 3rd PMI World Research Conference, Warsaw, Poland, 2008.

10.

Lin

S.P.

, et al., The Fuzzy-AI Modeling for Optimization of Long-Term Metro Vehicle Repair, Proceedings of the 6th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) 2009 vol. 6, Tianjin, China, 2009, pp. 459–463.

11.

Lin

S.P.

, et al., RCM-Fuzzy Model and its Applications to Maintenance with Cost/Serviceability Optimization, Proceedings of the PACIIA Conference 2010, Wuhan, China, 2010, pp. 4–6.

12.

Lin

S.P.

and Zhang

W.Y.

, Decision traps and fuzzy quantitative risk management of overseas projects, Civil Engineering Innovation 5(1) (2011), 29–40.

13.

Lin

S.P.

, et al., Optimization Studies of Metro repair based on “FEMA-Fuzzy” Model (I) — Theoretical Studies, Journal of Optimization in Infrastructure Management 24(2) (2012a), 22–28. (in Chinese).

14.

Lin

S.P.

, et al., Optimization Studies of Metro repair based on “FEMA-Fuzzy” Model (II) — Software Implementation, Journal of Optimization in Infrastructure Management 24(3) (2012b), 14–19. (in Chinese).

15.

Lin

S.P.

, Principles and Applications of Quantitative Management — Fuzzy-AI Model in Managerial Science, Comprehensive PM Learner Co. Ltd., 2014.

16.

PMI, Project Management Body of Knowledge (PMI-PMBOK), PMI Press, 2013

17.

Saaty

T.L.

, The analytic hierarchy process, McGraw Hill, New York, 1980.

18.

Vasant

P.M.

, et al., Innovation in Power, Control, and Optimization: Emerging Energy Technologies, Hershey, PA, IGI Global, 2012, pp. 1–396.

19.

Vasant

P.M.

, Meta-Heuristics Optimization Algorithms in Engineering, Business, Economics, and Finance, Hershey, PA, IGI Global 2013, pp. 1–734.

20.

Vasant

P.M.

, Handbook of Research on Novel Soft Computing Intelligent Algorithms: Theory and Practical Applications, Hershey, PA, IGI Global, 2014, pp. 1–1018.

21.

Vasant

P.M.

, Handbook of Research on Artificial Intelligence Techniques and Algorithms, Hershey, PA, IGI Global, 2015, pp. 1–873.

22.

and Lin

S.P.

, Financing risk analysis and its Fuzzy-AI model solution, Journal of Project management Technology 12(6) (2014a), 101–103. (in Chinese).

23.

and Lin

S.P.

, Risk management in overseas project construction and investment by Fuzzy-AI model, Journal of Optimization in Infra-Structure Investment 26(1) (2014b), 2–7. (in Chinese).

24.

, et al., Quantitative Solution of Overseas Project Risk Management by Knowledge Engineering, Proceedings of International Conference on Sustainable Development of Critical Infrastructure (IC-SDCI) 2014, Shanghai, China, 2014, pp. 16–18.

25.

Yang

X.F.

and Lin

S.P.

, Long-term highway system investment decision based on fuzzy model of comprehensive pavement distress, Proceedings of ISUMA’95, 1995.

26.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.