**4. Fuzzy Monte Carlo simulation (FMCS)**

The proposed FMCS is a joint propagation method based on both the probability theory of MCS and the possibility theory of fuzzy arithmetic. A generalized problem in which we have both types of uncertainty, fuzzy and probabilistic. Here, we need to determine the output Y of a function (M) that has R1, R2, … , Rn being random

#### **Figure 4.**

*Converting fuzzy sets to PDF before performing Monte Carlo simulation.*

variables and represented by probabilistic distributions and F1 (triangular values), F2 (triangular values), … , Fm being fuzzy sets (**Figure 4**).

The first stage of the model is to defuzzify the fuzzy variables to get crisp values. The centroid method is one of the most common methods for defuzzification, in which the defuzzified value is calculated by finding the center of the area under the membership function.

The second stage of the model is to transfer the crisp sets of numbers that we got in step one to random variables. To that, the confidence level of the intervals is estimated using the probability of that interval. This probability is equal to the area under the PDF that is bounded within that interval. Among different intervals of the same confidence level, the most informative interval is the one with minimal length [8].

The third stage of the model is to use an optimization algorithm for finding a set of optimal values minimizing or maximizing the given object function [MIN F(X) or MAX F(X)] subjected to minimum-maximum intervals. Then, the uncertainty quantifications propagated using the obtained optimal values are represented as a plausibility distribution and a belief distribution. **Figure 5** shows Fuzzy Monte Carlo simulation (FMCS) process.

### **5. Case study**

To illustrate an implementation of the FMCS model, the researcher analyzes the behavior of FMCS framework in comparison with traditional Monte Carlo simulation using a time range estimating example. Consider a sample application by [12] of a time range estimating problem for an excavation project. The time and cost needed for the project equipment are shown in **Table 1**, and probabilistic distributions are used to express the uncertainty regarding those variables.

These uncertainties may result from uncertainty regarding different scenarios that may happen in the field during construction. For example, uncertainty in the activity duration may be a result of uncertainty associated with the productivity of workers or variability in weather conditions.

The case study evaluated seven alternatives using Monte Carlo simulation (MCS), to compare the presented FMCS model with the traditional MCS that is used by [12], we will evaluate the same alternatives to see if the outcomes of the new FMCS model will be different. The seven equipment configurations alternatives are shown in **Table 2**.

Using Fuzzy logic toolbox in MATLAB the FMCS generates the results shown in **Table 3**.

From **Table 3**, we can conclude that the best alternative is number seven with shift duration of 17 days and shift cost of \$79,050. By comparing this result with the result published by [12], we find out that [12] concluded that the best alternative is also alternative seven with shift duration of 18 days and shift cost of \$82,665.

**Activity Triangular duration (min) Triangular cost (\$/min)**

20 cu yd hauler: haul, unload, return (18,33,48) 2.39 15 cu yd hauler: haul, unload, return (15,28,41) 2.29

*Fuzzy Monte Carlo Simulation to Optimize Resource Planning and Operations*

*DOI: http://dx.doi.org/10.5772/intechopen.93632*

**Figure 5.**

**Table 1.**

**175**

*Activity time and cost data.*

*Fuzzy Monte Carlo simulation (FMCS) process.*

*Fuzzy Monte Carlo Simulation to Optimize Resource Planning and Operations DOI: http://dx.doi.org/10.5772/intechopen.93632*

#### **Figure 5.**

variables and represented by probabilistic distributions and F1 (triangular values),

The first stage of the model is to defuzzify the fuzzy variables to get crisp values. The centroid method is one of the most common methods for defuzzification, in which the defuzzified value is calculated by finding the center of the area under the

The second stage of the model is to transfer the crisp sets of numbers that we got in step one to random variables. To that, the confidence level of the intervals is estimated using the probability of that interval. This probability is equal to the area under the PDF that is bounded within that interval. Among different intervals of the same confidence level, the most informative interval is the one with minimal length [8]. The third stage of the model is to use an optimization algorithm for finding a set of optimal values minimizing or maximizing the given object function [MIN F(X) or MAX F(X)] subjected to minimum-maximum intervals. Then, the uncertainty quantifications propagated using the obtained optimal values are represented as a plausibility distribution and a belief distribution. **Figure 5** shows Fuzzy Monte

To illustrate an implementation of the FMCS model, the researcher analyzes the behavior of FMCS framework in comparison with traditional Monte Carlo simulation using a time range estimating example. Consider a sample application by [12] of a time range estimating problem for an excavation project. The time and cost needed for the project equipment are shown in **Table 1**, and probabilistic distribu-

These uncertainties may result from uncertainty regarding different scenarios that may happen in the field during construction. For example, uncertainty in the activity duration may be a result of uncertainty associated with the productivity of

The case study evaluated seven alternatives using Monte Carlo simulation (MCS), to compare the presented FMCS model with the traditional MCS that is used by [12], we will evaluate the same alternatives to see if the outcomes of the new FMCS model will be different. The seven equipment configurations alterna-

Using Fuzzy logic toolbox in MATLAB the FMCS generates the results shown in

tions are used to express the uncertainty regarding those variables.

F2 (triangular values), … , Fm being fuzzy sets (**Figure 4**).

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

*Converting fuzzy sets to PDF before performing Monte Carlo simulation.*

membership function.

**Figure 4.**

Carlo simulation (FMCS) process.

workers or variability in weather conditions.

tives are shown in **Table 2**.

**Table 3**.

**174**

**5. Case study**

*Fuzzy Monte Carlo simulation (FMCS) process.*


#### **Table 1.**

*Activity time and cost data.*

From **Table 3**, we can conclude that the best alternative is number seven with shift duration of 17 days and shift cost of \$79,050. By comparing this result with the result published by [12], we find out that [12] concluded that the best alternative is also alternative seven with shift duration of 18 days and shift cost of \$82,665.

### *Concepts, Applications and Emerging Opportunities in Industrial Engineering*


**Conflict of interest**

*DOI: http://dx.doi.org/10.5772/intechopen.93632*

tion of this chapter.

**Author details**

**177**

Mohammad Ammar Alzarrad

Marshall University, Huntington, USA

provided the original work is properly cited.

\*Address all correspondence to: alzarrad@marshall.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The authors declare that there are no conflicts of interest regarding the publica-

*Fuzzy Monte Carlo Simulation to Optimize Resource Planning and Operations*

#### **Table 2.**

*Alternatives of equipment configurations.*


**Table 3.** *FMCS results.*
