Fuzzy multi-objective decision making approach for nuclear power plant installation

Abstract

Due to the increase in energy demand, many countries suffer from energy poverty because of insufficient and expensive energy supply. Plans to use alternative power like nuclear power for electricity generation are being revived among developing countries. Decisions for installation of power plants need to be based on careful assessment of future energy supply and demand, economic and financial implications and requirements for technology transfer. Since the problem involves many vague parameters, a fuzzy model should be an appropriate approach for dealing with this problem. This study develops a Fuzzy Multi-Objective Linear Programming (FMOLP) model for solving the nuclear power plant installation problem in fuzzy environment. FMOLP approach is recommended for cases where the objective functions are imprecise and can only be stated within a certain threshold level. The proposed model attempts to minimize total duration time, total cost and maximize the total crash time of the installation project. By using FMOLP, the weighted additive technique can also be applied in order to transform the model into Fuzzy Multiple Weighted-Objective Linear Programming (FMWOLP) to control the objective values such that all decision makers target on each criterion can be met. The optimum solution with the achievement level for both of the models (FMOLP and FMWOLP) are compared with each other. FMWOLP results in better performance as the overall degree of satisfaction depends on the weight given to the objective functions. A numerical example demonstrates the feasibility of applying the proposed models to nuclear power plant installation problem.

Keywords

Project management nearest interval approximation method goal programming fuzzy multi-objective linear programming nuclear power plant

Nomenclature

(i, j)

Sequence of nodes, j will be processed after i

K _{D
_ij}

Direct cost of activity (i, j) under normal time

Y _ij

Crash time for activity (i, j)

s _i,j

Crashing cost per unit time for activity (i, j)

Penalty cost per unit time

E _i

Start time for node i

T _ij

Duration time for activity (i, j)

D _ij

Normal duration time for activity (i, j)

d _ij

Shortest duration time for activity (i, j)

Required completion time for the project

λ _k

Membership degree of the objective k

1 Introduction

In order to stabilize the growth of countries and the need for electricity, renewable, non-renewable and nuclear energy must be used. Nuclear energy is one of the most practical ways to meet energy demands. That is why nuclear power plant installation seems to be an important subject for the country’s development. Turkey is largely dependent on fossil fuels such as petroleum, natural gas, and coal which make up a significant part of energy intake. Then, renewable and nuclear energy resources can be considered as some of the most effective solutions for clean and sustainable energy. They have fewer effects on the environment as compared to fossil energy resources. It is also important to note that nuclear and renewable energy resources reduce CO₂ emissions and assist in protecting the environment. Nuclear energy is seen by government as an important way of diversifying energy types, cheap, sustainable, eco-friendly and reducing this dependence [28]. But the construction of a nuclear power plant is a very complex project management task that requires a huge investment, time and attention. Due to the nature of project management, uncertainties are very high and Fuzzy Multi-Objective Decision Making methods can be a useful tool. Zimmermann’s [30] linear programming methods and the fuzzy set theory of Bellman and Zadeh [4] combine to achieve effective results in project planning and develop strategies that are more relevant to the objectives.

A new approach to existing techniques is introduced in our proposed model. The main contribution of this article is the use of Nearest Interval Approximation and Goal Programming to obtain the importance weight of the objectives in FMWOLP models. Therefore, deriving the weights of criteria even from inconsistent fuzzy comparison matrix would be possible with Nearest Interval Approximation and Goal Programming approach.

The reminder of the paper is organized as follows: Section 2 illustrates the latest literature highlights. Two project management models are constructed in Section 3. A case study of the installation of nuclear power plant in Turkey is examined in Section 4. Some final comments are made in Section 5. Last section is devoted to conclusions.

2 Literature review

In recent years, fuzzy set theory have evolved to solve real world problems. In many studies, fuzzy logic was included in mathematical modeling. Arya and Singh [1] performed a fuzzy efficient iterative method for multi-objective linear fractional programming problems. They proposed an iterative fuzzy approach to search fuzzy efficient solution set for multi-objective linear fractional programming (MOLFP) problems. This approach is based on randomly generated fuzzy parametric preferences in the interval [0, 1] and the fuzzy efficient solution is obtained with the percentage of satisfaction for each objective. Bhati et al. [2] stressed about a fuzzy based branch and bound approach for multi-objective linear fractional (MOLF) optimization problems. They convert the original MOLF optimization problem into equivalent fuzzy multi-objective linear fractional (FMOLF) optimization problem. Then, they applied branch and bound techniques on FMOLF optimization problem. The feasible space of FMOLF optimization problem is bounded by triangular simplex space. Su [27] studied a fuzzy multi-objective linear programming model for solving re-manufacturing planning problems with multiple products and joint components. He developed a fuzzy multi-objective linear programming (FMOLP) model that simultaneously minimizes total costs, lead time and CO₂ emissions with reference to multiple products and joint components. His proposed model evaluates cost-effectiveness, lead time and CO₂ emissions, while integrating multi-products, multi-suppliers, multi-components, joint components and multi-machines into one re-manufacturing production system. Dubey and Mehra [11] applied a bipolar approach in fuzzy multi-objective linear programming. Their approach facilitate a natural fusion of bipolarity in FMOLPPs. The flexible constraints in a fuzzy multi-objective linear programming problem (FMOLPP) are viewed as negative preferences for describing what is somewhat tolerable while the objective functions of the problem are viewed as positive preferences for depicting satisfaction to what is desirable.

Shaw et al. [25] presented an integrated approach for selecting the appropriate supplier in the supply chain, addressing the carbon emission issue, using fuzzy-AHP and fuzzy multi-objective linear programming. They applied Fuzzy AHP (FAHP) for analyzing the weights of the multiple factors. The factors taken into consideration are cost, quality rejection percentage, late delivery percentage, green house gas emission and demand. They use these weights of the multiple factors in fuzzy multi-objective linear programming for supplier selection and quota allocation. Cheng et al. [7] solved fuzzy multi-objective linear programming problems using deviation degree measures and weighted max–min method. Using the deviation degree measures and weighted max–min method, they transformed the FMOLP problem into crisp linear programming (CLP) problem. If decision makers fix the values of deviation degrees of two side fuzzy numbers in each constraint, then the d-pareto-optimal solution of the FMOLP problems can be obtained by solving the CLP problem. Baky [3] proposed two new algorithms to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. His first algorithm groups the membership functions for the defined fuzzy goals of the objective functions at all levels and the decision variables for each level except the lower level of the multi-level problem. The second algorithm lexicographically solves MOLP problems of the ML-MOLP problem by taking into consideration the decisions of the MOLP problems for the upper levels. Kang et al. [17] studied project management for a Wind Turbine Construction by Applying Fuzzy Multiple Objective Linear Programming Models. They used the fuzzy weighted multi objective linear programming model (FMWOLP).

The desire to use nuclear energy to meet the demand for energy has motivated many academic researches. Erdoğan and Kaya [12] applied a combined fuzzy approach to determine the best region for a nuclear power plant in Turkey. They combined fuzzy multi criteria decision making (MCDM) methodology that consists of Interval type-2 and fuzzy analytical hierarchy process (AHP). They ranked alternatives with interval type-2 fuzzy TOPSIS to determine the best location alternative for the nuclear power plant. Shahi et al. [24] studied the development of nuclear power plants by means of modified model of Fuzzy DEMATEL and GIS in Bushehr/Iran. They present a model to build a nuclear power plant in order to demonstrate both the cause and effect relationship among effective criteria in the decision making process and the ultimate weight for each criterion. Erol et al. [13] stressed about fuzzy MCDM framework for locating a nuclear power plant in Turkey. They proposed a tool based on fuzzy Entropy and t norm based fuzzy compromise programming to deal with the vagueness of human judgments. Purba [22] considered fuzzy probability on reliability study of nuclear power plant probabilistic safety assessment. His study has categorized two types of fuzzy probability approaches, i.e. fuzzy based fault tree analyses (FTA) and fuzzy hybrid FTA. His study confirms that the fuzzy based FTA should be used when the uncertainties are the main focus of the FTA. Kunsch and Teghem [18] proposed nuclear fuel cycle optimization using multi-objective stochastic linear programming. An interactive multicriteria approach using stochastic linear programming is provided by the code Strange, which has already been applied to different energy strategy problems, also outside the nuclear field. The literature that addresses this research issue is still growing. Some studies have discussed in the field, e.g. [8 , 29].

Our proposed model brings a new approach to existing models already present in the literature. For the first time, two subject such as FMOLP-FMWOLP and project planning are combined together. In addition, Goal Programming and Nearest Interval Approximation methodology are used to generate the weights of the objective functions without the need for consistency check.

3 Proposed models

In this section, a new approach to existing models is introduced in the light of literature review. The features added to the existing models are explained in the relevant sections.

3.1 Fuzzy multi-objective linear programming (FMOLP)

For the problem, a Fuzzy Multi-Objective Linear Programming (FMOLP) model is proposed [17]. The steps are as follows:

3.1.1 Multi-objective linear programming model

By constructing balance between the time and the price, a compromise project implementation plan is found. Three objectives are considered: least total project price, shortest total project period and longest total crash time.

First Objective Function: Minimize the total project cost TC. Min TC = ∑_i∑_jK_{D
_ij} + ∑_i∑_jY_ijs_ij $+ [l * Max {0, E_{N} - TP}]$ (1)

The purpose is to minimize the total cost (TC) to complete the project. This cost includes the direct cost (K_{D
_ij}), the crashing cost (s_ij) and the penalty cost (l) which depends on the total project duration (TP). (i.e. the cost to be incurred if the project is delayed during the normal period). Second Objective Function: Minimize the total project duration TP. $Min TP = E_{N} - E_{1}$ (2) This equation indicates the difference between the start and end times of the project, i.e. the total duration of the project.

Third Objective Function: Maximize the total crash time TR. $Max TR = \sum_{i} \sum_{j} Y_{ij}$ (3) This equation gives the sum of the crash times of each activity in the project.

The constraints are: $E_{i} + T_{ij} \leq E_{j} \forall i, j$ (4) $T_{ij} + Y_{ij} = D_{ij} \forall i, j$ (5) $Y_{ij} + d_{ij} \leq D_{ij} \forall i, j$ (6) $E_{1} = 0$ (7) $E_{N} \leq F$ (8)i, j = 1,...,N and all variables are non-negative.

Equation (4) shows that the sum of the start time (E_i) and the process time (T_ij) of activity i should be equal to or smaller than the start time (E_j) of activity j.

Equation (5) is the sum of the process time (T_ij) and the crashing time (Y_ij) between i and j activities equal to the planned normal duration (D_ij) of the (i, j) node.

Equation (6) shows that the sum of the crashing time (Y_ij) and the shortest possible process time (d_ij) of the node could be smaller than the planned normal process time (D_ij) of the node. That is, the process time (D_ij) of a node should not go below the shortest specified process time (d_ij).

Equation (7) assumes that the initial moment of the first activity (E₁) is zero.

Equation (8) ensures that the last activity is completed before or on the required project completion time (F).

3.1.2 Positive and negative ideal solutions

Calculate the positive ideal solution (PIS) and the negative ideal solution (NIS) of the three objective functions. ${TC}^{PIS} = Min TC, {TC}^{NIS} = Max TC$ (9)

Equation (9) assumes that positive ideal solution of total cost is minimization of total cost and negative ideal solution of total cost is maximization of total cost. ${TP}^{PIS} = Min TP, {TP}^{NIS} = Max TP$ (10)

Equation (10) assumes that positive ideal solution of total time is minimization of total time and negative ideal solution of total time is maximization of total time. ${TR}^{PIS} = Max TR, {TR}^{NIS} = Min TR$ (11)

Equation (11) assumes that positive ideal solution of total crash time is maximization of total crash time and negative ideal solution of total crash time is minimization of total crash time.

So, the membership function for each of the three objective functions is established as follows: $λ_{TC} = {\begin{matrix} 1, & TC \leq {TC}^{PIS} \\ \frac{{TC}^{NIS} - TC}{{TC}^{NIS} - {TC}^{PIS}}, & {TC}^{PIS} < TC \leq {TC}^{NIS} \\ 0, & TC > {TC}^{NIS} \end{matrix}$ (12) $λ_{TP} = {\begin{matrix} 1, & TP \leq {TP}^{PIS} \\ \frac{{TP}^{NIS} - TP}{{TP}^{NIS} - {TP}^{PIS}}, & {TP}^{PIS} < TP \leq {TP}^{NIS} \\ 0, & TP > {TP}^{NIS} \end{matrix}$ (13) $λ_{TR} = {\begin{matrix} 1, & TR \geq {TR}^{PIS} \\ \frac{TR - {TR}^{PIS}}{{TR}^{NIS} - {TR}^{PIS}}, & {TR}^{NIS} \leq TR < {TR}^{PIS} \\ 0, & TR < {TR}^{NIS} \end{matrix}$ (14)

3.1.3 Fuzzy programming method

After the membership function values are obtained, apply the Fuzzy Programming Method proposed by Zimmermann to add auxiliary variable λ, which might take into account the three objective functions at the same time [30]. The initial Fuzzy Multi-Objective Linear Programming problem will be remodeled into a Crisp Single-Goal Linear Programming problem. By maximizing λ, a compromise resolution will be obtained. $Max λ$ (15)subject to: $λ \leq λ_{TP}$ (16) $λ \leq λ_{TC}$ (17) $λ \leq λ_{TR}$ (18) $E_{i} + T_{ij} \leq E_{j} \forall i, j$ (19) $T_{ij} + Y_{ij} = D_{ij} \forall i, j$ (20) $Y_{ij} + d_{ij} \leq D_{ij} \forall i, j$ (21) $E_{1} = 0$ (22) $E_{N} \leq F$ (23) and all variables are non-negative.

The time constraints in Equation (19)-(20)-(21)-(22) are accepted to be crisp numbers.

3.1.4 FMOLP model result

The results include the whole project cost, the whole project completion time and also the total crash time and therefore the three objectives will be satisfied.

3.2 Fuzzy multiple weighted-objective linear programming (FMWOLP)

The decision makers could take into account that the importance of each objective is completely different. That is, some objectives are more vital than others. As a result, the weights of the objectives should be determined by the management first. We propose to integrate the Nearest Interval Approximation and Goal Programming used to find weights into the FMOLP model. Thus, the relative weights of the multiple objectives are calculated by the Nearest Interval Approximation and Goal Programming and by using these weights, the FMOLP will be recreated as FMWOLP to reach best TC, TP, TR values.

3.2.1 Integrated fuzzy pairwise comparison matrix

By collecting the fuzzy judgment matrices from all decision makers, this matrices can be aggregated by using the Fuzzy Geometric Mean Method of Buckley [5, 6]. The aggregated triangular fuzzy numbers of n decision makers’ judgment in an certain case ${\tilde{u}}_{ij} = (l_{ij}, m_{ij}, u_{ij})$ is: ${\tilde{u}}_{ij} = {(\prod_{i = 1}^{n} {\tilde{a}}_{ijk})}^{\frac{1}{n}}$ (24) where ${\tilde{a}}_{ijk}$ is the relative importance in form of triangular fuzzy numbers of the k^th decision makers’ view and n is the total number of decision makers.

3.2.2 Importance weights using the nearest interval approximation and goal programming

The Nearest Weighted Possibilistic Interval Approximation:

It is an interval operator of a fuzzy number. First we introduce an f-weighted distance quantity on the fuzzy numbers and then we obtain the Interval Approximations for a fuzzy number [16].

Definition: A weighting function is a function as $f = (\underline{f}, \bar{f}) : ([0, 1], [0, 1]) \to (R, R)$ such that the functions are non-negative, monotone increasing and satisfies the following normalization condition: $\int_{0}^{1} \underline{f} (α) d α = \int_{0}^{1} \bar{f} (α) d α = 1$

Definition: Let $\tilde{A}$ be a fuzzy number with ${\tilde{A}}_{α} = [\underline{a} (α), \bar{a} (α)]$ and $f (α) = (\underline{f} (α), \bar{f} (α))$ being a weighted function [23]. Then the interval ${NWIA}_{f} (\tilde{A}) = \int_{0}^{1} \underline{f} (α) \underline{a} (α) d α$ (25) $= \int_{0}^{1} \bar{f} (α) \bar{a} (α) d α$

is the Nearest Weighted Interval Approximation to fuzzy number $\tilde{A}$ .

The function f (α) can be understood as the weight of our Interval Approximation; the property of monotone increasing function f (α) means that the higher the cut level is, the more important its weight is in determining the Interval Approximation of fuzzy numbers. In applications, the function f (α) can be chosen according to the actual situation.

Corollary: Let $\tilde{A} = (a, b, c)$ be a triangular fuzzy number and let f (α) = (nα^n-1, nα^n-1) be a weighting function. Then ${NWIA}_{f} (\tilde{A}) = [\frac{a + nb}{n + 1}, \frac{nb + c}{n + 1}]$ (26)

Goal Programming: In the conventional case, if a Pairwise Comparison Matrix A be reciprocal and consistent then the weights of each criterion are simply calculated as $w_{i} = \frac{a_{ij}}{\sum_{k = 1}^{n} A_{kj}}, i = 1, . . ., n$ . In the case of inconsistent matrix, we must obtain the importance weights w_i, i = 1, . . . , n such that $a_{ij} = \frac{w_{i}}{w_{j}}$ or equivalently a_ijw_j - w_i = 0. Therefore in the case of uncertainty, for deriving the weights of criteria from inconsistent fuzzy comparison matrix we follow the following procedure [16].

Firstly by Equation (26) we convert each fuzzy element ${\tilde{a}}_{ij} = (a_{ij}^{L}, a_{ij}^{M}, a_{ij}^{U})$ of the PCM to the Nearest Weighted Interval Approximation $\bar{a_{ij}} = [{\bar{a}}_{ij}^{L}, {\bar{a}}_{ij}^{U}]$ . Hence the Fuzzy Pairwise Comparison Matrix $\tilde{A}$ is converted to an Interval Pairwise Comparison Matrix $\bar{A}$ . After, we must calculate the weight vector w_i, i = 1, . . . , n such that ${\bar{a}}_{ij}^{L} \leq \frac{w_{i}}{w_{j}} \leq {\bar{a}}_{ij}^{U}$ ; therefore we must have ${\bar{a}}_{ij}^{L} w j \leq w_{i} \leq {\bar{a}}_{ij}^{U} w j$ . Hence we introduce deviation variables $p_{ij}^{-}, p_{ij}^{+}$ and $q_{ij}^{-}, q_{ij}^{+}$ which lead to ${\bar{a}}_{ij}^{L} w_{j} - w_{i} + p_{ij}^{-} - p_{ij}^{+} = 0$ (27) $- {\bar{a}}_{ij}^{L} w_{j} + w_{i} + q_{ij}^{-} - q_{ij}^{+} = 0$ (28) where deviation variables $p_{ij}^{-}, p_{ij}^{+}$ and $q_{ij}^{-}, q_{ij}^{+}$ are non-negative real numbers but cannot be positive at the same time, that is, $p_{ij}^{-} p_{ij}^{+} = 0$ and $q_{ij}^{-} q_{ij}^{+} = 0$ . Now we apply the Goal Programming method. It is desirable that the deviation variables $p_{ij}^{+}$ and $q_{ij}^{+}$ are kept to be as small as possible, which leads to the following Goal Programming model: $Min \sum_{i = 1}^{n} \sum_{j = 1}^{n} (p_{ij}^{+} + q_{ij}^{+})$ (29)subject to: ${\bar{a}}_{ij}^{L} w_{j} - w_{i} + p_{ij}^{-} - p_{ij}^{+} = 0$ (30) $- {\bar{a}}_{ij}^{L} w_{j} + w_{i} + q_{ij}^{-} - q_{ij}^{+} = 0$ (31) $\sum_{i = 1}^{n} w_{i} = 1$ (32) $w_{i}, p_{ij}^{-}, p_{ij}^{+}, q_{ij}^{-}, p_{ij}^{+} \geq 0$

By solving this model the optimal weight vector W = (w₁, . . . , w_n) which shows the importance of each criterion will be obtained. We can use these weights in the process of solving a multiple criteria decision-making problem. Also, these weights show which criterion is more important than others. To rank these criteria, we assign rank 1 to the criterion with the maximal value of w_i and so forth, in a decreasing order of w_i. The proposed method is able to derive the weights of criteria when the elements of the PCM (consistent or inconsistent) are fuzzy in any form.

This Goal Programming model we use in practice is always feasible. Here is the little evidence for its feasibility. Consider $\hat{W} = ({\hat{w}}_{1}, . . ., {\hat{w}}_{n})$ which has the condition $\sum_{i = 1}^{n} {\hat{w}}_{i} = 1, {\hat{w}}_{i} \geq 0, i = 1, . . ., n$ . Then we can define ${\hat{p}}_{ij}^{-} = \max {\begin{matrix} - ({\bar{a}}_{ij}^{L} {\hat{w}}_{j} - {\hat{w}}_{i}) & , 0 \end{matrix}}$ (33) ${\hat{p}}_{ij}^{+} = \max {\begin{matrix} ({\bar{a}}_{ij}^{L} {\hat{w}}_{j} - {\hat{w}}_{i}) & , 0 \end{matrix}}$ (34) ${\hat{q}}_{ij}^{-} = \max {\begin{matrix} - (- {\bar{a}}_{ij}^{U} {\hat{w}}_{j} + {\hat{w}}_{i}) & , 0 \end{matrix}}$ (35) ${\hat{q}}_{ij}^{+} = \max {\begin{matrix} (- {\bar{a}}_{ij}^{U} {\hat{w}}_{j} + {\hat{w}}_{i}) & , 0 \end{matrix}}$ (36) It is clear that $(\hat{W}, {\hat{p}}_{ij}^{-}, {\hat{p}}_{ij}^{+}, {\hat{q}}_{ij}^{-}, {\hat{p}}_{ij}^{+})$ is a feasible solution [16].

3.2.3 FMWOLP model

The overall aim is to maximize the λ_TC, λ_TP, λ_TR which are the satisfaction rating, using the weights obtained. Thus, the best possible TC, TP, TR values are obtained, with the objectives of minimizing the total project cost, minimizing the total project duration time and maximizing the total crash time. $Max λ = w_{TC} * λ_{TC} + w_{TP} * λ_{TP} + w_{TR} * λ_{TR}$ (37)subject to: $λ_{TP} \leq \frac{{TP}^{NIS} - TP}{{TP}^{NIS} - {TP}^{PIS}}$ (38) $λ_{TC} \leq \frac{{TC}^{NIS} - TC}{{TC}^{NIS} - {TC}^{PIS}}$ (39) $λ_{TR} \leq \frac{TR - {TR}^{PIS}}{{TR}^{NIS} - {TR}^{PIS}}$ (40) $E_{i} + T_{ij} \leq E_{j} \forall i, j$ (41) $T_{ij} + Y_{ij} = D_{ij} \forall i, j$ (42) $Y_{ij} + d_{ij} \leq D_{ij} \forall i, j$ (43) $E_{1} = 0$ (44) $E_{N} \leq F$ (45) and all variables are non-negative.

where w_TC, w_TP and w_TR are the normalized importance weights for the total project cost, the total project duration time and the total crash time, respectively; λ_TC, λ_TP, λ_TR are the degrees of satisfaction for the total project cost, the total project duration time and the total crash time, respectively [30].

3.2.4 FMWOLP model result

The results include the whole project cost, the whole project completion time and also the total crash time and therefore the three objectives will be satisfied. Briefly, the steps of the proposed methodology are given in Fig. 1.

Fig. 1

Flowchart of the proposed methodology.

4 Application

Nuclear energy technology is an energy production method in the form of electricity generation from turbine connected generator and steam turbine rotation using the heat energy produced by the division of certain heavy atoms such as uranium and plutonium. It was first developed in the 1940s and after the World War II it was used for commercial electricity production. As a result of accidents, nuclear power plant conditions have been tightened, making their design and construction more expensive. Following new regulations, safety measures and strict licensing terms, nuclear energy has become more expensive and risky investment. In many countries, the problem of nuclear waste, which is not resolved due to political reasons and problems of interest with people’s acceptance, has limited the spread of nuclear power plants. Following the massive nuclear stagnation in the western world in the 1990s, the Fukushima accident hampered the nuclear renaissance and re-enactment for the post-2010 period. Many countries have changed or questioned their future nuclear predictions.

Today, about 16 countries provide at least 25 percent of their electricity needs from nuclear energy. In recent years, nuclear plant installation in Turkey has become an important issue and research on the subject has increased.

The nuclear power plant installation is utilized for instance to look at the common sense of the proposed models. The fundamental development information of installation nuclear power plant in Turkey are shown in Table 1. The project has been planned for 20 years.

Table 1
Construction data for nuclear power plant project

Activity code Activity Duration (Year) Crash time (Year) Cost (TL) Crash cost (TL)

(a) Pre-project 3 1 33,210,601.6 4,427,765.3

(b) Decision-making 7 3 349,090,909.091 23,271,072.19

(c) Project management 16 7 628,363,636.36 18,616,557.75

(d) Preparation of site infrastructure 4 1 768,000,000 68,261,811.76

(e) Detailed design engineering 6 2.5 698,181,818.18 53,191,022.14

(f) Equipment and component 5 2 5,026,909,090.91 446,804,586.04

(g) Construction erection and installation 5 2 6,004,363,636.36 533,683,255.55

(h) Commissioning and plant organization 4 1.5 488,727,272.72 52,127,201.704

Activity code	Activity	Duration (Year)	Crash time (Year)	Cost (TL)	Crash cost (TL)
(a)	Pre-project	3	1	33,210,601.6	4,427,765.3
(b)	Decision-making	7	3	349,090,909.091	23,271,072.19
(c)	Project management	16	7	628,363,636.36	18,616,557.75
(d)	Preparation of site infrastructure	4	1	768,000,000	68,261,811.76
(e)	Detailed design engineering	6	2.5	698,181,818.18	53,191,022.14
(f)	Equipment and component	5	2	5,026,909,090.91	446,804,586.04
(g)	Construction erection and installation	5	2	6,004,363,636.36	533,683,255.55
(h)	Commissioning and plant organization	4	1.5	488,727,272.72	52,127,201.704

4.1 Fuzzy multi-objective linear programming

4.1.1 Multi-objective linear programming model

Based on Equations (1)-(8), the Multi-Objective Linear Programming Model is constructed as follows:

Min TC =(33.210.601, 6 + 4.427.765, 3 Y₁₂) + (349.090.909, 091 + 23.271.072, 19 Y₂₄) + (628.363.636, 36 + 18.616.557, 75 Y₂₃) + (768.000.000 + 68.261.811, 76 Y₄₅) + (698.181.818, 18 + 53, 191, 022.14 Y₄₆) + (5.026.909.090, 91 + 446.804.586, 04 Y₄₇) + (6.004.363.636, 63 + 553.683.255, 55 Y₅₈) + (488.727.272, 72 + 52.127.201, 704 Y₆₉) Min TP = E₉ - E₁ Max TR = Y₁₂ + Y₂₃ + Y₂₄ + Y₄₅ + Y₄₆ + Y₄₇ + Y₅₈ + Y₆₉

subject to: E₁ + T₁₂ = E₂

E₂ + T₂₃ = E₃

E₂ + T₂₄ = E₄

E₄ + T₄₅ = E₅

E₄ + T₄₆ = E₆

E₄ + T₄₇ = E₇

E₅ + T₅₈ = E₈

E₆ + T₆₉ = E₉

T₁₂ + Y₁₂ = 3

T₂₃ + Y₂₃ = 16

T₂₄ + Y₂₄ = 7

T₄₅ + Y₄₅ = 4

T₄₆ + Y₄₆ = 6

T₄₇ + Y₄₇ = 5

T₅₈ + Y₅₈ = 5

T₆₉ + Y₆₉ = 4

Y₁₂ - 1 ≤0

Y₂₃ - 7 ≤0

Y₂₄ - 3 ≤0

Y₄₅ - 1 ≤0

Y₄₆ - 2.5 ≤ 0

Y₄₇ - 2 ≤0

Y₅₈ - 2 ≤0

Y₆₉ - 1.5 ≤ 0

and all variables are non-negative.

The linear programming code of this model for online linear programming solver [31] is in [9].

4.1.2 Positive and negative ideal solutions

By applying the Equations (9)-(11), the PIS and the NIS values of the three objectives are calculated under each objective function. The values are shown in Table 2.

Table 2
The PIS and the NIS values of the three objectives

Objective function Min TC Min TP Max TR (PIS, NIS)

TC (TL) 13,996,846,965.491 16,199,790,853.687 16,441,809,704.457 (13,996,846,965.491; 16,441,809,704.457)

TP (Year) 20 12 12 (12; 20)

TR (Year) 0 18.5 20 (20; 0)

Objective function	Min TC	Min TP	Max TR	(PIS, NIS)
TC (TL)	13,996,846,965.491	16,199,790,853.687	16,441,809,704.457	(13,996,846,965.491; 16,441,809,704.457)
TP (Year)	20	12	12	(12; 20)
TR (Year)	0	18.5	20	(20; 0)

Based on the values of the PIS and NIS, the membership functions of the objective functions are established using Equations (1)-(8) as follows:

$λ_{TC} = {\begin{matrix} 1, & TC \leq 13 . 10^{9} \\ \frac{16 . 10^{9} - TC}{16 . 10^{9} - 13 . 10^{9}}, & 13 . 10^{9} < TC \leq 16 . 10^{9} \\ 0, & TC > 16 . 10^{9} \end{matrix}$

$λ_{TP} = {\begin{matrix} 1, & TP \leq 12 \\ \frac{20 - TP}{20 - 12}, & 12 < TP \leq 20 \\ 0, & TP > 20 \end{matrix}$

$λ_{TR} = {\begin{matrix} 1, & TR \geq 20 \\ \frac{0 - TR}{0 - 20}, & 0 \leq TR < 20 \\ 0, & TR < 0 \end{matrix}$

4.1.3 Fuzzy programming method

The auxiliary variable λ is added to linear programming model with the membership values for the objective functions. By applying Equations (9)-(11), the initial FMOLP problem is remodeled into a Crisp Single-Goal Linear Programming problem.

Max λ

subject to: $λ \leq \frac{16, 441, 809, 704.457 - TC}{16, 441, 809, 704.457 - 13, 996, 846, 965.491}$ $λ \leq \frac{20 - TP}{20 - 12}$ $λ \leq \frac{0 - TR}{0 - 20}$

TC = (33, 210, 601.6 + 4, 427, 765.3 Y₁₂) + (349, 090, 909.091 + 23, 271, 072.19 Y₂₄) + (628, 363, 636.36 + 18, 616, 557.75 Y₂₃) + (768, 000, 000 + 68, 261, 811.76 Y₄₅) + (698, 181, 818.18 + 53, 191, 022.14 Y₄₆) + (5, 026, 909, 090.91 + 446, 804, 586.04 Y₄₇) + (6, 004, 363, 636.63 + 553, 683, 255.55 Y₅₈) + (488, 727, 272.72 + 52, 127, 201.704 Y₆₉)

TP = E₉ - E₁

TR = Y₁₂ + Y₂₃ + Y₂₄ + Y₄₅ + Y₄₆ + Y₄₇ + Y₅₈ + Y₆₉

Constraints Equation (19)-(20)-(21)-(19xxx)-(22) and all variables are non-negative.

The linear programming code of this model is in [9].

4.1.4 FMOLP model result

By using online linear programming solver [31], the results shows that λ_TC = 0.802, λ_TP = 1 and λ_TR = 0.8. Other results are shown in Table 3. Activities (a), (b), (c), (d), (e) and (h) should be crashed. Thus, the duration time (TP) for the activities will reduce from the original 20 years, as shown in Table 3, to 12 years. As a result, total cost (TC) is 14,480,834,021.227 TL, total project duration time (TP) is 12 years and the total crash time (TR) is 16 years. The resulting satisfaction degree of the problem (λ=0.8) appears to be a fairly high value.

Table 3
Results of FMOLP model

Activity code Duration (Year) Crash time (Year) Crash cost (TL)

(a) 2 1 4,427,765.3

(b) 4 3 69,813,216,57

(c) 9 7 130,315,904.25

(d) 3 1 68,261,811,76

(e) 3.5 2.5 132,977,555.35

(f) 5 0 -

(g) 5 0 -

(h) 2.5 1.5 78,190,802.556

TR = 16 Total Crash Cost = 483,987,055.786

Activity code	Duration (Year)	Crash time (Year)	Crash cost (TL)
(a)	2	1	4,427,765.3
(b)	4	3	69,813,216,57
(c)	9	7	130,315,904.25
(d)	3	1	68,261,811,76
(e)	3.5	2.5	132,977,555.35
(f)	5	0	-
(g)	5	0	-
(h)	2.5	1.5	78,190,802.556
		TR = 16	Total Crash Cost = 483,987,055.786

4.2 Fuzzy multiple weighted-objective linear programming

The same case applies to the proposed FMWOLP model. The main difference in both models is that the weights of the objectives are taken into account in the FMWOLP model. Five decision makers are selected and an Integrated Fuzzy Pairwise Comparison Matrix is generated.

Decision makers gave priority to three objectives. By collecting the fuzzy judgment matrices from all decision makers, Integrated Fuzzy PCM is constructed in Table 4.

Table 4
Integrated fuzzy pairwise comparison matrix

TC TP TR

TC 1 (1.34, 1.62, 1.99) (1.6, 1.92, 2.35)

TP ( $\frac{1}{1.99}, \frac{1}{1.62}, \frac{1}{1.34}$ ) 1 (1.64, 2.35, 2.99, 3.59)

TR ( $\frac{1}{2.35}, \frac{1}{1.92}, \frac{1}{1.6}$ ) ( $\frac{1}{3.59}, \frac{1}{2.99}, \frac{1}{2.35}, \frac{1}{1.64}$ ) 1

	TC	TP	TR
TC	1	(1.34, 1.62, 1.99)	(1.6, 1.92, 2.35)
TP	( $\frac{1}{1.99}, \frac{1}{1.62}, \frac{1}{1.34}$ )	1	(1.64, 2.35, 2.99, 3.59)
TR	( $\frac{1}{2.35}, \frac{1}{1.92}, \frac{1}{1.6}$ )	( $\frac{1}{3.59}, \frac{1}{2.99}, \frac{1}{2.35}, \frac{1}{1.64}$ )	1

4.2.1 Importance weights

We use two step to find degrees of importance of three objectives. First step is to convert Integrated Fuzzy PCM to Interval Approximation PCM by using the Nearest Interval Approximation. The second step is to construct the Goal Programming model. As stated in Section 3.2.2, this method can be used to derive weight from consistent and inconsistent matrices.

Step 1 - Nearest Interval Approximation: Let $\tilde{A} = (a, b, c)$ be a triangular fuzzy number and let f (a) = (na^n-1, na^n-1) be a weighted function. NWIA $f (A) = [\begin{matrix} \frac{a + nb}{n + 1}, \frac{nb + c}{n + 1} \end{matrix}]$

We set n = 5 to increase sensitivity and apply the formula accordingly. So, f (a) = (5a⁴, 5a⁴) NWIA $f (A) = [\begin{matrix} \frac{a + 5 b}{6}, \frac{5 b + c}{6} \end{matrix}]$

Using the above statements, the Integrated Fuzzy PCM in Table 4 is converted to Interval Approximation PCM in Table 5. Thus, the decisions are ready for use in the Goal Programming model.

Table 5
Interval approximation pairwise comparison matrix

TC TP TR

TC [1.0 1.0] [1.573 1.6812] [1.86 1.992]

TP [0.598 0.638] [1.0 1.0] [2.231 3.00]

TR [0.505 0.538] [0.325 0.456] [1.0 1.0]

	TC	TP	TR
TC	[1.0 1.0]	[1.573 1.6812]	[1.86 1.992]
TP	[0.598 0.638]	[1.0 1.0]	[2.231 3.00]
TR	[0.505 0.538]	[0.325 0.456]	[1.0 1.0]

Step 2 - Goal Programming: We construct the Goal Programming model for the above Interval Approximation PCM:

$Min p_{12}^{+} + q_{12}^{+} + p_{13}^{+} + q_{13}^{+} + p_{21}^{+} + q_{21}^{+} + p_{23}^{+} + q_{23}^{+} + p_{31}^{+} + q_{31}^{+} + p_{32}^{+} + q_{32}^{+}$

subject to: $1.573 w_{2} - w_{1} + p_{12}^{-} - p_{12}^{+} = 0$ $w_{1} - 1.6812 w_{2} + q_{12}^{-} - q_{12}^{+} = 0$ $1.86 w_{3} - w_{1} + p_{13}^{-} - p_{13}^{+} = 0$ $w_{1} - 1.992 w_{3} + q_{13}^{-} - q_{13}^{+} = 0$ $0.598 w_{1} - w_{2} + p_{21}^{-} - p_{21}^{+} = 0$ $w_{2} - 0.638 w_{1} + q_{21}^{-} - q_{21}^{+} = 0$ $2.231 w_{3} - w_{2} + p_{23}^{-} - p_{23}^{+} = 0$ $w_{2} - 3.09 w_{3} + q_{23}^{-} - q_{23}^{+} = 0$ $0.505 w_{1} - w_{3} + p_{31}^{-} - p_{31}^{+} = 0$ $w_{3} - 0.538 w_{1} + q_{31}^{-} - q_{31}^{+} = 0$ $0.325 w_{2} - w_{3} + p_{32}^{-} - p_{32}^{+} = 0$ $w_{3} - 0.456 w_{2} + q_{32}^{-} - q_{32}^{+} = 0$ w₁ + w₂ + w₃ = 1 $w_{i}, p_{ij}^{-}, p_{ij}^{+}, q_{ij}^{-}, q_{ij}^{+} \geq 0$

By solving this model by an online linear optimization solver application [31], we obtained the results as weights of objective functions, w_TC = 0.467288, w_TP = 0.2981297, w_TR = 0.2345823. The linear programming code of this model is in [9].

4.3 FMWOLP model

By applying "Max λ = w_TC * λ_TC + w_TP * λ_TP + w_TR * λ_TR" as in Section 3.2.3, the FMWOLP model is formulated as follows: Max λ = 0.467288λ_TC + 0.2981297λ_TP + 0.2345823λ_TR

subject to: $λ_{TC} \leq \frac{16, 441, 809, 704.457 - TC}{16, 441, 809, 704.457 - 13, 996, 846, 965.491}$

$λ_{TP} \leq \frac{20 - TP}{20 - 12}$ $λ_{TR} \leq \frac{0 - TR}{0 - 20}$

TP = E₉ - E₁

TR = Y₁₂ + Y₂₃ + Y₂₄ + Y₄₅ + Y₄₆ + Y₄₇ + Y₅₈ + Y₆₉

Constraints Equation (41)-(42)-(43)-(44)-(45) and all variables are non-negative.

The linear programming code of this model is in [9].

4.4 FMWOLP model result

By solving FWMOLP model, the results show that TC = 0.83, TP = 1 and TR = 0.75. Other results are shown in Table 6. Activities (a), (b), (c), (e) and (h) should be crashed. Thus, the duration time for the activities will reduce from the original 20 years, as shown in Table 6, to 12 years. As a result, total cost (TC) is 14,412,572,209.517 TL, total project duration time (TP) is 12 years and the total crash time (TR) is 15 years. The resulting degree of the problem (λ = 0.86) is a fairly high value to raise membership values.

Table 6
Results of FMWOLP model

Activity code Duration (Year) Crash time (Year) Crash cost (TL)

(a) 2 1 4,427,765.3

(b) 4 3 69,813,216,57

(c) 9 7 130,315,904.25

(d) 4 0 -

(e) 3.5 2.5 132,977,555.35

(f) 5 0 -

(g) 5 0 -

(h) 2.5 1.5 78,190,802.556

TR = 15 Total Crash Cost = 415,725,244.026

Activity code	Duration (Year)	Crash time (Year)	Crash cost (TL)
(a)	2	1	4,427,765.3
(b)	4	3	69,813,216,57
(c)	9	7	130,315,904.25
(d)	4	0	-
(e)	3.5	2.5	132,977,555.35
(f)	5	0	-
(g)	5	0	-
(h)	2.5	1.5	78,190,802.556
		TR = 15	Total Crash Cost = 415,725,244.026

5 Discussions

According to FMOLP model, activities (a), (b), (c), (d), (e) and (h) should be crashed. Thus, TP reduced from the original 20 years to 12 years. TC is 14,480,834,021.227 TL and TR is 16 years. According to FMWOLP model, activities (a), (b), (c), (e) and (h) should be crashed. TC is 14,412,572,209.517 TL, TP is 12 years and TR is 15 years. Based on data from DMs, TC becomes more important than other objectives. Therefore, TC is reduced by the changing in activity times, although TP does not change. The change in TR does not affect TP since it was not in the critical path. The comparison of the two models are shown in Table 7.

Table 7
Comparative results of FMOLP and FMWOLP Models

λ λ _TC λ _TP λ _TR TC (TL) TP (Year) TR (Year)

FMOLP 0.8 0.802 1 0.8 14,480,834,021.277 12 16

FMWOLP 0.86 0.83 1 0.75 14,412,572,209.517 12 15

	λ	λ _TC	λ _TP	λ _TR	TC (TL)	TP (Year)	TR (Year)
FMOLP	0.8	0.802	1	0.8	14,480,834,021.277	12	16
FMWOLP	0.86	0.83	1	0.75	14,412,572,209.517	12	15

6 Conclusions

Our proposed method uses fuzzy logic approach to make decisions for facility installation. A fuzzy mathematical model is proposed for the project and weighted fuzzy logic for the same problem is re-modeled. In this approach, three main objective functions have been identified for the facility installation project such as; total cost, total time and total crash time. These objectives are modeled by Fuzzy Multi Objective Linear Programming. The satisfaction degrees are determined for all objectives and these values were maximized with the established model. The goal is to achieve the lowest cost, the minimum time and the highest crash time optimization. The same problem is then used by weighting objective functions. First, with the help of decision makers, Nearest Interval Approximation and Goal Programming techniques are used for criterion weighting. Then, Fuzzy Multi-Weighted Objective Linear Programming model is established. The highest satisfaction is obtained. The results of the two models are evaluated. This approach has been applied on a case study of nuclear power plant installation in Turkey. It is shown that the model using weighting gives better results for all objectives than the first model. The main contribution of this article to the existing literature is the use of Nearest Interval Approximation and Goal Programming to obtain the importance weight of the objectives in FMWOLP models.

For further research, FMOLP and FMWOLP can be extended so as to work with hesitant fuzzy sets. Another type of integration of multi objective decision making methods such as interval type-2 fuzzy sets and pythagorean fuzzy sets can also be proposed. Non-linearity in membership functions can be considered. The results from this new integrations can be compared with the result of our study.

References

Arya

and Singh

, Fuzzy efficient iterative method for multi-objective linear fractional programming problems, Mathematics and Computers in Simulation 160 (2019), 39–54.

Arya

, Singh

and Bhati

, A fuzzy based branch and bound approach for multi-objective linear fractional (MOLF) optimization problems, Journal of Computational Science 24 (2018), 54–64.

Baky

I.A.

, Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach, Applied Mathematical Modelling 34(9) (2010), 2377–2387.

Bellman

and Zadeh

, Decision making in a fuzzy environment, Management Science 17(4) (2008), 141–164.

Buckley

J.J.

, Ranking alternatives using fuzzy numbers, Fuzzy Sets and Systems 15(1) (1985a), 21–31.

Buckley

J.J.

, Fuzzy hierarchical analysis, Fuzzy Sets and Systems 17(3) (1985b), 233–247.

Cheng

, Huang

, Zhou

and Caic

, Solving fuzzy multi-objective linear programming problems using deviation degree measures and weighted max–min method, Applied Mathematical Modelling 37(10–11) (2013), 6855–6869.

Çakır

and Ulukan

H.Z.

, Fuzzy Multi-Objective Decision Making Approach for Nuclear Power Plant Installation, In Intelligent and Fuzzy Techniques in Big Data Analytics and Decision Making, Berlin: Springer-Verlag, (2019), pp. 1258–1265.

Çakır

, GitHub repository, https://github.com/esrckr/Fuzzy-MODM-approach-for-nuclear-/tree/master, last accessed 2019/11/14.

10.

A.K.

, Dewan

and Biswas

, A unified approach for fuzzy multiobjective stochastic programming with Cauchy and extreme value distributed fuzzy random variables, In Intelligent Decision Technologies 12(1) (2018), 81–91.

11.

Dubey

and Mehra

, A bipolar approach in fuzzy multi-objective linear programming, Fuzzy Sets and Systems 246 (2014), 127–141.

12.

Erdogan

and Kaya

, A combined fuzzy approach to determine the best region for a nuclear power plant in Turkey, Applied Soft Computing 39 (2016), 84–93.

13.

Erol

, Sencer

, Özmen

and Searcy

, Fuzzy MCDM framework for locating a nuclear power plant in Turkey, Energy Policy 67 (2014), 186–197.

14.

Feylizadeh

M.R.

, Mahmoudi

, Bagherpour

and Li

D.F.

, Project crashing using a fuzzy multi-objective model considering time, cost, quality and risk under fast tracking technique: A case study, In Journal of Intelligent and Fuzzy Systems 35(3) (2018), 3615–3631.

15.

Hwang

B.C.

, Saif

and Jamshidi

, Fault Detection and Diagnosis of a Nuclear Power Plant Using Artificial Neural Networks, In Fuzzy Multi-Objective Decision Making Approach for Nuclear Power Plant Installation, Journal of Intelligent and Fuzzy Systems 3(3) (1995), 197–213.

16.

Izadikhah

, Deriving weights of criteria from inconsistent fuzzy comparison matrices by using the nearest weighted interval approximation, Hindawi Publishing Corporation Advances in Operations Research. (2012).

17.

Kang

H.Y.

, Lee

A.H.I.

and Huang

T.T.

, Project management for a wind turbine construction by applying fuzzy multiple objective linear programming models, MDPI. (2016).

18.

Kunsch

P.L.

and Teghem

Jr, , Nuclear fuel cycle optimization using multi-objective stochastic linear programming, European Journal of Operational Research 31(2) (1987), 240–249.

19.

, Zhang

and Yang

, Two algorithms for group decision making based on the consistency of intuitionistic multiplicative preference relation, In Journal of Intelligent and Fuzzy Systems 38(2) (2020), 2197–2210.

20.

Niroomand

, A multi-objective based direct solution approach for linear programming with intuitionistic fuzzy parameters, In Journal of Intelligent and Fuzzy Systems 35(2) (2018), 1923–1934.

21.

Ozkok

B.A.

, Finding fuzzy optimal and approximate fuzzy optimal solution of fully fuzzy linear programming problems with trapezoidal fuzzy numbers, In Journal of Intelligent and Fuzzy Systems 36(2) (2019), 1389–1400.

22.

Purba

J.H.

, Fuzzy probability on reliability study of nuclear power plant probabilistic safety assessment: A review, Progress in Nuclear Energy 76 (2014), 73–80.

23.

Saeidifar

, Application of weighting functions to the ranking of fuzzy numbers, Computers and Mathematics with Applications 62 (2011), 2246–2258.

24.

Shahi

, Alavipoor

F.S.

and Karimi

, The development of nuclear power plants by means of modified model of Fuzzy DEMATEL and GIS in Bushehr, Iran, Renewable and Sustainable Energy Reviews 83 (2018), 33–49.

25.

Shaw

, Shankar

, Yadav

S.S.

and Thakur

L.S.

, Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain, Expert Systems with Applications 39(9) (2012), 8182–8192.

26.

, Wu

and Yang

, An analysis of energy consumption and cost-effectiveness for overhead crane drive systems by using fuzzy multi-objective linear programming, In Journal of Intelligent and Fuzzy Systems 35(6) (2018), 6241–6253.

27.

T.S.

, A fuzzy multi-objective linear programming model for solving re-manufacturing planning problems with multiple products and joint components, Computers & Industrial Engineering 110 (2017), 242–254.

28.

Udum

, Turkey’s nuclear comeback, Non-prolif, Rev 17(2) (2010), s.365–377.

29.

J.Z.

, Xi

R.J.

and Zhu

, Correlative decision preference information consistency check and comprehensive dominance representation method, In Journal of Intelligent and Fuzzy Systems 38(2) (2020), 2009–2019.

30.

Zimmermann

H.J.

, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978), 45–55.

31.

Zwols

and Sierksma

, Linear and Integer Optimization: Theory and Practice, http://www.lio.yoriz.co.uk/.