A novel neutrosophic fuzzy programming model for multiobjective transportation problems with survival costs

1. Introduction

Mathematical programming utilizes linear programming problems (LPPs), which are widely recognized for their extensive application in numerous fields. Transportation problem (TP) is one of the fundamental optimization fields that fall within these applications. The TP is essential in logistics, supply chain, and supplier selection processes. Using a single objective to optimize TPs fails to deliver the best results when operating in highly competitive markets. Decision-makers (DMs) face situations that require analyzing numerous contradictory objectives, resulting in multi-objective transportation problems (MOTPs). The traditional MOTPs use linear objective functions, yet specific practical applications necessitate working with nonlinear functions. Introducing non-linearity into models delivers enhanced results that better serve the DM’s strategic choices when circumstances demand it.

Hitchcock (1941a) was the first to introduce the TP as a mathematical form. Later, Hitchcock (1941b) proposed a TP with nonlinear costs that was formulated and solved, thereby initiating a surge of research in this domain. A supply chain problem is considered incomplete without incorporating a transportation aspect. Many innovative concepts and mathematical techniques have been developed to address TPs in recent years. Akdemir and Tiryaki (2012) proposed a bi-level stochastic TP model, which optimizes transportation with stochastic demand in decentralized firms. Roy et al. (2012) proposed a technique for transforming cost coefficients with multiple choices or goals was proposed. Furthermore, a multi-choice stochastic TP with an exponential distribution was introduced. This problem was designed to address real-world transportation challenges involving uncertain factors. Javaid et al. (2013) presented a multipurpose bottleneck TP that is expanded to include randomness and flexibility in supply and demand. Moreover, a bi-criteria stochastic problem was also developed. Xu and Huang (2013) provided a new modeling approach to the traveling salesman problem (TSP) for the context of a transportation hotspot market for multiple bilateral swaps. This method examines the TSP considering a stochastic demand periodic sealed double auction. Rani and Gulati (2014) introduced an innovative approach to tackle the challenges of unbalanced TP. Quddoos et al. (2014) proposed a general form of multi-criteria stochastic TP with the equivalent deterministic model. Roy (2016) introduced Lagrange’s interpolating polynomial, which is used to minimize total cost. It also presents a method for transforming stochastic supply constraints into deterministic constraints using stochastic programming. Abdel-Basset et al. (2019) proposed a method for solving the neutrosophic linear programming models, whose parameters are represented by trapezoidal neutrosophic numbers. Angelov (1997) introduced the optimization method for the first time in an intuitionistic framework. Mahmoodirad et al. (2019) suggested a novel, efficient method for resolving the TPs by treating all parameters as triangular intuitionistic fuzzy values. Kour and Basu (2015) recommended using the extended fuzzy programming problem in a neutrosophic environment for real-world MOTPs.

Rizk-Allah et al. (2018) solved the MOTPs in a neutrosophic setting and used the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) methodology to measure the ranking degree to compare the findings with previous methods. Melethil et al. (2025) implemented neutrosophic programming in enhancing the state of Canada towards the achievement of Sustainable Development Goals (SDGs), noting that Canada has faced diverse difficulties in promoting sustainable economic development alongside sustainable development; they also noted uncertainties involved in decision-making systems. Ahmad and Adhami (2019) studied TPs that incorporate uncertainty using interval-based optimization, time-dependent cost functions, and probabilistic models, enhancing decision-making through robust mathematical and multiobjective optimization approaches. Ge and Ishii (2011) introduced the suggested bottleneck TP. Moreover, by integrating the assemblage of both the flexible supply side and the demand side, a new bi-directional stochastic criterion problem was produced as an extension towards overcoming the limitations of the bottleneck transit issue.

Adhami et al. (2022) proposed a neutrosophic compromise programming approach (NCPA) as the best compromise solution for the TP, which focused on uncertainties in multi-objective TPs and addressed fuzzy and stochastic parameter uncertainties efficiently. Biswas et al. (2019) investigated both crisp and interval settings in a multi-objective fixed-charge TP. Gupta et al. (2020b) proposed a multi-objective TP with capacitated restrictions. It assumes that the parameter of supply and demand follows a gamma distribution. Gupta et al. (2020a) conducted a case study on the issue of capacitated transportation difficulties with multiple objectives in an unpredictable setting. Soroush et al. (2020) studied problems associated with the transportation and inventory of maritime goods under three distinct daily demand scenarios. The research employs computational analysis to investigate the impact of gamma, exponential, and uniform demand patterns on the observed outcomes. Agrawal and Ganesh (2020) outlined a method to solve the resolution of fuzzy fractional TPs with stochastic exponential distribution parameter using fuzzy random variables (RVs), dual approach, and fuzzy programming to solve for the solutions. (Giri & Roy, 2024) proposed a fuzzy-random robust flexible programming model for a sustainable closed-loop renewable energy supply chain integrating PV, biogas, and incineration systems, offering optimized facility selection and reduced environmental impact under uncertainty. (Giri & Roy, 2025) proposed a distributionally robust, multi-objective conic goal programming model for India’s closed-loop hydrogen and energy network, demonstrating reduced emissions, improved waste utilization, and enhanced operational efficiency across integrated gas and electricity systems.

2. Literature review and thematic classification

To handle many sources of uncertainty, recent research on transportation optimization has moved from deterministic to stochastic, fuzzy, intuitionistic, and neutrosophic modelling frameworks. This evolution is summed up in the thematic synthesis that follows.

3. Fuzzy set (FS)

A fuzzy set $X$ within a universe $W$, where each element is denoted by $w$, is characterized by a membership function ${\mu }_X\left(w\right)$ that maps elements from $W$ to the interval $\left(0,1\right)$. This function assigns to each element $w\in W$ a membership degree ${\mu }_X\left(w\right)\in \left(0,1\right)$, indicating the extent of $w$’s membership in the fuzzy set $X$. Specifically, ${\mu }_X\left(w\right)=0$ signifies that $w$ is not a member of $X$, ${\mu }_X\left(w\right)=1$ indicates full membership, and values between 0 and 1 represent partial membership.

A parabolic fuzzy number is a triplet $X\left(x,y,z\right)$ that represents the smallest, middle, and greatest values of a membership function if its membership function is specified as follows:

For the parabolic fuzzy number $\tilde{X}\left(x,y,z\right)$, the de-fuzzified value function $d$ may be written as follows:

4. Intuitionistic fuzzy set (IFS)

5. Neutrosophic set (NS)

6. Transportation optimization with survival cost functions under fluctuating supply and demand

The normal day is significantly impacted by TPs. There are often objectives to meet when delivering partnerships or products. Understanding the optimal outcomes in real-world scenarios with relation to goal functions requires modeling and optimizing TPs under specific restrictions. When homogeneous commodities or products must be transported from different origins (or warehouses) to different sinks (or markets) at the lowest possible cost, in the shortest amount of time, or in order to optimize profit, among other limitations, this is referred to as a TPs.

Finding the total number of goods that must be delivered in order to achieve the recommended ideal goals is the goal.

Imagine a TP in which $I$ origins have a supply of $s_i$ units ($i=1,2,\dots ,I$) and $J$ destinations have a demand for $d_j$ units ($j=1,2,\dots ,J$). The cost of moving one unit from the $i$-th origin to the $j$-th destination is represented by $r_{ij}$.

To find the optimal solution, the decision variable $y_{ij}$ denotes the unknown number of units moved from the $i$-th origin to the $j$-th destination.

Now, the general mathematical model for TPs is given by

Model 1

$Z=\sum _{i=1}^I\sum _{j=1}^J\left(\frac{r_{ij}}{q_{ij}}\right)y_{ij}$

Subject to:

$\sum _{i=1}^I{y}_{ij}\ge d_j,j=1,2,\dots \mathrm{..},J$

$y_{ij}\ge 0,\forall i,j$

In most TPs, it is anticipated that the items will be delivered within the allotted period. Additional penalties, such as product damage or customer order cancellations owing to delays or lost time, may be levied if this requirement is not fulfilled. The distributor’s or transporter’s reputation may suffer because of these sanctions. Road conditions, traffic, weather, and other environmental factors are just a few of the variables that can affect the transportation system and result in product delivery delays. The earnings and transportation expenses might no longer be regarded as fixed values in such situations. Maity et al. (2016) introduced the concept of ‘Survival cost/profit,’ which is a probabilistic function of the cost and profit in the proposed study.

7. Survival function

In logistics, survival cost/profit is the likelihood that a product will be successfully transported and delivered within the specified time, subject to the constraints imposed by DMs. The fixed cost rises and the fixed profit falls when probabilistic cost is incorporated into TPs. This has a substantial effect on the profit made from these items/products and changes the initial value of the goods being delivered. This is the definition of the cost/profit probability function, ${\Psi }_{ij}\left(t\right)$, which is dependent on time $t$:

${\Psi }_{ij}\left(t\right)=\frac{\text{quantity of items that are still in good shape following delays in transit}}{\text{Total quantity of cargo shipped}}$

Likewise, the likelihood of failure, $F_{ij}\left(t\right)$, can be given as:

$F_{ij}\left(t\right)=\frac{\text{quantity of products harmed as a result of late or delayed shipment}}{\text{Total quantity of cargo shipped}}$

These probabilities satisfy the following equation:

The practical observation that delays, decay, and real-time disruptions directly impact transportation reliability is what drives the use of survival cost functions. The chance that a shipment will arrive at its destination within the allowed time frame is reflected in the survival probability. These probabilities are usually estimated using managerial expectations, past delivery performance, or expert judgment because historical data is frequently limited. DMs can explicitly account for risk and rapid losses that traditional fuzzy and neutrosophic models ignore by incorporating survival probability into the model.

The basic objective functions of profit and cost are changed into survival profit and survival cost when time is included as a factor, enabling the model to take delays and their effects into consideration. Determining whether the main goal should be to minimize total transportation cost or maximize overall profit is a significant difficulty in this approach Maity et al., 2016).

Case 1: In the event that the objective is to minimize transportation costs, the survival cost function is given as:

${\Psi }_{ij}\left(r_{ij}\right)=r_{ij}+r_{ij}\left(1-{\Psi }_{ij}\left(t\right)\right)$

Case 2: In the pursuit of maximal total profit, the survival profit function is given as:

${\Psi }_{ij}\left(q_{ij}\right)=q_{ij}-q_{ij}\left(1-{\Psi }_{ij}\left(t\right)\right)$

The addition of a survival cost/profit function to the conventional TP model may result in one of two possible outcomes.

Case 1: In the case of an objective function of the minimization type, the objective is to reduce the overall cost or damage while taking into account the probability of survival of the transported products.

Model 2

$MinZ=\sum _{i=1}^I\sum _{j=1}^J{\Psi }_{ij}\left(r_{ij}\right)y_{ij}$

Once the subsequent goal function has been simplified, the result is,

$MinZ=2\sum _{i=1}^I\sum _{j=1}^J{r}_{ij}{y}_{ij}-\sum _{i=1}^I\sum _{j=1}^J{r}_{ij}{\Psi }_{ij}\left(t\right)y_{ij}$

Subject to, Constraint (5)

Case 2: A maximization-type objective function aims to maximize overall profit while accounting for the probability of product survival.

Model 3

$MaxZ=\sum _{i=1}^I\sum _{j=1}^J{\Psi }_{ij}\left(q_{ij}\right)y_{ij}$

After simplifying the goal function, we obtain

$MaxZ=\sum _{i=1}^I\sum _{j=1}^J{q}_{ij}{\Psi }_{ij}\left(t\right)y_{ij}$

Subject to the constraint (5).

Models 2 and 3 represent the classical TPs with profit functions and survival costs, respectively.

Due to the implementation of various production planning strategies and prevailing market conditions, accurately quantifying the homogeneous product supply and demand in most TPs remains challenging. To ascertain the exact intervals within which the quantities will decrease, the DM must insert a change in the supply and demand quantities according to the circumstances.

Assumed to be variable in this instance, the quantity of supply $s_i$ and demand $d_j$ can be represented by ${\tilde{s}}_i$ and ${\tilde{d}}_j$, respectively,

The following represents the mathematical models.

Model 4

$MinZ=2\sum _{i=1}^I\sum _{j=1}^J{r}_{ij}{y}_{ij}-\sum _{i=1}^I\sum _{j=1}^J{r}_{ij}{\Psi }_{ij}\left(t\right)y_{ij}$

Subject to:

$\sum _{i=1}^I{y}_{ij}\ge {\tilde{d}}_j,\forall j=1,2,\dots \mathrm{..},J$

$y_{ij}\ge 0, \forall i, j.$

And the model 5 is given as,

$MaxZ=\sum _{i=1}^I\sum _{j=1}^J{\Psi }_{ij}\left(q_{ij}\right)y_{ij}$

Subject to the constraint (7).

8. Multi-objective transportation survival cost function with fuzzy parameters

It is acknowledged that uncertainty, whether probabilistic or fuzzy, is essential to creating decision-making models. The decision maker’s inaccurate estimates are the source of the parameter ambiguity. In some situations, the problem’s solution should take this ambiguity into account. The goal function’s cost parameter is assumed to be fuzzy in this study. Fuzzy set theory-based ranking function techniques are used to convert this fuzziness into a clear or predictable result. The defuzzified discussion is described as follows:

The general mathematical model for the transportation survival cost function is given as:

$MinZ=2\sum _{i=1}^I\sum _{j=1}^J{r}_{ij}{y}_{ij}-\sum _{i=1}^I\sum _{j=1}^J{r}_{ij}{\Psi }_{ij}\left(t\right)y_{ij}$

Subject to the constraint (7).

The following formula (2) is used to de-fuzzify the cost parameter, which is a fuzzy number in the objective function of the problem:

Model 6

$MinZ=2\sum _{i=1}^I\sum _{j=1}^{J}D\left(r_{ij}\right)y_{ij}-\sum _{i=1}^I\sum _{j=1}^{J}D\left(r_{ij}\right){\Psi }_{ij}\left(t\right)y_{ij}$

Subject to:

$\sum _{i=1}^I{y}_{ij}\ge D\left({\tilde{d}}_j\right),j=1,2,\dots \mathrm{..},J$

$y_{ij}\ge 0,\forall i,j.$

where $D\left(r_{ij}\right)$ is the defuzzified value of $\left(r_{ij}\right)$

9. Neutrosophic programming approach

A new method for solving multi-objective TPs with fuzzy factors has been presented. This approach uses three membership functions: the minimum of the falsity membership function, the highest degree of truth, and the maximum indeterminacy. Let $F$ stand for fuzzy judgments, $G$ for fuzzy aims, and $C$ for fuzzy constraints. The definition of the appropriate neutrosophic decision set, denoted by the notation $F_N$, is as follows:

Where,

Regarding the decision set $F_N$ of neutrosophic numbers, the truth, indeterminacy, and falsity degrees are provided by the functions $T_F\left(y\right)$, $I_F\left(y\right)$,$G_F\left(y\right)$. In order to determine the various membership functions for MOTPs, we have defined the limits for each objective function. The $L_p$ and $U_{p,}$ respectively, and can be obtained as follows: Initially, we addressed each objective function individually within the established limitations of the problem. Upon addressing p objectives independently, we obtain a collection of p solutions, $Y^1,Y^2,\mathrm{..}.Y^p.$ Subsequently, the derived solutions are substituted into each objective function to produce the lower and upper bounds for each objective, as outlined below:

At this point, the upper and lower boundaries are,

For truth membership:

For indeterminacy membership:

For falsity membership,

The parameters $s_k$ and $t_k$ correspond to the lower and upper aspiration levels of each objective function. They are selected directly from the minimum and maximum values observed in the payoff matrix to ensure that the membership functions are aligned with realistic operational bounds. In this case, the DMs have predetermined real values $s_k$ and $t_k\in \left(0,1\right)$. The linear membership functions are constructed using these lower and upper limits in a neutrosophic set as follows:

The condition $L_p^{\left(.\right)}\ne U_p^{\left(.\right)}$ holds for all objective functions. On the other hand, the equivalent membership function will be $1$ if $(L_p^{(.)}=U_p^{(.)})$. The following formulation of the problem may be made using the methodology described in Bellman and Zadeh (1970):

$\max mi{n}_{p=1,2,\dots ,P}{T}_p\left(Z_p\left(\underset{\_}{y}\right)\right)$

$\max mi{n}_{p=1,2,\dots ,P}{I}_p\left(Z_p\left(\underset{\_}{y}\right)\right)$

$\min ma{x}_{p=1,2,\dots ,P}{G}_p\left(Z_p\left(\underset{\_}{y}\right)\right)$

$y_{ij}\ge 0,\forall i,j$

With the use of auxiliary parameters, the issue may be changed into the following form.

Subject to:

$y_{ij}\ge 0, \forall i and j$

$\alpha \ge \beta ,\alpha \ge \gamma ,\alpha +\beta +\gamma \le 3,\alpha ,\beta ,\gamma \in \left(0,1\right)$

Using the above linear membership function, the problem can be written as,

Subject to,

$y_{ij}\ge 0,\forall iandj$

Fig. 1 illustrates the 3D surface plot that represents the neutrosophic membership function space, showing how each decision alternative is evaluated across the truth (T), indeterminacy (I), and falsity (F) dimensions, capturing uncertainty, inconsistency, and hesitation in decision modeling.

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Fig. 1.

Each decision alternative is evaluated across three independent membership dimensions representing truth, indeterminacy, and falsity in the neutrosophic environment.

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10. NPA stepwise algorithms

The stages that make up the suggested method may be summed up as follows:

• As described in Section (7), formulate the MOTP using fuzzy parameters.

• To turn the problem into a crisp form, use the de-fuzzification method described in Equation (2).

• Determine the goal functions for the set of restrictions at each level to create a payoff matrix.

• Establish the upper and lower bounds for each stage’s objective function.

• Determine the upper and lower bounds for the truth, indeterminacy, and falsity membership functions using Equations (16), (17), and (18).

• In a neutrosophic environment, define the linear membership function as shown by Equations (17) through (19).

• Equations (23) through (30) are used to create the neutrosophic problem, which is then transformed into a neutrosophic compromise programming problem using Equations (31) through (38).

• Utilizing the optimization software Lingo 16.0, find the compromise solution to the MOTP survival cost function.

11. Numerical problem

A pharmaceutical company is distributing its newly developed vaccine from four manufacturing plants in Maharashtra (India), located in Mumbai (S1), Pune (S2), Nagpur (S3), and Nashik (S4), to hospitals in four cities of Gujarat, namely Ahmedabad (D1), Surat (D2), Vadodara (D3), and Rajkot (D4). The DMs are unsure of the different transportation-related expenses because this is the first time the vaccine is being given in these areas. Naturally, because the distribution network is young, the exact amounts of supply and demand may not be known. The DMs thus use parabolic fuzzy numbers rather than fixed values to estimate all parameters because of the absence of accurate information. The transportation cost per unit and loss of time of the vaccine from different manufacturing plants to various hospital locations, denoted as ($r_{ij}{\Psi }_{ij}$), has been presented in Table 1 along with fuzzy supply and demand constraints. Additionally, the labor and safety costs per unit incurred in transportation, represented as ($r_{ij}{\Psi }_{ij}$) and ($r_{ij}{\Psi }_{ij}$), have been given in Tables 2 and 3, respectively. The company incurs a base cost for delivering the minimum required vaccine units to each hospital, and any excess supply beyond the agreed limit results in an additional charge per unit, in accordance with the company’s regulations. The DMs of the pharmaceutical company aim to determine the optimal number of vaccine units to be transported from different plants to various hospitals while minimizing the costs of each objective under uncertain cost, supply, and demand conditions.

In practical logistics settings, the survival cost reflects the risk of shipments failing to reach their destination within the acceptable time window, capturing delays, perishability, and reliability concerns. The labor and safety cost represents expenses related to workforce handling, accident prevention, and compliance with safety requirements, which are critical for operational sustainability. The time-loss cost measures delays and waiting times that increase operational inefficiency and reduce service quality. Together, these objectives capture the essential trade-offs faced by logistics managers in balancing reliability, safety, and timely delivery under uncertain conditions.

Step 1. The mathematical model of the above problem is formulated using the above information.

$\begin{array}{c}\text{min}{Z}_1=2(\left(2.2,2.4,2.1\right)y_{11}+\left(1.5,1.7,2.1\right)y_{12}+\left(2.3,2.5,2.6\right)y_{13}\\ +\left(2.5,3.5,4.5\right)y_{14}+\left(1.2,2.2,3.2\right)y_{21}+\left(1.4,1.8,2.2\right)y_{22}\\ +\left(2.8,3.7,4.8\right)y_{23}+\left(0,2,4\right)y_{24}+\left(0,2,4\right)y_{31}+\left(2,2.5,3.5\right)y_{32}\\ +\left(1.3,1.8,2.3\right)y_{33}+\left(1.3,1.5,1.7\right)y_{34}+\left(2,2.3,2.6\right)y_{41}\\ +\left(2.5,2.9,3.3\right)y_{42}+\left(2.2,2.8,3.4\right)y_{43}+\left(2.2,2.4,2.6\right)y_{44})\\ -(\left(2.2,2.4,2.1;0.3\right)y_{11}+\left(1.5,1.7,2.1;0.2\right)y_{12}\\ +\left(2.3,2.5,2.6;0.1\right)y_{13}+\left(2.5,3.5,4.5;0.4\right)y_{14}\\ +\left(1.2,2.2,3.2;0.4\right)y_{21}+\left(1.4,1.8,2.2;0.2\right)y_{22}\\ +\left(2.8,3.7,4.8;0.3\right)y_{23}+\left(0,2,4;0.1\right)y_{24}+\left(0,2,4;0.1\right)y_{31}\\ +\left(2,2.5,3.5;0.4\right)y_{32}+\left(1.3,1.8,2.3;0.3\right)y_{33}\\ +\left(1.3,1.5,1.7;0.2\right)y_{34}+\left(2,2.3,2.6;0.3\right)y_{41}\\ +\left(2.5,2.9,3.3;0.4\right)y_{42}\\ +\left(2.2,2.8,3.4;0.1\right)y_{43}+\left(2.2,2.4,2.6;0.2\right)y_{44})\end{array}$

$\begin{array}{c}\text{min}{Z}_2=2(\left(25,30,35\right)y_{11}+\left(17,22,27\right)y_{12}+\left(21,25,29\right)y_{13}+\left(22,25,28\right)y_{14}\\ +\left(15,20,25\right)y_{21}+\left(20,25,30\right)y_{22}+\left(14,18,22\right)y_{23}+\left(20,24,28\right)y_{24}\\ +\left(13,17,21\right)y_{31}+\left(14,18,22\right)y_{32}+\left(15,20,25\right)y_{33}+\left(22,26,30\right)y_{34}\\ +\left(24,28,32\right)y_{41}+\left(23,26,29\right)y_{42}+\left(22,25,28\right)y_{43}+\left(24,26,28\right)y_{44})\\ -(\left(25,30,35;0.2\right)y_{11}+\left(17,22,27;0.4\right)y_{12}+\left(21,25,29;0.3\right)y_{13}\\ +\left(22,25,28;0.1\right)y_{14}+\left(15,20,25;0.2\right)y_{21}+\left(20,25,30;0.4\right)y_{22}\\ +\left(14,18,22;0.2\right)y_{23}+\left(20,24,28;0.2\right)y_{24}+\left(13,17,21;0.3\right)y_{31}\\ +\left(14,18,22;0.2\right)y_{32}+\left(15,20,25;0.3\right)y_{33}+\left(22,26,30;0.2\right)y_{34}\\ +\left(24,28,32;0.3\right)y_{41}+\left(23,26,29;0.3\right)y_{42}+\left(22,25,28;0.15\right)y_{43}\\ +\left(24,26,28;0.25\right)y_{44})\end{array}$

$\begin{array}{c}\text{min}{Z}_3=2(\left(0.3,0.5,0.7\right)y_{11}+\left(0.2,0.4,0.6\right)y_{12}+\left(0.5,0.7,0.9\right)y_{13}\\ +\left(0.9,1.3,1.7\right)y_{14}+\left(0.4,0.8,1.2\right)y_{21}+\left(0.5,0.9,1.3\right)y_{22}\\ +\left(0.5,0.6,0.7\right)y_{23}+\left(1.2,1.5,1.8\right)y_{24}+\left(0.5,0.75,1\right)y_{31}\\ +\left(0.5,0.7,1\right)y_{32}+\left(0.5,0.8,1\right)y_{33}+\left(0.2,0.3,1\right)y_{34}+\left(0.8,0.9,1\right)y_{41}\\ +\left(0.3,0.7,1\right)y_{42}+\left(0.5,0.6,0.9\right)y_{43}+\left(0.4,0.8,1.2\right)y_{44})\\ -(\left(0.3,0.5,0.7;0.2\right)y_{11}+\left(0.2,0.4,0.6;0.3\right)y_{12}+\left(0.5,0.7,0.9;0.2\right)y_{13}\\ +\left(0.9,1.3,1.7;0.3\right)y_{14}+\left(0.4,0.8,1.2;0.4\right)y_{21}+\left(0.5,0.9,1.3;0.2\right)y_{22}\\ +\left(0.5,0.6,0.7;0.3\right)y_{23}+\left(1.2,1.5,1.8;0.1\right)y_{24}+\left(0.5,0.75,1;0.2\right)y_{31}\\ +\left(0.5,0.7,1;0.4\right)y_{32}+\left(0.5,0.8,1;0.3\right)y_{33}+\left(0.2,0.3,1;0.1\right)y_{34}\\ +\left(0.8,0.9,1;0.4\right)y_{41}+\left(0.3,0.7,1;0.2\right)y_{42}+\left(0.5,0.6,0.9;0.3\right)y_{43}\\ +\left(0.4,0.8,1.2;0.1\right)y_{44})\end{array}$

Step 2 : The intermediate defuzzified cost, labor, and safety parameters obtained from the parabolic fuzzy numbers have been presented in Equation (2). These values serve as the crisp equivalents used to construct the payoff matrix

$\begin{array}{c}\text{min}{Z}_1=2(2.28y_{11}+1.75y_{12}+2.48y_{13}+3.50y_{14}+2.20y_{21}\\ +1.80y_{22}+3.75y_{23}+2.00y_{24}+2.00y_{31}+2.63y_{32}\\ +1.80y_{33}+1.50y_{34}+2.30y_{41}+2.90y_{42}+2.80y_{43}\\ +2.40y_{44})-(0.684y_{11}+0.35y_{12}+0.248y_{13}+1.40y_{14}\\ +0.88y_{21}+0.36y_{22}+1.125y_{23}+0.20y_{24}+0.20y_{31}\\ +1.052y_{32}+0.54y_{33}+0.30y_{34}+0.69y_{41}+1.16y_{42}\\ +0.28y_{43}+0.48y_{44})\end{array}$

$\begin{array}{c}\text{Min }{Z}_2=2(30y_{11}+22y_{12}+25y_{13}+25y_{14}+20y_{21}+25y_{22}\\ +18y_{23}+24y_{24}+17y_{31}+18y_{32}+20y_{33}+26y_{34}\\ +28y_{41}+26y_{42}+25y_{43}+26y_{44})-6.0y_{11}+8.8y_{12}\\ +7.5y_{13}+2.5y_{14}+4.0y_{21}+10.0y_{22}+3.6y_{23}+4.8y_{24}\\ +5.1y_{31}+3.6y_{32}+6.0y_{33}+5.2y_{34}+8.4y_{41}+7.8y_{42}\\ +3.75y_{43}+6.5y_{44})\end{array}$

$\begin{array}{c}\text{Min }{Z}_3=2(0.50y_{11}+0.40y_{12}+0.70y_{13}+1.30y_{14}+0.80y_{21}\\ +0.90y_{22}+0.60y_{23}+1.50y_{24}+0.75y_{31}+0.73y_{32}\\ +0.78y_{33}+0.45y_{34}+0.90y_{41}+0.68y_{42}+0.65y_{43}\\ +0.80y_{44})-(0.10y_{11}+0.12y_{12}+0.14y_{13}+0.39y_{14}\\ +0.18y_{22}+0.18y_{23}+0.15y_{24}+0.15y_{31}+0.292y_{32}\\ +0.234y_{33}+0.045y_{34}+0.36y_{41}+0.136y_{42}+0.195y_{43}\\ +0.08y_{44})\end{array}$

$\begin{array}{cc}{y}_{11}+y_{12}+y_{13}+y_{14}& \le 29\\ y_{21}+y_{22}+y_{23}+y_{24}& \le 26\\ y_{31}+y_{32}+y_{33}+y_{34}& \le 32\\ y_{41}+y_{42}+y_{43}+y_{44}& \le 28\\ y_{11}+y_{21}+y_{31}+y_{41}& \ge 22\\ y_{12}+y_{22}+y_{32}+y_{42}& \ge 22\\ y_{13}+y_{23}+y_{33}+y_{43}& \ge 29\\ y_{14}+y_{24}+y_{34}+y_{44}& \ge 34\\ y_{ij}& \ge 0,\forall i,j\in \left(1,2,3,4\right)\end{array}$

Step 3: Following the conclusion of each goal separately, the payoff matrix has been shown in Table 4

Table 4 presents a portion of the payoff matrix, which reports the objective values obtained by optimizing each goal independently. These values form the basis for determining the upper and lower bounds required for computing neutrosophic membership functions.

Step 4: We obtained the upper bound values:

$U_1=489.80,U_2=4436.15,U_3=143.74$

and the lower bound values:

$L_1=372.32,L_2=3894,L_3=74.30$

for each objective function.

Steps 5 and 6: We derive the truth, indeterminacy, and falsity membership function upper and lower limits. Next, we build the three objectives membership functions.

For $Z_1$, the upper and lower bounds for the first objective and its membership function are:

$\begin{array}{c}{U}_1^T=U_1=489.80,U_1^I=L_1^T+s_1=372.32+s_1,U_1^F=U_1^T=489.80\\ L_1^T=L_1=372.32,L_1^I=L_1^T=372.32,L_1^F=L_1^T+t_1=372.32+t_1\end{array}$

$T_1\left(Z_1\left(y\right)\right)=\left(\begin{array}{c}1 if Z_1\left(\underset{\_}{y}\right)<372.32\\ 1-\frac{Z_1\left(\underset{\_}{y}\right)-372.32}{489.80-372.32} if372.30\le Z_1\left(\underset{\_}{y}\right)\le 489.80\\ 0 if{Z}_1\left(\underset{\_}{y}\right)>489.80\end{array}\right)$

$I_1\left(Z_1\left(\underset{\_}{y}\right)\right)=\left(1-\begin{array}{c}1 if Z_1\left(\underset{\_}{y}\right)<372.32\\ \frac{Z_1\left(\underset{\_}{y}\right)-372.32}{s_1} if372.30\le Z_1\left(\underset{\_}{y}\right)\le 372.32+s_1\\ 0 if{Z}_1\left(\underset{\_}{y}\right)>372.32+s_1\end{array}\right)$

$G_1\left(Z_1\left(\underset{\_}{y}\right)\right)=\left(\begin{array}{c}1if Z_1\left(\underset{\_}{y}\right)<489.80\\ 1-\frac{489.80-Z_1\left(\underset{\_}{y}\right)}{489.80-372.32-t_1}if372.32+t_1\le Z_1\left(\underset{\_}{y}\right)\le 489.80\\ 0if{Z}_1\left(\underset{\_}{y}\right)>372.32+t_1\end{array}\right)$

For $Z_2$, the upper and lower bounds for the second objective and its membership functions are:

$\begin{array}{cc}{U}_2^T=U_2=4436.15,U_2^I=L_2^T+s_2=3894+s_2,U_2^F=U_2^T=4436.15,& \\ L_2^T=L_2=3894,L_2^I=L_2^T=3894,L_2^F=L_2^T+t_2=3894+t_2.& \end{array}$

$T_2\left(Z_2\left(\underset{\_}{y}\right)\right)=\left(\begin{array}{c}1 if Z_1\left(\underset{\_}{y}\right)<3894\\ 1-\frac{Z_2\left(\underset{\_}{y}\right)-3894}{4436.15-3894} if3894\le Z_2\left(\underset{\_}{y}\right)\le 4436.15\\ 0 if{Z}_2\left(\underset{\_}{y}\right)>4436.15\end{array}\right).$

$I_2\left(Z_2\left(\underset{\_}{y}\right)\right)=\left(1-\begin{array}{c}1 if Z_2\left(\underset{\_}{y}\right)<3894\\ \frac{Z_1\left(\underset{\_}{y}\right)-3894}{s_2} if3894\le Z_2\left(\underset{\_}{y}\right)\le 3894+s_2\\ 0 if{Z}_1\left(\underset{\_}{y}\right)>3894+s_2\end{array}\right)$

$G_2\left(Z_2\left(\underset{\_}{y}\right)\right)=\left(\begin{array}{c}1 if Z_2\left(\underset{\_}{y}\right)<4436.15\\ 1-\frac{4436.15-Z_2\left(\underset{\_}{y}\right)}{4436.15-3894-t_2}if3894+t_2\le Z_2\left(\underset{\_}{y}\right)\le 4436.15\\ 0 if{Z}_2\left(\underset{\_}{y}\right)>3894+t_2\end{array}\right)$

For $Z_3$, the upper and lower bounds for the second objective and its membership functions are:

$\begin{array}{c}{U}_3^T=U_3=143.74,U_3^I=L_3^T+s_3=74.30+s_3,U_3^F=U_3^T=143.74,\\ L_3^T=L_3=74.30,L_3^I=L_3^T=74.30,L_3^F=L_3^T+t_3=74.30+t_3.\end{array}$

$T_3\left(Z_3\left(\underset{\_}{y}\right)\right)=\left(\begin{array}{c}1 if Z_3\left(\underset{\_}{y}\right)<74.30\\ 1-\frac{Z_3\left(\underset{\_}{y}\right)-74.30}{143.74-74.30} if74.30\le Z_3\left(\underset{\_}{y}\right)143.74\\ 0 if{Z}_3\left(\underset{\_}{y}\right)>143.74\end{array}\right)$