Carbon mechanism on sustainable multi-objective solid transportation problem for waste management in Pythagorean hesitant fuzzy environment

Solid waste management is a new concept in sustainability and transportation. This management relates with solid wastes which are irregular combinations of rejected and unwanted items that created bad smell, attracted to virus and bacteria, and pollute environment directly or indirectly. A part of solid wastes (e.g., paper, wood, cotton, vegetables, cardboard, leather, ash, crop, spoil fruit, etc.) finished their activities in first user or first time, but these may rework second time by proper way of recycling. After recycling process the generated items are resold in market, and SWM provides economical, environmental and social effect of sustainability. Starting process is to transport fresh items from agriculture field and forest department to cold storage for preservation and for future marketing. During this process, some waste items are found which are managed by decomposition, recycle (e.g., composting, vermi-composting, bio-methanation, pyrolysis processes) and reuse. In India, two creative mechanisms for recycling of solid wastes are approved to dispose the waste items that supply compost and waste to energy. The loop of SWM is completed by reaching all the recycled items to the customers or turns into biosphere in non-toxic way. Reuse of recycled waste items reduces environmental pollution and generates an income by trading these items. When woody biomass, fruit or vegetable wastes converts into bio-fuel [16] then that fulfil social satisfaction by increasing independency of fuel wood, improve environment and provide economical value, as bio-fuel is one type of renewable cleaner energy source used as alternative to fossil fuel. Compost used as organic substrates that protect plants, improve the soil with nutrient value and prevent water runoff by increasing plant cover.

There are several activities of SWM and some of them are provided from literature as: Abdullah and Gho [1] initiated an SWM problem by decision making process in Pythagorean fuzzy (PF) environment. A network for waste collection with sustainability and multiple objectives was designed by Asl et al. [5]. Mingaleva et al. [32] presented a case study from Russia on the base of waste management for green and smart cities. Recycle and cost reliability of SWM in uncertain situation were displayed by Muneeb et al. [33], and they proposed a decentralized decision making bi-level model. Nagarajan et al. [34] proved that vegetable wastes can be used as an alternative resource for bio-gas and bio-compost which are produced by lab scale process. Rabbani et al. [41] solved a multi-objective location-routing problem of industrial hazardous waste by metaheuristic algorithms. A case study of Bilaspur city (India) presented by Rathore and Sarmah [42] for the location of transfer station in urban area and to segregate the solid waste items. Use of vegetable wastes for bioenergy production was presented by Singh et al. [44]. Urban waste collection with robust bi-objective and arc routing problem performed in a multi-trip were presented by an invasive weed optimization by Tirkolaee et al. [46]. In [47] the authors Tirkolaee et al. provided a hybrid augmented ant colony optimization with multiple trip for SWM arc routing problem in urban area with fuzzy demand. In urban area, waste management was implemented by robust green location–allocation model with inventory in uncertain environment which was designed by Tirkolaee et al. [48]. Zaeimi and Rassafi [58] proposed municipal SWM using fuzzy chance-constrained programming in an uncertain environment for economical and environmental improvement.

Sustainability has a major area due to cooperate on environment, social and economical domain. Definition of “Sustainability” is that “Development which meets the demands of the present without compromising the capability of the future generations to meet their own demands” (Brundtland commission [8]). SD followed by sustainability and development which are concerned with three common parts as economical, social and environmental fact together. The three attributes of SD are fulfilled through transportation and SWM as follows.

A sustainable transportation with an integrated AHP–DEA multi-objective optimization model in mining industry was proposed by Gupta et al. [17]. A robust optimization model with resilient closed loop supply chain was designed by Lotfi et al. [27] for sustainability. Vafaei et al. [50] provided a case study for designing a distribution network on sustainable multi-channel supply chain. A green closed-loop supply chain network was designed by Zhen et al. [61] in an uncertain situation for sustainability.

TP is generated as linear programming problem initiated by Hitchcock [20] and it is renamed as Hitchcock–Koopmans TP. To transform the fresh items or waste items, closed type of vehicles are always used to prevent scattering or to control odour. The fresh items are transported from sources to destinations through several types of conveyances such as trucks, goods train, ships, etc. and dumper placer, hydraulic compactor vehicle, tripper, etc., are used for transporting waste items. Electric vehicles are suggested for sustainable transportation with economical and environmental improvement. Adding conveyance constraints on classical TP, the extended TP is called as solid transportation problem (STP), which was first defined by Haley [18]. Asim et al. [4] designed an integrated uncertain model on closed loop supply chain for transportation of production with cost reliability. An STP in fully intuitionistic fuzzy (IF) situation was initiated by Ghosh et al. [15], where the problem was expanded for multi-objective scenarios and fixed-charge.

Sustainable transportation on SWM includes transportation cost, job opportunity, carbon emission, etc., during shipment of items from distinct sources to several destinations by completing various steps. Traditional TP is not enough to control such situation that balance all the conflicting facts of sustainability. Therefore, a multi-objective transportation problem (MOTP) was initiated and extended as multi-objective solid transportation problem (MOSTP) by including conveyance constraint. TP with its several additions in various situations and different methodologies are incorporated from literature. Some related applications are selected here as: Dalman et al. [10] designed an MOSTP with multi-item and applied fuzzy approach in uncertain situation for finding a solution. An MOTP with truckload constraints was established by Ghosh and Roy [14] in fuzzy-rough environment. They analyzed fixed-charge by including product blending constraints and transfer station. Maity et al. [28] proposed an MOTP through time variation for SD with interval valued data. An MOTP with sustainability was introduced by Mehlawat et al. [30], where the problem was covered by three stages and fixed-charge. An MOSTP in IF environment was augmented by Midya et al. [31] for green supply chain with multiple stage and fixed-charge. Roy et al. [43] incorporated twofold uncertainty on MOSTP with fixed-charge that continued with multiple items. An MOTP with multiple items was discussed by Mardanya et al. [29] on the concept of just-in-time.

Nowadays, global warming, environment contamination, health implications and green house gas (GHG) emissions become a big challenge from transportation or from industrial plant due to burning of fossil fuel or disposal of waste items. For global development and to augment ozone layer, sustainable transportation is needed and carbon mechanism is introduced to encourage the reduction of CO\(_2\) emission. To reduce such CO\(_2\) emission there exists an agreement such as Kyoto Protocol, such that rate of emissions from fossil fuel does not outstrip the assimilative capacity of the environment. Government as the third party helps to curb the emission by introducing carbon regulation. There exist several policies of carbon mechanism, such as “carbon tax, carbon cap, carbon cap-and-trade, carbon subsidy, carbon offset” policy. For such different policies, there exist various facilities and their combinations give a better profit which are, defined in some of the following ways:

From literature we find various applications of carbon mechanism which are provided here as: Chiu et al. [9] launched carbon tax and trading theory on emissions for implementation of energy price. Das et al. [11] studied carbon emission tax, cap and offset policy on a green MOSTP in type-2 fuzzy environment and they found solution by fuzzy and non-fuzzy techniques with considering location and dwell time. Ghosh et al. [13] initiated carbon tax policy on a two-echelon supply chain and provided a collaborative model with uncertain demand. He et al. [19] represented cap-and-trade and carbon tax policy for a production with lot-sizing problem. Carbon subsidy policy for re-manufacturing closed-loop supply chain was defined by Li et al. [24]. In [25], the authors Li et al. analyzed single and multiple carbon policies for production and transportation in the supply chain by outsourcing the decisions. Trade-off between service time and carbon emissions on multi-echelon supply chains for safety stock was provided by Ni and Shu [36]. Palak et al. [37] applied the impacts of carbon mechanisms for bio-fuel supply chain with supplier and mode selection decision making problem. Cap-and-trade mechanism on production and pricing problems for supply chain was presented by Xu et al. [51]. Carbon emission constraint on global reverse supply chain was proposed by Xu et al. [52] for recycling waste items under uncertainties. Xu et al. [53] designed green supply chains and provided literature review of quantitative models based on carbon policies. A supply chain for energy saving and cost sharing was introduced by Yi and Li [55], where the emissions reduced by government subsidies and carbon tax policy. Yuyin and Jinxi [56] discussed the outcome of governmental policies as carbon taxes and energy-saving subsidies on a two-echelon supply chain to take enterprise decisions. Carbon pricing at variance with emissions trading on a supply chain was studied by Zakeri et al. [60]. Zhou et al. [62] showed the effects of carbon tariff by applying it on green supply chain design.

Sustainable MOSTP with SWM applied in realistic background, where all the data are not precisely defined for the presence of several hesitations, insufficient information, lack of evidence, competitive economic condition, fluctuations of financial market, etc. Some uncertain circumstances exist, where decision makers (DMs) cannot assign the precise value with their limited knowledge. These situations are evaluated by fuzzy, hesitant fuzzy, IF, neutrosophic, interval, rough, stochastic, randomness, PF, etc. Some uncontrolled realistic situations exist, where single uncertainty or traditional fuzzy set (FS) theory cannot sufficient to reflect the fact of decision making problem. In this state, we introduce a complex uncertainty as pythagorean hesitant fuzzy set (PHFS) which is the extension of PF set (PFS) [54] and hesitant fuzzy set (HFS) [49].

From literature we select some related works on HFS, PFS and PHFS as: Abbas et al. [2] presented cubic PFSs with unknown weight information and applied on multi-attribute decision making problem. Adhami and Ahmad [3] proposed an algorithm for solving MOTP in PHF uncertainty. Probabilistic HFSs for multiple criteria decision making problem was introduced by Farhadinia et al. [12]. Karaaslan and Ozlu [21] defined type-2 HFS with correlation coefficients and applied on clustering analysis. Pythagorean fuzzy approach applied by Kumar et al. [23] on TP. Peng and Selvachandran [39] discussed about the art and future directions of PFS. Nguyen et al. [35] explored PFS in decision making process by the application of exponential similarity measures. Qu et al. [40] implemented dual hesitant fuzzy linguistic uncertainty on stochastic multi-attribute method to measure the sustainability performance for supplier based and triple bottom line approach. Applications of correlation coefficients in PF situation was provided by Singh and Ganie [45]. Now, we place several research contributions in Table 1 which are connected with our work.

Table 1 highlights some research gaps and our current study provides some research contributions which are presented as follows:

The rest of this paper is outlined as follows: Motivation of our study are presented in the next section. The fundamental definitions of FS, HFS, PFS, PHFS with some basic properties are described in the following section. The following section depicts the notations with assumptions and the mathematical model of sustainable MOSTP with PHF uncertainty in connection of SWM. Two approaches, namely, FP and PHFP method with related to the models are illustrated in the next section. Two numerical examples are described in the next section. The following section contains computational results for optimal allocation and discussions about objective values. Sensitivity analysis for carbon emission is presented in the following section. Managerial insights are summarized in the next section. Concluding remarks with the outlook of future research are placed in the last section.

Abbreviations: Table 2 represents all the abbreviations which are used in this study.

Let X be a universal set, then TrPHFN \(\tilde{{\check{A}}}\) is defined as \(\tilde{{\check{A}}}=\{\big (\langle (a,b,c,d), \mu _{\tilde{{\check{A}}}}(x),\gamma _{\tilde{{\check{A}}}} \rangle ; \omega _1, \omega _2\big ): x\in X\}\), where \(\mu _{\tilde{{\check{A}}}}(x)\) and \(\gamma _{\tilde{{\check{A}}}}(x)\) are the sets of some values in [0, 1] denoting the possible Pythagorean hesitant membership and Pythagorean hesitant non-membership degrees of the element \(x \in X\) with \(a\le b\le c\le d\). Furthermore, \(\omega _1\), \(\omega _2\) \(\in [0,1]\) are the set of hesitant values for membership and non-membership functions, respectively, and \(0 \le \omega ^2_1+\omega ^2_2 \le 1\). The Pythagorean hesitant membership and non-membership function of \(\tilde{{\check{A}}}\) defined as

Arithmetic operations of TrPHFNs are required for the calculations of several data taken as TrPHFNs. Several ranking approaches are chosen to convert TrPHFNs into crisp numbers by utilizing these operations which are extended into different forms and applied in multi-attribute decision making problem. Let two TrPHFNs \( \tilde{{\check{A}}}_1\)= \(\big (\langle (a_{1}^1, a_{2}^1, a_{3}^1, a_{4}^1);\mu _{\tilde{{\check{A}}}_1},\gamma _{\tilde{{\check{A}}}_1} \rangle ; \omega _1, \omega _2\big )\) and \(\tilde{{\check{A}}}_2\) = \(\big (\langle (a_{1}^2, a_{2}^2, a_{3}^2,a_{4}^2);\) \(\mu _{\tilde{{\check{A}}}_2} \gamma _{\tilde{{\check{A}}}_2}\rangle \); \(\omega _3, \omega _4 \big )\), where \(\omega _1 \in \mu _{\tilde{{\check{A}}}_1}\), \(\omega _2 \in \gamma _{\tilde{{\check{A}}}_1}\), \(\omega _3 \in \mu _{\tilde{{\check{A}}}_2}\), \(\omega _4 \in \gamma _{\tilde{{\check{A}}}_2}\), are possible values in [0, 1] and \(0\le {\omega ^2_1}+{\omega ^2_2} \le 1\), \(0\le {\omega ^2_3}+{\omega ^2_4} \le 1\). The arithmetic operations are expounded as follows:

Addition: \( \tilde{{\check{A}}}_1 \oplus \tilde{{\check{A}}}_2 = \big (\langle (a_{1}^1+a_{1}^2, a_{2}^1+ a_{2}^2, a_{3}^1+a_{3}^2, a_{4}^1+a_{4}^2); \mu _{\tilde{{\check{A}}}_1}\wedge \mu _{\tilde{{\check{A}}}_2},\gamma _{\tilde{{\check{A}}}_1}\vee \gamma _{\tilde{{\check{A}}}_2} \rangle ; \theta _1, \theta _2 \big )\), where \(\theta _1 \in \{\mu _{\tilde{{\check{A}}}_1}\wedge \mu _{\tilde{{\check{A}}}_2}\}\), \(\theta _2 \in \{\gamma _{\tilde{{\check{A}}}_1}\vee \gamma _{\tilde{{\check{A}}}_2} \}\) and \(0\le {\theta ^2_1}+{\theta ^2_2} \le 1\).

Subtraction: \( \tilde{{\check{A}}}_1 \ominus \tilde{{\check{A}}}_2 = \big (\langle (a_{1}^1-a_{4}^2, a_{2}^1-a_{3}^2, a_{3}^1-a_{2}^2, a_{4}^1-a_{1}^2); \mu _{\tilde{{\check{A}}}_1}\wedge \mu _{\tilde{{\check{A}}}_2},\gamma _{\tilde{{\check{A}}}_1}\vee \gamma _{\tilde{{\check{A}}}_2} \rangle ; \theta _3, \theta _4 \big )\), where \(\theta _3 \in \{\mu _{\tilde{{\check{A}}}_1}\wedge \mu _{\tilde{{\check{A}}}_2}\}\), \(\theta _4 \in \{\gamma _{\tilde{{\check{A}}}_1}\vee \gamma _{\tilde{{\check{A}}}_2}\}\) and \(0\le {\theta ^2_3}+{\theta ^2_4} \le 1\).

Multiplication: \( \tilde{{\check{A}}}_1 \odot \tilde{{\check{A}}}_2 = \big (\langle (a_{1}^1 a_{1}^2, a_{2}^1 a_{2}^2, a_{3}^1 a_{3}^2, a_{4}^1 a_{4}^2); \mu _{\tilde{{\check{A}}}_1}\wedge \mu _{\tilde{{\check{A}}}_2},\gamma _{\tilde{{\check{A}}}_1}\vee \gamma _{\tilde{{\check{A}}}_2} \rangle ; \theta _5, \theta _6 \big )\), where \(\theta _5 \in \{\mu _{\tilde{{\check{A}}}_1}\wedge \mu _{\tilde{{\check{A}}}_2}\}\), \(\theta _6 \in \{\gamma _{\tilde{{\check{A}}}_1}\vee \gamma _{\tilde{{\check{A}}}_2}\}\) and \(0\le {\theta ^2_5}+{\theta ^2_6} \le 1\).

Scalar multiplication: Scalar multiplication for any real k is described as

To convert TrPHFNs to crisp numbers and to compare with TrPHFNs, defuzzification plays an important role. Different defuzzification methods exist in the literature, such as \(\alpha \)-cut, possibility concept, integration method, linguistic approach, etc. Here, we introduce an advanced and simple ranking approach through integration in terms of the TrPHFN that transforms TrPHFN into crisp number, which is applied in our proposed problem. Let \(\tilde{{\check{A}}}\) be a TrPHFN then ranking index is defined by \(\Re (\tilde{{\check{A}}})\), which maps TrPHFNs to real line, i.e., \(\Re : {\mathbb {F}}(\tilde{{\check{A}}}) \rightarrow {\mathbb {R}}\), where \({\mathbb {F}}(\tilde{{\check{A}}})\) is the collection of all TrPHFNs and \({\mathbb {R}}\) is the set of real numbers. Mathematically, the ranking index is defined for a TrPHFN \(\tilde{{\check{A}}}\) = \((\langle (a, b, c, d);\mu _{\tilde{{\check{A}}}},\gamma _{\tilde{{\check{A}}}}\rangle ; \omega _1, \omega _2)\), as follows:Let \( \tilde{{\check{A}}}_1 \) and \( \tilde{{\check{A}}}_2\) be TrPHFNs. Then, the ranking operation obeys the following comparisons as

We are to select an SWM and TP for agriculture field and forest department, such that sustainability criteria are followed up. At first the fresh fruits or grains or vegetables or other items are transported into cold storage for preservation and to use in future marketing. Various agriculture wastes are produced after harvesting and forestry wastes are produced from forest department as residue of main items, and some wastes are generated from cold storage by the deterioration of fresh items. These waste items are transported into bio-fuel production plant and compost production plant according to their nature. Various recycle processes produced bio-fuel and compost, and these are shifted to bio-energy supply centre and compost selling market for reuse and to achieve some revenues. This SWM is completed by performing several steps such that sustainability criteria (economical condition, social impact and environment improvement) are always fixed. For economical improvement, we try to minimize transportation cost, and for social impact we want to maximize the number of job opportunities. For environment protection, we incorporate carbon mechanism for reduction of carbon emission. Here, the object of carbon mechanism is to minimize total carbon emission based on “carbon tax with cap”, “carbon tax with cap-and-trade” and “carbon tax with offset” policy. Having these goals, we formulate an unpredicted mathematical model of MOSTP by including source, demand and conveyance constraints in the PHF environment for inconsistent and incomplete information. The performing stages of the formulated model are graphically described in Fig. 1.

A list of notations with their nature, meaning, and required assumptions are selected to design the proposed model. These notations are illustrated in Tables 3 and 4.

The transportation cost with other charges (emission charge) are fixed for each unit of transportation, and therefore, the cost is not of an imprecise nature. Similarly job opportunity, rate of carbon emission and tax per unit of item are fixed, not fluctuate, and considered in crisp nature. However, source, demand and vehicle capacity are effected due to several unpredictable conditions that are not previously detected. In such situations we consider these parameters as PHFNs and the corresponding model becomes in a PHF environment. The integrated optimization model of MOSTP with PHF environment is implemented in Model 1 to tackle real-life hesitation or uncertainty about SWM with sustainability. We strive to minimize transportation cost, maximize job opportunity and minimize carbon emission cost.

FP was initiated by Zimmermann [63] for solving any multi-objective linear programming problem. This is one of the simple fuzzy technique that provides Pareto-optimal solution in short elapsed time and simple way. To find the best Pareto-optimal solution, we utilize the facility of FP to solve our proposed MOSTP. The required steps are illustrated as follows:

For finding Pareto-optimal solution of the proposed MOSTP, a new modified programming approach PHFP based on PHFS is introduced here. PHFP maximizes Pythagorean hesitant membership grade and minimizes Pythagorean hesitant non-membership grade. The decision making concept through fuzzy set was first initialized by Bellman and Zadeh [6]. The fuzzy decision set \(({\tilde{D}})\) on a decision making problem can be defined by the intersection of fuzzy objective function \(({\tilde{O}})\) and fuzzy constraint (\({\tilde{C}}\)). Pythagorean hesitant fuzzy decision \((\tilde{{\check{D}}})\) is an extension of fuzzy decision that can be described as \(\tilde{{\check{D}}} = \tilde{{\check{O}}} \cap \tilde{{\check{C}}} = \{x, \mu _{\tilde{{\check{D}}}},\nu _{\tilde{{\check{D}}}}\}\), where \(\big \{\mu _{\tilde{{\check{D}}}}(x) \in (\mu _{\tilde{{\check{O}}}}(x)\cap \mu _{\tilde{{\check{C}}}}(x)): \mu _{\tilde{{\check{D}}}}(x) \le \min \{\max (\mu _{\tilde{{\check{O}}}}(x)\cap \mu _{\tilde{{\check{C}}}}(x))\} \big \}\) and \(\big \{\nu _{\tilde{{\check{D}}}}(x) \in (\nu _{\tilde{{\check{O}}}}(x)\cap \nu _{\tilde{{\check{C}}}}(x)): \nu _{\tilde{{\check{D}}}}(x) \ge \max \{\min (\nu _{\tilde{{\check{O}}}}(x)\cap \nu _{\tilde{{\check{C}}}}(x))\} \big \}\). \(\mu _{\tilde{{\check{D}}}}(x)\) and \(\nu _{\tilde{{\check{D}}}}(x)\) are the sets of membership value of acceptance and rejection of PHF solution under PHF decision set, respectively, and \(\tilde{{\check{O}}}\), \(\tilde{{\check{C}}}\) are the hesitant fuzzy objective function and hesitant fuzzy constraint, respectively. Using such PHF solution criteria we represent the stepwise procedure for solving the proposed model(s) as follows:

Now, we solve two examples with considering three policies of carbon mechanism by keeping fixed the carbon cap.

Here, we find the optimal allocations by evaluating two examples by regarding three policies and by the help of PHFP and FP. In Example 2, cap \(C=25000\) gm provides feasible solutions of three policies with all cases and solutions are presented in Table 10. For Example 3, when cap \(C=18000\) gm then only second case of second policy (i.e., tax with cap-and-trade policy) and second case of third policy (i.e., tax with offset policy) provide the feasible solutions defined in Table 11. If total emission is less than carbon cap then this example provides no feasible solution (NFS). The objective values in both examples with two methods are summarized in Table 12 and the preferable method, PHFP is highlighted by bold face for suitable comparison. Observing all over the objective values for the considered methods, we pick an appropriate solution to specify the final Pareto-optimal solution of both examples and star mark uses to select a better policy among three policies of carbon mechanism which declares as final result.

Tables 10 and 11 provide the Pareto-optimal solution of two numerical examples, respectively. From these tables, we find the objective values which define the main differences between the solutions. Table 10 shows that here the cap value is greater, and therefore, all the policies provide feasible solution in two methods. In Table 11, where cap value is lower and hence only Model 4B and Model 5B give solution. That is the two tables does not display the same type of solutions of two numerical examples, respectively.

From Table 12, we highlight the different optimal solutions in FP and PHFP for several reasons. FP supports only on the membership grades of the objective functions, but PHFP considers both the membership and non-membership grades of the objectives. FP maximizes the membership grade but PHFP maximizes the degree of satisfaction by finding highest value of Pythagorean hesitant membership grade and minimizes the degree of rejection by providing lowest value of Pythagorean hesitant non-membership grade. The hesitancy of membership and non-membership is tackled in PHFP, whereas FP cannot face to any hesitancy. PHFP overcomes different complexities, and displays more efficient Pareto-optimal solutions than FP as DMs have a privilege to choose the efficient solution in a technical way by setting the hesitant values for each membership and non-membership function.

Between FP and PHFP, we select PHFP as a better method which provides final Pareto-optimal solution of both examples. From Table 12, we note that Example 2 provides feasible solution in two methods with all of three policies. However, for Example 3, only Model 4B and Model 5B provide objective values, whereas the other cases have NFS in both methods. We compare three policies of both examples in two methods and notice that “carbon tax with cap-and-trade” policy (i.e., Model 4A and Model 4B) provides appropriate result than other two policies. In this policy we analyze that when total emission is less than carbon cap (i.e., Model 4A) then the organization can sell extra unused carbon units and find a profit that reduced emission cost 7214.13$. If total emission is greater than carbon cap (i.e., Model 4B) then organization purchases the deficit carbon unit and pays a penalty charge for extra emission. In such case total carbon emission cost is increased and it is 9283.47$. We find that carbon emission cost 7390.37$ (Model 3 and Model 5A) and 10527.32$ (Model 5B) are obtained for “carbon tax with cap” and “carbon tax with offset” policy, which indicate that emission cost is increased that of the previous policy. In this policy, when total emission is greater than carbon cap (i.e., Model 5B) then the organization pays a penalty charge but could not purchase their deficit carbon units. Whenever total emission is less than carbon cap (i.e., Model 3 and Model 5A) then the system cannot sell their surplus carbon unit and emission cost is not decreased. Therefore, Model 4A is better than Model 5A and Model 3, and Model 4B is better than Model 5B. Model 5A has a disadvantage that for low emission the system does not find any reward, whereas Model 4A finds such reward by selling their extra permission. Between Model 4A and Model 4B, we fix Model 4B of Example 2 and also Model 4B of Example 3, to provide final Pareto-optimal solution, as this model minimizes transportation cost and at the same time maximizes job opportunity. However, Model 4A only provides less emission cost than of Model 4B. For better comparison, we define three objective values of both example in Fig. 2. This graph defines the effect of carbon mechanism by the third objective value which variates for several cap values, i.e., these show that total emission cost of Example 2 is less than that of Example 3. Carbon market has made “carbon tax with cap-and-trade” mechanism that becomes a larger policy and attractive in many countries for reduction of carbon emission, and this policy measures the limit index of carbon emission for a sustainable transportation as well as for industrial problem. The case when the carbon cap is high (i.e., Example 2) then the system can choose high emission vehicles with minimum transportation cost, and therefore, carbon emission cost will be reduce, as at this time penalty charge is not included. If carbon cap is low (cf. Example 3), then the organization prefers low emission car with high transportation cost. Here carbon emission cost is increased for penalty charge as for exceed emission. The advantage of carbon mechanism is that it encourages the user to use alternative source of energy and always selects a better policy. Analyzing all the three policies of carbon mechanism we prefer “carbon tax with cap-and-trade” policy.

In this subsection, we inspect the affect of hesitant parameters \(\alpha _s\) and \(\beta _s\) on three objective functions (Model 4B and Example 2). We validate the proposed PHFP model by selecting 9 random achievements of the parameters under 10 violations. Each hesitant parameter generates a random number in [0, 1]. The optimal values of objective functions are resulted by PHFP model. Tables 13 and 14 display the objective values whenever four values of \(\alpha _s\) and \(\beta _s\) are fixed at 0.1, 0.5 and 1.0, whereas other two values of \(\alpha _s\) and \(\beta _s\) are violated at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0. All these cases are highlighted in Figs. 3, 4 and 5. According to the validation of the model, we find that the objective values are violated in the realization model for the cause of hesitant values and most selected optimal values appear in any middle or end or stating situation, where objectives are not increased or decreased corresponding to the increased or decreased valued of \(\alpha _s\) and \(\beta _s\). From this perspective, it can be decided that the proposed PHFP model offers an opportunity to the DMs for selecting a better decision in an uncertain situation. Important novelty of this approach is that the obtained solution of feasible region provides a spectrum of uncertainty which is different from fuzzy and IF.