Effect of variable carbon emission in a multi-objective transportation-p-facility location problem under neutrosophic environment
Introduction
The facility location problem (FLP) is a crucial integrant of strategic planning for a wide spectrum of the public as well as the private sector. In fact, it deals with locating facilities among existing sites with the goal of optimizing the economic criteria (e.g., transportation cost, transportation time, carbon emission cost and good service). The traditional FLP is described by four given sets, (i) a set of existing sites with capacity, (ii) a set of weights associated with the existing site, (iii) a set of potential facility sites with demand, and (iv) a set of objective functions. It can be cataloged into different categories depending on the assumptions. Industrial organizations locate assembly plants and depots. Warehouses are situated by the retailers. The performance of the manufacturing, productivity, and marketing of goods is dependent on the location of the facilities. Moreover, the government also selects the location of hospitals, offices, schools, fire stations, etc. Everywhere, the quality of service is dependent on the location of the facilities. The FLP was studied by several researchers. A few of them are depicted here. Farahani, SteadieSeifi, and Asgari (2010) made a comprehensive survey of the facility location problems in a multi-criteria environment. Then, Bieniek (2015) presented a note on the FLP where the demands follow the arbitrary distribution. Later, Chen, He, and Wu (2016) solved a single FLP with random weights. Moreover, the FLP can be applied in a broad area of transportation networks, supply chain management, plant location problem, and green logistics such as Mis̆ković et al., 2017, Melo et al., 2009, Amin and Baki, 2017, Saif and Elhedhli, 2016, and Harris, Mumford, and Naima (2014).
In the real scenario, the transportation problem (TP) plays a vital role in global competition for minimizing transportation cost, time and providing service. Generally, the classical TP consists of three major components: (a) a set of all sources, (b) a set of all destinations, and (c) single-objective function as total transportation cost. Mainly, in the TP, homogeneous goods are sent from sources to destinations, and the total transportation cost is directly proportional to the amount of goods to be transported. It was the first introduced by Hitchcock (1941). However, the traditional TP is not sufficient for handling real-life application problems. Due to this reason, the multi-objective environment is introduced here on the TP in which the objectives are conflicting and non-commensurable in nature. In fact, the multi-objective TP (MOTP) was analyzed by so many researchers in different environments. Some works are annexed here. Mahapatra, Roy, and Biswal (2013) solved a multi-choice stochastic TP where the supply and demand parameters follow extreme value distribution. Thereafter, Sabbagh, Ghafari, and Mousavi (2015) proposed a hybrid approach for the balanced TP. Maity, Roy, and Verdegay (2016) discussed a MOTP with cost reliability in an uncertain environment. Later on, Roy, Maity, Weber, and Gök (2017) described a MOTP where cost, demand, and supply parameters are in multi-choice nature. And they solved the problem using two approaches multi-choice goal programming and conic scalarizing function.
The FLP and TP are the core components of a tactical transportation planning system. Determining the best locations for the facilities (i.e., plants, depots, warehouses, offices, fire stations, railway stations, etc.) and minimizing the total transportation cost from existing sites to facilities can significantly affect the transportation planning system. Cooper (1972) first made a connection between the FLP and TP, and was also known as the transportation-location problem. Later, he (1978) studied the problem under stochastic environment. Recently, Carlo, David, and Salvat (2017) extended the problem with an unknown number of facilities. Afterward, several researchers made connections among the FLP and TP in many different ways. Klibi, Lasalle, Martel, and Ichoua (2010) studied a location-transportation problem delineated by multiple demand periods, multiple transportation options, and a stochastic demand. Gabrel, Lacroix, Murat, and Remli (2014) illustrated a robust location transportation problems under uncertain demands. Recently, Jaafari and Delage (2017) presented a capacitated fixed-charge multi-period location-transportation problem.
A fast-flowing of transportation emerges tremendous amounts of carbon, which is the fundamental explanation for global warming. To control carbon emanations, the government endorses several policies among all tax, cap and trade policy (TCTP) is widely accepted. Under TCTP, the companies are firstly allowed some emission cap with the usual tax basis from the government, and subsequently, they can also trade (i.e., buy or sell) the emission cap in the carbon trading market. This type of study was implemented by many scholars such as Benjaafar et al., 2013, Wu et al., 2017, Dua et al., 2016, Cao et al., 2017, Turken, Carrillo, and Verter (2017) and Elhedhli and Merrick (2012). Here, we consider variable carbon emission as it depends on the locations of facilities as well as the amounts of transported items. This concept is totally new which did not incorporate by the researcher(s).
From Table 1, we trace a gap for making a connection among the FLP, MOTP and carbon emission under TCTP. To fill the gap concretely, here, we flourish a formulation by integrating the FLP and TP in the light of a multi-objective optimization environment. Therefore, we refer to the proposed problem as the multi-objective transportation-p-facility location problem (MOT-p-FLP). In the MOT-p-FLP, one has to ask the locations of p-facilities in the Euclidean plane and the amounts of transported goods simultaneously with three objective functions. We believe that the proposed formulation will be more applicable than the traditional FLP and MOTP. In fact, it will be useful to the models of transportation systems, emergency services, and online-shopping systems.
Nowadays, the parameters of the MOT-p-FLP are conflicting and imprecise nature due to lack of proper information. In fact, this type of mathematical formulation is difficult to tackle by traditional approaches. To overcome this situation, Zadeh (1965) introduced the fuzzy set (FS). Thereafter, Zimmermann (1978) incorporated fuzzy programming to solve a multi-objective linear programming problem. But, there is a drawback of the FS, it could not manage the certain case of uncertainty. Because of that, the intuitionistic fuzzy set (IFS) was developed by Atanassov (1986) as a generalization of the FS. The IFS was applied in a multi-objective optimization problem like Roy, Ebrahimnejad, Verdegay, and Das (2018). Although the FS and IFS deal with all types of fuzzy uncertainty, still they cannot handle the indeterminate situation. For instance, a survey is done on a particular statement, then there are a few who said the possibility of the statement is true 0.7, the statement is false 0.4, and the statement is not sure 0.3. This issue is beyond the scope of the FS and IFS, and thus dealing with a kind of indeterminate situations of uncertain information indeed becomes a true challenge. Based on this instance, the neutrosophic set, an extended form of the FS and IFS was developed by Smarandache (1999). It provides a more general structure and suitable form to deal with the mentioned uncertainties. The neutrosophic set is formulated based on logic in which elements are represented by three degrees, explicitly, truth degree, indeterminacy degree, and falsity degree.
The main contributions of this study are as follows:
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An integrated nonlinear optimization model based on the FLP and MOTP is introduced.
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The model finds the decision regarding the assignment from multiple existing facilities to multiple potential facilities in the continuous planner surface with a hyperbolic approximation of Euclidean distance.
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The total transportation cost, total transportation time and total carbon emission cost are considered.
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The impact of variable carbon emission under TCTP due to transportation is also incorporated, a major contribution in the modern age.
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An improved hybrid approach is followed to find the optimal solution of the MOT-p-FLP.
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The nature of the obtained optimal solution is also studied.
The outline of this study is as follows: In the next section, the proposed problem is formulated. Section 3 presents the methodology of a hybrid approach along with its pros and cons. Then, Section 4 explores the non-dominated nature of the compromise solution. Moreover, the effectiveness of the stated model and the approach are evaluated with an example in Section 5. In Section 6, the obtained results for two cases are discussed. The sensitivity of the stated model is investigated in Section 7. Thereafter, Section 8 depicts the important managerial insights. At last, conclusions and future research directions based on our study are provided.
Section snippets
Mathematical description
In this section, we first define the proposed problem, i.e., MOT-p-FLP. Thereafter, the mathematical formulation is introduced on the following premises and notations. Moreover, the connection between the MOT-p-FLP and a MOTP, and some basic definitions are presented.
Methodology
In this section, a hybrid approach is presented to solve the proposed MOT-p-FLP. Thereafter, the advantages and disadvantages of the stated approach are also discussed.
Analysis of non-dominated solution
Here, we first demonstrate that if is a non-dominated solution of the MOT-p-FLP, then is a non-dominated solution of the unconstrained multi-objective FLPs of Eqs. (2.1), (2.2), (2.8) or Eqs. (2.1), (2.2), (2.10), where . Lemma 1 Let is a non-dominated solution of the MOT-p-FLP of Eqs. (2.1), (2.2), (2.8). Then is a non-dominated solution of the multi-objective FLP:
Experimental example
Herein, a real-life based example is presented to validate our model and methodology. In the example, an industrial association wishes to start a few new firms with the aim of minimizing the total transportation cost, time and carbon emission cost under tax, cap and trade policy. The association has 4 existing firms: and , and they want to establish 3 new firms: and . They transport goods by conveyances. In fact, we consider the weights of the conveyances depend on the
Computational results and discussion
An application example is provided to analyze the proposed model with the help of the hybrid approach. The approach first finds the initial locations, optimal feasible solutions, optimal locations, ideal solutions (individual minimum), and anti-ideal solutions (individual maximum), and then we determine the upper and lower bounds for truth, indeterminacy, and falsity. Thereafter, the neutrosophic models for two cases of the MOT-p-FLP are formulated to derive optimal compromise solutions. The
Sensitivity analysis
In this section, we investigate the resiliency of optimal compromise solutions in the MOT-p-FLP by varying the parameters. For the MOT-p-FLP, the difficulty arises when the range of parameters are chosen after small changes for which the optimal solution remains optimal. In fact, the complexity increases when the number of variables and constraints are in large size. Due to this reason, a simple procedure is adopted to evaluate the sensitivity of the proposed problem with the fact that the
Managerial insights
The fact that MOT-p-FLP is an especially application-based region, makes it essential to receive deep insights into the characteristics of optimal solutions. Herein, we gather information about the optimal solutions derived when employing Model 1 into two sub-problems. Observing the outcomes, the management’s discretion can easily pick the optimal solution between two sub-problems. A brief discussion of the effect of carbon emission under TCTP is depicted. From that discussion, the managements
Conclusions and future research directions
This study has been presented a practical formulation for planning and transportation system with the objectives of minimizing the total transportation cost, total transportation time, and total carbon emission cost under TCTP on the entire transportation chain, and at the same time it also asks the potential facility sites along with the amounts of transported goods simultaneously. To the best of authors’ knowledge, the problem of designing an MOT-p-FLP, considering variable carbon emission
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