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Abstract

This article aims to present research conducted on the literature regarding Multi-objective Optimization (MOO) for routing problems with environmental considerations (EC), referred to here as Multi-objective Optimization for the Green Vehicle Routing Problem (MOOGVRP). A Brazilian database, CAPES (Coordination for the Improvement of Higher Education Personnel), was used to collect articles of general application, case studies and reviews in English starting from, since 2012. The terms “green vehicle routing problem” (GVRP), “pollution routing problem” (PRP), “vehicle routing problem in reverse logistics” (VRPRL) and “multi-objective” were used in the research protocol. Consequently, this study obtained 1,744 research results that, following the application of the filtering criterion, resulted in a sample of 76 articles from 38 journals, for which a bibliometric data (bibliometric review) survey was conducted. When dealing with the bibliometric data of the sample, it was possible to identify information such as the number of publications per year and types of published works. Information was also identified regarding the most frequently used journals and the countries and institutions that published the most articles on the proposed theme. It was also possible to analyze the frequency of the protocol terms in the title, abstracts and keywords, the relationship between taxonomies, vehicle fleet types, multi-objective procedures, and VRP procedures. Information was also found regarding solution procedures, number of objectives, and programming languages for computational implementation, the frequency of the most used objectives, and the most cited articles in the sample. The originality of this article lies in how the research is presented, highlighting the results and particular details obtained through the survey, which may be considered of great academic importance in the sense of guiding the trends for future research.

Keywords: Innovative survey, Green Vehicle Routing Problem, pollution routing problem, vehicle routing problem in reverse logistic, multi-objective optimization

1. Introduction

In 2015, the UN proposed an agenda with 17 sustainable development goals to be implemented by all countries by the year 2030. The present study especially focuses on Goals 12 and 13. It addresses Goal 12 (“ensure sustainable consumption and production patterns”) by seeking an intelligent and optimized alternative to perform activities, and 13 (“take urgent action to combat climate change and its impacts”) by seeking to reduce greenhouse gas emissions [1].

According to the report published by the United States Environmental Protection Agency [2], the sources of greenhouse gas (GHG) emissions include five main sectors: Transportation; Electricity; Industry; Commercial and Residential; and Agriculture, as shown in Figure 1, which presents their respective percentages. In this study, the emphasis will be on the “share” related to Transportation (28.5%). Regarding this “share”, we found the following results: CO${\displaystyle _{2}}$ from Fossil Fuel Combustion (27.4%), Substitution of Ozone Depleting Substances (0.7%), Mobile Combustion (0.3%) and Non-Energy Use of Fuels (0.1%). Of these, this study will address the issues related to CO${\displaystyle _{2}}$ from Fossil Fuel Combustion and Mobile Combustion.

 Figure 1. Emission sources according to the Inventory of U.S. Greenhouse Gas Emissions and Sinks in 2016. Source: US EPA [2]

The Vehicle Routing Problem (VRP), proposed by Dantzing and Ramser [3], is highly significant for an efficient logistic distribution [4,5]. The Green Vehicle Routing Problem (GVRP) directs routing activities through a perspective with environmental considerations [6,7]. Thus, a Multi-objective approach aids this process of complex decision-making in order to meet the objectives to be achieved [8,9,10].

In this context, we identified several works in the literature that address the optimization of fuel consumption [5,11], CO${\displaystyle _{2}}$ emission [12,13], both cases simultaneously [14,15] and other approaches [16,17] applied to the Vehicle Routing Problem (VRP).

The aim of this study is to present research conducted on the literature regarding Multi-objective optimization (MOO) for vehicle routing problems with EC, referred to here as MOO for the Green Vehicle Routing Problem (MOOGVRP). Thus, this work aims to detect gaps in the literature for development and advances in this field of knowledge. This has been recognized as an important theme in a wide range of previous literature reviews, such as that of [18].

The originality of this article lies in present an innovative research, showing the results and peculiarities obtained through the content analysis (systematic review), with the bibliometric data collection (bibliometric review) regarding the published articles related to MOOGVRP from 2012 to 2018. Highlights of paper are structured overview of recent studies of VRP with multi-objective approaches and environmental considerations, more specifically, Pollution Routing Problem, Green Vehicle Routing Problem and VRP in Reverse Logistics.The main objectives addressed are cost, fuel consumption and CO${\displaystyle _{2}}$ emission minimization. It has research opportunities regarding electric vehicles, social issues and others.

The present work is organized as follows. After this introductory section, the methodological procedures are presented in Section 2. Section 3 contains a brief presentation of the Multi-objective approach (the VRP approach) and its variations, as well as the VRP with EC. In Section 4, the results are explained and discussed. The conclusions of the study are summarized in Section 5.

2. Methodological procedures

For the methodology of this research, the following steps were established as a search protocol:

1. First, the timeline cut, covering the period between 2012 and 2018, in order to identify the recent discussions analyzed by academia. To the protocol used was not found papers before 2012.
2. Database selection: the study used the Brazilian database of CAPES (Coordination for Improvement of Higher Education Personnel), which gathers and makes available international scientific content for educational and research institutions in Brazil, composed of 38,000 journals, 532 reference databases including Cambridge Journals Online, Emerald Insight Emerald, IEEE Xplore, Scopus (Elsevier), Science Direct, SpringerLink, Taylor & Francis, as well as books, encyclopedias, technical standards, statistics and audiovisual content [19].
3. The English language was adopted to select the articles.
4. Only journals (articles) were searched.
5. The keywords, as well as the logical operators used were: “green vehicle routing problem” OR “pollution routing problem” OR “vehicle routing problem in reverse logistics”, which were necessarily linked to “multiobjective” OR “multi-objective” OR “bi-objective”. The possible suffix variations in the terms used were also considered. The terms related to routing with EC (environmental considerations) were obtained from the literature review proposed by [20].
6. As an acceptance criterion, it was established that: (i) the research result should present the “green vehicle routing problem”, “pollution routing problem” or “vehicle routing problem in reverse logistics”, all of them being “multi-objective”; (ii) they could be literature “review”, “general” application (mathematical modeling and/or algorithm improvement articles), as well as “case” studies related to the proposed theme. As exclusion criteria, it was established that the results would be excluded if: (i) they were not related to the proposed theme; (ii) did not have full text available; or (iii) were duplicate articles.

From the protocol presented, we obtained 1,744 research results that, after the protocol filter, resulted in 76 articles for the research sample, from 38 journals. From this selection of articles, this study conducted a critical analysis of the research scope and sought to answer two questions:

1. What are the most relevant studies regarding MOOGVRP?
2. What are the trends regarding MOOGVRP that may guide future research?

3. Theoretical framework

In this section, we briefly present the problems of MOO, the VRP and its variations, as well as the VRP with EC, which will be subdivided into the Green Vehicle Routing Problem, Pollution Routing Problem and Vehicle Routing Problem in Reverse Logistics.

3.1 Multi-objective optimization

MOO addresses the process of simultaneously optimizing two or more conflicting objectives subject to restrictions [10,15]. Its mathematical representation is presented using the objective function in Eq. (1) and restrictions in (2)

 ${\displaystyle min\left(f_{1}(x),f_{2}(x),\cdots ,f_{k}(x)\right)}$ (1)
 ${\textstyle {\hbox{subject to }}x\in X}$ (2)

where ${\displaystyle k\geq 2}$ defines the number of objectives to be optimized and X represents a set of non-empty decision vectors (solution). When we optimize a MOO problem, we arrive at an x-optimal solution for a given objective and, when trying to improve it, the other objectives are affected. In other words, given two x and y solutions, so that ${\displaystyle f_{i}(x)\leq f_{i}(y)\,\forall \,i}$, ${\displaystyle i\in \{1,2,\cdots ,K\}}$ and ${\displaystyle f_{j}(x) for at least one objective ${\displaystyle j\in \{1,2,\cdots ,K\}}$. In this case, it is said that x is a non-dominated solution, i.e., that it belongs to the optimal Pareto frontier. Therefore, a solution will be considered Pareto efficient (Pareto optimal or non-dominated) if it cannot be eliminated from consideration due to another solution that improves one of the objectives without worsening the others. Thus, the goal of an MOO problem is to find non-dominated solutions and quantify trades-offs to meet the different objectives established [9,21].

Figure 2 presents a bi-objective academic example about the concept of domination through the Pareto front for the case of minimizing the objective functions 1 and 2. It is observed that z4 dominates z2, z5 and z3. Note that, z4 and z6 are non-dominated solutions, because z4 is worse than z6 in f2 but z6 is worse than z4 in f1. In this context the solutions z1, z4 and z6 are non-dominated by any other solution presented in the population. Thus, these solutions belong to the Pareto front.

 Figure 2. Example of Pareto front

MOO methods can be classified into generative methods and preference-based methods. The generative methods seek to generate one or more Pareto optimal points and can take into account the “non-use” for preferences, a scale approach or a multi-objective approach. The preference-based methods use information provided by the decision maker a priori or iteratively, as part of their solution process.

The classic methods commonly used for solving MOO problems can be organized into four categories. The first addresses the methods that do not articulate the given information of preference, such as the Method of Global Criterion. In the second one, we have “a priori” preferences, such as the Methods for Cardinal Information Given (Utility Function and Bounded Objective) and the Methods for Mixed Ordinal and Cardinal Information Given (Lexicographic Method and Goal Programing). In the third category, we have the methods for progressive articulation, most notably the Methods for Explicit Trade-Off Information Given (iterative methods) and implicit methods. Finally, we have “a posteriori” methods, among which we find the Parametric Method, є-constraint, Multi-objective Linear Programming (MOLP) and Adaptative Search [15,18].

According to Heilig et al. [22], the most frequently used Multi-objective techniques are: weighted sum scalarization, the є-constraint method, goal programming, lexicographic ordering and Pareto optimization. In relation to weighted sum scalarization, we must assign weights and treat the problem as a linear combination of established objectives. In the case of the є-constraint method, only one objective must be optimized, and the others must be considered as problem restrictions. In goal programming, the purpose is to meet all objectives, taking account of the goals established for each criterion. Lexicographic ordering considers an order of importance from the sequence in which the proposed objectives were listed. Finally, in Pareto optimization, an “a posteriori” approach provides the Pareto frontier with the intention of performing the trade-off of the solutions of the different objectives.

3.2 Vehicle Routing Problem

VRPs have many variations in the literature, and some of them are presented in Table 1 (in alphabetical order), as proposed by [20,23], followed by reference paper and brief description. It is worth mentioning that two or more of these variations can be combined, generating new variations for the VRP.

Table 1. Some VRP variations
 VRP variations Introduced by Brief Description Capacitated VRP Dantzig and Ramser [3] Considers the capacity of the vehicle. Dynamic VRP Speidel [24] This is a dynamic procedure where customer orders are organized during planning and assigned to the appropriate vehicle in real time. Fleet Size and Mix VRP Golden et al. [25] Determines the economic combination of vehicles in the fleet by considering the trade-offs between the vehicle’s fixed costs and the proportional variable costs. Fuzzy VRP Cheng et al. [26] Formulates the subjective, uncertain, vague and ambiguous elements. Generalized VRP Tsiligirides [27] Each customer has a different score (or profit) and is divided into unit groupings. Inventory Routing Problem Bell et al. [28] Ensures that there are no stocks on each customer. Location Routing Problem Watson-Gandy and Dohm [29] Addresses a single or a set of deposits and, subsequently, a series of routes to the respective open deposits to minimize the total cost. Multi-compartment VRP Christophides [30] In this case, the problem involves heterogeneous and qualified vehicles. A customer's demand for a particular product cannot be divided. Multi-depot VRP Tillman [31] Has more than one deposit. Each script starts and ends in the same deposit. Multi-echelon VRP Crainic et al. [32] Easily applied to true needs, aims to minimize the total cost of transportation of the vehicles involved at all levels, i.e., freight delivery covers the entire chain and is required to be carried out through an intermediary deposit; therefore, several stages are performed. Open VRP Sariklis and Powell [33] Each route is a Hamiltonian path instead of a Hamiltonian cycle. The vehicles are not required to return to the depot after finalizing their activities. Periodic VRP Beltrami and Bodin [34] Seeks feasible routing so that the cost associated with the time horizon is minimized. Pickup and Delivery Problem Wilson and Weissberg [35] Considers the collection and/or delivery of goods. Large variations are available in the literature. Site-dependent VRP Nag [36] There is compatible independence between clients (sites) and types of vehicles. Each customer can be visited by only a predefined group of vehicle types. Split-delivery VRP Dror and Trudeau [37] Each customer can be served by one or more vehicle. Stochastic VRP Tillman [31] Occurs in non-randomness of elements such as travel times, customer demand or customer groups. Time-dependent VRP Cooke and Halsey [38] The travel time between any pair of deposits and customers depends on the distance between the points or alternatively the time of day. VRP with Loading Constraints Iori et al. [39] Simultaneously determines (as a set) the routes and the packaging standards (loading restrictions). VRP with Time Windows Russell [40] This problem can be of two types: hard or soft time windows. For Hard Time Windows, the vehicle cannot delay, must arrive on time or early. For Soft Time Windows, there may be a violation if there is a penalty.

3.3 VRP with environmental considerations

When addressing the VRP with EC, the classification proposed by Lin et al. [20] was used here, which sticks to the Pollution Routing Problem, Green VRP and VRP in Reverse Logistics. The Pollution Routing Problem (PRP) search vehicle routing plan produces a lesser amount of pollution, in particular with a reduction in greenhouse gases (GHG) and may include broader objectives that reflect the environmental cost. The Green VRP considers the optimization of energy consumption in transport and reduction of fuel consumption.The objective of the VRP in Reverse Logistics (VRPRL) focuses on aspects of reverse logistics distribution. The latter category, VRPRL, can be divided into sub-categories: Selective Pickups with Pricing, Waste Collection, End-of-life Goods Collection and Simultaneous Distribution and Collection.

Selective Pickups with Pricing selects only profitable collection points to visit. Waste Collection (waste management) includes waste reuse, prevention and recycling. End-of-life Goods Collection is useful for remanufacturing. Finally, the Simultaneous Distribution and Collection addresses applied modeling in the context of reverse logistics [7,20,41].

Additionally, Demir et al. [18], in their review article, present, systematize and compare techniques for measuring GHG emission and fuel consumption applied to the road transportation of green loads.

Environmental considerations (EC) can be addressed in one or more Objective Functions (OF) subject to restraints through mathematical modeling, be it a linear, non-linear, integer, mixed or dynamic programming model or other approach. In this study, emphasis will be placed on the OFs focusing on the environmental, social and economic contexts, reverse logistics, fuel consumption and CO2 emissions.

Thus, it can be said that many authors are concerned with minimizing costs, be they operational, general, financial or for transport [11,42,43] or maximizing revenues or profit [44,45], as well as minimizing the number of vehicles [46,47]. Regarding the social aspect, a point in question is maximizing work opportunities [48], in addition to working hours [8] and social responsibility [49,50]. When it comes to reverse logistics, this is more difficult to measure, as not all problems found are clearly attributed to an OF, but rather to the consequent stratifications of its results [51,52]. On the other hand, GHG emissions and fuel consumption are easily measurable [18,53,54].

4. Results and discussion

In this section, we present the bibliometric data. This study identified information such as the quantity of publications per year, type of works published in the sample and percentage of journals that focus on environmental considerations. The study also identified the countries and institutions that published the most works on the proposed theme, a frequency analysis of protocol terms in the title, abstracts and keywords and the relation between the taxonomies employed. Other information includes the fleets that are used and the most commonly used Multi-objective procedures, VRP procedures, solution procedures, number of objectives and most frequently used metaheuristic methods. The software and programming languages for computational implementation and the frequency of the most used objectives were also identified. The study also includes the works related to the theme in question and a table summarizing the most frequently cited articles in the sample.

As presented in Section 2 (Methodology), this research sample was composed of 76 articles. Figure 3 shows that the number of articles on the proposed theme has increased over the years. It should be highlighted that the sample of 76 articles was collected from 2012 to 2018.

 Figure 3. Distribution per year of the 76 publications in the sample on MOOGVRP

According to Figure 4 (a), the articles with two, three and four authors correspond to 65 papers in the sample, demonstrating the existence of research groups or works developed in academic environments. As shown in Figure 4 (b), the articles were classified as General (27 articles), Case studies (43 articles) and Reviews (6 articles). As also shown in Figure 4 (c), it was verified that only 10 of the 38 journals in the sample articles focus on issues related to environmental considerations (more specifically, they show interest in environmental aspects, sustainability, ecology, green manufacturing, the environment and related contexts), namely: Applied Energy, Computers & Operations Research, Decision Science Letters, Ecological Indicators, International Energy & Environment, Journal of Cleaner Production, Journal of Manufacturing Systems, Journal of Manufacturing Technology Management, Sustainability and Transportation Research Part D.

 Figure 4. (a) Number of authors. (b) Number of article types. (c) Number of articles on environmental considerations among the 76 articles in the sample on MOOGVRP

Figure 5 shows the publication frequency of the 32 participating countries in the sample of 76 articles. It is worth noting that China and Iran contributed with 19 articles each.

 Figure 5. Countries that have contributed to the MOOGVRP Where China and Iran (19); the United Kingdom and United States of America (7); Canada and India (5); Australia, Brazil, Denmark, France and Greece (4); Colombia, Hong Kong and the Netherlands (3); Germany, Ireland, Malaysia, Norway, Sweden and Taiwan (2); Belgium, Chile, Indonesia, Italy, Japan, Morocco, New Zealand, Portugal, Singapore, Turkey and the United Arab Emirates (1).

Table 2 contains a list of affiliations related to the research sample, considering the institutions, research centers and companies with which the authors are associated. Note that several articles cover more than one affiliation. We can particularly highlight three universities in Iran, Iran University of Science and Technology [51,52,53,55,56,57], Islamic Azad University [7,11,16,44,45] and University of Tehran [13,51,56,58,59], with 6, 5 and 5 articles, respectively.

Table 2 Institutions that contributed most to MOOGVRP
 Affiliation Quantity Iran University of Science and Technology 6 Islamic Azad University and University of Tehran 5 University of Southern Denmark 4 Aristotle University of Thessaloniki, Hubei University, Tongji University and Universidade Federal de Minas Gerais 3 Aalborg University, Amirkabir University of Technology, China University of Mining and Technology, Chongqing Jiaotong University, Eindhoven University of Technology, Emlyon Business School, Oregon State University, The University of Hong Kong, Universidad Tecnológica de Pereira, Universidade Estadual Paulista Júlio de Mesquita Filho, University of Electronic Science and Technology, University of Southampton, University of Washington and Vienna University of Economics and Business 2 Others 96

Figure 6 presents the numerical frequency of the terms of the research protocol in the titles, abstracts, keywords and the three previous ones (All). This relationship is made clearer in Figure 7. This analysis is extremely important to the study because it is through these terms that the researchers achieved the sample for their works. It can be seen that the greater frequency of terms is concentrated in the abstract of articles and that there are few works that contain the search terms in the three fields.

 Figure 6. Frequency of the terms of the research protocol in the titles, abstracts and keywords of the 76 articles

Figure 7 shows the relationship between the taxonomies of the MOO-VRP with the EC presented here. We can observe (yellow circle on the left) that 12 articles in the sample address the MOO-GVRP [5,11,41,43,54,58,60,61,62,63,64,65]. Ten articles address the MOO-GVRP in conjunction with MOO-PRP [6,14,15,45,66,67,68,69,70,71]. Five articles address the MOO-GVRP in conjunction with the MOO-PRP and with the MOO-VRPRL [7,20,52,72,73]. Finally, 1 article addresses the MOO-GVRP together with the MOO-VRPRL [74]. Thus, a total of 28 articles are related to the MOO-GVRP. Likewise, we can interpret the other circles (left and right). It should be highlighted that the goal of the GVRP is different from that of the PRP, since the objective of the former is to reduce fuel or energy consumption (or the battery, alternatively) and the latter seeks to minimize GHG emissions, mainly CO2. It was observed that in some articles the authors implicitly used their own taxonomy, as in [13,22,75,76], generalizing the MOO-PRP as the MOO-GVRP. Furthermore, the opposite occurred in [41], who referred to the MOO-GVRP as the MOO-PRP.

 Figure 7. Representation of the research sample on MOO-VRP taxonomies with EC

Figure 8 to 13 are related to articles classified as General and Case Studies (70 articles). The review articles (6 articles) will not be considered in these figures because of their wide coverage. Figure 8 shows the fleet classification (homogeneous or heterogeneous) through the sample articles.

 Figure 8. Fleet type used by the sample of 70 articles

Regarding the MOO procedures presented in Figure 9, we can verify the predominance of the Pareto to analyze the dominance or non-dominance of the obtained solutions. We then have the classification of the use of heuristic, є-constraint and weighted procedures. It is worth noting that several articles include more than one procedure. A subtle description of the techniques Pareto, є-constraint, weighted, goal programing and lexicographic were presented at the end of Section 3.1. In the case of Heuristic, a heuristic or metaheuristic procedure such as NSGA-II was performed. The linearization process deals with the case of a nonlinear model that is converted to linear [7]. Finally, the L-P metrics method, the difference between any present solutions and the optimal solution, is minimized [55].

 Figure 9. Most frequently used multi-objective approaches MOOGVRP in the sample of 70 articles

As for the VRP procedures shown in Figure 10, it must be taken into account that the same problem may contain more than one variant, in accordance with the works conducted by [8,16,49,59]. The Location Routing Problem (LRP) and the classical case (Capacitated VRP) were the most frequently used in the sample, followed by Time Windows, Multi-echelon, and Pickup and Delivery.

 Figure 10. Solution procedures most used in the MOOGVRP Legend: CVRP (Capacitated VRP), DVRP (Dynamic VRP), FVRP (Fuzzy VRP), IRP (Inventory Routing Problem), LRP (Location Routing Problem), MDVRP (Multi-depot VRP), MPVRP (Multi-period VRP), MEVRP (Multi-echelon VRP), SVRP (Scheduling in VRP), TDVRP (Time-dependent VRP), VRPPD (VRP with Pickup and Delivery), VRPS (VRP Stochastic) and VRPTW (VRP with Time Windows)

Figure 11 (a) presents the use and quantity of exact and heuristic procedure, or both, for the 70 articles. It should be noted that 42 (60%) of the articles only used heuristic procedures. Figure 11 (b) shows that 45 (64%) of the articles analyzed presented an approach with only two objectives and 22 (31%) are tri-objective. Figure 11 (c) shows the use proportion of heuristic procedures in the MOO context. We can observe that the techniques derived from population algorithms are predominant, where GA and PSO correspond to 54%.

 Figure 11. (a) Solution approaches. (b) Approach. (c) Meta-heuristics used Legend: ABC (Artificial Bee Colony), ACO (Ant Colony Optimization), CW (Clarke and Wright Heuristic), GA (Genetic Algorithm), LS (Local Search), PSO (Particle Swarm Optimization), SA (Simulated Annealing), VNS (Variable Neighborhood Search)

Figure 12 presents the most frequently used software and the respective programming languages for the 70 articles. It should be mentioned that an article may have used more than one software or language, and that some software accepts different languages. Some authors reported only the software, others only the programming language, and there were some works lacking this information.

 Figure 12. (a) Software. (b) Most frequently used programming languages Legend: AMPL (A Mathematical Programming Language), CPLEX (Simplex method as implemented in the C programming language), GAMS (General Algebraic Modeling System), LINGO (Language for Interactive General Optimizer) and MATLAB (MATrix LABoratory)

Figure 13 presents the main objective used in the 70 analyzed works. They were grouped into GVRP, PRP and VRPRL, with all of them being multi-objective. We can observe that all prioritize the minimization of costs among the main objectives. In the case of the GVRP, the minimization of fuel consumption is also among the main objectives. The same occurs with the PRP with CO2 minimization.

 Figure 13. Most frequent objectives in MOO problems, applied to the GVRP, PRP

We now present brief reports of the sample of 76 articles in this study and a description of the most frequently cited works in the MOO context.

A summary of the 76 articles related to MMGVRP is presented in Table 3. This Table is organized by references, presented in alphabetical and chronological order. The type of research was classified as a case study (c), general application (g) and review (r). The fleet was classified as homogeneous (ho), heterogeneous (he) or not informed (ni). The VRP with EC was classified by the taxonomy used in this study: GVRP, PRP and VRPRL. It was determined whether the procedures used were exact, heuristic or, alternatively, both approaches. The number of objectives was organized as bi-objective (bi), tri-objective (tri) or multi-objective (multi). The desired outcomes of the study, the proposed application of the work, the solution techniques and number of citations were then specified (May 2019 through SCOPUS).

As an example, one can see [77] and [21]. Thus, [77] is a review article that addresses the PRP with the objective of integration of environmental aspects in logistics, applied to green logistics and with 353 citations. Ehrgott et al. [21], the case study type, with homogeneous fleet, which also addresses the PRP, used heuristic procedure with bi-objective mathematical modeling that proposed minimized journey time and maximized adaptation to the routes used; applied in cyclist route, used the technique bi-objective shortest path problem and had 36 citations.

Table 3. Summary of the 76 articles related to MOOGVRP

Legend: ABC (artificial bee colony); ALNS (adaptive large neighborhood search algorithm); AMOSA (achieved multi-objective simulated annealing); AMOVNS (adapted multi-objective variable neighborhood search); AUGMENCON (augment constraint algorithm); BEG-NSGA-II (bee evolutionary algorithm guiding nondominated sorting genetic algorithm II); BIOBAB (bi-objective branch-and-bound); CA (Cultural algorithm); CW_NSGA-II (Clarke and Wright savings method and the non-dominated sorting genetic algorithm-II); EABC (enhanced artificial bee colony); ESMPO (evolution-strategy-based memetic pareto optimization); HQIA (hybrid quantum immune algorithm); MIP (mixed integer programming); MILP (mixed integer linear programming); MNIP (mixed non-linear integer programming); MOACO (multi-objective ant colony optimization); MODE (multi-objective differential evolution); MOGA (multi-objective genetic algorithm); MOGA-II (multi-objective genetic algorithm of kind II); MOLS (multi-objective Local Search); MOPSO (multi-objective particle swarm optimization); MOSPEA (metaheuristic optimization algorithm); MOSPOOLS (multi-objective smart pool search metaheuristic); MOSPP (multi objective shortest path problem); MOSSA (multi-objective salp swarm algorithm); NRGA (non-dominated ranking genetic algorithm); NSGA-II (non-dominated sorting genetic algorithm II); PFGA (pareto front genetic algorithms); PMS-NSDEs (parallel multi-start non-dominated sorting differential evolution algorithms); SLPSO (hybrid self-learning particle swarm optimization); TOPSIS (technique for order preference by similarity to ideal solution); VNSGA-II (multi-objective variable neighborhood search with non-dominated sorting genetic algorithm II).

The most cited works in the sample are presented in Table 4. The table is structured by references, journals, article type and number of citations obtained through SCOPUS. A description of these works, listed according to their number of citations, is given below.

Table 4 The 11 most cited articles in the sample to MOOGVRP
 References Journals Types No. of Citations Dekker et al., [77] European Journal of Operational Research Review 353 Lin et al., [20] Expert Systems with Applications Review 314 Demir et al., [18] European Journal of Operational Research Review 260 Govindan et al., [80] International Journal of Production Economics Case 201 Demir et al., [41] European Journal of Operational Research Case 156 Validi et al., [82] International Journal of Production Economics Case 92 Ramos et al., [8] Omega (United Kingdom) Case 73 Chen et al., [79] Transportation Research Part E Case 53 Govindan et al., [49] Ecological Indicators Case 50 Ehrgott et al., [21] Transportation Research Part A Case 36 Kumar et al., [15] Computers and Industrial Engineering Case 36

The three most frequently cited works were literature reviews, conducted by Dekker et al. [77], Lin et al. [20] and Demir et al. [18]. Dekker et al. [77] proposed a contribution to the green logistic, promoting the integration of environmental aspects into the logistic. They addressed facilities, transport and inventory, considering the decision process through design, planning and control. Lin et al. [20] listed the main variations of the VRP, solution procedures (algorithms) and a proposed taxonomy for the VRP with environmental considerations (GVRP, PRP and VRPRL). A study on green road freight transportation was presented by Demir et al. [18], making use of different ways to measure CO2 emissions and fuel consumption. They identified the criticality of factors to explain consumption, type of vehicle, freight, speed and road gradient, among others.

A case study addressing perishable food distribution was conducted by [80]. For this purpose, the authors used multi-objective particle swarm optimization (MOPSO), adapted multi-objective variable neighborhood search (AMOVNS) and the multi-objective hybrid approach (MHPV), comparatively. Seeking to optimize fuel consumption and journey time, Demir et al. [41] used the є-constraint together with the weighting method for the adaptive large neighborhood search algorithm (ALNS), in addition to a hybrid method (HM). With a view to obtaining a sustainable distribution by minimizing costs and CO2 emission for milk transportation in Ireland, Validi et al. [82] presented a mixed integer linear programming (MILP) model and solved the problem through procedures derived from the GA: multi-objective GA of kind II (MOGA-II), NSGA-II (non-dominated sorting GA) and a Hybrid combining GA and sequential quadratic programming.

Ramos et al. [8] resolved a case study on a recyclable waste collection system in Portugal, more specifically, in 19 rural municipalities. This reverse logistics problem was modeled as a multi-depot periodic vehicle routing problem with inter-depot routes (MDPVRPI), using the є-constraint technique to solve it. The authors wished to comply with the triple bottom line of sustainability, considering minimizing costs and CO2 emissions and maximizing working hours. Chen et al. [79] proposed a case involving a bi-objective model based on queues to reduce long lines of trucks at ports. The authors sought to minimize the number of truck arrivals and waiting time. For this purpose, they used the Pareto front genetic algorithms based heuristic (PFGA) and compared it with the NSGA type II. A fuzzy multi-objective model was proposed by Govindan et al. [49] to address the multi-echelon VRP and multi-period reverse logistics. The authors compared a PSO approach with the є-constraint. They applied the model to a case of medical syringe recycling for the purpose of minimizing costs and environmental impact and maximizing social responsibility.

Ehrgott et al. [21] presented a case study on the routing of bicycles in New Zealand, seeking to minimize journey time and maximize the adequacy of the routes that were used. Finally, Kumar et al. [15] presented production and pollution routing problems with time windows (PPRP-TW), in which they implemented the hybrid Self-Learning Particle Swarm Optimization (SLPSO). The authors sought to minimize costs and fuel consumption.

5. Final considerations and suggestions for future works

This study presented bibliometric data involving MOO for the Green Vehicle Routing Problem (MOOGVRP) from 2012 to 2018. This study adopted a protocol that resulted in 1,744 articles that, after a filtering definition, provided a sample of 76 articles from 38 journals. Its main contribution that it highlights the synergy between the VRP, MOO and the approach with EC, which is a relatively new trend that has grown in recent years. The bibliometric analysis was essential to understanding the profiles of published works.

To answer the first question of this survey (What are the most relevant studies regarding MOOGVRP?) the most relevant studies during the analyzed period were presented in Table 4, in which the three most frequently cited articles were reviews. Dekker et al. [77] and Demir et al. [18] addressed aspects related to logistics, while Lin et al. [20] sought to demonstrate the evolution and relationship of the VRP with environmental considerations. The other most frequently cited articles sought to improve techniques and apply them to real contexts.

To answer the second question (What are the trends regarding MOOGVRP that may guide future research?) the stratifications of Section 4 are of great importance. In this context, Table 3 summarized the proposal of the 76 articles focused on reviews, general application and case studies. The sample authors are associated with 96 institutions, research centers and companies, and the largest contributions, so far, have come from Iran, China, the USA and the UK. Figure 7 proved to be very interesting, as it shows the amount of work developed considering GVRP, PRP and VRPRL, as well as their interactions in the analyzed sample.

It was possible to verify the dominance of bi-objective models, which have been solved through heuristic, є-constraint and weighted procedures, with “a posteriori” Pareto optimization, based on Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The most used VRP procedures in the analyzed sample were the Location Routing Problem (LRP) and the Capacitated VRP (CVRP). As shown in Figure 13, the main objectives addressed in the problems were to minimize costs, fuel consumption and CO2 emissions. There was great concern over improving the studies related to the reduction of GHG emission and the considerations of reverse logistics.

Many opportunities to explore applications and techniques related to MOOGVRP remain. We hope that the information presented here can aid those interested in their future research. Thus, by “going against the grain” of the most frequently researched themes, there is a great opportunity to make unprecedented contributions, of which the following may be mentioned:

1. Seeking heuristic procedures that have not often been used for multi-objective problems, duly adapted to the contexts in question, such as open, pickup time, multi-product and multi-modal (variations of the MOO-VRP least used in the sample of this study);
2. Although already widespread, seeking hybrid heuristic procedures to improve algorithms used in the context of the MOO-VRP;
3. Exploring applications that involve 4 or more objectives due to the serious lack of such applications in the literature;
4. Developing studies with large instances given the computational difficulty;
5. Presenting works that address the relationships between the MOO-GVRP, MOO-PRP and MOO-VRPRL or all of them simultaneously;
6. Use other types of documents, such as conference papers, book, dissertation or thesis;
7. Use other types of vehicles;
8. Use other databases to compare and complement the sample;
9. Other bibliometric (or review) techniques to expand the analysis;
10. Evaluate the relationship with the triple bottom line.

Acknowledgements

This study was financed in part by PUCPR and the Coordination for the Improvement of Higher Education Personnel - Brazil (CAPES; 1st author), Finance code 001 and by the National Council for Scientific and Technological Development – Brazil (CNPq; 2nd and 3rd authors).

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Document information

Published on 02/03/20
Accepted on 27/02/20
Submitted on 01/12/19

Volume 36, Issue 1, 2020
DOI: 10.23967/j.rimni.2020.03.001


text: Would it be safe to make the following generalization? We present a unifying metric formalism for connectedness, … Consider the topological space $(\mathbb{Z}, \tau)$ where $\tau$ is the cofinite topology. (iii) Give an example of two disjoint closed subsets of R2 such that inf{d(x,x0) : x ∈ E,x0 ∈ F} = 0. Then f: X!Y that maps f(x) = xis not continuous. Topology is one of the basic fields of mathematics.The term is also used for a particular structure in a topological space; see topological structure for that.. Topological spaces We start with the abstract deﬁnition of topological spaces. Give an example where f;X;Y and H are as above but f (H ) is not closed. How is it possible for this NPC to be alive during the Curse of Strahd adventure? (3) A function from the space into a topological space is continuous if and only if it preserves limits of sequences. Metric and Topological Spaces. A topological space is an A-space if the set U is closed under arbitrary intersections. (3)Any set X, with T= f;;Xg. Show that the sequence 2008,20008,200008,2000008,... converges in the 5-adic metric. the topological space axioms are satis ed by the collection of open sets in any metric space. Topological spaces with only ﬁnitely many elements are not particularly important. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. In nitude of Prime Numbers 6 5. 1.Let Ube a subset of a metric space X. Topology of Metric Spaces 1 2. This abstraction has a huge and useful family of special cases, and it therefore deserves special attention. (a) Let X be a compact topological space. 3.Find an example of a continuous bijection that is not a homeomorphism, di erent from This is called the discrete topology on X, and (X;T) is called a discrete space. The subject of topology deals with the expressions of continuity and boundary, and studying the geometric properties of (originally: metric) spaces and relations of subspaces, which do not change under continuous … It turns out that a great deal of what can be proven for ﬁnite spaces applies equally well more generally to A-spaces. Example 3. As I’m sure you know, every metric space is a topological space, but not every topological space is a metric space. Then is a topology called the trivial topology or indiscrete topology. Mathematics Subject Classi–cations: 54A20, 40A35, 54E15.. yDepartment of Mathematics, University of Kalyani, Kalyani-741235, India 236. Prove that f (H ) = f (H ). Determine whether the set $\{-1, 0, 1 \}$ is open, closed, and/or clopen. Suppose H is a subset of X such that f (H ) is closed (where H denotes the closure of H ). Schaefer, Edited by Springer. Basis for a Topology 4 4. 4 Topological Spaces Now that Hausdor had a de nition for a metric space (i.e. A Theorem of Volterra Vito 15 9. In general topological spaces, these results are no longer true, as the following example shows. METRIC AND TOPOLOGICAL SPACES 3 1. Let X be any set and let be the set of all subsets of X. An excellent book on this subject is "Topological Vector Spaces", written by H.H. One measures distance on the line R by: The distance from a to b is |a - b|. Give an example of a metric space X which has a closed ball of radius 1.001 which contains 100 disjoint closed balls of radius one. Lemma 1.3. 5.Show that R2 with the topology induced by the British rail metric is not homeomorphic to R2 with the topology induced by the Euclidean metric. The elements of a topology are often called open. Thank you for your replies. We refer to this collection of open sets as the topology generated by the distance function don X. Prove that fx2X: f(x) = g(x)gis closed in X. Example 1.1. (T3) The union of any collection of sets of T is again in T . 11. TOPOLOGICAL SPACES 1. a set together with a 2-association satisfying some properties), he took away the 2-association itself and instead focused on the properties of \neighborhoods" to arrive at a precise de nition of the structure of a general topological space… In general topological spaces do not have metrics. Such open-by-deﬂnition subsets are to satisfy the following tree axioms: (1) ?and M are open, (2) intersection of any finite number of open sets is open, and (2)Any set Xwhatsoever, with T= fall subsets of Xg. A topological space which is the image of a metric space under a continuous open and closed mapping is itself homeomorphic to a metric space. 2. 3. Examples show how varying the metric outside its uniform class can vary both quanti-ties. Jul 15, 2010 #5 michonamona. 6.Let X be a topological space. In fact, one may de ne a topology to consist of all sets which are open in X. Introduction When we consider properties of a “reasonable” function, probably the ﬁrst thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. Exercise 206 Give an example of a metric space which is not second countable from MATH 540 at University of Illinois, Urbana Champaign The natural extension of Adler-Konheim-McAndrews’ original (metric- free) deﬁnition of topological entropy beyond compact spaces is unfortunately inﬁnite for a great number of noncompact examples (Proposition 7). a Give an example of a topological space X T which is not Hausdor b Suppose X T from 21 127 at Carnegie Mellon University In compact metric spaces uniform connectedness and connectedness are well-known to coincide, thus the apparent conceptual difference between the two notions disappears. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. 4E Metric and Topological Spaces Let X and Y be topological spaces and f : X ! The family Cof subsets of (X,d)deﬁned in Deﬁnition 9.10 above satisﬁes the following four properties, and hence (X,C)is a topological space. The prototype Let X be any metric space and take to be the set of open sets as defined earlier. Prove that diameter(\1 n=1 Vn) = inffdiameter(Vn) j n 2 Z‚0g: [Hint: suppose the LHS is smaller by some amount †.] p 2;which is not rational. every Cauchy sequence converges to a limit in X:Some metric spaces are not complete; for example, Q is not complete. There are examples of non-metrizable topological spaces which arise in practice, but in the interest of a reasonable post length, I will defer presenting any such examples until the next post. Let βNdenote the Stone-Cech compactiﬁcation of the natural num-ˇ bers. of metric spaces. (3) Let X be any inﬁnite set, and … Metric and topological spaces, Easter 2008 BJG Example Sheet 1 1. is not valid in arbitrary metric spaces.] Examples of non-metrizable spaces. A topological space M is an abstract point set with explicit indication of which subsets of it are to be considered as open. 1.4 Further Examples of Topological Spaces Example Given any set X, one can de ne a topology on X where every subset of X is an open set. A Topological space T, is a collection of sets which are called open and satisfy the above three axioms. Topological Spaces Example 1. We give an example of a topological space which is not I-sequential. Definitions and examples 1. Let (X,d) be a totally bounded metric space, and let Y be a subset of X. 3. In mathematics, a metric or distance function is a function which defines a distance between elements of a set.A set with a metric is called a metric space.A metric induces a topology on a set but not all topologies can be generated by a metric. Some "extremal" examples Take any set X and let = {, X}. However, under continuous open mappings, metrizability is not always preserved: All spaces satisfying the first axiom of countability, and only they, are the images of metric spaces under continuous open mappings. Let me give a quick review of the definitions, for anyone who might be rusty. 1 Metric spaces IB Metric and Topological Spaces 1.2 Examples of metric spaces In this section, we will give four di erent examples of metrics, where the rst two are metrics on R2. This particular topology is said to be induced by the metric. Non-normal spaces cannot be metrizable; important examples include the Zariski topology on an algebraic variety or on the spectrum of a ring, used in algebraic geometry,; the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. On the other hand, g: Y !Xby g(x) = xis continuous, since a sequence in Y that converges is eventually constant. 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The set of open sets as the following generalization... converges in the metric. Be safe to make the following example shows is called the discrete topology on X d... The closure of H ) is closed ( where H denotes the closure of a to! Examples show how varying the metric outside its uniform class can vary both quanti-ties who might be rusty X! Only ﬁnitely many elements are not particularly important Euclidean metric, but not in the Euclidean,... And take to be alive during the Curse of Strahd adventure subset of a topology to consist of all which! ( Revision of real analysis ) Contents: Next page ( Revision of real analysis Contents... That can be described by a metric space X = f ( H ) topology that can proven. However, it is quite standard Y and H are as above but (.: the distance from a to b is |a - b| can be proven for ﬁnite spaces applies equally more. 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The topological space has a topology called the trivial topology or indiscrete topology open sets as defined earlier is! Particular topology is not I-sequential only ﬁnitely many elements are not particularly important abstract deﬁnition of topological spaces these! Finitely many elements are not particularly important refer to this collection of sets of is... For this example of topological space which is not metric to be induced by d. prove that f ( H ) is a! 2008,20008,200008,2000008,... converges in the Euclidean metric, we say that a great deal of can! H ) and/or clopen had a de nition for a metric space X prototype let be!

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## Main Question or Discussion Point

Find the laplace transform $${\cal L} (1 + t)$$
Without using the table.

$$\mathcal{L}[f(t)](s) := \int_0^\infty e^{-st}f(t)dt$$


text:
# Metric space

1. Dec 9, 2007

### ehrenfest

[SOLVED] metric space

1. The problem statement, all variables and given/known data

If x and y are two points in a metric space and d(x,y) = 1, is it always true that the closure of B(x,1/2) does not contain y?

In general, is $$closure( B(x,r)) = \{z | r \geq d(x,z)\}$$

2. Relevant equations

3. The attempt at a solution

Last edited: Dec 9, 2007
2. Dec 9, 2007

### ehrenfest

Never mind. The answer is of course because closed balls are closed in a metric space. Why are closed balls in a metric space closed. Because their complement is open. Why is their complement open? Because, for any point x in the complement, let d be the distance from x to the closed ball. Then B(x,d/2) is a nbhd of x that lies in the complement. d is always nonzero and well-defined because otherwise x would be a limit point of the closed ball.

3. Dec 10, 2007

### morphism

Hold on - this is not true. In the reals with the discrete metric, closure(B(x,1))={x}, while {z : 1>=d(x,z)}=R.

But what is true is that the closure of B(x,r) will always sit inside {z : r>=d(x,z)}.

4. Dec 10, 2007

### ehrenfest

I see. Thanks.


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resources.math.binary values are a long form and can be clumsy to keep writing out. Hexadecimal allows us to represent 4 binary bits (a "nibble") in each place.

000000
000111
001022
001133
010044
010155
011066
011177
100088
100199
101010A
101111B
110012C
110113D
111014E
111115F


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## Classical geometries in modern contexts. Geometry of real inner product spaces.(English)Zbl 1084.51001

Basel: Birkhäuser (ISBN 3-7643-7371-7/hbk). xii, 242 p. (2005).
The volume under review can be seen as the third part of an opus magnum, the first two being [Geometrische Transformationen (B. I. Wissenschaftsverlag, Mannheim) (1992; Zbl 0754.51005)] and [Real geometries (B. I. Wissenschaftsverlag, Mannheim) (1994; Zbl 0819.51002)], in which the author’s aim is to present (i) models of classical geometries in a dimension-free setting, (ii) the relevance of functional equations in geometry, (iii) very general forms of characterizations of mappings under mild hypothesis (of the Mazur-Ulam, Alexandrov-Zeeman, Beckman-Quarles type), (iv) a very precise conceptual framework for Klein’s Erlangen programme, with constant emphasis on the interplay between geometries and groups of transformations.
In Chapter 1, the author introduces the notion of a translation group associated to a real inner product space $$X$$, defines Euclidean and hyperbolic geometry over $$X$$ (in the case of hyperbolic geometry, we are presented with the Weierstrass-inspired model put forward by the author in [N. K. Artémiadis (ed.) et al., Proceedings of the 4th international congress of geometry, Thessaloniki, Greece, May 26–June 1, 1996. Athens: Aristotle University of Thessaloniki. 12–21 (1997; Zbl 0888.51020)], and presented in his textbook [Ebene Geometrie, Spektrum, Heidelberg (1997; Zbl 0870.51001)]), and proves an astonishing characterization, of a kind that brought the result of H.-C. Wang [Pac. J. Math. 1, 473–480 (1951; Zbl 0044.19602)] to the reviewer’s mind, of the Euclidean and hyperbolic geometries as the only geometries over $$X$$ satisfying rather general conditions involving the existence of a function that has only some of the properties the metric would have in geometries with a mildly special kind of translation group (first published by the author in [Publ. Math. 63, 495–510 (2003; Zbl 1052.39022)]).
Chapter 2 is devoted to an in-depth analysis of the models of Euclidean and hyperbolic geometry that are defined in the previous chapter, the emphasis being naturally on the hyperbolic case. Among the topics: determining the equations of lines, as defined in three different ways (by Blumenthal, Menger, and the author), hyperplanes, subspaces, equidistant surfaces, ends, parallelism, angles of parallelism, horocycles, the Cayley-Klein model, various characterizations of isometries and of translations.
Chapter 3 is devoted to the sphere geometries of Möbius (first presented in [Aequationes Math. 66, 284–320 (2003; Zbl 1081.51500)]) and Lie, in which we encounter Poincaré’s model of hyperbolic geometry, the main notions of these geometries (such as Lie and Laguerre cycles, Lie and Laguerre transformations), with their remarkable characterization under the very “mild hypothesis” that they are bijections preserving the cycle contact relation (but not necessarily the negation of this relation), which was the subject of the author’s [Result. Math. 40, 9–36 (2001; Zbl 0995.51003)].
Chapter 4 looks at space-times and their groups of transformations, Minkowski space-time and Lorentz transformations, de Sitter’s world, Einstein’s cylinder world, again with many characterizations of the corresponding transformations under mild hypotheses, as well as a characterization of a general notion of Lorentz-Minkowski distance, and a theorem that uncovers the very strong connection existing between hyperbolic motions and Lorentz transformations. The author’s model of hyperbolic geometry may thus be said to be the unifying thread, and the various results proved in this book which rely very heavily on this model testify to the significance of the discovery of this model, which turns out to be not “just another model of hyperbolic geometry”, but one that allows, by the very fact that its point-set is that of Euclidean geometry or of Minkowski space-time, a fruitful comparison.
The mathematical prerequisites are minimal – the rudiments of linear algebra suffice – and all theorems are proved in detail. Following the proofs does not involve more than following the lines of a computation, and the author makes every effort to avoid referring to a synthetic geometric understanding, given that he aims at attracting readers with a distaste for synthetic geometry, which, given the academic curricula of the past decades, represent the overwhelming majority of potential readers of any mathematical monograph. One of the lessons of this monograph is that there is a coordinate-free analytic geometry, which significantly simplifies computations and frees the mind from redundant assumptions. There is only minimal overlap with the first two volumes, the aim of the few repetitions being that of ensuring the volume’s independent readability.
In the realm of synthetic (axiomatic) geometry, the dimension-free approach can be traced back to H. N. Gupta [Contributions to the axiomatic foundations of Euclidean geometry (Ph. D. Thesis, University of California, Berkeley) (1965)] and W. Schwabhäuser [J. Reine Angew. Math. 242, 134–147 (1970; Zbl 0199.55002)].

### MSC:

 51-02 Research exposition (monographs, survey articles) pertaining to geometry 51B10 Möbius geometries 51B25 Lie geometries in nonlinear incidence geometry 51M05 Euclidean geometries (general) and generalizations 51M10 Hyperbolic and elliptic geometries (general) and generalizations 83A05 Special relativity 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory 39B52 Functional equations for functions with more general domains and/or ranges


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Two ultraviolet-sensitive cell lines, designated S-1M and S-2M, were isolated from the original HeLa S3 cells by the N-methyl-N′-nitro-N-nitrosoguanidine (MNNG)-ultraviolet light (UV) and 5-bromodeoxyuridine (BUdR)-visible light method. The D0 values were 87, 68, and$45\ {\rm ergs}/{\rm mm}^{2}$, and the extrapolation numbers (n) were 1.4, 1.0, and 1.0 for HeLa S3, S-1M, and S-2M cells, respectively. The average initial excision rate of UV-induced thymine dimers in the UV-sensitive cells is approximately one fifth that in HeLa S3 cells. The UV-sensitive cells may appear in the original HeLa S3 population due to the mutagenic action of MNNG. S-2M cells are more sensitive to the carcinogen 4-nitroquinoline 1-oxide (4-NQO) than HeLa S3 cells. However, there are no significant differences in sensitivity to x-irradiation among the three cell lines. The findings indicate that the 4-NQO-induced DNA damage and its repair are similar to that found after UV-irradiation, but different from that found after x-irradiation.

This content is only available as a PDF.


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discretize.base.BaseTensorMesh.average_edge_to_cell#

property BaseTensorMesh.average_edge_to_cell#

Averaging operator from edges to cell centers (scalar quantities).

This property constructs a 2nd order averaging operator that maps scalar quantities from edges to cell centers. This averaging operator is used when a discrete scalar quantity defined on mesh edges must be projected to cell centers. Once constructed, the operator is stored permanently as a property of the mesh. See notes.

Returns
(n_cells, n_edges) scipy.sparse.csr_matrix

The scalar averaging operator from edges to cell centers

Notes

Let $$\boldsymbol{\phi_e}$$ be a discrete scalar quantity that lives on mesh edges. average_edge_to_cell constructs a discrete linear operator $$\mathbf{A_{ec}}$$ that projects $$\boldsymbol{\phi_e}$$ to cell centers, i.e.:

$\boldsymbol{\phi_c} = \mathbf{A_{ec}} \, \boldsymbol{\phi_e}$

where $$\boldsymbol{\phi_c}$$ approximates the value of the scalar quantity at cell centers. For each cell, we are simply averaging the values defined on its edges. The operation is implemented as a matrix vector product, i.e.:

phi_c = Aec @ phi_e

Examples

Here we compute the values of a scalar function on the edges. We then create an averaging operator to approximate the function at cell centers. We choose to define a scalar function that is strongly discontinuous in some places to demonstrate how the averaging operator will smooth out discontinuities.

We start by importing the necessary packages and defining a mesh.

>>> from discretize import TensorMesh
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> h = np.ones(40)
>>> mesh = TensorMesh([h, h], x0="CC")

Then we create a scalar variable on edges,

>>> phi_e = np.zeros(mesh.nE)
>>> xy = mesh.edges
>>> phi_e[(xy[:, 1] > 0)] = 25.0
>>> phi_e[(xy[:, 1] < -10.0) & (xy[:, 0] > -10.0) & (xy[:, 0] < 10.0)] = 50.0

Next, we construct the averaging operator and apply it to the discrete scalar quantity to approximate the value at cell centers.

>>> Aec = mesh.average_edge_to_cell
>>> phi_c = Aec @ phi_e

And plot the results:

Expand to show scripting for plot
>>> fig = plt.figure(figsize=(11, 5))
>>> mesh.plot_image(phi_e, ax=ax1, v_type="E")
>>> ax1.set_title("Variable at edges", fontsize=16)
>>> mesh.plot_image(phi_c, ax=ax2, v_type="CC")
>>> ax2.set_title("Averaged to cell centers", fontsize=16)
>>> plt.show()

Below, we show a spy plot illustrating the sparsity and mapping of the operator

Expand to show scripting for plot
>>> fig = plt.figure(figsize=(9, 9))


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1. Confidence interval problems?

Hi all,

I really hate Statistics since the problems are very difficult

For the 9th problem:

The sample mean is (20+27+15+20+16+22+13)/7=19

The sample variance = E[X²] - 19²

= (400 + 729 + 225 + 400 + 256 + 484 + 169) / 7 - 361

= 19.428.

Bessel's correction >> 19,428 . 7/6 = 22.666

standard deviation = 4.76

... but how to go on?

For 8th problem:

Distribution binomial

mean = np = 13

standard deviation = [np (1-p)] (square root) = 3.486

... but how to go on?

I could not solve the 7th one.

2. Re: Confidence interval problems?

You are taking statistics- don't you know what a normal distribution [b]is[b]. First consult a table of the standard normal distribution to see how many standard deviations from the mean lie within 95% or 99%. Then use the mean and standard deviation you have calculated to find the numbers you want. The table at Standard Normal Distribution Table tells me that 95% of the standard normal distribution is within 1.91 standard deviations of the mean. Since the standard deviation, in the last problem, is 4.76, 95% will be within (1.91)(4.76), or approximately 9, of the mean. Since the mean is 19, that is from 19- 9= 10 to 19+ 9= 28.

3. Re: Confidence interval problems?

Originally Posted by HallsofIvy
You are taking statistics- don't you know what a normal distribution [b]is[b]. First consult a table of the standard normal distribution to see how many standard deviations from the mean lie within 95% or 99%. Then use the mean and standard deviation you have calculated to find the numbers you want. The table at Standard Normal Distribution Table tells me that 95% of the standard normal distribution is within 1.91 standard deviations of the mean. Since the standard deviation, in the last problem, is 4.76, 95% will be within (1.91)(4.76), or approximately 9, of the mean. Since the mean is 19, that is from 19- 9= 10 to 19+ 9= 28.
Problems 7 and 9 are small sample questions so the critical values from the t-distribution should be used with $\nu=n-1$, also the sd should be estimated with the $\frac{1}{n-1}$ formula not the $\frac{1}{n}$ formula (which is what Bessel's correction does).

.

4. Re: Confidence interval problems?

7) For a sampling distribution the mean is the same and the standard deviation becomes 64/sqrt(15)
a 99% CI for the mean would be 912+/-2.9768*64/sqrt(15)=
912+/-49.1908=(862.81, 961.19) (2.9768 is t_alpha/2 with 14 degrees of freedom)

8)13/200=.065
a 95% CI for the proportion of defective light bulbs would be
.065+/-1.96*sqrt(.065(1-.065)/200)=
.065+/-.034167=
(.0308, .0992)

9) Assuming your standard deviation and mean are correct, 2.4469 is the t value corresponding to alpha/2 with 6 degrees of freedom
since the sample size is small and the population standard deviation is unknown, we must use the t distribution (even though the sample came from a normal distribution).
19+/-2.4469*4.76/sqrt(7)=
19+/-4.4022=
(14.60, 23.40)


text:
TensorExpressions::NumberAsExpression Struct Reference

Defines an expression representing a double More...

#include <NumberAsExpression.hpp>

## Public Types

using type = double

using symmetry = tmpl::list<>

using index_list = tmpl::list<>

using args_list = tmpl::list<>

## Public Member Functions

NumberAsExpression (const double number)

template<typename... TensorIndices, Requires< sizeof...(TensorIndices)==0 > = nullptr>
double get (const std::array< size_t, 0 > &) const noexcept
Returns the number represented by the expression. More...

## Static Public Attributes

static constexpr auto num_tensor_indices = 0

## Detailed Description

Defines an expression representing a double

## ◆ get()

template<typename... TensorIndices, Requires< sizeof...(TensorIndices)==0 > = nullptr>
 double TensorExpressions::NumberAsExpression::get ( const std::array< size_t, 0 > & ) const
inlinenoexcept

Returns the number represented by the expression.

### Details

While a NumberAsExpression does not store a rank 0 Tensor, it does represent one. This is why this template is only defined for the case where TensorIndices is empty.

Template Parameters
 TensorIndices the TensorIndexs of the LHS tensor and RHS tensor expression

Returns: the number represented by this expression

The documentation for this struct was generated from the following file: