International Journal of Civil Engineering and Technology (IJCIET) Volume 9, Issue 8, August 2018, pp. 879–886, Article ID: IJCIET_09_08_089 Available online at http://iaeme.com/Home/issue/IJCIET?Volume=9&Issue=8 ISSN Print: 0976-6308 and ISSN Online: 0976-6316

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OPTIMIZING RURAL DEALERS LOCATION – A VORONOI APPROACH

Dr. K. Umadevi Assistant Professor, PG & Research Department of Commerce, Guru Nanak College, Velachery, Chennai, Tamilnadu, India

ABSTRACT To optimize the location of dealers of Fast Moving Consumer Goods Companies in rural areas and to check the areas where brand supply has to be maximized for better reach to consumers, Voronoi analysis was carried out. The Alber map projection was used and the Voronoi cells were derived based on major cities latitude and longitude in Thiruvannamalai District. The resulting Voronoi diagram can help marketing department to optimize their product delivery and can help in optimizing placement of brand exclusive outlets or can help in optimizing supply while increasing reach to consumers in the Voronoi cell area. Key words: Voronoi diagram, Delaunay tessellation, Optimizing Location, Logistics. Cite this Article: Dr. K. Umadevi, Optimizing Rural Dealers location – A Voronoi Approach. International Journal of Civil Engineering and Technology, 9(8), 2018, pp. 879-886. http://iaeme.com/Home/issue/IJCIET?Volume=9&Issue=8

1. INTRODUCTION India‘s 6,50,000 villages have 850 million consumers which make up 70 % of population and they contribute half of country‘s Gross Domestic Product (GDP). India‘s per capita GDP in rural regions has grown at a Compound Annual Growth Rate (CAGR) of 6.2 per cent since 2000. The Fast Moving Consumer Goods (FMCG) sector in rural and semi-urban India is expected to cross US$ 20 billion mark by 2018 and reach US$ 100 billion by 2025. These figures show the extent of opportunity available to FMCG companies to grow. One of the key determinants of growth is large network of dealers in the villages. It is imperative to be judicious when setting up dealer outlets. Minimizing dealer network while ensuring optimum coverage of products in rural areas is holy grail of rural marketing. This research uses Voronoi diagram to achieve optimum dealer network allocation in rural Tiruvanamalai district in State of Tamilnadu, India. The research will help Fast Moving Consumer Goods (FMCG) companies to adapt similar approach to optimize their dealer network in rural areas across India.

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2. LITERATURE REVIEW Biedl, Therese; Grimm, Carsten; Palios, Leonidas; Shewchuk, Jonathan; Verdonschot, Sander in their 2016 paper titled "Realizing farthest-point Voronoi diagrams basically tackled the Inverse Voronoi Problem which found the farthest point in a Vornoi diagram. This helps in understanding the largest radius of influence of a vornoi cell site. Zhi Yu, Can Wang, Jiajun Bu, Mengni ZhangZejun Wu and Chun Chen in their 2015 research paper titled ―Reduce the Shopping Distance: Map Region Search Based on High Order Voronoi Diagram‖ demonstrated the use of High order Voronoi diagram to reduce the time complexity of Region-of-Interests generation. Experimental results showed that their method is both efficient and effective. Krivacsy, Kevin Russell in their 2009 paper "Retail s market area analysis using a transportation network with consideration of population mobility" examined and implemented a methodology and conducted a case study for constructing a retail analysis along a road network rather than using Euclidean distance. The study looked at whether the difference between the two methods is significant. The results of the study was mixed. There does not exist a significant difference between the two approaches in many of the case.

3. METHODOLOGY 3.1. Sample and Sampling Method Tiruvannamalai District is divided into 3 sub districts - Arani, Tiruvannamalai and Cheyyar. Fifteen cities in these districts were chosen using random sampling from list of cities having population of 20,000 and above as per 2011 Census. All the chosen cities latitude and longitude was also recorded. the cities considered for this research are Tiruvannamalai, Arani, Vandavasi, Polur, Tiruvethipuram, Chengam, Chetpet, Kalasapakkam, Gandhinagar- Lakshmipuram, Kalambur, Vettavalam, Pudupalayam, Peranamallur, Adamangalam-Pudur and Kizh-Pennathur,.

3.2. Procedure In order to determine the optimum coverage for FMCG products while minimizing the dealer network in Tiruvannamalai District, Voronoi diagram was computed for the cities under consideration in the research. Voronoi Diagram: Voronoi diagram is named after Georgy Voronoi, and is also called a Dirichlet tessellation, Voronoi tessellation or Voronoi partition. It is a partitioning of a plane into regions based on distance to points in a specific subset of a plane or region. In simple terms, Voronoi diagrams are a geometrical method for determining the area surrounding a certain point that is closer to that point than any other point in the study area. These areas or regions are called Voronoi cells. To find the ideal shop locations in Thiruvannamalai District, the distance between points (latitude, longitude) of major cities in the district were taken and   they are measured using the Euclidean distance formula :          , where a and b are longitude and latitudes respectively.

Mathematically, if K is a set of indices and  is the tuple of non-empty sites in the map, the Voronoi cell region is just the tuple of cell  where             . Pk is site such that the set of all points in x metric space whose distance to Pk is not greater than their distance to the other sites Pj where j is any index different from k. To compute the Voronoi diagram, Open Source R Software and open source

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R package ‗‗deldir', a Delaunay Triangulation and Dirichlet (Voronoi) Tessellation package was used. Map Projection: A Voronoi diagram has to be projected on map of Tiruvannamalai District to put the Voronoi cell in perspective. However there are multitude of map projections. A map projection can be defined as a systematic transformation of the longitudes and latitudes of locations from the surface of a sphere into locations on a plane. Maps cannot be created without map projections. All map projections necessarily distort the surface in some fashion. Based on the cities data of Tiruvannamalai District, Conic Projection is the most suited projection. The conical map projection has Lambert, Albers and Polyconic projection. In a polyconic projection, except the central line, all meridians have curved lines. In Polyconic projection, the distances, direction, shape and areas are true only along the central meridian. Distortion increases away from its central meridian hence this projection is not used commonly. This research depicts the projection as academic exercise only. Lambert Conformal Conic map projection retains conformity. Although distances are reasonably accurate and retained along standard parallels, it is not ‗equal area’ as distortion increases from standard parallels. In this research, further computations and mapping are based on Albers map projection . The Albers projection was used as it is equal-area conic projection. In this map projection, scale and shape are not preserved but distortion is minimal between the standard parallels.

4. DATA ANALYSIS The major cities in Thiruvannamalai District which are considered for Voronoi diagram and their coordinates are tabulated below

Table 1 Coordinates of major cities in Thiruvannamalai District ID City Latitude Longitude 1 Tiruvannamalai 12.24085 79.03423 2 Arani 12.6704 79.26406 3 Vandavasi 12.50525 79.59612 4 Polur 12.5138 79.10814 5 Tiruvethipuram 12.66416 79.51946 6 Chengam 12.30977 78.78463 7 Chetpet 12.46079 79.33159 8 Kalasapakkam 12.43606 79.09906 9 Gandhinagar-Lakshmipuram 12.6925 79.09193 10 Kalambur 12.62752 79.20555 11 Vettavalam 12.10894 79.23462 12 Pudupalayam 12.36359 78.87102 13 Peranamallur 12.56684 79.42467 14 Adamangalam-Pudur 12.47791 78.98346 15 Kizh-Pennathur 12.24064 79.21064 Based on the above data, Voronoi Diagram was computed using R Software and ‗deldir' package. The three conical map projections were computed.

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Polyconic map projection was used to map the coordinates and is depicted in Figure 1 below.

Figure 1 Voronoi Diagram with Cells – Polyconic Projection The above figure depicts the Voronoi diagram with Polyconic Projection. The red dots represents the cities considered in the research and the cells shows the optimum area where dealers can be located to serve the consumers residing near the cities represented by the dot. This projection can be used in limited cases by the FMCG companies. Lambert Conformal Conic map projection too was used to map the coordinates and is depicted in Figure 2 below.

Figure 2 Voronoi Diagram with Cells – Lambert Conformal projection The figure 2 above depicts the Voronoi diagram with Lambert Projection. Here too the red dots represents the cities considered in the research and the cells area depicts the optimum area where dealers can be located to serve the consumers residing around the cities represented by the dot. This projection can be used in limited cases by the FMCG companies.

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The better option and the most reliable projection for small areas is Albers projection. The computation of Albers projection Voronoi diagram is depicted below:

Figure 3 Voronoi map with Cells The above figure depicts the Voronoi diagram with cell using Albers projection. The red dots represents the cities considered in the research and the cells shows the optimum area where shops can be located to serve the consumers residing near the cities represented by the dot. It should be noted that although the projected Voronoi diagrams at first glance looks similar for all types of projections, there are difference in cell perimeters and cell areas in Voronoi cells. All further computations are based on Albers projection Voronoi. The Voronoi diagram is overlaid on Thiruvannamalai District and depicted below for better marketing management planning.

Figure 4 Voronoi diagram overlaid on Thiruvannamalai District For better mapping of Voronoi diagram on real world map for marketing purpose, further computation is done . The computational results include the number of Delaunay triangles emanating from the Voronoi point, 1/3 of the total area of all the Delaunay triangles

http://iaeme.com/Home/journal/IJCIET 883 [email protected] Dr. K. Umadevi emanating from the Voronoi point, the corresponding entry of the area column divided by the sum of the column, the number of sides — within the rectangular window — of the Dirichlet tile surrounding the point , the number of points in which the Dirichlet tile intersects the boundary of the rectangular window, the area of the Dirichlet tile surrounding the point and the corresponding entry of the area column divided by the sum of the column. The result of the analsysis is tabukated below

Table 2 Summary of Voronoi Diagram Sl .No x y n.tri del.area del.wts n.tside nbpt dir.area dir.wts 1 -0.00294 -1.61295 6 8.00E-06 0.090382 6 2 1.80E-05 0.088846 2 0.00138 -1.60615 5 4.00E-06 0.047391 5 2 7.00E-06 0.035629 3 0.007613 -1.60875 3 7.00E-06 0.07932 4 2 2.80E-05 0.136849 4 -0.00154 -1.60863 5 5.00E-06 0.058716 5 0 6.00E-06 0.027952 5 0.006165 -1.60623 3 3.00E-06 0.030952 3 2 1.10E-05 0.052919 6 -0.00763 -1.61184 4 3.00E-06 0.039249 2 2 1.90E-05 0.095586 7 0.002651 -1.60947 8 1.30E-05 0.159907 8 0 1.70E-05 0.08448 8 -0.00171 -1.60986 5 7.00E-06 0.079452 5 0 7.00E-06 0.035089 9 -0.00184 -1.60579 5 4.00E-06 0.04708 4 2 1.10E-05 0.054046 10 0.000284 -1.60683 4 4.00E-06 0.043275 4 0 6.00E-06 0.027385 11 0.000833 -1.61504 4 8.00E-06 0.089282 4 2 1.90E-05 0.09504 12 -0.006 -1.611 3 3.00E-06 0.032746 3 2 1.50E-05 0.071834 13 0.004393 -1.60778 4 5.00E-06 0.057902 4 0 9.00E-06 0.044307 14 -0.00388 -1.60919 6 6.00E-06 0.069628 5 2 1.80E-05 0.089606 15 0.000381 -1.61296 4 6.00E-06 0.074717 4 0 1.20E-05 0.060431 The description of the columns are given below:  x (the x-coordinate of the point)  y (the y-coordinate of the point)  n.tri (the number of Delaunay triangles emanating from the point)  del.area (1/3 of the total area of all the Delaunay triangles emanating from the point)  del.wts (the corresponding entry of the del.area column divided by the sum of this column)  n.tside (the number of sides — within the rectangular window — of the Dirichlet tile surrounding the point)  nbpt (the number of points in which the Dirichlet tile intersects the boundary of the rectangular window)  dir.area (the area of the Dirichlet tile surrounding the point)  dir.wts (the corresponding entry of the dir.area column divided by the sum of the column).

5. FINDINGS AND DISCUSSION The potential Fast Moving Consumer Goods (FMCG) dealers‘ optimal location was found by Voronoi diagram computation. The optimal location as depicted by each of the 15 Voronoi cell will help in optimal placement of dealers for each city under study this research. This will help the FMCG companies cover the cities at minimum cost. For ease of projecting the Voronoi diagram and to help in better real world coordinates mapping, number of Delaunay triangles emanating from the Voronoi point many factors were computed which include 1/3rd of the total area of all the Delaunay triangles emanating from the Voronoi point, the corresponding entry of the area column divided by the sum of the column, the number of

http://iaeme.com/Home/journal/IJCIET 884 [email protected] Optimizing Rural Dealers location – A Voronoi Approach sides of the Dirichlet tile surrounding the point , the number of points in which the Dirichlet tile intersects the boundary of the rectangular window, the area of the Dirichlet tile surrounding the point and the corresponding entry of the area column divided by the sum of the column. The computation can be further improved by using Manhattan Distance instead of Euclidean distance where there is significant difference in road network and cities direct distances. Incorporating factors such as availability of public transport, availability of customers‘ private transport and nearby area congestion index can also help in improving the dealer network optimization. The FMCG distribution personnel can also try optimizing using different map projections instead of ‗Albers‘ to suit their objectives.

6. CONCLUSIONS The research highlights a novel Voronoi approach to optimize the placement of dealers and to concentrate on the areas where brand supply has to be maximized for better reach to consumers. This approach is generalized approach and can be implemented across various industries and across geographical region where the Euclidian distance does not differ significantly from Manhattan Distance. This research is also a reference point for further study of optimization for production and logistics.

REFERENCES

[1] Aurenhammer, Franz (1991). "Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure". ACM Computing Surveys. 23 (3): 345–405.

[2] Biedl, Therese; Grimm, Carsten; Palios, Leonidas; Shewchuk, Jonathan; Verdonschot, Sander (2016). "Realizing farthest-point Voronoi diagrams". Proceedings of the 28th Canadian Conference on Computational Geometry (CCCG 2016).

[3] India Brand Equity Foundation report. (2017, August). Retrieved August 20, 2018, from https://www.ibef.org/industry/indian-rural-market.aspx

[4] Okabe, Atsuyuki; Boots, Barry; Sugihara, Kokichi; Chiu, Sung Nok (2000). Spatial Tessellations – Concepts and Applications of Voronoi Diagrams (2nd ed.). John Wiley. ISBN 0-471-98635-6.

[5] Yu Z., Wang C., Bu J., Zhang M., Wu Z., Chen C. (2015) Reduce the Shopping Distance: Map Region Search Based on High Order Voronoi Diagram. In: Wang H. et al. (eds) Intelligent Computation in Big Data Era. ICYCSEE 2015. Communications in Computer and Information Science, vol 503. Springer, Berlin, Heidelberg

APPENDIX RStudio (Ver 1.1.419) Software Input for Voronoi Calculations fmcg <- read.csv("E://fmcg.csv",stringsAsFactors=FALSE) library(mapproj) *fmcg_projected <- mapproject(fmcg$Longitiude,fmcg$Latitude,"albers", param=c(39,45)) par(mar=c(0,0,0,0)) plot(fmcg_projected, asp=1, type="n", bty="n", xlab="", ylab="", axes=FALSE) points(fmcg_projected, pch=20, cex=0.1, col="red") # Voronoi

http://iaeme.com/Home/journal/IJCIET 885 [email protected] Dr. K. Umadevi library(deldir) vtess <- deldir(fmcg_projected$x,fmcg_projected$y) plot(vtess, wlines="tess", wpoints="none", number=FALSE, add=TRUE, lty=1) vtess <- deldir(fmcg_projected$x, fmcg_projected$y) plot(vtess, wlines="triang", wpoints="none", number=FALSE, add=TRUE, lty=1) vtess$summary * The projection type has to be changed for different projection types Library packages used : mapproj (Ver 1.2.6) and deldir Ver (0.115)

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