# Iterative methodology on locating a cement plant

**Part of the following topical collections:**

## Abstract

In this study, a cement plant location was determined by considering essential parameters such as the locations of resources and their importance in the manufacturing process. A crucial mathematical problem, named Weber problem, reinforced the decision of the method of allocating the factory. Additionally, not only the limitations of the cement production but also the importance weights of goods used in the manufacturing were taken into account in the iterative methodology in order to answer the engineering question via the mathematical problem. As a result, by optimizing the case through the iterations introduced in the paper, the location of the cement plant was set. Hence several losses such as extra travel distances and time wasting in transportation were minimized.

## Keywords

Cement plant Iterative technique Optimization Transportation cost Weber problem## 1 Introduction

For many decades, mathematical techniques, models, and methodologies have been essential in order to provide the understanding of countless engineering processes and implementations. Using mathematical models in many academic researches, it was demonstrated that numerous infrastructure problems can be solved, for instance, locating facilities or infrastructure related to a specific industry such as cement production.

The location of a facility is the place where the business will be established and will continue to work throughout service life. In determining the business strategy, the company determines what goods or services will be offered and in which market it will compete. After making the product and process design decisions, the next step is the choice of location. Location decision affects costs of supplying inputs, production, and distribution, thus influencing productivity. The location of the establishment is in question not only in new establishments, but also in decisions expanding the facility, producing new products, or adding a new section to the plant.

Therefore, the scope of this work is to make right, healthy, and sustainable location determination for a selected specific industry, namely cement manufacturing facility by using a method mentioned later on, which can be applied to any kind of facility construction by taking the parameters affecting the cost into account. Cement production industry is vital for the economic development of a country and cannot be totally removed. Thus eliminating the damaging effects of such facilities can be a more preferable way in order to determine the most convenient location of these plants. This study will offer a route for determination of location for similar industrial activities rather than only cement production.

The remainder of this paper is organized as follows. In the next section (Sect. 2), a literature review is provided. Section 3 describes the methodology, including the models applied for the implementation of our proposed approach. In Sect. 4, we present a typical case study, followed by the main results in Sect. 5 with discussions. Finally, conclusions are delivered in Sect. 6.

## 2 Literature review

Using mathematical techniques and models in many academic researches for many decades, it was shown that several infrastructure and transportation problems can be solved when locating facilities related to a particular industry. For instance, research on the problems of optimizing transportation costs, network analysis, and regulating the road capacities on a transportation network can be described via inequalities [1, 2, 3]. After employing an optimization technique, solutions with an adequate performance of traffic, location analysis, including transportation, can be found. Another specific area could be determining a transportation activity location such as that of a cement plant, which is one of the most common concerns of local or regional authorities. For such a purpose, many researches have been conducted in the literature. Researchers [4] studied the problem of locating undesirable facilities such as landfills, creating potential effects of soil anomalies. They utilized different maps such as those of land use, soil, rivers, road networks, etc., in order to overlay them in a geographical information system (GIS). There are musts and needs in order to locate facilities. For example, a facility could be located a few kilometers away from cultivable lands and within a specific distance from an asphalt road. In addition, specific facilities such as cement plants, which may cause undesirable effects to the environment, must keep a certain distance to special places such as residential areas. Therefore, due to the negative impact of cement plants on the environment, alternative materials instead of cement have been sought by some investigators in their research [5, 6]. Alternative areas can be revealed using the tools of computer software including mathematical models.

Another study [7], in order to situate an alumina–cement plant, suggested a multicriteria estimation method that was to compare a couple of alternative sites analyzing the important factors of accessibility to raw materials in terms of distance, water, and power supply, as well as land concerns. The model proposed a location of a plant by integrating typical inputs of a facility such as transport, water, power, fuel consumption, and land.

The authors of [8] proposed an approach of type-2 fuzzy sets and compared their method with a few other fuzzy approaches in order to solve a single-facility location problem. Additionally, several criteria of fixed and variable costs such as the costs of land, transportation, raw material, energy, environment, and insurance were considered in their study.

A couple of recent studies [9, 10] utilized analytical hierarchy processes and GIS techniques to develop decision making processes. An example can be for a site selection of an industry such as cement production. While the candidate areas were weighted by the criteria related to the industry, the characteristics of the areas were evaluated according to their sustainability concerns.

Recently, some other studies [11, 12, 13, 14] proposed methodologies of optimization models including decision making goals, as well as multi-objective decision making problems having conflicting objectives. While a couple of them looked for the optimization of properties of materials through models, the rest sought for the set of Pareto solutions rather than optimal solutions to determine a facility location.

## 3 Methodology

This study focuses on finding and determining the most suitable location for an intervention of a facility. Hence, an approach of locating a single point facility location is constructed; thus, a couple of techniques and methodologies completed in the related literature can be implemented to solve the case in this study. For example, an approach to solve the central median location problem known in the literature can be applied for determining the location [15, 16]. However, when placing a cement plant, we must consider more parameters. Therefore, scientifically a more advanced technique should be constructed for competent results.

### 3.1 The technique of Weber problem

The description in (1) uses \(\bar{x}=(x,y)\), that is, the optimum location coordinate which is wanted and preferred, and \(\bar{x}_{j}=(x _{j},y_{j})\) which represents the particular demand point location of *j* where \(j=1,\dots ,n\). Therefore, by subtracting *x̄* from the location information of \(\bar{x}_{j}\), the optimization process can detect the distance measure between two locations. Further, the weight of a specific demand point is presented by \(w_{j}\) in the description of (1). Put it differently, \(w_{j}\) symbolizes the significance or rank of the particular location point of \(\bar{x}_{j}\) in comparison to the other location points. In addition, all of the location points (i.e., *x̄* and \(\bar{x}_{j}\)) in (1) are the points at the surface of \(E^{2}\), that is, the Euclidean plane. It is the two-dimensional space of real numbers (i.e., commonly denoted by \(\mathbb{R}^{2}\)). Lastly, the minimization of the products of the certain distances and the related weights is finished. Consequently, by optimizing (1) we select the most appropriate position or location of *x̄*.

### 3.2 The application of Weber problem to Weiszfeld algorithm

Because the weights of demand points will be considered in this study, the most proper approach to answer our location question is to solve a particular WP. Weiszfeld has proven in his investigation that an iterative procedure can be implemented in order to solve WP [22, 23, 24]. This iterative technique, called Weiszfeld method or algorithm (WM), is accepted as a very powerful process in practice to identify the optimal facility location of WP [15, 25]. On the market, there are computer programs or software which are used for running the methodology of WM and achieving solutions through WM. Because the computer program LINGO can achieve the solution of WP by applying WM [25, 26], it is used as a tool for analyzing the location, and thus optimizing the location. The reason of solving the optimization problem in this study by using the computer software LINGO is that it is able to analyze and detect the optimum location of a solution point using the weighted Euclidean distances of demand points in a predetermined surface of a topography [27]. The program continues the iterations until all the demands in the analysis are satisfied in terms of their weights [28]. As soon as the location of solution does not change when running further iterations, the iterations are stopped and the final decision of the optimization is reported by the program.

*n*is the number of demand points; \(i=1,2,\dots ,n\) indexes the demand points from 1 to

*n*; \((x_{i} , y_{i})\) are the spatial coordinates of a demand point

*i*, which runs from 1 to

*n*; \(a_{i}\) is the total demand of point

*i*, again from 1 to

*n*; \(D_{i}\) is the Euclidean distance to \((X_{f} , Y_{f})\):

*n*is the number of resources around the facility. Also the position of the location must be subject to the inequality \((X_{f} , Y_{f}) \geq 0\). Basically, as can be seen in (5), the goal is to minimize the cost which is the distance for all resource points within the scope of the research [15]. The following section describes the scope of the input data, as well as the use of data including the conversions and the processes required.

## 4 A case study

*good*-1 has a total of 63 resources at different locations with their individual weights (Fig. 1),

*good*-2 has 58 resources at various locations with their distinguishing weights. Moreover,

*good*-3 and

*good*-4 are to be collected from 20 and 14 resource points, respectively, having particular weights as well. At last,

*good*-5 is another must-have ingredient for the production at the factory, which is found at 7 locations with their peculiar weights.

*good*-1 has the weight of 0.090 for the manufacturing process,

*good*-2 and

*good*-3 keep the weights of 0.200 and 0.310, respectively. Additionally,

*good*-4 is required to have a weight of 0.165, while

*good*-5 holds the weight of 0.235.

The weights of all five goods

Data | Number of resources | Weight |
---|---|---|

| 63 | 0.090 |

| 58 | 0.200 |

| 20 | 0.310 |

| 14 | 0.165 |

| 7 | 0.235 |

For this computation, *x* (the rectangular coordinate on the *x*-axis), *y* (the rectangular coordinate on the *y*-axis) can be found by using *R* (the radius of the sphere), \(\lambda ^{\circ }\) (the latitude in degrees), \(\lambda _{0}^{\circ }\) (the central latitude in degrees), and \(\phi ^{\circ }\) (the longitude in degrees). With the help of the conversion, WM can be applied for the location analysis on a rectangular projection.

## 5 Main results and discussion

For assessing the results of the optimization, the data were processed in LINGO 16.0 x64 [26]. The data as input and the objective function (2) were entered into the software. Then, the solver was run by a computer with an Intel®Core^{TM} i7-2640M CPU @ 2.80 GHz.

*good*-1 and the inequality \((X_{f} , Y_{f}) \geq 0\), the coordinates of \((X_{f_{1}}, Y_{f_{1}})\), which was the optimum solution for

*good*-1 resources, were determined as (0.6144099, 0.5994187). For the Mercator projection, the spatial result for the resources of good-1 corresponds to \(35^{\circ }12'11.1''\) N and \(32^{\circ }27'17.7''\) E, respectively. Likewise, the coordinates of \((X_{f_{2}}, Y_{f_{2}})\), \((X_{f_{3}}, Y_{f_{3}})\), \((X_{f_{4}}, Y _{f_{4}})\), and \((X_{f_{5}}, Y_{f_{5}})\) also satisfy the inequality \((X_{f} , Y_{f}) \geq 0\) (Table 2). These were the solutions of the optimal locations for

*good*-2,

*good*-3,

*good*-4, and

*good*-5 resources, respectively, which were assessed by the solver. As can be seen in Table 2, they were found on a rectangular projection. Corresponding coordinates were also computed with the help of the Mercator projection. Additionally, each of them was the local optimal solution of each cluster, and no infeasibility was reported by the solver. Because each cluster was formed by a convex problem, the local minimum was actually the global minimum in our case. Thus, the results were verified.

The optimal locations of five goods

Data | Number of resources | Location | Latitude | Longitude |
---|---|---|---|---|

| 63 | \((X_{f_{1}}, Y_{f_{1}})\) | 35 | 32 |

| 58 | \((X_{f_{2}}, Y_{f_{2}})\) | 35 | 34 |

| 20 | \((X_{f_{3}}, Y_{f_{3}})\) | 36 | 31 |

| 14 | \((X_{f_{4}}, Y_{f_{4}})\) | 35 | 30 |

| 7 | \((X_{f_{5}}, Y_{f_{5}})\) | 38 | 36 |

The optimal locations of every resource were computed by the iterative methodology of WM (Table 2). An output of location represents the best location of the facility for a particular resource by considering the resource points. For example, the resource of good-2 has a total of 58 resource points, and the optimal location for placing a facility for this particular one is at \(35^{\circ }11'49.8''\) N and \(34^{\circ }27'07.0''\) E. Also, the best location of placing a facility for good-5 is determined at \(39^{\circ }10'31.3''\) N and \(36^{\circ }54'10.3''\) E among those particular 7 points of resources. Consequently, as can be seen in Table 2, all the location outputs differ from each other. The main reason of possessing several optimal locations at the beginning is that every good has a particular optimal location because the resources of all of the goods are at different spatial positions (Fig. 1). Likewise, many researchers [8, 25, 30] demonstrate similar approaches for many other facility placement problems, rather than for a cement production plant, related to the resources as in our study. Therefore, the location outputs in Table 2 do vary.

Furthermore, as mentioned previously, every good has a specific weight of importance to be included in the cement production process. Therefore, the optimal location of all goods is required for locating the cement production facility. When the program was run using the inputs of the weights of goods (Table 1) and the location information recently acquired (Table 2), the coordinates of \((X_{f}, Y_{f})\), which was the final optimal solution for all goods, were determined as (0.6287260, 0.5890077). Related to Mercator projection, the spatial result of \((X_{f}, Y_{f})\) for all goods corresponded to \(36^{\circ }01'24.1''\) N and \(31^{\circ }57'00.6''\) E, meeting the inequality \((X_{f} , Y_{f}) \geq 0\). As discussed before, the result was verified, because the data cluster created a convex problem, delivering a local minimum which was actually a global minimum in a convex problem. Thus, the convexity in our study simplifies the search for a global optimum location, as also indicated by other researchers [25, 31, 32].

## 6 Conclusions

Several infrastructure problems, including the locations of infrastructure, buildings, or even facilities, can be solved by using mathematical models and algorithms. Using mathematical techniques and models in several academic studies completed thus far, researchers demonstrated that many infrastructure problems could be solved by applying mathematical techniques. For instance, particular industries such as that of cement production can be looking for a solution in order to select a cement plant location, which is one of the most common concerns of entrepreneurs or consultants at the level of local or regional authorities. In this study, a cement production facility location was determined using a common iterative methodology of WM by considering some parameters of the weights and the transportation costs, such as distances. For this case, a mathematical problem of WP supported the model of the application in this investigation. Therefore, according to the mathematical evaluation in this paper, the location of the cement production facility is proposed with the help of the spatial results. This study can contribute to not only regional planning, but also to state planning, which is vital for the economic development of a country. As an outcome, important losses for many industries such as time wasting, extra travel distances, and unnecessary expenditures due to distances traveled can be minimized by applying the technique used in this study.

## Notes

### Acknowledgements

This article is dedicated to Professor Gradimir V. Milovanović on the occasion of his 70th anniversary.

### Availability of data and materials

The datasets used and/or analyzed during the current study are derived synthetically. They are available from the corresponding author upon reasonable request.

### Authors’ contributions

Both authors participated in every stage of the research, and both authors read and approved the final manuscript.

### Funding

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

## References

- 1.Noor, M.A., Noor, K.I.: Some algorithms for equilibrium problems on Hadamard manifolds. J. Inequal. Appl.
**2012**, 230 (2012). https://doi.org/10.1186/1029-242X-2012-230 MathSciNetzbMATHGoogle Scholar - 2.Msigwa, R.E., Lu, Y., Ge, Y., Zhang, L.: A smoothing approach for solving transportation problem with road toll pricing and capacity expansions. J. Inequal. Appl.
**2015**, 237 (2015). https://doi.org/10.1186/s13660-015-0759-4 MathSciNetzbMATHGoogle Scholar - 3.Matisziw, T.C., Demir, E.: Inferring network paths from point observations. Int. J. Geogr. Inf. Sci.
**26**(10), 1979–1996 (2012) Google Scholar - 4.Fatta, D., Saravanos, P., Loizidou, M.: Industrial waste facility site selection using geographical information system techniques. Int. J. Environ. Stud.
**56**(1), 1–14 (1998) Google Scholar - 5.Kockal, N.U., Beycan, O., Gulmez, N.: Physical and mechanical properties of silica fume and calcium hydroxide based geopolymers. Acta Phys. Pol. A
**131**, 530–533 (2017) Google Scholar - 6.Kockal, N.U., Beycan, O., Gulmez, N.: Effect of binder type and content on physical and mechanical properties of geopolymers. Sadhana. Acad. Proc. Eng. Sci.
**43**, 1–9 (2018) Google Scholar - 7.Ataei, M.: Selection of alumina–cement plant location with application of multicriteria estimation method. J. Min. Sci.
**41**(2), 185–194 (2005) Google Scholar - 8.Cebi, F., Otay, I.: Multi-criteria and multi-stage facility location selection under interval type-2 fuzzy environment: a case study for a cement factory. Int. J. Comput. Intell. Syst.
**8**(2), 330–344 (2015) Google Scholar - 9.Mirhadi Fard, M., Kibert, C.J., Chini, A.R.: Decision-making for sustainable location of a cement plant in the state of Florida. Int. J. Sustain. Eng.
**9**(2), 76–92 (2015) Google Scholar - 10.Matisziw, T.C., Demir, E.: Measuring spatial correspondence among network paths. Geogr. Anal.
**48**(1), 3–17 (2016) Google Scholar - 11.Kockal, N.U., Ozturan, T.: Optimization of properties of fly ash aggregates for high-strength lightweight concrete production. Mater. Des.
**32**(6), 3586–3593 (2011) Google Scholar - 12.Kockal, N.U.: Optimizing production parameters of ceramic tiles incorporating fly ash using response surface methodology. Ceram. Int.
**41**(10), 14529–14536 (2015) Google Scholar - 13.Anvari, S., Turkay, M.: The facility location problem from the perspective of triple bottom line accounting of sustainability. Int. J. Prod. Res.
**55**(21), 6266–6287 (2017) Google Scholar - 14.Shen, P.-P., Lu, T.: Regional division and reduction algorithm for minimizing the sum of linear fractional functions. J. Inequal. Appl.
**2018**, 63 (2018). https://doi.org/10.1186/s13660-018-1651-9 MathSciNetGoogle Scholar - 15.Demir, E.: Approach for siting a support facility for transporting supplies in emergency cases in the republic of Bulgaria. MATEC Web Conf.
**234**, 06001 (2018). https://doi.org/10.1051/matecconf/201823406001 Google Scholar - 16.Demir, E.: Havalimanlarında kalkış öncesi, acil durumlarda, yardım alınabilecek en uygun lokasyonun Weber problemine uyarlanarak belirlenmesi. Türk Coğrafya Dergisi
**70**, 81–85 (2018) Google Scholar - 17.Weber, A.: Alfred Weber’s Theory of the Location of Industries. University of Chicago Press, Chicago (1929) Google Scholar
- 18.Tellier, L.N.: The Weber problem: solution and interpolation. Geogr. Anal.
**4**(3), 215–233 (1972) Google Scholar - 19.Cooper, L., Katz, I.N.: The Weber problem revisited. Comput. Math. Appl.
**7**(3), 225–234 (1981) MathSciNetzbMATHGoogle Scholar - 20.Brimberg, J.: The Fermat–Weber location problem revisited. Math. Program.
**71**(1), 71–76 (1995) MathSciNetzbMATHGoogle Scholar - 21.Venceslau, H.M., Karam Venceslau, M.B., Xavier, A.E., Maculan, N.: A geometric perspective of the Weiszfeld algorithm for solving the Fermat–Weber problem. RAIRO Oper. Res.
**50**(1), 157–173 (2016) MathSciNetzbMATHGoogle Scholar - 22.Weiszfeld, E.: Sur le point pour lequal la somme des distances de n pointes donnes est minimum. Tohoku Math. J.
**43**, 335–386 (1937) zbMATHGoogle Scholar - 23.Eckhardt, U.: Weber’s problem and Weiszfeld’s algorithm in general spaces. Math. Program.
**18**(1), 186–196 (1980) MathSciNetzbMATHGoogle Scholar - 24.Chen, R.: Location problems with costs being sums of powers of Euclidean distances. Comput. Oper. Res.
**11**(3), 285–294 (1984) zbMATHGoogle Scholar - 25.Church, R., Murray, A.: Business Site Selection, Location Analysis, and GIS. Wiley, Hoboken (2009) Google Scholar
- 26.LINDO Systems Inc.: LINGO and optimization modeling (2019) http://www.lindo.com/index.php/products/lingo-and-optimization-modeling. Accessed 16 Jan 2019
- 27.Aftab, K., Hartley, R., Trumpf, J.: Generalized Weiszfeld algorithms for \(L_{q}\) optimization. IEEE Trans. Pattern Anal. Mach. Intell.
**37**(4), 728–745 (2015) zbMATHGoogle Scholar - 28.Uster, H., Love, R.F.: The convergence of the Weiszfeld algorithm. Comput. Math. Appl.
**40**(4), 443–451 (2000) MathSciNetzbMATHGoogle Scholar - 29.Snyder, J.: Map projections—a working manual. U.S. Geological Survey Professional Paper 1395 (1987) Google Scholar
- 30.Murray, A.T., Grubesic, T.H.: Locational planning of health care facilities. In: Kanaroglou, P., Delmelle, E., Paez, A. (eds.) Spatial Analysis in Health Geography, pp. 243–259. Ashgate, New York (2016) Google Scholar
- 31.Murat, A., Laporte, G., Verter, V.: A global shooting algorithm for the facility location and capacity acquisition problem on a line with dense demand. Comput. Oper. Res.
**71**, 1–15 (2016) MathSciNetzbMATHGoogle Scholar - 32.Berman, O., Sanajian, N., Wang, J.: Location choice and risk attitude of a decision maker. Omega
**66**, 170–181 (2017) Google Scholar

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