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Location of Farmers Warehouse at Adaklu Traditional Area, Volta Region, Ghana

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Location of Farmers Warehouse at Adaklu Traditional Area, Volta Region, Ghana Hindawi Publishing Corporation Journal of Optimization Volume 2016, Article ID 2459734, 10 pages http://dx.doi.org/10.1155/2016/2459734 Research Article Location of Farmers Warehouse at Adaklu Traditional Area, Volta Region, Ghana Vincent Tulasi,1 Isaac Kwasi Adu,2 and Elikem Kofi Krampa1 1 Department of Mathematics and Statistics, Ho Polytechnic, Ho, Ghana 2Department of Mathematics, Valley View University, Techiman Campus, Techiman, Ghana Correspondence should be addressed to Vincent Tulasi; [email protected] Received 2 December 2015; Revised 29 March 2016; Accepted 14 April 2016 Academic Editor: Manlio Gaudioso Copyright © 2016 Vincent Tulasi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Postharvest loss is one major problem farmers in Adaklu Traditional Area that most Ghanaian farmers face. As a result, many farmers wallow in abject poverty. Warehouses are important facilities that help to reduce postharvest loss. In this research, Beresnev pseudo-Boolean Simple Plant Location Problem (SPLP) model is used to locate a warehouse at Adaklu Traditional Area, Volta Region, Ghana. This model was used because it gives a straightforward computation and produces no iteration as compared with other models. The SPLP is a problem of selecting a site from candidate sites to locate a plant so that customers can be supplied from the plant at a minimum cost. The model is made up of fixed cost and transportation cost. Location index ordering matrix was developed from the transportation cost matrix and we used it with the fixed cost and differences between variable costs to formulate the Beresnev function. Linear term developed from the function which was partial is pegged to obtain a complete solution. Of the 14 notable communities considered, Adaklu Waya is found most suitable for the setting of the warehouse. The total cost involved is GhL 78,180.00. 1. Introduction materials and finished goods [1, 2]. They balance production schedule and demand, consolidate products from several Food crop production is the major occupation of the people different manufacturers and suppliers, and reduce inventory of Adaklu. About 85% of the citizenry depend on these costs [3]. food crops for their well-being. Major food crops like maize, Quite a number of studies dealing with location of cassava, groundnut, and yam are produced. warehouses and similar facilities have been carried out The farmers largely depend on the natural rainfall for the in many different countries [3, 4] using the Simple Plant cultivation of their crops. Food crops are cultivated annually. Location Problem (SPLP) otherwise known as Uncapacitated During bumper harvest, however, food crops are left exposed Facility Location Problem (UFLP) [5–7]. The UFLP is the to bad weather and animals for destruction. Bushfire most of problem in which an arbitrary number of customers can be thetimealsodestroysfoodcropsstoredinfarmbyfarmers. connected to a facility. If each facility has a limit on the Farm produce most of the time is stolen by thieves as there is number of customers it can serve, the problem becomes a no proper storage of the crops. capacitated facility location problem (CFLP) Ling-Yun et al. To prevent food crops going waste, most farmers do sell [1]; Cornuejols et al. [8]; and Tragantalerngsak et al. [9]. their food crops at very cheap prices, thereby making huge On problem of Mathematical Standardization Theory, losses. Beresnev [10] developed Simple Plant Location Problem The objective of this study is to help locate a place where (SPLP) model which was used to locate site for building a warehouse could be built to store farm produce. It is also to plant. The model is made up of fixed cost and transportation help farmers minimize cost of transporting farm produce to cost. He developed location index ordering matrix from the market centers. Warehouses are basically used for storing raw transportation cost matrix and used it with the fixed cost and 2 Journal of Optimization 18000 38000 B 5.0 12.0 C 360000 2.5 A 30500 F D 8.0 2.0 Ho 4.0 1.0 21000 E 4600 6.0 Mafi Kumasi 10.0 L 15000 2.5 9.0 62000 J K 12.5 G 5.0 37000 5.0 10.0 32000 H 7.0 I 4.0 3700 51000 Ziope M 7.0 59500 6.5 N 10150 Figure 1: The figure shows the coded values of the fourteen communities with the distances linking them. The distances are shown onthe arcs of the network and the fixed costs (in GhL) of each community are indicated in the boxes. differences between variable costs to formulate the Beresnev The second data deals with the estimated cost for estab- function. lishing a modern warehouse in each community in the area. Hajiaghayi et al. [11]; Hindi and Pienkosz´ [12]; Fisher Masons, carpenters, and dealers in building materials in the [13]; and Geoffrion and McBride [14] studied the UFLP with communities were engaged in providing estimates for putting general setup cost functions. Their problem was motivated up the warehouse. by an application in placing servers on the Internet. The Adaklu Traditional Area is located at the eastern part of setup cost of a server is a nondecreasing function of the Ho,theVoltaRegionalCapital.Itismadeupof46smalltowns number of clients connected to it. When the number of and villages with Adaklu Abuadi as its traditional capital. The clients increases, the cost per client decreases since they share area is not opened to commercial activities as the communi- some common expenses. Therefore, the setup cost function exhibits concavity due to economies of scale. This problem ties are widely dispersed and the road network is in a very wasformulatedasageneralizedUFLPandaMixedInteger deplorablestate.Foodcropfarmingisthemajoroccupation Problem (MIP) was applied to solve the resulting model. They for the people. Unfortunately, due to its inaccessibility and concluded that an efficient result was obtained. noncommercial nature, food crops obtained are spoilt over time as a result of postharvest losses (a sketch map of the 2. Research Methodology traditional area is shown in Figure 2). Fourteen notable communities (see Appendix C for the The data obtained for this project is from two main sources. communities with their coded values) were selected for the The first data is the distances between various (notable) project as shown in Figure 1. towns in Adaklu Traditional Area. (refer to Appendix B). This data was obtained from the Department of Feeder Roads, Ho. 2.1. Model Formulation. The Beresnev pseudo-Boolean SPLP The distances were measured in kilometers. is formulated as follows [10, 15]: −1 [/] () = ∑ (1 − )+∑ {Δ [0, ] + ∑ Δ [, ]. ∏ } =1 =1 =1 =1 (1) Journal of Optimization 3 Xelekpe Kpetsu Agblefe Mountain Goefe Adaklu Ho Ahunda Kpodzi Tokor KodiabeMountain Tsriefe Ahunda Boso Abuadi Dawanu Mafi Kumasi Dorkpo River Waya Bludokofe Segbale Akoete Gbagbedeve Amuzudeve Anfoe Amuvu Waya Kpetoe Adidome Hlihave River Tordze Kpeleho Kpatove Kpodzi Ablornu Avedzi Torda Sofa Akatsixoe Tsyeme river River Torda Wudzedeke Ziope River Klikpa Kpeve Keyime Akatsi Aflao Towns not part of the traditional area Paramountcy of the traditional area First-class road Bridge A third-class road, most vehicles can ply Culvert Major river Minor rivers Footpath (only bicycles, motorbikes, and tractors can ply) Footpath Figure 2 with the following conditions: of the coefficients of all nonlinear terms containing ∗ for each site index . (a) if ≥0, then there is an optimal solution in which ∗ of is 0, else In (1), is the fixed cost, is the smallest number in each column of the transportation cost matrix, is the differences (b) if <0and + ≤0for each ∈in [/]() in the elements in each column of the transportation cost [/] for the SPLP instance , then there is solution matrix, and is the products of the values that is (1,1,...,1,0) assuming that the site indices are maintained after computation of each column. , ,and + ∑ arranged in decreasing order of ∈ values. formthevariablecostsofthemodel. That is, the optimal solution occurs at the minimum ,else 2.2. Algorithm for Computing Beresnev Pseudo-Boolean SPLP (c) if ∈ R and + ≤0, then there is an optimal ∗ ∗ ∗ =1̸ Step 1. Input edge matrix and the vector of facility setup cost solution in which of is 1, provided that (Appendix D). for some ≠ ,where =(∑:1= Δ[1, ]) + is the coefficient of the linear terms corresponding to 14×14 −1 Step 2. Do all pairs shortest path for the transportation and =∑;∈[1,...,] +1∑=2 Δ[, ] is the sum cost matrix ( ) (Table 1). 4 Journal of Optimization Table 1: All pairs shortest path for edge matrix (see Appendix G) using MATLAB code. 3600 0.0 12.0 17.0 19.5 20.5 22.5 13.0 18.0 25.0 30.0 32.5 33.5 34.0 40.5 1800 12.0 0.0 5.0 7.5 8.5 10.5 25.0 30.0 23.5 18.5 21.0 21.5 22.5 29.0 3800 17.0 5.0 0.0 2.5 3.5 5.5 30.0 25.5 18.5 13.5 16.0 16.5 17.5 24.0 2100 19.5 7.5 2.5 0.0 1.0 3.0 28.0 23.0 16.0 11.0 13.5 14.0 15.0 21.5 4600 20.5 8.5 3.5 1.0 0.0 2.0 27.0 22.0 15.0 10.0 12.5 15.0 14.0 20.5 3050 22.5 10.5 5.5 3.0 2.0 0.0 29.0 24.0 17.0 12.0 14.5 17.0 16.0 22.5 3200 13.0 25.0 30.0 28.0 27.0 29.0 0.0 5.0 12.0 17.0 19.5 32.0 21.0 27.5 5100 18.0 30.0 25.5 23.0 22.0 24.0 5.0 0.0 7.0 12.0 14.5 27.0 16.0 22.5 3700 25.0 23.5 18.5 16.0 15.0 17.0 12.0 7.0 0.0 5.0 7.5 20.0 9.0 15.5 6200 30.0 18.5 13.5 11.0 10.0 12.0 17.0 12.0 5.0 0.0 2.5 15.0 4.0 10.5 3700 32.5 21.0 16.0 13.5 12.5 14.5 19.5 14.5 7.5 2.5 0.0 12.5 6.5 13.0 1500 33.5 21.5 16.5 14.0 15.0 17.0 32.0 27.0 20.0 15.0 12.5 0.0 17.0 23.5 5950 34.0 22.5 17.5 15.0 14.0 16.0 21.0 16.0 9.0 4.0 6.5 17.0 0.0 6.5 1015 40.5 29.0 24.0 21.5 20.5 22.5 27.5 22.5 15.5 10.5 13.0 23.5 6.5 0.0 Step 3.
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