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The collection of solid waste from third class communities in most devel-oping countries is by skip containers, however, the location of these facilities has been done arbitrary without any mathematical considerations as to the number of customers the facility is serving, the distance one has to travel to access it and thereby making some of these residences to dump their refuse in gutters, streams and even burn them. In this paper we proposed an improved probabilistic distance, capacity clustering location model which takes into consideration the weight of solid waste from a customer and the capacity of the skip container to locate the skip container to serve a required number of customers based on the capacity constraint of the container. The model was applied on a real world situation and compared with the existing practice in terms of average distance customers had to travel to access the facility. Our results gave a well shorter average travel distance by customers, gave a number of skip containers needed in an area based on their waste generation per capita.

A Facility Location Problem (FLP) is “defined as in the positioning of a set of points (facilities in our case) within a given location space on the basis of the distribution of demand points (users) to be allocated to the facilities” [

The research was conducted in Tafo, one of the nine sub-metropolitan assemblies in Kumasi, the second populated city in Ghana, West Africa. The nine sub-metropolitan areas are of Kumasi are: Suame, Manhyia, Bantama, Kwadaso, Nkyiaeso, Asokwa, Oforikrom, Subin and Tafo. The study area is the smallest of the nine sub metropolitan areas in terms of land area but it is the second highest generator of solid waste after Subin. The study area has a population of 157,226 with eleven communities within its domain. Four of these communities are categorized under class two (areas where we have fairly good road network but difficult for compactor trucks to collect waste) whiles the remaining seven are under class three (areas where we have bad road network and therefore impossible for proper collection of waste by heavy vehicles). The study area shares boundary with Manhyia to the east, Suame to the west and Subin to the south. The area generates about eighty-eight tones of solid waste a day. Our study considered five of the seven third class zones namely Old Tafo, Pankrono Dome, Pankrono West, Tafo Adompom and Ahenbronum constituting about 52% of the area population. The area has the second largest market in the Kumasi metropolis, has a number of government assisted schools from basic to secondary levels and one government hospital.

Location of facilities be it semi or obnoxious has had its fair share in terms of research in literature, that notwithstanding there are still many areas that needs attention to improve the difficulties service customers go through to access a facility. This section takes a look into various research works that has gone into

facility location of obnoxious and semi-obnoxious alike. A semi-obnoxious facility is a facility which is useful to the end users but has some level of unwelcome effect that produces environmental concerns to the same end user. [

[

A location problem is to locate a facility, or facilities, to serve optimally a given set of customers. The customers are given by their coordinates and demands. The coordinates are points in R^{n} (usually n = 2), and the demands are positive numbers q_{i}. Assuming N customers, the data of the problem is a set of points (coordinates) X = { X 1 , X 2 , ⋯ , X N } in R n and a corresponding set of positive weights (demands) { q 1 , q 2 , ⋯ , q N } . Using the Euclidean distance d ( X , Y ) = ‖ X − Y ‖ between two points X, Y in R^{n}. If the customers are served by K facilities for a given K, we denote X K to be the set of customers assigned to the K^{th}-facility. The weighted sum of distance travelled by these customers is ∑ X i ∈ χ K 1 q i ‖ X i − c K ‖ where c_{K} is the location of the K^{th}-facility. The customers X = { X 1 , X 2 , ⋯ , X N } , their demands { q 1 , q 2 , ⋯ , q N } is to be located to facilities with centres { c 1 , c 2 , ⋯ , c K } so as to minimize the weighted sum of distances travelled by all the customers to access the bin. min c 1 , ⋯ , c K min X 1 , ⋯ , X K ∑ k = 1 K ∑ X 1 ∈ χ K 1 q i ‖ X i − c K ‖ and each container has a capacity Q, such that the sum of demands assigned to the K^{th}—facility cannot exceed it, i.e.

∑ X i ∈ χ K q i ≤ Q K (3.1)

Let a data set D ⊂ R p be partitioned into K clusters { c k : k = 1 , 2 , ⋯ , K } , D = ∪ k = 1 K C k and let C_{k} be the centre of the cluster C_{k} with capacity Q_{K}.

With each data point X ∈ D and a cluster centre C k , we denote:

· a distance d k ( X , c k ) by d k (X)

· a probability of membership in c k by P k ( X ) for each X ∈ D and each cluster C_{k},

P k ( X ) d k ( X ) Q k = Constant, (3.2)

depending on X and independent of the cluster k.

Given the cluster centers { C 1 , C 2 , ⋯ , C k } , let X be a data point and let { d k ( X ) : k = 1 , 2 , ⋯ , k } be its distances from the given centers.

From Equation (3.2) using i and k P i ( X ) d i ( X ) Q i = P k ( X ) d k ( X ) Q k

Since ∑ i = 1 K P i ( X ) = 1 , P k ( X ) ∑ i = 1 K ( d k ( X ) Q k d i ( X ) Q i ) = 1

P k ( X ) = ∏ j ≠ k d j ( X ) Q j ∑ i = 1 K ∏ j ≠ i d j ( X ) Q j , k = 1 , 2 , ⋯ , K

For two clusters, i.e. K = 2

P 1 = d 2 ( X ) Q 2 d 1 ( X ) Q 1 + d 2 ( X ) Q 2 , P 2 = d 1 ( X ) Q 1 d 1 ( X ) Q 1 + d 2 ( X ) Q 2 (3.3)

For three clusters i.e. K = 3

P 1 = d 2 ( X ) d 3 ( X ) Q 2 × Q 3 d 1 ( X ) d 2 ( X ) Q 1 × Q 2 + d 1 ( X ) d 3 ( X ) Q 1 × Q 3 + d 2 ( X ) d 3 ( X ) Q 2 × Q 3 (3.4)

Denoting the constant term in Equation (3.2) by D(X), function in X, P k ( X ) d k ( X ) Q k = D (X)

P k ( X ) = D ( X ) d k ( X ) Q k , k = 1 , 2 , 3 , ⋯ , K Since ∑ i = 1 K P k ( X ) = 1

D ( X ) = ∏ j = 1 K d j ( X ) Q j ∑ i = 1 K ∏ j ≠ i d j ( X ) Q j (3.5)

For two clusters, i.e. K = 2,

D ( X ) = d 1 ( X ) d 2 ( X ) Q 1 Q 2 d 1 ( X ) Q 1 + d 2 ( X ) Q 2 (3.6)

For three clusters, i.e. K = 3;

D ( X ) = d 1 ( X ) d 2 ( X ) d 3 ( X ) Q 1 Q 2 Q 3 d 1 ( X ) Q 1 + d 2 ( X ) Q 2 + d 3 ( X ) Q 3 (3.7)

Therefore for the entire data set D is the sum of (3.6) over all points, and is a function of the K cluster centers,

F ( c 1 , c 2 , ⋯ , c k ) = ∑ i = 1 N ∏ k = 1 K d k ( X i , c k ) ∑ t = 1 K ∏ j ≠ t d j ( X i , c j ) (3.8)

Consider a case of two clusters, using X to be the given data point with distances d 1 ( X ) , d 2 ( X ) to the cluster centers and a known cluster sizes Q 1 and Q 2 known. The probabilities in (3) are the optimal solution P 1 , P 2 of the extremal problem.

Minimize { d 1 ( x ) P 1 2 Q 1 + d 2 ( x ) P 2 2 Q 2 } Subjectto P 1 + P 2 = 1 P 1 , P 2 ≥ 0 (3.9)

The Lagrangian of the problem is:

L ( P 1 , P 2 , λ ) = d 1 ( x ) P 1 2 Q 1 + d 2 ( x ) P 2 2 Q 2 − λ ( P 2 + P 1 − 1 ) (3.10)

Setting the partial derivatives with respect to P 1 , P 2 to zero, we have P 1 d 1 ( x ) = P 2 d 2 ( x ) .

Substitute the probabilities in (3.4) in (3.10) we have L ∗ ( P 1 ( x ) , P 2 ( x ) , λ ) = d 1 ( x ) d 2 ( x ) Q 1 Q 2 d 1 ( x ) Q 1 + d 2 ( x ) Q 2 which is the same as the joint distance function obtained in Equation (3.9).

The extremal problem for the entire data set D = { X 1 , X 2 , ⋯ , X N } ⊂ R n

Minimize

∑ i = 1 N ( d 1 ( x i ) p 1 ( x i ) 2 Q 1 + d 2 ( x i ) p 2 ( x i ) 2 Q 2 ) (3.11)

Subjectto: p 1 ( x i ) + p 2 ( x i ) = 1 p 1 ( x i ) , p 2 ( x i ) ≥ 0 , i = 1 , 2 , 3 , ⋯ , N

where p 1 ( x i ) , p 2 ( x i ) are the cluster probabilities at x i and d 1 ( x i ) , d 2 ( x i ) are the corresponding distances. The problem separates into N as in (3.11) and its optimal value is ∑ i = 1 N d 1 ( x i ) d 2 ( x i ) Q 1 Q 2 d 1 ( x i ) Q 1 + d 2 ( x i ) Q 2 the sum of the joint distance function of all points.

Writing Equation (3.11) as a function of the cluster centers C 1 , C 2

f ( C 1 , C 2 ) = ∑ i = 1 N ( d 1 ( x i , C 1 ) p 1 ( x i ) 2 Q 1 + d 2 ( x i , C 2 ) p 2 ( x i ) 2 Q 2 ) .

For Euclidean distance d k ( x , c k ) = ‖ x − c k ‖ , k = 1 , 2 so that

f ( C 1 , C 2 ) = ∑ i = 1 N ( ‖ x i − C 1 ‖ p 1 ( x i ) 2 Q 1 + ‖ x i − C 2 ‖ p 2 ( x i ) 2 Q 2 ) (3.12)

and look for centers C 1 , C 2 that minimizes f and the probabilities p 1 ( x i ) , p 2 ( x i ) are given for i = 1 , 2 , 3 , ⋯ , N and with an assumption that the cluster centers C 1 , C 2 do not coincide with any of the points i = 1 , 2 , 3 , ⋯ , N

From the assumption above, the gradient of Equation (3.12) with respect to c k is

∇ c f ( c 1 , c 2 ) = − ∑ i = 1 N x i − c k ‖ x i − c k ‖ × p k ( x i ) 2 = − ∑ i = 1 N x i − c k d k ( x i , c k ) × p k ( x i ) 2 , k = 1 , 2

Setting the gradient to zero, and grouping like terms, we have

∑ i = 1 N ( p k ( x i ) 2 d k ( x i , c k ) ) x i = ( ∑ i = 1 N p k ( x i ) 2 d k ( x i , c k ) ) c k

C k = ∑ i = 1 N ( u k ( x i ) ∑ j = 1 N u k ( x j ) ) x i (3.13)

where u k = p k ( x i ) 2 d k ( x i , c k ) for k = 1 , 2

Giving the minimizers as

C 1 = ∑ i = 1 N ( u 1 ( x i ) ∑ j = 1 N u 1 ( x j ) ) x i , C 2 = ∑ i = 1 N ( u 2 ( x i ) ∑ j = 1 N u 2 ( x j ) ) x i (3.14)

where

u 1 = p 1 ( x i ) 2 d 1 ( x i , c 1 ) , u 2 = p 2 ( x i ) 2 d 2 ( x i , c 2 ) (3.15)

For a function of K cluster centers f ( C 1 , C 2 , ⋯ , C K ) = ∑ k = 1 K ∑ i = 1 N ( d k ( x i , C k ) p k ( x i ) 2 Q k ) an analog of (3.13) then by the results of Equation (3.2) the minimizers of f is

C k = ∑ i = 1 N ( u k ( x i ) ∑ j = 1 N u k ( x j ) ) x i with u k = p k ( x i ) 2 d k ( x i , c k ) for k = 1 , 2 , ⋯ , K (3.16)

The major steps involved in the formation of the algorithm are described below.

1) Calculation of number of skip containers

The number of skip containers (k) needed in each zone is dependent on demands (q_{i}) of the customers and the capacity of the (Q) of the container; k = ∑ i = 1 N q i Q .

2) Selection of initial cluster centers

· The x-coordinates of all the customers are arranged in ascending order. The highest x-coordinates is selected with its demand recorded, the next highest x-coordinates value is recorded with its demand until the capacity constraints of the container is satisfied.

· The centroid of these points is found by taken the average of the extreme x-values and that of the y-values.

· The process is continued until all the initial centres are assigned.

3) Updating of the cluster centers

i) Compute distances from C_{i} for all X ∈ D .

ii) Update the centre C ′ i using Equation (16)

iii) If the difference between ∑ i = 1 K | C i n − 1 − C i n − 2 | and ∑ i = 1 K | C i n − C i n − 1 | is less than ε stop else return to ii.

4) Data Analysis & Result Using the Proposed Model

The data for the study was obtained jointly from Waste management division of Kumasi Metropolitan Assembly (KMA), Physical Planning unit of KMA, Ghana Statistical service (regional office) and a six (6) month continuous field work. The study area was divided into four zones due to physical boundaries such as streams, huge bridges which cannot be used by vehicles. Zone one 1) has seven hundred and forty-nine (749) houses with and has 1100 bins, zone two 2) with five hundred and sixty-one (561) houses has 828 bins, zone three 3) has five hundred and forty-two (542) houses with 792 bins and zone four had six hundred and twenty-three (623) houses with 789 bins of 140 liters respectively. The model was coded in C++ and ran on Intel core 470 Gb computer.

Cluster Centre from the ZonesA 14 m^{3} skip containers were used for the location centers, with a compaction factor of 0.4, a container can hold 166 of 140 litre bins. With the number of bins in zone one, the number of bins needed is calculated as ∑ i = 1 n q i v j = 1100 166 = 6.62 ≅ 7 .

Coordinates of all the houses obtained by a Geographic Information System were taken and implemented using the probabilistic model. The final cluster (location centers) and their coordinates for each of the seven location centers by the model after 500 iterations is given in the tables below.

The maps mainly indicate the clusters obtained by the maximum insertion with their cluster centers where the waste collected should be sent to. The red color spots among the clusters of points represent the cluster centers for each cluster (which is differentiated by colors).

From

From

Zone 1 | ||||
---|---|---|---|---|

Sub-cluster | Cluster center | Number of customers | Number of bins (140l) | Average distance (m) from cluster center |

1 | (2836.52, 3542.69) | 83 | 165 | 83.42 |

2 | (2806.72, 3212.50) | 99 | 161 | 92.78 |

3 | (2995.70, 3822.21) | 85 | 165 | 79.88 |

4 | (3188.21, 3653.63) | 113 | 166 | 85.71 |

5 | (3566.97, 3458.67) | 136 | 166 | 90.89 |

6 | (3578.23, 3068.46) | 130 | 152 | 96.80 |

7 | (3936.45, 3067.07) | 103 | 125 | 88.08 |

Total | 749 | 1100 | 88.22 |

sub-cluster center. The detail of

From

From

This table contains the summary of sub-cluster centers, number of customers in a sub-cluster and the number of bins used by customers in each sub-cluster in Zone 1. It is associated to

Zone 2 | ||||
---|---|---|---|---|

1 | (3636.30, 3874.95) | 141 | 165 | 108.43 |

2 | (3962.67, 4287.58) | 134 | 166 | 87.43 |

3 | (3589.01, 4240.61) | 99 | 166 | 94.32 |

4 | (3399.84, 4593.46) | 84 | 166 | 79.28 |

5 | (3186.11, 4016.00) | 103 | 165 | 85.43 |

Total | 561 | 828 | 90.97 |

Zone 3 | ||||
---|---|---|---|---|

1 | (3533.30, 4751.13) | 84 | 165 | 112.35 |

2 | (4061.14, 5251.59) | 134 | 139 | 98.56 |

3 | (3680.54, 5224.59) | 98 | 166 | 90.21 |

4 | (3944.08, 4603.17) | 124 | 164 | 97.18 |

5 | (3690.92, 4966.87) | 102 | 158 | 96.33 |

Total | 542 | 792 | 98.92 |

users or customers in the bounded area. Thus, wastes collected in each bounded area in Zone 1 are sent to the sub-cluster center. There are seven sub-clusters in Zone 1 and labeled 1 - 7 in the table and in

This table contains the summary of sub-cluster centers, number of customers in a sub-cluster and the number of bins used by customers in each sub-cluster in Zone 2. It is associated to

Zone 4 | ||||
---|---|---|---|---|

1 | (4313.19, 6400.44) | 102 | 158 | 88.65 |

2 | (4470.51, 5961.12) | 156 | 166 | 95.22 |

3 | (4116.35, 5790.22) | 119 | 135 | 93.76 |

4 | (3792.16, 6061.22) | 106 | 166 | 96.59 |

5 | (3705.55, 5700.23) | 140 | 164 | 92.47 |

Total | 623 | 789 | 93.34 | |

Overall Total | 2475 | 3509 |

users or customers in the bounded area. Thus, wastes collected in each bounded area in Zone 2 are sent to the sub-cluster center. There are five sub-clusters in Zone 2 and labeled 1 - 5 in the table and in

This table contains the summary of sub-cluster centers, number of customers in a sub-cluster and the number of bins used by customers in each sub-cluster in Zone 3. It is associated to

This table contains the summary of sub-cluster centers, number of customers in a sub-cluster and the number of bins used by customers in each sub-cluster in Zone 4. It is associated to

The adoption of the improved probabilistic distance location model to the study area of 2475 houses and about 3509 of 140 litre bins has shown that the area needs twenty-two (22) skip containers, seven in zone one (1), five skip containers each in zones two (2), three (3), and four (4) respectively as compared to the current 15 skip containers in the area. The model has also clearly identified the required number of customers to be assigned to each skip container based on the capacity of each customer and the capacity of the skip container. Our results have given an average walking distance of 92.86 m for a customer to access the skip container as compared to the existing average of 234.71 m to access a skip container.

The authors declare no conflicts of interest regarding the publication of this paper.

Otoo, D., Sebil, C., Kessie, J.A. and Larbi, E. (2019) Probabilistic Distance, Capacity Clustering Location Model of a Semi-Obnoxious Facility, a Real Case of Tafo, Kumasi, Ghana. American Journal of Operations Research, 9, 146-160. https://doi.org/10.4236/ajor.2019.93009