Linear Programming Model for Waste Disposal by B&R Oil and Gas Company

The article discusses the linear programming model used by B&R Oil and Gas Company to minimize the number of tanks used for waste disposal while maximizing profit. It provides a step-by-step guide on how to solve the model using sampling analysis technique. The model aims at obtaining the minimum tanks that can be used to dispose of waste from the sites of the 20 counties. The article concludes that to obtain optimization that is minimizing the cost and maximizing the profit, we will require to use 6 tanks to dispose the wastes from the sites for the 20 counties.

Running head: DATA MINING 1
Topic
Name of Student
Institution Affiliation

DATA MINING 2
Introduction
The aim of the assignment is to model the data such that we come up with the minimum
number of tanks that can cover all the 20 counties. The model aims at maximizing the profit
while minimizing the cost. B&R Oil and Gas company works at maintaining the waste disposal
from the sites in 20 counties. The 20 counties are Ashtabula, Lake, Cuyahoga, Lorain, Huron,
Richland, Ashland, Wayne, Medina, Summit, Stark, Geauga, Portage, Columbiana,
Mahoning, Trumbull, Knox, Holmes, Tuscarawas, , and Carroll. The below map has all
counties mentioned.
Now, the tanks used for disposal can serve one or two counties either from residing and
adjacent counties. The data for the above information is provided in the excel sheet. The data
are set such that counties that share tanks are labeled 1where their cells intersect on the table
and the one which doesn't share the disposal tanks are labeled 0 in the cell intersection. The
data forms a square 11 by 11 matrix. We will model this matrix and come up with valuable
insights and method to answer our question which aims at obtaining the minimum tanks that
can be used to dispose of waste from the sites of the 20 counties.

DATA MINING 3
We will commence by identifying each objective and decision variables. These
variables are of utter-most importance when conducting optimization in the linear
programming model. To start with, let's discuss decision variables and how it applies in our
data. Decision variables measure the number of resources used and the level of the activities
being conducted (Hwang, 2012). In our case,, we have 11 columns and 11 rows. To determine
the decision variable, we will multiply the number of rows and columns. Therefore, our
decision variables are 20 x 20 = 400. Decision maker depends on the decision variables to
make a decision. They are the mathematical unknowns which are used to model our data. Our
first part of modeling is determining the number of details that are required during the
modeling.
Our next step is to define the objective variables after defining the decision variables.
We will have to assign the 11 variables using a simpler name say X1, X2 etc. (Amin, 2013)
Ashtabula as X1, Lake as X2, Cuyahoga as X3, Lorain as X4, Huron as X5, Richland as X6,
Ashland as X7, Wayne as X8, Medina as X9, Summit as X10, Stark as X11, Geauga as X12,
Portage as X13, Columbiana as X14, Mahoning as X15, Trumbull as X16, Knox as X17,
Holmes as X18, Tuscarawas as X19 and Carroll as X20. We fill form such kind of equations:
A = X1 + X2 + X4 + X12 + X16
B = X1 + X2 + X3 + X12
C = X2 + X3 + X4 + X9 + X10 + X12 + X13
D = X3 + X4 + X5 + X7 + X9
E = X4 + X5 + X6 + X7
F = X5 + X6 + X7 + X12
G = X4 + X5 + X6 + X7 + X8 + X9 + X17 + X18

DATA MINING 4
H = X7 + X8 + X9 + X10 + X11 + X18
I = X3 + X4 + X7 + X8 + X9 + X10
J = X3 + X8 + X9 + X10 + X11 + X12 + X13
K = X8 + X10 + X11 + X13 + X14 + X15 + X18 + X19 + X20
L = X1 + X2 + X3 + X10 + X12 + X13 + X16
M = X3 + X10 + X11 + X12 + X13 + X15 + X16
N = X11 + X14 + X15 + X20
O = X11 + X13 + X14 + X15 + X16
P = X1 + X12 + X13 + X15 + X20
Q = X6 + X7 + X17 + X18
R = X7 + X8 + X11 + X17 + X18 + X19
S = X11 + X18 + X19 + X20
T = X11 + X14 + X19 + X20
We will form 20 similar equations of this type to create a model of objective variables.
We can form an objective function from the above equations.
Z(x) = X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 + X12 + X13 + X14 +
X15 + X16 + X17 + X18 + X19 + X20
We will conduct an analysis to obtain the minimum values for the optimization (Deb, 2014).
There are many analysis tools but we will solve this task using the sampling analysis
technique. Sampling is so useful when handling a big data. Sampling reveals small samples
form the big data set.

DATA MINING 5
Here is the procedure for solving the model using sampling:
 Go to Data
 Click on data analysis
 Choose the sampling options
 On the input range, select the whole matrix
 On sample size, we choose 20
 Select any output options
 Then click ok
We will obtain the sampling variables, then we sum them up as shown on the excel sheet. We
will obtain 6. This is the value of the minimum numbers of tanks that are required to serve all
the 20 counties.
Conclusion
Linear programming applies mostly in companies and industries. The aim of applying linear
programming model is to minimize the cost of production and maximize the profit. We have
applied the same knowledge of this model to minimize the number of tanks being used to
dispose waste by B&R Oil and Gas company. We will conclude that to obtain optimization
that is minimizing the cost and maximizing the profit we will require to use 6 tanks to dispose
the wastes from the sites for the 20 counties.

DATA MINING 6
Reference
Hwang, C. L., & Masud, A. S. M. (2012). Multiple objective decision making—methods and
applications: a state-of-the-art survey (Vol. 164). Springer Science & Business Media.
Deb, K. (2014). Multi-objective optimization. In Search methodologies (pp. 403-449). Springer,
Boston, MA.
Amin, S. H., & Zhang, G. (2013). A multi-objective facility location model for closed-loop supply chain
network under uncertain demand and return. Applied Mathematical Modelling, 37(6), 4165-4176.