like this?

edit:
- your table has to wide column headers. to make them shorter i suggest to broke them into two lines by help of macro
makecell
from the package makecell
- for better formatting of numbers (aligning at decimal point) i suggest to use the column type
S
from package siunitx
- the packages
url
and hyperref
should be the loaded the last in preamble
- in the second example i show an alternative in which is:
- reorganized your first column in three columns with headers "case", "$\alpha$" and "$\beta$"
- for part of columns headers are in
multirow
cells
- removed vertical lines
- instead of
\hline
s used rules from ˙the package booktabs
\documentclass[preprint]{elsarticle}
%\documentclass[preprint, authoryear]{elsarticle}
%\usepackage{lineno}
%\modulolinenumbers[5]
\journal{Journal of Transportation Research, Part E}
\usepackage{graphicx}
\usepackage{natbib}
\usepackage{textcomp}
\usepackage{amstext}
\usepackage{amssymb}
\usepackage{amsmath}
%% \numberwithin{equation}{subsection}
\usepackage{multicol}
\setlength{\columnsep}{1cm}
\usepackage{tikz}
\usepackage{lettrine}
\allowdisplaybreaks
\usepackage[export]{adjustbox}
\usepackage{ragged2e}
\usepackage{subfig}
\usepackage{float}
\usepackage{booktabs}
\usepackage{makecell} % new
\usepackage{tabularx}
%\usepackage{array} % loaded by makecell and/or by tabularx
\usepackage{siunitx}
\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}}
\usepackage{url}
\urlstyle{same}
\usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref}
%% `Elsevier LaTeX' style
\bibliographystyle{elsarticle-harv}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\section{Introduction}\label{sec: intro}
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\begin{table}[htb]
\caption{Computational Results: Case I, $P_{1}$}
\label{tab: Case I, P1}
\centering
\begin{tabular}{|c| S[table-format=2.2]|
*{3}{S[table-format=1.2]|}
S[table-format=2.0]|}
\hline
Weight Legend
& {\makecell{AVG.\\ Time $(s)$}}
& {\makecell{Cost\\ ($\$1E6$)}}
& \multicolumn{2}{c|}{\makecell{Risk\\ (persons $1E6$)}}
& {\makecell{No. of\\ Trains}} \\ \hline
Cases & & & {Propane} & {Butane} & \\ \hline
A: $\alpha = 1.00$ , $\beta = 0.00$ & 2.06 & 0.96 & 1.80 & 2.16 & 19 \\ \hline
B: $\alpha = 0.90$ , $\beta = 0.10$ & 39.8 & 0.97 & 1.75 & 2.13 & 19 \\ \hline
C: $\alpha = 0.80$ , $\beta = 0.20$ & 53.06 & 0.97 & 1.75 & 2.13 & 19 \\ \hline
D: $\alpha = 0.70$ , $\beta = 0.30$ & 44.72 & 1.01 & 1.75 & 1.93 & 23 \\ \hline
E: $\alpha = 0.60$ , $\beta = 0.40$ & 40.92 & 1.12 & 1.75 & 1.93 & 23 \\ \hline
F: $\alpha = 0.50$ , $\beta = 0.50$ & 39.8 & 1.12 & 1.67 & 1.93 & 23 \\ \hline
G: $\alpha = 0.40$ , $\beta = 0.60$ & 36.22 & 1.12 & 1.67 & 1.93 & 23 \\ \hline
H: $\alpha = 0.30$ , $\beta = 0.70$ & 41.21 & 1.12 & 1.67 & 1.93 & 23 \\ \hline
I: $\alpha = 0.20$ , $\beta = 0.80$ & 34.43 & 1.20 & 1.60 & 1.93 & 26 \\ \hline
J: $\alpha = 0.10$ , $\beta = 0.90$ & 32.87 & 1.25 & 1.60 & 1.91 & 28 \\ \hline
K: $\alpha = 0.00$ , $\beta = 1.00$ & 17.96 & 1.62 & 1.60 & 1.91 & 41 \\ \hline
\end{tabular}
\end{table}
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\end{document}
addendum:
or like this (as i prefer):

\documentclass[preprint]{elsarticle}
%\documentclass[preprint, authoryear]{elsarticle}
%\usepackage{lineno}
%\modulolinenumbers[5]
\journal{Journal of Transportation Research, Part E}
\usepackage{graphicx}
\usepackage{natbib}
\usepackage{textcomp}
\usepackage{amstext}
\usepackage{amssymb}
\usepackage{amsmath}
%% \numberwithin{equation}{subsection}
\usepackage{multicol}
\setlength{\columnsep}{1cm}
\usepackage{tikz}
\usepackage{lettrine}
\allowdisplaybreaks
\usepackage[export]{adjustbox}
\usepackage{ragged2e}
\usepackage{subfig}
\usepackage{float}
\usepackage{booktabs}
\usepackage{makecell, multirow}
\usepackage{tabularx}
%\usepackage{array}
\usepackage{siunitx}
\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}}
\usepackage{url}
\urlstyle{same}
\usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref}
%% `Elsevier LaTeX' style
\bibliographystyle{elsarticle-harv}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\section{Introduction}\label{sec: intro}
Text text Text textText textText textText textText textText text
Text textText textText textText textText text Text textText textText textText textText textText textText textText textText textText textText textText textText textText text
\begin{table}[htb]
\caption{Computational Results: Case I, $P_{1}$}
\label{tab: Case I, P1}
\centering
\begin{tabular}{c
*{2}{S[table-format=1.2]}
S[table-format=2.2]
*{3}{S[table-format=1.2]}
S[table-format=2.0]
}
\toprule
\multicolumn{3}{c}{Weight Legend}
& {\multirow{2}{*}{\makecell{AVG.\\ Time $(s)$}}}
& {\multirow{2}{*}{\makecell{Cost\\ ($\$1E6$)}}}
& \multicolumn{2}{c}{\makecell{Risk (persons $1E6$)}}
& {\multirow{2}{*}{\makecell{No. of\\ Trains}}} \\
\cmidrule{1-3}
\cmidrule{6-7}
Case & {$\alpha$} & {$\beta$}& & & {Propane} & {Butane} & \\
\midrule
A & 1.00 & 0.00 & 2.06 & 0.96 & 1.80 & 2.16 & 19 \\
B & 0.90 & 0.10 & 39.8 & 0.97 & 1.75 & 2.13 & 19 \\
C & 0.80 & 0.20 & 53.06 & 0.97 & 1.75 & 2.13 & 19 \\
D & 0.70 & 0.30 & 44.72 & 1.01 & 1.75 & 1.93 & 23 \\
E & 0.60 & 0.40 & 40.92 & 1.12 & 1.75 & 1.93 & 23 \\
F & 0.50 & 0.50 & 39.8 & 1.12 & 1.67 & 1.93 & 23 \\
G & 0.40 & 0.60 & 36.22 & 1.12 & 1.67 & 1.93 & 23 \\
H & 0.30 & 0.70 & 41.21 & 1.12 & 1.67 & 1.93 & 23 \\
I & 0.20 & 0.80 & 34.43 & 1.20 & 1.60 & 1.93 & 26 \\
J & 0.10 & 0.90 & 32.87 & 1.25 & 1.60 & 1.91 & 28 \\
K & 0.0 & 01.00 & 17.96 & 1.62 & 1.60 & 1.91 & 41 \\
\bottomrule
\end{tabular}
\end{table}
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\end{document}
\begin{document}
/\end{document}
and other errors). Also, please indent the code by four spaces in order to display it in a code listing.