$5.$ Computer systems consulting company CSC is under contract to carry out five projects, all with deadlines, measured in days from now. The consultants are a small group and must work together on each project, so that the projects will be started and completed sequentially. Under the terms of contract, the consultants will receive $\backslash (\backslash$$ $800\backslash )$ for each project completed on time or before its deadline, but they will incur $\backslash (\backslash$$ $500\backslash )$ in penalties for each project completed late. Each project has an associated duration, which is the anticipated number of days required to carry the project out, as shown below.

Consider the following mathematical programming model developed for this problem and answer the following parts.

Parameters and indices:

$\backslash ($ n$=\backslash )$ Number of jobs

$\backslash ($ j$=\backslash )$ Index for jobs $\backslash (($j$=1,\backslash$ldots$,$ n$)\backslash )$

$\backslash ($ t$=\backslash )$ Index for positions of the job sequence $\backslash (($t$=1,\backslash$ldots$,$ n$)\backslash )$

$\backslash ($ q$\_\{$e$\}=\backslash )$ Penalty for each job $($project$)$ completed on time or before its deadline

$\backslash ($ q$\_\{$l$\}=\backslash )$ Penalty for each job completed late

$\backslash ($ p$\_\{$j$\}=\backslash )$ Processing time of job $\backslash ($ j $\backslash )$

$\backslash ($ d$\_\{$j$\}=\backslash )$ Deadline for job $\backslash ($ j $\backslash )$

$\backslash ($ M$=\backslash )$ Sufficiently large positive number