Venture Systems is a consulting firm that develops e$-$commerce systems and Web sites for its clients. It has six available consultants and eight client projects under contract. The consultants have different technical abilities and experience, and as a result, the company charges different hourly rates for its services. Also, the consultants' skills are more suited for some projects than others, and clients sometimes prefer some consultants over others. The suitability of a consultant for a project is rated according to a $5-$point scale, in which $1$ is the worst and $5$ is the best. The following table shows the rating for each consultant for each project, as well as the hours available for each consultant and the contracted hours and maximum budget for each project.

$\backslash$table$[[$Consultant$,\backslash$table$[[$Hourly$],[$Rate$]],$Project,$\backslash$table$[[$Available$],[$Hours$]]],[1,2,3,4,5,6,7,8],[$A$,$$$155,3,3,5,5,3,3,3,3,450],[$B$,140,3,3,2,5,5,5,3,3,600],[$C$,165,2,1,3,3,2,1,5,3,500],[$D$,300,1,3,1,1,2,2,5,1,300],[$E$,270,3,1,1,2,2,1,3,3,710],[$F$,150,4,5,3,2,3,5,4,3,860],[$Project Hours,$500,240,400,475,350,460,290,200,],[$Contract Budget $($$$1,000$s$),100,80,120,90,65,85,50,55,]]$

The company wants to know how many hours to assign each consultant to each project in order to best utilize the consultants' skills while meeting the clients' needs.

a$.$ Formulate a linear programming model for this problem.