UDT is a consulting firm that develops e$-$commerce project, systems and websites for its clients. It has six available consultants and eight clients project is 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 consultant's skill 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$[[,,$Project$],[$Consultant$,\backslash$table$[[$Hourly$],[$wage$]],1,2,3,4,5,6,7,8,\backslash$table$[[$Available$],[$hours$]],],[$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,],[\backslash$table$[[$Project$],[$Hours$]],,500,240,400,475,350,460,290,200,,],[\backslash$table$[[$Contract$],[$budget$],[($x$1000],[$USD$)]],,100,80,120,90,65,85,50,55,,]]$

The company wants to know how many hours to assign each consultant to each project in order to best utilize their skill while meeting clients needs

a$.$ Formulate a linear programming model and write down the mathematical model for this problem.

b$.$ Solve this problem using QM and SOLVER

c$.$ If the company want to maximize revenue while ignoring client preferences and consultant compatibility, will this change the solution in b$.?$

d$.$ Create a sensitivity report. What is the shadow price in this case?

e$.$ If consultant A and E change their hourly wage from $$155$ to $$200($A$)$ and from $$270$ to $$200,$ will the solution change?

f$.$ By exprerience, consultant B and E is getting better at their ability, which mean their capacity for every project now minimum start from $3$ instead of $1$ or $2,$ will the shadow