how do you formulate the problem as a linear program

Problem $10-05$

Premier Consulting$$s two consultants, Avery and Baker, can be scheduled to work for clients up to a maximum of $160$ hours each over the next four weeks. A third consultant, Campbell, has some administrative assignments already planned and is available for clients up to a maximum of $140$ hours over the next four weeks. The company has four clients with projects in process. The estimated hourly requirements for each of the clients over the four$-$week period are as follows:

Client:

A $-180$ hours

B $-75$ hours

C $-100$ hours

D $-85$ hours

Hourly rates vary for the consultant$$client combination and are based on several factors, including project type and the consultant$$s experience. The rates $($dollars per hour$)$ for each consultant$$client combination are as follows:

Consultant:

Avery $-100,125,115,100$

Baker $-120,135,115,120$

Campbell $-155,150,140,130$

Formulate the problem as a linear program, with the optimal solution providing the hours each consultant should be scheduled for each client to maximize the consulting firm$$s billings. What is the schedule and what is the total billing?

Let xij $=$ number of hours from consultant i assigned to client j$.$

Max fill in the blank $2$

x$11+$ fill in the blank $3$

x$12+$ fill in the blank $4$

x$13+$ fill in the blank $5$

x$14+$ fill in the blank $6$

x$21+$ fill in the blank $7$

x$22+$ fill in the blank $8$

x$23$

$+$ fill in the blank $9$

x$24+$ fill in the blank $10$

x$31+$ fill in the blank $11$

x$32+$ fill in the blank $12$

x$33+$ fill in the blank $13$

x$34$

s$.$t$.$ fill in the blank $14$

x$11+$ fill in the blank $15$

x$12+$ fill in the blank $16$

x$13+$ fill in the blank $17$

x$14<=$ fill in the blank $18$

fill in the blank $19$

x$21+$ fill in the blank $20$

x$22+$ fill in the blank $21$

x$23+$ fill in the blank $22$

x$24<=$ fill in the blank $23$

fill in the blank $24$

x$31+$ fill in the blank $25$

x$32+$ fill in the blank $26$

x$33+$ fill in the blank $27$

x$34<=$ fill in the blank $28$

fill in the blank $29$

x$11+$ fill in the blank $30$

x$21+$ fill in the blank $31$

x$31=$ fill in the blank $32$

fill in the blank $33$

x$12+$ fill in the blank $34$

x$22+$ fill in the blank $35$

x$32=$ fill in the blank $36$

fill in the blank $37$

x$13+$ fill in the blank $38$

x$23+$ fill in the blank $39$

x$33=$ fill in the blank $40$

fill in the blank $41$

x$14+$ fill in the blank $42$

x$24+$ fill in the blank $43$

x$34=$ fill in the blank $44$

xij $>=0$ for all i$,$ j

Hours Assigned Billing

Avery$-$Client A fill in the blank $45$

$fill in the blank $46$

Avery$-$Client B fill in the blank $47$

fill in the blank $48$

Avery$-$Client C fill in the blank $49$

fill in the blank $50$

Avery$-$Client D fill in the blank $51$

fill in the blank $52$

Baker$-$Client A fill in the blank $53$

fill in the blank $54$

Baker$-$Client B fill in the blank $55$

fill in the blank $56$

Baker$-$Client C fill in the blank $57$

fill in the blank $58$

Baker$-$Client D fill in the blank $59$

fill in the blank $60$

Campbell$-$Client A fill in the blank $61$

fill in the blank $62$

Campbell$-$Client B fill in the blank $63$

fill in the blank $64$

Campbell$-$Client C fill in the blank $65$

fill in the blank $66$

Campbell$-$Client D fill in the blank $67$

fill in the blank $68$

Total Billing $ fill in the blank $69$

New information shows that Avery doesn$$t have the experience to be scheduled for client B$.$ If this consulting assignment is not permitted, what impact does it have on total billings? What is the revised schedule?

Hours Assigned Billing

Avery$-$Client A fill in the blank $70$

$fill in the blank $71$

Avery$-$Client B fill in the blank $72$

fill in the blank $73$

Avery$-$Client C fill in the blank $74$

fill in the blank $75$

Avery$-$Client D fill in the blank $76$

fill in the blank $77$

Baker$-$Client A fill in the blank $78$

fill in the blank $79$

Baker$-$Client B fill in the blank $80$

fill in the blank $81$

Baker$-$Client C fill in the blank $82$

fill in the blank $83$

Baker$-$Client D fill in the blank $84$

fill in the blank $85$

Campbell$-$Client A fill in the blank $86$

fill in the blank $87$

Campbell$-$Client B fill in the blank $88$

fill in the blank $89$

Campbell$-$Client C fill in the blank $90$

fill in the blank $91$

Campbell$-$Client D fill in the blank $92$

fill in the blank $93$

Total Billing $fill in the blank $94$