Question

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 dients over the four-week period are as follows: Client 180 75 100 85 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 Client Consultant Avery 100 125 135 150 115 115 140 120 155 120 130 b. 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 xnumber of hours from consultant i assigned to clientj. Max x12+ X13 X22 + x23 x32 X33+ X34 s.t. x12+ x13 x14 X23 X24 x33+ X32+ x21 x11 x31 X12 X32 X22 X13 X23 x33 x14 X24 X34 2 0 for all ,j Xij Billing Hours Assigned Avery-Client A Avery-Client B Avery-Client C Baker-Client A Baker-Client B Baker-Client C Baker-Client D Campbell-Client A Il-Client B t c Campbell-Client D Total Billing c. 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? Numeric field Billing Hours Assigned Avery-Cllent A Avery-Client B Avery-Client C Avery-Client D Baker-Client A Baker-Cient B Baker-Client C Baker-Client D Campbell-Client A Campbell-Client B Campbell-Client C Campbell-Client D Total Billing 180 100 125 Avery 160 115 100 75 120 135 Baker 160 115 120 100 3 155 150 Campbell 140 140 130 85

Answer 1

b) Linear program is following:

Max 100X11+125X12+115X13+100X14+120X21+135X22+115X23+120X24+155X31+150X32+140X33+130X34

s.t.

X11+X12+X13+X14 <= 160

X21+X22+X23+X24 <= 160

X31+X32+X33+X34 <= 140

X11+X21+X31 = 180

X12+X22+X32 = 75

X13+X23+X33 = 100

X14+X24+X34 = 85

Xij >= 0

Solution using LINGO is following:

Lingo 170-Solution Report . Lingo! File Edit Solver Window Help Solution Report-Lingo Lindo Model-Lingol Global optimal solution found Objective value: Infeasibilities: Max 100X11+125X12+115X13+100X14+120X21+135X22+115X23 57925.00 +120X24+155X31+150X32+140X33+130X34 0.000000 Total solver iterations: 3.03 Model Class LP X21+X22+X23+X24 <= 160 Total variables: 12 Nonlinear variables: X31+X32+X33+X34 140 Integer variables: <= X11+X21+X31180 Total constraintS: Nonlinear constraints: X12+X22+X32-75 Total nonzeros: 36 X13+X23 X33-100 Nonlinear nonzeros: X14+X24+X34 85 = Variable Value Reduced Cost xil 0.000000 10.00000 x12 40.00000 0.000000 x13 100.0000 0.000000 x14 0.000000 10.00000 X21 40.00000 0.000000 X22 35.00000 0.000000 10.00000 0.000000 X23 0.000000 X24 85.00000 x31 140.0000 0.000000 X32 0.000000 20.00000 X33 0.000000 20.00000 X34 0.000000 25.00000 Row Slack or Surplus 57925.00 Dual Price 1.000000 20.00000 0.000000 0.000000 10.00000 0.000000 45.00000 For Help, press F1 NUM Ln 1, Col 1 4:28 pm

	Hours Assigned	Billing
Avery-Client A	0	0
Avery-Client B	40	5000
Avery-Client C	100	11500
Avery-Client D	0	0
Baker-Client A	40	4800
Baker-Client B	35	4725
Baker-Client C	0	0
Baker-Client D	85	10200
Campbell-Client A	140	21700
Campbell-Client B	0	0
Campbell-Client C	0	0
Campbell-Client D	0	0

Total Billing = 57,925

c)

A new constraint X12 = 0 is added in the LP model.

Revised model and solution is as follows:

Lingo 17.0- Solution Report-Lingo1 File Edit Solver Window Help Solution Report-Lingo Lindo Model-Lingol Global optimal solution found Objective value: Infeasibilities: Max 100X11+125X12+115X13+100X14+120X21+135X22+115X23 57525.00 +120X24+155X31+150X32+140X33+130X34 0.000000 Total solver iterations: Elap ed runtime econd ; 0.23 Model Class LP X21+X22+X23+X24 <= 160 Total variables: Nonlinear variables: X31+X32+X33+X34 140 Integer variables: <= X11+X21+X31180 Total constraintS: Nonlinear constraints: X12+X22+X32-75 Total nonzeros: X13+X23 X33-100 Nonlinear nonzeros: X14+X24+X34 85 = X12 Variable Value Reduced Cost xil 0.000000 0.000000 x12 0.000000 0.000000 x13 100.0000 0.000000 x14 40.00000 0.000000 X21 40.00000 0.000000 X22 75.00000 0.000000 20.00000 X23 0.000000 X24 45.00000 0.000000 x31 140.0000 0.000000 X32 0.000000 20.00000 X33 0.000000 30.00000 X34 0.000000 25.00000 Row Slack or Surplus 57525.00 Dual Price 1.000000 20.00000 0.000000 0.000000 20.00000 0.000000 55.00000 For Help, press F1 CAP NUMLn 1, Col 14:34 pm

	Hours Assigned	Billing
Avery-Client A	0	0
Avery-Client B	0	0
Avery-Client C	100	11500
Avery-Client D	40	4000
Baker-Client A	40	4800
Baker-Client B	75	10125
Baker-Client C	0	0
Baker-Client D	45	5400
Campbell-Client A	140	21700
Campbell-Client B	0	0
Campbell-Client C	0	0
Campbell-Client D	0	0

Total billing = 57,525