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Problem I


Premier Consulting has two consultants, Avery and Baker, who 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


Client Hours
A 180
B 75
C 100
D 85


Hourly rates vary for the consultant-client combinations and are based on several factors, including project type and consultant’s experience.  The rates (dollar per hour) for each consultant-client combination are


Consultant Client A Client B Client C Client D
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 to work for each client in order to maximize the consulting firm’s billings.  What is the schedule and what is the total billing?


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



Problem II


In a job shop operation, four jobs may be performed on any of four machines.  The number of hours required for each job on each machine is summarized in the table.  Formulate a linear program to minimize the total time job-machine assignment.


Job A B C D
1 22 18 32 26
2 22 24 12 16
3 24 30 26 24
4 26 30 28 20



Problem III


Adirondack Paper Mills, Inc., has paper plants in Augusta, Maine and Tupper Lake, New York.  Warehouse facilities are located in Albany, New York and Portsmouht, New Hampshire.  Distributors are located in Boston, New York and Philadelphia.  The plant capacities and distributor demands for next month are as follows:


Plant Capacity (units)   Distributor Demand (units)
Augusta 300   Boston 150
Tupper Lake 100   New York 100
      Philadelphia 150


The unit transportation costs ($) for shipments from the two plants to the two warehouses and from the two warehouses to the three distribution centers are as follows:


Plant Albany Portsmouth  
Augusta 7 5  
Tupper Lake 3 4  
Warehouse Boston New York Philadelphia
Albany 8 5 7
Portsmouth 5 6 10


Formulate the Adirondack Paper Mills problem as a linear programming problem.  What is the minimum-cost shipping schedule?