BUS 660 GCU Tri County Natural Gas Distribution Problems

Problem 10-03

Tri-County Utilities, Inc., supplies natural gas to customers in a three-county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to provide the following quantities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows:

1. Choose the correct network representation of this problem.
   (i)

(ii)

(iii)

(iv)

Model (i)Model (ii)Model (iii)Model (iv)Model (iii)

1. Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs.

   Let xij = amount shipped from supply node i to demand node j.
   Minfill in the blank 2×11+fill in the blank 3×12+fill in the blank 4×13+fill in the blank 5×21+fill in the blank 6×22+fill in the blank 7x23s.t.fill in the blank 8×11+fill in the blank 9×12+fill in the blank 10×13?fill in the blank 11fill in the blank 12×21+fill in the blank 13×22+fill in the blank 14×23?fill in the blank 15fill in the blank 16×11+fill in the blank 17×21=fill in the blank 18fill in the blank 19×12+fill in the blank 20×22=fill in the blank 21fill in the blank 22×13+fill in the blank 23×23=fill in the blank 24xij ? 0 for all i, j

2. Describe the distribution plan and show the total distribution cost.
   AmountCostSouthern – Hamiltonfill in the blank 25$fill in the blank 26Southern – Butlerfill in the blank 27fill in the blank 28Southern – Clermontfill in the blank 29fill in the blank 30Northwest – Hamiltonfill in the blank 31fill in the blank 32Northwest – Butlerfill in the blank 33fill in the blank 34Northwest – Clermontfill in the blank 35fill in the blank 36Total Cost$fill in the blank 37
3. Recent residential and industrial growth in Butler County has the potential for increasing demand by as much as 100 units. Which supplier should Tri-County contract with to supply the additional capacity?
   AmountCostSouthern – Hamiltonfill in the blank 38$fill in the blank 39Southern – Butlerfill in the blank 40fill in the blank 41Southern – Clermontfill in the blank 42fill in the blank 43Northwest – Hamiltonfill in the blank 44fill in the blank 45Northwest – Butlerfill in the blank 46fill in the blank 47Northwest – Clermontfill in the blank 48fill in the blank 49Total Cost$fill in the blank 50
   From the new solution we see that Tri-County should contract with SouthernNorthwestSouthern Gas for the additional 100 units. 

Problem 10-21 (Algorithmic)

United Express Service (UES) uses large quantities of packaging materials at its four distribution hubs. After screening potential suppliers, UES identified six vendors that can provide packaging materials that will satisfy its quality standards. UES asked each of the six vendors to submit bids to satisfy annual demand at each of its four distribution hubs over the next year. The following table lists the bids received (in thousands of dollars). UES wants to ensure that each of the distribution hubs is serviced by a different vendor. Which bids should UES accept, and which vendors should UES select to supply each distribution hub?Distribution HubBidder1234Martin Products190165120240Schmidt Materials140240150200Miller Containers215210145235D&J Burns175185190290Larbes Furnishings200185155230Lawler Depot230190140230BidderDecisionBidMartin Products

SelectRejectReject

190165120240not applicable as bidder is not selected190Schmidt Materials

SelectRejectSelect

140240150200not applicable as bidder is not selected240Miller Containers

SelectRejectSelect

215210145235not applicable as bidder is not selected210D&J Burns

SelectRejectReject

175185190290not applicable as bidder is not selected185Larbes Furnishings

SelectRejectReject

200185155230not applicable as bidder is not selected200Lawler Depot

SelectRejectSelect

230190140230not applicable as bidder is not selected230

The value of the objective function is fill in the blank 13 thousands of dollars. 

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:

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:

1. Choose the correct network representation of the problem.
   (i)

(ii)

(iii)

(iv)

Model (i)Model (ii)Model (iii)Model (iv)Model (iii)

1. 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.
   Maxfill in the blank 2×11+fill in the blank 3×12+fill in the blank 4×13+fill in the blank 5×14+fill in the blank 6×21+fill in the blank 7×22+fill in the blank 8×23+fill in the blank 9×24+fill in the blank 10×31+fill in the blank 11×32+fill in the blank 12×33+fill in the blank 13x34s.t.fill in the blank 14×11+fill in the blank 15×12+fill in the blank 16×13+fill in the blank 17×14?fill in the blank 18fill in the blank 19×21+fill in the blank 20×22+fill in the blank 21×23+fill in the blank 22×24?fill in the blank 23fill in the blank 24×31+fill in the blank 25×32+fill in the blank 26×33+fill in the blank 27×34?fill in the blank 28fill in the blank 29×11+fill in the blank 30×21+fill in the blank 31×31=fill in the blank 32fill in the blank 33×12+fill in the blank 34×22+fill in the blank 35×32=fill in the blank 36fill in the blank 37×13+fill in the blank 38×23+fill in the blank 39×33=fill in the blank 40fill in the blank 41×14+fill in the blank 42×24+fill in the blank 43×34=fill in the blank 44xij ? 0 for all i, jHours AssignedBillingAvery-Client Afill in the blank 45$fill in the blank 46Avery-Client Bfill in the blank 47fill in the blank 48Avery-Client Cfill in the blank 49fill in the blank 50Avery-Client Dfill in the blank 51fill in the blank 52Baker-Client Afill in the blank 53fill in the blank 54Baker-Client Bfill in the blank 55fill in the blank 56Baker-Client Cfill in the blank 57fill in the blank 58Baker-Client Dfill in the blank 59fill in the blank 60Campbell-Client Afill in the blank 61fill in the blank 62Campbell-Client Bfill in the blank 63fill in the blank 64Campbell-Client Cfill in the blank 65fill in the blank 66Campbell-Client Dfill in the blank 67fill in the blank 68Total Billing$ fill in the blank 69

2. 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 AssignedBillingAvery-Client Afill in the blank 70$fill in the blank 71Avery-Client Bfill in the blank 72fill in the blank 73Avery-Client Cfill in the blank 74fill in the blank 75Avery-Client Dfill in the blank 76fill in the blank 77Baker-Client Afill in the blank 78fill in the blank 79Baker-Client Bfill in the blank 80fill in the blank 81Baker-Client Cfill in the blank 82fill in the blank 83Baker-Client Dfill in the blank 84fill in the blank 85Campbell-Client Afill in the blank 86fill in the blank 87Campbell-Client Bfill in the blank 88fill in the blank 89Campbell-Client Cfill in the blank 90fill in the blank 91Campbell-Client Dfill in the blank 92fill in the blank 93Total Billing$fill in the blank 94