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Problem 6-03

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:

To
From Hamilton Butler Clermont
Southern Gas 10 20 15
Northwest Gas 12 15 18

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

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

  2. Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs. If the constant is “1” it must be entered in the box.
    Let xij = amount shipped from supply node i to demand node j.
    Min fill in the blank 2x11 + fill in the blank 3x12 + fill in the blank 4x13 + fill in the blank 5x21 + fill in the blank 6x22 + fill in the blank 7x23
    s.t.
    fill in the blank 8x11 + fill in the blank 9x12 + fill in the blank 10x13 = fill in the blank 11
    fill in the blank 12x21 + fill in the blank 13x22 + fill in the blank 14x23 = fill in the blank 15
    fill in the blank 16x11 + fill in the blank 17x21 = fill in the blank 18
    fill in the blank 19x12 + fill in the blank 20x22 = fill in the blank 21
    fill in the blank 22x13 + fill in the blank 23x23 = fill in the blank 24
    xij = 0 for all i, j
  3. Describe the distribution plan and show the total distribution cost. If your answer is zero enter “0”.
    Amount Cost
    Southern – Hamilton fill in the blank 25 $ fill in the blank 26
    Southern – Butler fill in the blank 27 fill in the blank 28
    Southern – Clermont fill in the blank 29 fill in the blank 30
    Northwest – Hamilton fill in the blank 31 fill in the blank 32
    Northwest – Butler fill in the blank 33 fill in the blank 34
    Northwest – Clermont fill in the blank 35 fill in the blank 36
    Total Cost $ fill in the blank 37
  4. 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? If your answer is zero enter “0”.
    Amount Cost
    Southern – Hamilton fill in the blank 38 $ fill in the blank 39
    Southern – Butler fill in the blank 40 fill in the blank 41
    Southern – Clermont fill in the blank 42 fill in the blank 43
    Northwest – Hamilton fill in the blank 44 fill in the blank 45
    Northwest – Butler fill in the blank 46 fill in the blank 47
    Northwest – Clermont fill in the blank 48 fill in the blank 49
    Total 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.

 
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