[SOLVED] A linear programming model

a linear programming model 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:

 

Choose the correct network representation of this problem.

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

 

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.

Min fill in the blank 2
x11 + fill in the blank 3
x12 + fill in the blank 4
x13 + fill in the blank 5
x21 + fill in the blank 6
x22 + fill in the blank 7
x23
s.t.
fill in the blank 8
x11 + fill in the blank 9
x12 + fill in the blank 10
x13 ≤ fill in the blank 11
fill in the blank 12
x21 + fill in the blank 13
x22 + fill in the blank 14
x23 ≤ fill in the blank 15
fill in the blank 16
x11 + fill in the blank 17
x21 = fill in the blank 18
fill in the blank 19
x12 + fill in the blank 20
x22 = fill in the blank 21
fill in the blank 22
x13 + fill in the blank 23
x23 = fill in the blank 24
xij ≥ 0 for all i, j

Describe the distribution plan and show the total distribution cost.

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

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?

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
Gas for the additional 100 units.