Download Book Problem – Chapter 10 Self Test #3

Michael Cavender
Team Highlander
Book Problem – Chapter 10 Self Test #3
3) 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 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 cost 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
Southern Gas
Northwest Gas
Demand (units)
Hamilton
Butler
Clermont
10
12
400
20
15
200
15
18
300
Supply Capacity (units)
500
400
a. Develop a network representation of this problem.
b. Develop a linear programming model that can be used to determine the plan that will
minimize the total distribution costs.
c. Describe the distribution plan and show the total distribution cost.
d. 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?
a)
b) Variable
xyz = amount
node y
x11 = Units shipped from
x12 = Units shipped from
x13 = Units shipped from
of x units shipped from supply
to demand node z.
supply node "Southern" to demand node "Hamilton."
supply node "Southern" to demand node "Butler."
supply node "Southern" to demand node "Clermont."
x21 = Units shipped from supply node "Northwest" to demand node "Hamilton."
x22 = Units shipped from supply node "Northwest" to demand node "Butler."
x23 = Units shipped from supply node "Northwest" to demand node "Clermont."
Objective Function - Minimize total distribution costs;
Min = 10*x11 + 20*x12 + 15*x13 + 12*x21 + 15*x22 + 18*x23;
!Constraints;
x11 + x12 + x13 <= 500;
x21 + x22 + x23 <= 400;
!Southern supply capacity;
!Northwest supply capacity;
x11 + x21 = 400;
x12 + x22 = 200;
x13 + x23 = 300;
!Hamilton demand forecast;
!Butler demand forecast;
!Clermont demand forecast;
x11
x12
x13
x21
x22
x23
>=
>=
>=
>=
>=
>=
0; !Cannot be less than 0;
0;
0;
0;
0;
0;
c)
Global optimal solution found.
Objective value:
Infeasibilities:
Total solver iterations:
Variable
X11
X12
X13
X21
X22
X23
Southern – Hamilton
Southern – Clermont
Northwest – Hamilton
Northwest – Butler
11900.00
0.000000
0
Value
200.0000
0.000000
300.0000
200.0000
200.0000
0.000000
Amount
200
300
200
200
Total Cost
Reduced Cost
0.000000
7.000000
0.000000
0.000000
0.000000
1.000000
Cost
$2000
4500
2400
3000
$11,900
d)
Butler County demand is increasing by 100 units, so we change our Butler demand in LINGO
from 200 to 300.
x12 + x22 = 300;
!Butler demand forecast;
To decide what company (Southern Gas or Northwest Gas) is best suited to provide this
increase, we increase the supply at each company in LINGO by 100.
x11 + x12 + x13 <= 600;
x21 + x22 + x23 <= 500;
!Southern supply capacity;
!Northwest supply capacity;
When we run LINGO again with the new numbers in place, we get the following results:
Global optimal solution found.
Objective value:
Infeasibilities:
Total solver iterations:
Variable
X11
X12
X13
X21
X22
X23
Southern – Hamilton
Southern – Clermont
Northwest – Hamilton
Northwest – Butler
Total Cost
13200.00
0.000000
1
Value
300.0000
0.000000
300.0000
100.0000
300.0000
0.000000
Amount
300
300
100
300
Reduced Cost
0.000000
7.000000
0.000000
0.000000
0.000000
1.000000
Cost
$3000
4500
1200
4500
$13,200
As we can see from the optimal solution, Northwest Gas is going to handle the increase in 100
demanded units from Butler County because it has a cheaper distribution rate to Butler.
Due to the increase in 100 units of supply capacity at Southern Gas and Northwest Gas, we
notice a couple other differences in the supplied units. Firstly, Southern Gas will now be
providing an additional 100 units to Hamilton and it will be maxing out its new supply
availability of 600 units. Southern Gas is cheaper than Northwest Gas at supplying to Hamilton
and Clermont, so that is why it is maxing out in these counties. Northwest Gas will now only be
providing 100 units to Hamilton.
The reason Southern Gas is the sole provider to Clermont County is because the distribution
rate difference between Southern and Northwest to Clermont is $5. The difference between
the rates to Hamilton is only $2, so it makes sense to for Tri-County to minimize cost by using
Southern to provide Clermont County’s entire demand of natural gas.