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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 Hamilton Butler Clermont Supply Capacity(units)Southern Gas 10 20 15 500

Northwest Gas 12 15 18 400

Demand(units) 400 200 300

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?


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a)

b) Variable xyz = amount of x units shipped from supplynode y to demand node z.

x11 = Units shipped from supply node "Southern" to demand node "Hamilton."

x12 = Units shipped from supply node "Southern" to demand node "Butler."

x13 = Units shipped from 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 = 0;

x13 >= 0;

x21 >= 0;

x22 >= 0;

x23 >= 0;


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c)

Global optimal solution found.

Objective value: 11900.00

Infeasibilities: 0.000000

Total solver iterations: 0

Variable Value Reduced Cost

X11 200.0000 0.000000

X12 0.000000 7.000000

X13 300.0000 0.000000

X21 200.0000 0.000000

X22 200.0000 0.000000

X23 0.000000 1.000000

Amount Cost

Southern Hamilton 200 $2000

Southern

Clermont 300 4500Northwest Hamilton 200 2400

Northwest Butler 200 3000

Total Cost $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


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When we run LINGO again with the new numbers in place, we get the following results:

Global optimal solution found.

Objective value: 13200.00

Infeasibilities: 0.000000

Total solver iterations: 1

Variable Value Reduced Cost

X11 300.0000 0.000000

X12 0.000000 7.000000

X13 300.0000 0.000000

X21 100.0000 0.000000

X22 300.0000 0.000000

X23 0.000000 1.000000

Amount Cost

Southern Hamilton 300 $3000

Southern Clermont 300 4500

Northwest

Hamilton 100 1200

Northwest Butler 300 4500

Total Cost $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 Countys entire demand of natural gas.


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