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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 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.