Please answer all parts clearly. Thanks for the help!!

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 a. Choose the correct network representation of this problem. (1) ( (ii) 10 12 Hamilton 1 Hamilton 1 300 12 400 10 Southern Southern 500 500 15 15 Butler 2 200 Butler 2 200 20 20 Northwest 2 15 400 Northwest 2 18 400 400 300 Clermont 3 Clermont 3 18 15 (iv) Hamilton 1 400 Hamilton 1 300 10 10 500 Southern 1 20 500 Southern 1 20 15 12 Butler 2 200 Butler 2 12 200 15 15 15 Northwest 2 Northwest 400 400 18 18 Clermont 3 300 Clermont 3 400 b. 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 X11 + X12 + X13 + X21 + X22 + X23 s.t. X11 + X12 + X13 VI X21 + X22+ X235 X11 + X21 X12 + X22 = X13 + X231 = Xij = 0 for all i, j c. Describe the distribution plan and show the total distribution cost. If your answer is zero enter "O". Amount Cost Southern - Hamilton $ Southern - Butler Southern - Clermont Northwest - Hamilton Northwest - Butler Northwest - Clermont Total 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? If your answer is zero enter "O". Amount Cost Southern - Hamilton Southern - Butler Southern - Clermont Northwest - Hamilton Northwest - Butler 1 Northwest - Clermont Total Cost From the new solution we see that Tri-County should contract with Gas for the additional 100 units