Question 2 The Canterbury City Council is faced with a severe budget shortage due to unexpected social costs from the Covid-19 pandemic. The council is now reconsidering an earlier proposal submitted by a property developer. This proposal seeks to demolish existing council buildings in the Wincheap area and replace them with modern residential developments that can attract higher council tax rates. The developer can build one- bedroom, two-bedroom, three-bedroom or four-bedroom houses. The council must, therefore, determine how many existing buildings to demolish and how many units of the new buildings to put up. The following information is available about the project: 1. The Wincheap area has about 250 units of council buildings that can be demolished. The cost of demolishing a building is budgeted to be 2500. Each building occupies a 0.30-acre lot. 2. The table below shows the construction cost and the budgeted lot size required for each type of new house. It also shows the expected annual council tax that could be generated from each type of property. House Type Unit cost One-bedroom Two-bedroom Three-bedroom Four-bedroom Lot size (acre) 0.15 0.30 0.35 0.40 45,000 65,000 120,000 150,000 Council tax () 1,292.63 1,508.87 1,723.51 1,938.95 3. Construction policies require that 10% of the available acreage after demolishing be reserved for streets, utility instalments and other emergency facilities. Additionally, at least 10% of the new units should be four-bedroom houses. At least 5% of the new units should be two-bedroom houses. Finally, at most 45% of the new units should be two- and three-bedroom houses. 4. The council can access a maximum of 12 million financing facility with a High Street bank. The council will want to determine the maximum annual council tax that could be generated after this project. Use this information and the sensitivity report provided to answer the questions that follow: Variable Cells Reduced Cost 0 Name One-bedroom Two-bedroom Three-bedroom Four-bedroom Demolished Final Value 175.29 10.31 0 20.62 139.39 Objective Coefficient 1292.63 1508.87 1723.51 1938.95 0 -1375.95 Allowable Increase 1E+30 291.99 1375.95 2301.08 60.53 Allowable Decrease 218.39 20352.34 1E+30 7797.80 2024.00 0 0 0 Constraints Name Available for Demolishing Lot size Project cost Min number of four-beds Min number of two-beds Max two- and three-beds Final Value 139.39 0.00 12000000 0.00 0.00 -82.49 Shadow Price 0 217.69 0.02 -1876.71 -286.62 0 Constraint R.H. Side 250 0 12000000 Allowable Increase 1E+30 38.76 9522260.27 68.27 71.92 1E+30 Allowable Decrease 110.61 30.76 12000000 25.29 10.50 82.49 0 0 0 a. Formulate a linear programming model to determine the total number of each type of new housing unit to be built and the number of old buildings to be demolished to maximise council tax generated. [10 marks] b. Why are there no three-bedroom houses to be built in the current plan? Advise the council on two ways to achieve positive units of three-bedroom units in the optimal plan. [8 marks] c. How much in tax will the current plan generate? What will be the effect on the optimal solution and tax if the council tax for a four-bedroom unit is set equal to that for a three-bedroom house? [10 marks] d. The council is considering increasing council taxes by 10%. What will be the impact of this policy on the optimal solution and total tax generated? [10 marks] e. The council would want to determine the impact of different financing projections on the optimal value. How much council tax will be generated for 5%, 10% and 50% change in the financing available? [12 marks] Question 2 The Canterbury City Council is faced with a severe budget shortage due to unexpected social costs from the Covid-19 pandemic. The council is now reconsidering an earlier proposal submitted by a property developer. This proposal seeks to demolish existing council buildings in the Wincheap area and replace them with modern residential developments that can attract higher council tax rates. The developer can build one- bedroom, two-bedroom, three-bedroom or four-bedroom houses. The council must, therefore, determine how many existing buildings to demolish and how many units of the new buildings to put up. The following information is available about the project: 1. The Wincheap area has about 250 units of council buildings that can be demolished. The cost of demolishing a building is budgeted to be 2500. Each building occupies a 0.30-acre lot. 2. The table below shows the construction cost and the budgeted lot size required for each type of new house. It also shows the expected annual council tax that could be generated from each type of property. House Type Unit cost One-bedroom Two-bedroom Three-bedroom Four-bedroom Lot size (acre) 0.15 0.30 0.35 0.40 45,000 65,000 120,000 150,000 Council tax () 1,292.63 1,508.87 1,723.51 1,938.95 3. Construction policies require that 10% of the available acreage after demolishing be reserved for streets, utility instalments and other emergency facilities. Additionally, at least 10% of the new units should be four-bedroom houses. At least 5% of the new units should be two-bedroom houses. Finally, at most 45% of the new units should be two- and three-bedroom houses. 4. The council can access a maximum of 12 million financing facility with a High Street bank. The council will want to determine the maximum annual council tax that could be generated after this project. Use this information and the sensitivity report provided to answer the questions that follow: Variable Cells Reduced Cost 0 Name One-bedroom Two-bedroom Three-bedroom Four-bedroom Demolished Final Value 175.29 10.31 0 20.62 139.39 Objective Coefficient 1292.63 1508.87 1723.51 1938.95 0 -1375.95 Allowable Increase 1E+30 291.99 1375.95 2301.08 60.53 Allowable Decrease 218.39 20352.34 1E+30 7797.80 2024.00 0 0 0 Constraints Name Available for Demolishing Lot size Project cost Min number of four-beds Min number of two-beds Max two- and three-beds Final Value 139.39 0.00 12000000 0.00 0.00 -82.49 Shadow Price 0 217.69 0.02 -1876.71 -286.62 0 Constraint R.H. Side 250 0 12000000 Allowable Increase 1E+30 38.76 9522260.27 68.27 71.92 1E+30 Allowable Decrease 110.61 30.76 12000000 25.29 10.50 82.49 0 0 0 a. Formulate a linear programming model to determine the total number of each type of new housing unit to be built and the number of old buildings to be demolished to maximise council tax generated. [10 marks] b. Why are there no three-bedroom houses to be built in the current plan? Advise the council on two ways to achieve positive units of three-bedroom units in the optimal plan. [8 marks] c. How much in tax will the current plan generate? What will be the effect on the optimal solution and tax if the council tax for a four-bedroom unit is set equal to that for a three-bedroom house? [10 marks] d. The council is considering increasing council taxes by 10%. What will be the impact of this policy on the optimal solution and total tax generated? [10 marks] e. The council would want to determine the impact of different financing projections on the optimal value. How much council tax will be generated for 5%, 10% and 50% change in the financing available? [12 marks]