As new areas develop around major cities, entire networks of collector sewers and treatment plants must be constructed to service growing population. Figure 1, displays wastewater network structure.

Nodes 1 to 8 of the network represent population centers, where smaller sewer feed into the main regional network, and locations where treatment plants might be built. Wastewater loads are roughly proportional to the population, so inflows indicated at nodes represent population units (in thousands). Arcs joining nodes 1 to 8 show possible routes for main collector sewers. Most follow the topology in gravity flow, but one pumped line (4,3) is included. A large part of construction cost for either type of line is fixed: right -of-way acquisition, trenching, and so on. Still the cost of a line also grows with the number of population units carried, because greater flows imply larger diameter pipes. Table 1, shows the fixed and variable cost for each arc in thousands of dollars. Treatment plant costs actually occur at nodes –here nodes 3,7, and 8. Figure 1, illustrates, however, that such costs can modeled on arcs by introducing an artificial “super sink” node 9. Costs shown for arcs (3,9), (7,9), and (8,9) capture the fixed and variable expenses of plant construction as flows depart the network. Required: Formulate integer linear programming model to determine which arcs (with its capacity)that should be included within wastewater network. 

As new areas develop around major cities, entire networks of collector sewers and treatment plants must be constructed to service growing population. Figure 1, displays wastewater network structure. Table 1, branches fixed cost, variable cost, and capacity 3 Branch acity Arc Fixed Cost Variable Cost population units 36 pumped 3 14 (1,2) 240 21 27 gravity (1,3) 350 30 27 (2,3) 200 13 22 30 7 (2,4) 750 58 30 21 (3,4) 610 43 44 (3,9) 3800 122 1 plant arcs (4,3) 1840 49 8 108 Fig.1 Wastewater Network (4,8) 780 63 122 Nodes 1 to 8 of the network represent population centers, where smaller sewer feed into the main regional network, and locations where treatment plants might be built. Wastewater loads are roughly proportional to the population, so inflows indicated at nodes represent population units (in thousands). (5,6) | 620 44 21 (5,7) 800 51 21 (6,7) 500 56 29 (6,8) 630 94 29 (7,4) 1120 82 42 Arcs joining nodes 1 to 8 show possible routes for main collector sewers. Most follow the topology in gravity flow, but one pumped line (4,3) is included. A large part of construction cost for either type of line is fixed: right -of-way acquisition, trenching, and so on. Still the cost of a line also grows with the number of population units carried, because greater flows imply larger diameter pipes. Table 1, shows the fixed and variable cost for each arc in thousands of dollars. (7,9) 3800 1. 42 (8,9) 2500 2 122 Treatment plant costs actually occur at nodes -here nodes 3,7, and 8. Figure 1, illustrates, however, that such costs can modeled on arcs by introducing an artificial "super sink" node 9. Costs shown for arcs (3,9), (7,9), and (8,9) capture the fixed and variable expenses of plant construction as flows depart the network. Required: Formulate integer linear programming model to determine which arcs (with its capacity)that should be included within wastewater network.