I need help in formulating the constraints for this problem. So, help with part (a) is all I need. The image presents a problem of optimizing flows through a sewage treatment plant using a flow network graph. The nodes represent junctions in the system, and the directed arcs between them represent the maximum possible sewage flow in tons per hour. The numbers on the arcs indicate this maximum flow capacity.### Graph Details:- **Nodes**: The graph consists of 8 nodes labeled from 1 to 8.- **Arcs and Capacities**: - Node 1 to Node 2: Capacity of 8 - Node 1 to Node 4: Capacity of 5 - Node 1 to Node 3: Capacity of 10 - Node 2 to Node 4: Capacity of 6 - Node 2 to Node 5: Capacity of 6 - Node 4 to Node 3: Capacity of 6 - Node 4 to Node 5: Capacity of 9 - Node 4 to Node 6: Capacity of 9 - Node 3 to Node 7: Capacity of 10 - Node 5 to Node 6: Capacity of 7 - Node 6 to Node 8: Capacity of 10 - Node 6 to Node 7: Capacity of 4 - Node 7 to Node 8: Capacity of 5 - Node 5 to Node 8: Capacity of 9### Tasks:a) **LP Model**: Formulate a linear programming model that includes the objective function and constraints based on the flow capacities.b) **Solution**: Use Analytic Solver to find the optimal flow configuration that maximizes the flow through the network and provides an efficient solution to the problem.The goal is to determine the flow capacities on each path to ensure that the maximum amount of sewage can be treated while adhering to the given constraints.

2) The graph below represents various flows that can occur through a sewage treatment plant, with the numbers on the arcs representing the maximum flow in tons of sewage per hour that can be accommodated. a) What is the LP model for this problem? (Objective function and constraints). b) Solve this problem with Analytic Solver. 1 8 10 2 5 3 6 6 6 9 10 9 5 5 6 7 10 5 9 8

2) The graph below represents various flows that can occur through a sewage treatment plant, with the numbers on the arcs representing the maximum flow in tons of sewage per hour that can be accommodated. a) What is the LP model for this problem? (Objective function and constraints). b) Solve this problem with Analytic Solver. 1 8 10 2 5 3 6 6 6 9 10 9 5 5 6 7 10 5 9 8

Practical Management Science

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Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

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I need help in formulating the constraints for this problem. So, help with part (a) is all I need.

The image presents a problem of optimizing flows through a sewage treatment plant using a flow network graph. The nodes represent junctions in the system, and the directed arcs between them represent the maximum possible sewage flow in tons per hour. The numbers on the arcs indicate this maximum flow capacity.

### Graph Details:
- **Nodes**: The graph consists of 8 nodes labeled from 1 to 8.
- **Arcs and Capacities**:
- Node 1 to Node 2: Capacity of 8
- Node 1 to Node 4: Capacity of 5
- Node 1 to Node 3: Capacity of 10
- Node 2 to Node 4: Capacity of 6
- Node 2 to Node 5: Capacity of 6
- Node 4 to Node 3: Capacity of 6
- Node 4 to Node 5: Capacity of 9
- Node 4 to Node 6: Capacity of 9
- Node 3 to Node 7: Capacity of 10
- Node 5 to Node 6: Capacity of 7
- Node 6 to Node 8: Capacity of 10
- Node 6 to Node 7: Capacity of 4
- Node 7 to Node 8: Capacity of 5
- Node 5 to Node 8: Capacity of 9

### Tasks:
a) **LP Model**: Formulate a linear programming model that includes the objective function and constraints based on the flow capacities.
b) **Solution**: Use Analytic Solver to find the optimal flow configuration that maximizes the flow through the network and provides an efficient solution to the problem.

The goal is to determine the flow capacities on each path to ensure that the maximum amount of sewage can be treated while adhering to the given constraints.

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