**Project Expansion of Horizon Cable in Smalltown**Horizon Cable is planning to expand its cable TV offerings in Smalltown by adding MTV and other exciting stations. To ensure a successful expansion, all activities listed in Table 22 must be completed prior to the launch.**Task Descriptions:****a. Project Network Development**- Draw the project network to visualize the sequence of activities.- Determine the critical path, which is the longest path through the network, indicating the shortest time to complete the project.- Calculate the total float for each activity, which is the amount of time that the activity can be delayed without delaying the project.- Identify the free float for each activity, the time an activity can be delayed without affecting the subsequent activity.**b. Linear Programming Setup**- Establish the Linear Programming (LP) model to be used in identifying the project’s critical path efficiently.By completing these tasks, Horizon Cable will ensure timely and efficient service expansion in Smalltown. **Table 22: Project Activity Schedule**| Activity | Description | Immediate Predecessors | Duration (Weeks) ||----------|-----------------------------------------------------|------------------------|------------------|| A | Choose stations | — | 2 || B | Get town council to approve expansion | A | 4 || C | Order converters needed to expand service | B | 3 || D | Install new dish to receive new stations | B | 2 || E | Install converters | C, D | 10 || F | Change billing system | B | 4 |This table outlines the activities involved in expanding a service, along with the sequence and duration for each task. Each activity is represented by a letter and described in terms of its immediate predecessors and the time required to complete it in weeks.- **Activity A** involves choosing stations and does not have any predecessors, taking 2 weeks.- **Activity B** requires approval from the town council and depends on Activity A, with a duration of 4 weeks.- **Activity C** involves ordering necessary converters, following Activity B, and takes 3 weeks.- **Activity D** is for installing a new dish to receive new stations, also depending on Activity B, with a duration of 2 weeks.- **Activity E**, which requires installing converters, is contingent upon the completion of Activities C and D and takes 10 weeks.- **Activity F** involves changing the billing system and follows Activity B, taking 4 weeks to complete.

*a The objective is to draw the project network diagram and compute the critical path, the total float and free float for each activity. Here activity A is the starting activity. Activity A is the immediate predecessor of activity B. Activity B is also the immediate predecessors of activities C, D and F. Activity E is the immediate predecessor of and activities C and D. Node 1 represents the start of the project. An arc should start from node 1 to represent each activity that has no predecessors. Activity A must be completed before Activity B can begin which is represented below. Activity B can be completed before Activities C and D which can be represented below: Activities C and D can be completed before Activity E which is represented below: Activity B can be completed before Activity C. The project network is represented as follows: Compute the Early event time on each node. Node1 has no predecessors. The early time at Node1 is, Node1 is the only immediate predecessor of node 2. The early time at Node2 is, Node2 is the only immediate predecessor of node3. The early time at Node3 is, Node3 is the only immediate predecessor of node4. The early time at Node4 is, Node3 and Node4 are the immediate predecessors of node5. The early time at Node5 is, Node5 and node3 are the immediate predecessors of node6. The early time at Node6 is, Node 6 is the completion of the project. The earliest time at node 6 represents the project completion time. Therefore, the project completion time is 19 weeks. Compute the Late event time on each node. The late times are computed beginning with the finish node moving backward until we determine LT(1). The project completion time is 19 weeks, which represents the late time at Node6. The late time at Node6 is, Node 6 is the only immediate successor of node 5. The late time at Node5 is, Node 5 is the only immediate successor of Node 4. The late time at Node 4 is, Node 6, Node 5 and Node 4 are the immediate successors of Node 3. The late time at Node 3 is, Node 6 and Node 3 are the immediate successors of Node2. The late time at Node2 is, Node2 is the only immediate successor of Node1. The late time at Node1 is, The early and late times at each node is represented below: The formula for computing total is, The total floats of each activity is computed as follows: Activity A: Activity B: Activity C: Activity D: Activity E: Activity F: Dummy activity: The formula for computing total is, The total floats of each activity is computed as follows: Activity A: Activity B: Activity C: Activity D: Activity E: Comment Step 13 of 23 Activity F: Dummy activity: The total and free floats of each activities is displayed below: An activity with a total float zero is called as a critical activity. The critical activities are A, B, C and E. A path of node 1 to the finish node that consists entirely of critical activities is called a critical path. From the project diagram, the critical path is, . Therefore, the critical path having a length of 19 weeks. b) The objective is to construct a Linear programming model. Let denote the time that the event corresponding to node j occurs. For each arc in the project network, . Let F be the node that represents completion of the project. The objective is to minimize the time required to complete the project. The objective function is, and The above LPP is solved by using excel solver. Enter the data into the spreadsheet as follows: Enter the coefficients of the objective function in the cell range D6 to I6. Enter zero’s as the by changing variables in the cell range D7 to I7. Enter the coefficients of the constraints in the cell range C9 to I 15. Enter the right-hand side of the constraints in the cell range L9 to L 15. Use the function and update it in the cell J9. Use the function and update it in the cell J10. Use the function and update it in the cell J11. Use the function and update it in the cell J12. Use the function and update it in the cell J13. Use the function and update it in the cell J14. Use the function and update it in the cell J15. The optimal solution is obtained by using the function, and update it in the cell D17. The screenshot is displayed below: All the inputs have been done; hence click on Solve icon. A dialog box will appear. In the set objective box , input D17 , where final result will appear. Also, select the MIN option box in the text box “By Changing variable Cells”, update the cell names where the actual numbers are to be updated and the minimax variable. The screenshot is shown below: Now, define the constraints in the text field named “Subject to the Constraints”. The constraint can be input by clicking on Add button, and a dialog box will appear. Define the first constraint Click Add button. The screenshot is as shown below, Now add all the constraint by clicking on add button, the screen shot as shown below