Let's break down the solution step-by-step for both parts (A and B): --- ### **A. Which (if any) predecessors shown could be eliminated without affecting the network logic?** #### **Step 1: Understanding Redundant Predecessors** A predecessor is **redundant** if it is already implied by another predecessor through the network logic. For example, if J has predecessors K and L, and K itself has L as a predecessor, then listing both K and L for J is redundant. #### **Step 2: Analyze Each Job’s Predecessors** Let’s check for each job if any predecessor can be removed: Let’s consider job-by-job: #### **For Job F:** - Predecessors: A, B, C Check if any of A, B, or C is a predecessor to another among them. - F depends on A, B, and C directly. None of A, B, or C is a predecessor of another, so **no redundancy**. #### **For Job G:** - Predecessor: C No redundancy. #### **For Job H:** - Predecessors: D, E Check if D is a predecessor of E or vice versa. Looking at D: predecessors are A, B. For E: predecessors are A, B, C. No overlap/redundancy. #### **For Job I:** - Predecessors: D, E, F Check if any one is a predecessor of another: - D: A, B - E: A, B, C - F: A, B, C E and F have similar predecessors, but neither is a predecessor of the other. No redundancy. #### **For Job J:** - Predecessors: F, G F: A, B, C G: C So, G is a predecessor of F through C but since F already requires A, B, C, G (C) is already included. **Redundant: G** for J. #### **For Job K:** - Predecessor: G No redundancy. #### **For Job L:** - Predecessors: F, G, K F: A, B, C G: C K: G - Since K requires G, and G is already a predecessor, **K is redundant** for L. #### **For Job M:** - Predecessors: H, I H: D, E I: D, E, F No redundancy. #### **For Job N:** - Predecessors: H, I, J, L Check if any is a predecessor of another: - H: D, E - I: D, E, F - J: F, G - L: F, G, K J and L both require F and G, so listing both may be redundant. However, L adds K, which is not present in J; check if K is covered elsewhere. - L requires K, which itself requires G, which is already present. Thus, **J is redundant** for N because L already covers all of J's predecessors (F, G, K), and H/I are independent. #### **For Job O:** - Predecessors: J, K, L J: F, G K: G L: F, G, K - L includes F, G, K, so listing J and K is redundant. **J and K are redundant** for O. #### **For Job P:** - Predecessors: M, N No redundancy. #### **For Job Q:** - Predecessors: O, P No redundancy. #### **For Job R:** - Predecessors: J, K, L, O J: F, G K: G L: F, G, K O: J, K, L O covers J, K, L, so listing J, K, L is redundant. **J, K, L are redundant** for R. #### **For Job S:** - Predecessors: N, Q, R No redundancy; each is a different branch. #### **For Job T:** - Predecessors: O, S No redundancy. --- ### **Summary Table of Redundant Predecessors** | Job | Redundant Predecessors | |-----|------------------------| | J | G | | L | K | | N | J | | O | J, K | | R | J, K, L | --- ### **B. Draw an Arrow Diagram for this Project** This is a **Project Network Diagram** (or Activity-on-Arrow diagram). Since I cannot draw images directly here, I will describe the steps and structure: #### **Step 1: Start Node** - Draw a start node. From this node, draw arrows to A, B, and C (since these have no predecessors). #### **Step 2: Progressively Add Jobs** - From A and B, draw arrows to D. - From A, B, C, draw arrows to E and F. - From C, draw to G. - From D, draw to H. - From D, E, F, draw to I. - From F, draw to J. - From G, draw to K. - From F, G, draw to L. - From H, I, draw to M. - From H, I, L, draw to N. - From J, K, L, draw to O. - From M, N, draw to P. - From O, P, draw to Q. - From O, draw to R. - From N, Q, R, draw to S. - From O, S, draw to T. #### **Step 3: Clean Up Redundant Arrows** - Remove any connections identified as redundant in part A. For example, for O, only draw from L (not from J or K), for R only from O (not J, K, L), etc. #### **Step 4: Final Diagram** - You can now draw the diagram as described, ensuring all dependencies are properly represented. --- ## **Final Answers** ### **A. Redundant Predecessors** - **J:** G is redundant. - **L:** K is redundant. - **N:** J is redundant. - **O:** J and K are redundant. - **R:** J, K, and L are redundant. ### **B. Arrow Diagram** - Draw nodes for each job (A to T). - Draw arrows to reflect the predecessor relationships, omitting the redundant predecessors identified above. If you need an actual hand-drawn diagram, let me know, and I can provide a sketch or step-wise drawing instructions!