Let's solve the problem step-by-step as requested: --- ## (a) Construct a Table for Critical Path, Duration, & Cost ### **Step 1: Draw the Project Network Diagram** #### **Activities and Predecessors** From the second table: - A (Investigate demand): Start - B (Develop pricing): Start - C (Design product): Start - D (Promotional cost analysis): after A - E (Manufacture prototype): after C - F (Product cost analysis): after E - G (Final pricing): after B, D, F - H (Market test): after G #### **Dependencies:** - Start → A, B, C - A → D - C → E → F - B, D, F → G - G → H Let's list all possible paths: 1. **A → D → G → H** 2. **B → G → H** 3. **C → E → F → G → H** --- ### **Step 2: Calculate Path Durations (Normal and Crash)** #### **Normal Durations:** - A → D → G → H: 3 + 1 + 2 + 8 = **14 weeks** - B → G → H: 1 + 2 + 8 = **11 weeks** - C → E → F → G → H: 5 + 6 + 0.5 + 2 + 8 = **21.5 weeks** #### **Crash Durations:** - A → D → G → H: 1 + 0.2 + 1 + 5 = **7.2 weeks** - B → G → H: 0.5 + 1 + 5 = **6.5 weeks** - C → E → F → G → H: 3 + 3 + 0.2 + 1 + 5 = **12.2 weeks** --- ### **Step 3: Find Critical Path** - **Normal:** C → E → F → G → H (21.5 weeks) - **Crash:** C → E → F → G → H (12.2 weeks) --- ### **Step 4: Calculate Direct Costs** #### **Direct Costs for Critical Path (Normal):** - C: $9k, E: $20k, F: $2.2k, G: $4k, H: $100k - **Total:** $9 + $20 + $2.2 + $4 + $100 = **$135.2k** #### **Direct Costs for Critical Path (Crash):** - C: $18k, E: $30k, F: $3k, G: $9k, H: $150k - **Total:** $18 + $30 + $3 + $9 + $150 = **$210k** #### **Add Direct Cost for All Activities (Project-wide):** - Normal: A ($3.5k) + B ($1.2k) + D ($1k) + [above] = $3.5 + $1.2 + $1 + $135.2 = **$140.9k** - Crash: A ($10k) + B ($2k) + D ($2k) + [above] = $10 + $2 + $2 + $210 = **$224k** --- ### **Step 5: Calculate Total Cost (Direct + Indirect + Opportunity)** #### **Indirect Cost:** - Fixed: $5,000 - Variable: $1,000 × Project Duration (weeks) #### **Opportunity Cost:** - If duration > 17 weeks: $2,000 × (weeks over 17) --- #### **Normal Schedule (21.5 weeks):** - Direct: $140,900 - Indirect: $5,000 + $1,000 × 21.5 = $26,500 - Opportunity: $2,000 × (21.5 - 17) = $9,000 - **Total:** $140,900 + $26,500 + $9,000 = **$176,400** #### **Crash Schedule (12.2 weeks):** - Direct: $224,000 - Indirect: $5,000 + $1,000 × 12.2 = $17,200 - Opportunity: $0 (no penalty) - **Total:** $224,000 + $17,200 + $0 = **$241,200** --- ### **Step 6: Table Summary** | Strategy | Critical Path | Duration (weeks) | Direct Cost ($1,000) | Indirect ($1,000) | Opportunity ($1,000) | **Total ($1,000)** | |----------|----------------------|------------------|----------------------|-------------------|----------------------|--------------------| | Normal | C-E-F-G-H | 21.5 | 140.9 | 26.5 | 9.0 | **176.4** | | Crash | C-E-F-G-H | 12.2 | 224.0 | 17.2 | 0.0 | **241.2** | --- ## (b) Gantt Chart for Minimum Total Cost Schedule The minimum total cost schedule is the **normal** schedule (21.5 weeks, $176,400). - **Early Start:** Each activity starts as soon as predecessors are finished. - **Late Start:** Each activity starts as late as possible to not delay the project. **Gantt Chart Example:** - Week 0: Start A, B, C - Week 3: A ends, D starts - Week 5: C ends, E starts - Week 6: D ends - Week 11: E ends, F starts - Week 11.5: F ends - Week 12: B, D, F all done, G starts - Week 14: G ends, H starts - Week 21.5: H ends (project complete) *(You can sketch this out for visualization.)* --- ## (c) Analysis: Peak Cost & Variance - **Lowest Peak Cost:** The normal schedule yields the lowest peak cost since the crash schedule incurs much higher direct costs to save time. - **Variance:** The crash schedule has the **highest variance** in cost, as it spends much more up-front to reduce time. - **Recommendation:** **Use the normal schedule** unless reducing time is mission-critical, as it minimizes total cost. The crash schedule is only justified if the opportunity cost of delay is higher than the additional direct and indirect costs. --- ## **Final Answers (Summary Tables)** ### (a) Table | Strategy | Critical Path | Duration (weeks) | Direct ($1,000) | Indirect ($1,000) | Opportunity ($1,000) | Total ($1,000) | |----------|--------------|------------------|-----------------|-------------------|----------------------|----------------| | Normal | C-E-F-G-H | 21.5 | 140.9 | 26.5 | 9.0 | 176.4 | | Crash | C-E-F-G-H | 12.2 | 224.0 | 17.2 | 0.0 | 241.2 | ### (b) Gantt Chart - Early Start: Activities start immediately after predecessors finish. - Late Start: Activities start as late as possible without delaying project. ### (c) Recommendation - **Normal schedule** yields lowest peak cost, lowest variance, and is recommended unless time savings outweigh extra cost. --- If you need the Gantt chart drawn or want to see how crashing specific activities (not the whole project) affects costs and time, let me know!

Certainly! Let's proceed with a detailed, step-by-step solution, including all calculations, explanations, and final answers. --- ## **Part (a): Construct a table that enumerates the critical path, duration, and costs for each possible funding strategy** ### **Step 1: Understand the Activities and Dependencies** From the problem statement and table: | Activity | Description | Duration (weeks) | Cost ($) | Predecessors | |------------|-------------------------------------|------------------|----------|--------------| | A | Investigate demand | 3 | 3.5K | Start | | B | Develop promotional activities | 1 | 1.2K | Start | | C | Design product | 5 | 9K | Start | | D | Conduct promotional cost analysis | 1 | 1K | A | | E | Manufacture prototype models | 6 | 20K | C | | F | Perform product cost analysis | 0.5 | 2.2K | E | | G | Perform final pricing analysis | 2 | 4K | B, D, F | | H | Conduct market test | 8 | 100K | G | --- ### **Step 2: Identify Paths and Critical Path** Possible paths: 1. **Path 1:** A → D → G → H Duration: 3 + 1 + 2 + 8 = **14 weeks** 2. **Path 2:** B → G → H Duration: 1 + 2 + 8 = **11 weeks** 3. **Path 3:** C → E → F → G → H Duration: 5 + 6 + 0.5 + 2 + 8 = **21.5 weeks** The **critical path** is the longest: **C → E → F → G → H** (21.5 weeks). --- ### **Step 3: Determine Crash Durations and Costs** Crash durations are given or can be inferred: | Activity | Normal Duration | Crash Duration | Cost (Normal) | Cost (Crash) | Cost Increase per Week | |------------|-------------------|------------------|--------------|--------------|------------------------| | A | 3 | 1 | $3.5K | $10K | $6.5K per week crash | | B | 1 | 0.5 | $1.2K | $2K | $0.8K per week crash | | C | 5 | 3 | $9K | $18K | $9K per week crash | | D | 1 | 0.2 | $1K | $2K | $1K per week crash | | E | 6 | 3 | $20K | $30K | $10K per week crash | | F | 0.5 | 0.2 | $2.2K | $3K | $0.8K per week crash | | G | 2 | 1 | $4K | $9K | $5K per week crash | | H | 8 | 5 | $100K | $150K | $50K per week crash | --- ### **Step 4: Calculate Total Duration and Cost for "Normal" and "Crash" Strategies** #### **Normal Schedule (using normal durations):** - Duration: **21.5 weeks** (critical path) - Direct costs: Sum of all activity costs: \( \text{Total activity costs} = \$3.5K + \$1.2K + \$9K + \$1K + \$20K + \$2.2K + \$4K + \$100K = \$140.9K \) - Indirect costs: \( \$5,000 + (\$1,000 \times 21.5) = \$5,000 + \$21,500 = \$26,500 \) - Opportunity costs (project exceeds 17 weeks): \( (\text{duration} - 17) \times \$2,000 = (21.5 - 17) \times 2,000 = 4.5 \times 2,000 = \$9,000 \) - **Total Cost (Normal):** \[ \text{Total} = \text{Direct} + \text{Indirect} + \text{Opportunity} = 140.9K + 26.5K + 9K = \boxed{\$176.4K} \] --- #### **Crash Schedule (reducing duration to minimum):** - For the critical path (C → E → F → G → H), crash activities to their shortest durations: | Activity | Crash Duration | Cost (Crash) | Cost Difference per Week | |------------|------------------|--------------|-------------------------| | C | 3 weeks | $18K | $9K/week | | E | 3 weeks | $30K | $10K/week | | F | 0.2 weeks | $3K | $0.8K/week | | G | 1 week | $9K | $5K/week | | H | 5 weeks | $150K | $50K/week | - Total crash duration: \( 3 + 3 + 0.2 + 1 + 5 = 12.2 \) weeks - Sum of activity costs during crash: \( \$18K + \$30K + \$3K + \$9K + \$150K = \$210K \) - For activities not on critical path, assume normal durations (A, B, D): | Activity | Duration | Cost | Cost difference | |------------|------------|-------|----------------| | A | 1 week | $10K | $6.5K/week | | B | 0.5 weeks | $2K | $0.8K/week | | D | 0.2 weeks | $2K | $1K/week | - Sum activity costs: \[ \text{Total} = \$10K + \$2K + \$2K + \$210K = \$224K \] - Indirect costs: \( \$5,000 + (\$1,000 \times 12.2) = \$5,000 + \$12,200 = \$17,200 \) - Opportunity cost: since duration is less than 17 weeks, **no opportunity cost**. - **Total Cost (Crash):** \[ 224K + 17.2K = \boxed{\$241.2K} \] --- ### **Final Table:** | Funding Strategy | Critical Path | Duration | Direct Cost ($K) | Indirect Cost ($K) | Opportunity Cost ($K) | Total Cost ($K) | |--------------------|-----------------|----------|------------------|--------------------|------------------------|-----------------| | Normal | C-E-F-G-H | 21.5 | 140.9 | 26.5 | 9.0 | **176.4** | | Crash | C-E-F-G-H | 12.2 | 210.0 | 17.2 | 0.0 | **241.2** | --- ## **Part (b): Construct Gantt charts for early start and late start schedules** ### **Normal Schedule (Early Start):** - Week 0: Activities A, B, C start simultaneously. - Week 3: A finishes; D can start. - Week 5: C finishes; E starts. - Week 6: D finishes. - Week 11: E finishes; F starts. - Week 11.5: F finishes. - Week 12: B, D, F finished; G starts. - Week 14: G ends. - Week 14-21.5: H runs. - **Project completes at week 21.5.** *(Sketch or use project management software for detailed Gantt.)* ### **Late Start Schedule:** - Start activities as late as possible without delaying project completion at week 21.5. - For activities on the critical path, late start times are aligned with early start times, but activities on non-critical paths are delayed to save resources. --- ## **Part (c): Which schedule yields the lowest peak cost? Which has more variance? Which schedule is recommended?** - **Peak cost:** The **crash schedule** has higher activity costs, leading to higher peak costs, but it reduces overall project duration. - **Variance in cost:** The **crash schedule** has more variance due to high activity crashing costs. - **Recommendation:** - Use the **normal schedule** for cost savings and lower variance unless project time is critical. - If early completion is vital, consider crashing activities on the critical path selectively rather than the entire project. --- ## **Final Summary:** | Aspect | Normal Schedule | Crash Schedule | |---------|------------------|----------------| | Duration | 21.5 weeks | 12.2 weeks | | Total Cost | \$176,400 | \$241,200 | | Peak Cost | Lower | Higher | | Variance | Lower | Higher | | Recommended | Yes, for cost savings | Only if time is critical | --- This completes a detailed, step-by-step calculation and analysis. If you'd like, I can help you create diagrams or further analyze partial crashing strategies!