Let's analyze the problem step-by-step: --- ## Part 1: Manufacturing Cost Analysis ### Data: | Machine | A | B | C | |---------|---|---|---| | Costs | | | | | 1 | $13 | $9 | $10 | | 2 | $11 | $12 | $8 | ### Hours needed per unit: | Machine | A | B | C | |---------|---|---|---| | 1 | .4 | 1.1 | .9 | | 2 | .5 | 1.2 | 1.3 | **Total cost per unit for each machine:** - **Machine 1:** - Product A: $13 * .4 = $5.20 - Product B: $9 * 1.1 = $9.90 - Product C: $10 * .9 = $9.00 - **Total per unit for Machine 1:** $5.20 + $9.90 + $9.00 = **$24.10** - **Machine 2:** - Product A: $11 * .5 = $5.50 - Product B: $12 * 1.2 = $14.40 - Product C: $8 * 1.3 = $10.40 - **Total per unit for Machine 2:** $5.50 + $14.40 + $10.40 = **$30.30** --- ## Part 2: Setup Cost Analysis ### Setup costs: | Machine | Cost | |---------|-------| | 1 | $55 / $65 | | 2 | $93 / $58 | **Total setup costs for all jobs:** - **Machine 1:** $55 + $65 = **$120** - **Machine 2:** $93 + $58 = **$151** --- ## Part 3: Job Times | Job | Machine 1 (min) | Machine 2 (min) | Machine 3 (min) | Machine 4 (min) | Machine 5 (min) | Setup Time (min) | |-------|----------------|----------------|----------------|----------------|----------------|------------------| | 1 | 42 | | | | | 30 | | 2 | | 85 | | | 60 | 40 | | 3 | 58 | 45 | 37 | | | 42 | | 4 | | | | 58 | 54 | 75 | | 5 | | | | 38 | | 55 | ### Goal: Minimize total time (sum of processing and setup times). --- ## Solution Approach: Use **Johnson's Algorithm** (for two-machine flow shop) or heuristic methods to find an optimal sequence. Here, since multiple machines are involved, a heuristic or simulation approach might be best. ### Key insights: - Jobs with the shortest setup times should be scheduled to reduce total setup time. - Jobs with the shortest processing times should be prioritized on the respective machines. --- ## Step-wise Scheduling: ### Assign jobs with minimal setup times first: - Job 1 (30 min setup, Machine 1) - Job 2 (40 min setup, Machine 2) - Job 5 (55 min setup, Machine 4) - Job 3 (42 min setup, Machine 3) - Job 4 (75 min setup, Machine 4) --- ### Final Scheduling and Calculation: Given the complexity, an approximate optimal schedule involves: - Starting with Job 1 on Machine 1 (since it has the shortest setup time). - Scheduling other jobs considering dependencies and minimal total time. - Sum of all setup times: **$120 + $151 = $271** --- ## **Total Minutes Needed to Complete All Jobs** The total time includes: - Sum of all processing times considering the sequence. - Sum of all setup times. **Approximate total:** - Sum of processing times (based on job-machine assignment and order). - Add total setup time: **$271 minutes**. --- ## **Final Summary:** - **Minimize total time by sequencing jobs to reduce setup and processing times.** - **Total setup time for all jobs: $271 minutes.** - **Total processing time** (roughly estimated based on processing durations): approximately **~400 minutes**. - **Estimated total time to complete all jobs:** **around 671 minutes**. --- ## **Answer:** **Total number of minutes needed to complete the jobs = approximately 671 minutes.** --- *Note:* This is an estimation based on heuristic scheduling. Precise optimization would require detailed sequencing and possibly software tools or linear programming methods.