Let's break down the production planning step by step. We'll use the data and requirements provided in the image and problem statement. --- # **Given Data** **Forecast Demand:** - Spring: **21,000** - Summer: **9,200** - Fall: **14,400** - Winter:18,000** **Starting Info** - Starting workers: **64- Starting inventory: **910 units** - Productivity: **.5 unit/worker-hour** - Work: **8 hours/day, 60 days/quarter** **Costs:** - Hiring: **$100/new worker** - Layoff: **$200/worker** - Holding: **$18/unit/quarter** - Backorder: **$9/unit** - Regular time: **$11/hour** - Overtime: **$17/hour** **Union Rules:** - Layoff only at start of summer. - Hire only at end of summer (start of fall). - Use overtime only in spring (if needed). - Number of workers laid off at start of summer and hired at end of summer should match planned production needs for summer and fall. --- # **Step 1: Calculate Regular Workforce Production Capacity** ## **Per Worker, Per Quarter:** - \( 8 \) hours/day × \( 60 \) days = **480 hours/quarter** - Productivity: \( .5 \) units/hour × \( 480 \) hours = **240 units/worker/quarter** ## **Initial Total Capacity (Spring):** - 64 workers × 240 units = **15,360 units** --- # **Step 2: Fill Out Production Table** ## **Spring** ### **Production Required** \[ 21,000 - 910 (\text{starting inventory}) = 20,090 \] But since we may use overtime or backorder, let's proceed: - **Regular Production:** 64 × 240 = **15,360 units** - **Production shortfall:** \( 21,000 - (15,360 + 910) = 4,730 \) units ### **Overtime Needed** - Overtime units needed: **4,730 units** - Overtime hours needed: \( 4,730 \div .5 = 9,460 \) hours ### **Ending Inventory** - Units produced: 15,360 (regular) + 4,730 (overtime) = **20,090** - Total available: 20,090 + 910 = 21,000 - Ending inventory: **** ### **Ending Backorders** - None (demand met) --- ## **Summer** - **Demand:** 9,200 - **Beginning inventory:** - **Workforce adjustment:** Layoffs at start of summer. - Let workforce = \( x \) - \( x \) workers × 240 units = Must cover summer demand: 9,200 - \( x = 9,200 / 240 = 38.33 \) → **39 workers** (round up) - **Regular production:** 39 × 240 = 9,360 - **Ending inventory:** 9,360 - 9,200 = **160** - **Workers laid off:** 64 - 39 = **25** --- ## **Fall** - **Demand:** 14,400 - **Beginning inventory:** 160 - **New hires at start of fall.** - Let workforce = \( y \) - \( y \) × 240 = 14,400 - 160 = 14,240 - \( y = 14,240 / 240 = 59.33 \) → **60 workers** (round up) - **Regular production:** 60 × 240 = 14,400 - **Ending inventory:** (160 + 14,400) - 14,400 = **160** - **Workers hired:** 60 - 39 = **21** --- ## **Winter** - **Demand:** 18,000 - **Beginning inventory:** 160 - **Workforce = 60** - Regular production: 60 × 240 = **14,400** - Shortfall: 18,000 - (14,400 + 160) = **3,440** units (backorder) - Ending backorders: **3,440** --- # **Step 3: Calculate Hours and Costs** ### **Summary Table** | Quarter | Reg. Workforce | Reg. Prod | OT Prod | Total Prod | End Inv | Backorders | Hired | Laid Off | |---------|----------------|-----------|---------|------------|---------|------------|-------|----------| | Spring | 64 | 15,360 | 4,730 | 20,090 | | | | | | Summer | 39 | 9,360 | | 9,360 | 160 | | | 25 | | Fall | 60 | 14,400 | | 14,400 | 160 | | 21 | | | Winter | 60 | 14,400 | | 14,400 | | 3,440 | | | --- ## **Cost Calculations by Quarter** ### **Spring** - Regular time: 64 × 480 = **30,720 hours** × $11 = **$337,920** - Overtime: 9,460 hours × $17 = **$160,820** - Inventory holding: - Backorder: - Hiring/Layoff: ### **Summer** - Regular time: 39 × 480 = **18,720 hours** × $11 = **$205,920** - Overtime: - Inventory: 160 × $18 = **$2,880** - Backorder: - Hiring: - Layoff: 25 × $200 = **$5,000** ### **Fall** - Regular time: 60 × 480 = **28,800 hours** × $11 = **$316,800** - Overtime: - Inventory: 160 units × $18 = **$2,880** - Backorder: - Hiring: 21 × $100 = **$2,100** - Layoff: ### **Winter** - Regular time: 60 × 480 = **28,800 hours** × $11 = **$316,800** - Overtime: - Inventory: - Backorder: 3,440 × $9 = **$30,960** - Hiring/Layoff: --- # **Final Cost Summary** | Quarter | Regular Time | Overtime | Inventory | Backorder | Hiring | Layoff | **Total** | |---------|-------------|-----------|-----------|------------|--------|--------|----------------| | Spring | $337,920 | $160,820 | $ | $ | $ | $ | $498,740 | | Summer | $205,920 | $ | $2,880 | $ | $ | $5,000 | $213,800 | | Fall | $316,800 | $ | $2,880 | $ | $2,100 | $ | $321,780 | | Winter | $316,800 | $ | $ | $30,960 | $ | $ | $347,760 | | | | | | | | | | | **Total Cost** | | | | | | | **$1,382,080** | --- # **Final Summary** - **Production plan** uses regular time, overtime in spring, layoffs at start of summer, hiring at start of fall, and backorders in winter. - **Total cost for the year:** **$1,382,080** - **Workforce** varies per quarter to match planned production. - **No backorders** until winter, where demand exceeds regular capacity. - **Inventory** is carried over from summer to fall, balancing production. --- **If you need the results filled in the tables exactly as per your worksheet, let me know and I can format the answers in that way.**

# **Production Planning Summary** ## **Step 1: Calculate Regular Workforce Production Capacity** In this step, we determined the production capacity of each worker per quarter, which is 240 units. With 64 workers, the initial production capacity for spring is calculated to be 15,360 units. ## **Step 2: Fill Out Production Table** We calculated the production requirements for each season, considering demand, starting inventory, and workforce adjustments. This resulted in planned production for each quarter, detailing regular and overtime needs while managing inventory and backorders. ## **Step 3: Calculate Hours and Costs** In this step, we outlined the labor hours and costs associated with production for each quarter. A summary table was created to capture total costs across all quarters, leading to a final total cost of $1,382,080 for the year.