Let's break down the problem and then summarize the findings. --- ## **Restating the Problem:** - **Company:** Reboot Incorporated (makes hiking boots) - **Demand per quarter (pairs):** - Q1: 3,000 - Q2: 4,000 - Q3: 8,000 - Q4: 7,000 - **Production capacity:** Max 6,400 pairs per quarter - **Profit per pair sold:** $20 - **Inventory cost per pair per quarter:** $8 - **Initial inventory:** 1,000 pairs at start of Q1 **Goal:** Meet all demand, decide optimal production plan for 4 quarters, and calculate net profit. --- ## **Analysis & Solution Approach:** ### 1. **Production Plan:** You need to: - Meet all demand (can use inventory and production) - Cannot produce more than 6,400 pairs in any quarter The suggested (incorrect) solution in the image was to simply produce exactly the demand each quarter. This does **not** account for capacity limits in Q3 and Q4 (when demand exceeds 6,400). ### 2. **Inventory Tracking:** At any point: - Inventory at start of quarter + produced in quarter - sold in quarter = inventory at end of quarter ### 3. **Profit Calculation:** - **Total sales profit:** (Total units sold) x $20 - **Total inventory holding cost:** (Units held in inventory at end of Q1, Q2, Q3) x $8 per quarter --- ## **Corrected Production & Inventory Plan:** ### **Step-by-Step Calculation:** #### **Quarter 1:** - **Demand:** 3,000 - **Start inventory:** 1,000 - **Production?** Let’s maximize inventory for later high-demand: - If produce 6,400 (max): Inventory end = 1,000 + 6,400 - 3,000 = **4,400** - **Holding cost:** 4,400 x $8 = $35,200 #### **Quarter 2:** - **Demand:** 4,000 - **Start inventory:** 4,400 - **Production:** 6,400 (max): Inventory end = 4,400 + 6,400 - 4,000 = **6,800** - **Holding cost:** 6,800 x $8 = $54,400 #### **Quarter 3:** - **Demand:** 8,000 - **Start inventory:** 6,800 - **Production:** 6,400 (max): Inventory end = 6,800 + 6,400 - 8,000 = **5,200** - **Holding cost:** 5,200 x $8 = $41,600 #### **Quarter 4:** - **Demand:** 7,000 - **Start inventory:** 5,200 - **Production:** Now, produce only what is needed: - Need to have 5,200 + X = 7,000 → X = 1,800 - But can produce up to 6,400; only need 1,800. - **End inventory:** 0 (all used up) - **Holding cost:** $0 --- ### **Total Production Plan:** | Quarter | Production | |------------|------------| | Quarter 1 | 6,400 | | Quarter 2 | 6,400 | | Quarter 3 | 6,400 | | Quarter 4 | 1,800 | --- ### **Calculate Net Profit:** - **Total sales:** 3,000 + 4,000 + 8,000 + 7,000 = **22,000 pairs** - **Sales profit:** 22,000 x $20 = **$440,000** - **Total holding costs:** $35,200 + $54,400 + $41,600 = **$131,200** - **Net profit:** $440,000 - $131,200 = **$308,800** --- ## **Final Summary:** - **Optimal production schedule:** - Q1: 6,400 - Q2: 6,400 - Q3: 6,400 - Q4: 1,800 - **Net profit:** **$308,800** --- **Explanation:** Producing to the maximum allowed in early quarters builds up inventory to meet higher demand later, avoiding missed sales and maximizing profit. Always consider both production constraints and inventory costs for best results. Let me know if you'd like a step-by-step spreadsheet layout!

Certainly! Here's a clear, detailed explanation and final summary of the problem and its solution: --- ### **Problem Restatement:** Reboot Incorporated manufactures hiking boots with the following parameters: - **Demand per quarter:** - Q1: 3,000 pairs - Q2: 4,000 pairs - Q3: 8,000 pairs - Q4: 7,000 pairs - **Production capacity:** Max 6,400 pairs per quarter - **Initial inventory:** 1,000 pairs at the start of Q1 - **Profit per pair sold:** $20 - **Inventory holding cost per pair per quarter:** $8 - **Objective:** Develop a production schedule over four quarters to meet all demand, maximize profit, and determine the net profit. --- ### **Approach & Reasoning:** 1. **Capacity constraints:** Can't produce more than 6,400 pairs in any quarter. 2. **Inventory management:** Use inventory to cover high demand in Q3 and Q4. 3. **Goal:** Maximize profit by producing the maximum allowable in earlier quarters to build inventory for later, higher demand quarters. --- ### **Step-by-step Production & Inventory Plan:** **Quarter 1:** - Starting inventory: 1,000 - Demand: 3,000 - Produce max: 6,400 - Ending inventory: 1,000 + 6,400 - 3,000 = **4,400** **Quarter 2:** - Starting inventory: 4,400 - Demand: 4,000 - Produce max: 6,400 - Ending inventory: 4,400 + 6,400 - 4,000 = **6,800** **Quarter 3:** - Starting inventory: 6,800 - Demand: 8,000 - Produce max: 6,400 - Ending inventory: 6,800 + 6,400 - 8,000 = **5,200** **Quarter 4:** - Starting inventory: 5,200 - Demand: 7,000 - Produce only what's needed: 1,800 (since inventory + production covers demand) - Ending inventory: 0 (all demand met exactly) --- ### **Summary of Production Schedule:** | Quarter | Production | Ending Inventory | |------------|--------------|-------------------| | Q1 | 6,400 | 4,400 | | Q2 | 6,400 | 6,800 | | Q3 | 6,400 | 5,200 | | Q4 | 1,800 | 0 | --- ### **Profit Calculation:** - **Total units sold:** 3,000 + 4,000 + 8,000 + 7,000 = **22,000 pairs** - **Gross profit:** 22,000 x $20 = **$440,000** - **Total inventory holding costs:** (Q1 end): 4,400 x $8 = $35,200 (Q2 end): 6,800 x $8 = $54,400 (Q3 end): 5,200 x $8 = $41,600 **Total holding costs:** $35,200 + $54,400 + $41,600 = **$131,200** - **Net profit:** $440,000 (sales profit) - $131,200 (holding costs) = **$308,800** --- ### **Final Results:** **a. Production schedule for the next four quarters:** - Q1: **6,400 pairs** - Q2: **6,400 pairs** - Q3: **6,400 pairs** - Q4: **1,800 pairs** **b. Net profit:** **$308,800** --- ### **Summary Statement:** By producing the maximum capacity of 6,400 pairs in the first three quarters, Reboot Inc. builds sufficient inventory to meet the high demand in Q3 and Q4 without missing sales. This strategy maximizes profit after accounting for inventory holding costs, resulting in a total net profit of **$308,800**. --- Let me know if you want a detailed spreadsheet layout or further clarifications!

Certainly! Here's a comprehensive explanation and final summary of the problem and its solution: --- ### **Restatement of the Problem** **Company:** Reboot Incorporated (manufactures hiking boots) **Demand per quarter:** - Q1: 3,000 pairs - Q2: 4,000 pairs - Q3: 8,000 pairs - Q4: 7,000 pairs **Production constraints:** - Maximum production capacity per quarter: 6,400 pairs - Initial inventory at start of Q1: 1,000 pairs **Financial parameters:** - Profit per pair sold: $20 - Inventory holding cost per pair per quarter: $8 **Objective:** Develop an optimal production schedule over four quarters to meet all demand, maximize profit, and compute the net profit. --- ### **Approach & Key Considerations** - **Capacity constraints:** Cannot produce more than 6,400 pairs in any quarter. - **Inventory management:** Use production in early quarters to build inventory for later high-demand quarters. - **Goal:** Maximize profit by producing at capacity early, minimizing lost sales and inventory costs. --- ### **Step-by-Step Production & Inventory Plan** **Quarter 1:** - **Starting inventory:** 1,000 - **Demand:** 3,000 - **Production:** max capacity = 6,400 - **Ending inventory:** 1,000 + 6,400 - 3,000 = **4,400** **Quarter 2:** - **Starting inventory:** 4,400 - **Demand:** 4,000 - **Production:** 6,400 - **Ending inventory:** 4,400 + 6,400 - 4,000 = **6,800** **Quarter 3:** - **Starting inventory:** 6,800 - **Demand:** 8,000 - **Production:** 6,400 (max capacity) - **Ending inventory:** 6,800 + 6,400 - 8,000 = **5,200** **Quarter 4:** - **Starting inventory:** 5,200 - **Demand:** 7,000 - **Production needed:** 1,800 (to meet demand) - **Ending inventory:** 0 (all demand met) --- ### **Final Production Schedule** | Quarter | Production | Ending Inventory | |-----------|--------------|------------------| | Q1 | 6,400 | 4,400 | | Q2 | 6,400 | 6,800 | | Q3 | 6,400 | 5,200 | | Q4 | 1,800 | 0 | --- ### **Profit Calculation** - **Total units sold:** 3,000 + 4,000 + 8,000 + 7,000 = **22,000 pairs** - **Gross profit from sales:** 22,000 x $20 = **$440,000** - **Total inventory holding costs:** - End of Q1: 4,400 units x $8 = $35,200 - End of Q2: 6,800 units x $8 = $54,400 - End of Q3: 5,200 units x $8 = $41,600 - **Total holding costs:** $131,200 - **Net profit:** $440,000 (sales profit) - $131,200 (holding costs) = **$308,800** --- ### **Final Summary** - **Production schedule for maximum profit:** - **Q1:** Produce 6,400 pairs - **Q2:** Produce 6,400 pairs - **Q3:** Produce 6,400 pairs - **Q4:** Produce 1,800 pairs - **Expected net profit:** **$308,800** This plan leverages maximum early production to build inventory, ensuring demand in high-demand quarters is met without missing sales, thus optimizing overall profit after inventory costs. --- Let me know if you'd like a detailed spreadsheet model or further clarifications!