Let's solve this step by step for each question. --- ## **1. Construct the Total Operating Cost Curve (0 to 5,000 lbs/hr)** **Data from Table:** | Station | Min Crew | Max Crew | Station Capacity (lbs/hr) | Wage Rate ($/hr) | Labor Standard (lbs/hr/crew) | |-----------------|----------|----------|---------------------------|------------------|------------------------------| | Receiving | 2 | 4 | 5,000 | 8.00 | 1,500 | | Sorting | 2 | 5 | 6,000 | 7.50 | 1,000 | | Chopping | 1 | 6 | 6,000 | 8.00 | 3,000 | | Mixing | 2 | 4 | 5,000 | 8.50 | 1,500 | | Bottling | 2 | 4 | 5,000 | 9.50 | 2,500 | | Boxing/Shipping | 2 | 4 | 9,000 | 7.00 | 1,500 | | Administration | - | 2 | --- | 20.00 | --- | Utilities: **$50/hour** Let’s consider quantities: 0, 1,000, 2,000, 3,000, 4,000, 5,000 lbs/hr. ### **Step 1.1: Calculate Crew Needed per Station at Each Throughput** Crew needed = ceil(Throughput / Labor Standard). Use max crew if crew needed exceeds max. #### **Example Calculation for 3,000 lbs/hr:** - **Receiving:** ceil(3,000 / 1,500) = 2 (min crew 2) - **Sorting:** ceil(3,000 / 1,000) = 3 (min crew 2) - **Chopping:** ceil(3,000 / 3,000) = 1 (min crew 1) - **Mixing:** ceil(3,000 / 1,500) = 2 (min crew 2) - **Bottling:** ceil(3,000 / 2,500) = 2 (min crew 2) - **Boxing/Shipping:** ceil(3,000 / 1,500) = 2 (min crew 2) - **Administration:** 2 (fixed) ### **Step 1.2: Calculate Labor Cost per Station** Labor cost per hour = Crew needed × Wage Rate #### **3,000 lbs/hr Example:** | Station | Crew Needed | Wage Rate | Labor Cost/hr | |-----------------|-------------|-----------|--------------| | Receiving | 2 | 8.00 | $16.00 | | Sorting | 3 | 7.50 | $22.50 | | Chopping | 1 | 8.00 | $8.00 | | Mixing | 2 | 8.50 | $17.00 | | Bottling | 2 | 9.50 | $19.00 | | Boxing/Shipping | 2 | 7.00 | $14.00 | | Administration | 2 | 20.00 | $40.00 | | **Total Labor** | | | **$136.50** | ### **Step 1.3: Add Utilities** - Utilities per hour = $50 - **Total Operating Cost/hr = Labor Cost/hr + Utilities** #### **3,000 lbs/hr Example:** - Total = $136.50 + $50 = **$186.50/hr** --- ### **Step 1.4: Repeat for All Throughputs** #### **Summary Table** | Throughput (lbs/hr) | Receiving | Sorting | Chopping | Mixing | Bottling | Boxing | Admin | Labor Total | Utilities | **Total Cost/hr** | |---------------------|-----------|---------|----------|--------|----------|--------|-------|-------------|-----------|-------------------| | 0 | 2 × 8 = 16| 2 × 7.5 = 15| 1 × 8 = 8| 2 × 8.5 = 17| 2 × 9.5 = 19| 2 × 7 = 14| 2 × 20 = 40| $129 | $50 | **$179** | | 1,000 | 2 × 8 = 16| 2 × 7.5 = 15| 1 × 8 = 8| 2 × 8.5 = 17| 2 × 9.5 = 19| 2 × 7 = 14| 2 × 20 = 40| $129 | $50 | **$179** | | 2,000 | 2 × 8 = 16| 2 × 7.5 = 15| 1 × 8 = 8| 2 × 8.5 = 17| 2 × 9.5 = 19| 2 × 7 = 14| 2 × 20 = 40| $129 | $50 | **$179** | | 3,000 | 2 × 8 = 16| 3 × 7.5 = 22.5| 1 × 8 = 8| 2 × 8.5 = 17| 2 × 9.5 = 19| 2 × 7 = 14| 2 × 20 = 40| $136.5 | $50 | **$186.5** | | 4,000 | 3 × 8 = 24| 4 × 7.5 = 30| 2 × 8 = 16| 3 × 8.5 = 25.5| 2 × 9.5 = 19| 3 × 7 = 21| 2 × 20 = 40| $175.5 | $50 | **$225.5** | | 5,000 | 4 × 8 = 32| 5 × 7.5 = 37.5| 2 × 8 = 16| 4 × 8.5 = 34| 2 × 9.5 = 19| 4 × 7 = 28| 2 × 20 = 40| $206.5 | $50 | **$256.5** | --- ### **Step 1.5: (Graph)** - X-axis: Throughput (lbs/hr) [0, 1,000, ..., 5,000] - Y-axis: Total Operating Cost per Hour --- ## **2. Optimal Quantity per Hour for 75,000 lbs in One Week (40 hours)** Order = 75,000 lbs Max operation = 40 hours/week Required production rate = 75,000 lbs / 40 hrs = **1,875 lbs/hr** From the table above, at 2,000 lbs/hr (closest value), total operating cost/hr = **$179** --- ## **3. Calculate AOCH and AOCIb at q = 3,500 lbs/hr** From table, interpolate between 3,000 and 4,000 lbs/hr: - 3,000 lbs/hr: $186.5/hr - 4,000 lbs/hr: $225.5/hr Interpolate for 3,500 lbs/hr: \[ \text{Increment per 1,000 lbs/hr} = 225.5 - 186.5 = 39 \] \[ \text{Increment per 100 lbs/hr} = 39/10 = 3.9 \] \[ \text{Half increment} = 3.9 \times 5 = 19.5 \] So, \[ \text{Total Cost/hr at 3,500 lbs/hr} \approx 186.5 + 19.5 = \$206/hr \] #### **AOCH (Average Operating Cost per Hour)** - **AOCH = $206/hr** #### **AOCIb (Average Operating Cost per lb)** - **AOCIb = $206 / 3,500 = $0.0589/lb** --- ## **4. Calculate TVCH and AVCIb for q = 5,000 lbs/hr (Material Costs)** **Inputs:** - Onion cost: $0.50/lb - Spices cost: $0.25/lb - Jar cost: $0.05 - Case costs: $0.25 for 12 jars Each jar = 1 lb relish, needs 2 lbs onion. ### **Material Cost per 5,000 lbs/hr:** - Onion: 2 lbs/lb × 5,000 lbs = 10,000 lbs × $0.50 = $5,000/hr - Spices: 5,000 lbs × $0.25 = $1,250/hr - Jars: 5,000 × $0.05 = $250/hr - Cases: (5,000/12) × $0.25 = 416.67 × $0.25 = $104.17/hr \[ \text{Total Variable Cost per Hour (TVCH)} = \$5,000 + \$1,250 + \$250 + \$104.17 = \$6,604.17/hr \] ### **AVCIb (Average Variable Cost per lb):** \[ \text{AVCIb} = \$6,604.17 / 5,000 = \$1.3208/\text{lb} \] --- ## **Summary Table of Answers** | Part | Answer | |------|--------| | 1. | See cost curve table above | | 2. | Optimal rate = **1,875 lbs/hr** at $179/hr | | 3. | AOCH = **$206/hr**, AOCIb = **$0.0589/lb** | | 4. | TVCH = **$6,604.17/hr**, AVCIb = **$1.3208/lb** | --- **Let me know if you need the graph or any further breakdown!**

Certainly! Let's go through each problem step-by-step with detailed calculations and clear explanations. --- ## **Part 1: Construct the Total Operating Cost Curve (0 to 5,000 lbs/hr)** **Step 1.1: Extract data for each station** | Station | Min Crew | Max Crew | Capacity (lbs/hr) | Wage Rate ($/hr) | Labor Standard (lbs/hr/crew) | |--------------------|----------|----------|-------------------|------------------|------------------------------| | Receiving | 2 | 4 | 5,000 | 8.00 | 1,500 | | Sorting | 2 | 5 | 6,000 | 7.50 | 1,000 | | Chopping | 1 | 6 | 6,000 | 8.00 | 3,000 | | Mixing | 2 | 4 | 5,000 | 8.50 | 1,500 | | Bottling | 2 | 4 | 5,000 | 9.50 | 2,500 | | Boxing/Shipping | 2 | 4 | 9,000 | 7.00 | 1,500 | | Administration | Fixed at 2 | - | - | 20.00 | - | Utilities cost = **$50/hour** --- **Step 1.2: Determine crew needed at each throughput** Calculate crew needed = ceil(throughput / labor standard). Use max crew if required. **Example for 3,000 lbs/hr:** - Receiving: ceil(3000/1500)=2 → min crew 2 - Sorting: ceil(3000/1000)=3 → min crew 2 (since max is 5, 3 is acceptable) - Chopping: ceil(3000/3000)=1 → min crew 1 - Mixing: ceil(3000/1500)=2 → min crew 2 - Bottling: ceil(3000/2500)=2 → min crew 2 - Boxing: ceil(3000/1500)=2 → min crew 2 - Admin: 2 (fixed) Repeat for all throughput levels. --- **Step 1.3: Calculate crew wages at each throughput** Multiply crew needed by wage rate per station. **Example at 3000 lbs/hr:** | Station | Crew Needed | Wage Rate ($/hr) | Labor Cost ($/hr) | |--------------------|--------------|------------------|-------------------| | Receiving | 2 | 8.00 | 2×8=16 | | Sorting | 3 | 7.50 | 3×7.50=22.50 | | Chopping | 1 | 8.00 | 8.00 | | Mixing | 2 | 8.50 | 17.00 | | Bottling | 2 | 9.50 | 19.00 | | Boxing/Shipping | 2 | 7.00 | 14.00 | | Admin | 2 | 20.00 | 40.00 | **Total labor cost:** 16 + 22.50 + 8 + 17 + 19 + 14 + 40 = **$136.50** --- **Step 1.4: Add utilities** Utilities are constant at $50/hour. **Total operating cost per hour:** Labor cost + utilities = **$136.50 + $50 = $186.50** --- **Step 1.5: Repeat for all throughput levels** | Throughput | Crew for each station | Total labor cost | Total cost (labor + utilities) | |--------------|-------------------------|------------------|------------------------------| | 0 lbs/hr | All at min crew (or fixed) | Sum of minimal crew wages | $50 (utilities only) + fixed costs | | 1,000 lbs/hr | Similar calculation | Approximately same as below | **$179** | | 2,000 lbs/hr | Same process | **$179** | | 3,000 lbs/hr | As above | **$186.50** | | 4,000 lbs/hr | Higher crew needed | **$225.50** | | 5,000 lbs/hr | Max crew needed | **$256.50** | *(Exact numbers can be refined with precise crew calculations, but these are close estimates based on the labor standards and capacities.)* **Final step:** Plot the total costs against throughput from 0 to 5,000 lbs/hr to visualize the cost curve. --- ## **Part 2: Determine Optimal Operating Quantity for 75,000 lbs in 1 Week** **Step 2.1: Calculate required throughput per hour** Total order = 75,000 lbs Time available = 40 hours/week \[ \text{Required rate} = \frac{75,000}{40} = 1,875\, \text{lbs/hr} \] **Step 2.2: Find closest cost-effective rate** From the table, at 2,000 lbs/hr, total cost per hour ≈ $179. **Conclusion:** Operate at **around 1,875 lbs/hr**, but since the cost curve is near 2,000 lbs/hr, the optimal operational rate is approximately **2,000 lbs/hr** with a cost of about **$179/hour**. --- ## **Part 3: At q = 3,500 lbs/hr — Calculate AOCH and AOCIb** **Step 3.1: Interpolate total cost at 3,500 lbs/hr** From previous data: - 3,000 lbs/hr: $186.50 - 4,000 lbs/hr: $225.50 Difference: 225.50 - 186.50 = $39 over 1,000 lbs/hr. Per 100 lbs/hr: \[ \frac{39}{10} = 3.90 \] For 500 lbs (from 3,000 to 3,500): \[ 3.90 \times 5 = 19.50 \] Estimate: \[ \text{Cost at 3,500 lbs/hr} \approx 186.50 + 19.50 = \boxed{\$206} \] **Step 3.2: Calculate AOCH** \[ \text{AOCH} = \frac{\text{Total hourly cost}}{\text{Quantity}} = \frac{\$206}{3,500} \approx \boxed{\$0.0589 \text{ per pound}} \] --- ## **Part 4: Material Cost Analysis at q = 5,000 lbs/hr** **Material costs per pound:** - Onion: $0.50 per lb - Spices: $0.25 per lb - Jar: $0.05 per jar (per lb relish) - Case: $0.25 per case of 12 jars **Step 4.1: Calculate total material costs per hour** - Onion: 2 lbs per pound of relish \[ 2 \times 5,000 = 10,000\, \text{lbs onion} \\ \$0.50 \times 10,000 = \$5,000 \] - Spices: \[ \$0.25 \times 5,000 = \$1,250 \] - Jars: \[ \$0.05 \times 5,000 = \$250 \] - Cases: \[ \frac{5,000\, \text{jars}}{12} \approx 416.67\, \text{cases} \\ \$0.25 \times 416.67 \approx \$104.17 \] **Total material cost per hour:** \[ \$5,000 + \$1,250 + \$250 + \$104.17 = \boxed{\$6,604.17} \] --- **Step 4.2: Calculate TVCH and AVCib** - **TVCH:** Total Variable Cost per Hour \[ \boxed{\$6,604.17} \] - **AVCIb:** Average variable cost per pound \[ \frac{\$6,604.17}{5,000} \approx \boxed{\$1.3208 \text{ per pound}} \] --- # **Summary of Final Results** | Part | Result | |---------|--------------| | **1. Cost Curve** | Estimated costs from ~$179 to ~$256.50 as throughput increases from 0 to 5,000 lbs/hr (see above table). | | **2. Optimal Rate** | Approximately **1,875 lbs/hr** (close to 2,000 lbs/hr) with cost ~$179/hour. | | **3. At 3,500 lbs/hr** | AOCH ≈ **$0.0589/lb**, AOCIb ≈ **$0.059/lb**. | | **4. Material Costs at 5,000 lbs/hr** | TVCH ≈ **$6,604.17/hour**, AVCIb ≈ **$1.32/lb**. | --- Let me know if you'd like me to prepare the actual graph or clarify any step further!