# Economic Production Quantity (EPQ) Calculation ## 1. **Given Data** | Parameter | Symbol | Value | |------------------------|----------------|-----------------------------| | Annual demand | \( D \) | 15,000 units/year | | Production rate | \( P \) | 48 units/hour | | Operating hours/day | - | 8 | | Days/week | - | 5 | | Weeks/year | - | 50 | | Setup cost per batch | \( S \) | \$150 | | Holding cost/unit/year | \( H \) | \$7 | | Production cost/unit | \( C \) | \$20 | | Setup time per batch | - | 3 hours | | Shortages allowed? | - | No | --- ## 2. **EPQ Model Formula** The Economic Production Quantity (EPQ) formula is: \[ Q^* = \sqrt{ \frac{2DS}{H} \cdot \frac{P}{P - d} } \] Where: - \( Q^* \): Optimal production lot size - \( D \): Annual demand - \( S \): Setup cost per batch - \( H \): Holding cost per unit per year - \( P \): Production rate (units/year) - \( d \): Demand rate (units/year) ### **Step 1: Calculate Production and Demand Rates** - **Production rate (\( P \)), in units/year:** - \( 48 \) units/hour × \( 8 \) hours/day × \( 5 \) days/week × \( 50 \) weeks/year - \( P = 48 \times 8 \times 5 \times 50 = 96,000 \) units/year - **Demand rate (\( d \)):** - \( D = 15,000 \) units/year --- ## 3. **Calculate EPQ** \[ Q^* = \sqrt{ \frac{2 \times 15,000 \times 150}{7} \cdot \frac{96,000}{96,000 - 15,000} } \] - \( 2DS = 2 \times 15,000 \times 150 = 4,500,000 \) - \( 4,500,000 / 7 = 642,857.14 \) - \( P/(P-d) = 96,000 / (96,000 - 15,000) = 96,000 / 81,000 \approx 1.185 \) - Multiply: \( 642,857.14 \times 1.185 \approx 761,785.21 \) - Take square root: \( Q^* = \sqrt{761,785.21} \approx 872.93 \) **So, the optimal production lot size is:** \[ \boxed{873 \text{ units (rounded to nearest unit)}} \] --- ## 4. **Calculate Production-Related Costs** ### **a. Number of Production Runs per Year** \[ \text{Number of runs} = \frac{D}{Q^*} = \frac{15,000}{873} \approx 17.19 \approx 18 \text{ runs (round up for planning)} \] But for cost calculation, use \( 17.19 \). ### **b. Setup Cost per Year** \[ \text{Annual Setup Cost} = \left(\frac{D}{Q^*}\right) \times S = 17.19 \times \$150 = \$2,578.50 \] ### **c. Average Inventory Level** \[ \text{Average Inventory} = Q^* \left(1 - \frac{d}{P}\right) / 2 \] \[ \frac{d}{P} = \frac{15,000}{96,000} = .15625 \] \[ 1 - \frac{d}{P} = 1 - .15625 = .84375 \] \[ \text{Average Inventory} = \frac{873 \times .84375}{2} = \frac{736.50}{2} = 368.25 \text{ units} \] ### **d. Annual Holding Cost** \[ \text{Annual Holding Cost} = \text{Average Inventory} \times H = 368.25 \times \$7 = \$2,577.75 \] ### **e. Annual Production Cost** \[ \text{Annual Production Cost} = D \times C = 15,000 \times 20 = \$300,000 \] --- ## 5. **Total Annual Cost** \[ \text{Total Cost} = \text{Setup Cost} + \text{Holding Cost} + \text{Production Cost} \] \[ = 2,578.50 + 2,577.75 + 300,000 = \$305,156.25 \] --- ## 6. **Summary Table** | Cost Component | Amount ($) | |----------------------|--------------| | Setup Cost | 2,578.50 | | Holding Cost | 2,577.75 | | Production Cost | 300,000 | | **Total Cost** | **305,156.25** | --- ## 7. **Final Summary** - **Optimal production lot size (EPQ):** **873 units** per batch. - **Number of production runs per year:** ~17.2 - **Total annual production-related cost:** **\$305,156.25** - **Cost breakdown:** Setup and holding costs are nearly equal (~\$2,578 each), with production cost dominating. - **Interpretation:** Producing batches of 873 units balances setup and holding costs, minimizing the total cost for the company under the given conditions. --- ###  *Alt text: Assembly line in a factory, symbolizing batch production*

# Summary of Steps for EPQ Calculation ## 1. Given Data This step outlines the relevant parameters necessary for the Economic Production Quantity (EPQ) calculation, including annual demand, production rate, setup and holding costs, and operational hours. These values form the foundation for the subsequent calculations. ## 2. EPQ Model Formula The EPQ formula is introduced, which helps determine the optimal production lot size that minimizes costs. Key variables in the formula include annual demand, setup costs, holding costs, production rates, and demand rates. ## 3. Calculate EPQ Using the provided data, the optimal production lot size \( Q^* \) is computed. The result, approximately 873 units, indicates the ideal batch size for production to balance setup and holding costs. ## 4. Calculate Production-Related Costs This section calculates the number of production runs per year and the associated costs, including setup, holding, and production costs. The calculations yield annual costs for setup and holding, which will be summed to find the total cost. ## 5. Total Annual Cost The total annual production-related cost is calculated by summing setup, holding, and production costs. The total cost amounts to \$305,156.25, reflecting the combined impact of these expenses. ## 6. Summary Table A concise table presents the breakdown of costs associated with the production process, including setup, holding, and production costs. This visual representation helps clarify the financial implications of the production strategy. ## 7. Final Summary The final summary reiterates the optimal production lot size of 873 units and the total annual cost of \$305,156.25. It emphasizes the balance achieved between setup and holding costs, highlighting the efficiency of the production strategy.