### **Step 1: Introduction / Overview** This problem deals with **production lot sizing** using the Economic Production Quantity (EPQ) model. The EPQ model helps determine the optimal batch size (run quantity) to minimize total setup and inventory holding costs when replenishment is not instantaneous (i.e., units are produced over time). We are provided with data on machine production rate, demand rate, setup cost, holding cost, and operational constraints, and asked to compute optimal run quantity, run and cycle times, inventory build-up rates, max inventory, and cost implications, including adjustments to accommodate a new job. --- ### **Step 2: Calculations for the Current Product** #### **A. Gather Data and Formulas** - **Demand rate (D):** 80 units/day - **Production rate (p):** 230 units/day - **Setup cost (S):** $260 - **Holding cost (H):** $14/year = $14/(250 days) = $0.056/day (since 5 days/week × 50 weeks = 250 operating days/year) - **Operating days/year:** 250 - **Formulas:** - EPQ (run quantity): \( Q^* = \sqrt{ \frac{2DS}{H[1 - (d/p)]} } \) - Run time: \( t_{run} = Q / p \) - Cycle time: \( t_{cycle} = Q / D \) - Inventory buildup rate: \( p - d \) - Max inventory: \( Q \times \left(1 - \frac{d}{p}\right) \) - Total annual cost: \( TC = \frac{DS}{Q} + \frac{Q}{2} \times H \times \left(1 - \frac{d}{p}\right) \) #### **B. Plug in the Numbers** ##### **Blank 1: Run Quantity (Q\*)** \[ Q^* = \sqrt{ \frac{2 \cdot 80 \cdot 250 \cdot 260}{14 \cdot [1 - (80/230)]} } \] First, let's make all units per year for consistency: - Annual demand \( D = 80 \text{ units/day} \times 250 \text{ days} = 20,000 \text{ units/year} \) - \( p = 230 \text{ units/day} \) - \( d = 80 \text{ units/day} \) So, \[ Q^* = \sqrt{ \frac{2 \cdot 20000 \cdot 260}{14 \cdot \left[1 - \frac{80}{230}\right]} } \] Calculate denominator: \[ 1 - \frac{80}{230} = 1 - 0.3478 = 0.6522 \] \[ 14 \times 0.6522 = 9.1308 \] Calculate numerator: \[ 2 \times 20000 \times 260 = 10,400,000 \] \[ Q^* = \sqrt{ \frac{10,400,000}{9.1308} } = \sqrt{1,139,236.9} = 1,067.25 \approx 1067 ~(\text{rounded to integer}) \] ##### **Blank 2: Length of Run Time (in days)** \[ t_{run} = \frac{Q^*}{p} = \frac{1067}{230} = 4.64 \text{ days} \] ##### **Blank 3: Length of Cycle Time (in days)** \[ t_{cycle} = \frac{Q^*}{d} = \frac{1067}{80} = 13.34 \text{ days} \] ##### **Blank 4: Inventory Build-Up Rate (units/day)** \[ \text{Inventory Build-Up Rate} = p - d = 230 - 80 = 150 \text{ units/day} \] ##### **Blank 5: Maximum Inventory Level** \[ \text{Max Inventory} = Q^* \times \left(1 - \frac{d}{p}\right) = 1067 \times (1 - 0.3478) = 1067 \times 0.6522 = 695.70 \] ##### **Blank 6: Total Annual Cost** \[ TC = \frac{DS}{Q} + \frac{Q}{2} \times H \times \left(1 - \frac{d}{p}\right) \] Where: - \( D = 20,000 \) - \( S = 260 \) - \( Q = 1067 \) - \( H = 14 \) - \( 1 - \frac{d}{p} = 0.6522 \) Plug in: \[ \text{First term: } \frac{20000 \times 260}{1067} = \frac{5,200,000}{1067} = 4874.79 \] \[ \text{Second term: } \frac{1067}{2} \times 14 \times 0.6522 = 533.5 \times 14 \times 0.6522 = 7469 \times 0.6522 = 4874.82 \] \[ TC = 4874.79 + 4874.82 = 9749.61 \] --- ### **Step 3: Feasibility for New Product and Adjustments** #### **A. Feasibility: Can new product be produced with current setting?** - The minimum cycle time needed = **9 days** - Current cycle time = **13.34 days** (> 9 days) - **YES**, it can be produced. #### **B. What would the run size need to be for a 9-day cycle?** \[ Q' = d \times t_{cycle} = 80 \times 9 = 720 \text{ units} \] #### **C. Total Annual Cost for Adjusted Run Size** \[ TC' = \frac{DS}{Q'} + \frac{Q'}{2} \times H \times \left(1 - \frac{d}{p}\right) \] \[ \text{First term: } \frac{20000 \times 260}{720} = \frac{5,200,000}{720} = 7222.22 \] \[ \text{Second term: } \frac{720}{2} \times 14 \times 0.6522 = 360 \times 14 \times 0.6522 = 5040 \times 0.6522 = 3286.99 \] \[ TC' = 7222.22 + 3286.99 = 10509.21 \] #### **D. Increase in Total Cost** \[ \text{Increase} = TC' - TC = 10509.21 - 9749.61 = 759.60 \] --- ### **Step 4: Final Answers** - **Blank 1:** 1067 - **Blank 2:** 4.64 - **Blank 3:** 13.34 - **Blank 4:** 150 - **Blank 5:** 695.70 - **Blank 6:** 9749.61 - **Blank 7:** YES - **Blank 8:** 720 - **Blank 9:** 10509.21 - **Blank 10:** 759.60