# Explanation of the Inventory Management Problem Let's break down the scenario and the data provided, step by step, to help you understand how to approach minimizing the combined cost of ordering and holding inventory. --- ## 1. **Scenario Overview** - **Steady Demand:** The factory produces a single product, and the demand is constant all year. - **No Backlog:** All units made are immediately sold—no leftover demand. - **Goal:** Find the best way to order raw materials so that the total cost of ordering and holding inventory is minimized. --- ## 2. **Key Data from the Table** | Parameter | Value | |-------------------------------|------------------------| | **Monthly Demand** | 9,000 units | | **Annual Demand** | 108,000 units | | **Unit Production Cost** | ₹500 | | **Annual Holding Cost Rate** | 20% of unit cost | | **Holding Cost per Unit/Year**| ₹100 | | **Ordering Cost per Order** | ₹2,250 | | **Lead Time** | 4 working days | | **Working Days per Month** | 25 days | --- ### **What Each Parameter Means:** - **Monthly/Annual Demand:** How many units are needed each month/year. - **Unit Production Cost:** Cost to make one unit (₹500). - **Annual Holding Cost Rate:** The yearly cost to store one unit, as a percentage of its cost (20% of ₹500 = ₹100). - **Holding Cost per Unit/Year:** Directly given as ₹100; this matches the calculation above. - **Ordering Cost:** Fixed cost incurred every time you place an order (₹2,250). - **Lead Time:** Time between placing an order and receiving materials (4 working days). - **Working Days per Month:** For planning, there are 25 days in a month. --- ## 3. **Your Objective** - **Minimize Total Inventory Cost**, which has two components: 1. **Ordering Cost:** Total cost from placing orders throughout the year. 2. **Holding Cost:** Cost for storing (holding) inventory. --- ## 4. **How to Approach the Problem** This is a classic **Economic Order Quantity (EOQ)** problem. EOQ tells you: - How much to order each time (order quantity, \( Q^* \)) - How many orders to place per year ### **EOQ Formula:** \[ EOQ = Q^* = \sqrt{\frac{2DS}{H}} \] Where: - \( D \) = Annual demand (units/year) - \( S \) = Ordering cost per order (₹) - \( H \) = Annual holding cost per unit (₹) --- ## 5. **What’s Next?** - **Calculate EOQ using the data.** - **Find the number of orders per year and the order cycle time.** - **Calculate total annual cost (ordering + holding).** - **Consider lead time for planning reorder point (when to order).** --- ## 6. **Key Points to Remember** - **EOQ minimizes the sum of ordering and holding costs.** - **Lead time helps you decide when to place the next order so you don’t run out.** - **No backlog means inventory should always be available to meet demand.** --- ### **Would you like to see the step-by-step EOQ calculation with these numbers?**

# Complete Solution to the EOQ Problem Let's calculate the Economic Order Quantity (EOQ) step by step using the provided data. --- ## 1. **Given Data** - **Annual Demand (\(D\))**: 108,000 units - **Ordering Cost (\(S\))**: ₹2,250 per order - **Holding Cost per Unit per Year (\(H\))**: ₹100 --- ## 2. **EOQ Calculation** Using the EOQ formula: \[ EOQ = Q^* = \sqrt{\frac{2DS}{H}} \] ### **Plugging in the values:** \[ Q^* = \sqrt{\frac{2 \times 108,000 \times 2,250}{100}} \] ### **Calculating the numerator:** \[ 2 \times 108,000 \times 2,250 = 486,000,000 \] ### **Calculating the EOQ:** \[ Q^* = \sqrt{\frac{486,000,000}{100}} = \sqrt{4,860,000} \approx 2204.12 \] Thus, the **optimal order quantity (EOQ)** is approximately **2,204 units**. --- ## 3. **Number of Orders Per Year** To find how many orders are placed in a year: \[ \text{Number of Orders} = \frac{D}{Q^*} \] \[ \text{Number of Orders} = \frac{108,000}{2204.12} \approx 49 \] --- ## 4. **Order Cycle Time** To determine how often orders are placed: \[ \text{Order Cycle Time} = \frac{12 \text{ months}}{\text{Number of Orders}} \] \[ \text{Order Cycle Time} = \frac{12}{49} \approx 0.245 \text{ months} \approx 7.4 \text{ days} \] --- ## 5. **Total Annual Cost** The total annual cost consists of the ordering cost and holding cost. ### **Ordering Cost Calculation:** \[ \text{Total Ordering Cost} = \text{Number of Orders} \times S \] \[ \text{Total Ordering Cost} = 49 \times 2,250 = ₹110,250 \] ### **Holding Cost Calculation:** \[ \text{Average Inventory} = \frac{Q^*}{2} = \frac{2204.12}{2} \approx 1102.06 \] \[ \text{Total Holding Cost} = \text{Average Inventory} \times H \] \[ \text{Total Holding Cost} = 1102.06 \times 100 = ₹110,206 \] ### **Total Annual Cost:** \[ \text{Total Cost} = \text{Total Ordering Cost} + \text{Total Holding Cost} \] \[ \text{Total Cost} = 110,250 + 110,206 = ₹220,456 \] --- ## 6. **Reorder Point Calculation** To ensure that inventory is ordered just in time before running out, calculate the reorder point based on lead time. ### **Daily Demand:** \[ \text{Daily Demand} = \frac{D}{\text{Working Days per Year}} = \frac{108,000}{300} = 360 \text{ units/day} \] ### **Reorder Point:** \[ \text{Reorder Point} = \text{Daily Demand} \times \text{Lead Time} \] \[ \text{Reorder Point} = 360 \times 4 = 1,440 \text{ units} \] --- ## **Summary of Results** - **EOQ**: 2,204 units - **Number of Orders**: 49 orders/year - **Order Cycle Time**: 7.4 days - **Total Annual Cost**: ₹220,456 - **Reorder Point**: 1,440 units Feel free to ask if you have any questions or need further clarification on any step!