Consider An Inventory System For Replenishment

Consider an inventory system whereby an inventory replenishment order

Consider an inventory system whereby an inventory replenishment order for Q units is placed whenever inventory reaches ROP units. The lead time for the order is LT days and daily demand is d units. Beginning on-hand inventory at time=0 is OH units. Use these parameters: Q = 1,500; ROP = 2,200; LT = 7; d = 300; OH = beginning on-hand inventory = 2,800. and provide an answer for each of the following: Long-run average inventory on-hand (note – don't include the inventory in the first order cycle, since it depends on the beginning inventory, but determine what the long run average amount of on-hand inventory will be once order cycles and inventory amounts fall into a regular pattern): Inventory on hand at time=10: Number of orders outstanding (in the “pipeline†but not yet received) at time=10 (your number will be 0,1,2, or 3): Order cycle length (time between orders): days. (Draw yourself a “sawtooth diagram†for analyzing this, like we did in class and posted in an example on Blackboard. The diagram "maps out" inventory at discrete points in time t=0, 1 day, 2 days, etc., as though demand occurs between two points in time and order placement and order receipt occur at a specific point in time. So, inventory at time 1 = starting on-hand inventory – d units. If the result is equal to the reorder point (ROP), then an order for Q units should be placed at time 1 and will arrive at time 1+LT).

Paper For Above instruction

The management of inventory in supply chain systems is critical for maintaining operational efficiency and customer satisfaction. A key component of inventory management is understanding the order replenishment cycle, which involves determining optimal reorder points, order quantities, and the resulting inventory levels over time. In this analysis, we examine a specific inventory system with given parameters to evaluate important metrics such as long-run average inventory on-hand, current inventory levels, orders outstanding, and the cycle length between orders. This comprehensive evaluation provides insights into optimizing inventory replenishment strategies in a just-in-time environment.

The parameters for this system are: order quantity (Q) = 1500 units; reorder point (ROP) = 2200 units; lead time (LT) = 7 days; daily demand (d) = 300 units; initial on-hand inventory (OH) = 2800 units. The goal is to understand the behavior of the inventory under these parameters and to derive key performance indicators that inform optimal operational practices.

Long-Run Average Inventory On-Hand

The long-run average inventory on-hand can be approximated by analyzing the cycle stock during repeated replenishment cycles, excluding initial stock. Since demand is consistent at 300 units per day over a 7-day lead time, the system follows a sawtooth pattern where inventory decreases linearly from a maximum following replenishment until reaching the ROP, prompting a new order. The maximum inventory before new orders are placed is Q units, and the average inventory over a cycle, considering uniform depletion, is Q/2. However, since initial stock is not part of the regular pattern, we subtract it for a precise estimate.

Amortized over multiple cycles, the long-run average inventory is computed as:

\[

\text{Average inventory} = \frac{Q}{2} + \text{average safety stock}

\]

In systems with deterministic demand and lead times, the safety stock accounts for variability, but here demand is assumed constant, making safety stock minimal. Therefore, the long-run average on-hand inventory is approximately Q/2, equal to 750 units.

Inventory at Time=10

To evaluate inventory at time t=10 days, we analyze the depletion from the last replenishment cycle. Starting with the maximum inventory, inventory decreases by 300 units per day. Since the last replenishment occurs when inventory drops to ROP at about day 7, a new order is triggered and arrives on day 14. Between day 7 and 14, inventory declines linearly, reaching near zero by day 11, but since the order arrives on day 14, at day 10 the inventory is approximately:

Initial inventory at last order: Q units (1500 units)

Demand since last order: 10 days * 300 units/day = 3000 units

But since Q units are replenished when inventory drops to ROP, and the depletion since last order is 10 days * 300 units/day = 3000 units, which exceeds the replenishment size, the current inventory at day 10 is approximately:

Maximum inventory after last order: 1500 units

Inventory at day 10: 1500 - (300 * 10) + some safety consideration (negligible here), but since initial stock exceeds demand, inventory still remains above zero, approximately 1500 - 3000 + adjustments, leading to an estimate of around 500 units. More precise calculations would be based on actual cycle times, but roughly, inventory would be around 500 units at day 10.

Number of Orders Outstanding at Time=10

By day 10, the number of outstanding orders (in the pipeline) depends on their placement relative to demand and lead time. Since orders are placed after inventory drops to ROP and arrive after 7 days, and considering the current cycle, there would be:

- A pending order placed around day 7, arriving around day 14.

- Since the current date is day 10, only the order placed at day 7 is in transit, so the number of outstanding orders at day 10 is 1.

Order Cycle Length

The order cycle length can be estimated based on demand and order size. Each cycle begins when inventory is replenished to Q units, and depletes to ROP before the next order. The total demand during a cycle is Q units, and with a daily demand of 300 units, the cycle length is:

Cycle length = Q / d = 1500 / 300 = 5 days.

In practice, considering the lead time of 7 days, the cycle length may be set slightly longer to account for safety stock and demand variability, but under deterministic assumptions, the basic cycle is 5 days.

Conclusion

Understanding this inventory system's behavior enables managers to optimize reorder points and quantities, minimizing costs while maintaining service levels. The sawtooth pattern analysis demonstrates the importance of balancing demand with replenishment timing to sustain smooth operations. Accurate calculations of inventory at specific times assist in planning and understanding the system's responsiveness.

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