How Is Quantity Determined In Inventory Control Cover EOQ

How Is Quantity Determined In Inventory Control Cover EOQ Includi

How is quantity determined in inventory control? Cover EOQ (including quantity discounts and EPQ), FOI, and the Single Period Model. In addition, give simple examples with input information, tabulation, graphs, and calculations of each of the following making up your own data. Explain as you proceed. (80 points) a. EOQ with quantity discounts using a constant holding cost b. EPQ c. FOI d. Single Period Model with discrete measures using the Poisson distribution 2. Discuss when inventory is replenished. Give a simple example of an ROP with variable demand with constant lead time. Make up your own data. Explain as you proceed. (20 points)

Paper For Above instruction

Introduction

Inventory control is a fundamental aspect of operations management that ensures organizations maintain optimal stock levels to meet demand while minimizing costs. Various models exist to determine the appropriate order quantities, each suited to specific circumstances and demand patterns. This paper explores several core inventory control models, including Economic Order Quantity (EOQ) with quantity discounts, Economic Production Quantity (EPQ), Fixed Order Interval (FOI), and the Single Period Model, along with a discussion on replenishment policies such as Reorder Point (ROP). Through detailed explanations, examples, and calculations, we aim to elucidate the methods used to determine inventory quantities under different operational scenarios.

Economic Order Quantity (EOQ) with Quantity Discounts

The EOQ model seeks to identify the optimal order quantity that minimizes total inventory costs, including ordering and holding costs. When quantity discounts are available, the model adjusts to account for unit price reductions at specific order quantities, potentially affecting the optimal order quantity. The basic EOQ formula without discounts is expressed as:

EOQ = sqrt( (2DS) / H )

where D is annual demand, S is the ordering cost per order, and H is the holding cost per unit annually.

Incorporating Quantity Discounts

When discounts are available, the unit price (P) decreases at certain quantity thresholds, which impacts the purchase cost and overall costs. The total cost function becomes a piecewise function, where the cost structure may change at each discount level. The decision involves calculating the EOQ for each discount level and then comparing total costs to identify the most economical order quantity.

Example: EOQ with Quantity Discounts

Suppose annual demand (D) is 10,000 units, the ordering cost (S) is $50 per order, and the holding cost (H) is $2 per unit per year. The supplier offers the following discounts:

  • Up to 3000 units: $10 per unit
  • 3001-7000 units: $9 per unit
  • above 7000 units: $8 per unit

Calculating EOQ for each discount level involves adjusting for the unit costs and total costs at each threshold—this includes ordering costs, holding costs, and purchase costs. The aim is to choose the order quantity that minimizes total inventory cost, considering the quantity discounts.

Economic Production Quantity (EPQ)

The EPQ model extends EOQ to incorporate production batches, accounting for regular output rates that replenish inventory over time. The formula for EPQ is:

EPQ = sqrt( (2DS) / (H) * (p / (p - d)) )

where p is the production rate and d is the demand rate. This model is suitable for manufacturing environments where items are produced internally rather than ordered externally.

Example: EPQ Calculation

Assuming annual demand (D) is 12,000 units, ordering cost (S) of $100, holding cost (H) of $3 per unit, production rate (p) is 600 units/day, and demand rate (d) is 50 units/day. Calculations involve substituting these into the EPQ formula to determine the optimal batch size.

Fixed Order Interval (FOI)

The FOI model orders inventory at fixed intervals, regardless of current stock levels. It simplifies replenishment scheduling, with order quantities varying based on demand during the interval. Pseudo-constants like replenishment cycle length are used to determine order size and timing.

Example: FOI Application

If a company reviews inventory weekly with average weekly demand of 200 units, and the initial stock is 1,500 units, the order quantity is calculated based on demand during the interval minus current stock, accounting for safety stock if needed.

Single Period Model Using Poisson Distribution

This model applies to perishable or one-time purchase scenarios, where only one ordering period exists. Demand during the period is discrete and probabilistic, often modeled using Poisson distribution. The model helps decide whether to stock out or overstock based on the probability distribution of demand.

Example: Single Period Model with Poisson Demand

Suppose the expected demand is 4 units per period, modeled by a Poisson distribution. The decision rule involves calculating the service level or fill rate, considering the probability of demand exceeding stock levels, to determine optimal order quantity.

Replenishment Timing and Reorder Point (ROP)

Replenishment occurs when inventory levels reach a specified threshold called the Reorder Point (ROP). The ROP depends on demand variability and lead time. For a constant lead time and variable demand, the ROP can be calculated using:

ROP = demand during lead time + safety stock

where safety stock accounts for demand fluctuations during the lead time.

Example: ROP with Variable Demand and Constant Lead Time

Assume average weekly demand is 100 units with a standard deviation of 20 units, and the lead time is 2 weeks. Safety stock is calculated based on the desired service level (e.g., 95%), using the demand variability and lead time, ensuring sufficient stock to prevent stockouts.

Conclusion

Effective inventory management critically depends on selecting suitable models based on demand patterns, cost structure, and operational constraints. EOQ with discounts optimizes order sizes considering price breaks, EPQ addresses production environments, FOI simplifies scheduling, and the Single Period Model caters to perishables or one-time purchases. Proper replenishment policies like ROP ensure continuous availability, balancing service levels and costs. Combining these models enhances inventory efficiency and operational resilience.

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