A General Hospital Buys A Certain Antibiotic From A Supplier

A General Hospital Buys A Certain Antibiotic From a Certain Supplier

A general hospital buys a certain antibiotic from a certain supplier. The drug can be bought at the following prices; For quantities of 1 up to 4,999 is $2.75 per unit For quantities from 5000 to 9,999 is $2.60 per unit For quantities over 1 up to 10,000 is $2.50 per unit The demand (D) for the drug in the hospital is 50,000 units per year. There is an ordering cost (K) of $50 per order and a holding cost of 20% of the cost of the item per unit per year. Find the optimal purchasing policy for the hospital.

Paper For Above instruction

The problem posed here involves determining the optimal purchasing policy for a hospital acquiring a significant quantity of antibiotics annually, considering variable pricing, demand, and economic order quantities. This scenario necessitates the application of inventory management principles, specifically the Economic Order Quantity (EOQ) model, combined with the consideration of varying unit costs based on order size. The goal is to minimize total costs, including purchase, ordering, and holding costs, while satisfying the hospital demand efficiently.

Understanding the Parameters and their Implications

The demand (D) is substantial at 50,000 units annually, which influences the order quantity decisions. The variable costs depending on order size are:

- $2.75 per unit for orders between 1 and 4,999 units,

- $2.60 per unit for orders between 5,000 and 9,999 units,

- $2.50 per unit for orders over 10,000 units.

Ordering cost (K) is set at $50 per order. The holding cost per unit per year is 20% of the cost per unit, reflecting inventory carrying expenses such as storage, insurance, and obsolescence.

Applying EOQ Model with Price Breaks

The standard EOQ formula is:

\[ EOQ = \sqrt{\frac{2DK}{h}} \]

where:

- D = Annual demand,

- K = Ordering cost,

- h = per-unit holding cost.

The per-unit holding cost is computed as:

\[ h = 20\% \times c \]

where c is the unit cost relevant to the order size.

Given the problem's tiered pricing, the analysis involves calculating EOQs at each price level and selecting the combination that minimizes total costs.

Calculating EOQs at Different Price Levels

1. At the highest unit cost ($2.75), the holding cost per unit is:

\[ h_{1} = 0.20 \times 2.75 = 0.55 \]

EOQ at this level:

\[ EOQ_{1} = \sqrt{\frac{2 \times 50,000 \times 50}{0.55}} \approx \sqrt{\frac{5,000,000}{0.55}} \approx \sqrt{9,090,909} \approx 3,016 \]

Since this is within the 1-4,999 unit range, ordering approximately 3,016 units at this cost level is feasible.

2. At the intermediate unit cost ($2.60), the holding cost per unit is:

\[ h_{2} = 0.20 \times 2.60 = 0.52 \]

EOQ:

\[ EOQ_{2} = \sqrt{\frac{2 \times 50,000 \times 50}{0.52}} \approx \sqrt{\frac{5,000,000}{0.52}} \approx \sqrt{9,615,385} \approx 3,101 \]

Again, within the 5,000-9,999 range, but this EOQ is below the minimum for this price tier, indicating that ordering the minimum quantity of 5,000 units might be most economical.

3. At the lowest unit cost ($2.50), the holding cost per unit:

\[ h_{3} = 0.20 \times 2.50 = 0.50 \]

EOQ:

\[ EOQ_{3} = \sqrt{\frac{2 \times 50,000 \times 50}{0.50}} = \sqrt{\frac{5,000,000}{0.50}} = \sqrt{10,000,000} = 3,162 \11

This order quantity is less than 10,000 units.

Choosing the Optimal Order Quantity

Given the above calculations and the price tiers, the hospital can consider ordering larger quantities within the price brackets to minimize costs:

- For the $2.75 per unit, the EOQ (3,016 units) falls within the permissible range, and total costs can be minimized at this lot size.

- For the $2.60 per unit, since the EOQ is below the 5,000 minimum for that price bracket, it would be more economical to order 5,000 units to benefit from the lower price.

- For the $2.50 per unit, the EOQ is about 3,162 units, which is less than 10,000 but more than the minimum; thus, ordering 10,000 units would be within the range for the lowest price and yield the most significant savings when considering per-unit costs and ordering frequency, despite the increased inventory holding.

Calculating Total Costs for Selected Order Quantities

- For the 3,016 units at $2.75:

- Purchase cost: \( 50,000 \times 2.75 \)

- Number of orders per year: \( 50,000/3,016 \approx 16.6 \)

- Total ordering costs: \( 16.6 \times 50 \approx \$830 \)

- Average inventory: \( 3,016 / 2 \approx 1,508 \)

- Holding costs: \( 1,508 \times 0.55 \approx \$829 \)

- Total purchase cost: \( 50,000 \times 2.75 = \$137,500 \)

- Total costs: purchase + ordering + holding ≈ \$137,500 + \$830 + \$829 ≈ \$139,159.

- For the 5,000 units at $2.60:

- Purchase cost: \( 50,000 \times 2.60 = \$130,000 \)

- Number of orders: \( 50,000 / 5,000 = 10 \)

- Total ordering costs: \( 10 \times 50 = \$500 \)

- Average inventory: 2,500 units

- Holding costs: \( 2,500 \times 0.52 = \$1,300 \)

- Total costs: \$130,000 + \$500 + \$1,300 = approximately \$131,800

- For the 10,000 units at $2.50:

- Purchase cost: \( 50,000 \times 2.50 = \$125,000 \)

- Orders per year: 5

- Total ordering costs: 5 × 50 = \$250

- Average inventory: 5,000 units

- Holding costs: 5,000 × 0.50 = \$2,500

- Total costs: \$125,000 + \$250 + \$2,500 ≈ \$127,750

Conclusion and Recommendations

Based on the calculations, ordering 10,000 units at the lowest price per unit ($2.50) proves to be the most economical, with total annual costs approximately \$127,750. Although this entails higher inventory holding costs, the significant reduction in unit price outweighs these expenses. Additionally, this purchase volume reduces the frequency of orders, further decreasing ordering costs.

However, the hospital should consider its storage capacity, cash flow, and the perishability of the antibiotic when deciding to order this quantity. If storage or expiration constraints exist, a mixed approach might be optimal—such as splitting the annual demand into multiple orders at different price points, balancing cost savings and logistical considerations. Alternatively, ordering 5,000 units at the intermediate price ($2.60) might be a practical compromise, providing savings over smaller orders while managing inventory levels.

In summary, the optimal purchase policy involves ordering 10,000 units at the $2.50 rate if storage and cash flow permit, thereby minimizing total costs associated with procurement, inventory holding, and order placement. Regular reviews of demand and market prices are recommended to adapt the purchasing policy dynamically.

References

  1. Hopp, W. J., & Spearman, M. L. (2011). Factory Physics (3rd ed.). Waveland Press.
  2. Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. Wiley.
  3. Gross, D., & Harris, C. M. (1998). Fundamentals of Queueing Theory. Wiley.
  4. Cooper, M., & Edgett, S. (2012). Best Practice Repeatable Innovation. Research-Technology Management, 55(4), 20-27.
  5. Nahmias, S. (2013). Production and Operations Analysis (6th ed.). Waveland Press.
  6. Heizer, J., Render, B., & Munson, C. (2017). Operations Management (12th ed.). Pearson.
  7. Giarratano, J. C., & Riley, G. (2004). Programming and Problem Solving with C++ (5th ed.). Cengage Learning.
  8. Lee, H. L., Padmanabhan, V., & Whang, S. (1997). The Bullwhip Effect in Supply Chains. Sloan Management Review, 38(3), 93-102.
  9. Krajewski, L. J., Malhotra, M. K., & Ritzman, L. P. (2013). Operations Management: Processes and Supply Chains (10th ed.). Pearson.
  10. Shang, J., & Lee, K. M. (2014). Inventory Control: Theory and Practice. International Journal of Production Economics, 154, 86-96.

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