County Hospital Orders Syringes From A Supply Firm

County Hospital Orders Syringes From A Hospital Supply Firm The Hospi

County hospital orders syringes from a hospital supply firm. The hospital expects to use 38,000 syringes per year. The order cost to deliver the syringes is AED 1.85 per syringe. The supplying company offers the following quantity discount-pricing schedule:

- 0 – 9,999 units at $3.99 each

- 10,000 – 19,999 units at $3.20 each

- 20,000 – 29,999 units at $3.00 each

- 30,000 – 39,999 units at $2.80 each

- 40,000 – 49,999 units at $2.60 each

- 50,000+ units at $2.40 each

Determine the optimal order size for the hospital based on economic order quantity (EOQ) principles, considering the discount schedule and ordering costs.

Paper For Above instruction

The efficient management of inventory is essential for healthcare institutions like county hospitals, especially when procuring consumables such as syringes. Determining the optimal order quantity involves balancing ordering costs against holding costs, minimizing total inventory costs while ensuring adequate supply to meet demand. The presence of quantity discounts further complicates this decision, as purchasing in larger quantities may reduce unit costs but increase inventory holding charges. This paper explores the calculations and reasoning involved in establishing the most cost-effective order quantity for the hospital's syringe procurement, considering both economic order quantity (EOQ) models and discount schedules.

Understanding the Context and Data: The hospital's annual usage rate (demand) for syringes is 38,000 units. The fixed cost per order (ordering cost) is AED 1.85 per syringe, which likely refers to the setup or administrative expense incurred each time an order is placed, not per individual syringe, but the problem seems to suggest a per-syringe order cost. The pricing schedule provides six distinct price brackets corresponding to different order quantities, with decreasing unit costs as order size increases:

- 0–9,999 units: $3.99

- 10,000–19,999 units: $3.20

- 20,000–29,999 units: $3.00

- 30,000–39,999 units: $2.80

- 40,000–49,999 units: $2.60

- 50,000+ units: $2.40

In practice, to find the optimal order quantity, the EOQ model must be adapted to account for these piecewise price reductions.

Calculating the Basic EOQ

The classic EOQ formula is:

\[

EOQ = \sqrt{\frac{2DS}{H}}

\]

where:

- \( D \) = annual demand = 38,000 units

- \( S \) = ordering cost per order, which in this case is AED 1.85

- \( H \) = holding cost per unit per year, typically a percentage of the unit cost. For this calculation, assuming a standard holding cost rate of 20%, the per-unit holding cost varies with the unit price.

Since the unit price affects the holding cost (typically a percentage of the unit cost), the calculations for each price bracket involve updating the per-unit cost and consequently the holding cost. The calculation proceeds by estimating EOQ for each price bracket, then selecting the order quantity that minimizes total costs, considering both purchase and ordering costs.

Estimating EOQ for Each Price Bracket

For simplicity, assume the holding cost rate is 20% of the unit cost; thus, the per-unit holding cost \( H \) varies with price.

- For $3.99: \( H = 0.2 \times 3.99 = AED 0.798 \)

- For $3.20: \( H = 0.2 \times 3.20 = AED 0.640 \)

- For $3.00: \( H = 0.6 \times 3.00 = AED 0.600 \)

- For $2.80: \( H = 0.2 \times 2.80 = AED 0.560 \)

- For $2.60: \( H = 0.2 \times 2.60 = AED 0.520 \)

- For $2.40: \( H = 0.2 \times 2.40 = AED 0.480 \)

Next, compute EOQ for each price bracket:

\[

EOQ = \sqrt{\frac{2 \times D \times S}{H}}

\]

Plugging values:

- For $3.99: \( EOQ = \sqrt{\frac{2 \times 38,000 \times 1.85}{0.798}} \approx 590 \) units

- For $3.20: \( EOQ = \sqrt{\frac{2 \times 38,000 \times 1.85}{0.640}} \approx 664 \) units

- For $3.00: \( EOQ = \sqrt{\frac{2 \times 38,000 \times 1.85}{0.600}} \approx 702 \) units

- For $2.80: \( EOQ = \sqrt{\frac{2 \times 38,000 \times 1.85}{0.560}} \approx 744 \) units

- For $2.60: \( EOQ = \sqrt{\frac{2 \times 38,000 \times 1.85}{0.520}} \approx 784 \) units

- For $2.40: \( EOQ = \sqrt{\frac{2 \times 38,000 \times 1.85}{0.480}} \approx 828 \) units

However, each EOQ must be checked against the relevant order quantity ranges to ensure validity within each price bracket.

Adjustments for Quantity Discounts

Given the demand and the computed EOQ estimates, the next step involves aligning these quantities with the discount brackets:

- The estimated EOQs are below 900 units, which fall into the initial discount bracket (up to 9,999 units).

- Since the EOQ calculations suggest order quantities around 590–828 units, the optimal order quantities likely lie within the first few brackets, possibly near the EOQ values.

Evaluating Cost for Each Bracket

To determine the most economical choice, calculate the total annual cost (TC) for each relevant quantity in the applicable bracket:

\[

TC = \left( \frac{D}{Q} \times S \right) + \left( \frac{Q}{2} \times H \right) + D \times \text{unit price}

\]

Using the average EOQ and price within each bracket:

- For example, at \( Q=590 \) units and price = $3.99:

\[

TC = \left( \frac{38,000}{590} \times 1.85 \right) + \left( \frac{590}{2} \times 0.798 \right) + 38,000 \times 3.99

\]

Calculating each component yields total costs to compare across brackets.

Similarly, the total cost is recalculated for \( Q=664, 702, 744, 784, 828 \) units at respective prices, selecting the lowest total cost as the optimal order quantity.

Conclusion

The analysis indicates that the optimal order quantity falls within the 30,000–39,999 units range at $2.80 each, given the proximity of the EOQ estimates and total cost considerations. Orders slightly larger than the EOQ estimate would leverage the unit discount and reduce overall costs. However, since demand is 38,000 units annually, purchasing in larger quantities, such as 39,000 units, may be optimal to maximize discounts and reduce total procurement costs. Real-world considerations, such as storage capacity and cash flow, should also influence the final decision.

In summary, based on EOQ analysis adjusted for quantity discounts, the hospital should consider ordering approximately 39,000 units per order, obtaining the $2.80 price. This approach balances ordering costs, holding costs, and unit prices, resulting in minimized total annual procurement costs.

References

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