A Refinery In Houston Replaces All Of The 3 Full Open Ball V

A Refinery In Houston Replaces All Of The 3 Full Open Ball Valves In

A refinery in Houston replaces all of the 3" full-open ball valves in its facility according to a very strict maintenance policy which spaces replacements fairly uniformly throughout the year. The refinery has approximately 1000 of these valves in place and safety regulations call for annual replacement. Ball valves are priced according to the schedule below. It costs $30 to place an order for valves and spare parts carrying costs are estimated at $7 per valve per year.

AMOUNT PRICE

1 to 75 $22.50

76 to 119 $20.00

120 + $18.25

What order quantity will enable the refinery to achieve the lowest total cost for the year in purchasing the 3" ball valves?

Paper For Above instruction

The process of determining the optimal order quantity, especially in a scenario involving price breaks and multiple price levels, aligns with the principles of the Economic Order Quantity (EOQ) model. The EOQ aims to minimize total inventory costs, which comprise ordering costs, holding costs, and the purchasing costs affected by price breaks. This case particularly involves tiered pricing, which complicates the straightforward application of classic EOQ formulas.

Understanding the problem: The refinery needs to purchase and replace approximately 1000 valves annually, following a strict maintenance schedule. The ordering costs are fixed at $30 per order, regardless of order size, up to a certain volume; carrying costs are $7 per valve per year. The goal is to identify the ordering quantity that minimizes total costs, considering the tiered pricing based on order quantity.

Tiered Pricing Analysis: The unit price of the valves decreases with larger order quantities, structured into three ranges:

• 1 to 75 units at $22.50 each
• 76 to 119 units at $20.00 each
• 120 or more units at $18.25 each

This tiered structure suggests the optimal order quantity might fall within or at the boundary of one of these ranges to benefit from lower prices while balancing holding and ordering costs.

Calculating EOQ in Tiered Pricing

Traditional EOQ calculations assume a single constant unit price. To accommodate tiered pricing, the analysis should evaluate EOQ within each price range and consider the impact of buying at the edge of each tier, thus comparing total costs for each scenario.

1. EOQ at the highest price ($22.50):

Using the EOQ formula:

EOQ = √(2DS / H)

Where:

D = annual demand = 1000 valves

S = ordering cost = $30

H = holding cost per unit = $7

EOQ = √(2 1000 30 / 7) ≈ √(60,000 / 7) ≈ √8,571.43 ≈ 92.59 units

This EOQ (~93 units) is within the first tier (1-75 units), but slightly exceeds it, suggesting a need to evaluate ordering at the boundary (75 units) for the most economical strategy at this price level.

2. EOQ at the intermediate price ($20.00)

Similarly, EOQ remains roughly the same (~93 units), but as the price tier begins at 76 units, ordering around 75-76 units would be at the boundary, potentially reducing purchase price while maintaining near-optimal total costs.

3. EOQ at the lowest price ($18.25)

Again, EOQ is approximately 93 units, which exceeds the 75-unit upper boundary of the first tier, but is close to the 76-119 units range where the lower price applies.

Evaluating the optimal order quantity

Given the previous calculations, ordering exactly at the boundary of the price tiers seems promising. Specifically, ordering 75 units to capitalize on the higher price may increase costs, but the EOQ suggests ordering around 93 units to minimize combined ordering and holding costs. Since 93 units exceeds 75 but falls within the 76-119 range, it might be more cost-effective to order 119 units to benefit from the lower price of $20.00 per valve, especially if the total costs balance the inventory holding and ordering costs effectively.

Total Cost Calculation: Let's analyze the total annual cost for each likely order quantity—75, 93, 119, and possibly 120—considering purchase price, ordering cost, and holding cost.

Case 1: Ordering 75 units at $22.50

• Number of orders per year: 1000 / 75 ≈ 13.33, rounded up to 14 orders
• Total ordering cost: 14 * $30 = $420
• Average inventory: 75 / 2 = 37.5 units
• Holding cost: 37.5 * $7 = $262.50
• Total purchase cost: 1000 * $22.50 = $22,500
• Total annual cost: $22,500 + $420 + $262.50 = $23,182.50

Case 2: Ordering 93 units at $20.00 (assuming we can buy at this rate for this quantity)

• Number of orders: 1000 / 93 ≈ 10.75, rounded to 11
• Total ordering cost: 11 * $30 = $330
• Average inventory: 93 / 2 = 46.5 units
• Holding cost: 46.5 * $7 = $325.50
• Total purchase cost: 1000 * $20.00 = $20,000
• Total cost: $20,000 + $330 + $325.50 = $20,655.50

Case 3: Ordering 119 units at $20.00 or the next tier at $18.25

• Number of orders: 1000 / 119 ≈ 8.40, rounded to 9
• Order cost: 9 * $30 = $270
• Average inventory: 119 / 2 ≈ 59.5 units
• Holding cost: 59.5 * $7 ≈ $416.50
• Total purchase cost: 1000 * $18.25 = $18,250
• Total cost: $18,250 + $270 + $416.50 = $18,936.50

Recommendation

Comparing these total costs indicates that ordering approximately 119 units at the lowest price tier yields the minimal annual cost. This aligns with inventory principles: larger orders at the lower price tier reduce purchase costs markedly, while the slight increase in holding costs is offset by savings on the purchase price and minimized ordering frequency. To optimize, the refinery should order 119 units per order, so the total annual expenditure is minimized, thus achieving the lowest total cost in purchasing the valves annually.

Conclusion

Considering the tiered pricing and EOQ principles, the optimal order quantity for the refinery in Houston is approximately 119 units per order. This quantity aligns with the lowest unit price of $18.25, minimizes total ordering and holding costs, and complies with the demand of approximately 1000 valves annually. By adhering to this ordering strategy, the refinery can ensure cost-effective procurement while maintaining its strict maintenance schedule and safety regulations.

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