Hi! There Are Three Problems To Be Completed For T

Hi! There are three problems that have to be completed for this assignment

Hi! There are three problems that have to be completed for this assignment. It relates to math & information systems. Due date is 4/19/16 at 5:00 pm New York Eastern Time. Any questions or extra info (ex. course powerpoints) I can provide, please message if needed. Thanks in advance!

Paper For Above instruction

This assignment encompasses three distinct problems centered around inventory management, probability, and decision-making strategies within business contexts, specifically focusing on a bakery, a clothing retailer, and an electronics manufacturer. These problems require application of quantitative methods such as economic order quantity calculations, probability assessments, expected profit optimization, and risk analysis to real-world scenarios involving stock levels, order quantities, demand variability, and profit maximization.

Problem 2: Inventory Management for a Bakery

A large bakery operates every day of the year, purchasing flour in 25-pound bags and using approximately 48,600 bags annually. Each order and shipment process incurs a $100 cost, and the monthly inventory holding cost is $1 per bag. The bakery seeks to optimize its inventory management through economic order quantity (EOQ), determine reorder points considering lead times, and assess inventory cost differences with capacity constraints.

a. Determine the economic order quantity (EOQ).

The EOQ is calculated to minimize the total costs associated with ordering and holding inventory. The EOQ formula is:

EOQ = √(2DS / H)

Where: D = annual demand = 48,600 bags, S = ordering cost = $100, H = holding cost per unit per year = $12 ($1 per month × 12).

Calculating, EOQ = √(2 × 48,600 × 100 / 12) ≈ √(9,720,000 / 12) ≈ √810,000 ≈ 900 bags.

b. Average number of bags on hand

The average inventory level when using EOQ is half of the order quantity: 900 / 2 = 450 bags.

c. Number of orders per year

Number of orders = D / EOQ = 48,600 / 900 ≈ 54 orders annually.

d. Total cost of ordering and carrying flour

Total ordering cost = number of orders × S = 54 × $100 = $5,400.

Total holding cost = average inventory × H = 450 × $12 = $5,400.

Total cost = $5,400 + $5,400 = $10,800.

e. Re-order point considering a 3-day lead time

Demand during lead time = (Annual demand / 365) × lead time in days = (48,600 / 365) × 3 ≈ 133 bags.

The re-order point is approximately 133 bags to ensure timely replenishment.

f. Additional cost for inventory if warehouse capacity limits to 500 bags

Maximum inventory with capacity constraint = 500 bags, which exceeds EOQ of 900 bags. This means the company must order less often or hold less inventory, which can increase total costs. The extra cost relates to the inability to always hold EOQ, potentially increasing ordering frequency and costs. Quantitatively, the difference in inventory costs depends on actual ordering policies adopted, but given maximum capacity of 500 bags, the average inventory drops to 250 bags, reducing holding costs to 250 × $12 = $3,000, saving $2,400 compared to EOQ level. Therefore, the additional cost due to capacity constraint is approximately $2,400 per cycle, or about $129,600 annually (considering 54 orders). However, holding less inventory could lead to stockout risks not addressed here.

Problem 3: Demand and Inventory Decisions for Teddy Bower

Teddy Bower sells parkas at $22 each, sourcing at $10 per unit from TeddySports. Demand is uncertain, modeled as a normal distribution with mean 2,100 units and standard deviation of 1,200 units. The problem involves probability calculations, optimal order quantities, and risk assessments to maximize profit and ensure service levels.

a. Probability demand exceeds 1,800 units

Z = (X - μ) / σ = (1,800 - 2,100) / 1,200 = -300 / 1,200 = -0.25

Using standard normal tables, P(Z > -0.25) = 0.5987. Thus, approximately 59.87% chance demand exceeds 1,800 units.

b. Probability demand is between 1,800 and 2,500 units

Z for 2,500 = (2,500 - 2,100) / 1,200 ≈ 0.33. Using standard normal distribution, P(1,800

c. Optimal order quantity to maximize expected profit

This involves applying the Newsvendor model. The critical ratio (CR) = (selling price - cost price) / (selling price - salvage value) = (22 - 10) / (22 - 0) = 12 / 22 ≈ 0.5455.

The corresponding z-score for CR = 0.5455 is approximately 0.11. Therefore, optimal order quantity Q* = μ + z × σ = 2,100 + 0.11 × 1,200 ≈ 2,100 + 132 ≈ 2,232 units.

d. Order quantity for a 98.5% in-stock probability

Using the standard normal table, z-score for 98.5% = 2.17. Thus, Q = μ + z × σ = 2,100 + 2.17 × 1,200 ≈ 2,100 + 2,604 = 4,704 units.

e. Expected profit for ordering 3,000 parkas

The expected sales, profit calculations, and remaining inventory depend on demand distribution. Expected sales = ∫ min(Q, D)f(D)dD. Using properties of the normal distribution and assuming demand Q = 3,000, expected sales approximately equal to demand's mean plus a small adjustment considering the demand variability, roughly 2,100 units given the mean.

Expected profit = (expected sales × profit per unit) - (order quantity × cost per unit). Assuming expected sales close to 2,100 units: Profit ≈ 2,100 × ($22 - $10) - 3,000 × $10 = 2,100 × $12 - $30,000 = $25,200 - $30,000 = -$4,800, indicating a loss in expectation. Precise calculation involves integrating the demand distribution further, but this approximation shows potential overstocking risks.

f. Inventory sold in secondary market

The expected unsold units = Q - expected sales ≈ 3,000 - 2,100 = 900 units. These can sell in secondary market at $50 each, yielding additional revenue.

g. Expected total profit with order of 1,200 units

Calculations involve integrating demand distribution and considering sales, costs, salvage value, and secondary market revenue. The detailed expected profit accounts for sales, leftovers, and costs, which typically yields positive expected profit when Q is near the critical ratio point.

h. Expected profit for Solectrics

Solectrics' profit is based on the units sold to Flextrola minus production costs: (units shipped) × (selling price - production cost). If all ordered units are delivered, profit = order quantity × ($72 - $52) = order quantity × $20.

i. Stockout probability for Flextrola

The probability demand exceeds order quantity Q = 1,200 units is P(D > 1,200). Calculate Z = (1,200 - 2,100) / 600 ≈ -1.5. P(Z > -1.5) ≈ 0.9332, so approximately a 6.68% chance of stockout.

Problem 4: Inventory and Order Optimization for Flextrola

Flextrola’s demand during the season is normally distributed with mean 1,000 and standard deviation 600. The procurement cost per unit is $52, selling at $121, with salvage at $50. The company has a single order opportunity, and unsold units can be sold at a secondary market. The goal is to determine order quantities that maximize expected profit considering demand uncertainty, salvage options, and market conditions.

a. Probability demand between 800 and 1,200 units

Z for 800 = (800 - 1,000) / 600 ≈ -0.33; P(Z

Z for 1,200 = (1,200 - 1,000) / 600 ≈ 0.33; P(Z

Probability demand is between 800 and 1,200 units = 0.6293 - 0.3707 ≈ 0.2586.

b. Optimal order quantity to maximize expected profit

Using the Newsvendor critical ratio: CR = (price - cost) / (price - salvage) = (121 - 52) / (121 - 50) = 69 / 71 ≈ 0.9718.

Corresponding z-score ≈ 2.00, so Q* = μ + z × σ = 1,000 + 2.00 × 600 = 2,200 units.

Expected profit calculation involves accounting for expected sales, leftover units, salvage values, and revenues from secondary markets, with detailed integration over demand distribution. Approximate expected profit can be assessed via standard formulas or simulation, but generally, ordering near 2,200 units maximizes profit given the high critical ratio.

d. Expected sales, leftovers, and profit for 1,200 units order

Expected sales approximate the demand’s mean when order quantity is 1,200 units; however, variance affects actual sales. Typically, expected sales are around 1,000 units, with excess inventory sold at salvage value, yielding partial profit.

e. Secondary market sales estimate

Expected leftover units = max(0, Q - demand). For Q=1,200, expected leftover roughly (Q - μ) + (standard deviation × some factor), often around 200-300 units, sold at salvage for profit.

f. Expected profit analysis

Expected profit combines revenues from sales, salvages, and costs, generally optimized at the calculated order quantity, with detailed calculations requiring integrating demand distribution.

g. Summary

Overall, these problems demonstrate applying inventory decision models, probability analysis, and risk management to optimize profits and service levels across various business sectors dealing with demand variability and capacity constraints.

References

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