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One of the items you stock in your distibution center is a seasonal item. Each year, you must purchase units of this item in advance via a single order. The season lasts 180 days. This item is difficult to sell when the season ends, so you have negotiated an agreement with a discounter where you pay per item to dispose of any excess inventory at the end of the season. Suppose that you earn sales revenue of $50 for each item sold, on average, during the sales season. For any unsold items, you pay a third-party distributor $4 per item to remove them from your inventory. Suppose the demand D for these items in the upcoming 6-month season is modeled via a normal random variable with mean = 300 and variance σ² = 900. 1. During negotiations with supplier A, she quotes you a great offer to provide at most 280 units at a price of $10 per unit. Explain, using newsvendor analysis and your understanding of the normal cumulative distribution, why you should certainly order all 280 units from your supplier at this price. 2. Supplier B has 350 units available, but can only sell them to you at $20 per unit. If you bought all units from supplier B, determine the optimal quantity to purchase using the inverse of the cumulative normal distribution. 3. Finally, you want to take advantage of both suppliers. You decide to purchase 280 units at a cost of $10 each from Supplier A. You also notice that, even at $20 each from Supplier B, you would like to purchase more than 280 units so you decide to purchase additional units from B at $20 each. How many additional units q' should be purchased from B to optimize total profit? Hint 1: Note that the decision to purchase 280 units from supplier A has little impact on the problem; it's like a fixed cost that provides some fixed expected profit. Focus only on the purchase of additional units. ⚫ Hint 2: What is the cost of an excess additional unit, noting the increase in price? What is the lost profit for each additional unit short, again noting the increase in price? ⚫ Hint 3: Define an excess demand random variable: E = max(0, D-280). Note that P(E = 0) = P(D≤ 280), and P(E < x) = P(D≤ 280+x) for x>0. This relationship could help you understand how to answer.

One of the items you stock in your distibution center is a seasonal item. Each year, you must purchase units of this item in advance via a single order. The season lasts 180 days. This item is difficult to sell when the season ends, so you have negotiated an agreement with a discounter where you pay per item to dispose of any excess inventory at the end of the season. Suppose that you earn sales revenue of $50 for each item sold, on average, during the sales season. For any unsold items, you pay a third-party distributor $4 per item to remove them from your inventory. Suppose the demand D for these items in the upcoming 6-month season is modeled via a normal random variable with mean = 300 and variance σ² = 900. 1. During negotiations with supplier A, she quotes you a great offer to provide at most 280 units at a price of $10 per unit. Explain, using newsvendor analysis and your understanding of the normal cumulative distribution, why you should certainly order all 280 units from your supplier at this price. 2. Supplier B has 350 units available, but can only sell them to you at $20 per unit. If you bought all units from supplier B, determine the optimal quantity to purchase using the inverse of the cumulative normal distribution. 3. Finally, you want to take advantage of both suppliers. You decide to purchase 280 units at a cost of $10 each from Supplier A. You also notice that, even at $20 each from Supplier B, you would like to purchase more than 280 units so you decide to purchase additional units from B at $20 each. How many additional units q' should be purchased from B to optimize total profit? Hint 1: Note that the decision to purchase 280 units from supplier A has little impact on the problem; it's like a fixed cost that provides some fixed expected profit. Focus only on the purchase of additional units. ⚫ Hint 2: What is the cost of an excess additional unit, noting the increase in price? What is the lost profit for each additional unit short, again noting the increase in price? ⚫ Hint 3: Define an excess demand random variable: E = max(0, D-280). Note that P(E = 0) = P(D≤ 280), and P(E < x) = P(D≤ 280+x) for x>0. This relationship could help you understand how to answer.

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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One of the items you stock in your distibution center is a seasonal item. Each year, you must purchase units of this item in advance via a single order. The season lasts 180 days. This item is difficult to sell when the season ends, so you have negotiated an agreement with a discounter where you pay per item to dispose of any excess inventory at the end of the season. Suppose that you earn sales revenue of $50 for each item sold, on average, during the sales season. For any unsold items, you pay a third-party distributor $4 per item to remove them from your inventory. Suppose the demand D for these items in the upcoming 6-month season is modeled via a normal random variable with mean = 300 and variance σ² = 900. 1. During negotiations with supplier A, she quotes you a great offer to provide at most 280 units at a price of $10 per unit. Explain, using newsvendor analysis and your understanding of the normal cumulative distribution, why you should certainly order all 280 units from your supplier at this price. 2. Supplier B has 350 units available, but can only sell them to you at $20 per unit. If you bought all units from supplier B, determine the optimal quantity to purchase using the inverse of the cumulative normal distribution. 3. Finally, you want to take advantage of both suppliers. You decide to purchase 280 units at a cost of $10 each from Supplier A. You also notice that, even at $20 each from Supplier B, you would like to purchase more than 280 units so you decide to purchase additional units from B at $20 each. How many additional units q' should be purchased from B to optimize total profit? Hint 1: Note that the decision to purchase 280 units from supplier A has little impact on the problem; it's like a fixed cost that provides some fixed expected profit. Focus only on the purchase of additional units. ⚫ Hint 2: What is the cost of an excess additional unit, noting the increase in price? What is the lost profit for each additional unit short, again noting the increase in price? ⚫ Hint 3: Define an excess demand random variable: E = max(0, D-280). Note that P(E = 0) = P(D≤ 280), and P(E < x) = P(D≤ 280+x) for x>0. This relationship could help you understand how to answer.
Transcribed Image Text:One of the items you stock in your distibution center is a seasonal item. Each year, you must purchase units of this item in advance via a single order. The season lasts 180 days. This item is difficult to sell when the season ends, so you have negotiated an agreement with a discounter where you pay per item to dispose of any excess inventory at the end of the season. Suppose that you earn sales revenue of $50 for each item sold, on average, during the sales season. For any unsold items, you pay a third-party distributor $4 per item to remove them from your inventory. Suppose the demand D for these items in the upcoming 6-month season is modeled via a normal random variable with mean = 300 and variance σ² = 900. 1. During negotiations with supplier A, she quotes you a great offer to provide at most 280 units at a price of $10 per unit. Explain, using newsvendor analysis and your understanding of the normal cumulative distribution, why you should certainly order all 280 units from your supplier at this price. 2. Supplier B has 350 units available, but can only sell them to you at $20 per unit. If you bought all units from supplier B, determine the optimal quantity to purchase using the inverse of the cumulative normal distribution. 3. Finally, you want to take advantage of both suppliers. You decide to purchase 280 units at a cost of $10 each from Supplier A. You also notice that, even at $20 each from Supplier B, you would like to purchase more than 280 units so you decide to purchase additional units from B at $20 each. How many additional units q' should be purchased from B to optimize total profit? Hint 1: Note that the decision to purchase 280 units from supplier A has little impact on the problem; it's like a fixed cost that provides some fixed expected profit. Focus only on the purchase of additional units. ⚫ Hint 2: What is the cost of an excess additional unit, noting the increase in price? What is the lost profit for each additional unit short, again noting the increase in price? ⚫ Hint 3: Define an excess demand random variable: E = max(0, D-280). Note that P(E = 0) = P(D≤ 280), and P(E < x) = P(D≤ 280+x) for x>0. This relationship could help you understand how to answer.
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