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The local supermarket buys lettuce each day to ensure really fresh produce. Each morning, any leftover lettuce from the previous day is sold to a dealer who resells it to farmers to feed their animals. This week, the supermarket can buy fresh lettuce for $6.00 per box. The lettuce is sold for $10.00 per box, and the dealer who buys old lettuce is willing to pay $3.50 per box.

Past history indicates that tomorrow's demand for lettuce averages 350 boxes with a standard deviation of 45 boxes.

Question: How many boxes of lettuce should the supermarket purchase tomorrow? (Round the answer to the nearest integer)

Hint: Apply the single-period inventory model to solve this problem.

Answer :

The supermarket buys fresh lettuce for $6.00 a box and sells it for $10.00 a box. In case there is any leftover lettuce from the previous day, it is sold to a dealer for $3.50 a box.

Given that the demand for lettuce for tomorrow averages 350 boxes, the supermarket needs to determine how many boxes of lettuce should it purchase tomorrow. We can use the single-period inventory model to solve this problem.The formula for single-period inventory model is as follows:Expected shortage cost = (probability of a shortage) x (shortage cost)Expected excess cost = (probability of an excess) x (excess cost)Expected total cost = expected shortage cost + expected excess costWhere,Expected excess cost = (cost per unit) x (expected number of units left over)Expected shortage cost = (shortage cost per unit) x (expected number of units short)Given that the lettuce is bought for $6.00 a box and sold for $10.00 a box, we have,Cost per unit = $6.00Revenue per unit = $10.00Shortage cost per unit = $0.00 since the dealer is willing to purchase the leftover lettuce for $3.50 a boxExpected demand = 350Standard deviation of demand = 45Probability of shortage can be calculated using z-score:Z-score = (Reorder point - Expected demand) / Standard deviation of demand= (Reorder point - 350) / 45.

Let us assume that the supermarket purchases "x" boxes of lettuce tomorrow, then we have,Expected excess cost = (10 - 6) x (x - 350) x P(x > 350)Expected shortage cost = 0 x P(x < 350) = 0Expected total cost = expected shortage cost + expected excess costLet us assume that the reorder point is "r" boxes, then we have,P(x > r) = 0.05 (assuming alpha = 0.05, 2-tailed test)Then, the value of z-score = 1.645Putting the values in the z-score formula, we get,1.645 = (r - 350) / 45r = 419.

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