Stock Markets Exposed
Probability Distributions?
Members of the management team recommended order quantities of 15,000, 18,000, 24,000, and 28,000. Considerable disagreement concerning the market potential is evidenced by the different order quantities suggested. The product management team has asked you for an analysis of the stock-out probabilities for various order quantities, an estimate of the profit potential, and to help make an order quantity recommendation. Specialty expects to sell Weather Teddy for $24, and the cost is $16 per unit. If inventory remains after the holiday season, Specialty will sell all surplus inventory for $5 per unit. After reviewing the sales history of similar products, Specialty's senior sales forecaster predicted an expected demand of 20,000 units with a 0.90 probability that demand would be between 10,000 units and 30,000 units. 1. Compute the probability of a stock-out for the order quantities suggested by members of the management team.
Public Comments
- Assume that the demand is distributed normally. The mean is 20 (thousand units) and the standard deviation can be determined from the prediction. There is 1 - 0.90 = 0.10 = 2 x 0.05 probability that the demand will be more than 10 more or less than the mean. The Z-score corresponding to a 0.05 probability is ... Z = (-)1.645, which means that the quantities 10 and 30 lie 1.645 standard deviations from the mean. Therefore, the standard deviation is ... s = 10 / 1.645 = 6.080 (thousand units). Once we know this, the rest is easy. The first recommendation, 15 thousand units, lies 5 below the mean, so its Z-score is ... Z = -5 / 6.080 = -0.8224. The probability that the demand is *less* than this can be found from the cumulative normal distribution, ... Phi(-0.8224) = 0.2054. The complement, 1 - 0.2054 = 0.7946, is the probability of stock-out: almost 80%. Likewise, for the second suggestion, ... Z = -2 / 6.080 = -0.3290. ... Phi(Z) = 0.3711 ... 1 - Phi(Z) = 0.6289, almost 63%. For the third suggestion, ... Z = 4 / 6.080 = 0.6579 ... Phi(Z) = 0.7447 ... 1 - Phi(Z) = 0.2553, close to 25%. Finally, for the last suggestion, ... Z = 8 / 6.080 = 1.3159 ... Phi(Z) = 0.9059 ... 1 - Phi(Z) = 0.0941, close to 10%.
