Quiz 12: Betting on Uncertain Demand: the Newsvendor Model

Business

In a single-period inventory model, the order size decision is made by a marginal analysis. In this analysis, the optimal order quantity occurs at the point where the marginal benefit of stocking one additional unit is just less than its expected marginal cost. In a single-period inventory model, increase the order size as long as the probability of selling what is ordered in less than or equal to the critical ratio ( CR ) defined as: img Where img and img are the cost per unit of demand underestimated and that of demand overestimated respectively. a. Compute the probability of having the demand, D ~ N (200, 80) more than or equal to 400 units. img * Note: In MS-Excel, img is computed as img . So, the probability of having more than 400 units of demand is img b. Compute the probability of having the demand, D ~ N (200, 80) less than or equal to 50% of the mean forecast. img So, the probability of the book becoming a "dog" is img c. Compute the probability of having the demand, D ~ N (200, 80) within 20% of the mean forecast. img So, the required probability is img d. img Compute the critical ratio using the following formula: img For optimal order quantity, Q , the distribution function, F(Q) must be equal to the critical ratio. F(Q) = 0.667. Compute the standard normal variable z as follows: img Compute corresponding order quantity using the following method. img Since a fractional order quantity is infeasible, the optimal order quantity is img e. Compute the order quantity given the in-stock probability as follows: The in-stock probability or F(Q) = 0.95 Compute the standard normal variable z as follows: img Compute corresponding order quantity using the following method. img Since a fractional order quantity is infeasible, the order quantity is img f. If the in-stock probability is 95%, the stock-out probability must be img So, the probability that some customers won't be able to purchase a copy is img g. Given the order size, compute the expected profit using the following method. Order size, Q = 300 img For z = 1.25, the corresponding loss function from the Exhibit 12.4 is img Expected lost sales, img So, img img img So, the expected profit is img

In a single-period inventory model, the order size decision is made by a marginal analysis. In this analysis, the optimal order quantity occurs at the point where the marginal benefit of stocking one additional unit is just less than its expected marginal cost. In a single-period inventory model, increase the order size as long as the probability of selling what is ordered in less than or equal to the critical ratio ( CR ) defined as: img Where img and img are the cost per unit of demand underestimated and that of demand overestimated respectively. a. Use the following method to find the probability of some demand not being satisfied. Note that Demand, D ~ img The stock-out happens when the demand is more that the order quantity which is 3 units. img So, the stock-out probability or the probability of some demand not being satisfied is img b. The mark down requirement in 3 or more when the demand is 7 or less given the purchase quantity is 10. img So, the probability of marking down 3 or more units is img c. img Compute the critical ratio using the following formula: img For optimal order quantity, Q , the distribution function, F(Q) must be equal to the critical ratio. F(Q) = 0.657. From the Poisson distribution table, F(4) = 0.5321 and F(5) = 0.7029. Since F(5) 0.657, select order quantity of 5 units Therefore, the required order quantity is img d. For four baskets, using the Poisson loss function table, the loss function value is 1.088. Therefore, the expected sales img So, the expected sold units are img e. For six baskets, using the Poisson loss function table, the loss function value is 0.323. Therefore, the expected sales img So, the left-over units for mark-down are img f. Given the in-stock probability requirement of 90%, find the required purchase quantity using the following method: Using the Poisson distribution table, F(6) = 0.831 and F(7) = 0.913. Since F(7) 0.90, select order quantity of 7 units. Therefore, the required order quantity is img g. Given the order size, compute the expected profit using the following method. For eight baskets, using the Poisson loss function table, the loss function value is 0.0676. Therefore, the expected sales, img So, the left-over units for mark-down are img img So, the expected profit is img

In a single-period inventory model, the order size decision is made by a marginal analysis. In this analysis, the optimal order quantity occurs at the point where the marginal benefit of stocking one additional unit is just less than its expected marginal cost. In a single-period inventory model, increase the order size as long as the probability of selling what is ordered in less than or equal to the critical ratio ( CR ) defined as: img Where img and img are the cost per unit of demand underestimated and that of demand overestimated respectively. a. The liquidation requirement in 10,000 or more when the demand is 30,000 or less given the production quantity is 40,000. For Q = 30,000, F(Q) = 0.7852 So, the probability of having 1,000 units or more being liquidated is img b. img Compute the critical ratio using the following formula: img For optimal order quantity, Q , the distribution function, F(Q) must be equal to the critical ratio. F(Q) = 0.6316. From the empirical distribution table, for Q = 30,000, F(Q) = 0.7852 which is greater than the optimality requirement of F(Q). So, the optimal production quantity is img c. Given the in-stock probability requirement of 90%, find the required production quantity using the following method: F(Q) = 0.90 From the empirical distribution table, for Q = 40,000, F(Q) = 0.9489 which is greater than the optimality requirement of F(Q). Therefore, the required production quantity is img d. With the given production quantity, use the following method to compute the left-over units eligible for discount. Production quantity, Q = 50,000 Corresponding loss function value from the empirical distribution table , L(Q) = 63 So, the Expected sales, img img So, the left-over units eligible for discount are img e. The 100% in-sock probability occurs when the production quantity, Q = 75,000. Corresponding loss function value from the empirical distribution table , L(Q) = 2 So, the Expected sales, img img img So, there will be no profit. The expected loss is img