# Quiz 12: Betting on Uncertain Demand: the Newsvendor Model

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: Where and 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. * Note: In MS-Excel, is computed as . So, the probability of having more than 400 units of demand is b. Compute the probability of having the demand, D ~ N (200, 80) less than or equal to 50% of the mean forecast. So, the probability of the book becoming a "dog" is c. Compute the probability of having the demand, D ~ N (200, 80) within 20% of the mean forecast. So, the required probability is d. Compute the critical ratio using the following formula: 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: Compute corresponding order quantity using the following method. Since a fractional order quantity is infeasible, the optimal order quantity is 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: Compute corresponding order quantity using the following method. Since a fractional order quantity is infeasible, the order quantity is f. If the in-stock probability is 95%, the stock-out probability must be So, the probability that some customers won't be able to purchase a copy is g. Given the order size, compute the expected profit using the following method. Order size, Q = 300 For z = 1.25, the corresponding loss function from the Exhibit 12.4 is Expected lost sales, So,   So, the expected profit is 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: Where and 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 ~ The stock-out happens when the demand is more that the order quantity which is 3 units. So, the stock-out probability or the probability of some demand not being satisfied is b. The mark down requirement in 3 or more when the demand is 7 or less given the purchase quantity is 10. So, the probability of marking down 3 or more units is c. Compute the critical ratio using the following formula: 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 d. For four baskets, using the Poisson loss function table, the loss function value is 1.088. Therefore, the expected sales So, the expected sold units are e. For six baskets, using the Poisson loss function table, the loss function value is 0.323. Therefore, the expected sales So, the left-over units for mark-down are 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 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, So, the left-over units for mark-down are  So, the expected profit is 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: Where and 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 b. Compute the critical ratio using the following formula: 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 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 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,  So, the left-over units eligible for discount are 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,   So, there will be no profit. The expected loss is 