Abstract:

The decision-making of single-period optimal ordering quantity under uncertain setting is a critical issue which has been widely applied. The classic newsvendor model considers uncertain demand that follows an available demand distribution， to optimize the expected benefit or cost. However， not only demand， but also price， are uncertain in practice. Because of the rapidly changing market conditions and shorter product life cycles， demand and price fluctuations becomes more significant. Thus， the data that decisions require is scarce in such setting. Therefore， the scarce-data-driven newsvendor problem under two random variables， i.e.， price and demand， is studied here. Both the data scarcity and the risk attitude of decision makers are taken into account.The Conditional Value-at-Risk （CVaR） criteria is adopted to capture the risk tolerance， and two scarce-data-driven robust newsvendor models are proposed. The first model， namely the robust-Copula model， is developed by considering possible Copula functions. The tractable linear programming equivalence is developed by discretizing the proposed model. The second one is a moment-based distributionally robust model which is based on the ambiguity set under the given mean and variance. The equivalent semi-definite programming is reformulated by dualization.A simulation experiment is performed and the two robust models are compared with the classic Copula-CVaR model. The results show that， as the decision maker’s risk tolerance decreases， the optimal ordering quantity of all three models decreases. Under the same risk tolerance， the optimal ordering quantity given by the distributionally robust model is the most conservative. Moreover， to simulate future markets， three sets of out-of-sample are generated by using Normal， Gamma and Uniform distributions， and an out-of-sample test is conducted. It is found that the two robust models perform better than the Copula-CVaR model in terms of market performance. To be specific， when the decision maker’s risk tolerance is high， the distributionally robust model can generate the better objective value. In other setting， the robust Copula model is better. It is also shown that the distributionally robust model can generate the smallest deviation between the objective values under the in-sample and out-of-sample.The above results reveal the following managerial insights. At first， decision makers with lower risk tolerance would make smaller ordering quantities because of the rise in risk aversion. The more possibilities of future market are taken into account， the less ordering quantities would be made. Second， when decision makers can forecast the future market conditions， a ‘bolder’ decision can be made. However， in the setting of data scarcity， decision makers should have the bottom-line thinking. Specifically， when they have a relatively bigger risk tolerance， the decisions can be made by considering typical market possibilities. Instead， when the risk tolerance is smaller， more market possibilities should be taken into account. Finally， if the conditions of future markets are unknown， it is suggested that more possible market should be considered within the decision-making. It can help to estimate the performance in thefuture market better.

Key words: scarce data, newsvendor model, conditional value at risk （CVaR）, distributionally robust, robust copula