Essays on Decision Problems Under Uncertainty

One of the fundamental questions in many operations and decision problems is how to incorporate available information into the decision-making process in uncertain environments. In two essays, we develop tools for operations decision problems. In the first essay, we consider the case when only partial information (mode and a few percentiles) about the probability distribution of the variable of interest is available. In the second essay, we consider the case when a sample observation from the variable of interest and its predictors, along with prior distributions about the parameters of the probability model, are available. Essay 1. Maximum entropy distributions with mode and quantiles for applications to operations problem We use maximum entropy (ME) modeling to derive probability distributions for management science applications when partial information consists of quantiles and mode. In Bajgiran et al. (2021), we developed ME modeling when partial information includes quantiles and moments. Previous research has shown that respondents are able to assess the mode and quantiles more accurately than the mean and variance. The ME methodology, given mode and other partial information, was developed in the 1990s. This essay provides formulas for implementing this methodology, which depends on the relationships between the mode and each pair of given consecutive quantiles. This essay presents applications that include two management science problems. The first application is a model for stochastic project planning using Project Evaluation and Review Technique (PERT). Recently, there was an attempt to use ME distribution for PERT with mode information. We present a more extensive information-theoretic modeling for PERT. We find distributions of the duration of the paths in a network of multiple tasks by convolutions of the ME models of the duration of activities at each path. Given the distributions of the paths, the minimal and maximal distributions of project completion time are derived. We also develop a new probabilistic approach for computing the distribution of the expected completion time by incorporating Dirichlet prior distribution. We report applications of our methodologies using real-world data for two pre-construction projects based on information provided by a national construction firm. The second application develops ME models for demand distribution in the Newsvendor (NV) problem. In NV, a profit-maximizing solution is a quantile of the demand distribution. When the demand distribution is unknown, some rules (e.g. minimax regret) are used to derive optimal order quantity based on partial information. Then some arbitrary probability models are chosen for the demand distribution where its profit-maximizing quantile can be different from the proposed optimal order quantity. This essay illustrates the perfectly robust ME models for the demand distribution given by the NV minimax regret rule when partial information includes the mode with and without the median. Essay 2. Bayesian prescriptive framework for complementary products: tariff strategies and channels inefficiencies This essay considers the problem faced by an omnichannel retailer of complementary products that encountered on average a 35 percent increase in tariffs and attempted to pass a portion of tariff cost to customers through price and freight charge. The omnichannel retailer offers various integrated channels and touchpoints, including media, purchase, and delivery channels. We develop a new Bayesian predictive and prescriptive model for omnichannel retailers to study revenue management strategies of complementary products for the tariff when demand is not observable. We define retailer sales inefficiency as the deviation of demand (maximum potential sales) from observed sales. Inefficiency can be due to various reasons, including shortage of products and salespersons’ performance; occurrences of these cases are not recorded. We develop a new Stochastic Frontier Model (SFM) to model sales in the presence of inefficiency. The basic SFM in the econometrics literature assumes that firms are inefficient in meeting the frontier. We combine two existing extensions of the basic SFM to model sales of complementary products by a system of zero-inflated SFMs. We use copula to model the dependency of models for the different products. The use of a zero-inflated distribution for the inefficiency relaxes the restrictive assumption of the basic SFM to allow for the possibility of an efficient firm. We implement the model using data provided by a retailer that operates nationwide. In our empirical model, the demands for each complementary product are specified as a stochastic function of price, freight charge, discounts, tariff, and media channels (advertising, catalogs, and website visits). The inefficiency is a stochastic function of delivery channels and purchase channels characteristics, including salesperson’s performance and contract types. Our results provide the probability of the firm’s full efficiency and probability distribution for the inefficiency of each purchase channel. We use the Bayesian approach for the inference, which includes posterior intervals and the Bayes Factor for evidence about directional hypotheses for predictors of the demands and inefficiencies of the delivery and purchase channels. Moreover, the Bayesian approach enables deriving a predictive probability distribution of the profit. This approach provides more insightful decision-making tools for managers in complex environments. These tools are more versatile than the common approach of making decisions based on estimates of the expected profit. We show applications of our model to study various scenarios of passing a portion of tariff cost to customers. The results enable examine stochastic ordering between profit distributions of various scenarios to select the best scenario