In this thesis, we consider contextual newsvendor problems where one seeks to determine ordering quantities of perishable products based on the observations of past demands and some features (such as seasonality, weather forecasts, economic indicators, etc.) related to the demand. We propose solving the problems via a single-step optimal decision-tree approach. Unlike the traditional two-step approach that first predicts a demand distribution based on given features and then optimizes the order quantity, our approach seeks to determine a tree-based ordering policy that directly maps given features to optimal order quantities. We show how the optimal policies can be found by solving mixed-integer programming (MIP) problems. The tree structure overcomes the black-box nature of most machine learning algorithms while reaching better performance than simple solutions such as linear regression. In addition to risk-neutral newsvendor problems, we further extend the method to address risk-averse newsvendor problems formulated based on Conditional Value-at-Risk (CVaR). Numerical experiments on synthetic and real-world data suggest that our approach outperforms existing approaches with the same objective function, such as the ERM-based convex optimization model which is referred to as Ban and Rudin's big data newsvendor model, and quantile regression decision trees.

The thesis studies the problem well-known in literature as the newsvendor problem. After summarizing the basic model we pay attention to two extensions of this problem and their combination in single model. The first extension concerns the possibility of the vendor to choose his selling price. The second extension is creation of market with several vendors. We describe both situations in the first chapter of the thesis. In the second chapter we study the combination of both extensions, which means the market with several vendors who can choose their selling prices. We touched several models of such market and we found that the problem is very complex. However we found the optimal reaction of one vendor on the strategy of the other vendor in case of special market with two vendors. That enabled us to create a programme that examines such market, mainly the dependence of the optimal decision of one vendor on the strategy of the second vendor and presence of the Nash equilibriums. 1

An optimal ordering scheme for the capacitated, finite-horizon, multi-period, multiproduct newsvendor problem was proposed by cite {shao06} using a hedging point policy. This solution requires the calculation of a central curve that divides the different ordering regions and a vector that defines the target inventory levels. The central curve is a nonlinear curve that determines the optimal order quantities as a function of the initial inventory levels. In this paper we propose a method for calculating this curve and vector using spline functions, infinitesimal perturbation analysis (IPA), and convex optimization. Using IPA the derivatives of the cost with respect to the variables that determine the spline function are efficiently calculated. A convex optimization algorithm is used to optimize the spline function, resulting in a optimal policy. We present the mathematical derivations and simulation results validating this solution.

The presence of substitution among multiple similar products plays an important role in inventory management. It has been observed in the literature that incorporating the impact of substitution among products can substantially improve the profit and reduce the understock or overstock risk. This thesis focuses on exploring and exploiting the impact of substitution on inventory management problems by theoretically analyzing mathematical models and developing efficient solution approaches. To that end, we address four problems. In the first problem, we study different pricing strategies and the role of substitution for new and remanufactured products. Our work presents a two-stage model for an original equipment manufacturer (OEM) in this regard. A closed-form one-to-one mapping of product designs onto the optimal product strategies is developed, which provides useful information for the retailer. Our second problem is a multi-product newsvendor problem with customer-driven demand substitution. We completely characterize the optimal order policy when the demand is known and reformulate this nonconvex problem as a binary quadratic program. When the demand is stochastic, we formulate the problem as a two-stage stochastic program with mixed integer recourse, derive several necessary optimality conditions, prove the submodularity of the profit function, develop polynomial-time approximation algorithms, and show their performance guarantees. Our numerical investigation demonstrates the effectiveness of the proposed algorithms and, furthermore, reveals several useful findings and managerial insights. In the third problem, we study a robust multi-product newsvendor model with substitution (R-MNMS), where both demand and substitution rates are uncertain and are subject to cardinality-constrained uncertainty set. We show that for given order quantities, computing the worst-case total profit, in general, is NP-hard, and therefore, address three special cases for which we provide closed-form solutions. In practice, placing an order might incur a fixed cost. Motivated by this fact, our fourth problem extends the R-MNMS by incorporating fixed cost (denoted as R-MNMSF) and develop efficient approaches for its solution. In particular, we propose an exact branch-and-cut algorithm to solve small- or medium-sized problem instances of the R-MNMSF, and for large-scale problem instances, we develop an approximation algorithm. We further study the effects of the fixed cost and show how to tune the parameters of the uncertainty set. / Doctor of Philosophy / In a multi-product supply chain, the substitution of products arises if a customer's first-choice product is out-of-stock, and she/he have to turn to buy another similar product. It has been shown in the literature that the presence of product substitution reduces the assortment size, and thus, brings in more profit. %and reduce the inventory level. However, how to quantitatively study and analyze substitution effects has not been addressed in the literature. This thesis fills this gap by developing and analyzing the profit model, and therefore, providing judicious decisions for the retailer to make in order to maximize their profit. In our first problem, we consider substitution between new products and remanufactured products. We provide closed-form solutions, and a mapping that can help the retailer in choosing optimal prices and end-of-life options given a certain product design. In our second problem, we study multi-product newsvendor model with substitution. We first show that, when the probability distribution of customers' demand is known, we can tightly approximate the proposed model as a stochastic integer program under discrete support. Next, we provide effective solution approaches to solve the multi-product newsvendor model with substitution. In practice, typically, there is a limited information available on the customers' demand or substitution rates, and therefore, for our third problem, we study a robust model with a cardinality uncertainty set to account for these stochastic demand and substitution rates. We give closed-form solutions for the following three special cases: (1) there are only two products, (2) there is no substitution among different products, and (3) the budget of uncertainty is equal to the number of products. Finally, similar to many inventory management problems, we include a fixed cost in the robust model and develop efficient approaches for its solution. The numerical study demonstrates the effectiveness of the proposed methods and the robustness of our model. We further illustrate the effects of the fixed cost and how to tune the parameters of the uncertainty set.

This thesis studies the allocation of stock in a two-level inventory system with stochastic demand. The system consists of one central warehouse which supplies N non-identical retailers with one single product. Customer demand occurs solely at the retailers and follows independent Poisson processes. The purpose is to investigate the value of using a more advanced allocation policy than First Come-First Serve at the central warehouse. The focus is on evaluating how well the simple First Come-First Serve assumption works in a system where the warehouse has access to real-time point-of-sale data, and where shipments are time based and consolidated for all retailers. The considered allocation policy is a myopic policy where the solution to a minimization problem, formulated as a constrained newsvendor problem, determines how the warehouse allocates its stock to the retailers. The minimization problem is solved using (a heuristic method based on) Lagrangian relaxation, and simulation is used to evaluate the average inventory holding costs and backorder costs per time unit when using the considered policy. The simulation study shows that cost savings around 1-4 percent can be expected for most system configurations. However, there were cases where savings were as high as 5 percent, as well as cases where the policy performed worse than First Come-First Serve. The study also shows that the highest cost savings are found in systems with relatively low demand, few retailers, short transportation times and a short time interval between shipments.

Studies in newsvendor decision-making have shown that human decisions systematically deviate from analytical solutions found in many utility models of the single period problem (SPP). Yet for the most part the impacts of this human decision bias in systems of newsvendor type products have not been investigated. We study bias in human decision-making to determine how different factors affect the performance of systems of newsvendor type products. We extended the state of the arts utility models of SPP to analyze the effects of individuals wealth on individual decision-making. Our theoretical and empirical results proved that individuals wealth significantly affected individual decision-making. Specifically, our analysis concluded that wealthier individual ordered more than poorer individual did when presented with the same investment opportunity. We created a human decision bias (HDB) model to include different newsvendor ordering policies that individuals could use to determine their order quantities. This model is set up to investigate individuals reliance on different ordering policies under different experimental conditions. We designed multi period newsvendor experiments to study effects of factors such as item profit margin, wealth, value of learning, and salvage value on decision-maker's order quantity. We found that wealth and profit margin factors significantly affected individual newsvendor decision-making. Learning, gender, and salvage value factor did not exhibit significant effects in our empirical studies. We designed multi period multi echelon newsvendor experiments to study effects of factors such as the relationship between newsvendors, item profit margin, and newsvendors' wealth on the performance of two-echelon newsvendors system. We found item profit margin, wealth, and relationship between supplier and retailer to significantly affect newsvendor decision-making. Finally, we present a case study of US fresh produce industry to illustrate the impacts of human decision bias on the performance of a supply chain system.

<p>This thesis studies the allocation of stock in a two-level inventory system with stochastic demand. The system consists of one central warehouse which supplies N non-identical retailers with one single product. Customer demand occurs solely at the retailers and follows independent Poisson processes. The purpose is to investigate the value of using a more advanced allocation policy than First Come-First Serve at the central warehouse. The focus is on evaluating how well the simple First Come-First Serve assumption works in a system where the warehouse has access to real-time point-of-sale data, and where shipments are time based and consolidated for all retailers. The considered allocation policy is a myopic policy where the solution to a minimization problem, formulated as a constrained newsvendor problem, determines how the warehouse allocates its stock to the retailers. The minimization problem is solved using (a heuristic method based on) Lagrangian relaxation, and simulation is used to evaluate the average inventory holding costs and backorder costs per time unit when using the considered policy. The simulation study shows that cost savings around 1-4 percent can be expected for most system configurations. However, there were cases where savings were as high as 5 percent, as well as cases where the policy performed worse than First Come-First Serve. The study also shows that the highest cost savings are found in systems with relatively low demand, few retailers, short transportation times and a short time interval between shipments.</p>

The thesis deals with market models with random inputs represented by the newsvendor problem for which the randomness is given through a random number of customers. Presented work is divided into three chapters. In the first chapter we present the elementar newsvendor problem as stochastic programming problem with a fixed recourse. In the second chapter we present the multiplayer game theory adapted to the newsvendors problem. Moreover, in the second chapter we extend the problem by the second newsvendor on the market and in the third chapter we generalize the problem for n newsvendors on the market. We deal with the situations that arise in the chapters two and three from the game theory point of view and we study characteristics of a Nash equilibrium. Presented theory is demonstrated on illustrative examples in the ends of the two last chapters. 1

The Bertrand Duopoly model for demand in economics is a well-used model. Although this model has important insights towards pricing strategy, it does not accurately depict true market behaviors. In this paper, we will examine the advantages and disadvantages of the current model and its assumptions.We then take a whole new approach towards modeling this phenomena, using Poisson processes to model the demand of goods. We will discuss why this is a better approach and explain how we can extend this to better understand pricing strategies and market dynamics. We then apply our findings to the newsvendor problem, a commonly used problem in inventory management. Using non-homogeneous Poisson processes we explain how to find an optimal pricing strategy and an optimal inventory level for the newsvendor problem.In this paper we explain how to extend the newsvendor problem to a newsvendor duopoly problem. Again we show how to find the optimal pricing strategies and inventory levels for multiple goods in a market. Having found the optimal pricing strategy and inventory level, we then examine the market dynamics in more details. We explore monopolistic and duopolistic markets where the goods range from complements to substitutes and homogeneous to differentiated goods. We discuss how to model the progression of the inventory probabilities and then explain how to price inventory options.

The diploma thesis deals with the modeling of selected risks in healthcare. Motivated by the current pandemic situation, it focuses on analysis of risks associated with the vaccination center in Brno. The theoretical part is mainly devoted to the issue of risk management with a focus on risks in healthcare, where the methods that are used in the practical part are defined. Furthermore, the thesis presents selected topics of mathematical programming. Especially, the newsvendor problem is introduced as inspiring case for further modelling. The brief description of the covid-19 pandemic situation later serves as one of the data sources. The practical part deals with the description and risk analysis of the vaccination process using the methods "What If?" and the FMEA method. Appropriate decisions are then proposed for selected risk situations using the GAMS optimization system. Based on the results of the calculations, specific recommendations are proposed.