Multi-item Newsvendor Problem Research Articles

Robust knapsack ordering for a partially-informed newsvendor with budget constraint

This paper studies a robust version of the multi-item newsvendor problem with limited budget. The demand distribution belongs to an ambiguity set that contains all distributions that share the same range, mean and mean absolute deviation. The resulting optimization problem turns out to be solvable by a method reminiscent of the greedy algorithm for continuous knapsack problems, purchasing items in order of marginal effect on the total cost until the budget is spent.

Robust concave utility maximization over chance constraints

This paper first studies an expected utility problem with chance constraints and incomplete information on a decision maker’s utility function. The model maximizes the worst-case expected utility of random outcome over a set of concave functions within a novel ambiguity set, while the underlying probability distribution is known with the assumption of a discretization of possible realizations for the uncertainty parameter. To obtain computationally tractable formulations, we employ a discretization approach to derive a max–min chance-constrained approximation of this problem. This approximation is further reformulated as a mixed-integer program. We show that the discrete approximation converges to the true counterpart under mild assumptions. We also present a row generation algorithm for optimizing the max–min program. A computational study for a bin-packing problem and a multi-item newsvendor problem is conducted to demonstrate the benefit of the proposed framework and the computational efficiency of our algorithm. We find that the row generation algorithm can significantly reduce the computational time, and our robust policy could achieve a better out-of-sample performance when compared with the non-robust policy and the one without the chance constraints.

Stochastic First-Order Algorithms for Constrained Distributionally Robust Optimization

We consider distributionally robust optimization (DRO) problems, reformulated as distributionally robust feasibility (DRF) problems, with multiple expectation constraints. We propose a generic stochastic first-order meta-algorithm, where the decision variables and uncertain distribution parameters are each updated separately by applying stochastic first-order methods. We then specialize our results to the case of using two specific versions of stochastic mirror descent (SMD): (i) a novel approximate version of SMD to update the decision variables, and (ii) the bandit mirror descent method to update the distribution parameters in the case of [Formula: see text]-divergence sets. For this specialization, we demonstrate that the total number of iterations is independent of the dimensions of the decision variables and distribution parameters. Moreover, the cost per iteration to update both sets of variables is nearly independent of the dimension of the distribution parameters, allowing for high-dimensional ambiguity sets. Furthermore, we show that the total number of iterations of our algorithm has a logarithmic dependence on the number of constraints. Experiments on logistic regression with fairness constraints, personalized parameter selection in a social network, and the multi-item newsvendor problem verify the theoretical results and show the usefulness of the algorithm, in particular when the dimension of the distribution parameters is large. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: This work was supported by the National Science Foundation [Grant 2112533]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0167 .

A Column Generation Scheme for Distributionally Robust Multi-Item Newsvendor Problems

This paper studies a distributionally robust multi-item newsvendor problem, where the demand distribution is unknown but specified with a general event-wise ambiguity set. Using the event-wise affine decision rules, we can obtain a conservative approximation formulation of the problem, which can typically be further reformulated as a linear program. In order to efficiently solve the resulting large-scale linear program, we develop a column generation-based decomposition scheme and speed up the computational efficiency by exploiting a special column selection strategy and stopping early based on a Karush-Kuhn-Tucker condition test. Focusing on the Wasserstein ambiguity set and the event-wise mean absolute deviation set, a computational study demonstrates both the computational efficiency of the proposed algorithm, which significantly outperforms a commercial solver and a Benders decomposition method, and the out-of-sample superiority of distributionally robust solutions relative to their sample average approximation counterparts. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [492997-2016, RGPIN-2016-05208], the National Natural Science Foundation of China [71972012], Alliance de recherche numérique du Canada, and Canada Research Chairs [CRC-2018-00105]. It was also supported by Groupe d’études et de recherche en analyse des décisions (GERAD). Finally, this research was enabled in part by support provided by Digital Research Alliance of Canada ( https://alliancecan.ca/en ). Supplemental Material: The software that supports the findings of this study is available within the paper and its supplemental information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0010 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0010 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Supermodularity in Two-Stage Distributionally Robust Optimization

In this paper, we solve a class of two-stage distributionally robust optimization problems that have the property of supermodularity. We exploit the explicit worst case expectation of supermodular functions and derive the worst case distribution for the robust counterpart. This enables us to develop an efficient method to obtain an exact optimal solution to these two-stage problems. Further, we provide a necessary and sufficient condition for checking whether any given two-stage optimization problem has the supermodularity property. We also investigate the optimality of the segregated affine decision rules when problems have the property of supermodularity. We apply this framework to several classic problems, including the multi-item newsvendor problem, the facility location problem, the lot-sizing problem on a network, the appointment-scheduling problem, and the assemble-to-order problem. Whereas these problems are typically computationally challenging, they can be solved efficiently under our assumptions. Finally, numerical examples are conducted to illustrate the effectiveness of our approach. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the National Natural Science Foundation of China [Grant 71971187], and Hong Kong Research Grants Council [Collaborative Research Fund (C6103-20GF), General Research Fund (14210821, 16204521, 16212419)]. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4748 .

예산제약을 고려한 다기간 Newsvendor 문제에서의 Q-learning 기법 적용

This paper considers a multi-period, multi-item Newsvendor problem under budget constraints in which a decision-maker orders items with aims to minimize the total inventory cost including inventory holding cost and backlog cost. In this decision process, the order quantities are constrained by two types of budget constraint: periodic budget and flexible budget. The problem is formulated as an action-constrained Markov Decision Process (MDP). To overcome the dimensionality and ambiguity, we employed a Q-learning method for solving the MDP model. In particular, we modified the conventional Q-learning procedure to handle a constrained action space by imposing penalties for constraint violations or incentives for constraint satisfactions on Q-values. The penalties and incentives are obtained by solving a quadratic optimization problem included in the learning procedure. Numerical analysis compares the performance of the proposed Q-learning method with others such as EOQ (Economic Order Quantity), Q-learning without the budget constraint, and a heuristic method. The experimental results showed that the proposed Q-learning method lowers the total inventory cost while increasing the chance of satisfying the budget constraint.

From predictive to prescriptive analytics: A data-driven multi-item newsvendor model

This paper considers a multi-item newsvendor problem with a capacity constraint (Z). The problem has already been addressed in the literature using the classical newsvendor problem. However, provided solutions made assumptions for demand distributions, which are often incorrect and led to errors in the inventory optimization. This research proposes a distribution-free and completely data-driven solution approach to Z. The proposed approach uses sample demand data as input, and machine (and deep) learning methods with empirical risk minimization principle to find order quantities. A heuristic is developed using hierarchies of the retail products to perform multi-item inventory optimization when a capacity constraint is active. The proposed approach is tested on a real-world dataset of retail products. The results from the proposed method are compared with data-driven max-min and empirical inventory optimization methods, and it outperformed all of them. The machine (and deep) learning-based demand forecasting methods (part of the proposed approach) providing better results than neural networks, multiple regression, arima, arimax, etc. Finally, a comparison of total inventory cost from the proposed, max-min, and empirical inventory optimization methods are carried out, and it is observed that the proposed data-driven approach leads to a significant reduction in inventory cost.

On the multi-product newsvendor with bounded demand distributions

We consider a multi-item newsvendor problem with side constraints and common continuous demand distributions that are bounded implying that the items’ profit functions are nondifferentiable. In particular we focus on the cases of uniform and triangular distributions. These distributions naturally describe demand that is guaranteed to exceed a certain threshold – for example, consumption of basic food products or electric power consumption over any given day. Moreover, in practice, it is often difficult to estimate the demand distribution. Accordingly, the uniform and triangular distributions become default modeling choices when only information regarding the bounds and possibly the mode of the distribution is known. We generalize a previous quadratic programming model for uniformly distributed demand on [a,b] to allow a to be nonzero and to allow the order quantity to be smaller than a. We study the corrected model and propose an efficient algorithm for determining an optimal solution. The algorithm is motivated by a structural result of an upper bound on the number of guaranteed shortage products, which typically appear in multiproduct settings with a positive demand distribution lower bound. The performance of our specialized algorithm is compared to that achieved when solving our formulation with a piecewise quadratic objective using a state-of-the-art standard solver. We also extend the modeling technique to propose a nonlinear programming formulation for triangular demand distributions. A similar approach can be adopted to approximate other demand distributions with a possibly non-finite support, such as truncated normal with strictly positive lower bounds.

Ordering behavior in a newsstand experiment

We study ordering behavior in a multi-item newsvendor problem with resource constraints (the newsstand problem). We run a laboratory experiment of a newsstand problem with 95 undergraduate students, using a two-product portfolio while manipulating a budget constraint according to Abdel-Malek and Montanari (2005a)'s newsstand problem analysis. We observe level and poor demand chasing behaviors, as documented in newsvendor experiments. We also observe that more budget availability favors profit performance for a high profitability item and service level performance for a low profitability item. These results are supported by estimates of how participants assess leftover and shortage costs as well as their willingness to pay for additional resources. The results also highlight the role that service level preferences and the tradeoff between increased resources and profit performance play in ordering decisions. We discuss implications for inventory management and behavioral experiments of ordering problems.

On reduced semidefinite programs for second order moment bounds with applications

We show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points $$\{{1 \atopwithdelims (){\varvec{x}}}{1 \atopwithdelims (){\varvec{x}}}' \ | \ {\varvec{x}} \in {\mathcal {X}}\}$${1x1xź|xźX} where $${\mathcal {X}}$$X is the feasible region. In fact, we can replace the completely positive programming formulation for the moment bound on $${\mathcal {X}}$$X, with an associated semidefinite program, provided we have a linear or a semidefinite representation of this convex hull. As an application of the result, we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi-item newsvendor problem by exploiting results from the Boolean quadric polytope. For $${\mathcal {X}}$$X described explicitly by a finite set of points, our formulation leads to a reduction in the size of the semidefinite program. We illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best---worst choice models.

Robust Optimization of Sums of Piecewise Linear Functions with Application to Inventory Problems

Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be “robustified” is not concave (or linear) with respect to the perturbing parameters. In this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set. Given that these problems are known to be intractable, we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed-integer linear program and relate this scheme to methods that are based on exploiting affine decision rules. Our new scheme gives rise to two tractable models that, respectively, take the shape of a linear program and a semidefinite program, with the latter having the potential to provide solutions of better quality than the former at the price of heavier computations. We present conditions under which our approximation models are exact. In particular, we are able to propose the first exact reformulations for a robust (and distributionally robust) multi-item newsvendor problem with budgeted uncertainty set and a reformulation for robust multiperiod inventory problems that is exact whether the uncertainty region reduces to a L1-norm ball or to a box. An extensive set of empirical results will illustrate the quality of the approximate solutions that are obtained using these two models on randomly generated instances of the latter problem.

Optimal resource acquisition policy for a newsvendor under supply risk

We consider a single-period manufacturing problem involving uncertainty in the availability of a production resource. The resource is stochastically available at the regular cost, but by paying a premium it is possible to reserve and hence guarantee any desired level of the resource in advance. Given the resource consumption rates for a number of products, the manufacturer needs to determine the optimal forward purchase quantity of the resource such that expected profit from selling the products is maximized. The problem is formulated as an extension of the traditional multi-item newsvendor problem. A computational optimization procedure is developed for solving the problem. We find that depending on the profit margins associated with the products, the optimal reservation amount of the resource may increase or decrease as the supply variability increases. The demand volatilities of products are observed to influence the forward purchase quantity of the resource in a similar manner.

Technical Note—Deriving Robust and Globalized Robust Solutions of Uncertain Linear Programs with General Convex Uncertainty Sets

We propose a new way to derive tractable robust counterparts of a linear program based on the duality between the robust (“pessimistic”) primal problem and its “optimistic” dual. First we obtain a new convex reformulation of the dual problem of a robust linear program, and then show how to construct the primal robust solution from the dual optimal solution. Our result allows many new uncertainty regions to be considered. We give examples of tractable uncertainty regions that were previously intractable. The results are illustrated by solving a multi-item newsvendor problem. We also apply the new method to the globalized robust counterpart scheme and show its tractability.

MULTI-ITEM NEWSVENDOR PROBLEM WITH AN EQUALITY RESOURCE CONSTRAINT

This research addresses a multi-item newsvendor problem with an equality resource constraint. We focus on retail industries with insufficient development power, where shelves must be filled with a variety of goods at the beginning of every period to make the selling space attractive. Technically, the total occupation volume of the goods must be adjusted to the shelf space. Using resource terminology, these elements correspond to resource consumption and resource capacity, respectively. Thus, we consider a constraint in which the total resource consumption is adjusted to a given resource capacity. We derive explicit solutions for several scenarios regarding resource capacity. These give important managerial implications that are helpful for both managers and front-line workers. We also consider two approaches for obtaining an approximate optimal solution for an arbitrary resource capacity, one of which can be implemented with ease in an ordering support system.

A New Method for Deriving Robust and Globalized Robust Solutions of Uncertain Linear Conic Optimization Problems Having General Convex Uncertainty Sets

We propose a new way to derive tractable robust counterparts of a linear conic optimization problem by using the theory of Beck and Ben-Tal on the duality between the robust (“pessimistic”) primal problem and its “optimistic” dual. First, we obtain a new convex reformulation of the dual problem of a robust linear conic program, and then show how to construct the primal robust solution from the dual optimal solution. Our result allows many new uncertainty regions to be considered. We give examples of tractable uncertainty regions that were previously intractable. The results are illustrated by solving a multi-item newsvendor problem. We also propose a new globalized robust counterpart that is more flexible, and is tractable for general convex uncertainty sets and any convex distance function.

An In-Class Competition Introducing Inventory Management Concepts

Encouraging interest in inventory management necessitates that instructors overcome concerns that the subject is too abstract or conceptual. To aid in this process, we describe a competition engaging students in concepts, including demand estimation, demand uncertainty, and costs of inventory and shortages. The competition simulates a multi-item newsvendor problem employing participant-generated data. We present results from use of the exercise in multiple class settings over the past decade. A number of possible extensions of the basic competition are discussed. Data collection and analysis materials are available to interested readers.

Introduction to the Special Issue: Teaching Service and Retail Operations Management

I the May 2010 and September 2010 issues, we explore leading-edge thinking on service operations management education. A growing number of business and engineering schools have courses dedicated to the management of service operations. This growth reflects the increasing interest among faculty and students in understanding the contribution of services to modern economies. Although service operations courses have become quite standard in curricula, a newer trend of more focused service operations courses has emerged. We highlight teaching in retail operations management as a leading example of such a focus and provide examples of teaching materials used in this context. The May 2010 issue presents two games, one to teach global dual sourcing and the other to teach newsvendor-related concepts in a retail setting. In addition, this issue provides another example of an industry-specific service operations course, namely, healthcare operations management. Retailing is a compelling service setting in which to apply operations management theory. In the first paper of this issue, Allon and Van Mieghem provide an experiential learning opportunity pertaining to retail operations. Students play the role of sourcing managers who must make a strategic sourcing decision between two suppliers, one in Mexico and the other in China, with different costs and lead times. This allocation decision must be made dynamically as random demand is revealed. Through this learning tool, Allon and Van Mieghem demonstrate the complexity of these decisions while also offering insight into how such decisions substantially impact a firm’s financial performance. By making the link to research findings on the dual sourcing setting being simulated in the game, the authors provide an opportunity to illustrate how academic research can aid managers in making these difficult decisions. Another in-class game, described by Robb, Johnson, and Silver, focuses on a multi-item newsvendor problem frequently encountered by retail managers. The game highlights issues of demand estimation and tries to build student intuition about demand uncertainty and the role that shortage and procurement costs play in such ordering decisions. The set-up allows elicitation of demand from students, thus making the frequently abstract idea of uncertain demand much more tangible and accessible to students. Michael Lapre, in his article on teaching healthcare operations in MBA programs, introduces us to a specific service operations context. Healthcare, Lapre notes, represents 17% of GDP in the United States, and there is growing interest among MBA students, hospital administrators, and providers themselves in understanding process improvement techniques. With this article, we see how the management of healthcare operations is taught at Vanderbilt. Lapre describes what he found to be effective and ineffective, and he identifies the topical themes used in his course, as well as the cases associated with each theme. A further aim of this issue is to highlight the multidisciplinary aspect of teaching service operations. Kimes, Verma, and Hart present a case study on revenue management that draws connections between marketing issues such as brand image formation and human resources issues such as employee satisfaction to illustrate the interrelatedness of such concepts within the service sector. The case describes a loyalty program offered by a hotel chain for dining in hotel restaurants. Students explore the potential drawbacks of a fee-based loyalty program that offers customers

A multi-item newsvendor problem with preseason production and capacitated reactive production

Given items with short life cycles or seasonal demands, one can potentially improve profits by producing during the selling season, especially when its production capacity is substantial. We develop a two-stage, multi-item model incorporating reactive production that employs a firm’s internal capacity . Production occurs in an uncapacitated preseason stage and a capacitated reactive stage . Demands occur in the reactive stage. Reactive capacities are pre-allocated to each item in the preseason stage and cannot be changed during the reactive stage. Reactive production occurs during the selling season with full knowledge of demands. The objective is expected profit maximization. Unsatisfied demand is lost. The revenue, salvage value, and production and lost sales costs are proportional. Assuming no fixed costs, we present a simple algorithm for computing optimal policies. For a model with fixed costs for allocating preseason stage production and reactive stage capacity to product families, we characterize optimal policies and develop optimal and heuristic algorithms.

The Multi-Item Newsvendor Problem with a Budget Constraint and Fixed Ordering Costs

The Multi-Item Newsvendor Problem with a Budget Constraint and Fixed Ordering Costs

The multi-item newsvendor problem with a budget constraint and fixed ordering costs

This paper deals with a multi-item newsvendor problem subject to a budget constraint on the total value of the replenishment quantities. Fixed costs for non-zero replenishments have been explicitly considered. Dynamic programming procedures are presented for two situations: (i) where the end item demand distributions are assumed known (illustrated for the case of normally distributed demand) and (ii) a distribution free approach where only the first two moments of the distributions are assumed known. In addition, simple and efficient heuristic algorithms have been developed. Computational experiments show that the performance of the heuristics are excellent based on a set of test problems.