The data-driven multi-item newsvendor problem with resource constraints

Ordering behavior in a newsstand experiment

The multi(or single-) product newsvendor problem is a well-known classical problem in inventory management. The general setting analyzed is as follows. There are multiple (or single) perishable products with random demand in a single-selling season. Given a purchase cost and resale price, the decision maker (aka the newsvendor) chooses the optimal ordering quantity for each product at the beginning of the selling season. If the newsvendor orders too much of any product, all leftovers are sold at salvage value; if the newsvendor orders too little, it incurs lost opportunity of sales. Given uncertain demand, it is obvious that the final profit for the newsvendor is random. Thus prior researchers have focused on maximizing expected profits for the newsvendor in making the ordering decision. Unless there are resource constraints and/or demand substitution effects, most multiproduct newsvendor problems can be decomposed into multiple independent single-product problems. The general solution (for expected profit maximization) is a simple closed-form ratio of the overage and underage ”costs” for the newsvendor’s (cumulative) marginal demand distribution. Given such a straightforward result, this approach has been applied in numerous industry settings to address problems of revenue management and/or overbooking. From a decision-maker’s perspective, maximizing expected profits implies risk neutrality. However, risk neutrality guarantees the best decision only on average. Although the use of this model may be justified by the Law of Large Numbers, one cannot expect that a single realization will be sufficiently close to the expected value. In fact, when actual outcomes deviate greatly from their expected values due to their randomness, it may cause an unacceptably large loss to the newsvendor.

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

本文主要係針對多階段訂購之報童模式，提出一具多元限制條件下之數學規劃模型，以進行最適訂購週期與最適訂購量之決策分析探討。為符合實務運作之考量，本研究將次要市場之價格函數納入總期望利潤Tπ(T,Q) 模式中，建構含有限制條件之報童模式，並利用數值分析演算法，求解總期望利潤最大化下之最適訂購週期與最適訂購量。接著，進行各相關參數對總期望利潤函數之敏感度分析，以進一步了解相關參數對總期望利潤函數之影響，且以一數值範例闡述本研究主題的意義及對推論做一驗證，最後，列出六點結論供後續研究及實務應用之參考。

An application of bi-level newsboy problem in two substitutable items under capital cost

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

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.

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.

Robust concave utility maximization over chance constraints

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.

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

On the multi-product newsvendor with bounded demand distributions

The newsvendor problem is fundamental to many operations management models. The problem focuses on the trade-off between the gains from satisfying demand and losses from unsold products. The newsvendor model and its extensions have been applied to various areas, such as production plan and supply chain management. This chapter examines the study about newsvendor problem. In this research, there is a review of the contributions for the multiproduct newsvendor problem. It focuses on the current literature concerning the mathematical models and the solution methods for the multiitem newsvendor problems with single or multiple constraints, as well as with the risks. The objective of this research is to go over the newsvendor problem and bring into comparison different newsvendor models applied to the flower industry. A few case studies are described addressing topics related to the newsvendor problem such as discounting and replenishment policies, inventory inaccuracies, or demand estimation. Three newsvendor models are put into practice in the field of flower selling. A full database of the flowers sold by an anonymous retailer is available for the study. Computational experiments for practical example have been conducted with use of the CPLEX solver with AMPL programming language. Models are solved, and an analysis of different circumstances and cases is accomplished.

The co-production newsvendor problem is motivated by two-stage production processes that simultaneously yield a set of output products of different grades from the same input stocks. Co-production is a characteristic feature of processes such as semiconductor manufacturing and crude oil distillation. In the first stage, the newsvendor executes the order quantities for the input stocks prior to learning the actual demands and grading fractions of the products. In the second stage, the available production is allocated to satisfy the realized demands. Downward substitution is allowed in the allocation; i.e., demands for lower grades can always be filled by higher grades but not vice versa. The co-production newsvendor seeks to achieve maximum demand service level, subject to resource or budget constraints. This article proposes the use of the aspiration level approach to model the decision problem. Furthermore, it is assumed that only the means and supports of the uncertain demands and grading fractions are available, and the model is extended using robust optimization techniques. The resulting model is a linear program and can be solved very efficiently. Computational tests show that the proposed model performs favorably compared to other stochastic optimization approaches for the same problem.

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.