We study a two-stage newsvendor model where the initially uncertain mean demand is revealed midstream, allowing for a second, costlier order to be placed. The twostage process is relevant to many retailers who have access to two supply options: a longlead, low-cost option where orders need to be placed under much demand uncertainty, and a short-lead, high-cost option after a signal is revealed that updates the mean demand. We introduce a forecast evolution model that describes how the initial forecast for mean demand varies in response to the signal, which generalizes the popular additive and multiplicative martingale model of forecast evolution (MMFE). We show that the optimal firststage order solves a simple ordinary differential equation (ODE), whereas the second-stage order is analytically available. We characterize asymptotics for the first-stage order and expected profit as the forecasted mean demand grows, and propose a simple, asymptotically optimal heuristic. We apply our study to data from a national retail chain, where we calibrate our model by maximum likelihood estimation (MLE). The comparison of several heuristic policies, as well as two benchmarks, shows the efficacy of our method. When the signal is more uncertain (e.g., for product lines with impulse-and trend-driven purchases), our heuristic with a simple adjusted critical fractile outperforms benchmarks; otherwise, the classic newsvendor solution performs well and is asymptotically optimal. We also extend the approach to distribution-free settings and capacitated systems.

On the Almost Threshold Policy for Multisourcing Under Uncertain Supplies In “On the Almost Threshold Policy for Multisourcing Under Uncertain Supplies,” Federgruen, Feng, and Shanthikumar examine optimal multisourcing strategies in the presence of supply uncertainty. The authors develop a general framework based on the concept of stochastic midpoint linearity and demonstrate that a simple almost threshold policy is optimal under broad and realistic conditions. Their findings provide both theoretical insights and practical guidance for designing resilient and data-driven multisourcing systems.

Problem definition: We study a dual sourcing problem in an increasingly volatile world. We consider two types of volatilities. External volatilities reflect fluctuating economic conditions via an underlying Markov-modulated state-of-the-world that affects the two suppliers’ cost structures, capacity limits, supply mechanisms, and demands. Internal volatilities affect the actual outputs resulting from random supply processes. Demand distributions are impacted by both types of volatilities. Methodology/results: We show how the optimal combined ordering strategy from the two suppliers, along with a salvaging policy, can be efficiently computed under the conventional assumption of consecutive lead times. We characterize the relatively simple structure of the optimal policies and systematically compare the two types of volatilities. Managerial implications: By exploiting dual sourcing options, we find that the firm can benefit from external volatilities; indeed, benefits increase as volatilities increase in specific ways. Numerical studies illustrate these results and reject other reasonable conjectures. Funding: This work was supported by the Guangdong Key Lab of Mathematical Foundations for Artificial Intelligence and the National Natural Science Foundation of China [Grants 72192805 and 72401245]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2024.0987.

We analyze an inventory system with an arbitrary number I of items sharing a limited storage capacity or inventory budget in each of T periods of the planning horizon. Demands for the different items follow a general multivariate Normal distribution allowing for general correlation structures. Inventories may be adjusted by placing orders which arrive after a given lead time, or by salvaging part of the inventory. The capacity constraints are modeled as chance constraints that impose an upper bound on the overflow probability in each period.We design a heuristic which is asymptotically optimal in I when demands are correlated only among items within a common product line. The complexity grows quadratically in I and like O(T^{\frac{3}{2}}) in T.Thereafter, we design a practical heuristic which we recommend for moderate values of I, and which is of similar worst case complexity as the asymptotically optimal heuristic.An extensive numerical study involving more than 28,000 instances with up to 40 items, shows that the average gap between the upper bound and lower bound is 1.05\%, with 98.2\% exhibiting a gap smaller than 5\%. Empirically, we observe that the runtime of the inventory reduction algorithm grows linearly with I. We provide a scalable methodology to identify near optimal procurement strategies for a problem that is central to most brick-and-mortar and online retailers and distributors. Additionally, this methodology can be used to guide capacity planning as well as assortment decisions.

We analyze a stochastic inventory model with $I$ items. At the beginning of each period, the inventory position of each item can be adjusted. Orders arrive and salvage batches deplete the inventory after a lead time. There are variable order and salvaging costs and convex holding and backlogging costs. In each period, there are multiple constraints that bound the expected value of aggregate measures of the products' inventory positions and orders, or the likelihood of these aggregate measures exceeding given thresholds. These constraints couple the inventory levels of the different products. Under specific demand correlation patterns, a simple structured policy is asymptotically optimal in $I$. The policy manages each item by itself, with coordination achieved through Lagrange multipliers. Additional structure arises, depending on the properties of the constraint measures. We show that the problem, specified with chance constraints, can be sandwiched between two problems with expected value constraints only.

We study a general finite horizon, periodic review combined inventory and pricing model with N suppliers and T periods, where both the demands and the supply mechanisms are random. The random supply mechanisms are of a general type that includes most structures encountered in practice. Demands are price dependent according to general, stochastic demand functions. We characterize the optimal combined pricing and ordering policies to all N suppliers. The general results pertain to general independent supply mechanisms. Under random capacities—one of the special random supply mechanisms—they also extend to suppliers that are positively dependent on each other.

We study a finite horizon, single product, periodic review inventory system with two supply sources and a salvage option. These supply sources are typically capacitated and capacity levels often need to be reserved or installed in advance of the operational planning horizon. The supply sources may thus be differentiated by their lead times, capacities, and fixed and variable order costs. Salvage options allow for inventory reductions and incur fixed cost and variable revenues. We first analyze the tactical problem of determining an optimal procurement strategy under given capacity profiles at the two suppliers. We then address the strategic model in which optimal capacity profiles, both static and dynamically adjusted, are obtained based on two‐part capacity contracts. We characterize the structure of optimal procurement strategies when the lead times of the two suppliers differ by a single period and the lead time for salvage opportunities matches that of one of the suppliers. For general lead time combinations, we show that the optimal procurement strategies satisfy monotonicity and limited sensitivity properties, and construct effective heuristics and upper and lower bounds based on our structural results.

We analyze an inventory system with an arbitrary number of items sharing a limited storage capacity or inventory budget in each period. Demands for the different items follow a general joint distribution allowing for arbitrary correlation structures. Inventories may be adjusted by placing orders which arrive after a given lead time, or by salvaging part of the inventory. The capacity constraint is modeled as a chance constraint that imposes an upper bound on the overflow probability in each period.We successively apply two different relaxations to the complex nonconvex feasible action spaces in the exact dynamic program to obtain a polyhedral action space, followed by a Lagrangian relaxation to decouple the products. The optimal solution of the Lagrangian dual is given by a pair of modified base stock policies for each item. Next, we craft a heuristic that takes the optimal strategies from the relaxation as the starting point, but reduces the prescribed inventory positions in order to achieve feasibility with minimal additional expected costs. This set of reductions is identified by polymatroidal optimization and a scaling algorithm. An extensive numerical study involving more than 28,000 instances with up to 40 items, shows that the average gap between the upper bound and lower bound is 1.05%, with 98.2% exhibiting a gap smaller than 5%. Most of the computational effort is spent on computing the inventory reduction. Empirically, we observe that the runtime of the inventory reduction algorithm grows linearly with the number of items.We provide a scalable methodology to identify near optimal procurement strategies for a problem that is central to most brick-and-mortar and online retailers and distributors. Additionally, this methodology can be used to guide capacity planning as well as assortment decisions.

Executive Director, WomenLift Health, Seattle, Washington, USA Executive Vice President, Center for Global Development, Washington, DC, USA Chair of the Decision, Risk, and Operations (DRO) Division, Columbia University, New York, New York, USA Coordinator of Polio Eradication Advocacy Task Force, UK National Advocacy Adviser for Polio, Rotary International, London, UK Former Director General, Indian Council of Medical Research (ICMR), Delhi, India Former Minister, Ministry of Finance and Economic Development, Harare, Zimbabwe Emeritus Professor at the Wistar Institute, Vaccinologist, University of Pennsylvania, Philadelphia, Pennsylvania, USA

We address a two-stage Newsvendor model in which the mean demand -- but not the actual demand -- at first random itself, gets revealed in midstream, in time to place a second order, albeit that the unit cost price of this second order is higher than that of the original one. The two-stage process is most relevant to many retail organizations, where the retailer has access to two supply options: one with a relatively long lead time where orders need to be placed with much uncertainty about even the mean demand for the season, and a second more expensive option with a much smaller lead time that can be exercised after a signal is revealed, which provides the decision maker with an update of the mean demand. We show that the optimal first-round order can be found by solving a simple ordinary differential equation, while the second-round order is analytically available. We characterize the asymptotic behavior of the initial order, derive analytical upper and lower bounds for this initial procurement, and extend our results to the case where procurement capacities prevail. A numerical study shows the benefits of postponed procurements. We also show necessary and sufficient conditions under which a simple heuristic, suggested by the asymptotic analysis, outperforms the optimal single-stage Newsvendor solution.

We analyze a periodic review stochastic inventory model with T periods and I items. At the beginning of each period, the inventory position of each item can be adjusted by placing an order or by salvaging some of the inventory. There is a limited capacity for the total inventory held at the end of each period. Orders arrive and salvage batches deplete the inventory after a given lead time. In addition to variable order and salvaging costs, there are linear or convex holding and backlogging costs. In every period, the probability of the aggregate inventory level exceeding the prevailing inventory capacity must be smaller than a given tolerance.In this paper we show that when demands are independent across items or when each item is correlated with at most O(1) other items, a simple structured policy can be found which is asymptotically optimal when the number of items I grows to infinity. We achieve these results by showing that the problem, specified with chance constraints on the overflow probability in each period, can be sandwiched in between two problems with sets of expected value constraints.

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the “regular” extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models, such as the multinomial logit model or any of its variants. We show that a special equilibrium in this model has global robust stability. This means that, from any starting point, the market converges to this equilibrium when firms use a particular response mapping to dynamically adjust their own prices in response to their competitors’ prices. The mapping requires each firm to only know the demand function and cost structure for its own products (but not for other firms’ products).

ObjectiveWith an eye toward possible public policy implications, our objective is to identify the socio-economic and demographic factors that drive the large variation in COVID-19 incidence rates observed within relatively compact geographic regions, and to quantify the relative impact of each of these factors. We use international comparisons as a starting point.MethodsNew York City, consisting of some 175 zip codes, is an ideal arena to pursue the above study given the large variation in case incidence rates across zip codes. We conducted systematic regression studies employing data with zip code granularity. Our model specifications are based on a well-established epidemiologic model that explains the effects of household sizes on R0.ResultsAverage household size emerges as the single most important driver behind the large variation in COVID-19 incidence rates. It independently explains 62% of the variation. The percentage of the population above the age of 65 and the percentage below the poverty line are also strongly positively associated with zip code incidence rates. As to ethnic/racial characteristics, the percentages of African Americans, Hispanics and Asians within the population are significantly associated, but the magnitude of the impact is smaller. (The proportion of Asians within a zip code has a negative association.) Contrary to common belief, population density, by itself, does not have a significantly positive impact (other than when a high population is driven by large household sizes).ConclusionOur findings support implemented and proposed policies to quarantine patients and separate infected individuals from families or dormitories; they also support newly revised nursing home admission policies.

Correspondence to Ms Amie Batson; aebatson@ gmail. com © Author(s) (or their employer(s)) 2021. Reuse permitted under CC BYNC. No commercial reuse. See rights and permissions. Published by BMJ. POLIO ERADICATION IS IN SIGHT As of 24 August 2021 there has not been a case of wild poliovirus (wPV) anywhere in the world for more than 7 months. Perhaps this is one of the longest periods, if not the longest, without a case of wPV in the world. We may be much closer to polio eradication than any of us had dared hope. If we are, indeed, entering the final stages of polio eradication, planning for posteradication is essential to keep the world poliofree. Since the beginning of 2021, there have been only two reported cases of wPV, compared with 102 cases for the same period in 2020 (January to August). There have been 62 wPVpositive environmental samples reported so far this year till August 2021, compared with 304 for the same period in 2020. There have been no wPVpositive samples from Afghanistan for more than 6 months. While the polio eradication efforts have made an enormous contribution to COVID19 control, it may be that the restrictions imposed by the pandemic have brought us this much closer to polio eradication. It is an unexpected synergy. Of course, the reduction in wPV cases in Pakistan may not be sustainable as suggested by Shaikh et al, noting their belief that the COVID19 pandemic may have led to underreporting in 2020. However, the surveillance networks (including environmental sampling) in both Afghanistan and Pakistan continue to function within the parameters set in the Global Polio Eradication Initiative (GPEI) surveillance indicators. Some have also noted that the global goal should not focus on just zero cases of wPV, but also on circulating vaccinederived poliovirus (cVDPV) eradication. Chumakov et al wrote in June 2021 that the strategy of the GPEI had always been based on stopping transmission of wPV, but:

The number of confirmed COVID-19 cases, relative to population size, has varied greatly throughout the United States and even within the same city. In different zip codes in New York City, the epicentre of the epidemic, the number of cases per 100,000 residents has ranged from 437 to 4227, a 1:10 ratio. To guide policy decisions regarding containment and reopening of the economy, schools, and other institutions, it is vital to identify the factors that drive this large variation. This paper reports on a statistical study of incidence variation by zip code across New York City. Among many socio-economic and demographic measures considered, the average household size emerges as the single most important explanatory variable: an increase in average household size by one member increases the zip code incidence rate, in our final model specification, by at least 876 cases, 23% of the range of incidence rates, at a 95% confidence level. The percentage of the population above the age of 65, the percentage below the poverty line, and their interaction term are also strongly positively associated with zip code incidence rates, In terms of ethnic/racial characteristics, the percentages of African Americans, Hispanics, and Asians within the population, are significantly associated, but the magnitude of the impact is considerably smaller. (The proportion of Asians within a zip code has a negative association.) These significant associations may be explained by comorbidities, known to be more (less) prevalent among the black and Hispanic (Asian) population segments. In turn, the increased prevalence of these comorbidities among the black and Hispanic population, is, in large part, the result of poorer dietary habits and more limited access to healthcare, themselves driven by lower incomes Contrary to popular belief, population density, per se, does not have a significantly positive impact. Indeed, population density and zip code incidence rates are negatively correlated, with a -33% correlation coefficient. Our model specification is based on a well-established epidemiologic model that explains the effects of household sizes on R0, the basic reproductive number of an epidemic. Our findings support implemented and proposed policies to quarantine pre-acute and post-acute patients, as well as nursing home admission policies.

Contract farming is a growing practice in developing countries and first-world economies alike. It generates necessary guarantees to sustain the continued operations of vulnerable farmers while enabling the manufacturers to manage the aggregate supply and price risk. We consider a single manufacturer who owns several manufacturing plants, each with a random demand for the crop. The manufacturer selects a set of farmers to offer a menu of contracts, which is exogenously specified or endogenously determined. Each “selected” farmer chooses a contract from this menu in advance of the growing season. After the growing season, under known demands and supplies, the manufacturer minimizes the distribution costs from the selected farmers to the production facilities. We formulate this problem as a Stackelberg game with asymmetric information, where the manufacturer is the leader and the farmers are the followers. The manufacturer’s problem is a two-stage stochastic planning program for which we develop two solution approaches. We have applied our model to problem instances anchored on data from a large manufacturer of potato chips contracting with thousands of small farmers in India. We report on the performance of the solution methods compared with a lower bound based on the Lagrangean dual of the problem and show that the optimality gap is below 1%, for problem instances with 1,000 potential farmers. We also show how our model can be used to gain various managerial insights. As an example, when constructing the contract menu endogenously, often a small number of contract options suffices, depending on the degree of heterogeneity among the farmer pool. Thus, relatively simple menus often suffice. The online appendices are available at https://doi.org/10.1287/msom.2018.0735 . This paper has been accepted for the Manufacturing & Service Operations Management Special Issue on Value Chain Innovations in Developing Economies.

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the "regular" extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models. We identify a fully best response operator which is monotonically increasing so that the market converges to a Nash equilibrium, when firms dynamically adjust their prices, as best responses to their competitors' prices, at least when starting in one of two price regions. Moreover, geometrically fast convergence to a common equilibrium can be guaranteed for an arbitrary starting point, under an additional condition for the price sensitivity matrix.

We consider a general two‐echelon distribution system consisting of a depot and multiple sales outlets, henceforth referred to as retailers, which face random demands for a given item. The replenishment process consists of two stages: the depot procures the item from an outside supplier, while the retailers' inventories are replenished by shipments from the depot. Both of the replenishment stages are associated with a given facility‐specific leadtime. The depot as well as the retailers faces a limited inventory capacity. Inventories are reviewed and orders are placed on a periodic basis. When a retailer runs out of stock, unmet demand is backlogged.We propose a new approach to the above class of dynamic programming models based on Lagrangian relaxation. Every choice of the vector of Lagrange multipliers generates a lower bound via the solution of a single dynamic program (DP) with a one‐dimensional state‐space. The best such bound is obtained by maximizing over the vector of multipliers. The strategy that is optimal for this (maximal) lower bound DP employs an (s, S) ordering policy (generally, with time‐dependent policy parameters). To arrive at an upper bound and an implementable heuristic, this (s, S) policy is paired with one of several possible allocation policies that allocate the system‐wide inventory across the different facilities.We report on an extensive numerical study with close to 14 000 instances which evaluates the accuracy of the lower bound and the optimality gap of the various heuristic policies. The study reveals that the lower bound and the heuristic strategy that is constructed on its basis perform exceedingly well, almost across the entire parameter spectrum, including instances where demands are rather volatile or the average cycle time between consecutive orders is relatively large. The exception arises when storage at the depot is as expensive as at the retailer level and the retailers have large storage capacities.

We address a general periodic review inventory control model with the simultaneous presence of the following complications: (a) bilateral inventory adjustment options, via procurement orders and salvage sales or returns to the supplier; (b) fixed costs associated with procurement orders and downward inventory adjustments (via salvage sales or returns); and (c) capacity limits associated with upward or downward inventory adjustments. We characterize the optimal adjustment strategy, both for finite and infinite horizon periodic review models, by showing that in each period the inventory position line is to be partitioned into (maximally) five regions. Our results are obtained by identifying a novel generalized convexity property for the value functions, which we refer to as strong (C 1 K 1 , C 2 K 2 )-convexity. To our knowledge, we recover most existing structural results for models with exogenous demands as special cases of a unified analysis.

We study a finite horizon, single product, periodic review inventory system with two supply sources and a salvage option. These supply sources are typically capacitated and capacity levels often need to be reserved or installed in advance of the operational planning horizon. The supply sources may thus be differentiated by their lead times, capacities and fixed and variable order costs. Salvage options allow for inventory reductions and incur fixed cost and variable revenues. We first analyze the tactical problem of determining an optimal procurement strategy under given capacity profiles at the two suppliers. We then address the strategic model in which optimal capacity profiles, both static and dynamically adjusted, are obtained based on two-part capacity contracts. We characterize the structure of optimal procurement strategies when the lead times of the two suppliers differ by a single period and the lead time for salvage opportunities matches that of one of the suppliers. For general lead time combinations, we show that the optimal procurement strategies satisfy monotonicity and limited sensitivity properties, and construct effective heuristics and upper and lower bounds based on our structural results.