Mahesh Nagarajan

Coalition Stability in Assembly Models

In this paper, we study dynamic supplier alliances in a decentralized assembly system. We examine a supply chain in which n suppliers sell complementary components to a downstream assembler, who faces a price-sensitive deterministic demand. We analyze alliance/coalition formation between suppliers, using a two-stage approach. In Stage 1, suppliers form coalitions that each agree to sell a kit of components to the assembler. In Stage 2, coalitions make wholesale price decisions, whereas the assembler buys the components (kits) from the coalitions and sets the selling price of the product. Stage 2 is modeled as a competitive game, in which the primary competition is vertical (i.e., supplier coalitions compete against the downstream assembler), and the secondary competition is horizontal, in that coalitions compete against each other. Here, we consider three modes of competition---Supplier Stackelberg, Vertical Nash, and Assembler Stackelberg models---that correspond to different power structures in the market. In Stage 1, we analyze the stability of coalition structures. We assume that suppliers are farsighted, that is, each coalition considers the possibility that once it acts, another coalition may react, and a third coalition might in turn react, and so on. Using this framework, we predict the structure of possible supplier alliances as a function of the power structure in the market, the number of suppliers, and the structure of the demand.

Inventory Models for Substitutable Products: Optimal Policies and Heuristics

In this paper, we examine the nature of optimal inventory policies in a system where a retailer manages substitutable products. We first consider a system with two products 1 and 2 whose total demand is D and individual demands are negatively correlated. A fixed proportion of the unsatisfied customers for an item will purchase the other item if it is available in inventory. For the single-period case, we show that the optimal inventory levels of the two items can be computed easily and follow what we refer to as “partially decoupled” policies, i.e., base stock policies that are not state dependent, in certain critical regions of interest both when D is known and random. Furthermore, we show that such a partially decoupled base-stock policy is optimal even in a multiperiod version of the problem for known D for a wide range of parameter values and in an N-product single-period model under some restrictive conditions. Using a numerical study, we show that heuristics based on the decoupled inventory policy perform well in conditions more general than the ones assumed to obtain the analytical results. The analytical and numerical results suggest that the approach presented here is most valuable in retail settings for product categories where the level of substitution between items in a category is not high, demand variation at the aggregate level is not high, and service levels or newsvendor ratios are high.

A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales

In this paper, we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst-case performance guarantees. In particular, we introduce dual-balancing policies for lost-sales models that are conceptually similar to dual-balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost. That is, in each period, we balance two opposing costs: the expected marginal holding costs against the expected marginal lost-sales cost. Specifically, we show that the dual-balancing policies for the lost-sales models provide a worst-case performance guarantee of two under relatively general demand structures. In particular, the guarantee holds for independent (not necessarily identically distributed) demands and for models with correlated demands such as the AR(1) model and the multiplicative autoregressive demand model. The policies and the worst-case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times. Our analysis has several novel elements beyond the balancing ideas for backorder models.

Stable Farsighted Coalitions in Competitive Markets

In this paper, we study dynamic alliance formation among agents in competitive markets. We look at n agents selling substitutable products competing in a market. In this setting, we examine models with deterministic and stochastic demand, and we use a two-stage approach. In Stage 1, agents form alliances (coalitions), and in Stage 2, coalitions make decisions (price and inventory) and compete against one another. To analyze the stability of coalition structures in Stage 1, we use two notions from cooperative games---the largest consistent set (LCS) and the equilibrium process of coalition formation (EPCF)---which allow players to be farsighted. Thus, in forming alliances, players consider two key phenomena: First, players trade off the size of the total profit of the system versus their allocation of this total pie, and second, they weigh the possibility that an immediate beneficial defection can trigger further counter defections that in the end may prove to be worse than the status quo. In particular, one such example is that of the grand coalition---which we show to be stable in the farsighted sense---even though players benefit myopically by defecting from it. We also provide conditions under which a situation of a few lone players competing against a large coalition is stable. We examine the impact of the size of the market (n), the degree of competition, the effect of cost parameters, and the variability of the demand process on the prices, inventory levels, and structure of the market. We discuss the possible strategic implications of our results to firms in a competitive market and for new entrants.

Prospect Theory and the Newsvendor Problem

The newsvendor problem is a fundamental decision problem in operations management. Various independent experimental studies in laboratory settings have shown similar deviations from the theoretical optimal order quantity. We clarify that Prospect Theory, a prevalent framework for decision making under uncertainty, cannot explain the consistent empirical findings. This paper was accepted by Yossi Aviv, operations management.

Dynamic Supplier Contracts Under Asymmetric Inventory Information

In this paper, we examine a supply chain in which a single supplier sells to a downstream retailer. We consider a multiperiod model with the following sequence of events. In period t the supplier offers a contract to the retailer, and the retailer makes her purchasing decision in anticipation of the random demand. The demand then unravels, and the retailer carries over any excess inventory to the next period (unmet demand is lost). In period t+1 the supplier designs a new contract based on his belief of the retailers inventory, and the game is played dynamically. We assume that short-term contracts are used, i.e., the contracting is dynamically conducted at the beginning of each period. We also assume that the retailers inventory before ordering is not observed by the supplier. This setting describes scenarios in which the downstream retailer does not share inventory/sales information with the supplier. For instance, it captures the phenomenon of retailers distorting past sales information to secure better contracting terms from their suppliers. We cast our problem as a dynamic adverse-selection problem and show that, given relatively high production and holding costs, the optimal contract can take the form of a batch-order contract, which minimizes the retailers information advantage. We then analyze the performance of this type of contract with respect to some useful benchmarks and quantify the value of prudent contract design and the value of inventory information to the supply chain. Markovian adverse-selection models, in which the state and action in a period affect the state in the subsequent period, are recognized as theoretically challenging and are relatively less understood. We take a nontrivial step towards a better understanding of such models under short-term contracting.

Technical note---Linear Inflation Rules for the Random Yield Problem: Analysis and Computations

In this paper, we propose a simple heuristic approach for the inventory control problem with stochastic demand and multiplicative random yield. Our heuristic tries to find the best candidate within a class of policies that are referred to in the literature as the linear inflation rule (LIR) policies. Our approach is computationally fast, easy to implement, and intuitive to understand. Moreover, we find that in a significant number of instances our heuristic performs better than several other well-known heuristics that are available in the literature.

Average Cost Single-Stage Inventory Models: An Analysis Using a Vanishing Discount Approach

An important problem in the theory of dynamic programming is that of characterizing sufficient conditions under which the optimal policies for Markov decision processes (MDPs) under the infinite-horizon discounted cost criterion converge to an optimal policy under the average cost criterion as the discount factor approaches 1. In this paper, we provide, for stochastic inventory models, a set of such sufficient conditions. These conditions, unlike many others in the dynamic programming literature, hold when the action space is noncompact and the underlying transition law is weakly continuous. Moreover, we verify that these conditions hold for almost all conceivable single-stage inventory models with few assumptions on cost and demand parameters. As a consequence of our analysis, we partially characterize, for the first time, optimal policies for the following inventory systems under the infinite-horizon average-cost criterion, which have thus far been a challenge: (a) capacitated systems with setup costs, (b) uncapacitated systems with convex ordering costs plus a setup cost, and (c) systems with lost sales and lead times.

Technical Note---A Multiperiod Model of Inventory Competition

This paper explores when it is important for firms to consider stockout-based substitution and competitors inventory levels in making inventory decisions in the context of a duopoly model. To address this question, we consider a model where two newsvendors sell substitutable products in a market with aggregate market demand D. The two firms get a proportion p and (1-p) of this demand, where p is random. We characterize the equilibrium inventory levels of the two firms in a single-period model and show the striking property that, under certain reasonable conditions on the cost parameters, the two firms ignore their competitors inventory levels and potential substitution demand, i.e., their inventory decisions are decoupled. Furthermore, we show under slightly more restrictive conditions on the cost parameters that the single-period results can be extended to the case where D is random. Finally, we extend the decoupling property to a multiperiod periodic review scenario and show that the resulting Nash equilibrium can be characterized simply as the solution to a single-product dynamic newsvendor problem that ignores substitution demand.

Technical Note---Capacitated Serial Inventory Systems: Sample Path and Stability Properties Under Base-Stock Policies

We study a periodically reviewed multiechelon serial inventory system with a capacity constraint on the order quantity at every stage. Under echelon base-stock policies, we demonstrate a simple sample-path result that maps the echelon shortfalls in the serial system to the shortfalls of suitably defined single-stage systems. Because the shortfall processes of single-stage systems are well understood, our result allows us to reinterpret results in the literature on the stability and regeneration times of such multiechelon systems in a simpler fashion with weaker assumptions.