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Article Excerpt
1. Introduction
This paper studies how resource allocation in networks can mitigate risk exposure. It presents theory and insight on how risk attitude and network configuration drive the strategic placement of safety capacity and inventory for operational hedging. The networks considered here are designed and managed by a single expected utility maximizer. Design involves the sizing of resources, which include inventories as well as capacities, and management means processing to best fill market demands. Timing follows a two-stage recourse model: Resources are sized ex ante when the demand vector is uncertain but its probability distribution is known, while processing occurs after observing demand.
Sections 2 and 3 present the model, theory, and general results in terms of statistical quantities that allow for computation by simulation. These propositions hold for any portfolio of real options with general network topology and utility functions. To bring that theory to life, however, the remaining sections focus on newsvendor networks, which are linear recourse models that feature parsimony, tractability, and effectiveness in yielding insights into planning under uncertainty. Sections 4 through 7 each analyze a specific network in two steps. First, observations are made from a numerical study and intuitive explanations are proffered. Second, the insights are generalized as properties, which are statements for a specific network under certain conditions that are specified and proved analytically in the appendix.
After reviewing the single-resource case in [section]4, special attention is devoted to the three canonical newsvendor networks shown in Figure 1. All three serve two markets and are building blocks for general networks. The dedicated network features a dedicated resource for each market and pure diversification benefits that provide a natural or passive hedge: While its two resources lack operational dependence, the network profit has less variability than the sum of the individual resource profits. Aversion to financial variability thus induces resource investment dependence. The other two networks add a third resource, either in series or in parallel, that features operational flexibility benefits: demand pooling and ex post revenue maximizing allocation provide an active hedge that increases value by exploiting upside variations. In serial networks, each market requires some upstream dedicated work that is followed by a shared resource requirement. Examples are disk drive and computer manufacturing, where a common set of computers perform final burn-in and test routines. The serial network also is a core model for pure component commonality. The parallel network is a dedicated network augmented with auxiliary flexible capacity; it also models inventory substitution and resource redundancy. It allows a tailored response to uncertainty where dedicated capacity mainly fills base demand, while flexible capacity supplies variable demand. Economically, the shared resource is a complementary real asset, while the flexible resource is a substitute.
[FIGURE 1 OMITTED]
Risk exposure (or simply risk) refers to the undesirable consequence of a random prospect. In newsvendor models, the random prospect is typically called demand and is modeled by an exogenous probability distribution. The operational consequence of random demand is a likely, but undesirable mismatch between supply and demand manifested as overage or underage. The expected cost of overages and underages is called mismatch cost, following Cachon and Terwiesch (2006). The financial consequence of random demand is profit risk: profit variability risk as well as a decrease in expected profit.
Risk attitude describes how a decision maker perceives risk. Risk-averse agents prefer the expected value over the random variable. While risk-neutral newsvendors only care about the mismatch cost, risk-averse agents also care about profit variability risk. Traditionally, an increase in risk is equated with an increased mean-preserving spread (e.g., see Gollier 2001 for general definitions). For univariate normal random variables, this is equivalent to an increase in variance while keeping the mean constant. While demand variance impacts risk exposure (and will be used in our graphs), the remainder will illustrate that profit variances are the natural descriptors of risk exposure in networks. They summarize market and network interactions and capture the important impact of demand correlations, an increase of which typically increases profit risk. (1)
This paper establishes that risk attitude and network structure fundamentally change resource allocation. In contrast to single-resource settings, risk-averse newsvendors may invest more in networks than risk-neutral newsvendors: some resources and even total spending may exceed risk-neutral levels. With normally distributed demand, risk-averse newsvendors change resource levels roughly proportionally to demand variance (i.e., levels are quadratic in standard deviation), while risk-neutral agents adjust only proportionally to standard deviation.
These findings are explained in terms of hedging. Hedging is the action of a decision maker to mitigate a particular risk exposure. Operational hedging is risk mitigation using operational instruments. This definition is deliberately general to include risk-neutral agents as well as univariate settings. Holding excess assets such as stock or capacity reserves by a risk-neutral newsvendor is interpreted as operational hedging because it mitigates mismatch costs. This illustrates that operational hedging impacts expected profits; operational flexibility can even exploit risk and add value. Yet, the standard financial motivation for hedging is mitigation of profit variability risk, (2) which is the main topic of investigation in this paper. Hedging by "betting on two horses" or "not holding all eggs in one basket" presumes access to at least two risks whose counterbalancing effect is to reduce total risk. With multiple risks (demands) and multiple baskets (resources), newsvendor networks are a natural vehicle to study how operational instruments reduce total risk and may create value.
The analysis of the three canonical networks identifies three types of operational hedging that are summarized in Table 1. Risk mitigation through pure diversification or demand pooling steers the portfolio mix toward assets supplying lower-profit variance markets. These need not be the lower demand variance markets, which highlights the importance of profit variance to understanding risk-averse network design. Capacity imbalance in the serial and parallel networks can remain even with perfectly positive correlations (i.e., in the absence of risk pooling). This isolates the contingent optimization option imbedded in shared and flexibile resources: they can steer and allocate production toward the higher-profit market. When markets differ in profitability, risk aversion rebalances capacity toward the (redundant) flexible resource, but away from the (unique) shared resource. (Given that both types of resources are "product-flexible," this means that the appropriate hedging action for product-flexible resources depends on their network position: increase when in parallel with dedicated resources, decrease when in series.) Capacity imbalance and allocation flexibility thus mitigate profit risk which confirms and refines their interpretation as operational hedges.
Section 8 concludes with managerial take-aways and discusses model limitations and extensions. The appendix contains one key proof; all other proofs are given in the online appendix (provided in the e-companion). (3)
Three research areas are most related to this article: risk-averse single-resource newsvendor models, newsvendor networks, and operational hedging. This article is a natural successor to the seminal work by Eeckhoudt et al. (1995), who prove that the optimal level of a single-resource newsvendor is always decreasing in risk aversion for general concave utility functions. This article extends their ingenious proof technique to a newsvendor network and shows that their unambiguous result does not hold for networks. Other studies of risk-averse single-resource newsvendor models include Atkinson (1979), Lau (1980), Spulber (1985), Anvari (1987), Lau and Lau (1999), Agrawal and Seshadri (2000a, b), Gan et al. (2005), Gaur and Seshadri (2005), Caldentey and Haugh (2006), and Chod et al. (2006), with multiperiod extensions in Bouakiz and Sobel (1992) and Chen et al. (2004).
This article is also a natural successor to Van Mieghem and Rudi (2002), who define and analyze newsvendor networks under expected profit maximization. Flexibility in risk-neutral parallel networks was first studied by Fine and Freund (1990) with a discrete math-programming model and by Van Mieghem (1998) with a newsvendor network model. Other related newsvendor network studies of the risk-neutral parallel network include Bassok et al. (1999), Hale et al. (2000), Rudi (2000), Netessine et al. (2002), Van Mieghem (2004), Bish and Wang (2004), and Goyal and Netessine (2007). The risk-neutral serial network was studied in Harrison and Van Mieghem (1999) and extended in Van Mieghem (2003). As far as we are aware, the only other paper on risk-averse newsvendor networks is Tomlin and Wang (2005), which complements this one in terms of research question and treatment of risk attitude. They investigate flexibility and dual sourcing in unreliable newsvendor networks and consider both loss aversion and conditional value-at-risk. With unreliable resources and risk aversion, inherent redundancy in a dedicated network can make it the preferred strategy to a flexible resource even if the latter is cheaper.
This article also relates to the literature on operational hedging, a term promulgated by Huchzermeier and Cohen (1996). They provided a valuation model and numerical evidence that embedded real options like contingent supply and production switching reduce downside risk in the presence of exchange-rate uncertainty. Ding et al. (2007) review and add recent analytical advances on joint operational and financial hedging of exchange-rate risk. Hedging in that setting of price uncertainty also may lead to an increase in capacity, similar to our finding under demand uncertainty. Operational hedging by means of flexibility and capacity imbalance in newsvendor networks under demand uncertainty was studied in Harrison and Van Mieghem (1999) and extended in Van Mieghem (2003). Boyabatli and Toktay (2004) survey and critically discuss papers on operational hedging, most of which assume expected profit maximization. Hedging obviously requires the presence of uncertainty but its standard objective is to reduce risk, not to make money. This paper shows that risk aversion magnifies these operational constructions, establishing that they mitigate risk and strengthening their interpretation as operational hedges.
2. Model
2.1. Decision Problem
Consider a firm that has n different real assets or "means of processing," which we will call resources. We adopt the notation of Van Mieghem (2003) and denote its resource portfolio by the nonnegative resource vector K [member of] [R.sub.+.sup.n], whose ith component represents the level of resource i available for processing during the period. The resulting operating profit gained at the end of the period is a random variable that is a function of the available resources. Let [pi](K, [omega]) denote this operating profit function, where [omega] is a sample point in the sample space [OMEGA]. The operating profit function is concave in the resource vector K, reflecting the natural assumption of decreasing marginal returns from investment. The financial investment cost to install resource levels K is denoted by [C.sub.0](K). As usual, [C.sub.0] is assumed to be convex to guarantee a well-behaved concave optimization problem. A typical economic assumption, however, is that [C.sub.0] exhibits economies of scale and slightly concave functions or the addition of a fixed cost often does not pose a problem.
The research problem is to decide on the resource vector K given a probability distribution P on the sample space [OMEGA]. The risk-averse decision maker...
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