Managing Demand Risk in Tactical Supply Chain Planning for a Global Consumer Electronics Company

1. Introduction

The consumer electronics industry is notorious for risk stemming from short product lifecycles and high demand uncertainty. However, a particular global consumer electronics company follows a deterministic planning process with a rolling 26-week horizon. We outline a process to manage this risk by suggesting two risk measures for demand- and inventory-related risk respectively and two linear-programming (LP) models: one for allocating the plants' replenishment schedule among the customers and the other to guide the request to plants for replenishment over the horizon. Our contribution is a model-aided process to manage demand and inventory risk in tactical supply chain planning using simple models and risk measures.

Of the two LP models, the first model is deterministic and allocates the plants' replenishment schedule for the coming 26 weeks among the different customers each week. This model can be used to automate the existing manual process. The other model is stochastic and provides the optimal replenishment from the plants under demand uncertainty. The two risk measures we propose are "demand-at-risk" (DaR) and "inventory-at-risk" (IaR). Another measure we propose is the expected cost of unmet demand and excess inventory at the retailers as a function of changing the magnitude of the supply continuously between ?20 percent. The models can be used to determine the values of the risk measures associated with the output. The values of these risk measures across products and the gap between the ideal and the actual values of these measures can be used by the company to real-locate capacity among different products.

The present work is exploratory and we have not implemented the models and the proposed process at the company although we do provide model solutions to illustrate the models and the risk measures. Our effort only explores modeling issues and the stochastic model we present is one of many possible with different motivations and different degrees of sophistication. It should not be seen as a definitive solution even for the company under consideration. Our focus is on managing demand risk in tactical supply chain planning; refer to Sodhi (2003) for a strategic supply chain planning application. We limit the discussion of risk to unmet demand and excess inventory within tactical supply-chain planning; see Chopra and Sodhi (2004) for other sources of uncertainty. Kleindorfer and Saad (2005) deal with disruption risk, something that does apply to tactical planning, but is not under consideration here. Game-theoretic issues or risk-sharing between the company and its customers do not apply to the present context.

2. Motivating Example

Consider the fulfillment process at the digital camera-and-videocamera division of a global consumer-electronics company. The company has regional offices throughout the world. These offices deal with customers in their region and coordinate with the company's headquarters for orders and fulfillment. Each region comprises one or more countries. Customers in a region include electronics retail chains like Circuit City or Best Buy in the US and Dixons in the UK as well as distributors. Every week, every customer provides a weekly "order" or forecast for each week of a 26-week rolling horizon for each of the division's products at the SKU-level to the appropriate regional office. Each regional office aggregates these orders by week and sends the aggregate weekly orders to headquarters. Headquarters allocates current and planned supplied to these orders and ships against the current week's orders from a central warehouse either to customer warehouses directly or to regional warehouses from where they can be dispatched to customers in the region.

There are restrictions on customers to revise their weekly orders relative to the ones they had provided a week earlier for the same period in the 26-week horizon. Likewise, plants are restricted by necessity to keep their production schedule and replenishment stable each week. For weeks 1-4, the time fence for order revisions, customer, and region orders are considered firm, while customers can change their "orders" for weeks 5-26 with restrictions. Likewise, the plants confirm their replenishment schedule for weeks 1-12, the time fence for production, but are more flexible for changes beyond that.

2.1. Existing Planning and Execution Processes

The headquarters including the central warehouse, regional offices, customers, and plants have the following two-tiered fulfillment process at the beginning of every week for each product:

1. Customers send their weekly orders for the next 26-weeks orders to the appropriate regional office. There, these orders are compared against the previous week's 26-week order to ensure any changes are within requirement. Unfilled orders from past weeks are also taken into account as backorders.

2. These orders are then aggregated at the region level. Headquarters compares these to the previous week's 26-week orders by region. Unfilled orders from past weeks are backordered in full.

3. Headquarters passes on the total orders for the 26-week horizon to the plants to help them schedule their production.

4. The plants provide headquarters (central warehouse) with their replenishment plan to the central warehouse for the 26-week period in consideration. The headquarters then allocates these current and planned supplies among the regions according to each region's "importance rating" and unfulfilled orders from previous weeks (backordered items are prioritized by the extent of delay and by the region's importance). The plants' supply schedule is considered "fixed" for the coming 12 weeks for the purpose of planning. However, in execution, supplies from plants to the central warehouse do vary from plan.

5. For each region, headquarters allocates the regional replenishment plan for the next 26 weeks to each of the region's customers according to each customer's importance rating and outstanding back-orders.

6. For the current week, headquarters ships out the orders to each region's warehouse or directly to the warehouses of large customers from the central warehouse using the replenishment scheduled for the current week. Likewise, the regional warehouses ship to their respective customers.

The above two-tiered process can be simplified to a single tier with the warehouse allocating shipments to each customer directly and we do so in the models we develop. We only need to incorporate a region's importance rating into the customer's individual importance rating, e.g., we can simply multiply the customer's rating by its region's rating.

2.2. A Deterministic Distribution Model

For any of this division's products, we can support the above process with a multi-period network flow problem that matches demand and supply. The network is the central warehouse as a source node that supplies to many "customers" (sink nodes) that represent retailers, regions, or a mix thereof. Below, we choose to model with customers, i.e., retailers and distributors, directly.

The company allocates the planned replenishments from the plants S^sub t^ among the customers c over the horizon that has T = 26 weeks. We index the weeks in the horizon by t = 0 . . . T - 1 with the present week denoted by t = 0. Additionally, t = -1 can be used to denote the past week so we can refer to values of quantities such as shipments made the previous week. In the current period (t = 0), each customer c makes an "order" or "forecast" F^sup c^^sub t^ for period t including the order for this week F^sup c^^sub 0^. The plants promise replenishment S^sub t^ to the warehouse for period f over the decision horizon including the actual replenishment S^sub 0^.

The main set of decision variables is s^sup c^^sub t^, the supply promised from the warehouse to customer c in period t including t = 0. A part of this supply, x^sup c^^sub t^ goes into meeting the customer's order for that week and the remainder y^sup c^^sub t^ to meeting the backorder. Given the order F^sup c^^sub t^, a part of the customer's order remains unmet and the decision variable to reflect that is a^sup c^^sub t^. This is backordered in full for the next period. Likewise, a part of the backorder for that customer remains unmet and this is denoted by b^sup c^^sub t^.

As regards parameters, we assume a discounting factor δ^sub t^ for shortage and holding penalties over time. Customer and backorder based on customer/region importance rating can be converted into customerspecific penalties, α^sup c^ and β^sup c^, for not meeting current orders and backorders respectively.

2.3. Numerical Example

Assume the company has three "customers" (c = 1, 2, 3) and uses a demand horizon of T = 10 weeks (instead of the actual 26 weeks). Customer orders F^sup c^^sub t^ for each of these weeks are with the requested replenishment r^sub t^ from the warehouse to the plants being simply the sum across customers (assuming no backorders from the previous week t = -1). However, the plants have promised a replenishment schedule S^sub t^ to the warehouse that is less than the total forecasted orders for most periods.

We can use ELP to allocate the replenishment schedule among the three customers and obtain s^sup c^^sub t^. Taking the penalties for each customer for unmet demand and unmet backorders respectively to be the discount factor δ to be 0.99, we can solve ELP to obtain the optimal customer shipments s*^sup c^^sub t^

Notice how S^sub t^ gets allocated among the three customers based on their relative importance and that of backorders. The optimal value is 11,418. (We obtained the solution within a tiny fraction of a second on a 1.2 GHz Pentium laptop running Dash Optimization's Xpress-MP solver on Windows 2000.)

Thus, the above model can be used to automate the existing process of allocation for the company under consideration. It is quite simple and can be integrated into an enterprise resource planning (ERP) system like the one running at the company. It can be easily implemented for all the products. We can also easily add other warehouses, e.g., those in the Far East, to the model-the existing process can handle only one warehouse.

3. Demand Risk

Even though its customers give their orders in advance to the company, the reality is that end-consumer demand is uncertain. The restrictions on the plants regarding changing the replenishment schedule and on the customers (i.e., retailers and distributors) changing their forecast or orders for future weeks mean that supply from the warehouse need not match end-consumer demand. If customers like Best Buy cannot meet consumer demand, it is bad not only for these customers (possibly even if consumers switch to competitors' products), but also for the company. An end-consumer not finding or seeing the product in the customer's store may purchase a competitor's product and stay with that competitor for accessories and upgrades. Likewise, excess inventory with the customer is also a risk. If the inventory keeps building up, the customer will eventually have to sell at a discount because the company does not allow returns from customers. Eventually, the customer may decrease shelf space for this product in favor of those selling better and order smaller quantities (or none) of this product as soon as it can.

3.1. Risk Measures

We propose two risk measures, one for not meeting end-consumer demand and the other for a customer having excess inventory. Both are taken across all customers by week for the 26-week horizon. Using the analogy of the so-called value-at-risk (VaR) measures in financial risk management, we can adopt a "demand-at-risk" (DaR) measure to quantify this unmet demand across customers. For a particular probability p for any week in the horizon, DaR^sub p^ can be defined as that value for which there is a p percent chance that unmet consumer demand across all customers can exceed this value for a particular week in question. The same applies to the total inventory across customers at the end of each period to give an "inventory-atrisk" (IaR^sub p^) measure. (Instead of VaR as analogy, we can use another measure, "conditional value-at-risk" or cVaR-mean excess loss-that is well suited as an objective for linear programming. We would then define cDar and cIaR similar to cVaR, but we have not explored these risk measures here because we are not minimizing the risk measures per se as can be seen from the model in the next sub-section.)

The risk measures, DaR and IaR, compete in one sense, but not in another. One could always reduce the DaR at the expense of increasing IaR or vice versa. So we need to balance these for a given uncertainty level and a given flexibility level of the plants in revising their production schedule and of the warehouse allowing customer order revisions. However, both risk measures can be reduced if uncertainty were reduced or if the plants' supply schedule or customers' order schedule were allowed greater flexibility.

Another risk measure useful for considering capacity reallocation at the plants among different products is the expected "cost" of unmet demand as well as excess inventory for each product summed over customers as a function of a multiple of the supply from the plants. This multiple, reflecting reallocation of capacity can be anywhere between 0.8 and 1.2 to show an increase or decrease in capacity. So, if the allocated capacity were to suddenly decrease by 15% resulting in a proportional drop in replenishment to the warehouse over the next 26 weeks, then we know the increased expected cost due to the increase in unmet demand across all customers. We can use this to compare the cost and benefit of allocating capacity among the different products using each product's importance rating as a proxy for its profitability. (At this level of planning, personnel do not have access to profits or selling price.)

3.2. Risk Management

Assume for the moment that we have a suitable way to model uncertain end-consumer demand at the customer level for the product we are considering. Given the uncertainty and the (in)flexibility of the plants and warehouse over the 26 weeks, one decision is to allow the warehouse to request an amount r^sub t^ that may be different from the aggregated customer orders or forecasts Σ^sub c^. F^sup c^^sub t^. This amount can be computed using a stochastic linear program putting different weights on shortage and on excess inventory at the customer. A simple baseline can be obtained by making two assumptions that (a) the plants have no restrictions in meeting this replenishment schedule in full, and (b) the customers can modify their orders without restriction at the last minute. The resulting value of r^sub t^ = Σ^sub c^ s^sup t^^sub c^ gives the "best" amount to request from the plants by period just as the newsvendor problem gives the optimal number of newspapers to order for one period.

Neither assumption is true in reality but the resulting baseline values of DaR and IaR as well as the cost measure discussed earlier can enable comparison between this ideal situation and the actual situation. The shipments promised to the customers using ELP are also an idealization but in the opposite way: they assume complete inflexibility of plants to change their supply schedule and the customers to change their orders over the horizon. In an environment with several products competing for capacity, the gap between the risk measures as well as between the actual and the ideal values of r^sub t^ for each product can help the plants prioritize across products.

To develop a stochastic programming model, we need discrete demand scenarios so that for any period t we get scenarios ζ^sub t^ each with probability φ^sub ζ^sub t^^. The main set of decision variables to determine is still s^sup c^^sub t^, the supply promised from the warehouse to customer c in period t including the current period. As with model ELP, part of this supply, χ^sup c^^sub ζ^sub t^^ goes into meeting the customer's order for that week and the remainder y^sup c^^sub ζ^sub t^^ to meeting the backorder except that now these variables are scenario-specific. Unlike ELP, only a portion θ (0 ≤ θ ≤ 1) of the unmet demand from the previous period carries over-the remaining fraction (1 - θ) is lost. Instead of F^sup c^^sub t^ as in ELP, we have scenario-specific consumer demand D^sup c^^sub ζ^sub t^^, which either remains remains "unmet," leaving a^sup c-^^sub ζ^sub t^^ for backorder or is exceeded by the supply leaving a^sup c+^^sub ζ^sub t^^ as excess inventory. Likewise, a part of the backorder for the customer remains unmet and this is denoted by b^sup c^^sub ζ^sub t^^.

The parameters of and β^sup c^ β^sup c^remain customer-specific penalties for not meeting the current order and backorders, respectively. In addition, we have another penalty γ^sup +^^sub c^ for excess inventory at the customer and γ^sup -^^sub c^ for demand that is permanently lost. Another new parameter is k as a multiple for total supply S^sub t^ to help us understand the implications of increasing or decreasing capacity for the product in question. (Even though the total capacity is not assumed to change in the next 26 weeks, capacity for an individual product can be increased or decreased by allocating capacity differently among the multiple products that the plants make.)

If we take k = 1 and consumer demand D^sup c^^sub ζ^sub t^^ = F^sup c^^sub t^ for all scenarios ζ^sub t^, this model can be used for deciding on the shipment level just like ELP. However, that is not the intent of this model. A better, if straightforward, use is to run it with value of s^sup c^^sub t^ fixed to be that of the optimal solution from ELP to obtain the values of DaR and IaR from the the output for the period in question.

Of greater benefit is to use this model assuming unlimited production capacity. When k [much greater than] 1, the total shipment schedule Σ^sub c^ s^sup c^^sub t^ provides the optimal balance between the excess inventory costs against that of losing (end-consumer) demand, both costs being to the company. We can also obtain the values of DaR and IaR under these conditions.

3.3. Numerical Example

Let us solve the stochastic program ESP. For a 10-week horizon, we have 2^sup 10^ - 1 = 1023 equiprobable scenarios with scenario probability φ^sub ζ^sub t^^ = 2^sup -t^. We consider all these scenarios, not just a sample, for reasons discussed later. To check the program against the deterministic model ELP, we first try it with the variation v = O so that Var[ε^sub t^] = O. Therefore, all scenarios for any given period are identical. We also zero out the effect of parameters that are in ESP, but not in ELP. Therefore, the penalty on excess inventory at the customer is taken as zero. Likewise, unmet demand is carried over in full so θ = 1. Then, we obtain the same solution as from ELP in about 10 seconds on the same computer with the same solution value of 11,418. (By contrast, the solution time for ELP was a tiny fraction of one second).

Next, we increase the variation v to 2.0 so that Var[ε^sub t^] = 2^sup 2^/4 = 1.0. Then, keeping the penalty on excess inventory zero, we obtain a slightly different solution regarding actual and planned shipments s^sup c^^sub t^ with a slightly lower expected cost at 9,303 in just over 10 seconds. (Solution has been rounded to zero decimal places.) So under demand uncertainty, the shipments to the customers can be improved using the stochastic solution even with the same supply.

For each model run, the scenario variables a^sup c-^^sub ζ^sub 0^^, a^sup c+^^sub ζ^sub 0^^, and b^sup c^^sub ζ^sub 0^^ can be used to compute the values of DaR^sub p^ and IaR^sub p^ for any p and for any time period. For instance, in the above run with k [much greater than] 1, for the period t = 8, we can obtain unmet demand and excessive inventory as a total across all customers (Figure 1) and use the fact that each scenario has probability 2^sup -8^ to obtain DaR^sub p^ and IaR^sub p^ values. In this case, there is a 10 percent probability that unmet demand will exceed about 50 units and excess inventory will exceed about 32 units. So DaR^sub 10%^ = 50 and IaR^sub 10%^ = 32. These measures can be tracked over time for all products to see which products have greater risk associated with them.

For the present case where S^sub t^ is that given by the plants, i.e., k = 1, we have a different situation with a lot more unmet demand and no any excessive inventory in any scenario (Figure 2). The values of the risk measures reflects this: DaR^sub 10%^ is about 195 and IaR^sub 10%^ is 0. As such, the company needs to consider increasing capacity for the product at hand at the expense of some other product.

One way for the company to look at how much capacity to increase is using another metric to show the change in the objective function of ESP as we vary the promised supply ?20 percent. Our computer runs with 0.8 ≤ k ≤ 1.2 show that an increase of 5 percent in the capacity allocated to the product (k = 1.05) in question can almost halve the cost relative to the current situation (k = 1) (Figure 3).

A number of interesting questions arise when we start thinking about demand being uncertain and we can attempt to answer these using the model ESP:

* How should we model uncertainty of demand? We used a simple model using deseasonalized AR(1) demand and then used a binomially distributed forecast error to build demand scenarios in the form of a binary scenario tree. However, we will revisit this issue in the next section.

* How many items should headquarters ask the plants to manufacture in each week of the rolling horizon? While the company (warehouse) can use the deterministic program ELP to promised supplies to the customers, it can use the stochastic program ESP with k [much greater than] 1 to request what to make assuming infinite supply is possible from the plants.

* What would be the benefit of increasing the flexibility for the plants and for the customers? This means reducing the period over which the production schedule cannot be revised from 12 weeks to, say, 4 weeks, and reducing the period over which customers cannot revise their orders from 4 weeks to 2 weeks? The model ESP can be easily modified to reflect that k = 1 for t < T^sub S^ and k = 1.2 for t ≥ T^sub S^ with T^sub S^ = 12. Then the impact of flexibility can be determined by reducing T^sub S^. As regards customers, ESP already assumes complete flexibility but we can modify the model to reflect how customer revisions are currently handled.

There are also questions regarding returns and lateral shipments but these are more strategic than the scope of the present work allows.

The above situations illustrate the use of the ESP model, not for scheduling that is best left to ELP, but for the company to understand and track the risks associated with different products. It can then use this information to reallocated production capacity among these products. ESP is a simple model and solves very quickly despite tens of thousands of variables. So, it can be run repeatedly for multiple products using different values of k as discussed above in deciding capacity reallocation and regarding where to show more flexibility. Still, our attempt at outlining a process using ESP or similar models is only a beginning on how to manage risk in this context.

4. Literature Survey and Modeling Considerations

There is a long history of using deterministic models for strategic and tactical planning (Geoffrion and Powers 1995). There are at least three ways to deal with demand uncertainty: (1) running demand scenarios separately in different computer runs with the same deterministic model or together in a single run in a stochastic programming model, (2) multi-echelon inventory management at the operational level, and (3) the newsvendor problem. Of these, we have taken the stochastic programming approach as this fits the present scope and answers many of the questions that arise from dealing with uncertainty. However, there are a number of modeling considerations for those choosing this particular route as will be seen from our review of the literature below; Sodhi (2005) deals with similar considerations in the asset-liability management literature in detail.

4.1. Overall Approaches

For strategic decisions like network redesign, we can run three (or more) different scenarios of "optimistic," "most likely," and "pessimistic" demand, and then create a plan through dialog as is done for more strategically aimed models. Geoffrion (1976) describes the approach at Hunt-Wesson for distribution system redesign. When we extend these deterministic models to include multiple demand (or other) scenarios, we get stochastic programming models for such uses as capacity planning (e.g., Eppen, Martin, and Schrage 1989; Alonso-Ayuso et al. 2003) and tactical planning (e.g., Escudero et al. 1999). A stochastic programming model can make specific recommendations at the detailed individual product level and time interval that we need rather than simply providing broad policies. Meeting operational requirements and incorporating managerial preferences is straightforward with stochastic programming. Recent advances in LP solver technology and in stochastic programming also help make the case for using such models. Still, modeling advantages of stochastic programming, as seen for instance in the excellent book by Birge and Louveaux (1997), have not translated into widespread use in practice despite reported successes of deterministic models and some early stochastic programming applications, except perhaps for financial applications like asset-liability management.

Multiechelon inventory theory is more operational in its scope in that it helps companies determine the right type of inventory to keep at various nodes in a supply chain (e.g., that of auto-repair parts) to handle day-to-day demand fluctuations. The flow of goods is provided by deterministic models for multicommodity network flows. For industries with short product cycles as in our case, multiechelon inventory theory may be difficult to apply. Also, the time window for high profits and establishing market presence may be small even if the product lifecycle is not short relative to the decision horizon.

Multiechelon inventory theory is also considered in decentralized supply chain management as for example in the seminal work of Clark and Scarf (1960). As supply chain research evolved from deterministic models for centralized supply chains to explicitly dealing with demand uncertainty in decentralized contexts, it also moved to strategic issues like contracts (e.g., Mart?nez-de-Alb?niz and Simchi-Levi 2005), sourcing, and channel coordination (e.g., Gan, Sethi, and Yan 2004; 2005). There is greater reliance on economic theory including incentives and game-theoretic modeling and well as on incorporating market information, e.g., using real options (e.g., Miller and Park 2005; Kleindorfer and Wu 2003; Huchzermeier and Cohen 1996; Kogut and Kulatilaka 1994). Market adjustment in the model has also made inroads into other areas of operations in particular with capacity investment (e.g., Birge 2000; Lederer and Mehta 2005) and R&D investment (Huchzermeier and Loch 2001). However, centralized planning is a fact for the consumer electronics company we describe and also for companies like Cisco as highlighted by the oft-quoted example of their having to writeoff $2 billion of inventory following the dot-com bust in 2001. Incentives may not apply if a company, as is the case with Cisco, owns the inventory at the supplier.

Another approach would be to consider the newsvendor problem and its variants possibly employing such simplifying assumptions as i.i.d. demand to adapt the newsvendor model for multiple time-periods. Such models are easy to solve and are easy to operationalize as has been done in apparel companies as L.L. Bean and World Company. But in the problem at hand, we have 26 periods with rules on how to prioritize backorders and current orders. Demand is also not i.i.d. over time. There are interactions across products of the company in question because demand for these may be positively correlated with the entire market and yet negatively with each other in that these products can compete with each other for the consumer's wallet. Consequently the newsvendor model may be difficult to use here although the stochastic program we employ is similarly motivated.

4.2. Modeling Demand Uncertainty

For the model ESP, we used a binary tree of scenarios, using deseasonalized AR(1) process, and assumed that demand of the single product under consideration is perfectly correlated across customers. One complication in our case is that demand uncertainty is experienced firsthand by only the customers and even these customers do not know lost demand for each product they carry. Moreover, if end-customers switch their purchase to a competitor's product, the customer (retailer) would not be able to detect the demand (e.g., Kraiselburd, Narayanan, and Raman 2004). So, we have to work with forecasted orders rather than with total expected consumer demand.

In finance, uncertainty pertaining to securities is modeled in many cases with binary scenario trees depicting the underlying stochastic process (e.g., Black, Derman, and Toy 1990). All 2^sup T^ scenarios need to be included in the stochastic program for portfolio optimization to avoid sample bias and even unbounded solutions so the number of periods T cannot be very large in practice. Recall we have 26 weeks in the example, although we used only 10 periods for illustration. To reduce T, we can take successive time periods of increasing length so that we could have T = 6 with two periods of one week, one period of a fortnight, two periods of one month each, and one period of one quarter, although doing so makes things trickier in a rolling horizon model like ours. It may also be argued that 26 weeks (with weekly periods) is too long for a fast moving industry like consumer electronics.

Multiple factors are interesting for supply chain researchers if we were to consider all our products as a handful of different product families and create demand scenarios jointly for these product families. When there are multiple factors, we could use an n-ary tree of scenarios (e.g., Bradley and Crane 1972 for bank portfolio optimization), but doing so comes at the expense of tractability challenges. We could use methods already developed in the asset-liability management literature: Kouwenberg (2001) as well as Gaivorinski and de Lange (2000) create event trees for ALM with a few asset classes (stocks, bonds, real estate, etc.) to match the first so many moments of the joint continuous probabilities obtained from history or future expectations; see also Hoyland and Wallace (2001). Demand history can be analyzed to obtain various moments at the product family level. Indeed, this may be a research opportunity for industries like consumer electronics.

Scenarios can be hand-created to take into account different possible futures with subjective probabilities attached. Yet another approach is to use continuous probability distributions for the uncertain factors. For example, Gupta and Maranas (2003) consider multiple sources of uncertainty, all modeled as continuous probability distributions, for determining a chemical plant production plan.

4.3. Objective Function

We sought to minimize the expected cost of unmet or delayed consumer demand and that of excessive customer inventory. This choice was based on the information available to the personnel involved in the tactical planning process. If we could have access to data on product margins, we could maximize expected profit as well. We could take risk-averse utility functions and attempt to maximize risk-averse utility u(.). A simplified version is the expected utility function Σ^sub ζ^sub T^^ [straight phi]^sub ζ^sub T^^ u^sub 1^(w^sub ζ^sub T^^) where w^sub ζ^sub T^^ is the profit level in scenario ζ^sub T^. Kallberg and Ziemba (1983) discuss the relevance of different forms of concave utility functions and show that they all give similar results if the average risk aversion is the same. Others attempted to avoid nonlinearity by adding constraints to model risk-aversion (Kusy and Ziemba 1986; Carino et al. 1998ab).

4.4. Constraints

Some researchers use chance constraints to limit the probability of a negative outcome, e.g., Dert (1999) and Drijver et al. (2000) use these to limit the possibility of underfunding in pension fund management, while Eppen et al. (1989) do so in a capacity planning context. For us, the negative outcome is not meeting the demand or having excess inventory. However, according to Kusy and Ziemba (1986), such models have difficulties in multiperiod situations and we do expect any model in the present context to have double-digit number of periods.

4.5. Frictional Losses

Our stochastic model does not include frictional losses like tariffs and excise duties. Cohen and Lee (1988) provide a deterministic model to take such frictional losses into account for a computer manufacturer. Note that dealing with these frictional costs means adding many more variables relative to a model that ignores these considerations.

4.6. Solution Approaches

Our ESP model solves quickly on an ordinary computer, because it is essentially a multiperiod network problem and because the main set of decision variables Sf is not scenario-specific. But, if the problem is extended to a full-fledged production planning problem with multiple products and joint-product capacitation, it could become truly challenging. For such problems, researchers have considered different solution approaches including decomposition, taking only a sample of scenarios, and aggregation.

As regards decomposition, Bradley and Crane (1972) obtain subproblems with a n-ary tree; Kusy and Ziemba (1986) use the algorithm in Wets (1984) for a stochastic LP with fixed recourse; Birge (1982) provides a solution method to tackle the large size of multistage stochastic LPs in general; Carino et al. (1998ab) use Bender's decomposition; and Mulvey and Vladimirou (1992) use the progressive hedging algorithm of Rockafellar and Wets (1991). For a Dutch pension funds application, Gondzio and Kouwenberg (2001) use Benders decomposition and a model "generator" to solve problems with T = 6 and upto 13^sup 6^ (≥4.8 million) scenarios on a parallel computer with 16 processors. On the other hand, Mulvey and Shetty (2004) describe challenges in solving for even a modest number of scenarios (4096) on a 128-processor machine. Thus, the large number of scenarios means that researchers have to approximate uncertainty in some way for optimal solution.

Taking only a sample of scenarios generated from an arbitrage-free stochastic process can result in poor quality solutions depending on how the scenarios were generated (Kouwenberg 2001). Also, the solution in each run can be quite different from the previous one with the same input because the number of randomly-generated scenarios is only a tiny percentage of a large population of highly diverse scenarios. To reduce variance, Mulvey and Thorlacius (1999), among others, use antithetic sampling, while Mulvey and Shetty (2004) stress robustness against perturbations in scenarios. Seshadri et al. (1998) combine simulation and optimization in asset-liability management that could work for tactical supply-chain planning as well.

Another way to handle the large number of scenarios is to use aggregation to approximate the uncertainty itself by combining scenarios. However, care must be taken to maintain primal and dual feasibility of the unaggregated problem as this has not always been the case (Klaassen 1997, 2002). We already have useful results in systematic aggregation for the general stochastic linear program (Birge 1985; Wright 1994) as well as for asset-liability management models specifically (Klaassen 1998). However, Gaivorinski and de Lange (2000) suggest that it is better to simplify decision variables-e.g., by using simple proportions after the first one or two periods-rather than to aggregate scenarios as the latter leads to worse solutions. For us, this could mean, for example, combining product demand across customers with similar importance ratings and then disaggregating based on the proportion of the forecasted orders.

5. Conclusion

We have shown how stochastic modeling can be useful in a tactical supply chain planning context for a particular electronics company. We presented a stochastic LP model that works in conjunction with a deterministic one, the former to determine the shipments promised to customers in for the next 26 weeks, and the latter to request the plants for replenishment over the same horizon. The models can therefore be used as part of the existing process. Moreover the risk metrics we provided can be used to guide the plants to reallocate capacity among the different products they manufacture although this would require a new process.

As we mentioned before, the present work is only a starting point in developing model-aided processes to manage risk. Therefore, much remains to be done on all three fronts: designing and solving models; changing the existing or devising new business processes; and devising and using risk measures. Regarding modeling, stochastic programming has a dearth of empirical findings as well as an absence of consensus to guide modeling choices even in such domains as asset-liability management where it has been used heavily. This leaves modelers with ad-hoc solution techniques. Besides the merits and limitations of the aggregation approaches, we need to determine how to create demand scenarios. We need models of uncertainty that are rich enough to capture the uncertainty of end-consumer demand for multiple products at multiple locations but not so complex as to render the model(s) intractable. This is especially important as the company can have many levers like reducing leadtime or increasing capacity for responding to demand uncertainty (Fisher et al. 1997).

Besides modeling, there are process issues to consider. Tactical planning typically does not include risk management. But with short product lifecycles and high margin products, we need to tackle uncertainty. In a tactical planning environment like the one we considered here, there are hundreds of products to deal with and we do not have information like product margins at this level. Processes have to be necessarily simple, not use much data, and be simple to implement if supported with software. We have presented simple models and risk measures that could fit in a tactical planning process. However, the process we outlined has not been tested in industry.

Finally, risk measures need to be considered carefully in light of operations. It is quite possible to adopt too many measures and then not use them because they are too many or too complicated. The measures we presented here are only a start and need to be implemented to see if they are useful in practice.

Thus, tactical supply chain planning under demand uncertainty is a fertile area of research with many modeling, process, and operational challenges. We hope that the present work will whet researchers' appetite.