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SLIM Technologies, LLC

Professor Shapiro explains the theory behind inventory modeling and describes the problems inherent in mixing optimization and classical inventory models.

Introduction

Supply chain managers in a growing number of companies are using optimization models to support fact-based decision-making. Such models assist them in making better decisions about sourcing, manufacturing, transportation, warehousing, customer service, and inventory management across the geographically dispersed facilities and markets of their companies' supply chains. Imbedded in easy-to-use systems, optimization models have helped many companies identify plans with significantly reduced supply chain costs.

Practitioners who develop these models harbor a secret that is rarely revealed to their clients. Classical inventory models, which compute safety stocks, replenishment quantities, and re-order points, are incompatible with optimization models, which compute holistic plans for minimizing total supply chain cost. Specifically, the techniques of probability theory underlying the construction of classical inventory models do not blend well with the techniques of linear and mixed integer programming underlying the construction of supply chain optimization models.

Thus, many—if not most—supply chain models developed to date are limited by one of two complementary deficiencies. Some emphasize inventory decisions with insufficient detail about other supply chain decisions, such as those relating to facility location, manufacturing processes, transportation, and warehousing. Conversely, other supply chain models emphasize these other decisions with insufficient detail about inventory management.

A guiding principle underlying integration of inventory decisions with other supply chain decisions is hierarchical planning. As shown in Figure 1, inventory decision-making as part of total supply chain management is divided into three areas of planning: (1) strategic planning involving long-term inventory deployment plans; (2) tactical planning involving aggregate inventory plans; and (3) operational planning involving detailed inventory management plans. Moreover, decision-making at the three levels of planning should be linked to ensure short, medium and long-term profitability of the firm.

Figure 1—Hierarchical Planning and Inventory Management

The sections below begin with a brief review of classical inventory theory. Following that, approximations of inventory costs and plans that can be incorporated in optimization models are presented. This discussion focuses primarily on strategic and tactical supply chain planning problems because models for these problems require new constructs to achieve integrated decision-making. Thus, segment planning models as:

• Strategic: Inventories in Snapshot Models
• Tactical: Inventories in Multiple-period Models
• Operational: Inventories in Multiple-period Scheduling Models

Concluding is a brief discussion of areas of applied research and new modeling approaches for combining inventory decisions with other supply decisions.

Classical Inventory Theory

Academics have studied inventory models for over 40 years, producing an extensive literature (Peterson and Silver, 1985). Obviously, we cannot do justice to the breadth and depth of these results. Still, it is instructive to review briefly the following model, which has found wide application (Winston, 1994):

(r, q) model: When inventory of the product falls to the re-order point r, order the replenishment quantity q.

This model is typically applied to fast moving products for which inventory is closely tracked.

Figure 2—Inventory Subject to Uncertain Demand (Source: Shapiro, 2001)

The model's strategy is depicted in Figure 2. A time t during which a random amount of demand occurs is associated with delivery of q items of the product; for simplicity, we assume t is known with certainty. In some re-ordering cycles, demand becomes negative before the order is received. The probability that this occurs is controlled by the quantity of safety stock held. As we have shown it, negative demand is treated as a backlog that must be filled once replenishment occurs.

Hedging against uncertainty in inventory over the delivery time t is the essence of inventory theory. For consumer products, inventory stockouts may lead to customer dissatisfaction or lost sales. For inventories of critical parts for machine tools, mainframe computers or commercial aircraft, inventory stockouts may cause costly equipment downtime. Managerial judgment is needed to decide on how much to spend on safety stock in avoiding, but not eliminating, these stockouts.

The requisite safety stock for the product is implicitly determined by selecting the reorder point r = r* in Figure 1 so as to meet the company's desired

order fulfillment rate = 1 - Probability (demand during lead time ³ r*),

which implies

safety stock = r* - Expected (demand during lead time).

Assuming demand is normally distributed, these quantities are illustrated in Figure 3. For fixed expected demand, it is an increasing function of the variance of demand, because a larger variance implies greater dispersion of the bell-shaped curve, and therefore larger values for r* and the safety stock.

Figure 3—Normal Density Function of Demand

Classical inventory theory takes a somewhat different approach in computing an optimal strategy for the (r, q) model. To describe it, we let E(D) denote the expected annual demand for the item, c_B denote the cost incurred for each unit backlogged, K the fixed cost of placing an order, and h the yearly inventory holding cost for one unit. It can be shown that the optimal values r* and q* must satisfy

In comparing this last equation with the earlier definition of the order fulfillment rate, we see that the unit stockout cost c_B in the classical model effectively implies an order fulfillment rate. Conversely, a given order fulfillment rate could be used to determine an implied stockout cost.

Clearly, safety stocks for individual products held in inventory are critical to the execution of an effective inventory policy. The issue at all levels of planning is how they should be computed. The simplicity of the (r,q) model, and others similar to it, is appealing, especially for short-term control of the firm's inventories. Still, practical considerations cast doubt on the validity of order fulfillment strategies based on the classical models. For example:

• Individual items may be shipped together, which affects delivery times.
• Storage space in the company's distribution centers may be limited, which may constrain the quantities that can be ordered.
• Availability of products at vendors may be limited, which also constrains the quantities that can be ordered.
• Inventories held in nearby distribution centers may be used to service customers when a stockout occurs in a given distribution center.
• Product demand may fluctuate over time.

Complications, such as these and others, reflect realities of modern supply chain management. In practice, managers can and must use combinations of judgment and classical analysis in determining safety stocks for their products.

Strategic Planning: Modeling Inventories in Snapshot Models

Strategic supply chain planning is concerned with long-term decisions faced by the firm about resource acquisition or divestment; for example, decisions about opening or closing new plants or distribution centers, implementing the supply chain of a new product, or acquiring another company. The majority of optimization models used today to analyze such planning problems are yearly snapshots of the company's supply chain. Scenarios for different years may be generated and optimized to examine the timing and phasing of major decisions.

To evaluate the effectiveness of potential strategic plans, optimization models must incorporate aggregate descriptions of how the supply chain will operate under them. Such descriptions often are based on product families that correspond to aggregations of similar products. Similarly, the models focus on demand in market zones that correspond to aggregations of customers in close geographical proximity.

The objective of many strategic supply chain models is to minimize the total cost of meeting forecasted demand for the given year subject to constraints on customer service. The total cost analysis should include sourcing, manufacturing, transportation, warehousing, as well as inventory costs. For situations where product mix decisions are allowed, the objective is to maximize net revenues, which equals gross revenues minus the cost of products sold. Customer service requirements are achieved by limiting links that connect inventory stocking points only to those markets that are sufficiently close.

Five types of inventory costs may be incorporated in a single-period optimization model: pipeline, handling, product losses, holding, and storage. Pipeline costs refer to holding costs associated with products tied up in transportation. If significant, these costs can be incorporated in an optimization model as increments to transportation costs associated with movements along links connecting origins to destinations. Handling costs and costs due to product losses can be incorporated as process costs at distribution centers.

Holding costs are often the most important inventory costs, as well as the most difficult to represent in an optimization model. In determining the level of effort appropriate for describing inventory decisions and associated holding costs, the modeling practitioner is guided by the relative magnitude of these costs. For commodities that are difficult to store, such as bulk paper for which inventory holding cost may be two percent or less of total supply chain cost, the holding costs may be ignored. For products with high intrinsic value that may not sell quickly, such as jewels for which inventory costs may be 20 percent or more of total supply chain cost, holding costs are important and must be included in the total cost analysis.

To model holding costs, we require empirical approximations that actualize the following:

Goal of Modeling Inventory Decisions for Strategic Planning: For each product family, determine an optimal inventory deployment plan by selecting the number and location of sites where the family will be stored, the yearly throughput of the family at these sites, and how markets will be served so that total supply chain cost for the year is minimized and customer service requirements are met.

The choice of sites where each family of products will be held may be linked to investment decisions associated with the family; for example, special equipment for frozen food products or flow-through equipment for fast-moving items.

Figure 4—Holding cost as a function of throughput.

Figure 4 depicts a typical empirical analysis of inventory holding cost for a product family as a function of yearly throughput of the family at a storage facility during the year. The marginal holding cost deceases with volume because higher volume implies faster turnover, which implies the product family is held for a shorter time. Letting V denote throughput volume, a functional representation of the inventory holding cost in Figure 4 is _xV^y where y usually lies between 0.5 and 0.8, and x depends on the average value of items in the product family (Ballou [1992]).

To incorporate holding cost curves of this type in a single-period, yearly model, we employ piecewise linear approximations. Such approximations allow us to use general-purpose methods of mixed integer programming in optimizing the model. Mixed integer programming is needed to describe these piecewise linear cost curves because they have economies of scale. Since such methods are needed to optimize facility location and other logical decisions, the approximations do not require the development of special purpose optimization techniques. Still, they do add to the complexity of the model. If we consider a distribution center location model involving 25 potential sites and 50 product families that might be held at each site, we would require 1250 zero-one variables to incorporate the simplest piecewise linear approximations. Models with this number of zero-one variables are solvable in reasonable run times, especially if the optimizer's search routines are customized to the structure of the model, but they will cause run times to increase.

Finally, storage costs, such as those paid to a third-party warehouse, may also be an important element of total supply chain cost. They can be modeled based on throughput in a manner similar to the one outlined for holding costs. However, depending on contractual arrangements or other sources of storage costs, the functional relationship between annual throughput and annual storage costs may be somewhat different in form.

Occasionally, multiple-period optimization models have been implemented to explicitly analyze the timing and phasing of supply chain investments or divestments. The constructions discussed here are still required to capture inventory holding costs within each of the long periods (e.g., years) in such models.

Illustration: A multi-national retailer, with more than 20 retail stores in the United States and Canada, applied a supply chain modeling system in studying strategic expansion plans. Over a five-year period, the company intended to significantly increase sales throughout the United States and Canada by constructing new retail stores. This growth would require expansion of the company's supply chain. In doing so, they wanted to limit the costs of inventories, warehousing and transportation, as well as capital investment in new distribution centers.

Historically, inventory holding costs for the company's product line were 15 to 20 percent of total supply chain costs. Regression analysis of historical data for the 75 product families used in the study was performed using a piecewise linear approximation. The R2 values measuring the goodness of the fits had an average value of .56 (an R2 = 1 corresponds to a perfect fit). Such approximations were used for each of the five years in the multi-period model generated and optimized by the modeling system.

The study team used optimization models created and solved by SLIM/2000 to analyze multiple distribution strategies, including the company's current strategy (Slim Technologies, 2002). The model considered fixed, operating and shutdown costs of company-owned distribution centers, inventory holding and transportation costs, and store service levels. Compared to current distribution strategy, the team identified a distribution strategy that would lower costs by roughly $50 million while improving service and increasing inventory turns, which lowered holding costs compared with historical figures.

Tactical Planning:

Modeling Inventories in Multiple-Period Models

Tactical supply chain planning is concerned with medium term decisions faced by the firm regarding the refinement and allocation of resources to support integration of functional and geographically dispersed activities. Analysis of tactical planning problems can best be carried out using multiple-period models where the periods may be weeks or months. For convenience, we assume all periods have the same length. The planning horizon of a model refers to the number T of periods in the model.

Inventory decisions and costs may be directly modeled using inventory balance equations of the form

ending inventory in period one
= beginning inventory in period T
+ quantity manufactured or received during period t
- quantity sold or shipped during period t

Such an equation must be specified for each product, each facility where the product may be held, and each period of the tactical model. Note that the ending inventory in period t becomes the beginning inventory in period t + 1. In applications where the number of distinct products is large, an aggregation of products into product families may be used in constructing the balance equations and other constructs of the model.

Each inventory balance equation is initialized using current inventory for the product at the facility, which equals beginning inventory in period one. Moreover, since the firm will almost always continue to sell the product after the end of the planning horizon, the tactical model should constrain ending inventory in period t to at least a minimal level.

Goal of Modeling Inventory Decisions for Tactical Planning: For all products, facilities, and periods in the planning horizon, determine an optimal aggregate inventory plan by selecting beginning and ending inventories that allow orders and forecasted demand to be met while minimizing the total delivered cost of all products and achieving desirable levels of customer service. According to the inventory balance equations, these inventory decisions are dependent on sourcing, manufacturing and transportation decisions.

The model determines an aggregate plan because the products may be grouped into product families. Moreover, the time periods in the tactical planning models are too long (weeks or months) to provide details about the timing of replenishment plans for individual products.

Inventory balance equations in tactical optimization models can be reconciled with classical inventory theory in computing holding costs and imposing safety stock restrictions to protect against stockouts. The holding costs are computed on ending inventories; that is, if h is the unit holding cost for a given product over the period length used by the model, the holding cost for that product for a single period is

h * ending inventory of period T

The parameter h may include storage cost for the product. The total inventory holding cost over the planning horizon of the model is the sum of such terms over all products, facilities and time periods.

A simple algebraic manipulation of this sum reveals that we are actually charging inventory holding costs on a close approximation to the average of starting and ending inventory of the product at the facility during each period; this approximation is exact if we require inventories at the end of period t to equal inventories at the start of period one. Thus, the validity of the computation depends on how accurately these averages describe actual inventories. If they are insufficiently accurate, the lengths of the planning periods in the model may be too long implying that the model should be revised.

Safety stocks for each product at each facility at the end of each period may be imposed by constraints of the form

ending inventory in period t is greater than or equal to safety stock for period t + 1

These constraints can be directly linked to the binary decision about whether or not to open or keep open a facility and which products to hold there; that is, if the facility is chosen to be open, each product that has a positive throughput there must maintain the indicated safety stocks. If the products have been aggregated into product families, the safety stock for the family will be a weighted sum of safety stocks for individual items in the family.

Computation of safety stocks depends on the scope of the application. For a manufacturing/distribution application where the company's facilities have been operating for a while, safety stocks for each product and at each facility can be derived using classical models and managerial judgment. However, optimization of future tactical supply chain plans might significantly change product inventories at facilities and the product flows to the markets, thereby casting doubt on historical safety stocks. Moderate adjustment of safety stocks based on initial model results should provide approximations that are sufficiently accurate for the purposes of tactical planning.

Illustration: A large beer manufacturer used SLIM/2000 to develop a 12-month tactical planning model to optimize mid-term production and distribution planning (Slim Technologies, 2002). As shown in Figure 5, the company's supply chain is comprised of five plants and 40 DCs. The DCs are organized in three levels: plant DCs, regional DCs and local DCs. In addition, the company has arrangements with third-party warehouses to handle surge inventory for peak demand during the summer season. Each surge warehouse is designated with an S in Figure 5. The company sells approximately 100 products nation-wide.

Figure 5—Supply Chain Network for a Brewery Company (Source: Shapiro, 2001)

The company's tactical planning model is large-scale with 12 monthly sub-models linked by inventories of the company's 100 products. The sub-models for each month capture the manufacturing plans at the five plants and the distribution plans linking the five plants to 40 DCs and warehouses. Forecasted market demand is backed up to the 40 DCs. In other words, assuming that the customers to be served by each DC are known and given, the sum total of their demand for each product is associated with that DC. The model has been successfully applied to tactical planning in the company for over two years.

Operational Planning:

Modeling Inventories in Multiple-Period Scheduling Models

Models to support operational planning of inventories are a vast topic that we cannot cover in any depth. The classical models discussed above were designed for and are relevant to these problems, particularly for pure distribution systems where capacity constraints and connections between products are largely absent. Nevertheless, supply chain complications linking inventory decisions to short-term manufacturing and/or transportation decisions implied by inventory balance equations suggest the need for and use of optimization models to help managers identify cost-effective plans. These balance equations provide the connection to capacity constrained production and transportation sub-models that describe how products are made and distributed.

Conclusions

Our discussion illustrated methods for integrating inventory decisions with other supply chain decisions in optimization models to support strategic and tactical planning. The methods relied on approximations of holding cost curves and safety stocks computed by classical inventory models. Additional applied research is needed to extend and improve analytical methods for integrating these decisions.

One applied research area is the construction of methods to combine optimization models, which identify minimal cost supply chain network designs, with Monte Carlo simulation methods, which evaluate the customer service performance of the inventory systems when faced with uncertainties in demand and other factors. The missing element in such a methodological integration is rigorous procedures for providing the optimization models with feedback from the simulation models about unsatisfactory or over satisfied customer service.

A second research area is the extension of linear and mixed integer programming models for tactical planning to stochastic programming models (Shapiro, 2001). A stochastic programming model explicitly considers multiple scenarios of an uncertain future, each with a probability of occurrence. It simultaneously determines an optimal contingency plan for each scenario and a short-term plan that optimally hedges against these contingencies. Such models are conceptually a very good fit to the uncertainties associated with tactical inventory planning. Demand uncertainties are translated into a number of scenarios that would be simultaneously optimized. Research is needed to discover efficient, practical schemes for generating such scenarios from demand forecasts and to invent efficient practical schemes for optimizing the resulting models when the number of scenarios is large.

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