Some Properties of Decentralized Supply Chains

This paper analyzes the bullwhip effect in decentralized, linear and time-invariant (LTI) supply chains. It generalizes existing results by broadening the class of policies and customer demand processes under consideration. The supply chain is modeled as a single-input, singleoutput control system driven by arbitrary demands. The paper discusses the appropriateness of various metrics for the bullwhip effect, and derives analytical conditions to predict its presence independently of the demand process. The paper also gives a formula for the variance of the order stream at any stage when the demand process is known and ergodic. Advance demand information (ADI) is shown to mitigate the bullwhip effect for general ordering policies. In the supply chain literature, the term “bullwhip effect” refers to a phenomenon where the fluctuations in order sequence are usually greater upstream than downstream of the chain. Figure 1 shows an empirical example where the orders placed by a supplier are more variable than the actual quantities sold. In multi-echelon chains, even very steady customer demand can generate wildly fluctuating supplier orders several stages upstream. The upstream suppliers feel as if they were at the end of a bullwhip, where small perturbations at the handle (customers) cause huge movements at the tip (upstream suppliers). The phenomenon is also evident in macroeconomic data [18, 2, 19, 26, 25]. The bullwhip effect is of much practical importance. The term was originally coined by the Procter & Gamble Corporation to describe their empirical observations. It has also been described 1 by Callioni1 as the “No.1 issue” faced by Supply Chain Services at Hewlett-Packard [4]. In business schools, “beer games” are widely used to demonstrate its existence and pernicious effects [29, 16, 20]. The bullwhip effect is important because it results in huge operating costs for upstream suppliers. These suppliers have to forecast demand, plan production and storage capacity, and control inventory based on the orders they receive. But these activities become inefficient with high order variability. With the bullwhip effect, a manufacturer would have to either: (i) set a high production capacity to satisfy its peak demands, wasting capacity during non-peak periods; (ii) set capacity at a lower level (e.g., slightly above the average demand rate), and either incur shortage in peak periods or carry large inventories; or (iii) adjust the capacity over time, incurring set-up costs. All these options imply either operating inefficiencies (high costs) or lack of responsiveness (poor customer service and loss of customer goodwill). Empirically, the bullwhip effect is estimated to inflate supply chain operating costs by 12.5− 25% [21, 22]. If the bullwhip effect is eliminated, the U.S. grocery industry alone could save on the order of 30 billion dollars each year [8, 22]. 1 Previous Work Not surprisingly, the body of research on the bullwhip effect is extensive. The bullwhip effect was first recognized in the 1950s [23, 14, 15, 24]. Later, simulations and games [29, 16] revealed that it arises persistently, even if the games are unstructured. These reports attributed the causes of the bullwhip effect to “players’ systematic irrational behavior”, or to “misperception of feedback”. Lee et al. [21, 22] looked for more satisfying explanations. They identified several operational causes and quantified their impacts for a single-echelon chain with an AR(1) customer demand process. The bullwhip effect was analyzed parametrically by comparing the variances of the orders placed by the supplier and the customer demand. Similar efforts, e.g. [1, 5, 6, 28], were later made to study variants of the problem for specific families of stationary demand processes. All these studies provide useful but, unfortunately, inconclusive insights because of their focus on single-echelons and their specific demand assumptions. More recently, [9, 10, 11] used harmonic analysis to obtain analytical results for multi-echelon Former director of Strategic Planning and Modeling (SPaM) at Hewlett-Packard 2 chains with general non-stationary demand inputs. These references show that all operationally efficient (rational) inventory control policies trigger the bullwhip effect, independently of the demand process. This is the essential reason for its prevalence. The references also show that if advance demand information and order commitments are allowed, the bullwhip effect can be eliminated while retaining efficiency. A decentralized ordering policy that achieves this goal for multi-echelon chains is presented. In related but independent work, [12, 13] used the “transfer function” concept of control theory to derive variance formulae for a generalized family of order-up-to policies and numerically illustrate the bullwhip effect. This analysis did not include analytical results. This paper uses the transfer function approach to generalize the results of [9, 10, 11, 12, 13] to a broader set of policies. Section 2 below formulates the problem; section 3 discusses the appropriateness of different metrics to measure the bullwhip effect; section 4 presents new analytical tests; section 5 gives some numerical examples; and section 6 discusses the effects of advance demand information.