An Analytical Investigation of the Bullwhip Effect

Warburton, Roger D.H.

doi:10.1111/j.1937-5956.2004.tb00151.x

Cited by 119 publications

(65 citation statements)

References 26 publications

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“…A detailed derivation of an equation of inventory balance (which will be used in this chapter) can be found in the following publications [6,19].…”

Section: An Equation Of Inventory Balancementioning

confidence: 99%

“…In paper [19] an equation of a replenishment policy is defined in which the size of an order is directly proportional to the shortage of inventory:…”

Section: An Equation Of Inventory Balancementioning

confidence: 99%

“…If delays in supplies are long, this might result in a significant increase in inventory, which is unwelcome [6]. Therefore, it is advisable to include equation (4) in the model, which adds constraints allowing for the suspension of any replenishment if its level exceeds the required value [12,[18][19].…”

Section: An Equation Of Inventory Balancementioning

confidence: 99%

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Numerical Solution of the Inventory Balance Delay Differential Equation

Novotná

2015

International Journal of Engineering Business Management

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Inventory represents an essential part of current assets, which are typically characterized by their transience. This paper aims to outline a numerical solution of the inventory balance equation supplemented by an order-upto replenishment policy for a case in which the problem is described by a differential equation with delayed argument. The results are demonstrated on a specific example and the behaviour of the model is presented using a computer simulation. The results are graphically shown in the Maple system. The solution makes use of the theory of functional differential equations, especially the part dealing with differential equations with delayed arguments.

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“…A detailed derivation of an equation of inventory balance (which will be used in this chapter) can be found in the following publications [6,19].…”

Section: An Equation Of Inventory Balancementioning

confidence: 99%

“…In paper [19] an equation of a replenishment policy is defined in which the size of an order is directly proportional to the shortage of inventory:…”

Section: An Equation Of Inventory Balancementioning

confidence: 99%

See 1 more Smart Citation

Numerical Solution of the Inventory Balance Delay Differential Equation

Novotná

2015

International Journal of Engineering Business Management

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“…References to original papers and further details of these derivations may be found in [1]. The Lambert W function has found applications in an astonishing diversity of disciplines including celestial mechanics [3], electromagnetism [4], supply chain dynamics [5], quantum theory [6,7], and solar physics [8]. Typically, the Lambert W function is used in these references to derive closed-form solutions where previously this was not possible.…”

Section: Origin Of the Lambert W Functionmentioning

confidence: 99%

Analytic Root Locus and Lambert W Function in Control of a Process with Time Delay

Cogan¹,

Paor²

2011

Journal of Electrical Engineering

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Recently, the Lambert W function has arisen in the analysis of many systems including a restricted class of time-delay systems. An alternative approach to this analysis, based on the well-established root locus method, is shown here to contain the Lambert W function as a special case.As a purely illustrative example of the equivalence between the Lambert W function and analytic root locus a system comprising a Proportional controller with a time-delay process is analysed. Controller designs based on rightmost eigenvalue location and the dominant eigenvalue method are described.K e y w o r d s: Lambert W function, root locus, time-delay system, stability, eigenvalues ORIGIN OF THE LAMBERT W FUNCTIONJohann Heinrich Lambert (1728-1777) was a multitalented scientist and philosopher who produced important work in fields such as number theory, optics, meteorology, and astronomy -to name but a few. Some notes on Lambert's life and work may be found in [1]. For further biographical information and details of Lambert's contributions to the mathematical theory of perspective see [2].In 1758 Lambert derived a series solution to the trinomial equationIn 1779 Euler studied the following transformed version of (1)When considering a special case of his series solution to (2) Euler introduced a function w that satisfiesA related function W (z) that solves ANALYTIC METHOD FOR DRAWING ROOT LOCUS PLOTS FOR SYSTEMS WITH TIME DELAYIf the process G(s) has time delay L > 0 the characteristic polynomial of the system in Fig. 1 iswhere N (s) and M (s) are polynomials of degree n and m respectively, with m ≤ n, s = σ + jω and the gain k is a real number. The roots of (5) are the eigenvalues of the system. In root locus terminology, the poles and zeros of the root locus are given by the roots of N (s) = 0 and M (s) = 0 , respectively. The root locus of (5) consists of the paths traced out in the (σ, ω) plane by the roots of (5) as k varies and these paths are called the root locus branches. As for the delay-free case, the root loci for time-delay systems are symmetrical about the real axis, but unlike the delay-free case, the number of root locus branches is infinite. Setting k = 0 , these branches start at the poles (roots of N (s) = 0 ) and at σ = −∞; setting k = ±∞, these branches terminate at the zeros (roots of M (s) = 0 ) and at σ = +∞. Root locus branches that do not terminate at a zero approach σ = +∞ along asymptotes. These asymptotes are infinite in number and parallel to the real axis. A geometric method, consisting of rules for drawing the root locus of (5) when L = 0 may be found in standard texts such as [12]. An extended set of these rules for drawing the root locus of (5) with L > 0 may be found in [13][14][15]. Alternatively, when L = 0 there is an analytic method [16,17] for drawing the root locus of (5). This method consists of deriving an equation for the root locus curves. A root locus equation for L > 0 will now be derived. In the following the real and imaginary parts of the polynomial P are written as ReP and ImP ...

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“…Other key papers were written by Simon (1952) and Deziel and Eilon (1967). More recent work can be found in Graves (1999), Disney and Towill (2003), Warburton (2004aWarburton ( , 2004b and Disney, Farasyn, Lambrecht, Towill and Van de Velde (2006).…”

Section: Literature Backgroundmentioning

confidence: 99%

The value of coordination in a two-echelon supply chain

et al. 2008

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We study a coordination scheme in a two-echelon supply chain. It involves sharing details of replenishment rules, lead times, demand patterns and tuning the replenishment rules to exploit the supply chain's cost structure. We examine four different coordination strategies: naïve operation; local optimization; global optimization; and altruistic behavior on behalf of the retailer. We assume the retailer and the manufacturer use an order-up-to policy to determine replenishment orders and the consumer's demand is a stationary independent and identically distributed random variable. We derive the variance of the retailer's order rate and inventory levels and the variance of the manufacturer's order rate and inventory levels. We initially assume that costs in the supply chain are directly proportional to these variances (and later the standard deviations) and investigate the options available to the supply chain members for minimizing costs. Our results show that if the retailer takes responsibility for supply chain cost reduction and acts altruistically by dampening his or her order variability, then the performance enhancement is robust to both the actual costs in the supply chain and to a naïve or uncooperative manufacturer. Superior performance is achievable if firms coordinate their actions and if they can find ways to reallocate the supply chain gain.

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An Analytical Investigation of the Bullwhip Effect

Cited by 119 publications

References 26 publications

Numerical Solution of the Inventory Balance Delay Differential Equation

Numerical Solution of the Inventory Balance Delay Differential Equation

Analytic Root Locus and Lambert W Function in Control of a Process with Time Delay

The value of coordination in a two-echelon supply chain

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