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Author: Michael N. Katehakis
Publisher:
ISBN:
Category :
Languages : en
Pages : 39
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Book Description
Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. In this paper, we shall be primarily interested in minimizing an inventory cost function with respect to a constant replenishment rate. Consider a single-product inventory system under continuous review with constant replenishment and compound Poisson demands subject to lost-sales. The system incurs inventory carrying costs and lost-sales penalties, where the carrying cost is a linear function of on-hand inventory and a lost-sales penalty is incurred per lost sale occurrence as a function of lost-sale size. We first derive an integro-differential equation for the expected cumulative cost until and including the first lost-sale occurrence. From this equation, we obtain a closed form expression for the time-average inventory cost, and provide an algorithm for a numerical computation of the optimal replenishment rate that minimizes the aforementioned time-average cost function. In particular, we consider two special cases of lost-sales penalty functions: constant penalty and loss-proportional penalty. We further consider special demand size distributions, such as constant, uniform and Gamma, and take advantage of their functional form to further simplify the optimization algorithm. In particular, for the special case of exponential demand sizes, we exhibit a closed form expression for the optimal replenishment rate and its corresponding cost. Finally, a numerical study is carried out to illustrate the results.
Author: Michael N. Katehakis
Publisher:
ISBN:
Category :
Languages : en
Pages : 39
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Book Description
Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. In this paper, we shall be primarily interested in minimizing an inventory cost function with respect to a constant replenishment rate. Consider a single-product inventory system under continuous review with constant replenishment and compound Poisson demands subject to lost-sales. The system incurs inventory carrying costs and lost-sales penalties, where the carrying cost is a linear function of on-hand inventory and a lost-sales penalty is incurred per lost sale occurrence as a function of lost-sale size. We first derive an integro-differential equation for the expected cumulative cost until and including the first lost-sale occurrence. From this equation, we obtain a closed form expression for the time-average inventory cost, and provide an algorithm for a numerical computation of the optimal replenishment rate that minimizes the aforementioned time-average cost function. In particular, we consider two special cases of lost-sales penalty functions: constant penalty and loss-proportional penalty. We further consider special demand size distributions, such as constant, uniform and Gamma, and take advantage of their functional form to further simplify the optimization algorithm. In particular, for the special case of exponential demand sizes, we exhibit a closed form expression for the optimal replenishment rate and its corresponding cost. Finally, a numerical study is carried out to illustrate the results.
Author: Jim (Junmin) Shi
Publisher:
ISBN:
Category :
Languages : en
Pages : 43
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Book Description
This paper considers a continuous-review, single-product, production-inventory system with a constant replenishment rate, compound Poisson demands, and lost sales. Two objective functions that represent metrics of operational costs are considered: (1) the sum of the expected discounted inventory holding costs and lost-sales penalties, both over an infinite time horizon, given an initial inventory level; and (2) the long-run time average of the same costs. The goal is to minimize these cost metrics with respect to the replenishment rate. It is, however, not possible to obtain closed-form expressions for the aforementioned cost functions directly in terms of positive replenishment rate (PRR). To overcome this difficulty, we construct a bijection from the PRR space to the space of positive roots of Lundberg's fundamental equation, to be referred to as the Lundberg positive root (LPR) space. This transformation allows us to derive closed-form expressions for the aforementioned cost metrics with respect to the LPR variable, in lieu of the PRR variable. We then proceed to solve the optimization problem in the LPR space and, finally, recover the optimal replenishment rate from the optimal LPR variable via the inverse bijection. For the special cases of constant or loss-proportional penalty and exponentially distributed demand sizes, we obtain simpler explicit formulas for the optimal replenishment rate.
Author: Junmin Shi
Publisher:
ISBN:
Category : Costs, Industrial
Languages : en
Pages : 99
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Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. This thesis considers a continuous-review, single-product Make-to-Stock production-inventory system with infinite base-stock level, compound Poisson demands and constant continuous replenishment under the lost-sales policy, in which the inventory is subject to a cost function consisting of holding costs and lost-sale penalties. The main objective is to minimize pertinent inventory cost functions (the expected discounted cost and the time average cost) with respect to the replenishment rate. For the expected discounted cost case, we first derive an integro-differential equation system for the expected discounted cost incurred up until the first loss occurrence, conditioned on an initial inventory level, from which we obtain the Laplace transform for the conditional expectation of the discounted cost over an infinite time horizon. For a system starting from an arbitrary initial inventory level, we obtain a closed form formula for the expected discounted cost via the inversion of its Laplace transform. For the special cases of constant or proportional penalty function and exponentially distributed demand sizes, we exhibit an explicit expression for the conditional expectation of the discounted cost. Finally, we minimize the cost function with respect to the replenishment rate and provide an algorithm to compute the attendant optimal replenishment rate. We further obtain a closed form formula for the time-average cost under a suitable stability condition. For exponentially distributed demand sizes, we exhibit explicit solutions for the optimal replenishment rate for both the expected discounted cost function conditioned on initial empty inventory, as well as the time-average cost function. For each case, numerical studies are conducted to illustrate our results and investigate further properties of the system.
Author: Hyo-Seong Lee, M.M. Srinivasan
Publisher:
ISBN:
Category :
Languages : en
Pages : 34
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Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Author: Jing-Sheng J. Song
Publisher: Edward Elgar Publishing
ISBN: 180037710X
Category : Technology & Engineering
Languages : en
Pages : 565
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Book Description
This comprehensive Handbook provides an overview of state-of-the-art research on quantitative models for inventory management. Despite over half a century’s progress, inventory management remains a challenge, as evidenced by the recent Covid-19 pandemic. With an expanse of world-renowned inventory scholars from major international research universities, this Handbook explores key areas including mathematical modelling, the interplay of inventory decisions and other business decisions and the unique challenges posed to multiple industries.
Author: George Hadley
Publisher:
ISBN:
Category :
Languages : en
Pages : 474
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Author: George Joseph Feeney
Publisher:
ISBN:
Category : Inventory control
Languages : en
Pages : 32
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This memorandum derives the simple analytic solution to the special, but important inventory problem in which the optimal policy is to reorder a unit whenever one is demanded. The demand distribution can be any compound Poisson; the resupply distribution is arbitrary. Both the backorder case and the lost sales case are solved by generalizing a queueing theorem due to Palm. The procedure is to calculate by item the steady state probabilities for the number of units in resupply (or repair). These probabilities completely describe the item's long term behavior, and are simply the normalized values of the compound Poisson demand distribution based on the mean of the resupply distribution but not on the distribution itself. Knowledge of these state probabilities enables us to compute several measures of item supply performance as a function of the spare stock, s. Traditional inventory analysis can then be applied to minimize total cost based on estimates of holding cost and supply performance cost. The appendices contain a description of the algorithm and the computer program for calculating stuttering Poisson state probabilities and the measures of effectiveness for the backorder case. Numerical illustrations are also provided. (Author).
Author: Karin Möllering
Publisher: Kölner Wissenschaftsverlag
ISBN: 3937404341
Category :
Languages : en
Pages : 196
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Author: Michael H. Isaac
Publisher:
ISBN:
Category : Inventory control
Languages : en
Pages : 330
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