Review of stochastic cost functions for production and inventory control (original) (raw)
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Average cost per unit time control of stochastic manufacturing systems: Revisited
Mathematical Methods of Operations Research (ZOR), 2001
An optimal production planning for a stochastic manufacturing system is considered. The system consists of a single, failure-prone machine that produces a ®nite number of di¨erent products. The objective is to determine a rate of production that minimizes an average cost per unit time criterion where the demand is random. The results given in this paper are based on some large deviation estimates and the Hamilton-Jacobi-Bellman equations for convex functions.
2010
The single-product, stationary inventory problem with set-up cost is one of the classical problems in stochastic operations research. Theories have been developed to cope with finite production capacity in periodic review systems, and it has been proved that optimal policies for these cases are not of the (modified) (s, S)-type in general, but more complex. In this paper we consider a production system such that the production rate is constrained, rather than the amount as is common in periodic review models. Thus, in our case the production rate is positive and finite when the system is on and zero when off, while a cost is incurred to switching on or off. We prove that a long-run optimal stationary policy exists for this single-item continuous review inventory problem with non-zero switching cost and finite production rate, and that this optimal policy has an (s, S)-structure. We also provide an efficient numerical procedure to compute the parameters of the optimal policy. Another, and perhaps more precise, way to include a capacity constraint is to constrain the production rate, rather then the production amount per review period. In this paper we follow this idea, and determine the structure of the optimal policy for the single-item inventory problem with set-up cost, backlogging, and finite production rate. Orders arrive according to a Poisson process and the i.i.d. demands follow some (rather) arbitrary distribution. The inventory is replenished at a constant rate only when production is on. It will be proved that a long-run optimal stationary policy exists and is described by two parameters: s and S. As soon as the inventory gets below s, production is switched on, while as soon as the inventory hits S, production is switched off. We also provide a dynamic-programming based numerical method to efficiently characterize the optimal policy. In passing we mention that in (s, S)-policy literature, see e.g. and references therein, an important objective of knowing that the optimal policy is of the (s, S)-type is to use this to find the optimal values for s and S. For our approach, however, it is not necessary to know the structure of the optimal policy; it is a corollary that the optimal stationary policy has an (s, S)-structure and can be found simply by iteration (bisection).
Total-Cost Stochastic Inventory Control Problems
2016
This paper describes the structure of optimal policies for discounted periodic-review single-commodity total-cost inventory control problems with fixed ordering costs for finite and infinite horizons. There are known conditions in the literature for optimality of (st, St) policies for finite-horizon problems and the optimality of (s, S) policies for infinitehorizon problems. The results of this paper cover the situation, when such assumption may not hold. This paper describes a parameter, which, together with the value of the discount factor and the horizon length, defines the structure of an optimal policy. For the infinite horizon, depending on the values of this parameter and the discount factor, an optimal policy either is an (s, S) policy or never orders inventory. For a finite horizon, depending on the values of this parameter, the discount factor, and the horizon length, there are three possible structures of an optimal policy: (i) it is an (st, St) policy, (ii) it is an (st,...
Stochastic optimal capacity limits with linear management costs
2003
Abstract An optimal capacity management policy, based on Markov decision theory, is presented for a firm facing stochastic market demand with linear management costs. The optimal policy is presented as optimal boundaries and their limits representing the optimal capacity expansion and reduction levels. These boundaries show over time, regions where capacity should be unchanged, increased, or decreased in response to market demand based on the available information on the demand forecasts.
Optimal production planning in a stochastic manufacturing system with long-run average cost
1997
This paper is concerned with the optimal production planning in a dynamic stochastic manufacturing system consisting of a single machine that is failure prone and facing a constant demand. The objective is to choose the rate of production over time in order to minimize the long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost.
On the Relationship between Price and Capacity Decisions in Inventory Systems with Stochastic Demand
SSRN Electronic Journal, 2000
We address the simultaneous determination of pricing and capacity investment strategies in a multi-period setting under demand uncertainty. In our model a monopolistic firm makes three decisions: capacity investment (or disinvestment), production (inventory), and price, all of which can be specified dynamically as a function of the state of the system. We analyze the optimal joint strategy and through that investigate the relationships between the main strategic decision variables: price and capacity. We consider models that allow for either bi-directional price changes or models with markdowns only, and in the latter case we prove that capacity and price are strategic substitutes.
On inventory control models with a convex cost function
An inventory model, used to control a one-type product stock with an A-convex cost function is analyzed. The replenishment amount is taken as a control parameter. Strategies that are optimal in the sense of an average expected cost and a total revalued cost with a revaluation coefficient β are found and examined.
Approximation Algorithms for Capacitated Stochastic Inventory Control Models
Operations Research, 2008
We develop the first algorithmic approach to compute provably good ordering policies for a multi-period, capacitated, stochastic inventory system facing stochastic non-stationary and correlated demands that evolve over time. Our approach is computationally efficient and guaranteed to produce a policy with total expected cost no more than twice that of an optimal policy. As part of our computational approach, we propose a novel scheme to account for backlogging costs in a capacitated, multi-period environment. Our cost-accounting scheme, called the forced marginal backlogging cost-accounting scheme, is significantly different from the period-by-period accounting approach to backlogging costs used in dynamic programming; it captures the long-term impact of a decision on system performance in the presence of capacity constrains. In the likely event that the per-unit order costs are large compared to the holding and backlogging costs, a transformation of cost parameters yields a significantly improved guarantee. We also introduce new semi-myopic policies based on our new cost-accounting scheme to derive bounds on the optimal base-stock levels. We show that these bounds can be used to effectively improve any policy. Finally, empirical evidence is presented that indicates that the typical performance of this approach is significantly stronger than these worst-case guarantees.
Capacity and Production Managment in a Single Product Manufacturing System
Annals of Operations Research, 2004
In planning and managing production systems, manufacturers have two main strategies for responding to uncertainty: they build inventory to hedge against periods in which the production capacity is not sufficient to satisfy demand, or they temporarily increase the production capacity by "purchasing" extra capacity. We consider the problem of minimizing the long-run average cost of holding inventory and/or purchasing extra capacity for a single facility producing a single part-type and assume that the driving uncertainty is demand fluctuation. We show that the optimal production policy is of a hedging point policy type where two hedging levels are associated with each discrete state of the system: a positive hedging level (inventory target) and a negative one (backlog level below which extra capacity should be purchased). We establish some ordering of the hedging levels, derive equations satisfied by the steady-state probability distribution of the inventory/backlog, and give a more detailed analysis of the optimal control policy in a two state (high and low demand rate) model.
On multi-stage production/inventory systems under stochastic demand
International Journal of Production Economics, 1994
This paper was presented at the 1992 Conference of the International Society of Inventory Research in Budapest, as a tribute to professor Andrew C. Clark for his inspiring work on multi-echelon inventory models both in theory and practice. It reviews and extends the work of the authors on periodic review serial and convergent multi-echelon systems under stochastic stationary demand. In particular, we highlight the structure of echelon cost functions which play a central role in the derivation of the decomposition results and the optimality of base stock policies. The resulting optimal base stock policy is then compared with an MRP system in terms of cost effectiveness, given a predefined target customer service level. Another extension concerns an at first glance rather different problem; it is shown that the problem of setting safety leadtimes in a multi-stage production-to-order system with stochastic lead times leads to similar decomposition structures as those derived for multi-stage inventory systems. Finally, a discussion on possible extensions to capacitated models, models with uncertainty in both demand and production lead time as well as models with an aborescent structure concludes the paper.