scholarly journals Optimal policies for a deterministic continuous-time inventory model with several suppliers

2020 ◽  
Author(s):  
Lakdere Benkherouf ◽  
Brian Gilding
Keyword(s):  
Unique Solution ◽  
Continuous Time ◽  
Infinite Horizon ◽  
Planning Horizon ◽  
Inventory Model ◽  
Inventory Level ◽  
Discounted Cost ◽  
Continuous State ◽  
Quasi Variational Inequality ◽  
Necessary And Sufficient

This paper is concerned with determining the optimal inventory policy for an infinite-horizon deterministic continuous-time continuous-state inventory model, where, in the absence of intervention, changes in inventory level are governed by a differential evolution equation. The decision maker has the option of ordering from several suppliers, each of which entails differing ordering and purchasing costs. The objective is to select the supplier and the size of the order that minimizes the discounted cost over an infinite planning horizon. The optimal policy is formulated as the solution of a quasi-variational inequality. It is shown that there are three possibilities regarding its solvability: it has a unique solution that corresponds to an $(s,S)$ policy; it does not admit a solution corresponding to an $(s,S)$ policy but does have a unique solution that corresponds to a generalized $(s,S)$ policy; or, it does not admit a solution corresponding to an $(s,S)$ policy or a generalized $(s,S)$ policy. A necessary and sufficient condition for each possibility is obtained. Examples illustrate their occurrence.

scholarly journals Optimal policies for a deterministic continuous-time inventory model with several suppliers: a hyper-generalized (s,S) policy

2021 ◽  
Author(s):  
Lakdere Benkherouf ◽  
Brian H Gilding
Keyword(s):  
Continuous Time ◽  
Control Policy ◽  
Planning Horizon ◽  
Inventory Model ◽  
Inventory Level ◽  
Purchasing Cost ◽  
Continuous State ◽  
Quasi Variational Inequality ◽  
New Type ◽  
Total Inventory

A deterministic continuous-time continuous-state inventory model is studied. In the absence of intervention, the level of stock evolves by a process governed by a differential equation. The inventory level is monitored continuously, and can be adjusted upwards at any time. The decision maker can order from several suppliers, each of which charges a different ordering and purchasing cost. The problem of selecting the supplier and the size of the order to minimize the total inventory cost over an infinite planning horizon is formulated as the solution of a quasi-variational inequality (QVI). It is shown that the QVI has a unique solution. This corresponds to a generalized $(s,S)$ policy under amenable conditions, which have been characterized in an earlier work by the present authors. Under the complementary conditions a new type of optimal control policy emerges. This leads to the concept of a hyper-generalized  (s,S) policy. The theory behind a policy of this type is exposed.

scholarly journals A stochastic inventory model with stock dependent demand items

2001 ◽  
Vol 14 (4) ◽  
pp. 317-328 ◽  
Author(s):  
Lakdere Benkherouf ◽  
Amin Boumenir ◽  
Lakhdar Aggoun
Keyword(s):  
Variational Inequality ◽  
Continuous Time ◽  
Infinite Horizon ◽  
Inventory Model ◽  
Stochastic Inventory ◽  
Stochastic Inventory Model ◽  
Stock Dependent Demand ◽  
Quasi Variational Inequality ◽  
Discounted Costs ◽  
Dependent Demand

In this paper, we propose a new continuous time stochastic inventory model for stock dependent demand items. We then formulate the problem of finding the optimal replenishment schedule that minimizes the total expected discounted costs over an infinite horizon as a Quasi-Variational Inequality (QVI) problem. The QVI is shown to have a unique solution under some conditions.

scholarly journals A Discounted-Cost Continuous-Time Flexible Manufacturing and Operator Scheduling Model Solved by Deconvexification Over Time

1995 ◽  
Vol 9 (1) ◽  
pp. 65-98
Author(s):  
B. Curtis Eaves ◽  
Uriel G. Rothblum
Keyword(s):  
Continuous Time ◽  
Flexible Manufacturing ◽  
Infinite Horizon ◽  
Solution Procedure ◽  
Linear Program ◽  
Discounted Cost ◽  
Scheduling Model ◽  
Availability Constraints ◽  
Operator Assignment ◽  
Over Time

A discounted-cost, continuous-time, infinite-horizon version of a flexible manufacturing and operator scheduling model is solved. The solution procedure is to convexify the discrete operator-assignment constraints to obtain a linear program and then to regain the discreteness and obtain an approximate manufacturing schedule by deconvexification of the solution of the linear program over time. The strong features of the model are the accommodation of linear inequality relations among the manufacturing activities and the discrete manufacturing scheduling, whereas the weak features are intra-period relaxation of inventory availability constraints and the absence of inventory costs, setup times, and setup charges.

scholarly journals Planning Horizon for Production Inventory Models with Production Rate Dependent on Demand and Inventory Level

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jennifer Lin ◽  
Henry C. J. Chao ◽  
Peterson Julian
Keyword(s):  
Production Rate ◽  
Optimal Solution ◽  
Planning Horizon ◽  
Inventory Model ◽  
Inventory Level ◽  
Inventory Models ◽  
New Production ◽  
Production Inventory ◽  
Infinite Planning Horizon ◽  
Finite Planning Horizon

This paper discusses why the selection of a finite planning horizon is preferable to an infinite one for a replenishment policy of production inventory models. In a production inventory model, the production rate is dependent on both the demand rate and the inventory level. When there is an exponentially decreasing demand, the application of an infinite planning horizon model is not suitable. The emphasis of this paper is threefold. First, while pointing out questionable results from a previous study, we propose a corrected infinite planning horizon inventory model for the first replenishment cycle. Second, while investigating the optimal solution for the minimization problem, we found that the infinite planning horizon should not be applied when dealing with an exponentially decreasing demand. Third, we developed a new production inventory model under a finite planning horizon for practitioners. Numerical examples are provided to support our findings.

Continuous-Time Q-Learning for Infinite-Horizon Discounted Cost Linear Quadratic Regulator Problems

2015 ◽  
Vol 45 (2) ◽  
pp. 165-176 ◽  
Author(s):  
Muthukumar Palanisamy ◽  
Hamidreza Modares ◽  
Frank L. Lewis ◽  
Muhammad Aurangzeb
Keyword(s):  
Continuous Time ◽  
Infinite Horizon ◽  
Linear Quadratic Regulator ◽  
Linear Quadratic ◽  
Discounted Cost ◽  
Q Learning

INFINITE-HORIZON OPTIMAL CONTROL BASED ON CONTINUOUS-TIME CONTINUOUS-STATE HOPFIELD NEURAL NETWORKS

2006 ◽  
Vol 04 (04) ◽  
pp. 707-719
Author(s):  
MING-AI LI ◽  
NAI-GONG YU ◽  
JUN-FEI QIAO ◽  
XIAO-GANG RUAN
Keyword(s):  
Optimal Control ◽  
Continuous Time ◽  
Control Method ◽  
Infinite Horizon ◽  
Hopfield Neural Network ◽  
Time Varying ◽  
Optimization Strategy ◽  
Linear Quadratic ◽  
Continuous State ◽  
Optimization Control

An optimal control method based on continuous-time continuous-state Hopfield neural network (CTCSHNN) is proposed for time-varying multivariable systems. The equivalence is built theoretically between receding-horizon linear quadratic (LQ) performance index and energy function of CTCSHNN, and the CTCSHNN is constructed to solve the above LQ optimization control problems. Moreover, the rolling optimization strategy is adopted to form closed-loop control structure that includes CTCSHNN so, the dynamic infinite-horizon optimization control is realized for multivariable time-varying systems. As an example, a second order time-varying system is simulated. Simulation results show the effectiveness and feasibility of the proposed method.

Ordering Policy for Deteriorating Items with Short Shelf Life Based on Lateral Transshipment

2011 ◽  
Vol 204-210 ◽  
pp. 915-918
Author(s):  
Xin Yi Bu ◽  
Fang Zhou Teng
Keyword(s):  
Supply Chain ◽  
Shelf Life ◽  
Conceptual Model ◽  
Planning Horizon ◽  
Inventory Model ◽  
Deteriorating Items ◽  
Inventory Level ◽  
Ordering Policy ◽  
Demand Rate ◽  
Lateral Transshipment

It is usually observed in the supermarkets that display of deteriorating items. These items are with short shelf life and difficult to be transported and stored. A supply chain conceptual model which consists of one supplier and two retailers is structured. Based on analysis of deteriorating inventory model in exist, the transshipment between the two retailers is added into the supply chain conceptual model and a deteriorating model is developed in this paper which can reflect more accurately the relation among inventory level, demand rate, deteriorating rate and lateral transshipment rate. The ordering policy taking into account the shelf life constraint is investigated with two last ordering opportunities on condition that the planning horizon can be modified based on the lateral transshipment. At last, the conclusion implies that the ordering policy for deteriorating items with short shelf life based on the lateral transshipment can improve the profit for the retailer.

scholarly journals Minimizing spectral risk measures applied to Markov decision processes

2021 ◽  
Author(s):  
Nicole Bäuerle ◽  
Alexander Glauner
Keyword(s):  
Minimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Planning Horizon ◽  
Insurance Company ◽  
Existence Of A Solution ◽  
Discounted Cost ◽  
Spectral Risk Measures ◽  
Markov Decision

AbstractWe study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (Math Methods Oper Res 74(3):361–379, 2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

1993 ◽  
Vol 25 (01) ◽  
pp. 82-102
Author(s):  
M. G. Nair ◽  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Necessary And Sufficient ◽  
Forward Equations ◽  
Quasi Stationary Distribution

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992). In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

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