We consider a production-inventory control model with finite capacity and two different production rates, assuming that the cumulative process of customer demand is given by a compound Poisson process. It is possible at any time to switch over from the different production rates but it is mandatory to switch-off when the inventory process reaches the storage maximum capacity. We consider holding, production, shortage penalty and switching costs. This model was introduced by Doshi, Van Der Duyn Schouten and Talman in 1978. Our aim is to minimize the expected discounted cumulative costs up to infinity over all admissible switching strategies. We show that the optimal cost functions for the different production rates satisfy the corresponding Hamilton-Jacobi-Bellman system of equations in a viscosity sense and prove a verification theorem. The way in which the optimal cost functions solve the different variational inequalities gives the switching regions of the optimal strategy, hence it is stationary in the sense that depends only on the current production rate and inventory level. We define the notion of finite band strategies and derive, using scale functions, the formulas for the different costs of the band strategies with one or two bands. We also show that there are examples where the switching strategy presented by Doshi et al. is not the optimal strategy.
In this paper, we present a simple microeconomic model with linear continuous-time dynamics that describes a production-inventory system with debt repayment. This model is formulated in terms of optimal control and its exact solutions are derived by prudent application of the maximum principle under different sets of initial conditions (scenarios). For a potentially profitable small firm, we also propose some alternative short-term control strategies resulting in a positive final profit and prove their optimality. Practical implementation of such strategies is also discussed.
We investigate the potential added value of being able to relocate production capacity, relative to fixed production capacity, in a network of multiple, geographically distributed manufacturing sites. There is a growing interest in production capacity that can be geographically relocated; e.g., modular units for pharmaceutical intermediates. It shows promise for enabling the fast fulfillment of a distributed network with a reduction in the total inventory and total production capacity of a distributed network with fixed production capacity without sacrificing customer service levels or total s…
In this paper, we consider joint drift rate control and impulse control for a stochastic inventory system under long-run average cost criterion. Assuming the inventory level must be nonnegative, we prove that a $\{(0,q^{\star},Q^{\star},S^{\star}),\{μ^{\star}(x): x\in[0, S^{\star}]\}\}$ policy is an optimal joint control policy, where the impulse control follows the control band policy $(0,q^{\star},Q^{\star},S^{\star})$, that brings the inventory level up to $q^{\star}$ once it drops to $0$ and brings it down to $Q^{\star}$ once it rises to $S^{\star}$, and the drift rate only depends on the …
Green Inventory Management: Leveraging Multiobjective Reverse Logistics
Queensland University of Technology (QUT)（昆士兰科技大学）；Institute of Innovation, Science and Sustainability, Federation University Australia（联邦澳大利亚大学创新、科学与可持续发展研究所）
We consider a general class of high-volume, fast-moving production-inventory systems based on both lost-sales and backorder inventory models. Such systems require a fundamental understanding of the asymptotic behavior of key performance measures under various supply strategies, as well as the pre-planning of these strategies. Our analysis relies on a thorough study of the asymptotic behavior of a random walk with power drift, which is of independent interest. In addition to providing key insights, our analysis leads to approximations of the corresponding optimization problem that yield simple …
School of Data Science, Fudan University（复旦大学数据科学学院）；School of Management and School of Data Science, Fudan University（复旦大学管理学院与数据科学学院）
Many large-scale production networks include thousands types of final products and tens to hundreds thousands types of raw materials and intermediate products. These networks face complicated inventory management decisions, which are often too complicated for inventory models and too large for simulation models. In this paper, by combing efficient computational tools of recurrent neural networks (RNN) and the structural information of production networks, we propose a RNN inspired simulation approach that may be thousands times faster than existing simulation approach and is capable of solving…
A classical inventory problem is studied from the perspective of embedded options, reducing inventory-management to the design of optimal contracts for forward delivery of stock (commodity). Financial option techniques à la Black-Scholes are invoked to value the additional `option to expand stock'. A simplified approach which ignores distant time effects identifies an optimal `time to deliver' and an optimal `amount to deliver' for a production process run in continuous time modelled by a Cobb-Douglas revenue function. Commodity prices, quoted in initial value terms, are assumed to evolve as a…
In this paper, an attempt has been made to develop a simple leakage inventory model without shortages with instantaneous or finite production rate under fuzzy environment. In the present day scenario, it is difficult to decide the exact annual demand rate and hence a major difficulty is faced by a decision maker to forecast the demand. Also, in any inventory system, goods in stock may subject to deterioration or leakage. Deterioration of goods refers to decrease in quality whereas quantity remains more or less the same. On the other hand, leakage refers to loss in quantity whereas the quality …
We consider a deterministic continuous time model of monopolistic firm, which chooses production and pricing strategies of a single good. Firm's goal is to maximize the discounted profit over infinite time horizon. The no-backlogging assumption induces the state constraint on the inventory level. The revenue and production cost functions are assumed to be continuous but, in general, we do not impose the concavity/convexity property. Using the results form the theory of viscosity solutions and Young-Fenchel duality, we derive a representation for the value function, study its regularity propert…
The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey auctions -- throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the op…
When imperfect quality products are produced in a production process, rework may be performed to make them become serviceable. In an inventory system, items may deteriorate. Selling deteriorated items to customers will create negative impact on corporate image. In this paper, two economic production quantity (EPQ) models are proposed for deteriorating items with rework process. A single production-rework plant system and a system consists of $n$ production plants and one rework plant are considered. Approximated analytic results are obtained and numerical examples are provided to illustrate th…
For central banks, managing the minting is one of the most important task since a shortage yields negative economic and social impacts, and the budget committed for minting is one of the largest within the central banks. Hence, the central bank requires to find the mixture of coins to be produced that satisfies the demand, inventory and production constraints while minimizing the cost. We propose a mixed-integer programming model that minimize the cost of minting by reducing the number of extra-shifts required while fulfilling the constraints. We also perform a simulation with data of a centra…
Inspired by a military context, we study a Stackelberg production game where a country's government, the leader, wants to maximize the production of military assets. The leader does so by allocating his resources among a set of production facilities. His opponent, the follower, observes this allocation and tries to destroy the associated production as much as possible by allocating his destructive resources, for example bombs, among these facilities. In this paper, we identify a follower's optimal strategy. For the leader, we show that an optimal production strategy can be found in the class o…
In manufacturing systems, setup time and production time play important parts which are reduced through human learning effect. Furthermore, defective products of the manufacturing lines are classified as imperfect (repairable) and scraps (non-repairable) products. To this end, purpose of this research is to determine the optimal run time for an Economic Production Quantity (EPQ) model with scrap, rework, setup time, and machine breakdown under learning effect assumption. Machine breakdown occurs after generating a specific quantity of defective items which follows by reworking process. In this…
This paper presents an application of mean field control to dynamic production optimization. Both noncooperative and cooperative solutions are considered. We first introduce a market of a large number of agents (firms) with sticky prices and adjustment costs. By solving auxiliary limiting optimal control problems subject to consistent mean field approximations, two sets of decentralized strategies are obtained and further shown to asymptotically attain Nash equilibria and social optima, respectively. The performance estimate of the social optimum strategies exploits a passivity property of the…
The paper deals with a class of parametrized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability is thoroughly investigated. This theory is then applied to evolution of a oligopolistic market in which the firms adopt their production strategies to changing input costs, while each change of the production is associated with some "costs of change". We examine both the Cournot-Nash equilibria as well as the two-leve…
The hierarchical structure of production planning has the advantage of assigning different decision variables to their respective time horizons and therefore ensures their manageability. However, the restrictive structure of this top-down approach implying that upper level decisions are the constraints for lower level decisions also has its shortcomings. One problem that occurs is that deterministic mixed integer decision problems are often used for long-term planning, but the real production system faces a set of stochastic influences. Therefore, a planned utilisation factor has to be include…
In this paper, we formulate an optimal ordering policy as a stochastic control problem where each firm decides the amount of input goods to order from their upstream suppliers based on the current inventory level of its output good. For this purpose, we provide a closed-form solution for the optimal request of the raw materials for given a fixed production policy. We implement the proposed policy on a 15-firm acyclic network based on a real product supply chain. We first simulate ideal demand situations, and then we implement demand-side shocks (i.e., demand levels outside of those considered …
