Long-run Average Profit Research Articles

Asymptotically Optimal Control of Make-to-Stock Systems

We consider multiclass make-to-stock production/inventory systems in which the manager makes three decisions, including pricing, outsourcing, and scheduling, to maximize the long-run average profit. For a sequence of systems in the heavy-traffic regime, with linear or strictly convex holding/waiting cost functions, we propose a sequence of policies and establish its asymptotic optimality. Our proof combines the lower bound approach and a thorough steady-state analysis of the systems. We also establish general results on the existence and tightness of the stationary distributions of the state processes under a more general family of policies. Funding: X. Gao’s research is supported in part by the Hong Kong Research Grants Council [Grants 14201520, 14201421, and 14212522]. J. Huang’s research is supported in part by the Hong Kong Research Grants Council [Grants 14500819 and 14505820] and the National Natural Science Foundation of China [Grant 72222023].

Asymptotically Optimal Control of Make-to-Stock Systems

We consider multiclass make-to-stock production/inventory systems in which the manager makes three decisions, including pricing, outsourcing, and scheduling, to maximize the long-run average profit. For a sequence of systems in the heavy-traffic regime, with linear or strictly convex holding/waiting cost functions, we propose a sequence of policies and establish its asymptotic optimality. Our proof combines the lower bound approach and a thorough steady-state analysis of the systems. We also establish general results on the existence and tightness of the stationary distributions of the state processes under a more general family of policies. Funding: X. Gao’s research is supported in part by the Hong Kong Research Grants Council [Grants 14201520, 14201421, and 14212522]. J. Huang’s research is supported in part by the Hong Kong Research Grants Council [Grants 14500819 and 14505820] and the National Natural Science Foundation of China [Grant 72222023].

A complete algebraic solution to the optimal dynamic rationing policy in the stock-rationing queue with two demand classes.

In this paper, we study a stock-rationing queue with two demand classes by means of the sensitivity-based optimization, and develop a complete algebraic solution to the optimal dynamic rationing policy. We show that the optimal dynamic rationing policy must be of transformational threshold type. Based on this finding, we can refine three sufficient conditions under each of which the optimal dynamic rationing policy is of threshold type (i.e., critical rationing level). To do this, we use the performance difference equation to characterize the monotonicity and optimality of the long-run average profit of this system, and thus establish some new structural properties of the optimal dynamic rationing policy by observing any given reference policy. Finally, we use numerical experiments to demonstrate our theoretical results of the optimal dynamic rationing policy. We believe that the methodology and results developed in this paper can shed light on the study of stock-rationing queue and open a series of potentially promising research.

Optimal control of supervisors balancing individual and joint responsibilities

AbstractWe consider a two-stage service system with two types of servers, namely subordinates who perform the first-stage service and supervisors who have their own responsibilities in addition to collaborating with the subordinates on the second-stage service. Rewards are earned when first- or second-stage service is completed and when supervisors finish one of their own responsibilities. Costs are incurred when impatient customers abandon without completing the second-stage service. Our problem is to determine how the supervisors should distribute their time between their joint work with the subordinates and their own responsibilities. Under the assumptions that service times at both stages are exponentially distributed and that the customers waiting for second-stage service abandon after an exponential amount of time, we prove that one of two policies will maximize the long-run average profit. Namely, it is optimal for supervisors to start collaborating with subordinates either when subordinates can no longer serve new customers or as soon as there is a customer ready for second-stage service. Furthermore, we show that the optimality condition is a simple threshold on the system parameters. We conclude by proving that pooling supervisors (and their associated subordinates) improves system performance, but with limited returns as more supervisors are pooled.

Production and energy mode control of a production-inventory system

Energy efficiency in manufacturing can be improved by controlling energy modes and production dynamically. We examine a production-inventory system that can operate in Working, Idle, and Off energy modes with mode-dependent energy costs. There can be a warm-up delay to switch between one mode to another. With random inter-arrival, production and warm-up times, we formulate the problem of determining in which mode the production resource should operate at a given time depending on the state of the system as a stochastic control problem under the long-run average profit criterion considering the sales revenue together with energy, inventory holding and backlog costs. The optimal solution of the problem for the exponential inter-arrival, production and warm-up times is determined by solving the Markov Decision Process with a linear programming approach. The structure of the optimal policy for the exponential case uses two thresholds to switch between the Working and Idle or Working and Off modes. We use the two-threshold policy as an approximate policy to control a system with correlated inter-event times with general distributions. This system is modelled as a Quasi Birth and Death Process and analyzed by using a matrix-geometric method. Our numerical experiments show that the joint production and energy control policy performs better compared to the pure production and energy control policies depending on the system parameters. In summary, we propose a joint energy and production control policy that improves energy efficiency by controlling the energy modes depending on the state of the system.

Integrated optimisation of pricing, manufacturing, and procurement decisions of a make-to-stock system operating in a fluctuating environment

ABSTRACT Manufacturers experience random environmental fluctuations that influence their supply and demand processes directly. To cope with these environmental fluctuations, they typically utilise operational hedging strategies in terms of pricing, manufacturing, and procurement decisions. We focus on this challenging problem by proposing an analytical model. Specifically, we study an integrated problem of procurement, manufacturing, and pricing strategies for a continuous-review make-to-stock system operating in a randomly fluctuating environment with exponentially distributed processing times. The environmental changes are driven by a continuous-time discrete state-space Markov chain, and they directly affect the system's procurement price, raw material flow rate, and price-sensitive demand rate. We formulate the system as an infinite-horizon Markov decision process with a long-run average profit criterion and show that the optimal procurement and manufacturing strategies are of state-dependent threshold policies. Besides that, we provide several analytical results on the optimal pricing strategies. We introduce a linear programming formulation to numerically obtain the system's optimal decisions. We, particularly, investigate how production rate, holding cost, procurement price and demand variabilities, customers' price sensitivity, and interaction between supply and demand processes affect the system's performance measures through an extensive numerical study. Furthermore, our numerical results demonstrate the potential benefits of using dynamic pricing compared to that of static pricing. In particular, the profit enhancement being achieved with dynamic pricing can reach up to 15%, depending on the problem parameters.

Reliability Evaluation and Optimization of a System with Mixed Run Shock

In this paper, we investigate a wear and mixed shock model in which the system can fail due to internal aging or external shocks. The lifetime of the system, due to internal wear, follows continuous phase-type (PH) distributions. The external random shocks arrive at the system according to a PH renewal process. The system will fail when the internal failure occurs or k1 consecutive external shocks, the size of at least d1 or k2 consecutive external shocks the size of at least d2 occur, where d1<d2, k1>k2. The failed system can be repaired immediately, and the repair times of the system are governed by continuous PH distributions. The system can be replaced by a new and identical one based on a bivariate replacement policy (L,N). The long-run average profit rate for the system is obtained by employing the closure property of the PH distribution. Finally, a numerical example is also given to determine the optimal replacement policy.

A hybrid manufacturing system with demand for intermediate goods and controllable make-to-stock production rate

This paper presents a two-stage hybrid manufacturing system in which the first stage produces semifinished products to store as intermediate inventory in a make-to-stock mode, whereas the second stage produces customized products from semifinished goods in a make-to-order mode. The system faces external demand for semifinished goods with a compound Poisson process and has the option of controlling the make-to-stock production rate. We study the problem of coordinating the controls of order admission, make-to-stock production rate, and inventory rationing for SP demand. The problem is formulated as a Markov decision process model. We identify the structural properties of optimal policies to maximize the long-run average profit. Using a numerical experiment, we examine how the flexibility in make-to-stock production rate and the inventory rationing for semifinished product demand can affect the system performance. We also propose a heuristic and evaluate its effectiveness using example problems.

Optimal Asynchronous Dynamic Policies in Energy-Efficient Data Centers

In this paper, we apply a Markov decision process to find the optimal asynchronous dynamic policy of an energy-efficient data center with two server groups. Servers in Group 1 always work, while servers in Group 2 may either work or sleep, and a fast setup process occurs when the server’s states are changed from sleep to work. The servers in Group 1 are faster and cheaper than those of Group 2 so that Group 1 has a higher service priority. Putting each server in Group 2 to sleep can reduce system costs and energy consumption, but it must bear setup costs and transfer costs. For such a data center, an asynchronous dynamic policy is designed as two sub-policies: The setup policy and the sleep policy, both of which determine the switch rule between the work and sleep states for each server in Group 2. To find the optimal asynchronous dynamic policy, we apply the sensitivity-based optimization to establish a block-structured policy-based Markov process and use a block-structured policy-based Poisson equation to compute the unique solution of the performance potential by means of the RG-factorization. Based on this, we can characterize the monotonicity and optimality of the long-run average profit of the data center with respect to the asynchronous dynamic policy under different service prices. Furthermore, we prove that a bang–bang control is always optimal for this optimization problem. We hope that the methodology and results developed in this paper can shed light on the study of more general energy-efficient data centers.

A two-stage hybrid manufacturing model with controllable make-to-order production rates

As an effective strategy to facilitate delivering customized products within short lead time, hybrid manufacturing via a two-stage process has received attention from academia and industry. In this paper, we study a two-stage hybrid manufacturing system in which semifinished products are manufactured in a make-to-stock fashion in the first stage and end-products are produced from semifinished goods in a make-to-order (MTO) mode in the second stage. The rate of MTO production can be controlled within given limits, depending on the status of the system. The primary goal of this paper is to study a policy for coordinating order admission, MTO production rate, and inventory replenishment controls. Formulating the problem as a Markov decision process model, we characterize the structure of optimal control policies to maximize the long-run average profit. Using a numerical experiment, we study how the flexibility in MTO production rate affects the optimal policy and the optimal profit. We also examine the effect of the number of alternative MTO production rates on the optimal profit. We propose three heuristic policies implementable for general cases. The first heuristic describes two linear switching functions for admission and production controls and a selection rule for MTO production rate control. The second heuristic specifies fixed thresholds for the control decisions using the local information. The third heuristic presents linear switching functions that approximate the optimal threshold curves. Unlike second and third heuristics, the first heuristic does not require a grid search to determine the control parameters. We implement numerical studies to examine the marginal impact of system parameters and the effect of the number of alternative MTO production rates on the performance of the heuristics. Compared to the optimal policy, the average percentage performance of the first and third heuristics is less than 1% for both numerical studies. On the other hand, the average percentage performance of the second heuristic is larger than 3%, and it exceeds 10% for a set of particular problem examples.

Double-tolerance design for manufacturing systems

Most production environments are stochastic in nature, due to the randomness inherent in the production processes. One important engineering problem commonly faced by practitioners is to determine optimal engineering tolerances to be used in production. This article develops optimization models for determining tolerance sets to maximize the long-run average net profit on a production line with processing and rework stations, as well as instantaneous inspection and scrap operations. We assume that only one server works at the rework station, and the service times at the processing and rework stations are uncertain, thus, a stochastic queueing system is embedded into the manufacturing process. We also consider the trade-off between the overall production cost and the cost associated with a quality loss in the final product. Our work is the first to introduce the concept of double-tolerance sets to the tolerance design optimization literature. By comparing the proposed double-tolerance model with a single-tolerance model, we investigate the impact of different parameter settings and modeling assumptions on the optimal tolerances through numerical examples and a sensitivity analysis.

Evolution Dynamics of Agricultural Internet of Things Technology Promotion and Adoption in China

Internet of Things is the core technology of smart agriculture and can reform and upgrade traditional agriculture for reducing cost, reducing pollution, and increasing productivity and quality in China. From government-led and market economy perspectives, promotion mechanisms and sustainable adoption of agricultural Internet of Things technology are analyzed. In the initial application phase, the promotion of Internet of Things requires government support. For investigating the relationship between the government and farmers, this study builds an evolutionary game model and finds that increases of cost subsidy, farmers’ negative feedback, government’s positive feedback, and chemical agriculture cost can make the model evolve toward the strategy set: farmer adoption and government support. For long-term development, a sustainable model in competitive market is built by competition game and exponential replication equation. This paper analyzes the equilibrium of adoption ratio, long-run profit, and the conversion between equilibrium points under capacity sharing strategy in competitive market. It is also found that the market will eventually evolve to the technology selection strategy whose long-run average profit dominates the market. The innovations are that evolutionary game is used for analyzing the initial stage and competitive game and asynchronous update mechanism are used for analyzing the sustainable development adoption. At last, references are provided for agricultural Internet of Things development policy from the perspectives of initial promotion and long-run sustainability.

Joint Pricing and Inventory Management with Strategic Customers

We consider a model wherein the seller sells a product to customers over an infinite horizon. At each time, the seller decides a set of purchase options offered to customers and the inventory replenishment quantity. Each purchase option specifies a price and a product delivery time. Customers are infinitesimal and arrive to the system with a constant rate. Customer product valuations are heterogenous and follow a stationary distribution. Customers’ arrival times and product valuations are their private information. Customers are forward looking; that is, they strategize their purchasing times. Customers incur delay disutility from postponing to place an order and waiting for the product delivery. Customers’ delay disutility rates are perfectly and positively correlated with their valuation. The seller has zero replenishment lead time. The seller incurs fixed ordering cost and inventory holding cost. The seller seeks a joint pricing, delivery, and inventory policy that maximizes the seller’s long-run average profit. Through a tractable upper bound constructed by solving a mechanism design problem, we derive an optimal joint pricing, delivery, and inventory policy, which is a simple cyclic policy. We also extend our policy to a stochastic setting and establish its asymptotic optimality.

Optimal energy-efficient policies for data centers through sensitivity-based optimization

In this paper, we propose a novel dynamic decision method by applying the sensitivity-based optimization theory to find the optimal energy-efficient policy of a data center with two groups of heterogeneous servers. Servers in Group 1 always work at high energy consumption, while servers in Group 2 may either work at high energy consumption or sleep at low energy consumption. An energy-efficient control policy determines the switch between work and sleep states of servers in Group 2 in a dynamic way. Since servers in Group 1 are always working with high priority to jobs, a transfer rule is proposed to migrate the jobs in Group 2 to idle servers in Group 1. To find the optimal energy-efficient policy, we set up a policy-based Poisson equation, and provide explicit expressions for its unique solution of performance potentials by means of the RG-factorization. Based on this, we characterize monotonicity and optimality of the long-run average profit with respect to the policies under different service prices. We prove that the bang-bang control is always optimal for this optimization problem, i.e., we should either keep all servers sleep or turn on the servers such that the number of working servers equals that of waiting jobs in Group 2. As an easy adoption of policy forms, we further study the threshold-type policy and obtain a necessary condition of the optimal threshold policy. We hope the methodology and results derived in this paper can shed light to the study of more general energy-efficient data centers.

Optimal Pricing for Tandem Queues with Finite Buffers

We consider optimal pricing for a two-station tandem queueing system with finite buffers and price-sensitive customers. The service provider quotes prices to customers using either static or dynamic pricing. The objective is to maximize either the infinite-horizon discounted profit or the long-run average profit of the provider. We show that there exists an optimal dynamic policy that exhibits an interesting monotone structure, in which the quoted prices have greater dependency on the queue length at station 1 than at station 2, for both the discounted and long-run average problems. We then study the long-run average problem and show that the optimal static policy performs as well as the optimal dynamic policy when the buffer size at station 1 becomes large, there are no holding costs, and the arrival rate is either small or large. We learn from numerical results that for systems with small arrival rates, the optimal static policy is near optimal even when the buffer at station 1 is small and there are positive holding costs. On the other hand, for systems with arrival rates that are not small, there are cases where the optimal dynamic policy performs much better than the optimal static policy.

Dynamic Pricing of a Multiclass Make-to-Stock Queue

We consider a make-to-stock manufacturing system selling multiple products to price-sensitive customers. The system manager seeks to maximize the long-run average profit by making dynamic pricing, outsourcing, and scheduling decisions: First, she adjusts prices dynamically depending on the system state. Second, when the backlog of work is judged excessive, she may outsource (or reject) new orders thereby incurring outsourcing costs. Third, she decides dynamically on which product to prioritize in the manufacturing process, i.e., she makes dynamic scheduling decisions. This problem appears analytically intractable. Thus, we resort to an approximate analysis in the heavy-traffic regime and consider the resulting Brownian control problem. We solve this problem explicitly by exploiting the solution to a particular Riccati equation. The optimal solution to the Brownian control problem is a two-sided barrier policy with drift rate control: Outsourcing and idling processes are used to keep the workload process above the lower reflecting barrier and below the upper reflecting barrier, respectively. Between the two barriers, a state-dependent drift rate is used to control the workload process. By interpreting this solution in the context of the original model, we propose a joint dynamic pricing, outsourcing, and scheduling policy, and demonstrate its effectiveness through a simulation study.

Resource allocation and routing in parallel multi-server queues with abandonments for cloud profit maximization

This paper considers a Markov decision model for profit maximization of a cloud computing service provider catering to customers submitting jobs with firm real-time random deadlines. Customers are charged on a per-job basis, receiving a full refund if deadlines are missed. The service provider leases computing resources from an infrastructure provider in a two-tier scheme: long-term leasing of basic infrastructure, consisting of heterogeneous parallel service nodes, each modeled as a multi-server queue, and short-term leasing of external servers. Given the intractability of computing an optimal dynamic resource allocation and job routing policy, maximizing the long-run average profit rate, the paper addresses the design, implementation and testing of low-complexity heuristics. The policies considered are a static policy given by an optimal Bernoulli splitting, and three dynamic index policies based on different index definitions: individually optimal (IO), policy improvement (PI) and restless bandit (RB) indices. The paper shows how to implement efficiently each such policy, and presents a comprehensive empirical comparison, drawing qualitative insights on their strengths and weaknesses, and benchmarking their performance in an extensive study.

Optimizing constant pricing and inventory decisions for a periodic review system with batch ordering

In business practice the misalignment between frequent inventory adjustments and a sticky price is common. Motivated by Pipi Milk, a diary manufacturer who commits its customer at a fixed price for a whole year meanwhile weekly replenishes its cheese product stock at a specific batch size, this paper develops an integrated constant pricing and inventory control model in a periodic-review system with batch order consideration. For such a system, our focus is on an (nQ, r)-p policy: the price p is determined and committed at the beginning of the planning horizon; inventories are managed based on an (nQ, r) policy with reorder level r and order quantities at non-negative integer multiples of batch size Q. With a fixed ordering cost included, we develop an efficient algorithm to compute the optimal policy parameters with the purpose to maximize the expected long-run average profit. We also numerically investigate the policy performance and verify its advantages by comparing to a sequential optimization of price and inventory decisions. Finally, managerial insights are generated.

Joint inspection and inventory control for deteriorating items with random maximum lifetime

In this paper, we consider inventory control problems for deteriorating items. Six practical models are developed for an inventory system with deterioration rate depending on the maximum lifetime of items. Due to the physical nature of items, the maximum lifetime of items is assumed to be random over a time period. Items exceeding the maximum lifetime are regarded as scarp and no longer serviceable. Two replenishment policies: (i) quantity-based policy and (ii) time-based policy, and two inspection scenarios: (i) one inspection and (ii) continuous monitoring have been discussed in these models. Examples and sensitivity analysis are given for each model. The results indicate that it may be beneficial for the inventory holder to adopt the quantity-based policy and drop the plan of continuous monitoring if the cost of continuous monitoring is too high. Otherwise, performing continuous monitoring is able to increase the optimal long-run average profit. These results provide useful insights for inventory holder in making managerial decisions.

Joint pricing and inventory control for a stochastic inventory system with Brownian motion demand

ABSTRACTIn this article, we consider an infinite horizon, continuous-review, stochastic inventory system in which the cumulative customers’ demand is price dependent and is modeled as a Brownian motion. Excess demand is backlogged. The revenue is earned by selling products and the costs are incurred by holding/shortage and ordering; the latter consists of a fixed cost and a proportional cost. Our objective is to simultaneously determine a pricing strategy and an inventory control strategy to maximize the expected long-run average profit. Specifically, the pricing strategy provides the price pt for any time t ⩾ 0 and the inventory control strategy characterizes when and how much we need to order. We show that an (s*, S*, p*) policy is optimal and obtain the equations of optimal policy parameters, where p* = {p*t: t ⩾ 0}. Furthermore, we find that at each time t, the optimal price p*t depends on the current inventory level z, and it is increasing in [s*, z*] and decreasing in [z*, ∞), where z* is a negative level.