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Abstract:
Firms frequently utilize multiple communications instruments as part of their marketing campaign. Interactions between these instruments suggest that firms should apply Integrated Marketing Communications (IMC) to benefit from the synergies. We review different IMC models and then present a stochastic IMC model for which explicit closed-loop solutions of the optimal advertising and market share are obtained. This enables us to understand the role of firm and market parameters, such as synergy, on the optimal advertising budget and allocation. For the proposed and existing IMC models we show that the budget and allocation decisions can be made independently, greatly simplifying the implementation of IMC. We also show that there is an optimal long-run market share that the firm should try to maintain through appropriate use of IMC. Finally, the model and results are generalized to multiple (>2) instruments and multiple competitors.

Advertising, Dynamics, Budgeting, Integrated Marketing Communications, Decisions under Uncertainty, Dynamic Optimization

Abstract:
Stackelberg differential game models have been used to study sequential decision making in non-cooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models to the supply chain management and marketing channels literatures. A common feature of these applications is the specification of the game structure: a decentralized channel composed of a manufacturer and independent retailers, and a sequential decision procedure with demand and supply dynamics and coordination issues. In supply chain management, Stackelberg differential games have been used to investigate inventory issues, wholesale and retail pricing strategies, and outsourcing in dynamic environments. The underlying demand typically has growth dynamics or seasonal variation. In marketing, Stackelberg differential games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. The demand dynamics are usually extensions of the classical advertising capital models or sales-advertising response models. We begin by explaining the Stackelberg differential game solution methodology and then provide a description of the models and results reported in the literature.

Stackelberg differential games, supply chain management, marketing channels, open-loop equilibria, feedback policies, channel coordination, optimal control

Abstract:
Firms have the choice of developing software as either open source or closed source. The open-source approach to software development has been advocated as a new and better method for developing high quality software than the traditional closed-source approach. In open source, volunteer programmers freely contribute code to develop and improve the software. This paper describes the key nonpecuniary motivations for these programmers. They are less motivated to contribute if they observe commercial marketing of the open-source software they helped create, leading to a reduction in improvements to the software. A primary concern for software firms seeking to develop and market open-source software is, thus, how the motivation of contributors should be managed. We examine optimal pricing strategies for open-source and closed-source software keeping in mind the distinct motivations of programmers in the two cases. We compare profits and software qualities from the two approaches and provide implications for firms in the software industry.

Open source, Pricing, altruism, optimal control, software management, the maximum principle, chattering control

Abstract:
We analyze optimal advertising spending in a duopolistic market where each firm's market share depends on its own and its competitor''s advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented.

Advertising, advertising budgeting, competitive strategy, stochastic differential game, stochastic calculus, duopoly, stochastic differential equations, Ito equations, dynamic programming, the Sethi model, Nash equilibrium

Abstract:
Companies spend hundreds of millions of dollars annually on advertising to build and maintain awareness for their brands in competitive markets. However, awareness formation models in the marketing literature ignore the role of competition. Consequently, we lack both the empirical knowledge and normative understanding of building brand awareness in dynamic oligopoly markets. To address this gap, we propose an N-brand awareness formation model, design an extended Kalman filter to estimate the proposed model using market data for five car brands over time, and derive the optimal closed-loop Nash equilibrium strategies for every brand. The empirical results furnish strong support for the proposed model in terms of both goodness-of-fit in the estimation sample and cross-validation in the out-of-sample data. In addition, the estimation method offers managers a systematic way to estimate ad effectiveness and brands as well as competitors' brands. Finally, the normative analysis reveals an inverse allocation principle that suggests - contrary to the proportional-to-sales or competitive parity heuristics - that large (small) brands should invest in advertising proportionally less (more) than small (large) brands.

marketing, competitive strategy, advertising, media, Kalman filter, dynamic games, differential games, Nash equilibrium, Optimal control, Sethi model, Feedback Nash, competative advertising model, car companies data from Italy

Abstract:
The problem of optimal consumption and investment is concerned with the decisions of a single agent endowed with some initial wealth who seeks to maximize total expected discounted utility of consumption. The decisions are the rate of consumption and the allocation of their wealth directed to risky and risk-free investments over time. The problem was first studied by Paul Samuelson and Robert Merton in 1969; however none of their formulations took into account the possibility that an agent might go bankrupt in the process. In a set of articles published in 1979 and 1983, Suresh Sethi and co-authors (Abel Cadenillas, Myron Gordon, Brian Ingham, Ioannis Karatzas, John Lehoczky, Ernst Presman, Steven Shreve, and Michael Taksar) explicitly introduced a bankruptcy value/penalty in the consumption/investment model. In addition, they also introduced a nonzero subsistence consumption level, which makes the consideration of bankruptcy even more important. This provided the ability to deal mathematically with the problems of bankruptcy in the study of consumption and investment. Optimal Consumption and Investment with Bankruptcy provides a useful frame for deepening our understanding of the consumption and portfolio selection behavior of individuals and households. Foreword by Harry M. Markowitz. Not included are Chapters 2, 3 and 13, which are available directly from the websites of the specified journals in which they first appeared.

Consumption and Investment problem, Portfolio and Consumption problem, bankruptcy, subsistence consumption, minimal consumption, borrowing constraints, stochastic optimal control, martingale problems, optimal stopping problems, Risk aversion measures, financial engineering

Abstract:
A model of new-product adoption is proposed that incorporates price and advertising effects. An optimal control problem that uses the model as its dynamics is explicitly solved to obtain the optimal price and advertising effort over time. The model has a great potential to be used in obtaining solutions and insights in a variety of differential game settings.

new-product adoption model, durable goods model, pricing, advertising, optimal control

Abstract:
We examine an oligopoly model of advertising competition where each firm's market share depends on its own and its competitors' advertising decisions. A differential game model is developed and used to derive the closed-loop Nash equilibrium under symmetric as well as asymmetric competition. We obtain explicit solutions under certain plausible conditions, and discuss the effects of an increase in the number of competing firms on advertising expenditure, market share and profitability.

Advertising, Oligopoly, Differential games, Optimization

Abstract:
Cooperative (co-op) advertising is an important instrument for aligning manufacturer and retailer decisions in supply chains. In this, the manufacturer announces a co-op advertising policy, i.e., a participation rate that specifies the percentage of the retailer's advertising expenditure that it will provide. In addition, it also announces the wholesale price. In response, the retailer chooses its optimal advertising and pricing policies. We model this supply chain problem as a stochastic Stackelberg differential game whose dynamics follows Sethi's stochastic sales-advertising model. We obtain the condition when offering co-op advertising is optimal. We provide in feedback form the optimal advertising and pricing policies for the manufacturer and the retailer. We contrast the results with the advertising and price decisions of the vertically integrated channel, and suggest a method for coordinating the channel.

Co-op Advertising, Cooperative advertising, Sales-Advertising Dynamics, Differential Games, Sethi Model, Distribution Channel, Stackelberg equilibrium, Feedback Stackelberg strategy, Sales-advertising model, advertising participation rate, optimal advertising, optimal pricing

Abstract:
This paper studies a supply chain consisting of two suppliers and one retailer in a spot market, where the retailer uses the newsvendor solution as its purchase policy, and suppliers compete for the retailer's purchase. Since each supplier's bidding strategy affects the other's profit, a game theory approach is used to identify optimal bidding strategies. We prove the existence and uniqueness of a Nash solution. It is also shown that the competition between the supplier leads to a lower market clearing price, and as a result, the retailer benefits from it. Finally, we demonstrate the applicability of the obtained results by deriving optimal bidding strategies for power generator plants in the deregulated California energy market.

demand allocation, game theory, information updates, dynamic programming, deregulated energy market

Abstract:
We solve an agent’s optimization problem of meeting demands for cash over time with cash deposited in bank or invested in stock. The stock pays dividends and uncertain capital gains, and a commission is incurred in buying and selling of stock. We use a stochastic maximum principle to obtain explicitly the optimal transaction policy.

Cash management, stochastic control, maximum principle, risky assets

Abstract:
Miller and Modigliani (1961) consider valuation of infinite horizon firms that may not engage in purchasing their own shares. While their fundamental valuation approach applies also to firms that purchase their own shares, their stream of dividends approach does not apply to a class of firms paying out "insufficient" dividends as characterized by an if and only if condition in the paper. The latter approach is modified so that it can be used for valuation of infinite horizon firms including those which may purchase their own shares. The modified approach is the natural extension of the traditional dividend stream approach used for valuing finite horizon firms. Moreover, it is proved to be equivalent to the fundamental valuation approach.

Miller and Modigliani Theory, MM theory, Valuation of firms, Dividend approach, Cash flow approach, Arbitrage, Infinite horizon firms, Share repurchase, Share price, Dividend policy

Abstract:
Over the last two decades, differential game (DG) models have been used extensively to study such issues in dynamic environments as competitive advertising and pricing for new products in the marketing literature, capacity investments in the energy industry, government's subsidy policy in new technology, and monetary and fiscal policies in economics. Recently, a number of papers have applied DGs to treat dynamic interactions between the channel members in decentralized supply chains. This review focuses on these applications. Specifically, we review papers that analyze dynamic retail-wholesale pricing strategies, joint slotting and pricing decisions to launch an innovative durable product, and investment in supply chain infrastructure. We consider Stackelberg equilibria as the solution concept for the games under consideration. We shall begin our review with an introduction to the basics of the Stackelberg DGs. We then summarize the important managerial insights obtained in each of the studies being reviewed. Finally, we point out future research avenues for applications of DGs in supply chain management.

differential games, Stackelberg equilibrium, supply chain management, coordination, pricing

Abstract:
The accumulated evidence indicates that pure revenue models, such as free-access models (where the revenue is solely dependent on advertisements) or pure subscription fee based models (where the revenue is solely based on subscription fees, and advertisements are not shown to customers), are not sufficient to support the survival of online information sellers. Hence, hybrid models based on a combination of subscription fees and advertising revenues have replaced the pure revenue models on many web sites including the popular online content provider sites such as Wall Street Journal and Classmates Inc. In response to increasing interest in hybrid models, we study the problem of dynamic pricing of web content on a site where revenue is generated from subscription fee as well as advertisements. Using the optimal control theory, we obtain a dynamic pricing strategy of subscription and the optimal level of advertisements shown to the subscribers. Since the decision in any one time period affects the decisions of all subsequent time periods, the proposed dynamic model provides a globally optimal solution. Our model shows that the subscription fee is reduced initially to attract more customers, and is subsequently increased once a large customer base is obtained. Even when the fee increases in later periods, the number of subscribers increases due to the value associated with the quality of the content. We present several analytical and numerical results which provide some important managerial implications.

Optimal Control Theory, E-Commerce, Web Services, Advertising, Subscription

Abstract:
This article is an attempt to survey the vast literature on flexibility in manufacturing that has accumulated over the last 10 to 20 years. The survey begins with a brief review of the classical literature on flexibility in economics and organization theory, which provides a background for manufacturing flexibility. Several kinds of flexibilities in manufacturing are then defined carefully along with their purposes, the means to obtain them, and some suggested measurements and valuations. Then we examine the interrelationships among the several flexibilities. Various empirical studies and analytical/optimization models dealing with these flexibilities are reported and discussed. The article concludes with suggestions for some possible future research directions.

Flexibility, Manufacturing flexibility, Operations management, Operations strategy

Abstract:
A stock market model is presented that advances our understanding of the portfolio-consumption policy of investors and the behaviour of capital market statistics. The model's building blocks are the Samuelson-Merton model of portfolio-consumption policy, the Gordon-Sethi extension of that model to recognize bankruptcy, the Gordon dividend growth model for pricing a share, and the assumption that the system is closed. The last assumption makes price and expected return adjust to persuade investors to hold the outstanding shares and bonds. Analysis and simulation of the model reveal, among other things, that (1) the market is more stable and it performs better when investors have increasing relative risk aversion; and (2) the average infinite horizon return on a share falls below the average realized holding period return to a degree that varies with the volatility in the latter's return. Further advances in knowledge should follow from withdrawal of the simplifying assumptions that were employed to make clear the model's basic structure.

stock market, portfolio-consumption policy, investment-consumption problem, simulation, Keynesian, neoclassical, bankruptcy, capital markets, financial engineering

Abstract:
The accumulated evidence indicates that pure revenue models, such as free-access models and pure subscription fee based models, are not sufficient to support the survival of online information sellers. Hence, hybrid models based on a combination of subscription fees and advertising revenues are replacing the pure revenue models. In response to increasing interest in hybrid models, we study the problem of dynamic pricing of web content on a site where revenue is generated from subscription fee as well as advertisements. We use the optimal control theory to solve the problem and obtain the subscription fee and the advertisement level over time. We first consider the case when the subscription fee can vary over time, but the advertisement level stays the same. Then we extend it by optimizing both the subscription fee and the advertisement level dynamically. We also present several analytical and numerical results that provide important managerial insights.

Pricing, revenue, advertising, optimal control theory, web contents

Abstract:
The extant supply chain management literature has not addressed the issue of coordination in supply chains involving risk-averse agents. We take up this issue and begin with defining a coordinating contract as one that results in a Pareto-optimal solution acceptable to each agent. Our definition generalizes the standard one in the risk-neutral case. We then develop coordinating contracts in three specific cases: (i) the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint; (ii) the supplier and the retailer each maximizes his own mean-variance trade-off; and (iii) the supplier and the retailer each maximizes his own expected utility. Moreover, in case (iii), we show that our contract yields the Nash Bargaining solution. In each case, we show how we can find the set of Pareto-optimal solutions, and then design a contract to achieve the solutions. We also exhibit a case in which we obtain Pareto-optimal sharing rules explicitly, and outline a procedure to obtain Pareto-optimal solutions.

supply chain management, Pareto-optimality, coordination, risk aversion, Nash Bargaining solution, operations management

Abstract:
This paper analyzes dynamic advertising and pricing policies in a durable-good duopoly. The proposed infinite-horizon model, while general enough to capture dynamic price and advertising interactions in a competitive setting, also permits closed-form solutions. We use differential game theory to analyze two different demand specifications - linear demand and isoelastic demand - for symmetric and asymmetric competitors. We find that the optimal price is constant and does not vary with cumulative sales, while the optimal advertising is decreasing with cumulative sales. Comparative statics for the results are presented.

Optimal Control, Dynamic programming, Game theory, Differential games, Pricing, Advertising, Nash equilibrium, the Sethi model, Durable goods, Innovative products

Abstract:
Information delays exist in an inventory system when it takes time to collect, process, validate, and transmit inventory/demand data. A general framework is developed in this paper to describe information flows in an inventory system with information delays. We characterize the suffcient statistics for making optimal decisions. When the ordering cost is linear, the optimality of a state-dependent base-stock policy is established even when information flows are allowed to cross over time. Additional insights into the problem are obtained via a comparison between our models and the models with stochastic order lead times. We also show that inventory can substitute for information and vice versa.

Information delay, partial observations, stochastic inventory problem, incomplete information, base-stock policy, dynamic programming, sufficient statistic, destination and origin determined dynamic delays, Information flow crossing

Abstract:
We study a supply chain consisting of a supplier and a retailer who faces a news vendor problem. The supplier has better knowledge of his unit production cost than the retailer has. We model this problem as a game of adverse selection. In this model, the retailer (principal) offers a menu of contracts, each of which consists of two parameters: the ordering quantity and the supplier's proportion of the channel profit. The supplier (agent), who has alternative opportunities and associated reservation profits, either chooses one from the menu of contracts or rejects them all. Generalizing the traditional contracts, we consider type-dependent reservation profits for low cost and high cost suppliers. We derive an optimal contract menu for the retailer under this generalization. Surprisingly, we find that in some cases the optimal contract coordinates the supply chain even with asymmetric information. Moreover, we warn the retailers who use the traditional contracts with type-independent reservation profits. With such a contract, a retailer may offer less profit to a low cost supplier than his reservation profit - unintentional disqualification of low cost suppliers from the contracts. By incorporating type-dependent reservation profits into contract design, we eliminate the possibility of this unintentional disqualification.

Asymmetric Information, Type-dependent reservation profits, Type-independent reservation profits, Coordination, Supply Contract, Adverse Selection, Contract theory

Abstract:
We develop a new, unified approach to treating continuous-time stochastic inventory problems with both the average and discounted cost criteria. The approach involves the development of an adjusted discounted cycle cost formula, which has an appealing intuitive interpretation. We show for the first time that an (s, S) policy is optimal in the case of demand having a compound Poisson component as well as a constant rate component. Our demand structure simultaneously generalizes the classical EOQ model and the inventory models with Poisson demand, and we indicate the reasons why this task has been a difficult one. We do not require the surplus cost function to be convex or quasi-convex as has been assumed in the literature. Finally, we show that the optimal s is unique, but we do not know if optimal S is unique.

(s, S) policy, stochastic inventory model, EOQ model, compound Poisson process, average discounted-cost formula, dynamic programming, quasi-variational inequality

Abstract:
We study a coordination contract for a supplier-retailer channel producing and selling a fashionable product exhibiting a stochastic price-dependent demand. The product's selling season is short, and the supply chain faces great demand uncertainty. We consider a scenario where the supplier reserves production capacity for the retailer in advance, and permits the retailer to place an order not exceeding the reserved capacity after a demand information update during a leadtime. We formulate a two-stage optimization problem in which the supplier decides the amount of capacity reservation in the first stage, and the retailer determines the order quantity and the retail price after observing the demand information in the second stage. We propose a three-parameter risk and profit sharing contract that coordinates the supply chain. The proposed contract is robust which permits any agreed-upon division of the supply chain profit between the channel members.

Supply chain coordination, lead-time, information updating, return policy, newsvendor problem, risk and profit sharing, price dependent demand

Abstract:
We investigate how a supply chain involving a risk-neutral supplier and a downside-risk-averse retailer can be coordinated with a supply contract. We show that the standard buy-back or revenue-sharing contracts may not coordinate such a channel. Using a definition of coordination of supply chains proposed earlier by the authors, we design a risk-sharing contract that offers the desired downside protection to the retailer, provides respective reservation profits to the agents, and accomplishes channel coordination.

supply chain management, coordination, risk sharing, downside risk, value at risk,operations management, risk management

Abstract:
We study a drop-shipping supply chain in which the retailer receives a customer's order and the supplier fills it. In such a chain, the supplier keeps inventory and bears inventory risks; the retailer focuses on marketing and customer acquisition, and forwards the orders to the supplier. The retailer usually has better customer demand information, and may send an over-estimated demand forecast to maximize her own interest, which may result in overstock for the supplier. On the other hand, since the retailer does not own inventory, the main concern of the retailer is that the acquired orders may not be fulfilled because of the supplier's shortage of stock. To cope with these challenges, we propose a menu of commitment-penalty contracts that can provide greater certainty of demand as well as greater certainty of supply. We focus our study in the asymmetric demand information case and we show that the supplier can obtain the retailer's demand information by offering a menu of commitment-penalty contracts. Under this mechanism, we find the solution that maximizes the supplier's expected profit.

contracts, dropshipping, supply chain, asymmetric information, commitment-penalty contract

Abstract:
We consider a supply chain in which a manufacturer sells an innovative durable product to an independent retailer over its life cycle. We assume that the product demand follows a Bass-type diffusion process, and it is determined by the market influences, retail price of the product, and shelf space allocated to it. We consider the following retailer profit optimization strategies: (i) the myopic strategy of maximizing the current-period profit and(ii) the far-sighted strategy of maximizing the life-cycle profit. We characterize the optimal dynamic shelf-space allocation and retail pricing policies for the retailer and wholesale pricing policies for the manufacturer. We also compute these policies. Surprisingly, we find that the manufacturer, and sometimes even the retailer, is better of with a myopic retailer strategy in some cases.

Stackelberg differential games, the Bass model, pricing, slotting, supply chain management, open-loop policies, durable products

Abstract:
We examine optimal control decisions regarding pricing, network size and hiring strategy in the context of open source software development. Opening the source code to a software product often implies that consumers would not pay for the software product itself. However, revenues may be generated from complementary products. A software firm may be willing to open the source code to its software if it stands to build a network for its complementary products. The rapid network growth is doubly crucial in open source development, where the users of the firm's products are also contributors of code that translates to future quality improvements. To determine whether or not to open the source, a software firm must jointly optimize prices for its various products while simultaneously managing its product quality, network size, and employment strategy. Whether or not potential gains in product quality, network size, and labor savings are sufficient to justify opening the source code depends on product and demand characteristics of both the software and the complementary product as well as on the cost and productivity of in-house developers relative to open source contributors. This paper investigates these crucial elements to allow firms to reach the optimal decision in choosing between the open and closed source models.

Pricing, Optimal control, Open Source, Network Externalities, complementary product or service

Abstract:
We develop variations of the M|G|1 queue to model the process of software maintenance within organizations and use these models to compute the optimal allocation of resources to software maintenance. User requests are assumed to arrive following a Poisson process and a binomial distribution is used to model duplication of requests. We obtain expressions for expected queue lengths with an exponential server using an N-policy for an integer N≥1. We also obtain the optimal batching size and mean service rate by minimizing the total cost consisting of the cost of the server, the cost of waiting, and the fixed cost of maintenance, if applicable.

MG1, software maintenance, Queuing Theory, N-policy, batching, resource allocation

Abstract:
We consider optimal consumption and portfolio investment problems of an investor who is interested in maximizing his utilities from consumption and terminal wealth subject to a random inflation in the consumption basket price over time. We consider two cases: (i) when the investor observes the basket price and (ii) when he receives only noisy observations on the basket price. We derive the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The compositions of the funds in the two cases are the same, but in general the investor's allocations of his wealth into these funds will differ. However, in the particular case when the investor has constant relative risk-aversion (CRRA) utility, his optimal investment allocations into these funds are also the same in both cases.

Abstract:
We consider optimal consumption and portfolio investment problems of an investor who is interested in maximizing his utilities from consumption and terminal wealth subject to a random inflation in the consumption basket price over time. We consider two cases: (i) when the investor observes the basket price and (ii) when he receives only noisy observations on the basket price. We derive the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The compositions of the funds in the two cases are the same, but in general the investor's allocations of his wealth into these funds will differ. However, in the particular case when the investor has CRRA utility, his optimal investment allocations into these funds are also the same in both cases.

Optimal Consumption and Investment, Inflation, Stochastic Control with partial observations, Separation Principle, Zakai Equation

Abstract:
We study an economy in which the rate of change of population depends on population policy decisions. This requires population as well as capital as state variables. By showing the algebraic relationship between the shadow price of the population and the shadow price of the per capita capital stock, we are still able to depict the optimal path and its convergence to the long-run equilibrium on a two-dimensional phase diagram. Moreover, we derive explicitly the expression of genuine savings in our model to evaluate the sustainability of the system.

savings, genuine savings, green gross national product, population policy, value of the population, economic growth, optimal control, phase diagram, dynamic programming, Maximum principle

Abstract:
In this paper, we use quasi-variational inequalities to provide rigorous proof of the familiar square root formula for the economic order quantity (EOQ) in the classical deterministic average cost inventory model.

EOQ, Square root formula, quasi-variational inequalities, inventory model, lotsize model, lotsize formula

Abstract:
We develop a general filtering framework for the problem of estimating the state of a system whose dynamics is governed by a discrete-time Markov process. We describe applications to inventory control systems with partial observations. We introduce conditional distributions and unnormalized conditional probabilities to transform nonlinear transition equations into linear ones. Moreover, this transformation greatly facilitates our study of the stochastic optimal control problems governed by nonlinear transition equations.

Filtering, inventory control, optimal control, adaptive control, unnormalized probability, Kalman filter, Zakai equation, partial observations, incomplete information

Abstract:
Managing customer satisfaction in a cost effective way has always been a major challenge faced by inventory managers. This paper studies long-term service performance of a two-stage newsvendor selling a perishable product with short-term demand patterns. We characterize the optimal inventory policy to minimize the expected inventory cost such that a long-term stock availability target is satisfied. Both in-stock probability and fill rate targets are examined and compared. In particular, we address the following questions: How should an inventory manager evaluate his long-term fill rate performance without observing the lost sales? How are in-stock probabilities and fill rates connected with respect to different demand patterns? How does the forecast update impact the evaluation of the long-term service performance? How do the short-term cost trade-offs under different long-term service targets depend on the monotone structures of the forecast signal?

Newsvendor Model, inventory model, service measures, forecast updates, Two-stage newsvendor model, fill rate

Abstract:
In many businesses, inventory levels are only incompletely observed. This may be due to non-observation, of demand, spoilage, misplacement, or theft of inventory. The non-observation of demand may be caused, e.g., by transaction errors or by discrepancies/delays in transmitting/processing sales data. We study a periodic review inventory system where the demand is not observed and the unmet demand is backordered. As a result, the inventory manager cannot tell the exact quantities of inventories or backorders. However, by looking at the shelf, he knows whether the inventory is positive or non-positive. Only with this information, the inventory manager must determine the order quantity in each period that would minimize the expected total discounted cost over an infinite-horizon. The dynamic programming formulation of this problem has an infinite-dimensional state space. We use the concept of unnormalized probability to establish the existence of an optimal feedback policy and the uniqueness of the solution of the dynamic programming equation when the periodic cost has linear growth.

Inventory model, partial observations, incomplete information, stochastic control, dynamic programming, infinite dimensioanl spaces, contraction mapping, fixed point

Abstract:
We study a single-period two-stage service-constrained supply chain with an information update. The buyer has two procurement opportunities with the second one after observing a market signal, which updates the demand forecast. He also commits to a service level after observing the market signal. We derive his optimal ordering decisions and show that the critical market signal, the optimal first-stage order quantity, and the optimal expected profit are monotone with respect to the target service level. We also discuss the impact of the forecast quality on the optimal decisions. We show that the optimal first-stage order quantity may not be monotone with respect to information accuracy, as is in the case without the service constraint. In addition, we extend our analysis to the situation when an order cancellation is allowed upon the observation of the market signal. We also compare the results obtained for the problems with and without an order cancellation. Finally, we discuss the supply chain coordination issue and find that a buyback contract can also coordinate the supply chain in the presence of the service constraint.

supply chain, information update, customer service level, dynamic programming, forecasting, signalling, coordiantion

Abstract:
A supplier provides several lead-time options to its customers in a periodic review inventory system. The replenishment lead time is a multiple of the inventory review cycle. Customers are classified into two groups: short lead-time customers requiring the product immediately and long lead-time customers to whom the supplier may deliver immediately or in the next cycle. We consider an inventory-commitment problem, in which the supplier allocates its on-hand inventory to these two groups of customers. When inventory runs out, the supplier backlogs orders to future cycles. Therefore, the supplier faces two problems: how the on-hand inventories are allocated between the two classes of customers and how the backlogged orders are cleared when replenishments arrive? We treat the former as an inventory-commitment problem and handle the latter with priority rules. We solve the former by dynamic programming and use three priority rules in clearing backlogs. We also explore the optimal inventory replenishment issue and evaluate the performance of each priority rule.

Inventory Models, Lead times, backloggging, priority rules, customer classes, FCFS, make-to stock, make-to-order, two critical number policies

Abstract:
This chapter examines the interaction between supply price uncertainty and demand uncertainty. We consider a manufacturer who sources a key component using different procurement options: a long-term order on a price-only contract, short-term orders on an adjustment contract, and short-term purchases directly from the market. At the beginning of the planning cycle, the manufacturer places a long-term order and reserves a certain amount of supply capacity for the purpose of adjusting the long-term order, if needed. Before the selling season, the manufacturer has multiple options to place supplementary orders from the reserved capacity or from the market.

We compare two types of capacity arrangements: dedicated capacity and overall capacity. Under a dedicated capacity arrangement, the manufacturer reserves capacities separately for different adjustment opportunities. On the overall capacity arrangement, she keeps the flexibility of using the reserved capacity within the given period for possibly multiple adjustments. We discuss the optimal procurement strategies and the criteria for capacity allocations, as well as the policy behavior and service performance in different situations.

Procurement Flexibility, Price Risk-Sharing, Capacity, Base-Stock Policy

Abstract:
We study a single-period two-stage service-constrained supply chain with an information update. The buyer has two purchasing instances: before and after the forecast update. The procurement cost at the second stage is uncertain at the first stage. We determine the optimal ordering policy with two service constraints (a service constraint is imposed for each procurement stage) and discuss the impact of the forecast quality. Finally, we extend our model to a multi-period problem with two ordering stages in each period.

supply chain, information update, customer service level,dynamic programming, forecasting, signalling,

Abstract:
This paper deals with optimal pricing of a personalized product such as a personal portrait or photo. A new model of the pricing structure inspired by two real-life cases is introduced to the literature and solved to obtain optimal photo sitting fees and the final product price. A sensitivity analysis with respect to the problem parameters is performed.

personalized product, pricing, salvage loss, customized product, sensitivity analysis

Abstract:
We prove that an (s, S)policy is optimal in a continuous-review stochastic inventory model with a fixed ordering cost when the demand is a mixture of (i) a diffusion process and a compound Poisson process with exponentially distributed jump sizes, and (ii) a constant demand and a compound Poisson process. The proof uses the theory of impulse control. The Bellman equation of dynamic programming for such a problem reduces to a set of quasi-variational inequalities (QVI). An analytical study of the QVI leads to showing the existence of an optimal policy as well as the optimality of an (s, S) policy. Finally, the combination of a diffusion and a general compound Poisson demand is not completely solved. We explain the difficulties and what remains open. We also provide a numerical example for the general case.

(s, S) policy, compound Poisson process, diffusion process, economic order quantity model, impulse control, quasi-variational inequalities, stochastic inventory model, EOQ Model

Abstract:
We present a periodic review inventory model with multiple delivery modes. We generalize the notion of the base-stock policy for inventory systems with multiple delivery modes. While base-stock policies are optimal for one or two consecutive delivery modes, it is not so otherwise. For multiple consecutive delivery modes, we show that only the fastest two modes have optimal base stocks, and provide simple counterexamples to show that the remaining ones do not in general. We investigate why the base-stock policy is not optimal through detailed analyses of two numerical examples.

inventory models, inventory/production, uncertainty, multiple delivery modes, base-stock policies,dynamic programming

Abstract:
Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. Such situations arise when it takes a while to process the demand data, count the inventory, and pass the results to the IM. We show that the optimal total inventory-related cost decreases when the length of the information delay decreases. The amount of the decrease is an important datum for an IM interested in considering whether or not to invest in reducing the delay. The investment is required to finance design and acquisition of an information (collection and dissemination) system that can reduce the information delay. Such systems include phone calls, business meetings, and the use of information collection mechanisms such as radio frequency identification tags.

economic evaluation, information delays,delay evaluation,base-stock policy, partially observed inventory, incomplete information, sufficient statistics, RFID, radio frequency identification tags

Abstract:
The open source paradigm is often defined as a collaborative effort, implying that firms and consumers come together in a non-competitive climate. We show here that open source development can arise from a competitive climate. Under competition, we find that open source is the surplus maximizing outcome and can be in equilibrium if cost asymmetries are small. However, when cost asymmetries are large, contradictions between equilibrium and welfare maximization result. Considerations typical to public good problems arise, with issues of asymmetric contributions and freeriding. These issues should guide the firm's as well as the society's decisions to implement open source in particular environments. We analyze this problem in the framework of a dynamic duopolistic competition, with firms controlling their investments in software.

Differential Games, Public Goods, Open source, Software, Nash equilibrium, Two point boundary Value Problems, Optimal control, welfare maximization

Abstract:
We examine an oligopoly model of advertising competition where each firm's market share depends on its own and its competitors' advertising decisions. A differential game model is developed and used to derive the closed-loop Nash equilibrium under symmetric as well as asymmetric competition. We obtain explicit solutions under certain plausible conditions, and discuss the effects of an increase in the number of competing firms on advertising expenditure, market share and profitability.

Sethi model, advertising, dynamic games, game theory, differential games, optimal control, close-loop Nash equilibrium, oligopoly, Feedback Nash equilibrium

Abstract:
This paper introduces continuous and discrete modern control theory, especially the free-end point versions of the maximum principle, to the field of finance. We shall not, however, go into the proofs and other mathematical details because they are available in the cited literature. Instead, we shall briefly review the continuous and discrete maximum principles and then model simple cash balance problems as problems in control theory. we shall be especially intersted in the financial interpretations of the various functions such as the Hamiltonian function and the adjoint functions that arise in the course of the situation. Thus, we illustrate how such problems can be solved by simply capitalizing on the available control theory literature.

control theory, cash balance problems, the maximum principle, free-end point problems, adjoints, economic interpretations, finacial engineering

Abstract:
In the paper Optimum Consumption and Portfolio Rules in a continuous-Time Model, by R. C. Merton (J. Econ. Theory 3 (1971), 373-413), solutions obtained in cases when marginal utility at zero consumption is finite are not feasible. While they do satisfy the Hamilton-Jacobi Bellman equations, they do not represent appropriate value functions because the boundary behavior near zero wealth is not satisfactorily dealt with. In this note, we specify the boundary behavior and characterize optimal solutions.

Consumption/portfolio problem, investment-consumption problem, dynamic programming, stochastic control, R. C. Merton

Abstract:
Firms that want to increase the sales of their brands through advertising have the choice of capturing market share from their competitors through brand advertising, or increasing primary demand for the category through generic advertising. In this paper, differential game theory is used to analyze the effects of the two types of advertising decisions made by firms offering a product in a dynamic duopoly. Each firm's sales depend not only on its own and its competitor's brand advertising strategies, but also on the generic advertising expenditures of the two firms. Closed-loop Nash equilibrium solutions are obtained for symmetric and asymmetric competitors in a finite-horizon setting. The analysis for the symmetric case results in explicit solutions, and numerical techniques are employed to solve the problem for asymmetric firms.

Advertising, Generic Advertising, Brand Advertising, duopoly, dynamic model, optimal control, differential games, Nash equilibrium. Feedback Nash, marketing mix

Abstract:
The newsvendor problem is relatively easy to solve when the distribution of demand for newspapers is known. When the demand is unknown, the newsvendor faces a dual problem in the sense of Feldbaum (1960): to choose a decision variable that maximizes profit in the present period, and choose a large enough "observation window" to be able to view the process correctly so that consistent parameter estimation can occur. This is a difficult problem in general. In this paper, we treat a special case when the newsvendor faces exponential demand, independently and identically distributed with unknown mean.

Consistent estimators, censored data, Bayesian control, adaptive control, Newsvendor model, unnormalized probability, optimal control, incomplete information, partial observations

Abstract:
The problem of cash management, in its simplest form, is to formulate decision rules which control the level of a firm's cash balance to meet its demands for cash at minimum total discounted cost. Control is achieved by transacting securities for cash. The cost of control is the commission expense. Optimality depends on balancing excess opportunity costs of holding balances which are too large and having excess buying and selling costs (to meet cash obligations) of balances which are too small.

cash management, cash management, decision horizon, forecast horizon, cash balance problem, financial engineering

Abstract:
We study a model of economic growth in which an exogenously changing population enters in the objective function under total utilitarianism and into the state dynamics as the labor input to the production function. We consider an arbitrary population growth until it reaches a critical level (resp. saturation level) at which point it starts growing exponentially (resp. it stops growing altogether). This requires population as well as capital as state variables. By letting the population variable serve as the surrogate of time, we are still able to depict the optimal path and its convergence to the long-run equilibrium on a two-dimensional phase diagram. The phase diagram consists of a transient curve that reaches the classical curve associated with a positive exponential growth at the time the population reaches the critical level. In the case of an asymptotic population saturation, we expect the transient curve to approach the equilibrium as the population approaches its saturation level. Finally, we characterize the approaches to the classical curve and to the equilibrium.

economic growth, optimal control, phase diagram, dynamic programming, optimal savings, genuine savings

Abstract:
In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to nonobservation of demand, spoilage, misplacement, or theft of inventory. We study a partially observed inventory system where the demand is not observed, inventory level is noticed when it reaches zero, the unmet demand is lost, and replenishment orders must be decided so as to minimize the total discounted costs over an infinite horizon. This problem has an infinite-dimensional state space, and for it we establish the existence of a feedback policy when single-period costs are bounded or when the discount factor is sufficiently small. We also provide an approximately optimal feedback policy that uses a finite state representation.

inventory models, incomplete information, partially observed systems, optima control, zero-balance walk, inventory control, feedback policy, lost sales case

Abstract:
Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. We introduce dynamic information delays as a Markov process into the standard multiperiod stochastic inventory problem with backorders. We develop the
concept of a reference inventory position. We show that this position observed delay and the age of this observation are sufficient statistics for finding the optimal order quantities. Furthermore, we establish that the optimal ordering policy is of state-dependent base-stock type with respect to the reference inventory position (or state-dependent (s, S) type if there is a fixed ordering cost). The optimal base stock and
(s, S) levels depend on the magnitude of the latest observed delay and the age of this observation. Finally, we study the sensitivity of the optimal base stock and the optimal cost with respect to the sufficient statistics.

dynamic information delays,partial observations, stochastic inventory problem, base-stock policy, (s, S) policy, inventory models, incomplete information, sufficient statistics

Abstract:
This paper introduces recent developments in the analysis of inventory systems with partial observations. The states of these systems are typically conditional distributions, which evolve in infinite dimensional spaces over time. Our analysis involves introducing unnormalized probabilities to transform nonlinear state transition equations to linear ones. With the linear equations, the existence of the optimal feedback policies are proved for two models where demand and inventory are partially observed. In a third model where the current inventory is not observed but a past inventory level is fully observed, a sufficient statistic is provided to serve as a state. The last model serves as an example where a partially observed model has a finite dimensional state. In that model, we also establish the optimality of the basestock policies, hence generalizing the corresponding classical models with full information.

Optimal Control, Inventory models, Partially Observed Systems, unnormalized probabilities, infinite dimensional systems, conditional distributions, incomplete information

Abstract:
In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to non-observation of demand, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system where the unmet demand is backordered. When inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur and the inventory manager issues rain checks to customers. The shortages are fully observable via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.

Partially Observed Systems, inventroy control, Rain Checks, incomplete information, lost sales, optimal control, incomplete information, feedback policy, dynamic programming, unnormalized probability, infinite dimensional space

Abstract:
This paper revisits the model of the censored newsvendor presented by Ding, Puterman and Bisi (2002). We analyze that model in an infinite-horizon context by using the interesting concept of unnormalized probabilities. The unnormalized probabilities considerably simplify the dynamic programming equation and facilitate the proof of the existence of an optimal policy. They can also be used to give a simple, alternative proof to Ding et al.'s claim that the myopic order quantity is always less than or equal to the optimal order quantity. Importantly, the concept of unnormalized probabilities can be used to treat other important operations research problems with partial observations.

Unnormalized probabilities, Unobserved unmet demand, Newsvendor problem, dynamic programming, infinite dimensional spaces, incomplete information, partially obsereved systems, optimal control

Abstract:
Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. In this paper, we establish that the average cost optimal policy is of state-dependent basestock type with respect to the reference inventory position. We show that the optimal base stock depends on the age and the magnitude of the latest observed delay. If there is a fixed ordering cost, the methodology of this paper can be adapted for proving that the optimal ordering policy is a state-dependent (s, S) policy.

dynamic information delays, partial observations, stochastic inventory, base stock policy, average cost, ergodic problem, vanishing discount approach

Abstract:
We present a model to determine the optimal point for maintaining a software application. We also address the question: given that a maintenance project has been initiated, should maintenance effort continue till the project is completed? Most previous literature has implicitly assumed that it is optimal to complete a maintenance project once it has been initiated. We analyze two policies: a work-based policy and a time-based policy. In the work-based policy, a fixed amount of work needs to be completed, and the time taken to accomplish the work is random. In the time-based policy, a fixed amount of time is allocated to maintenance, but a random amount of work is completed. We examine similarities and differences between the above two policies and provide insights into the management of software maintenance projects. A key insight of this study is that under a variety of situations, partial maintenance is suboptimal.

Software maintenance, Dynamic programming, Threshold policy, optimal timing

Abstract:
We extend the classical newsvendor problem by introducing a service constraint and a demand forecast update. The newsvendor orders an initial amount and has the possibility to adjust the initial order after she observes a demand updating signal. She also commits to a target service level before any forecast update is available. The resulting problem does not permit a dynamic programming formulation. We use the method of Lagrange multipliers to solve the problem, and we derive an analytical expression for the optimal ordering decisions. Various properties of the optimal policy are discussed, and numerical results are presented to provide further insights into the newsvendor's behavior.

Inventory, newsvendor problem, service constraint, forecast update, two-stage ordering

Abstract:
Flexible transfer lines or mixed-model assembly lines are capable of diversified small-lot production due to negligible switch-over costs. With these lines, it is possible to implement just-in-time (JIT) production, which involves producing only the necessary parts in the necessary quantities at the necessary times. The problem of sequencing flexible transfer lines according to the JIT philosophy can be formulated as a nonlinear integer programming problem. Heuristic algorithms to solve the problem have appeared in the literature. In this paper, we show that the problem can be explicitly reduced to an assignment problem. Thus, we provide an efficient algorithm for an optimal solution to the JIT sequencing problem.

just-in-time production, flexible transfer lines, mixed-model assembly lines, assignment problem,JIT production system

Abstract:
This paper considers a problem of optimal preventive maintenance and replacement schedule of equipment devoted to extracting resources from known deposits. Typical examples are oil drills, mine shovels, etc. At most one replacement of the existing machinery by a new one is allowed. The problem is formulated as an optimal control problem subject to the state constraint that the remaining deposit at any given time is nonnegative. We show that the optimal preventive maintenance, production rates, and the replacement and salvage times of the existing machinery and the new one, if required, can be obtained by solving sequentially a series of free-end-point optimal control problems. Moreover, an algorithm based on this result is developed and used to solve two illustrative examples.

maintenance and replacement, optimal control, the maximum principle, extraction, mining equipment, extraction machinery

Abstract:
We study the problem of optimal staged purchases of electricity in time-sequential deregulated electricity markets. In recent years, the electricity industry has been deregulated and multiple time-sequential auction markets, such as the block forward, the day-ahead and the hour-of, and the real-time electricity markets, are formed. Thus, a load serving entity need to purchase electricity in these markets economically to meet its demand in each settlement time interval. We use the stochastic dynamic programming approach to establish a condition for staged purchases of electricity to be optimal and study its properties. Two algorithms for computing the optimal staged purchases are also developed.

Energy markets, Elecricity Markets, Deregulation, demand allocation, auction, time-sequential electricity markets

Abstract:
We present a new paradigm of hierarchical decision making in production planning and capacity expansion problems under uncertainty. We show that under reasonable assumptions, the strategic level management can base the capacity decision on aggregated information from the shop floor, and the operational level management, given this decision, can derive a production plan for the system, without too large a loss in optimality when compared to simultaneous determination of optimal capacity and The results are obtained via an asymptotic analysis of a manufacturing system with convex costs, constant demand, and with machines subject to random breakdown and repair. The decision variables are purchase time of a new machine at a given fixed cost and production plans before and after the costs of investment, production, inventories, and backlogs. If the rate of change in machine states such as up and down is assumed to be much larger than the rate of discounting costs, one obtains a simpler limiting mean. We develop methods for constructing asymptotically optimal decisions for the original problem from the optimal decisions for the limiting problem. We obtain error estimates for these constructed decisions.

Hierarchical Decision Making, Capacity Expansion, Production Planning, Dynamic Progrramming, Markov Processes, Near-Optimal Decisions

Abstract:
This paper considers the case of partially observed demand in the context of a multi-period inventory problem with lost sales. Demand in a period is observed if it is less than the inventory level in that period and the leftover inventory is carried over to the next period. Otherwise, only the event that it is larger than or equal to the inventory level is observed. These observations are used to update the demand distributions over time. The state of the resulting dynamic program consists of the current inventory level and the current demand distribution, which is infinite dimensional. The state evolution equation for the demand distribution becomes linear with the use of unnormalized probabilities. We study two demand cases. First, the demands evolve according to a Markov chain. Second, the demand distribution has an unknown parameter which is updated in the Bayesian manner. In both cases, we prove the existence of an optimal feedback ordering policy.

inventory models, partially observed systems, incomplete information, lost sales

Abstract:
Recently, the production control problem in stochastic manufacturing systems has generated a great deal of interest. The goal is to obtain production rates to minimize total expected surplus and production cost. This paper reviews the research devoted to minimum average cost production planning problems in stochastic manufacturing systems. Manufacturing systems involve a single or parallel failure-prone machines producing a number of different products, random production capacity, and constant demands.

Dynamic programming, manufacturing systems, long-run average cost minimization, unreliable machines, production planning

Abstract:
This paper compares several different production control policies in terms of their robustness to random disturbances such as machine failures, demand fluctuations, and system parameter changes. Simulation models based on VLSI wafer fabrication facilities are utilized to test the performance of the policies. Three different criteria, namely, the average total WIP, the average backlog, and a cost function combining these measures, are used to evaluate performance. Among the policies tested, the Two-Boundary Control policy outperforms the others.

Simulation, Kanban, Two-Boundary Control, Push and Pull Production Policies, Re-Entrant Shop, Semiconductor Manufacturing, Robust Policies, Two-Boundry Control Policies, WIP, Wafer Fabrication

Abstract:
This paper revisits the finite-horizon model of a censored newsvendor by Ding et al. [Ding, X., M. L. Puterman, A. Bisi. 2002. The censored newsvendor and the optimal acquisition of information. Oper. Res. 50 517–527]. An important result claimed there without a proper proof is that the myopic order quantity is always less than or equal to the optimal order quantity. Lu et al. [Lu, X., J. S. Song, K. Zhu. 2008. Analysis of perishable inventory systems with censored demand data. Oper. Res. 56(4) 1034–1038.] supplied a correct proof of the result. We analyze the same model using the interesting concept of the unnormalized probability, which simplifies the dynamic programming equation considerably and facilitates the proof of the claim. Moreover, it produces the proof of the existence of an optimal solution for an infinite-horizon setting of the problem.

Inventory/production: unknown demand, lost sales, censoring, optimal policies, partial observations, incomplete demand information, Bayesian approach, myopic policy, unnormalized probability, dynamic programming, infinite dimensional state space, optimal control

Abstract:
As the competition in the software development business grows fiercer, the features provided in the software become more and more important. With the growing popularity of open source software and the advent of new software delivery models, such as software-as-a-service, the traditional software vendors cannot compete on price in the long run. Therefore, they need to compete by increasing the number of features provided to customers. Moreover, the new entrants and the open source software vendors also need to provide increased number of features continuously in order to penetrate the market. Hence, these software firms need to invest judiciously in the software enhancement effort such that their revenue is maximized at the minimum possible cost. In addition, this strategy should be dynamic in order to capture the dynamic nature of the market. Therefore, we propose a dynamic optimization approach to obtain the optimal software enhancement effort over time. In this paper, we consider the value of adding more features as well as the impact of more bugs introduced while adding new features. Most of the past literature ignore the fact that the value of adding more features is not realized immediately by users. Therefore, we consider that there is a lag between the addition of new features and the increase in system's value. We also present several interesting managerial insights that can be used by software vendors to efficiently allocate resources based on the values of related parameters.

Software enhancement, optimal control theory, software features, dynamic optimization

Abstract:
This paper is concerned with optimal production rates for a failure-prone machine that produces two distinct part types. We consider the problem of controlling production rates so as to minimize the expected long-run average cost of product surpluses over time. We assume constant per unit holding and shortage costs and constant demand rates for the par types. We provide an expression for the potentila function and prove the optimality of a zero-inventory policy, when the average capacity is sufficiently large.

Dynamic programming, manufacturing systems, long-run average cost minimization, zero-inventory policy, just-in-time policy

Abstract:
We generalize the concept of K-convexity to an n-dimensional Euclidean space. The resulting concept of K-convexity is useful in addressing production and inventory problems where there are individual product setup costs and/or joint setup costs. We derive some basic properties of K-convex functions. We conclude the paper with some suggestions for future research.

K-convexity, inventory models, (s, S) policy, supermodular functions

Abstract:
Markov-modulated processes have been used in stochastic inventory models with setup costs for modeling demand under the influence of uncertain environmental factors, such as fluctuating economic and market conditions. The analyses of these models have been carried out in the literature only under the assumption that unsatisfied demand is fully backlogged. The lost sales situation occurs in many retail establishments such as department stores and supermarkets. We use the analysis of the Markovian demand model with backlogging to analyze the lost sales case; in particular, we establish the optimality of an (s, S)-type policy under fairly general conditions.

dynamic inventory model, Markov chain, (s, S) policy, lost sales, optimization, dynamic programming, Markovian demand, Markov modulated demand

Abstract:
We develop a general model for software development process and propose a policy to manage system coordination using system fault reports (e.g., interface inconsistencies, parameter mismatches, etc.). These reports are used to determine the timing of coordination activities that remove faults. We show that under an optimal policy, coordination should be performed only if a "threshold" fault count has been exceeded. We apply the policy to software development processes and compare the management of those projects under different development conditions. A series of numerical experiments are conducted to demonstrate how the fault threshold policy needs to be adjusted to changes in system complexity, team skill, development environment, and project schedule. Moreover, we compare the optimal fault threshold policy to an optimal release-based policy. The release-based policy does not take into account fault data and is easier to administer. The comparisons help to define the range of project parameters for which observing fault data can provide significant benefits for managing a software project. We develop a general model for software development process and propose a policy to manage system coordination using system fault reports (e.g., interface inconsistencies, parameter mismatches, etc.). These reports are used to determine the timing of coordination activities that remove faults. We show that under an optimal policy, coordination should be performed only if a threshold fault count has been exceeded. We apply processes and compare the management of those projects under different development conditions. A series of numerical experiments are conducted to demonstrate how the fault threshold policy needs to be adjusted to environment, and project schedule. Moreover, we compare the optimal fault threshold policy to an optimal release-based policy. The release-based policy does not take into account fault data and is easier to administer. The comparisons help to define the range of project parameters for which observing fault data can provide significant benefits for managing a software project.

software development, threshold policy, fault threshold, release-based policy, fault growth model, dynamic programming

Abstract:
The presentation of Table 2 in the original version of this article ("A Survey of Stackelberg Differential Game Models in Supply and Marketing Channels", Journal of Systems Science and Systems Engineering, Vol. 16, No. 4, pp., 385-413, 2007) contained a few typos. The corrected Table 2 is given below.

Stackelberg differential games, supply chain management, marketing channels, open-loop equilibria, feedback policies, channel coordination, optimal control

Abstract:
Information delays exist when the most recent inventory information available to the inventory manager (IM) is dated; namely, the IM observes only the inventory level of an earlier period. We introduce information delays into the standard multiperiod stochastic inventory problem with backorders. We consider two types of information delays: (i) a constant delay and (ii) a random delay. We define an appropriate reference inventory position, which is a sufficient statistics for finding the optimal order quantity. We show that the optimal ordering policy is of base-stock type with respect to the reference inventory position and is of (s, S) type if there is also a fixed cost of ordering.

Information delay, partial observations, stochastic inventory problem, incomplete information, base-stock policy, dynamic programming, sufficient statistic

Abstract:
Under many circumstances, demand observations are often censored due to the lack of tracking lost sales caused by stockouts. To understand the impact of the lost sales information on the ordering decisions, a periodic-review inventory model is formulated in which only the sales information is obtained immediately upon the realization of the demand. This is equivalent to observing the demand when the sales are less than the available stock and to inferring that the demand is higher than the stock when there is a stockout. Subsequently, the lost sales information is obtained after a delay. In the resulting model, an optimal policy, if exists, reveals a very complex structure. By decomposing the derivative of the value function, we demonstrate two different roles of inventory in our model: satisfying the demand and extracting the demand information. We show that the optimal inventory levels under the delayed observation of the lost sales are always higher than those for which the demands are fully observed. Moreover, as illustrated in numerical examples, the optimal policy possesses a counterintuitive behavior with respect to the problem parameters. To understand the key drivers of the optimal decisions, we further compare the costs under different demand observations. Two important observations are made. First, a lower cost is obtained when the realized demand is observed than when the demand is only observed to be higher than the inventory level, and, furthermore, the cost difference represents the value of demand information. Second, while a higher inventory level induces a more accurate demand forecast, the value of exact demand observation is not monotone in the procurement cost. Consequently, the optimal ordering quantity is not always decreasing in the procurement cost.

Inventory, newsvendor problem, partial observation, lost sales, incomplete demand information, dynamic programming, delayed observations

Abstract:
This paper studies the issue of coordinating equipment maintenance operations with capital investment strategy in the presence of random equipment failures. This problem represents an important extension of the celebrated Kamien and Schwartz (KS) paper published in Management Science. The traditional KS approach is to formulate the problem as a deterministic optimal control problem with the probability of machine failure as the state variable. Consequently, a deterministic policy is derived. As a major departure from the KS approach, we explicitly model the underlying stochastic process of machine failures. Our analysis of the stochastic dynamic programming model offers new insights into the problem. Under a long planning horizon with a limited replacement opportunity, each individual machine serves as a revenue generator and contributes a significant amount to the profit of the system. On the other hand, when the replacement budget is quite generous in a relatively short planning horizon, adding one extra machine only helps as a backup for unexpected failures of the machines purchased before it. An interesting result derived from this comparison is that a deterministic policy turns out to be optimal for the former, while a state-contingent policy must be applied to the latter. In other words, the deterministic KS approach does not work when a chain of machine replacement is considered. We further discuss the implications of the discount rate, productivity deterioration, learning, decision delay, and technology advancement on the optimal policy.

Maintenance and replacement, optimal control, impulse control, equipment investment; stopping time; variational inequality, reliability, hazard rate, the Kamien-Schwartz model, dynamic programming, chain of machines model

Abstract:
This paper discusses an explicit necessary and sufficient condition on the dividend stream of a publicly traded company, under which the price of the company's share is equal to the present value of the future dividends that will accrue to it. When it is not, the share price equals the present value of the future per share dividend plus the limiting per share value of the company at infinity. It uses a well-accepted generalization of the Miller-Modigliani framework, and assumes that the firm is an infinite horizon firm which may engage in repurchasing its own shares. It develops a proper dividend approach that can value such a firm for any dividend stream. The paper concludes by clarifying some remarks in the Miller-Modigliani paper.

Dividend approch, financial valuation, Miller-Modigliani theory, share price, cash flow approach, partial equilibrium

Abstract:
Managing customer satisfaction in a cost-effective way has always been a major challenge faced by inventory managers. We investigate the ordering strategy to meet an aggregate service target when a forecast update is available. The resulting problem does not permit a dynamic programming formulation. Instead, the infinite dimensional Kuhn-Turker theory is deployed to solve the problem. As an important feature of the model, the aggregate service target cannot be represented merely by a shortage cost. Moreover, the ordering decisions are not monotone with respect to the surplus and shortage costs as one would usually expect. An important insight derived from our model is that the trade-off between cost efficiency and service performance depends critically on the value of the product. A relatively consistent post-update service level should be provided across all demand signals in managing a high-value product, while cost savings are achieved for a low-value product by differentiating the service levels according to the demand signals. In general, the optimal policy tends to maintain a low post-update service level if the observed signal implies a variable postupdate demand. Our analysis also suggests that the information obtained from the forecast update helps to reduce cost without affecting the aggregate service level.

Inventory models, newsvendor problem, service constraint, Kuhn-Tucker conditions, Two-stage newsvendor, forecast updates, aggregate service target, long-term service level, infinite-dimensional problem, Chance constraints

Abstract:
This paper is concerned with a periodic-review inventory system with three consecutive delivery modes (fast, medium, and slow) and demand forecast updates. At the beginning of each period, the inventory level and demand information are updated and decisions on how much to order using each of the three delivery modes are made. It is shown that there is a base-stock policy for fast and medium modes which is optimal. Furthermore, the optimal policy for the slow mode may not be a base-stock policy in general.

Inventory, multiple delivery modes, forecast updates, base-stock policy

Abstract:
This paper is concerned with the problem of production planning in a stochastic manufacturing system with serial machines that are subject to breakdown and repair. The machine capacities are modeled as Markov chains. Since the number of parts in the internal buffers between any two the problem is inherently a state constrained problem. The objective is choose the input rates at the various machines over time in order to meet formulated as a stochastic dynamic programming problem. We prove a verification theorem and derive the optimal feedback control policy in terms of the directional derivatives of the potential function.

Production Planning, Optimal Control, Stochastic Manufacturing System, Long-Run Average Cost, Dynamic Programming, Ergodic Problem, Feedback Controls, Flowshop

Abstract:
We consider a newsvendor problem with partially observed Markovian demand. Demand is observed if it is less than the inventory. Otherwise, only the event that it is larger than or equal to the inventory is observed. These observations are used to update the demand distribution from one period to the next. The state of the resulting dynamic programming equation is the current demand distribution, which is generally infinite dimensional. We use unnormalized probabilities to convert the nonlinear state transition equation to a linear one. This helps in proving the existence of an optimal feedback ordering policy. So as to learn more about the demand, the optimal order is set to exceed the myopic optimal order. The optimal cost decreases as the demand distribution decreases in the hazard rate order. In a special case with finitely many demand values, we characterize a near-optimal solution by establishing that the value function is piecewise linear.

unobserved unmet demand; Markovian demand; newsvendor problem, dynamic programming, inventroy models, partially observed systems, incomplete information, base stock policy, myopic policy

Abstract:
We consider the problem of finding an optimal financing mix of retained earnings and external equity for maximizing the value of a corporation in a stochastic environment. We formulate the problem as a singular stochastic control for a diffusion process. We show that the value function satisfies a free-boundary problem. We characterize the value function and show that the optimal policy can be characterized in terms of two threshold parameters. With asset level below the lower threshold, optimal policy is to finance the firm's growth by retaining all earnings and raising the required external equity financing. With asset level above the higher threshold, optimal policy is to pay all retained earnings as dividends and to bring in no new equity. Between the two thresholds, the optimal policy is to retain all earnings but not raise any external equity. We obtain an explicit solution for the value function when there is no brokerage commission in floating external equity. We provide economic interpretations of the results obtained in the paper.

Abstract:
We analyze the tradeoff between (demand) substitution costs and (production) changeover costs in a discrete-time production-inventory setting. For this purpose, we consider a two-product dynamic lot-sizing model with changeover costs, inventory carrying costs, and costs associated with product substitution. Based on practical considerations, we study a restricted class of solutions and first describe a polynomial-time algorithm for optimizing over this class. Then, we develop an effcient procedure for computing forecast horizons. Using these tools, we develop several insights for managing such systems. A key driver for the extent of substitution is the ratio of changeover cost to the substitution cost associated with mean demand. When than 0.5), substitution is typically non-existent and when it is high (greater than 2), the frequency of changeovers is typically low. Another important observation is that the length of the minimal forecast horizon increases considerably in the presence of substitution. Consequently, the length of an (effective) rolling horizon needs to be significantly higher under substitution. Finally, we make the interesting observation concerning decisions on investments aimed at reducing changeover and/or substitution costs: when the changeover cost is large, it cost and vice-versa.

Multiperiod Problems; Forecast Horizons; Rolling Horizons, Decision Horizons, Planning Horizons, Solution Horizons, Forecasting, Dynamic Lot Size Models, Operations Management, Production changeover, demand substitution, downward substitution, product substitution, polynomial algorithm

Abstract:
This paper is concerned with a generalization of classical inventory models (with fixed ordering costs) that exhibit (s, S) policies. In our model, the distribution of demands in successive periods is dependent on a Markov chain. The model includes the case of cyclic or seasonal demand. The model is further extended to incorporate some other realistic features such as no ordering periods and storage and service level constraints. Both finite and infinite horizon nonstationary problems are considered. We show that (s, S) policies are also optimal for the generalized model as well as its extensions.

dynamic inventory model, Markov chain, (s, S) policy, optimization, dynamic programming, Markovian demand, Markov modulated demand, production scheduling, overhrad costs

Abstract:
This paper deals with an asymptotic analysis of hierarchical production planning in stochastic manufacturing systems consisting of a single or parallel failure-prone machines producing a number of different products without attrition. The objective is to choose production rates over time in order to minimize the long-run average expected cost of production and surplus. As the rate of machine break-down and repair approaches infinity, the analysis results in a limiting problem in which the stochatic machine capacity is replaced by the equilibrium mean capacity. The optimal value for the original problem is proved to converge to the optimal value of the limiting problem. This suggests a heuristic to construct an open-loop control for the original stochastic problem from the open-loop control of the limiting deterministic problem. We as well as obtain error bound estimates for constructed open-loop controls.

Stochastic manufacturing system, hierarchical control, dynamic programming, viscosity solution, convergence rate, error bound, ergodic problem, long-run average cost

Abstract:
This is a Foreword to Handbook of Niche Marketing: Principles & Practice written by Tevfic Dalgic and published by Best Business Books, The Haworth Reference Press, New York, 2006.

Niche Marketing, buzz, marketing strategy

Abstract:
In this paper, we use a Markov decision process (MDP) to model the joint inventory-promotion decision problem. The state variable of the MDP represents the demand state brought about by changing environmental factors as well as promotion decisions. The demand state in a period determines the distribution of the random demand in that period. Optimal inventory and promotion decision policies in the finite horizon problem are obtained via dynamic programming. Under certain conditions, we show that there is a threshold inventory level P for each demand state such that if the threshold is exceeded, then it is desirable to promote the product. For the proportional ordering cost case, the optimal inventory replenishment policy is a base-stock type policy with the optimal base-stock level dependent on the promotion decision.

dynamic inventory model, promoton decisions, base stock type policy, optimization, dynamic programming, Markovian decision process, production scheduling

Abstract:
This paper studies stochastic inventory problems with unbounded Markovian demands, ordering costs that are lower semicontinuous, and inventory/backlog (or surplus) costs that are lower semicontinuous with polynomial growth. Finite-horizon problems, stationary and nonstationary discounted-cost infinite-horizon problems, and stationary long-run average-cost problems are addressed. Existence of optimal Markov or feedback policies is established. Furthermore, optimality of (s, S)-type policies is proved when, in addition, the ordering cost consists of fixed and proportional cost components and the surplus cost is convex.

Dynamic inventory models - Markov chains - dynamic programming, finite horizon, infinite horizon , cyclic demand, (s, S)-policy, inventory models, Markov process, cost with polynomial growth, average cost, ergodic problem, base-stockpolicy, incomplete information, partial observations

Abstract:
This paper is concerned with long-run average cost minimization of a stochastic inventory problem with Markovian demand, fixed ordering cost, convex surplus cost, and lost sales. The states of the Markov chain represent different possible states of the environment. Using a vanishing discount approach, a dynamic programming equation and the corresponding verification theorem are established. Finally, the existence of an optimal state-dependent (s; S) policy is proved.

Dynamic inventory model, lost sales, Markov chain, dynamic programming, infinite

Abstract:
We consider a problem of optimal production control of a single unreliable machine. The objective is to minimize a discounted convex inventory/backlog cost over an infinite horizon. Using the variational analysis methodology, we develop the necessary conditions of optimality in terms of the co-state dynamics. We show that an inventory-threshold control policy is optimal when the work and repair times are exponentially distributed, and demonstrate how to find the value of the threshold in this case. We consider also a class of distributions concentrated on infinite intervals and prove properties of the optimal trajectories, as well as properties of an optimal inventory threshold that is time dependent in this case.

Optimal Control, production control, inventory model, unreliable machine, threshold control, variational analysis

Abstract:
Sethi and Thompson (1970) illustrated the applications of the maximum principle to solve simple dynamic cash balance problems. There we introduced the idea of penalty function to solve the cash balance problem with bounded state variables arising out of disallowing overdrafts and short selling. This resulted in the adjoint equations containing terms in the state variables. We then stated the need for solving a two-point boundary value problem. In this note, we show that for the example under consideration we can get the optimal solution without solving a two-point boundary value problem. That we can do so depends on two observations: (1) the adjoint equations are independent of the terms in state variables if they are within their bounds, and (2) the Hamiltonian can be easily maximized in a way that keeps the state variables at nonnegative levels.

cash balance problems, the maximum principle, optimal control problems, two-point boundary value problem, state constraints, cash management, financial engineering

Abstract:
Most manufacturing systems are large and complex and operate in an uncertain environment. One approach to managing such systems is that of hierarchical decomposition. This paper reviews the research devoted to proving that a hierarchy based on the frequencies of occurrence of different types of events in the systems results in decisions that are asymptotically optimal as the rates of some events become large compared to those of others. The paper also reviews the research on stochastic optimal control problems associated with manufacturing systems, their dynamic programming equations, existence of solutions of these equations, and verification theorems of optimality for the systems. Manufacturing systems that are addressed include single machine systems, dynamic fowshops, and dynamic jobshops producing multiple products. These systems may also incorporate random production capacity and demands, and decisions such as production rates, capacity expansion, and promotional campaigns are also presented.

manufacturing, hierarchical systems, dynamic programming, dynamic jobshops, optimal control, stochastic optimal control, operations and marketing, hierarchical decomposition, asymptotic optimality, averaging principle, multi time scale systems

Abstract:
This paper studies multiproduct inventory models with stochastic demands and a warehousing constraint. Finite horizon as well as stationary and nonstationary discounted-cost infinite-horizon problems are addressed. Existence of optimal feedback policies is established under fairly general assumptions. Furthermore, the structure of the optimal policies is analyzed when the ordering cost is linear and the inventory/backlog cost is convex. The optimal policies generalize the base-stock policies in the single-product case. Finally, in the stationary infinite-horizon case, a myopic policy is proved to be optimal if the product demands are independent and the cost functions are separable.

Multiproduct inventory model, warehousing constraints, dynamic programming, finite and infinite horizons, generalized base-stock policies, myopic policies, substitute property

Abstract:
We are concerned with a discrete-time undiscounted dynamic lot size model in which demand and the production setup cost are constant for an initial few periods and the holding cost of inventory is an arbitrary nondecreasing function assumed to be stationary (i.e., explicitly independent of time) in the same initial few periods. We show that there exists a finite forecast horizon in our model and obtain an explicit formula for it. In addition, we obtain fairly general conditions under which the existence of a solution horizon in the model implies the existence of a forecast horizon. We also derive an explicit formula for the minimal solution horizon. These results extend the earlier ones obtained for the dynamic lot size model with linearly increasing holding costs.

Forecast Horizons, Solution Horizons, Decision Horizons, Dynamic Lot Size Model, Graph Theory, Wagner-Whitin Algorithm, Inventory Theory, Dynamic Programming, Fowrard Algorithm

Abstract:
This paper examines open source software development in a competitive environment. The quality of open source software improves over time based upon contributions by firms and users. A firm's decision to contribute is interesting because it also augments competitors' software quality in future periods subject to compatibility considerations with their existing software. A differential game model is developed to understand why firms are increasingly involved in open source software development by determining the optimal contributions and software quality over time. We obtain a closed-loop Nash equilibrium solution. Examples are given to derive insights from this model.

Open sourcd software, dynamic games, game theory, differential games, optimal control, close-loop Nash equilibrium, software quality

Abstract:
This paper presents an asymptotic analysis of a stochastic manufacturing system consisting of parallel machines subject to breakdown and repair and facing a constant demand, as the rates of change of the machine states approach infinity. This situation gives rise to a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. The long-run average cost for the original problem converges to the long-run average cost of the limiting problem. Open-loop and feedback controls for the original problem are constructed from optimal controls of the limiting problem in a way that guarantees their asymptotic optimality. The convergence rate of the long-run average cost for the original problem to that of the limiting problem is established. This helps in providing an error estimate for the constructed open-loop asymptotic optimal control.

Production Planning, Hierarchical Controls, Stochastic Manufacturing System, Long-Run Average Cost, Dynamic Programming, Ergodic Problem, Asymptotic Optimality, Open-Loop Controls, Feedback Controls

Abstract:
This paper considers a dynamic lot sizing problem faced by a producer who supplies a single product to multiple customers. Characterized by their backorder costs as well as shipping costs, a customer with a high backorder cost has a greater need for the product than a customer with a low backorder cost. We show that the general problem with time-varying customer-dependent backlogging and shipping costs is NP-hard in the strong sense. We then develop an efficient dynamic programming algorithm for an important instance of the problem when there is no speculative motive for backlogging. We also establish forecast horizon results for the case of stationary production and shipping costs, which help the decision maker determine a proper forecast horizon in a rolling-horizon planning process.

Dynamic lot sizing model, dynamic programming, forecast horizon, multiple customers

Abstract:
This paper considers a production-inventory problem in which the manufacturer participates in an energy buy-back program, which offers him probabilistic opportunities with rewards for not using electricity. That is, the manufacturer will get paid for stopping production to save on electricity. The amount rewarded in a period will depend on the electricity market condition at that time. The market condition in any given period is represented by three states: normal (i.e., non-peak), semi-peak, or peak, and the reward amount in the period will be 0, K1; and K2, respectively. The occurrence of each state in a period is dictated by a known probability distribution. The objective is to determine from the manufacturer's perspective, whether to take such an offer when it arises. Under a mild assumption, we show that in the normal market condition, the production decision is partly a base-stock policy, whereas under semi-peak and peak conditions, the manufacturer, upon accepting the offer, produces according to (s1; S) and (s2; S) policies, corresponding to K1 and K2; respectively. Two variants of the model are also discussed.

Production-inventory model, (s; S) policy, dynamic programming, electricity use in production, energy buy-back program

Abstract:
This paper surveys the research on optimal consumption and investment problem of an agent who is subject to bankruptcy that has a specified utility (reward or penalty). The bankruptcy utility, modeled by a parameter, may be the result of welfare subsidies, the agent's innnate ability to recover from bankruptcy, psychic costs associated with bankruptcy, etc. Modeled with non-negative consumption, positive subsistence consumption, risky assets modeled by geometric Brwonian motions or semimartingales are discussed. The paper concludes with suggestions for open research problems.

Consumption and Investment problem, Portfolio and Consumption problem, bankruptcy, subsistence consumption, minimal consumption, borrowing constraints, stochastic optimal control, martingale problems, optimal stopping problems, Risk aversion measures, Open Research Problems, financial engineering

Abstract:
This paper is concerned with a periodic review inventory system with fast and slow delivery modes and regular demand forecast updates. At the beginning of each period, on-hand inventory and demand information are updated. At the same time, decisions on how much to order using fast and slow delivery modes are made. Fast and slow orders are delivered at the end of the current and the next periods, respectively. It is shown that there exists an optimal Markov policy and that it is a modified base-stock policy.

forecast revisions, multiple delivery modes, stochastic optimization, base-stock policy, inventory models, Peeling Layers of an Onion, Forecast Updates, dynamic programming

Abstract:
This paper is concerned with a production planning problem in a two-machine flowshop subject to breakdown and repair of machines subject to nonnegativity constraints on work-in-process. The objective is to choose machine production rates over time to meet the demand facing the flowshop at a minimum long-run average cost. It si shown that the dynamic programming equation for the problem has a solution consisting of the minimal average cost and the so-called potentila function. The result helps in establishing the verification theroem and partial characterization of the optimal control policy if it exists.

Production planning, Dynamic programming, state constraints, feedback controls, optimal control theory, long-run average cost

Abstract:
This paper introduces the notion of mixed leadership in non-zero-sum differential games, where there is no fixed hierarchy in decision making with respect to the players. Whether a particular player is leader or follower depends on the instrument variable s/he is controlling, and it is possible for a player to be both leader and follower, depending on the control variable. The paper studies two-player open-loop differential games in this framework, and obtains a complete set of equations (differential and algebraic) which yield the controls in the mixed-leadership Stackelberg solution. The underlying differential equations are coupled and have mixed boundary conditions. The paper also discusses the special case of linear-quadratic differential games, in which case solutions to the coupled differential equations can be expressed in terms of solutions to coupled Riccati differential equations which are independent of the state trajectory.

Differential games, Stackelberg-Nash solution, mixed leadership games, two-point boundary value problems, Stackelberg games, multi-leader multi follower games, hierarchical games, open-loop solution, linear-qudratic differential games

Abstract:
This is an erratum on my paper "A Note on Modeling Simple Dynamic Cash Balance Problems" published in The Journal of Financial and Quantitative Analysis, Vol. 8, No. 4 (Sep., 1973), pp. 685-687.

Cash balance problems, cash management, optimal control, maximum principle, state constraints, financial engineering

Abstract:
This paper is concerned with the reduction of a class of optimal control problems to simpler problems by using decomposition and aggregation. Decomposition is shown to provide a good approximation when the system dynamics involve nearly decomposable matrices or variables with strong and weak interactions. Aggregation provides a good approximation if each of the decomposed matrices has one or more dominant eigenvalues. It is shown how one can construct nearly-optimal controls for the given system from the optimal solutions of the simpler reduced problems.

Optimal control, dynamic systems, decomposition, aggregation, near optimization

Abstract:
We consider a production planning problem in a two-machine flowshop subject to breakdown and repair of machines and subject to nonnegativity and upper bound constraints on work-in-process. The objective is to choose machine production rates over time to minimize the long-run average inventory/backlog and production costs. For sufficiently large upper bound on the work-in-process, the problem is formulated as a stochastic dynamic program. We then establish a verification theorem and a partial characterization of the optimal control policy if it exists.

Production Planning, Optimal Controls, Stochastic Manufacturing System, Long-Run Average Cost, Dynamic Programming, Ergodic Problem, Feedback Controls, flowshop, average cost optimality

Abstract:
A comparison of and the relationship between the Ito and Stratonovich formulations of stochastic integration is given. It is argued that generally the Ito formulation is appropriate for problems of finance and economics. The Black-Scholes option pricing problem is discussed in both frameworks with the Ito formulation being shown to be clearly superior. A finance example in which the Stratonovich approach is simplest is also presented.

Option pricing, Black-Scholes, Ito Calculus, Stratonovich Calculus, finance

Abstract:
This paper presents an asymptotic analysis of hierarchical production planning in a manufacturing system with two tandem machines that are subject to breakdown and repair. The system produces a single product, whose rate of demand over time is given to be constant. The problem is formulated as a continuous-time dynamic programming problem in which the objective is to minimize the cost of production, inventories, and backlogs. The size of the buffer between the two machines is assumed to be finite. As the rates of change in machines' states approach infinity, the analysis results in a limiting problem in which the stochastic machine capacity is replaced by the average capacity. The value function for the original problem is shown to converge to the value function of the limiting problem. Both open-loop and feedback controls for the original problem are constructed from near-optimal controls of the limiting problem in a way which guarantees their asymptotic optimality as the rates of changes in machines' states become large in comparison to the discount rate. The convergence rate of the value function for the original problem to that of the limiting problem together with the error estimate for the constructed asymptotic optimal controls are obtained. In addition, the constructed feedback control is compared to the Kanban control policy for the stochastic two-machine flowshop.

Production Planning, Hierarchical Controls, Stochastic Manufacturing System, Dynamic Programming, Asymptotic Optimality, Open-Loop Controls, Feedback Controls, Error Bounds, Discrete event systems, stochastic optimal control.

Abstract:
We consider a production planning problem for a general jobshop subject to breakdown and repair of machines and subject to lower and upper bound constraints on work-in-process. The machine capacities and demand processes are assumed to be finite state Markov chains. The problem is to choose the rates of production on the various machines over time to minimize the expected discounted costs of production and inventory/backlog over an infinite horizon. It is formulated as a stochastic dynamic programming problem. We show that the value function of the problem is locally Lipschitz and is a solution to a dynamic programming equation together with a certain boundary condition. We provide an interpretation of the boundary condition, provide a verification theorem, and derive the optimal feedback control policy in terms of the directional derivatives of the value function. The results are proved via reduction to a deterministic optimal control problem that is equivalent to the stochastic production planning problem under consideration.

Production planning, Optimal control, Dynamic programming, state constraints, feedback controls, inventory problem, jobshop, piecewise deterministic processes