154 research outputs found

    Conditions for the discovery of solution horizons

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    We present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable. These problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network. We provide general forward algorithms with stopping rules for their solution. The key condition required is that of weak reachability, which roughly requires that for any sequence of nodes or states, it must be possible from optimal states to reach states close in cost to states along this sequence. Moreover the costs to reach these states must converge to zero. Applications are considered in optimal search, undiscounted Markov decision processes, and deterministic infinite horizon optimization.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47927/1/10107_2005_Article_BF01581244.pd

    Solvability in Discrete, Nonstationary, Infinite Horizon Optimization

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    For several time-staged operations management problems, the optimal immediate decision is dependent on the choice of problem horizon. When that horizon is very long or indefinite, an appropriate modeling technique is infinite horizon optimization. For problems that have stationary data over time, optimizing system performance over an infinite horizon is generally no more difficult than optimizing over a finite horizon. However, restricting problem data to be stationary can render the models unrealistic, failing to include nonstationary aspects of the real world. The primary difficulty in nonstationary, infinite horizon optimization is that the problem to solve can never be known in its entirety. Thus, solution techniques must rely upon increasingly longer finite horizon problems. Ideally, the optimal immediate decisions to these finite horizon problems converge to an infinite horizon optimum. When finite detection of that optimal decision is possible, we call the underlying infinite horizon problem well-posed. The literature on nonstationary, infinite horizon optimization has generally relied upon either uniqueness of the optimal immediate decision or monotonicity of that decision as a function of horizon length. In this thesis, we require neither of these, instead developing a more general structural condition called coalescence that is equivalent to well-posedness. Chapters 2-4 study infinite horizon variants of three deterministic optimization applications: concave cost production planning, single machine replacement, and capacitated inventory planning. For each problem, we show that coalescence is equivalent to well-posedness. We also give a solution procedure for each application that will uncover an infinite horizon optimal immediate decision for any well-posed problem. In Chapter 5, we generalize the results of these applications to a generic classes of optimization problems expressible as dynamic programs. Under two different sets of assumptions concerning the finiteness of and reachability between states, we show that coalescence and well-posedness are equivalent. We also give solution procedures that solve any well-posed problem under each set of assumptions. Finally, in Chapter 6, we introduce a stochastic application: the infinite horizon asset selling problem, and again show that coalescence and well-posedness are equivalent and give a solution procedure to solve any such well-posed problem.Ph.D.Industrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60810/1/tlortz_1.pd

    Managing longevity risk

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    A simple forward algorithm to solve general dynamic lot sizing models with n periods in O(nlogn) or O(n) time

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    This paper is concerned with the general dynamic lot size model, or (generalized) WagnerWhitin model. Let n denote the number of periods into which the planning horizon is divided. We describe a simple forward algorithm which solves the general model in 0(n log n) time and 0(n) space, as opposed to the well-known shortest path algorithm advocated over the last 30 years with 0 (n 2) time. A linear, i.e., 0(n)-time and space algorithm is obtained for two important special cases: (a) models without speculative motives for carrying stock, i.e., where in each interval of time the per unit order cost increases by less than the cost of carrying a unit in stock; (b) models with nondecreasing setup costs. We also derive conditions for the existence of monotone optimal policies and relate these to known (planning horizon and other) results from the literature. (DYNAMIC LOT SIZING MODELS; DYNAMIC PROGRAMMING; COMPLEXITY) This paper is concerned with the dynamic lot size model, one of the most frequently employed deterministic single-item inventory planning models. This model was introduced by Wagner and Whitin (1958) and is therefore often referred to as the WagnerWhitin model ( W-W model): it specifies a horizon divided into finitely many (say n) periods each with a known demand which must be satisfied. An unlimited amount may be ordered (produced) in each period. The cost structure consists of fixed-plus-linear order (or production) costs and holding costs assumed to be proportional with the endof-the-period inventory levels. All parameters, i.e., demands, setup costs, variable replenishment and holding cost rates, may differ from period to period. Two distinct rationales prevail for maintaining inventories in systems with deterministic demands and unlimited replenishment opportunities: (I) the cycle stock motive: economies of scale in the replenishment costs provide an incentive for order quantities to cover more than a single period's demand; (II) the speculative motive (see Chand and Morton 1986): even in the absence of economies of scale, it may be advantageous to order some future period's demand in the current period, if the future cost of ordering a unit exceeds the cost of ordering this unit now and carrying it until the future period. In this paper we describe a simple algorithm which solves the general dynamic lot size model in 0(n log n) time and 0(n) space, as opposed to the well-known shortest path algorithm advocated over the last 30 years with 0 (n2) time. A linear, i.e., 0 (n)-time and space algorithm is obtained for two important special cases: (a) models without speculative motives for carrying stock, i.e., instances in which in each interval of time, the per unit order cost increases by less than the cost of carrying a unit in stock over this interval (constant variable order cost rates represent a special case of such models; Wagner and Whitin, for example, originally confined themselves to this case)

    The Recovery Theorem

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    We can only estimate the distribution of stock returns but we observe the distribution of risk neutral state prices. Risk neutral state prices are the product of risk aversion – the pricing kernel – and the natural probability distribution. The Recovery Theorem enables us to separate these and to determine the market’s forecast of returns and the market’s risk aversion from state prices alone. Among other things, this allows us to determine the pricing kernel, the market risk premium, the probability of a catastrophe, and to construct model free tests of the efficient market hypothesis.

    The Effect of Demand Forecasting on Production Policies and Coordination in Periodic Review Production Systems.

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    We consider a sequential decision process in a periodic review production system. At the beginning of each period, marketing decides how much money to be invested in demand forecasting. After its decision, marketing obtains the forecast and passes the forecast to production, which then determines production quantity. Centralized and decentralized models are formulated. In the centralized model, the two divisions make their decisions by considering the total profit of the company. In the decentralized model, marketing pays production a unit transfer price for every product sold. Three types of policies about the unit transfer price are considered. In the company with the marketing-oriented policy, the unit transfer price is decided by marketing; in the company with the production-oriented policy, it is decided by production; and, in the company with the coordination policy, it is determined by considering the overall profit of the company. We find that, in the decentralized companies, production favors a larger unit transfer price, but marketing prefers a lower unit transfer price. In the company with the marketing-oriented policy, marketing tends to spend more money on forecasting, but production becomes less likely to start a run. However, in the production-oriented company, production is more likely to produce, but marketing spends less money on forecasting. The results in the company with the coordination policy can be considered as outcomes of cooperation, and it has the best performance among the decentralized companies. Furthermore, we find that the centralized company has a larger total profit than do the decentralized companies. Most of the time, it spends more money on forecasting, and it is more likely to produce. We develop two multi-period models. For the multi-period centralized model with a finite planning horizon, we find that the learning effect can induce the system to adopt a forecasting method with high cost and precision. As to the other multi-period model, we study a system with a fixed batch size, a fixed lead time, and an infinite planning horizon. The important result is that there exists a threshold inventory level, which property is similar to that of an (s, Q) inventory management system

    Fiscal Policy, Past and Present

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    This paper begins by reviewing the current U.S. fiscal situation and the causes of its recent deterioration. As a guide to possible policy actions, it estimates past responses of revenue and expenditure both at the federal and at the state and local level. Federal fiscal policy is found to be responsive to both economic and fiscal conditions, and this responsiveness may have grown over time. For states, economic conditions are less important, but responses to budget gaps are swifter. Given current conditions, equations for federal revenue and expenditure predict tax cuts and expenditure increases, but of a considerably smaller magnitude than President Bush initially proposed. However, current circumstances are difficult to evaluate because of the enormous implicit entitlements liabilities, which are much more significant today than in the past. This difficulty is but one of the problems facing policy prediction and evaluation.macroeconomics, Fiscal Policy, Past fiscal policy, present fiscal policy

    Quality and production control with opportunities and exogenous random shocks

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    Cataloged from PDF version of article.In a production process, opportunities arise due to exogenous or indigenous factors, for cost reduction. In this dissertation, we consider such opportunities in quality control chart design and production planning for the lot sizing problem. In the first part of the dissertation, we study the economic design of X control charts for a single machine facing exogenous random shocks, which create opportunities for inspection and repair at reduced cost. We develop the expected cycle cost and expected operating time functions, and invoking the renewal reward theorem, we derive the objective function to obtain the optimum values for the control chart design parameters. In the second part, we consider the quality control chart design for the multiple machine environment operating under jidoka (autonomation) policy, in which the opportunities are due to individual machine stoppages. We provide the exact model derivation and an approximate model employing the single machine model developed in the first part. For both models, we conduct extensive numerical studies and observe that modeling the inspection and repair opportunities provide considerable cost savings. We also show that partitioning of the machines as opportunity takers and opportunity non-takers yields further cost savings. In the third part, we consider the dynamic lot sizing problem with finite capacity and where there are opportunities to keep the process warm at a unit variable cost for the next period if more than a threshold value has been produced. For this warm/cold process, we develop a dynamic programming formulation of the problem and establish theoretical results on the optimal policy structure. For a special case, we show that forward solution algorithms are available, and provide rules for identifying planning horizons. Our numerical study indicates that utilizing the undertime option results in significant cost savings, and it has managerial implications for capacity planning and selection.Toy, Ayhan Ă–zgĂĽrPh.D
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