A well-known result in the bayesian inventory management literature is: If lost sales are not observed, the bayesian optimal inventory level is larger than the myopic inventory level (one should "stock more"...
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A well-known result in the bayesian inventory management literature is: If lost sales are not observed, the bayesian optimal inventory level is larger than the myopic inventory level (one should "stock more" to learn about the demand distribution). This result has been proven in other studies under the assumption that inventory is perishable, so the myopic inventory level is equal to the bayesian optimal inventory level with observed lost sales. We break that equivalence by considering nonperishable inventory. We prove that with nonperishable inventory, the famous "stock more" result is often reversed to "stock less," in that the bayesian optimal inventory level with unobserved lost sales is lower than the myopic inventory level. We also prove that making lost sales unobservable increases the bayesian optimal inventory level;in this specific sense, the famous "stock more" result of other studies generalizes to the case of nonperishable inventory. When the product is out of stock, a customer may accept a substitute or choose not to purchase. We incorporate learning about the probability of substitution. This reduces the bayesian optimal inventory level in the case that lost sales are observed. Reducing the inventory level has two beneficial effects: to observe and learn more about customer substitution behavior and (for a nonperishable product) to reduce the probability of overstocking in subsequent periods. Finally, for a capacitated production-inventory system under continuous review, we derive maximum likelihood estimators (MLEs) of the demand rate and probability that customers will wait for the product. (Accepting a raincheck for delivery at some later time is a common type of substitution.) We investigate how the choice of base-stock level and production rate affect the convergence rate of these MLEs. The results reinforce those for the bayesian, uncapacitated, periodic review system.
We consider the optimization problem of a decision maker facing a sequence of coin tosses with an initially unknown probability Theta for heads. Before each toss she bets on either heads or tails and she wins one euro...
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We consider the optimization problem of a decision maker facing a sequence of coin tosses with an initially unknown probability Theta for heads. Before each toss she bets on either heads or tails and she wins one euro if she guesses correctly, otherwise she loses one euro. We investigate the effect of changes in the distribution of Theta on the expected optimal gain of the decision maker. Using techniques from bayesian dynamic programming we will show that under the assumption of a beta distribution for the prior a riskier prior implies higher expected gains. The rationale for this is that a riskier prior allows better learning and provides higher informational value to the observations. We will also consider the case of a risk-sensitive decision maker in a two-period model.
We consider a complex system consisting of N identical components that are expected to function properly during a given period (mission time). Due to the possibility of failure before completion of the mission time, o...
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We consider a complex system consisting of N identical components that are expected to function properly during a given period (mission time). Due to the possibility of failure before completion of the mission time, one allows for maintenance as well as burn-in, the latter in order to cope with the problem of "infant mortality". We present a two-level (bayesian) dynamicprogramming approach to determine the optimal timing of the maintenance interventions and durations of burn-in, including maintenance-induced burn-in. We also discuss ways to reduce the "curse of dimensionality".
Retailers are frequently uncertain about the underlying demand distribution of a new product. When taking the empirical bayesian approach of Scarf (1959), they simultaneously stock the product over time and learn abou...
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Retailers are frequently uncertain about the underlying demand distribution of a new product. When taking the empirical bayesian approach of Scarf (1959), they simultaneously stock the product over time and learn about the distribution. Assuming that unmet demand is lost and unobserved, this learning must be based on observing sales rather than demand, which differs from sales in the event of a stockout. Using the framework and results of Braden and Freimer (1991), the cumulative learning about the underlying demand distribution is captured by two parameters, a scale parameter that reflects the predicted size of the underlying market, and a shape parameter that indicates both the size of the market and the precision with which the underlying distribution is known. An important simplification result of Scarf (1960) and Azoury (1985), which allows the scale parameter to be removed from the optimization, is shown to extend to this setting. We present examples that reveal two interesting phenomena: (1) A retailer may hope that, compared to stocking out, realized demand will be strictly less than the stock level, even though stocking out would signal a stochastically larger demand distribution, and (2) it can be optimal to drop a product after a period of successful sales. We also present specific conditions under which the following results hold: (1) Investment in excess stocks to enhance learning will occur in every dynamic problem, and (2) a product is never dropped after a period of poor sales. The model is extended to multiple independent markets whose distributions depend proportionately on a single unknown parameter. We argue that smaller markets should be given better service as an effective means of acquiring information.
Our bayesian dynamic programming model builds on existing models to account for inspection delay, choice of keeping production going during inspection and/or restoration, and lot sizing. We focus on describing how dyn...
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Our bayesian dynamic programming model builds on existing models to account for inspection delay, choice of keeping production going during inspection and/or restoration, and lot sizing. We focus on describing how dynamic statistical process control (DSPC) rules can improve on traditional, static ones. We explore numerical examples and identify nine opportunities for improvement. Some of these ideas are well known and strongly supported in the literature, Other ideas may be less well, understood. Our list includes the following: Cancel some of the inspections called for by an (economically) optimal static rule when starting in control (such as at the beginning of a production run and following a restoration). Inspect more frequently than called for by an optimal static rule once inspections begin, and inspect even more frequently than that when negative evidence is accumulated. Utilize evidence from previous inspections to justify either restoration or another inspection. Cancel inspections and hesitate to restore the process at the end of a production run. Consider using scheduled restoration, in which restoration is carried out regardless of the results of any inspections. implementation, limitations, and extensions are addressed.
We consider the problem of making one choice from a known number of i.i.d. alternatives. It is assumed that the distribution of the alternatives has some unknown parameter. We follow a bayesian approach to maximize th...
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We consider the problem of making one choice from a known number of i.i.d. alternatives. It is assumed that the distribution of the alternatives has some unknown parameter. We follow a bayesian approach to maximize the discounted expected value of the chosen alternative minus the costs for the observations. For the case of gamma and normal distribution we investigate the sensitivity of the solution with respect to the prior distributions. Our main objective is to derive monotonicity and continuity results for the dependence on parameters of the prior distributions. Thus we prove some sort of bayesian robustness of the model.
We consider a non-stationary bayesiandynamic decision model with general state, action and parameter spaces. It is shown that this model can be reduced to a non-Markovian (resp. Markovian) decision model with complet...
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We consider a non-stationary bayesiandynamic decision model with general state, action and parameter spaces. It is shown that this model can be reduced to a non-Markovian (resp. Markovian) decision model with completely known transition probabilities. Under rather weak convergence assumptions on the expected total rewards some general results are presented concerning the restriction on deterministic generalized Markov policies, the criteria of optimality and the existence of Bayes policies. These facts are based on the above transformations and on results of Hindererand Schäl.
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