Continuous-Review under Uncertainty

Uncertainty in demand over lead time in a (Q, R) inventory system. 

Quoting the renowned statistician G. E. P. Box, AU models are wrong, but some are useful.  

Some time-honored texts on production and inventory management—e.g., Hax and Candea (1984) and Silver et al. (1998)—include explicit stockout costs in the annual cost expression. Our approach, however, will be to consider a service constraint in finding the (Q, R) solution that minimizes expression (3.12). This is consistent with our earlier discussion of the newsvendor model. 

Note that this model implies that expected inventory is Q/2 + SS. Technically, however, we should be careful about how demand is handled in the event of a stockout. If demand is backlogged, then expected inventory at any point in time is actually 0/2 + SS + E[BO], where E[BO] is the expected number of backorders. Silver et al. (1998, p. 258), however, indicate that since E[BO] is typically assumed to be small relativ e to inventory, it is reasonable to use 0/2 + SS for expected inventory. If demand is lost in the event of a stockout, then expected inventory is exactly 0/2 + SS. 

an important issue in our formulation will be how we specify the service constraint on our optimization problem. From our earlier discussion of in-stock probability and fill rate, recall that the probability distribution of demand figured prominently in measuring the service level. The most general case is the one in which both demand per unit time period (where the time period is typically specified as days or weeks) and replenishment lead time (correspondingly expressed in days or weeks) are random variables. Let us, therefore, assume that the lead time L follows a normal distribution with mean Pl variance al. Further, let us assume that the distribution of the demand also follows a normal distribution, such that its mean and variance, Po Od, are expressed in time units (i.e., demand per unit time) that are consistent with the time units used to express lead time L (e.g., days or weeks). 

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Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

The realization of the need and importance of petrochemical planning has inspired a great deal of research in order to devise different models to account for the overall system optimization. Optimization models include continuous and mixed-integer programming under deterministic or parameter uncertainty considerations. Related literature is reviewed at a later stage in this book, based on the chapter topic. 

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