An Optimal Step-Size Policy

Once an acceptable value for the step-size has been determined, we can continue and with only one additional evaluation of the objective function, we can obtain the optimal step-size that should be used along the direction suggested by the Gauss-Newton method. 

Essentially we need to perform a simple line search along the direction of Ak(j+I). The simplest way to do this is by approximating the objective function by a quadratic along this direction. Namely, 

Having estimated P) and P2, we can proceed and obtain the optimum step-size by using the stationary criterion 

The above expression for the optimal step-size is used in the calculation of the next estimate of the parameters to be used in the next iteration of the Gauss-Newton method, 

If we wish to avoid the additional objective function evaluation at p=qa/2, we can use the extra information that is available at p=0. This approach is preferable for differential equation models where evaluation of the objective function requires the integration of the state equations. It is presented later in Section 8.7 where we discuss the implementation of Gauss-Newton method for ODE models. 

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The main difference with differential equation systems is that every evaluation of the objective function requires the integration of the state equations, In this section we present an optimal step size policy proposed by Kalogerakis and Luus (1983b) which uses information only at g=0 (i.e., at k ) and at p=pa (i.e., at... 

In this section we first present an efficient step-size policy for differential equation systems and we present two approaches to increase the region of convergence of the Gauss-Newton method. One through the use of the Information Index and the other by using a two-step procedure that involves direct search optimization. 

In Table 15-5, we show the impact of different quantity flexibility contracts on profitability for the music supply chain when demand is normally distributed, with a mean of jx = 1,000 and a standard deviation of a- = 300 (see worksheet Examplel5-4 in spreadsheet Chapterl5-examples). We assume a wholesale price of c = 5 and a retail price of p = 10. All contracts considered are such that a = (3. The results in Table 15-5 are built in two steps. We first fix a and j8 (say a = j8 = 0.2). The next step is to identify the optimal order size for the retailer. This is done using Excel by selecting an order size that maximizes expected retailer profits given a and j8. For example, when a = j8 = 0.05 and c = 5, retailer profits are maximized for an order size of 0 = 1,017. For this order size, we obtain a supplier commitment to deliver up to g = (1 + 0 05) X 1,017 = 1,068 and a retailer commitment to buy at least q = ( - 0.05) X 1,017 = 966 discs. In our analysis, we assume that the supplier produces Q = 1,068 discs and sends the precise number (between 966 and 1,068) demanded by the retailer. Such a policy results in retailer profits of 4,038 and supplier profits of 4,006. 

For single separation duty, Diwekar et al. (1989) considered the multiperiod optimisation problem and for each individual mixture selected the column size (number of plates) and the optimal amounts of each fraction by maximising a profit function, with a predefined conventional reflux policy. For multicomponent mixtures, both single and multiple product options were considered. The authors used a simple model with the assumptions of equimolal overflow, constant relative volatility and negligible column holdup, then applied an extended shortcut method commonly used for continuous distillation and based on the assumption that the batch distillation column can be considered as a continuous column with changing feed (see Type II model in Chapter 4). In other words, the bottom product of one time step forms the feed of the next time step. The pseudo-continuous distillation model thus obtained was then solved using a modified Fenske-Underwood-Gilliland method (see Type II model in Chapter 4) with no plate-to-plate calculations. The... 

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