Introduction

Solutions of the examples of optimal investment problems considered in the previous chapter depend on predicted stock rates. In general, predictions are needed considering most of the investment problems, because the final result of an investment depends on future values of various parameters. Predictions are an important part of scheduling problems, too, because the optimal schedules depend on the predicted demand, and supply. Time series models are common prediction tools. We start by investigating a well known time series model- the autoregressive moving average (ARMA) model. We will briefly consider the Artificial Neural Network (ANN) and the Bilinear models, too.

In this chapter, mainly, the financial data will be predicted. The prediction of call rates, using the same models, will be briefly considered, for comparison. We return to the call rate prediction in the next chapter, regarding the call center optimization. The traditional ARMA will be supplemented by a model of expert predictions.

Modeling economic and financial time series by ARMA has attracted the attention of many researchers in recent years [3,1,20,2,10,15]. In estimating the parameters of the ARMA models, three approaches have been used: Maximum Likelihood (ML) [18], approximate ML [11,4,6,7], and two-step procedures [5,8]. In all the cases local optimization techniques were used. In such cases, the optimization results depend on the initial values, what implies that one cannot be sure if the global maximum is found.

The global optimization is very difficult in almost all the cases. The reason is a high complexity of multi-modal optimization problems in general. It is well known [9] that optimization of real functions cannot be done in polynomial-time, unless $P=NP$. In practice, this means that we need an algorithm of exponential time to obtain the $\epsilon$-exact solution. The number of operations in exponential algorithms grows exponentially with the accuracy of solution and dimensions of the optimization problem.

A popular approach in estimating the parameters of ARMA models is Least Squares (LS). We minimize the log-sum of square residuals using ARMA models and their extensions (see [15]). In this chapter the multi-modality problems are considered using different data. The data include daily exchange rates of $/£ and DM/$ , closing rates of stocks of AT&T , Intel Co, and some Lithuanian banks, a London stock exchange index [17], and call-rates of a commercial call-center. The graphical images and comparison of the average prediction results of ARMA and the Random Walk (RW) models are presented.