Abstract
The Least Absolute Deviation (LAD) method is the one of many methods used to estimate transition probability matrix of Markov Chain. It can be formu lated as a Linear Programming Problem (LP) and solved using its regular state of the art software. However, when the Markov Chain has a large number of states and historical state probabilities data, the corresponding LP size can be very large reaching c omputer hardware/software limitation. The aim of this study is to apply the Column Generation (CG) technique t o solve this large scale LP and to evaluate its extension b eyond direct hardware/software capabilities. In thi s study, the sample state probabilities data were simulated stat istically and two methods were used to solve the pr oblem. The first method was using ‘linprog’ function in MATLAB to solve the related LP that all decision vari ables were considered simultaneously while the others was the CG method expected to require a much less percentage of all variables. As result effectivenes s, both methods solved all test problems resulting equal LADs each. The CG method required more average time. Nevertheless, less than 30% of decision variables were considered in the CG method. The lesser percentages were found as the problem size grew. Moreover, larger size problems beyond direct use of software were solved using the proposed approach.
Highlights
Lin (2011) applied Markov chain to establish an adaptive production procurement system.Markov chain is widely used as model in many areas i.e., economics, marketing, capital theory, industrial structure, demography and social science which shown in the review of Dent and Ballintine (1971)
Markov model is used to describe the chance of an event occurring in the period when we know the chance of that event in the present period and transition probability matrix in the following manner: Sendi (2002) used Markov chain to study the changes of chronic diseases
The trials start on generating square transition probability matrix P order n and n×S matrix of data according to the Equation 1 Subsequently, prepare the matrices and vectors according to the requirements of the ‘linprog’ function in MATLAB
Summary
Lin (2011) applied Markov chain to establish an adaptive production procurement system. There are many researchers who proposed the methods to estimate probability statistics in transition matrices. The Restricted Least Squared (RLS) method is ignored because it is equivalent to solve the Quadratic Programming Problem (QPP). The MAD method called Least Absolute Deviation (LAD) under constraints in Equation 2 and 3 is used to estimate the elements of the transition matrix when the approximation of π (k) was known. The above model (A) consisted of Equation 4-6 has n linear constraints, n2 of non-negativity constraints and decision variables.
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