CEE Oct-Dec 2012

added to the left-hand side of the equations in Equation I. So for any given estimate, if the equality cannot be satisfied for one of the oxides, the deviation term will have a value as depicted below. m i=1 aiJ x; + s i = b i J I . . n [1a] If there was a perfect solution to the system that sol ution will be such that all deviations terms si, are zero. Because such a solution is not possible , the best fit solution is sought. The best fit solution solves the equation system of Equation I a and Equation 2 with the least amount of "st. A simple sum of the deviation terms to define the least deviation is not correct because the terms that have negative value and positive value would cancel each other. To find the best fit, there are two main methods for calculating the total deviation: one is the sum of the squares of the deviations and the other is the sum of the absolute values of the deviations. For the problem at hand, minimizing the sum of the absolute deviations is chosen. The main reason for minimizing the sum of absolute value of the deviations is that it is less susceptible to an outlier (Hill and Holland, 1977, Portnoy and Koenker, 1997). The Heterogeneous nature of the blended cement and different accuracies in the measurement of the oxides, may cause one of the equations to be an outlier. When the sum of the square of the deviations is minimized, the outlier equation carries and unequal weight, resulting in a non robust estimate. Minimization of the sum of the absolute deviations can be modelled as a linear optimization programme (Kiountouzi s 1973). A linear optimization problem can be solved by a spreadsheet software with a solver capability like Microsoft Excel. The linear optimization problem can be defined as such: Minimize [Function 0] deviations Satisfying Sum of all absolute value of [Equation 1 a] Amount of oxide in ingredient • ratio of ingredient + deviation = amount of oxide in the blended cement. [Equation 2 a] All ingredient ratios add up to I [Equation 2 b] All ingredient ratios are non- negative Function 0, which is called the objective function in linear programming literature, is the value that is minimized . Equations 1a, 2a and 2b are called constraints of the model. If the objective function and all the constraints can be written as linear functions, the model is called linear programming. In the model given above, absolute value function is not a linear function. Transforming the model by substituting two non-negative deviation terms in place of the deviation term, results in a fully linear model. One of the substituted deviation terms, Sj 1, is added to, while the other terms, si2, is subtracted from the left-hand side of the Equation 1 a. The transformed constraint written as follows. [1 b] The objective function of the transformed model becomes a summation of two deviation terms si1andsi2· without any need for the absolute value of function. As a result the transformed model shown below in Equations 3 and 4 becomes a linear optimization problem as follows: Minimize Subject to: Where m L i=l a,, x,+sil- s,:=b, Vj= l. .n ' "' X= I L ,,l I x,~ 0 Vi= l. .m si/ . s, 1 ~ 0 Vj= l. .n aii: percentage of oxide j in ingredient "i", bi : percentage of oxide "j" in the blend , Xj : Unknown ratio ingredient "i" in the blend , si 1: positive deviation term of for oxide "j", SJ2: negative deviation term for oxide "j". [3] [4] This new model returns the same result and same objective value as the model before the transformation. The transformed model is equivalent because for any oxide "j" at least one of the deviation terms (sj1, sj 2) is zero in the optimal solution. Courtesy: Cement and Concrete World, March, April, 2012, Pp72-77. 15