General Goal Programming Model

A general multiple criteria mathematical programming (MCMP) problem is given as follows  

S is called the decision space and Y is called the criteria or objective space in MCMP. 

Consider the general MCMP problem given in Equation 6.21. The assumption that there exists an optimal soluhon to the MCMP problem involving multiple criteria implies the existence of some preference ordering of the criteria by the DM. The goal programming (GP) formulation of the MCMP problem requires the DM to specify an acceptable level of achievement b for each criterion f, and specify a weight W (ordinal or cardinal) to be associated with the deviation between f, and bj Thus, the GP model of an MCMP problem becomes  

Equation 6.22 represents the objective function of the GP model, which minimizes the weighted sum of the deviational variables. The system of equations 

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With these five equations (Eqs. 23-42 to 23-46), two of them partial differential equations, the limits of the analytical approach and the goals of this book are clearly exceeded. However, at this point we take the occasion to look at how such equations are solved numerically. User-friendly computer programs, such as MAS AS (Modeling of Anthropogenic Substances in Aquatic Systems, Ulrich et al., 1995) or AQUASIM (Reichert, 1994), or just a general mathematical tool like MATLAB and MATHE-MATICA, can be used to solve these equations for arbitrary constant or variable parameters and boundary conditions. 

For both CPM and PERT, the overall project is viewed as a series of activities or operations performed in an optimum sequence to reach a desired objective. Each activity is considered as having a beginning and an end so that the overall project consists of a series of these events. The general technique, then, is to develop a mathematical model to give the best program or interrelated series of events to achieve a desired goal. The major difference in concept between CPM and PERT is that involved in estimating the time duration of activities. Thus, CPM may be relatively specific on time items, while PERT includes measures of the uncertainties involved. 

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