Mathematical models linear programming

Mathematical modeling, using digital computers, aids in performing a systems-type analysis for either the entire system or parts of it. By means of integer or linear-programming techniques, optimum systems can be identified. The dynamic performance of these can then be determined by simulation techniques. 

Many process engineers think of linear programming (L.P.) as a sophisticated mathematical tool, which is best applied by a few specialists extremely well grounded in theory. This is certainly true for your company s central linear program. The layman does not write a linear program, he only provides input that will model the process in which he is interested. 

The SNP optimizer is based on (mixed-integer) linear programming (MILP) techniques. For a general introduction into MILP we refer to [11], An SAP APO user has no access to the mathematical MILP model. Instead, the modeling is done in notions of master data of example products, recipes, resources and transportation lanes. Each master data object corresponds to a set of constraints in the mathematical model used in the optimizer. For example, the definition of a location-product in combination with the bucket definition is translated into inventory balance constraints for describing the development of the stock level over time. Additional location-product properties have further influence on the mathematical model, e.g., whether there is a maximum stock-level for a product or whether it has a finite shelf-life. For further information on the master data expressiveness of SAP SCM we refer to [9],... 

A large number of optimization models have continuous and integer variables which appear linearly, and hence separably, in the objective function and constraints. These mathematical models are denoted as Mixed-Integer Linear Programming MILP problems. In many applications of MILP models the integer variables are 0 - 1 variables (i.e., binary variables), and in this chapter we will focus on this sub-class of MILP problems. 

The computer program for the material balance contains several parts. First, a description ofeach item of equipment in terms of the input and output flows and the stream conditions. Quite complicated mathematical models may be required in order to relate the input and output conditions (i.e. performance) of complex units. It is necessary to specify the order in which the equipment models will be solved, simple equipment such as mixers are dealt with initially. This is followed by the actual solution of the equations. The ordering may result in each equation having only one unknown and iteration becomes unnecessary. It may be necessary to solve sets of linear equations, or if the equations are non-linear a suitable algorithm applying some form of numerical iteration is required. 

Fries and Marathe (1981) used mathematical models to determine the optimal variable-sized multiple-block appointment system. Calichman (1990) used a linear programming model to btiltmce the bed demand among various surgical services. 

Fouier, R. (1983), Modeling Languages versus Matrix Generators for Linear Programming, ACM Transactions on Mathematical Software, Vol. 9, pp. 143-183. 

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