Integer Programming Methods

An early application of this type of approach for reaction removal was carried out by Petzold and Zhu (1999) for several ignition problems. Although this method has its drawbacks, we discuss it in some detail here since it provides a useful UlustraticHi of how optimisation approaches are applied in practice. The rate of change of species mass fractions y, is given by 

The optimisation problem described by (7.22) is then solved using sequential quadratic programming, and in order to obtain the reduced model, the values of dj are rounded to 0 or 1. 

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Shih L (1997) Planning of fuel coal imports using a mixed integer programming method 51 243-249... 

Operations sire bound to functional units, and values are bound to registers, using a zero-one integer programming method, smd attempting to minimize the amount of interconnection. This method can silso perform tradeoffs between different-speed functionsil units that perform the same operation. 

Balinski, M. (1965). Integer programming methods, uses, computation. Management Science, 12, 254-313. 

The modern branch-and-bound algorithms for MILPs use branch-and-bound with integer relaxation, i.e., the branch-and-bound algorithm performs a search on the integer components while lower bounds are computed from the integer relaxation of the MILP by linear programming methods. The upper bound is taken from the best integer solution found prior to the actual node. 

Branch and bound (BB) is a class of methods for linear and nonlinear mixed-integer programming. If carried to completion, it is guaranteed to find an optimal solution to linear and convex nonlinear problems. It is the most popular approach and is currently used in virtually all commercial MILP software (see Chapter 7). 

Relaxation of hard constraints is critical for optimization-based planning models used in industry practice with more than even 100,000 constraints and specifically for hard integer programming problems (Fisher 2004). Hard constraints set hard minimum and maximum boundaries for decision variables that have to be fulfilled. It may occur that no solution exists fitting all constraints at the same time. Planners have difficulties to identify manually constraints leading to infeasibility. Value chain planning model infeasibility is mainly caused by volume-related constraints of material flows e.g. by bounding sales quantities, inventories, transportation quantities, production and procurement quantities. Examples in literature for relaxation methods to e.g. transportation problems is presented by Klose/Lidke (2005)... 

Fisher ML (2004) The Lagrangian Relaxation Method for Solving Integer Programming Problems. Management Science 50 (12) 1861-1871... 

M. L. Fisher. The Lagrangian relaxation method for solving integer programming problems. ManagementSci., 27 1,1981. 

J. A. Tomlin. An improved branch and bound method for integer programming. Oper. Res., 19 1070, 1971. 

McKinney, D. C., and Lin, M. D. (1995). "Approximate mixed-integer nonlinear programming methods for optimal aquifer remediation design." Water Resour. Res., 31(3), 731-740. 

In the steady state case example, because the data were sparse and there were more unknowns than constraining equations, an additional restriction to the linear programming method was needed. This was done by utilizing a mixed integer-programming model such as in a stepwise multiple regression solution. In the multiple regression method, the identification problem was formulated as follows ... 

The most widely used method for solving both integer and MIP problems is the branch-and-bound algorithm. Most commercial computer codes for solving integer programming problems use this approach. The method performs an efficient enumeration of a small fraction of the possible feasible integer solutions to locate the optimum. 

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