Abstract
This chapter contains several case studies with an industrial background which involve mixed integer programming techniques. We discuss real-world problems of increasing size and complexity. The first group of case studies considers a contract allocation problem, metal ingot production, and a project planning problem. This follows a more extensive scheduling problem in the carton industry.
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Notes
- 1.
In Sect. 10.4.5 we present a reformulated version of the model which is capable of handling an arbitrary number of modes and changes between them.
- 2.
Here we used the special case information that M i = N P.
- 3.
See also Patterson et al. (1989,[438]) for a general discussion of such problems.
- 4.
The term client is used to refer to one who asks for support from a mathematical consultant. The term customer is used to denote one who purchases goods from the client.
- 5.
The values for N P ≤ 4 have been computed by Colombani & Heipcke (1997,[127]) in less than 2 min using the constraint programming software package SchedEns.
- 6.
Take the following example: The inequality 3.2α 1 + 3.9α 2 ≤ 8 with integer variables α 1 and α 2 has the valid inequality 4α 1 + 4α 2 ≤ 8 which is equivalent to α 1 + α 2 ≤ 2. If we draw the feasible regions associated with the original constraint and α 1 + α 2 ≤ 2 we see that the latter gives a smaller feasible region and thus is the tighter constraint.
- 7.
ModelCuts are problem-specific cuts, i.e., valid inequalities that will cut-off unwanted fractional values of binary or integer variables and that are otherwise redundant constraints. They are added directly to the model formulation. In contrast, in B&C cuts are added dynamically in the tree to cut-off unwanted fractional variables.
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Kallrath, J. (2021). How Optimization Is Used in Practice: Case Studies in Integer Programming. In: Business Optimization Using Mathematical Programming. International Series in Operations Research & Management Science, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-73237-0_10
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