Abstract
In this chapter, we provide reflections on the material presented in this book. The chapter ends with the authors’ view of future developments and suggests some conclusions.
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Notes
- 1.
Not to be confused with standard “user interfaces” like reports in a browser, in Tableau, or Xpress Insight.
- 2.
It is sometimes possible to solve very large problems. But there are small problem instances leading to runtime issues.
- 3.
A classification into the class of nice hard and hopelessly hard would perhaps be more appropriate, because the computing time for the allegedly mild problems solvable in polynomial time can become quite large depending on the coefficients and powers occurring in this polynomial context. In addition, one should bear in mind that complexity theory deals with the determination of the runtime behavior as a function of the size of the problem. However, there are many problems, including those classified as \( \mathcal {N}\mathcal {P}\)-complete or \(\mathcal {N}\mathcal {P}\) difficult, which occur in practice in small instances and can easily be solved with B&B or B&C methods. If the problem is hopelessly serious, the situation is not hopeless, depending on the size of the problem.
- 4.
Prefactors or lower order polynomials are not considered here.
- 5.
However, it should again be noted here that complexity theory makes statements about the scalability of problems with regard to their solution behavior. In practice, the Simplex algorithm often performs better than the interior-point methods.
- 6.
Furthermore, there are the \(\mathcal {N}\mathcal {P}\)-hard problems. Every problem to which we can map or transform an NP-complete problem in polynomial time and which, regardless of whether or not this problem belongs to the class \( \mathcal {N}\mathcal {P}\), has the property that it cannot be solved in polynomial time, unless \(\mathcal {P}\) \(=\mathcal {N}\mathcal {P}\), means \(\mathcal {N}\mathcal {P}\)-hard. The \( \mathcal {N}\mathcal {P}\)-hard problems are therefore at least as hard as the \(\mathcal {N}\mathcal {P}\)-complete problems.
- 7.
Usually, optimization problems are NP-hard (objective function); and the underlying decision version is NP-complete.
- 8.
The webpage www.PlanNow.ai provides further details.
- 9.
Physician of antiquity, who lived from about 460 to 377 bc and laid the base of ancient Greek medicine. He formulated the Hippocratic Oath, which contains moral commandments still valid and binding today for a true physician. For natural scientists the equivalent of the Hippocratic oath would probably be “Measured values must not be faked”; in the case of persons with decision making responsibility, especially managers and politicians, perhaps the moral commandment “name all goals and intentions” openly and completely would make sense.
- 10.
In the truest sense of the word rest, such doctors will not worry about the end of the working day, weekends or other things that are often anchored in company agreements or labor law.
References
Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88, 411–424 (2000)
Bliek, C., Spellucci, P., Vicente, L., Neumaier, A., Granvilliers, L., Monfroy, E., Benhamouand, F., Huens, E., Hentenryck, P.V., Sam-Haroud, D., Faltings, B.: Algorithms for solving nonlinear constrained and optimization problems: The State of the Art. Report of the European Community funded project COCONUT, Mathematisches Institut der Universität Wien, http://www.mat.univie.ac.at/~neum/glopt/coconut/StArt.html (2001)
Carœ, C.C., Schultz, R.: Dual decomposition in stochastic integer programming. Oper. Res. Lett. 24, 37–45 (1999)
Eichner, T., Pfingsten, A., Wagener, A.: Strategisches Abstimmungsverhalten bei Verwendung der Hare-Regel. zfbv 48, 466–473 (1996)
Engell, S., Märkert, A., Sand, G., Schultz, R., Schulz, C.: Online scheduling of multiproduct batch plants under uncertainty. In: Online Optimization of Large Scale Systems, pp. 649–676. Springer, Berlin (2001)
Garey, M.R., Johnson, D.S.: Computers and intractability - a guide to the theory of NP completeness, 22nd edn. W. H. Freeman and Company, New York (2000)
Gollmer, R., Nowak, M.P., Römisch, W., Schultz, R.: Unit commitment in power generation - a basic model and some extensions. Ann. Oper. Res. 96, 167–189 (2000)
Harding, S.T., Floudas, C.A.: Locating heterogeneous and reactive azeotropes. Ind. Eng. Chem. Res. 39, 1576–1595 (2000)
Harjunkoski, I., Jain, V., Grossmann, I.E.: Hybrid mixed-integer/constrained logic programming strategies for solving scheduling and combinatorial optimization problems. Comput. Chem. Eng. 24, 337–343 (2000)
Heipcke, S.: Comparing constraint programming and mathematical programming approaches to discrete optimisation. The change problem. J. Oper. Res. Soc. 50(6), 581–595 (1999)
Hooker, J.: Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction. Wiley, Chichester (2000)
Ierapetriou, M.G., Floudas, C.A.: Effective continuous-time formulation for short-term scheduling. 1. Multipurpose batch processes. Ind. Eng. Chem. Res. 37, 4341–4359 (1998)
Ierapetriou, M.G., Floudas, C.A.: Effective continuous-time formulation for short-term scheduling. 2. Continuous and semicontinuous processes. Ind. Eng. Chem. Res. 37, 4360–4374 (1998)
Ierapetriou, M.G., Hene, T.S., Floudas, C.A.: Continuous time formulation for short-term scheduling with multiple intermediate due dates. Ind. Eng. Chem. Res. 38, 3446–3461 (1999)
Jain, V., Grossmann, I.E.: Algorithms for hybrid MILP/CP models for a class of optimization problems. IFORMS J. Comput. 13, 258–276 (2001)
Kallrath, J.: Mixed-integer nonlinear programming applications. In: Ciriani, T.A., Gliozzi, S., Johnson, E.L., Tadei, R. (eds.) Operational Research in Industry, pp. 42–76. Macmillan, Houndmills, Basingstoke (1999)
Kallrath, J.: Combined strategic, design and operative planning - two success stories in MILP and MINLP. In: Bulatov, V., Baturin, V. (eds.) Proceedings of 12th Baikal International Conference: Optimization Methods and Their Applications, pp. 123–128. Institute of System Dynamics and Control Theory, Irkutsk (2001)
Kallrath, J.: Combined strategic and operational planning - an MILP success story in chemical industry. OR Spectrum 24(3), 315–341 (2002)
Kallrath, J.: Planning and scheduling in the process industry. OR Spectrum 24(3), 219–250 (2002)
Kallrath, J.: Combined strategic design and operative planning in the process industry. Comput. Chem. Eng. 33, 1983–1993 (2009)
Kallrath, J.: Polylithic modeling and solution approaches using algebraic modeling systems. Optim. Lett. 5, 453–466 (2011). https://doi.org/10.1007/s11590-011-0320-4
Kallrath, J., Schreieck, A.: Discrete optimization and real world problems. In: Hertzberger, B., Serazzi, G. (eds.) High-Performance Computing and Networking. Lecture Notes in Computer Science, vol. 919, pp. 351–359. Springer, Berlin, Heidelberg, New York (1995)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Khachian, L.G.: A polynomial algorithm in linear programming. Sov. Math. Dokl. 20, 191–194 (1979)
Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Shisha, O. (ed.) Inequalities III, pp. 159–175. Academic Press, New York (1972)
Klosterhalfen, S.T., Kallrath, J., Frey, M.M., Schreieck, A., Blackburn, R., Buchmann, J., Weidner, F.: Creating cost transparency to support strategic planning in complex chemical value chains. Eur. J. Oper. Res. 279, 605–619 (2019)
Kurschl, W., Pimminger, S., Wagner, S., Heinzelreiter, J.: Concepts and requirements for a cloud-based optimization service. In: 2014 Asia-Pacific Conference on Computer Aided System Engineering (APCASE), pp. 9–18 (2014)
Lin, X., Floudas, C.A.: Design, synthesis and scheduling of multipurpose batch plants via an effective continuous-time formulation. Comput. Chem. Eng. 25, 665–674 (2001)
Lin, X., Floudas, C.A., Modi, S., Juhasz, N.M.: Continuous-time optimization approach for medium-range production scheduling of a multiproduct batch plant. Ind. Eng. Chem. Res. 41, 3884–3906 (2002)
Sand, G., Engell, S., Märkert, A., Schultz, R., Schulz, C.: Approximation of an ideal online scheduler for a multiproduct batch plant. Comput. Chem. Eng. 24, 361–367 (2000)
Schultz, R.: On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Math. Program. 70, 73–89 (1995)
Timpe, C.: Solving mixed planning & scheduling problems with mixed branch & bound and constraint programming. OR Spectrum 24, 431–448 (2002)
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Kallrath, J. (2021). Concluding Remarks and Outlook. 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_17
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