Abstract
In this chapter we shall look at ways in which problems may be formulated using a series of devices mainly involving integer variables for counting indivisible entities (people, living animals, air planes, etc.), and particularly binary variables. Among other things, binary variables can be used to model
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1.
states,
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logical conditions and logical expressions, and
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special nonlinear terms and expressions
In particular, we shall concentrate on the use of binary variables to model simple nonlinear features. Such features can be handled because binary variables allow us to model logical conditions. These approaches will then be illustrated by examples and later many of the approaches will appear in a series of short case studies.
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Notes
- 1.
Note that logical variables can only take the values true and false, or T and F, for short.
- 2.
We use the symbol ⇔ to mean “is equivalent to.”
- 3.
Sometimes, these equivalent MILP formulations are also called linearization. However, we avoid this term in this context, as we feel it is more appropriate to Taylor series expansions stopping after the linear term.
- 4.
It is not easy to define quantitatively what is meant by very large compared to other coefficients (e.g., is 1000 very large compared to 1?), without experience or knowledge of a solver’s capabilities. Thus we prefer to give an example: If all values of input data and expected values of the variables are between 1 and 10, M = 106 is probably very large compared to other coefficients — it would be a good idea to try smaller values, e.g., M = 100, and see how the solver performs with that choice.
- 5.
In general to interpolate means to compute intermediate values of a quantity between a series of given values. But in our case, since the variables λ i are elements in an SOS2, we interpolate between two adjacent values.
- 6.
SCICONIC is an example.
References
Balas, E.: Disjunctive Programming. Springer Nature, Cham (2018)
Beale, E.M.L., Daniel, R.C.: Chains of Linked Ordered Sets. Tech. Rep., Scicon, Wavendon Tower, Wavendon, Milton Keynes (1980)
Beale, E.M.L., Forrest, J.J.H.: Global optimization using special ordered sets. Math. Program. 10, 52–69 (1976)
Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In: J. Lawrence (ed.) OR ’69: Proceedings of the 5th International Conference on Operational Research, pp. 447–454. Tavistock, London (1970)
Belotti, P., Bonami, P., Fischetti, M., Lodi, A., Monaci, M., Nogales-Gómez, A., Salvagnin, D.: On handling indicator constraints in mixed integer programming. Comp. Opt. Appl. 65(3), 545–566 (2016)
Brearley, A.L., Mitra, G., Williams, H.P.: Analysis of mathematical programming problems prior to applying the simplex algorithm. Math. Program. 8, 54–83 (1975)
Grossmann, I.E., Trespalacios, F.: Systematic modeling of discrete-continuous optimization models through generalized disjunctive programming. AIChE J. 59, 3276–3295 (2013)
Hummeltenberg, W.H.: Implementations of special ordered sets in MP software. Eur. J. Oper. Res. 17, 1–15 (1984)
Jeroslow, R.G., Lowe, J.K.: Modelling with integer variables. Math. Program. Study 22, 167–184 (1984)
Jeroslow, R.G., Lowe, J.K.: Experimental results with the new techniques for integer programming formulations. J. Oper. Res. Soc. 36, 393–403 (1985)
McKinnon, K.I.M., Williams, H.P.: Constructing integer programming models by the predicate calculus. Ann. Oper. Res. 21, 227–246 (1989)
Miller, C.E.: The simplex method for local separable programming. In: Graves, R.L., Wolfe, P.L. (eds.) Recent Advances in Mathematical Programming, pp. 311–317. McGraw-Hill, London (1963)
Neumann, K., Morlock, M.: Operations Research. Carl Hanser, München, Wien (1993)
Rebennack, S., Kallrath, J.: Continuous piecewise linear delta-approximations for univariate functions: computing minimal breakpoint systems. J. Optim. Theor. Appl. 167, 617–643 (2015)
Vielma, J.P., Nemhauser, G.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. Ser. A 128, 49–72 (2011)
Wilson, J.M.: Generating cuts in integer programming with families of special ordered sets. Eur. J. Oper. Res. 46, 101–108 (1990)
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Kallrath, J. (2021). Modeling Structures Using Mixed 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_6
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