073607ba-7ec7-414d-a715-4985e4d6d91220211228064001395naun:naunmdt@crossref.orgMDT DepositInternational Journal of Pure Mathematics2313-057110.46300/91019http://www.naun.org/cms.action?id=6985292021292021810.46300/91019.2021.8https://www.naun.org/cms.action?id=23293ImdatKaraDepartment of Industrial Engineering, Department of Finance and Banking Baskent University Baglica Campus, 06790, Etimesgut, Ankara, Turkey TurkeyHalil IbrahimKarakasDepartment of Industrial Engineering, Department of Finance and Banking Baskent University Baglica Campus, 06790, Etimesgut, Ankara, Turkey TurkeyThe Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set of positive integers. Our first formulation is used to check if a given positive integer is the Frobenius number of a given set of positive integers. The second formulation aims at finding the Frobenius number directly. The third formulation involves the residue classes with respect to the least member of the given set of positive integers, where a residue table is computed comprising all values modulo that least member, and the Frobenius number is obtained from there. Based on the same approach underlying the third formulation, we propose our fourth formulation which produces the Frobenius number directly. We demonstrate how to use our formulations with several examples. For illustrative purposes, some computa-tional analysis is also presented.122820211228202160658https://www.naun.org/main/NAUN/puremath/2021/a162019-008(2021).pdf10.46300/91019.2021.8.8https://www.naun.org/main/NAUN/puremath/2021/a162019-008(2021).pdf10.1287/moor.1040.0099Aardal K., Lenstra J.K., Hard Equality Constrained Integer Knapsacks, Mathematics of Operations Research, Vol.29, No.3, 2004, pp. 724-738. Alfonsin J.L.R., The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and its applications, Vol.30, 1995. Alfonsin J.L.R., Complexity of the Frobenius Problem, Combinatorica, Vol.16, No.11, 1996, pp. 143-147. 10.1287/moor.1090.0393Aliev I., Henk M., Integer Knapsacks: Average Behavior of the Frobenius Numbers, Mathematics of Operations Research, Vol.34, No.3, 2009, pp. 698- 705. 10.1007/s10878-011-9411-5Aliev I., Henk M., LLL-reduction for integer knapsacks, Journal of Combinatorial Optimization, Vol.24, No.4, 2012, pp. 613-626. 10.37236/1924Beihoffer D., Hendry J., Nijenhuis A., Wagon S., Faster Algorithms for Frobenius Numbers, The electronic Journal of Combinatorics, Vol.12, 2005. 10.1007/s00453-007-0162-8Böcker S., Liptak Z., A Fast and Simple Algorithm for the Money Changing Problem, Algorithmica, Vol.48, 2007, pp. 413-432. Brauer A., On a problem of partitions, Amer J. Math, Vol.64, 1942, pp. 299-312. Brauer A., Shockley J.E., On a problem of Frobenius, J. reine angew. Math, Vol.211, 1962, pp. 215-200. 10.7146/math.scand.a-12330Curtis F., On formulas for the Frobenius number of a numerical semigroup, Math Scand, Vol.67, 1990, pp. 190-192. Delgado M., Garcia-Sanchez P.A., Morais J., “NumericalSgps”, A GAP Package for Numerical Semigroups Version 1.0.1. 2015, http://www.gapsystem.org/ Einstein D., Lichtblau D., Strzebonski A., Wagon S., Frobenius numbers by Lattice point Enumeration, Integers: Electronic Journal of Combinatorial Number Theory, 2007. 10.1016/0377-2217(94)00214-2Hansen P., Ryan J., Testing integer knapsacks for feasibility, European Journal of Ope-rational Research, Vol.88, 1996, pp. 578-582. 10.1016/0166-218x(89)90019-xKrawczyk H., Paz A., The Diophantine problem of Frobenius: A close Bound, Discrete Applied Mathematics, Vol.23, 1989, pp. 289-291. Lewin M., An algorithm for a solution of a problem of Frobenius, J. reine angew. Math, Vol.276, 1975, pp. 68-82. Mohammed D.H., Generalized Frobenius numbers: geometry of upper bounds, Frobenius graphs and exact formulas for arithmetic sequences, Dissertation, Cardiff University, 2017. Navarro A.S.R., Linear Diophantine equations and applications, Dissertation, Universidad de Granada. 2015. Ong D.C., Ponomarenko V., The Frobenius Number of Geometric Sequences. Integers: Electronic Journal of Combinatorial Number Theory, Vol.8, No.1, 2008. 10.1080/0025570x.2003.11953192Owens R.W., An Algorithm to solve the Frobenius Problem, Mathematics Magazine, Vol.76, No.4, 2003, pp. 264-275. Roberts J.B., Note on Linear forms, Proc. Amer. Math. Soc., Vol.7, 1956, pp. 465-469. 10.1016/j.jsc.2007.06.002Roune B.H., Solving thousand-digit Frobenius Problem using Gröbner bases, Journal of Symbolic Computation, Vol.43, 2008, pp. 1-7. 10.1016/0166-218x(90)90006-xRyan, J., The structure of an ıntegral monoid and Integer Programming feasibility, Discrete Applied Mathematics, Vol.28, 1990, pp. 251-263. Sylvester J.J., Mathematical questions with their solutions. Educational Times, Vol.41, 1884, 21. The On - Line Encyclopedia of Integer Sequences, http://oeis.org Trimm J.E., On Frobenius numbers in three variables, Dissertation, Auburn University – Alabama. 2006. 10.1016/j.jnt.2016.05.027Tripathi A., Formulae for the Frobenius number in three variables, Journal of Number Theory, Vol.170, 2017, pp. 368-389. 10.1007/bf01876632Vizvari B., An application of an optimal behaviour of the greedy solution in number theory, Periodica Mathematica Hungarica, Vol.27, No.2, 1993, pp. 69-83.