The problem with integer programming

Abstract

Integer Programming (IP), also known as Discrete Optimisation, is a way of modelling a very wide range of problems involving indivisibilities (eg. Yes/No investment decisions) and non-convexities (eg. economies of scale and fixed cost allocation). Such problems arise in many areas which will be mentioned. However IP demands ingenuity in both building models and solving them. A lot is still not properly understood. This paper investigates the question. 'Is IP like Linear Programming (LP)?' The mathematical and economic properties of IP will be contrasted with LP. It will be suggested that the mathematics and economics of IP are still not properly understood. Many of the results which apply to LP do not apply to IP. It will be asserted that this lack of understanding reveals inadequacies in both the mathematics and economics.It will be shown that in many (but not all) situations the rounding of an LP solution does not produce satisfactory (feasible or optimal) solutions to an IP. A topical example will be given of political apportionment leading to the Alabama Paradox.IP is essentially concerned with the intersection of two structures: (i) Linear inequalities giving rise to polytopes.(ii) Lattices of integer points.Mathematical and computational methods and results exist for both these structures on their own. However mixing them is like mixing oil and water. Problems arise in both the computation of optimal solutions and the economic interpretation of the results. It will be suggested that the appropriate mathematical structure is an integer monoid. This structure will be explained. Connected with this structure are Chvátal functions which are made up of non-negative combinations of the arguments together with the (nested) integer round-up operation. The use of these functions, in place of the conventional non-negative linear combinations of LP allows one to capture many of the classical LP, results eg. The Weyl-Minkowski theorems, 'pricing' of indivisible resources and the closing of the 'duality gap' leading to the optimal IP solution. It will be shown how optimal Chvátal functions can be interpretated as the (non-marginal) valuation of indivisible resources. However a major problem remains as to how to represent them in a transparent and compact way.

Implementasi Integer Programming dengan Algoritma Branch and Bound Menggunakan QM for Windows dalam Memaksimalkan Keuntungan di PT XYZ

“…Sesuai dengan namanya metode ini membatasi penyelesaian optimal yang akan menghasilkan bilangan pecahan dengan cara membuat cabang atas atau bawah bagi masing-masing variabel keputusan yang bernilai bulat sehingga setiap oembatasan akan menghasilkan cabang baru [2]. Pada penelitian ini dengan menggunakan algoritma Branch and Bound efektif dan sederhana untuk menyelesaikan model program integer [3], [4], [5], [10].…”