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The user interface and performance of commercial mixed integer programming software

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Summary

Today's mixed integer programming software packages are still not sophisticated enough to be treated as black boxes. Their efficiency can be augmented considerably by a deep knowledge of their functioning and the degrees of freedom they offer to the user. This paper describes at first the choices of the underlying branch and bound method in seven representative MIP-systems and demonstrates their usage, their profitability and their catches at illustrative examples. An account follows of the procedures and administrative facilities of commercial MIP-modules as far as they are submitted to user influence. An outlook is given on future developments, which can further improve the acceptance and the performance of MIP-software. Finally, some effort assessment criteria are proposed, which help judging the amount of work ahead before and at different stages during the solution process.

Zusammenfassung

Heute verfügbare Softwarepakete der gemischt-ganzzahligen Programmierung sind noch nicht fortgeschritten genug, um als Black Boxes behandelt zu werden. Ihre Effizienz kann durch eine gründliche Kenntnis ihrer Funktionsweise und der dem Benutzer angebotenen Freiheitsgrade beträchtlich gesteigert werden. Diese Veröffentlichung beschreibt zuerst die Auswahlmöglichkeiten der zugrundeliegenden Branch und Bound Methode in sieben repräsentativen MIP-Systemen und demonstriert anhand illustrativer Beispiele deren Anwendung, deren Nutzen und deren Fußangeln. Im Anschluß werden Proceduren und administrative Funktionen in kommerziellen MIP-Modulen, soweit sie dem Benutzereinfluß unterliegen, dargestellt. Ein Ausblick auf zukünftige Entwicklungen zeigt, wie sich Akzeptanz und Leistungvon MIP-Software steigern können. Schließlich werden einige Kriterien zur Aufwandsabschätzung vor und in verschiedenen Stadien während des Lösungsprozesses vorgeschlagen.

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Abbreviations

c :

vector of objective function coefficients of continuous variables

C :

matrix of coefficients of continuous variables

d :

vector of objective function coefficients of integer variables

D :

matrix of coefficients of integer variables

d 1 :

distance betweenF as and X *0

d 2 :

distance betweenF as andF bk

d −j :

estimated functional value degradation when lowering the value ofy j by one unit

d +j :

estimated functional value degradation when rising the value ofy j by one unit

E k :

estimation of best functional value of any integer solution contained in waiting nodek

F as :

aspiration level; nodes with functional value worse thanF as are postponed

F ase :

aspiration level; nodes with estimationE k worse thanF ase are postponed

F bI :

functional value of best integer solution known so far

F bk :

best functional value of all waiting nodes

F IT0 :

functional value of integer optimum

F k :

functional value of waiting nodek

F 0 :

functional value of continuous optimum

M :

big positiv real number

p :

maximum percentage deviation ofF ITO fromF bI

skj :

successor nodej

u :

distance by whichX *0 is lower thanF bI

X *0 :

cutoff value; nodes with functional value beyondX *0 are finally discarded

x :

vector of integer variables

y :

vector of integer variables

y j :

integer variablej

y j (k):

value of integer variablej at nodek

w j :

associated weight of integer variabley j in a Special Ordered Set

¯w :

average weight of a Special Ordered Set

[⋯ ]:

greatest integer smaller than or equal to the argument

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  1. FB Ökonomie, Quantitative Methoden der BWL, Johann Wolfgang Goethe-Universität, D-6000, Frankfurt/Main 1

    H. Peeters

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Peeters, H. The user interface and performance of commercial mixed integer programming software. OR Spektrum 2, 235–249 (1981). https://doi.org/10.1007/BF01721012

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