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<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">Thx :-)<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">This was my suggestion too. So, I assume that multiplying two binary variables in objective function not is legal
in COIN-OR MILP model.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">And using <o:p></o:p></span></p>
<p class="MsoNormal" style="text-indent:35.4pt"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">+ 12.3 * x<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">in the objective function should be OK, I assume.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:green"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:green"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:green">Best, Bjorn<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
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<p class="MsoNormal"><b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext">From:</span></b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext"> Tobias Stengel [mailto:Tobias.Stengel@locom.de]
<br>
<b>Sent:</b> onsdag 4. oktober 2017 10:37<br>
<b>To:</b> Bjørn Sigurd Johansen (Spider Solutions AS) <bjorn.sigurd.johansen@spidersolutions.no>; cbc@list.coin-or.org<br>
<b>Subject:</b> AW: Multiplying two binary variables in the objective function<o:p></o:p></span></p>
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<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">Dear Bjorn,<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">observation: B1 * B2 == 1 iff B1 == 1 and B2 == 1 and 0 otherwise. So you can use a binary helper variable x = B1 <and> B2.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">This can be modeled e.g. by<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">x >= B1 + B2 – 1<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">x <= B1<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">x <= B2<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">x \in {0, 1}<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">then use x instead of B1*B2 in the objective.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">Tobias<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
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<p class="MsoNormal"><b><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext">Von:</span></b><span lang="DE" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext"> Cbc [<a href="mailto:cbc-bounces@coin-or.org">mailto:cbc-bounces@coin-or.org</a>]
<b>Im Auftrag von </b>Bjørn Sigurd Johansen (Spider Solutions AS)<br>
<b>Gesendet:</b> Mittwoch, 4. Oktober 2017 10:26<br>
<b>An:</b> John Forrest <<a href="mailto:john.forrest@fastercoin.com">john.forrest@fastercoin.com</a>>;
<a href="mailto:cbc@list.coin-or.org">cbc@list.coin-or.org</a><br>
<b>Betreff:</b> [Cbc] Multiplying two binary variables in the objective function<o:p></o:p></span></p>
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<p class="MsoNormal"><span lang="DE"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">Dear CBC list / John,<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">Is it possible to multiply two binary variables in the objective function scaled with the same factor?<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">E.g.
<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">MINIMIZE Obj:<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">+12.3 B1 * 12.3 B2<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">or<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">+12.3 B1 * + 12.3 B2<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">If binary variables are set through some sort of branching algorithm, perhaps it is possible? But if the binary
variables are set/estimated through some other mechanism, it might not be possible?<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">If possible, what is the exact syntax for doing this?<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">If not possible, I guess the option is to introduce a third binary B3, and set up the following constraints:<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">B3 <= B1<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">B3 <= B2<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">B3 >= B1 + B2 - 1<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">And then in the objective function use:<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">+12.3 B3<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">Correct / best option?<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">Best regards,<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext;mso-fareast-language:EN-US">Bjorn
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext"><o:p> </o:p></span></p>
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