2

$a=k.B$, where $a$, $B$ (binary), $k$ all are scaler variables.

When $B=0$, $a=0$; and when $B=1$, $a=k$.

$B$ is binary, i.e., $0$ or $1$.

$k$ is $0$ or positive integer, i.e., $0, 1, 2, 3, 4...$

How to decompose it in Linear Programming so that two variables are not in multiplication format?

Is it even possible?

As the question is on hold and I got how to do it, I'm posting it here in the question itself. The link is given in the comment of this question by Erwin Kalvelagen.

$0 \le a \le k_{max}$

$a \le k_{max}.B$

$a \le k$

$a \ge k - k_{max}(1-B)$

Dr.PB
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    Are $a$, $k$, $B$ scalars, vectors, or matrices? Are they continuous variables or integer variables (or some of both)? – Brian Borchers Mar 16 '18 at 18:27
  • Why do you think this issue has something in common with Linear Programming ? On which basis (no intended pun) ? 2) What are the dimensions of $a,B,k$ ? Are $a$ and $k$ row vectors ?
    • – Jean Marie Mar 16 '18 at 18:37
  • all scalers. a, k are positive integer (eg: a=k=4), B is binary (eg: B=1 or B=0) – Dr.PB Mar 16 '18 at 18:52
  • Thus you want to express ''$a=k$ OR $a=0$, meaning that you have to implement an "or" condition, whereas Linear Programmign uses a set of conditions connected by (multiple) "and"s. Have a look at (https://cs.stackexchange.com/q/12102) – Jean Marie Mar 16 '18 at 21:45
  • @JeanMarie that's a helpful link, but the fact that $a$ and $k$ are not binary complicates things. – Michael Grant Mar 16 '18 at 22:55
  • a<=MB lower limits a=0 when B=0,where M is large number (constant). But when B=1, how to make a=k? Yes it is or conditions, but not in binary. – Dr.PB Mar 17 '18 at 04:36
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    See here how to model this with inequalities. – Erwin Kalvelagen Mar 17 '18 at 21:06
  • @Erwin Kalvelagen, the link is a great help. Thank you very much. – Dr.PB Mar 18 '18 at 07:06