How do I remove redundant variables from this formulation of a Mathematical Program

[Maalik_Serebryakova]

Firstly, I wasn't sure which section of this forum to post mathematical programming/optimization questions in so I just put it here because generally it is learned after Calculus where I am from. I think you should make a section of mathematical programming maybe.

So the problem I'm dealing with is I have formulated the design problem of a handpump mathematically. First I'll display the whole formulation then I will ask the question.

Notation explanation
[imath]x_{1,2,3,4....n}[/imath] means [imath]x_{1}[/imath] , [imath]x_{2}[/imath], [imath]x_{3}[/imath]... all the way to [imath]x_{n}[/imath] just means theres n different variables. the reason I separated the variables with different base letters is to denote that they are the variables for different parts of the handpump. C would mean the Cylindrical Top, h would mean the Handlebar. etc

d= very small number. For a part to fit inside its socket it needs to be slightly smaller.

Variable List(38)​

This is a total list of all design variables in the mathematical program
[imath]C_{1,2,3}[/imath] [imath]P_{1,2,3,4,5,6}[/imath] [imath]T_{1,2,3,4,5,6,7}[/imath] [imath]h_{1,2,3}[/imath] [imath]V_{1,2,3,4,5,6}[/imath] [imath]M_{1,2,3,4,5,6,7}[/imath] [imath]F_{1,2,3,4,5,6,7}[/imath]
​

System of Geometric Equations​

These are Equality constraints/relationships between the design variables I deduced by observing sketches of the Handpump.​

[imath]P_{3}=T_{4}[/imath]
[imath]T_{1}=C_{1}[/imath]
[imath]2h_{3}=h_{2}[/imath]
[imath]C_{3}=T_{2}[/imath]
[imath]M_{3}=h_{3}+d[/imath]
[imath]C_{3}=M_{1}+d[/imath]
[imath]T_{2}=M_{1}+d[/imath]
[imath]M_{4}=F_{6}+d[/imath]
[imath]V_{1}=P_{3}+d[/imath]
[imath]P_{4}=F_{2}+d[/imath]
[imath]T_{1}=P_{1}+d[/imath]
[imath]V_{1}=T_{4}+d[/imath]
[imath]P_{6}=\frac{M_{1}}{2}+F_{5}+F_{1}+F_{3}\cos(45)[/imath]
[imath]0.8(C_{2}+T_{3})=M_{2}+F_{3}+F_{7}+P_{2}[/imath]
[imath]T_{3}=T_{6}+1500[/imath]
[imath]h_{1}+h_{3}=160[/imath]
[imath]h_{2}=50[/imath]

Now of course I have a tonne of redundant variables here, and I would like to be rid of them to simplify my mathematical program. (I haven't wrote the constraints and objective function yet). But for right now, How do I use these equality constraints to reduce the number of Design Variables? sry if this is too basic for the advanced math section. I am a total Ape. Lol

 

[Maalik_Serebryakova]

I have this system of 17 Equations

[imath]P_{3}=T_{4}[/imath]
[imath]T_{1}=C_{1}[/imath]
[imath]2h_{3}=h_{2}[/imath]
[imath]C_{3}=T_{2}[/imath]
[imath]M_{3}=h_{3}[/imath]
[imath]C_{3}=M_{1}+0.2[/imath]
[imath]T_{2}=M_{1}+0.2[/imath]
[imath]M_{4}=F_{6}+0.2[/imath]
[imath]V_{1}=P_{3}+0.2[/imath]
[imath]P_{4}=F_{2}+0.2[/imath]
[imath]T_{1}=P_{1}+0.2[/imath]
[imath]V_{1}=T_{4}+0.2[/imath]
[imath]P_{6}=\frac{M_{1}}{2}+F_{5}+F_{1}+F_{3}\cos(45)[/imath]
[imath]0.8(C_{2}+T_{3})=M_{2}+F_{3}+F_{7}+P_{2}[/imath]
[imath]T_{3}=T_{6}+1500[/imath]
[imath]h_{1}+h_{3}=160[/imath]
[imath]h_{2}=50[/imath]

And the 25 variables mentioned in this system are:

[imath]C_{1,2,3}[/imath] [imath]P_{1,2,3,4,6}[/imath] [imath]T_{1,2,3}[/imath] [imath]h_{1,2,3}[/imath] [imath]M_{1,2,3,4}[/imath] [imath]V_{1}[/imath] [imath]F_{1,2,3,5,6,7}[/imath]

My question is, how can I determine the minimum number of Variables I need to solve the full system?
thx

 

[BigBeachBanana]

I have this system of 17 Equations

[imath]P_{3}=T_{4}[/imath]
[imath]T_{1}=C_{1}[/imath]
[imath]2h_{3}=h_{2}[/imath]
[imath]C_{3}=T_{2}[/imath]
[imath]M_{3}=h_{3}[/imath]
[imath]C_{3}=M_{1}+0.2[/imath]
[imath]T_{2}=M_{1}+0.2[/imath]
[imath]M_{4}=F_{6}+0.2[/imath]
[imath]V_{1}=P_{3}+0.2[/imath]
[imath]P_{4}=F_{2}+0.2[/imath]
[imath]T_{1}=P_{1}+0.2[/imath]
[imath]V_{1}=T_{4}+0.2[/imath]
[imath]P_{6}=\frac{M_{1}}{2}+F_{5}+F_{1}+F_{3}\cos(45)[/imath]
[imath]0.8(C_{2}+T_{3})=M_{2}+F_{3}+F_{7}+P_{2}[/imath]
[imath]T_{3}=T_{6}+1500[/imath]
[imath]h_{1}+h_{3}=160[/imath]
[imath]h_{2}=50[/imath]

And the 25 variables mentioned in this system are:

[imath]C_{1,2,3}[/imath] [imath]P_{1,2,3,4,6}[/imath] [imath]T_{1,2,3}[/imath] [imath]h_{1,2,3}[/imath] [imath]M_{1,2,3,4}[/imath] [imath]V_{1}[/imath] [imath]F_{1,2,3,5,6,7}[/imath]

My question is, how can I determine the minimum number of Variables I need to solve the full system?
thx

The only suggestion I can make is that start with the equations with most variables, then try to make substitutions using other relationships.

 

[Maalik_Serebryakova]

The only suggestion I can make is that start with the equations with most variables, then try to make substitutions using other relationships.

thanks ? finally somebody replied to me.

Should I keep substituting until I have only a few equations left and no more substitutions can be made ? Then I can count the variables in the equations and that is the minimum number of required variables to solve the system?


But imagine i have a bigger system Surely there must be a computer method of doing this

 

[BigBeachBanana]

I spotted a simple one:
If [imath]h_2=50,2h_3=h_2\implies h_3=25\\ \text{Furthermore},\\ h_1+h_3=h_1+25=160 \implies h_1=135\\ h_3=M_3=25[/imath]

 

[BigBeachBanana]

thanks ? finally somebody replied to me.

Should I keep substituting until I have only a few equations left and no more substitutions can be made ? Then I can count the variables in the equations and that is the minimum number of required variables to solve the system?

That's what I would do.

But imagine i have a bigger system Surely there must be a computer method of doing this

I would think so too, but I'm unfamiliar with other systematic approaches.
Lastly, I just noticed the equation from your OP is different than the ones in post#2. Namely, [imath]h_3=M_3 \quad vs \quad h_3=M_3+d[/imath].