Please attach your data as a JMP data table in the future. I spent time copying it to a JMP data table as a single character data column and then split it into three numeric data columns.
The simplest model is usually the best. You are interested in predicting the effect, not in explaining the effect. I plotted the data first.
The Speed and Pump Flow are highly, linearly correlated, so both of them cannot be used as predictors in any type of model. I chose Speed as the predictor of Power. The relationship is curvilinear.
Several models were fit to the data. The Cubic Polynomial model is the best by far based on AICc.
I attached the JMP data table with your data and several saved table scripts to reproduce my analysis.
Is the pump speed ordinal, or can you set something in between those values? If it is continuous then it seems the optimal value would be as slow as possible as the specific power always decreases with lower flow. You could find that optimal flow given your constraints using the profiler as @Mark_Bailey suggests. Here is how I did it, run the script below to recreate the table with saved scripts.
- Calculate specific power in a new column, or the power per unit of flow
- Calculate flow in gal/day in a new column (I assumed the flow is gpm)
- Model the flow based on the pump speed. I used a neural network because it captured the visual curve in the data while giving smooth response between points and without responding to what looks like noise around 2500-3000 RPM, save the profiler prediction formula. (Script 1)
- Model the specific power, again a neural networked and saved the profiler formula (Script 2)
- Check the predicted versus actual values for reasonableness (Script 3)
- Use column properties to set a lower spec on the flow and a a response limit to minimize specific power
- Found optimal conditions using the profiler, using both predicted columns in the y and rpm in the X. Used Maximize Desirability to find the optimal pump setting. (Script 5)
New Table( "Maximize Pump",
Add Rows( 13 ),
New Script(
"1 Model Flow (gal/day) by RPM with NN",
Neural(
Y( :"Flow (gal/day)"n ),
X( :RPM ),
Informative Missing( 0 ),
Validation Method( "Holdback", 0.3333 ),
Fit(
NTanH( 3 ),
Profiler( 1, Term Value( RPM( 2637, Lock( 0 ), Show( 1 ) ) ) )
)
)
),
New Script(
"2 Model Specific Power by RPM with NN",
Neural(
Y( :Specific Power ),
X( :RPM ),
Informative Missing( 0 ),
Validation Method( "Holdback", 0.3333 ),
Fit( NTanH( 3 ) )
)
),
New Script(
"3 Check for reasonable models using Graph Builder",
Graph Builder(
Size( 534, 456 ),
Show Control Panel( 0 ),
Variables(
X( :RPM ),
Y( :"Predicted Flow (gal/day)"n ),
Y( :"Flow (gal/day)"n, Position( 1 ) ),
Y( :Predicted Specific Power ),
Y( :Specific Power, Position( 2 ) )
),
Elements(
Position( 1, 1 ),
Points( X, Y( 2 ), Legend( 3 ) ),
Formula( X, Y( 1 ), Legend( 5 ) )
),
Elements(
Position( 1, 2 ),
Points( X, Y( 2 ), Legend( 6 ) ),
Formula( X, Y( 1 ), Legend( 7 ) )
),
SendToReport(
Dispatch(
{},
"graph title",
TextEditBox,
{Set Text( "Predicted and Actual Flow and Power per Flow" )}
)
)
)
),
New Script( "4 Added spec and desirability function", . ),
New Script(
"5 Optimize with the Profiler",
Profiler(
Y( :"Predicted Flow (gal/day)"n, :Predicted Specific Power ),
Profiler(
1,
Desirability Functions( 1 ),
"Predicted Flow (gal/day)"n <<
Response Limits(
{Lower( 43000, 0.066 ), Middle( 43005, 0.5 ),
Upper( 43010, 0.9819 ), Goal( "Maximize" ), Importance( 1 )}
),
Term Value( RPM( 1039.13398714365, Lock( 0 ), Show( 1 ) ) )
)
)
),
New Column( "RPM",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Values(
[500, 750, 1000, 1250, 1500, 1750, 2000, 2250, 2500, 2750, 3000, 3250,
3450]
)
),
New Column( "Flow",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Values( [17, 24, 29, 35, 41, 50, 57, 65, 75, 79, 90, 98, 107] )
),
New Column( "Power",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Values(
[63, 92, 132, 185, 260, 370, 515, 700, 926, 1210, 1540, 1920, 2290]
)
),
New Column( "Specific Power",
Numeric,
"Continuous",
Format( "Best", 12 ),
Formula( :Power / :Flow )
),
New Column( "Flow (gal/day)",
Numeric,
"Continuous",
Format( "Best", 12 ),
Formula( :Flow * (60 * 24) )
),
New Column( "Predicted Flow (gal/day)",
Numeric,
"Continuous",
Format( "Fixed Dec", 12, 1 ),
Set Property( "Notes", "Model NTanH(3) - Save Profile Formulas" ),
Formula(
(-4986263.94864684) + 12250231.3478716 *
TanH( 0.5 * ((-0.195770498135355) + 0.00013382112616832 * :RPM) )
+54339329.8694451 * TanH(
0.5 * (0.227571869547046 + -0.0000279518693191086 * :RPM)
) + -40377.1104869192 * TanH(
0.5 * ((-1.66726266024003) + 0.00154547501582029 * :RPM)
)
),
Set Property(
"Predicting",
{:"Flow (gal/day)"n, Creator( "Neural" ), Std Dev( 1578.4156263483 )}
),
Set Property( "Expand Intermediates", 1 ),
Set Property( "Spec Limits", {LSL( 43000 ), Show Limits( 0 )} )
),
New Column( "Predicted Specific Power",
Numeric,
"Continuous",
Format( "Fixed Dec", 12, 2 ),
Set Property( "Notes", "Model NTanH(3) - Save Profile Formulas" ),
Formula(
99.4343260017528 + 5.9968762233528 * TanH(
0.5 * ((-2.70046669266303) + 0.00107500481416692 * :RPM)
) + -70.0855096923245 * TanH(
0.5 * (0.536433235085508 + 0.000419373832905438 * :RPM)
) + 191.605900251259 * TanH(
0.5 * ((-0.790641306358042) + 0.000145038719985201 * :RPM)
)
),
Set Property(
"Predicting",
{:Specific Power, Creator( "Neural" ), Std Dev( 0.225420351211502 )}
),
Set Property( "Expand Intermediates", 1 ),
Set Property(
"Response Limits",
{Goal( Minimize ), Importance( 1 ), Show Limits( 0 )}
),
Set Selected
)
);