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Re: [Help-glpk] [Fwd: help on interpreting sensitivity analysis]

From: Andrew Makhorin
Subject: Re: [Help-glpk] [Fwd: help on interpreting sensitivity analysis]
Date: Wed, 13 Jul 2011 02:43:14 +0400

> I have just started learning LP and glpk so apologies if I'm missing 
> something 
> important.
> I setup a problem (using the c interface), composed of the three following 
> constraints, and no objective function:
> 4.0*x0 - x1 <= 3.0
> 3.0*x0 + x1 >= 8.0
> 2.0*x0 + x1 <= 7.5
> where x0 and x1 are unbounded, and correspond to column 1 and 2.
> The pdf file in attachment shows the feasible region of this system (x0 is 
> vertical axis). From the docs I understood that the column Activity Range 
> allows me to obtain (among other things) the ranges that each variable may 
> take in order for the problem to stay feasible and optimal. Please see the 
> output of the print_ranges function attached. It shows a correct interval for 
> x0, but the upper bound of x1 is not correct, it should be almost 7 as shown 
> in the chart.
> I suppose I'm assuming something that I shouldn't. Could someone please point 
> me in the right direction?

You didn't assign names to rows and columns, so some information
in the analysis report was lost.

I rewrote your instance in cplex lp format:

   z: 0 x0
subject to
   r1: 4.0 x0 - x1 <= 3.0
   r2: 3.0 x0 + x1 >= 8.0
   r3: 2.0 x0 + x1 <= 7.5
   x0 free
   x1 free

and solved it with glpsol as follows:

   glpsol --lp test.lp --ranges test.txt -o test.sol

In particular, for column x1 the analysis report (test.txt) says:

Column name:         x1
St:                  BS
Activity:            3.28571
Obj.coef:            0.00000
Marginal:            0.00000
Lower bound:         -Inf
Upper bound:         +Inf

This means that in the current basis (which is optimal) x1 is basic, and
its value is 3.28571. 

If the objective coefficient at x1 is decreasing (first line):
Activity range:      4.00000
Obj.coef.range:      0.00000
Obj.value at
break point:         0.00000
Limiting variable:   r2

This means that if we would make the objective coefficient at x1 a bit
less than zero (Obj.coef.range), the limiting variable, which is
auxiliary variable r2, would enter the basis (that is, constraint r2
would become inactive), and in the new adjacent optimal basis x1 would
take on value 4.0.

If the objective coefficient at x1 is increasing (second line):
Activity range:      3.28571
Obj.coef.range:      +Inf
Obj.value at
break point:         +Inf
Limiting variable:   (none)

This means that the current basis remain optimal for any positive
objective coefficient at x1.

Note that the analysis has different meaning for basic and non-basic
rows/columns. For more details please see the glpk reference manual. And
check your plot; it looks incorrect.

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