Differences in Linear Programming models

With my mind back on to the Science of Decision Making book, I have two different ways to build the same linear programming model. The book describes how to do so using Excel. So below is a screen shot of the model in Excel before it is resolved. E3:E8 uses the function of SUMPRODUCT of the related values in columns B, C, and D.

Seeking to maximize E8, it is subject to 

$B$9:$D$9 >= 0

$E$3:$E$7 <= $G$3:$G$7

with the changing cells $B$9:$D$9. 

This results in the answer below.

Produce 

20 of S

30 of F

0 of L

Now to do the same problem using the GNU Linear Programming Kit (GLPK), the model is built as follows.

--------

# This finds the optimal solution for maximizing the RV plants's profit

#

/* Decision variables */

var x1 >=0;  /* Standard */

var x2 >=0;  /* Fancy */

var x3 >=0;  /* Luxury */

/* Objective function */

maximize z: 840*x1 + 1120*x2 + 1200*x3;

/* Constraints */

s.t. Engine_Shop : 3*x1 + 2*x2 + 1*x3 <= 120;

s.t. Body_Shop   : 1*x1 + 2*x2 + 3*x3 <= 80;

s.t. Standard_Fin : 2*x1 + 0*x2 + 0*x3 <= 96;

s.t. Fancy_Fin : 0*x1 + 3*x2 + 0*x3 <= 102;

s.t. Luxury_Fin : 0*x1 + 0*x2 + 2*x3 <= 40;

end;

--------

While spreadsheet fans may find this slightly harder to follow, the format in this tool is closer to how a student defines a linear programming problem in class, only slightly more complex given scripting variables. GLPK produces the following report after running "glpsol -m glpk_Science_of_decision_Ch001.mod -o test.sol"

--------

Problem:    glpk_Science_of_decision_Ch001

Rows:       6

Columns:    3

Non-zeros:  12

Status:     OPTIMAL

Objective:  z = 50400 (MAXimum)

   No.   Row name   St   Activity     Lower bound   Upper bound    Marginal

------ ------------ -- ------------- ------------- ------------- -------------

     1 z            B          50400                             

     2 Engine_Shop  NU           120                         120           140

     3 Body_Shop    NU            80                          80           420

     4 Standard_Fin B             40                          96 

     5 Fancy_Fin    B             90                         102 

     6 Luxury_Fin   B              0                          40 

   No. Column name  St   Activity     Lower bound   Upper bound    Marginal

------ ------------ -- ------------- ------------- ------------- -------------

     1 x1           B             20             0               

     2 x2           B             30             0               

     3 x3           NL             0             0                        -200

Karush-Kuhn-Tucker optimality conditions:

KKT.PE: max.abs.err. = 0.00e+000 on row 0

        max.rel.err. = 0.00e+000 on row 0

        High quality

KKT.PB: max.abs.err. = 0.00e+000 on row 0

        max.rel.err. = 0.00e+000 on row 0

        High quality

KKT.DE: max.abs.err. = 1.14e-013 on column 1

        max.rel.err. = 1.35e-016 on column 1

        High quality

KKT.DB: max.abs.err. = 0.00e+000 on row 0

        max.rel.err. = 0.00e+000 on row 0

        High quality

End of output

--------

While the report introduces a greater amount of information, we can still see the same results. 

The company should produce

20 of the standard

30 of the fancy

0 of the luxury

The Google Docs spreadsheet looks almost the same as Excel for the purposes of this post so a screenshot of it is excluded. The only difference between the two for this example is how the constraints are setup. See the previous post for the details.