6.3

Habitat Suitability Index Problem.

We need to formulate this problem to maximize habitat 
suitability index scores, and require some timber volume
to be produced over a 30-year time period.  After examining
the tables on page 315 in the book, we developed the
following problem formulation.

The LINDO input file contained the following:


max + 102.79 X11
    + 103.17 X12
    + 102.60 X13
    +  97.80 X21
    +  98.40 X22
    +  97.40 X23
    +  92.69 X31
    +  95.09 X32
    +  96.76 X33
    +  91.56 X41
    +  93.03 X42
    +  92.19 X43
subject to
2) X11 + X12 + X13 = 1
3) X21 + X22 + X23 = 1
4) X31 + X32 + X33 = 1
5) X41 + X42 + X43 = 1
6)  893 X11 + 1020 X21 + 1128.5 X31 + 1197 X41 - H1 = 0
7) 1026 X12 + 1060 X22 + 1128.5 X32 + 1239 X42 - H2 = 0
8) 1064 X13 + 1080 X23 + 1147 X33   + 1260 X43 - H3 = 0
9) H1 >= 1
10) H2 >= 1
11) H3 >= 1
end
inte X11
inte X12
inte X13
inte X21
inte X22
inte X23
inte X31
inte X32
inte X33
inte X41
inte X42
inte X43


Here, we are maximizing HSI values.  To obtain the objective function
value for variable X11, multiply the number of acres in Stand A by 
the HSI score for Stand A during period 1 (0.541 x 190).

Rows 2-4 indicate that each stand will only be entered once during
the three time periods, since X11-X43 will take on integer values
(as indicated by "inte X11" at the end of the LINDO formulation).

Rows 6-7 sum the volume produced during each time period.  To
obtain the value for X11, multiply the number of acres in Stand A
by the volume produced from Stand A during time period 1 (4.7 x
190).  H1, H2, and H3 are the total harvest volumes for periods
1, 2, and 3, respectively.

Rows 9-11 indicate that the total harvest volume for periods 1-3
must be greater than (or equal to) 1, or that there must be "some"
volume produced in each time period.


The LINDO output contained the following:


 LP OPTIMUM FOUND AT STEP     20
 OBJECTIVE VALUE =   391.359589    
 FIX ALL VARS.(    7)  WITH RC >  0.220001    

 NEW INTEGER SOLUTION OF    390.980011     AT BRANCH      0 PIVOT      42

        OBJECTIVE FUNCTION VALUE

        1)      390.9800    

  VARIABLE        VALUE          REDUCED COST
       X11         1.000000          0.000000
       X12         0.000000          0.000000
       X13         0.000000         23.697525
       X21         0.000000          0.000000
       X22         1.000000          1.000000
       X23         0.000000        488.940002
       X31         0.000000      37682.738281
       X32         0.000000          0.000000
       X33         1.000000          0.000000
       X41         0.000000          0.000000
       X42         1.000000          0.000000
       X43         0.000000          0.000000
        H1       893.000000          0.000000
        H2      2299.000000          0.000000
        H3      1147.000000          0.000000


       ROW   SLACK OR SURPLUS     DUAL PRICES
        2)         0.000000        103.169998
        3)         0.000000         98.180000
        4)         0.000000         96.760002
        5)         0.000000         92.190002
        6)         0.000000          0.000000
        7)         0.000000          0.000000
        8)         0.000000          0.000000
        9)       892.000000          0.000000
       10)      2298.000000          0.000000
       11)      1146.000000          0.000000
       12)         0.000000         -0.379997

 NO. ITERATIONS=      42
 BRANCHES=    0 DETERM.=  1.000E    0
 BOUND ON OPTIMUM:  390.9800    
 ENUMERATION COMPLETE. BRANCHES=     0 PIVOTS=      42

 LAST INTEGER SOLUTION IS THE BEST FOUND
 RE-INSTALLING BEST SOLUTION...
                                              

After interpreting the LINDO output, we arrived at the 
following answers to the problem:

a) The optimum management schedule is for Stand A to be 
thinned during period 1, producing 893 cords; Stands B 
and D during period 2, producing 2299 cords; and Stand C 
during period 3, producing 1147 cords.  Stand A actually
would have had a higher HSI score if thinned during period
2, yet one of the stands had to be thinned during period
1 to produce "some" timber volume during this period.

b) The total HSI score was 390.98, or a 0.498 HSI score
per acre.  

c) Over the three time periods, the owners could expect
to harvest 4339 cords, or an average of 5.5 cords per
acre.