Let x = ì1 if investment alternative i is selected i íî0 otherwise
6000×2 + 10500×3 2500×2 + 6000×3
3500×2 + 4000×3 3500×2 + 5000×3
+ 4000×4 + 2000×4
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+ 1500×4 + 1800×4
+ 8000×5 + 3000×6
+ 5000×5 + 1000×6 £ 10,500
+ 1000×5 + 500×6 £ 7,000 + 4000×5 + 900×6 £ 8,750
max 4000×1 + s.t.
3000×1 + 1000×1 + 4000×1 +
The optimal solution is
x3 = 1 x4 = 1 x6 = 1
Value = 17,500
The following mutually exclusive constraint must be added to the model. x1 + x2 £ 1 No change in optimal solution.
The following co-requisite constraint must be added to the model in b. x3 – x4 = 0. No change in optimal solution.
x1, x2, x3, x4, x5, x6 =0,1
Let x = ì1 if a principal place of business in in county i i íî0 otherwise
y = ì1 if county i is not served i íî0 if county i is served
The objective function for an integer programming model calls for minimizing the population not served.
min 195y1+96y2+•••+175y13
There are 13 constraints needed; each is written so that yi will be forced to equal one whenever it is
not possible to do business in county i.
Constraint 1: x1 + x2 + x3 + y1 3 1 Constraint 2: x1 + x2 + x3 + x4 + x6 + x7 + y2 3 1 ••• ••• ••• Constraint13: x11 + x12 + x13 + y13 3 1
One more constraint must be added to reflect the requirement that only one principal place of business may be established.
x1 + x2 + • • • + x13 = 1
The optimal solution has a principal place of business in County 11 with an optimal value of 739,000. A population of 739,000 cannot be served by this solution. Counties 1-5 and 10 will not be served.
The only change necessary in the integer programming model for part a is that the right-hand side of the last constraint is increased from 1 to 2.
x1 + x2 + • • • + x13 = 2.
The optimal solution has principal places of business in counties 3 and 11 with an optimal value of
76,000. Only County 10 with a population of 76,000 is not served.
c. It is not the best location if only one principal place of business can be established; 1,058,000 customers in the region cannot be served. However, 642,000 can be served and if there is no opportunity to obtain a principal place of business in County 11, this may be a good start. Perhaps later there will be an opportunity in County 11.
Integer Linear Programming
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