Constrained Optimization with constraints on Sum of Columns conditional on selection of Rows

[thepushkarsingh]

For a simplified case, let's assume, I have following data:

id	cost	profit_in_case1	profit_in_case2
A	-1	10	11
B	-0.6	10	5
C	-0.75	10	10.5
D	-0.75	10.5	10

And I have to maximize my profit with decisions,so if I just see the profits as individual rows, I'll have decisions like - 

A - case2, B - case1, C - case2, D-case1.

But I have constraints like, I can only avail costs up to 1.5, so I can't choose A-case2 and B-case1 or A-case2 and C-case2(D-case1), instead I have to choose : C-case2 and D-case1. The final decision for maximum profit will look like:

decision_Cost	decision_profit_case1	decision_profit_case2
1	0	0
1	0	0
0	0	1
0	1	0

This way, we have minimized cost (Total = 1.33) and maximized profit (Total = 21) in the constraints that we can chose only one of the decisions to be 1 and total cost doesn't breach the boundary of 1.5.

My original problem has other constraints as well, but if I can get a solution to this one in PROC IML or PROC OPTMODEL, I think I can build from there. Any help is deeply appreciated. I am using SAS 9.4

[RobPratt · Accepted solution]

I think the following does what you want, using PROC OPTMODEL in SAS/OR:

data have;
   input id $ cost profit_in_case1 profit_in_case2;
   cost = -cost;
   datalines;
A	-1	10	11
B	-0.6	10	5
C	-0.75	10	10.5
D	-0.75	10.5	10
;

proc optmodel;
   set <str> ITEMS;
   num cost {ITEMS};
   set CASES = 1..2;
   num profit {ITEMS, CASES};
   read data have into ITEMS=[id] cost {j in CASES} <profit[id,j]=col('profit_in_case'||j)>;

   var X {ITEMS, CASES} binary;
   max TotalProfit = sum {i in ITEMS, j in CASES} profit[i,j] * X[i,j];
   con ChooseOne {i in ITEMS}:
      sum {j in CASES} X[i,j] <= 1;
   con Budget:
      sum {i in ITEMS, j in CASES} cost[i] * X[i,j] <= 1.5;

   solve;
   print X;
quit;

[thepushkarsingh]

Hi @RobPratt, thanks very much, this works perfectly. Many thanks.

Is there a source where I can read more about the features PROC OPTMODEL have with examples? 

[thepushkarsingh]

Hi @RobPratt any idea on how to output the final X table as it appears in PRINT statement. My efforts are resulting in one column for X with different cases in rows. I know I can transpose it, but I am asking if any direct way is there? Many thanks in adavnce.

[RobPratt]

Try this CREATE DATA statement:

create data want from [id]=ITEMS {j in CASES} <col('X'||j)=X[id,j]>;

[RobPratt]

The official documentation is a good source:

http://support.sas.com/documentation/onlinedoc/or/

See especially these two:

• SAS/OR 15.1 User's Guide: Mathematical Programming
• SAS/OR 15.1 User's Guide: Mathematical Programming Examples

[thepushkarsingh]

Thank you very much. Works like a charm.😊