HELP!!! An integer nonlinear problem - proc optmodel

[claridad123]

I am trying to solve an integer nonlinear problem like this:

• There are five providers (p1, p2, .. p5), at three locations (g, s, t), seeing patients with 2 different types of visits: office visits and video visits.
• For the next 10 business days, at each location, the demand for these two types of visits are:

	location g	location s	location t
Office Visit	80	70	90
Video Visit	25	31	27

• For each provider, they can have as many as 7 office visits, and 3 video visits per day.
• For each provider, they can work no more than 6 days within the next 10 business days.

I am trying to see how to allocate these providers in these locations, to meet the demand most.

Here are my codes, and I have 15 variables, represent the number of days for each provider to stay in one location for the next 10 days. For example, p1g means the number of days that provider p1 at location g.

proc optmodel;
var p1g<=6 >=0 , p2g<=6>=0, p3g<=6>=0, p4g<=6>=0, p5g<=6>=0 integer;
var p1s<=6>=0, p2s<=6>=0, p3s<=6>=0 ,p4s<=6>=0, p5s<=6>=0 integer;
var p1t<=6>=0,  p2t<=6>=0, p3t<=6>=0, p4t<=6>=0, p5t<=6>=0 integer;

/* declare objective */
impvar OfficeVisit=min(7*(p1g+p2g+p3g+p4g+p5g),80)
  +min(7*(p1s+p2s+p3s+p4s+p5s),70)
  +min(7*(p1t+p2t+p3t+p4t+p5t),90);
impvar VideoVisit=min(3*(p1g+p2g+p3g+p4g+p5g),25)
  +min(3*(p1s+p2s+p3s+p4s+p5s),31)
  +min(3*(p1t+p2t+p3t+p4t+p5t),27);

/*declare constraints*/
con p1g +p1s +p1t<=6;
con p2g +p2s +p2t<=6;
con p3g +p3s +p3t<=6;
con p4g +p4s +p4t<=6;
con p5g +p5s +p5t<=6;

max bts = OfficeVisit + VideoVisit;
solve ;
print p1g p2g p3g p4g p5g p1s p2s p3s p4s p5s p1t p2t p3t p4t p5t;
quit;

[RobPratt · Accepted solution]

If you have SAS Viya, you can use the LINEARIZE option to automatically linearize the problem:

solve linearize;

Problem Summary
Objective Sense	Maximization
Objective Function	bts
Objective Type	Nonlinear
Number of Variables	15
Bounded Above	0
Bounded Below	0
Bounded Below and Above	15
Free	0
Fixed	0
Binary	0
Integer	3
Number of Constraints	5
Linear LE (<=)	5
Linear EQ (=)	0
Linear GE (>=)	0
Linear Range	0
Constraint Coefficients	15

Solution Summary
Solver	MILP
Algorithm	Branch and Cut
Objective Function	bts
Solution Status	Optimal
Objective Value	292
Relative Gap	0
Absolute Gap	0
Primal Infeasibility	0
Bound Infeasibility	0
Integer Infeasibility	0
Best Bound	292
Nodes	1
Solutions Found	3
Iterations	10
Presolve Time	0.03
Solution Time	0.04

p1g	p2g	p3g	p4g	p5g	p1s	p2s	p3s	p4s	p5s	p1t	p2t	p3t	p4t	p5t
1.25	1.25	1.25	1.25	6	2.5	2.5	2.5	2.5	0	2.25	2.25	2.25	2.25	0

But notice that some variables are fractional.  That is because the INTEGER keyword applies only to three of the variables in your VAR statements.  What you intended was instead:

var p1g<=6>=0 integer, p2g<=6>=0 integer, p3g<=6>=0 integer, p4g<=6>=0 integer, p5g<=6>=0 integer;
var p1s<=6>=0 integer, p2s<=6>=0 integer, p3s<=6>=0 integer, p4s<=6>=0 integer, p5s<=6>=0 integer;
var p1t<=6>=0 integer, p2t<=6>=0 integer, p3t<=6>=0 integer, p4t<=6>=0 integer, p5t<=6>=0 integer;

And this yields integer values for all variables:

p1g	p2g	p3g	p4g	p5g	p1s	p2s	p3s	p4s	p5s	p1t	p2t	p3t	p4t	p5t
6	2	0	0	3	0	4	0	6	0	0	0	6	0	3

A more compact and less error-prone approach is to use index sets:

proc optmodel;
   set PROVIDERS = 1..5;
   set LOCATIONS = /g s t/;
   var NumDays {PROVIDERS, LOCATIONS} >= 0 <= 6 integer;

   /* declare objective */
   num officeDemand {LOCATIONS} = [80, 70, 90];
   impvar OfficeVisit = sum {l in LOCATIONS} min(7*sum {p in PROVIDERS} NumDays[p,l], officeDemand[l]);
   num videoDemand {LOCATIONS} = [25, 31, 27];
   impvar VideoVisit = sum {l in LOCATIONS} min(3*sum {p in PROVIDERS} NumDays[p,l], videoDemand[l]);

   /*declare constraints*/
   con ProviderLimit {p in PROVIDERS}: sum {l in LOCATIONS} NumDays[p,l] <= 6;

   max bts = OfficeVisit + VideoVisit;
   solve linearize;
   print NumDays;
quit;

NumDays
	g	s	t
1	6	0	0
2	2	4	0
3	0	0	6
4	0	6	0
5	3	0	3

[claridad123]

@RobPratt thank you so much. I don't have Viya but "solve with LSO" works after I add 'INTEGER' keyword to all variables.