Facility location model Q&A

set i /i1*i50/
set j /facility1,facility10/

parameter maxdist /40/;

set c /x,y/

Table dloc(i,c)
           x         y
i1         6584
i2         872
i3         5728
i4         7079
i5         8280
i6         8795
i7         1498
i8         286
i9         6222
i10        2280
i11        3194
i12        8774
i13        8032
i14        9248
i15        2274
i16        042
i17        8090
i18        10056
i19        731
i20        8284
i21        715
i22        319
i23        9989
i24        5694
i25        1835
i26        830
i27        7119
i28        1066
i29        771
i30        1366
i31        6814
i32        9543
i33        2350
i34        9543
i35        463
i36        5113
i37        5626
i38        4156
i39        7572
i40        4168
i41        5791
i42        1663
i43        7582
i44        4494
i45        7932
i46        725
i47        452
i48        1620
i49        3160
i50        9263
;

scalar M /1000000/

variable
   floc(j,c)
   n

binary variables
  isopen(j)
  assign(i,j)

equations
   distance,assigndemand,closed,numfacilities,order;

distance(i,j)..          (sum(c,dloc(i,c)- floc(j,c)))**2 =l= Maxdist**2 + M*(1-assign(i,j));
assigndemand(i)..         Sum(j,assign(i,j)) =e= 1;
closed(i,j)..             assign(i,j) =l= isopen(j) ;
Numfacilities..           n =e= sum(j,isopen(j));
order(j+1)..              isopen(j) =g= isopen(j+1);

model location /all/

option limcol = 180;
option limrow =180;

solve location minimizing n using MIQCP;

display n.l,floc.l;

1. Find the number of facilities \(n\) needed, so all demand locations are within MAXDIST kilometers from the closest facility.
2. Given that we know \(n\), place these \(n\) facilities such that some total distance measure is minimized. 

The first model is presented as:

What is wrong with the GAMS model in the e-mail?

1. The set \(j\) only has two elements. This is not sufficient. We should have more elements in \(j\) than the optimal value of \(n\).  The optimal value for this data set is \(n^* = 3\), so just having two elements in \(j\) will cause the model to be infeasible.
2. The value for \(M\) needs to get much more attention. The value needs to be large enough that the distance equation becomes non-binding when \(\mathit{assign}_{i,j}=0\). A simple value would be \[M=\sum_c \left(\max_i \{ \mathit{dloc}_{i,c}\} -\min_i \{\mathit{dloc}_{i,c}\}\right)^2 \] This is the square of the length of the diagonal of the box containing all demand locations. For this data this leads to \(M=19409\). 
   We can assume facilities will be placed in the convex hull of the demand locations (sorry, I don't know how to prove that). That means we can use the maximum pairwise squared distance between demand locations \((i,i')\) as a good value for \(M\): \[M= \max_{i,i'} \sum_c \left(\mathit{dloc}_{i,c}-\mathit{dloc}_{i',c}\right)^2\] This sets \(M=14116\).
   With some effort better values can be derived by using individual values \(M_{i,j}\) instead of one \(M\). Big-M values with some computation and just having assigned some large value, are always a point of attention.
   Using just very big values for big-\(M\) constants, is something I see very often. This can lead to all kind of problems when solving the MIP problem. 
3. The expression (sum(c,dloc(i,c)- floc(j,c)))**2 takes the square of the sum. This is not the same as a sum of squares:\[\sum_i x_i^2 \ne \left(\sum_i x_i\right)^2\] This expression is just malformed.
4. In GAMS x**y is a general non-linear function, and x**2 is not recognized as quadratic function (GAMS could be much smarter here). This will make the model an MINLP model and not a MIQCP. This will lead to the somewhat cryptic error:
   
     --- Executing SBB: elapsed 0:00:00.140
     Could not load data
     Detected 100 general nonlinear rows in model
   
   Of course you can solve it with an MINLP solver. (Confusingly SBB is an MINLP solver).  Of course it is better to use the sqr(x) function to help GAMS. The left-hand side of the distance constraint should be written as: sum(c,sqr(dloc(i,c)- floc(j,c))).
   Side note: if we would try to solve the original expression (sum(c,dloc(i,c)- floc(j,c)))**2 as a MINLP model, we would still be in a heap of problems. The function x**y assumes \(x\ge 0\). This is obviously not the case here. We would see a ton of domain errors. 
5. For an optimal solution, you will need to add: option optcr=0; GAMS has a default relative gap tolerance of 10%. With this options we set this tolerance to zero.