The Mixed Integer Linear Programming Solver

Example 7.3 Facility Location

Consider the classic facility location problem. Given a set L of customer locations and a set F of candidate facility sites, you must decide on which sites to build facilities and assign coverage of customer demand to these sites so as to minimize cost. All customer demand $d_ i$ must be satisfied, and each facility has a demand capacity limit C. The total cost is the sum of the distances $c_{ij}$ between facility j and its assigned customer i, plus a fixed charge $f_ j$ for building a facility at site j. Let $y_ j = 1$ represent choosing site j to build a facility, and 0 otherwise. Also, let $x_{ij} = 1$ represent the assignment of customer i to facility j, and 0 otherwise. This model can be formulated as the following integer linear program:

\[  \begin{array}{llllll} \min &  \displaystyle \sum _{i \in L} \displaystyle \sum _{j \in F} c_{ij} x_{ij} + \displaystyle \sum _{j \in F} f_ j y_ j \\ \mr {s.t.} &  \displaystyle \sum _{j \in F} x_{ij} &  = &  1 &  \forall i \in L &  \mr {(assign\_ def)} \\ &  x_{ij} &  \leq &  y_ j &  \forall i \in L, j \in F &  \mr {(link)}\\ &  \displaystyle \sum _{i \in L} d_ i x_{ij} &  \leq &  Cy_ j &  \forall j \in F &  \mr {(capacity)} \\ &  x_{ij} \in \{ 0,1\}  & & &  \forall i \in L, j \in F \\ &  y_{j} \in \{ 0,1\}  & & &  \forall j \in F \end{array}  \]

Constraint (assign_def) ensures that each customer is assigned to exactly one site. Constraint (link) forces a facility to be built if any customer has been assigned to that facility. Finally, constraint (capacity) enforces the capacity limit at each site.

Consider also a variation of this same problem where there is no cost for building a facility. This problem is typically easier to solve than the original problem. For this variant, let the objective be

\[  \begin{array}{lllll} \min &  \displaystyle \sum _{i \in L} \displaystyle \sum _{j \in F} c_{ij} x_{ij} \end{array}  \]

First, construct a random instance of this problem by using the following DATA steps:

title 'Facility Location Problem';
                                                                                                         
%let NumCustomers  = 50;
%let NumSites      = 10; 
%let SiteCapacity  = 35;
%let MaxDemand     = 10; 
%let xmax          = 200;
%let ymax          = 100; 
%let seed          = 938; 

/* generate random customer locations */                                                                                             
data cdata(drop=i);                                                                                                                  
   length name $8;                                                                                                                   
   do i = 1 to &NumCustomers;                                                                                                        
      name = compress('C'||put(i,best.));                                                                                            
      x = ranuni(&seed) * &xmax;                                                                                                     
      y = ranuni(&seed) * &ymax;                                                                                                     
      demand = ranuni(&seed) * &MaxDemand;                                                                                           
      output;                                                                                                                        
   end;                                                                                                                              
run;                                                                                                                                 

/* generate random site locations and fixed charge */                                                                                
data sdata(drop=i);                                                                                                                  
   length name $8;                                                                                                                   
   do i = 1 to &NumSites;                                                                                                            
      name = compress('SITE'||put(i,best.));                                                                                         
      x = ranuni(&seed) * &xmax;                                                                                                     
      y = ranuni(&seed) * &ymax;                                                                                                     
      fixed_charge = 30 * (abs(&xmax/2-x) + abs(&ymax/2-y));                                                                
      output;                                                                                                                        
   end;                                                                                                                              
run;   

The following PROC OPTMODEL statements first generate and solve the model with the no-fixed-charge variant of the cost function. Next, they solve the fixed-charge model. Note that the solution to the model with no fixed charge is feasible for the fixed-charge model and should provide a good starting point for the MILP solver. Use the PRIMALIN option to provide an incumbent solution (warm start).


proc optmodel;
   set <str> CUSTOMERS;
   set <str> SITES init {};
   /* x and y coordinates of CUSTOMERS and SITES */
   num x {CUSTOMERS union SITES};
   num y {CUSTOMERS union SITES};
   num demand {CUSTOMERS};
   num fixed_charge {SITES};
    
   /* distance from customer i to site j */
   num dist {i in CUSTOMERS, j in SITES}
       = sqrt((x[i] - x[j])^2 + (y[i] - y[j])^2);
    
   read data cdata into CUSTOMERS=[name] x y demand;
   read data sdata into SITES=[name] x y fixed_charge;
    
   var Assign {CUSTOMERS, SITES} binary;
   var Build {SITES} binary;
    
   min CostNoFixedCharge
       = sum {i in CUSTOMERS, j in SITES} dist[i,j] * Assign[i,j];
   min CostFixedCharge
       = CostNoFixedCharge + sum {j in SITES} fixed_charge[j] * Build[j];
    
   /* each customer assigned to exactly one site */
   con assign_def {i in CUSTOMERS}:
      sum {j in SITES} Assign[i,j] = 1;
    
   /* if customer i assigned to site j, then facility must be built at j */
   con link {i in CUSTOMERS, j in SITES}:
      Assign[i,j] <= Build[j];
    
   /* each site can handle at most &SiteCapacity demand */
   con capacity {j in SITES}:
      sum {i in CUSTOMERS} demand[i] * Assign[i,j] <= 
         &SiteCapacity * Build[j];
    
   /* solve the MILP with no fixed charges */      
   solve obj CostNoFixedCharge with milp / logfreq = 500;      
    
   /* clean up the solution */
   for {i in CUSTOMERS, j in SITES} Assign[i,j] = round(Assign[i,j]);
   for {j in SITES} Build[j] = round(Build[j]);
    
   call symput('varcostNo',put(CostNoFixedCharge,6.1));
    
   /* create a data set for use by GPLOT */
   create data CostNoFixedCharge_Data from
      [customer site]={i in CUSTOMERS, j in SITES: Assign[i,j] = 1}
      xi=x[i] yi=y[i] xj=x[j] yj=y[j];
      
   /* solve the MILP, with fixed charges with warm start */
   solve obj CostFixedCharge with milp / primalin logfreq = 500;
    
   /* clean up the solution */
   for {i in CUSTOMERS, j in SITES} Assign[i,j] = round(Assign[i,j]);
   for {j in SITES} Build[j] = round(Build[j]);
    
   num varcost = sum {i in CUSTOMERS, j in SITES} dist[i,j] * Assign[i,j].sol;
   num fixcost = sum {j in SITES} fixed_charge[j] * Build[j].sol;
   call symput('varcost', put(varcost,6.1));
   call symput('fixcost', put(fixcost,5.1));
   call symput('totalcost', put(CostFixedCharge,6.1));
    
   /* create a data set for use by GPLOT */
   create data CostFixedCharge_Data from
      [customer site]={i in CUSTOMERS, j in SITES: Assign[i,j] = 1}
      xi=x[i] yi=y[i] xj=x[j] yj=y[j];
quit; 
 

The information printed in the log for the no-fixed-charge model is displayed in Output 7.3.1.

Output 7.3.1: OPTMODEL Log for Facility Location with No Fixed Charges

Facility Location Problem

NOTE: Problem generation will use 2 threads.                                    
NOTE: The problem has 510 variables (0 free, 0 fixed).                          
NOTE: The problem has 510 binary and 0 integer variables.                       
NOTE: The problem has 560 linear constraints (510 LE, 50 EQ, 0 GE, 0 range).    
NOTE: The problem has 2010 linear constraint coefficients.                      
NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).      
NOTE: The MILP presolver value AUTOMATIC is applied.                            
NOTE: The MILP presolver removed 10 variables and 500 constraints.              
NOTE: The MILP presolver removed 1010 constraint coefficients.                  
NOTE: The MILP presolver modified 0 constraint coefficients.                    
NOTE: The presolved problem has 500 variables, 60 constraints, and 1000         
      constraint coefficients.                                                  
NOTE: The MILP solver is called.                                                
          Node  Active    Sols    BestInteger      BestBound      Gap    Time   
             0       1       2    972.1737321              0    972.2       0   
             0       1       2    972.1737321    961.2403449    1.14%       0   
             0       1       3    966.4832160    966.4832160    0.00%       0   
             0       0       3    966.4832160    966.4832160    0.00%       0   
NOTE: The MILP solver added 11 cuts with 596 cut coefficients at the root.      
NOTE: Optimal.                                                                  
NOTE: Objective = 966.483216.                                                   


The results from the warm start approach are shown in Output 7.3.2.

Output 7.3.2: OPTMODEL Log for Facility Location with Fixed Charges, Using Warm Start

Facility Location Problem

NOTE: Problem generation will use 2 threads.                                    
NOTE: The problem has 510 variables (0 free, 0 fixed).                          
NOTE: The problem uses 1 implicit variables.                                    
NOTE: The problem has 510 binary and 0 integer variables.                       
NOTE: The problem has 560 linear constraints (510 LE, 50 EQ, 0 GE, 0 range).    
NOTE: The problem has 2010 linear constraint coefficients.                      
NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).      
NOTE: The MILP presolver value AUTOMATIC is applied.                            
NOTE: The MILP presolver removed 0 variables and 0 constraints.                 
NOTE: The MILP presolver removed 0 constraint coefficients.                     
NOTE: The MILP presolver modified 0 constraint coefficients.                    
NOTE: The presolved problem has 510 variables, 560 constraints, and 2010        
      constraint coefficients.                                                  
NOTE: The MILP solver is called.                                                
          Node  Active    Sols    BestInteger      BestBound      Gap    Time   
             0       1       3  16070.0150023              0    16070       0   
             0       1       3  16070.0150023   9946.2514269   61.57%       0   
             0       1       3  16070.0150023  10928.9752350   47.04%       0   
             0       1       3  16070.0150023  10935.9357070   46.95%       0   
             0       1       3  16070.0150023  10939.3645882   46.90%       0   
             0       1       3  16070.0150023  10939.8308022   46.89%       0   
             0       1       3  16070.0150023  10940.6691108   46.88%       0   
             0       1       6  12678.8372464  10941.0776158   15.88%       0   
             0       1       6  12678.8372464  10941.0776158   15.88%       0   
NOTE: The MILP solver added 16 cuts with 405 cut coefficients at the root.      
            28       6       7  10948.4603380  10941.6896516    0.06%       0   
            38      15       8  10948.4603380  10941.6896516    0.06%       0   
            66       3       8  10948.4603380  10947.6054588    0.01%       1   
NOTE: Optimal within relative gap.                                              
NOTE: Objective = 10948.4603.                                                   


The following two SAS programs produce a plot of the solutions for both variants of the model, using data sets produced by PROC OPTMODEL:

title1 h=1.5 "Facility Location Problem";                                                                                                 
title2 "TotalCost = &varcostNo (Variable = &varcostNo, Fixed = 0)";   

data csdata;                                                                                                                        
   set cdata(rename=(y=cy)) sdata(rename=(y=sy));                                                                                  
run; 

/* create Annotate data set to draw line between customer and assigned site */                                                      
%annomac;                                                                                                                           
data anno(drop=xi yi xj yj);                                                                                                        
   %SYSTEM(2, 2, 2);                                                                                                               
   set CostNoFixedCharge_Data(keep=xi yi xj yj);                                                                                    
   %LINE(xi, yi, xj, yj, *, 1, 1);                                                                                                  
run;

proc gplot data=csdata anno=anno;                                                                                                   
   axis1 label=none order=(0 to &xmax by 10);                                                                                       
   axis2 label=none order=(0 to &ymax by 10);                                                                                       
   symbol1 value=dot interpol=none                                                                                                  
      pointlabel=("#name" nodropcollisions height=1) cv=black;                                                                   
   symbol2 value=diamond interpol=none                                                                                              
      pointlabel=("#name" nodropcollisions color=blue height=1) cv=blue;                                                                    
   plot cy*x sy*x / overlay haxis=axis1 vaxis=axis2;                                                                                
run;
quit;

The output of the first program is shown in Output 7.3.3.

Output 7.3.3: Solution Plot for Facility Location with No Fixed Charges

Solution Plot for Facility Location with No Fixed Charges


The output of the second program is shown in Output 7.3.4.

title1 "Facility Location Problem";                                                                                                 
title2 "TotalCost = &totalcost (Variable = &varcost, Fixed = &fixcost)";    

/* create Annotate data set to draw line between customer and assigned site */                                                      
data anno(drop=xi yi xj yj);                                                                                                        
   %SYSTEM(2, 2, 2);                                                                                                                
   set CostFixedCharge_Data(keep=xi yi xj yj);                                                                                      
   %LINE(xi, yi, xj, yj, *, 1, 1);                                                                                                  
run;

proc gplot data=csdata anno=anno;                                                                                                   
   axis1 label=none order=(0 to &xmax by 10);                                                                                       
   axis2 label=none order=(0 to &ymax by 10);                                                                                       
   symbol1 value=dot interpol=none                                                                                                  
      pointlabel=("#name" nodropcollisions height=1) cv=black;                                                                    
   symbol2 value=diamond interpol=none                                                                                              
      pointlabel=("#name" nodropcollisions color=blue height=1) cv=blue;                                                                    
   plot cy*x sy*x / overlay haxis=axis1 vaxis=axis2;                                                                                
run;
quit;   

Output 7.3.4: Solution Plot for Facility Location with Fixed Charges

Solution Plot for Facility Location with Fixed Charges


The economic trade-off for the fixed-charge model forces you to build fewer sites and push more demand to each site.

It is possible to expedite the solution of the fixed-charge facility location problem by choosing appropriate branching priorities for the decision variables. Recall that for each site j, the value of the variable $y_ j$ determines whether or not a facility is built on that site. Suppose you decide to branch on the variables $y_ j$ before the variables $x_{ij}$. You can set a higher branching priority for $y_ j$ by using the .priority suffix for the Build variables in PROC OPTMODEL, as follows:

   for{j in SITES} Build[j].priority=10;

Setting higher branching priorities for certain variables is not guaranteed to speed up the MILP solver, but it can be helpful in some instances. The following program creates and solves an instance of the facility location problem, giving higher priority to the variables $y_ j$. The LOGFREQ= option is used to abbreviate the node log.

   
%let NumCustomers  = 45;    
%let NumSites      = 8;    
%let SiteCapacity  = 35;    
%let MaxDemand     = 10;    
%let xmax          = 200;    
%let ymax          = 100;    
%let seed          = 2345;

/* generate random customer locations */
data cdata(drop=i);
   length name $8;
   do i = 1 to &NumCustomers;
      name = compress('C'||put(i,best.));
      x = ranuni(&seed) * &xmax;
      y = ranuni(&seed) * &ymax;
      demand = ranuni(&seed) * &MaxDemand;
      output;
   end;
run;

/* generate random site locations and fixed charge */
data sdata(drop=i);
length name $8;
   do i = 1 to &NumSites;
      name = compress('SITE'||put(i,best.));
      x = ranuni(&seed) * &xmax;
      y = ranuni(&seed) * &ymax;
      fixed_charge = (abs(&xmax/2-x) + abs(&ymax/2-y)) / 2;
      output;
   end;
run;     
proc optmodel;                                                                                                                                                                                                                        
   set <str> CUSTOMERS;
   set <str> SITES init {};
   
   /* x and y coordinates of CUSTOMERS and SITES */
   num x {CUSTOMERS union SITES};
   num y {CUSTOMERS union SITES};
   num demand {CUSTOMERS};
   num fixed_charge {SITES};
       
   /* distance from customer i to site j */
   num dist {i in CUSTOMERS, j in SITES} 
       = sqrt((x[i] - x[j])^2 + (y[i] - y[j])^2);
       
   read data cdata into CUSTOMERS=[name] x y demand;
   read data sdata into SITES=[name] x y fixed_charge;
   
   var Assign {CUSTOMERS, SITES} binary;
   var Build {SITES} binary;
   
   min CostFixedCharge   
       = sum {i in CUSTOMERS, j in SITES} dist[i,j] * Assign[i,j] 
         + sum {j in SITES} fixed_charge[j] * Build[j];
   
   /* each customer assigned to exactly one site */
   con assign_def {i in CUSTOMERS}:
      sum {j in SITES} Assign[i,j] = 1;
      
   /* if customer i assigned to site j, then facility must be built at j */
   con link {i in CUSTOMERS, j in SITES}:
      Assign[i,j] <= Build[j];
       
   /* each site can handle at most &SiteCapacity demand */
   con capacity {j in SITES}:
      sum {i in CUSTOMERS} demand[i] * Assign[i,j] <= &SiteCapacity * Build[j];
       
   /* assign priority to Build variables (y) */
   for{j in SITES} Build[j].priority=10;
   
   /* solve the MILP with fixed charges, using branching priorities */
   solve obj CostFixedCharge with milp / logfreq=1000;
quit;

The resulting output is shown in Output 7.3.5.

Output 7.3.5: PROC OPTMODEL Log for Facility Location with Branching Priorities

Facility Location Problem
TotalCost = 10948 (Variable = 1329.8, Fixed = 9619)

NOTE: There were 45 observations read from the data set WORK.CDATA.             
NOTE: There were 8 observations read from the data set WORK.SDATA.              
NOTE: Problem generation will use 2 threads.                                    
NOTE: The problem has 368 variables (0 free, 0 fixed).                          
NOTE: The problem has 368 binary and 0 integer variables.                       
NOTE: The problem has 413 linear constraints (368 LE, 45 EQ, 0 GE, 0 range).    
NOTE: The problem has 1448 linear constraint coefficients.                      
NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).      
NOTE: The MILP presolver value AUTOMATIC is applied.                            
NOTE: The MILP presolver removed 0 variables and 0 constraints.                 
NOTE: The MILP presolver removed 0 constraint coefficients.                     
NOTE: The MILP presolver modified 0 constraint coefficients.                    
NOTE: The presolved problem has 368 variables, 413 constraints, and 1448        
      constraint coefficients.                                                  
NOTE: The MILP solver is called.                                                
          Node  Active    Sols    BestInteger      BestBound      Gap    Time   
             0       1       3   2823.1827978              0   2823.2       0   
             0       1       3   2823.1827978   1727.0208789   63.47%       0   
             0       1       3   2823.1827978   1758.4959444   60.55%       0   
             0       1       3   2823.1827978   1777.9581456   58.79%       0   
             0       1       3   2823.1827978   1786.2487641   58.05%       0   
             0       1       3   2823.1827978   1789.8831106   57.73%       0   
             0       1       3   2823.1827978   1791.4930241   57.59%       0   
             0       1       3   2823.1827978   1793.9911665   57.37%       0   
             0       1       3   2823.1827978   1795.5152566   57.24%       0   
             0       1       3   2823.1827978   1796.6395103   57.14%       0   
             0       1       3   2823.1827978   1797.2196277   57.09%       0   
             0       1       3   2823.1827978   1798.7116308   56.96%       0   
             0       1       6   1867.2953460   1799.1463919    3.79%       0   
             0       1       6   1867.2953460   1799.4020954    3.77%       0   
             0       1       6   1867.2953460   1799.9500748    3.74%       0   
             0       1       6   1867.2953460   1800.1075050    3.73%       0   
             0       1       6   1867.2953460   1800.1075050    3.73%       0   
NOTE: The MILP solver added 32 cuts with 1013 cut coefficients at the root.     
            16      16       7   1841.9436945   1801.7219901    2.23%       0   
           148     136       8   1831.7800505   1805.5957623    1.45%       1   
           238     191       9   1826.0904114   1806.4101894    1.09%       1   
           307     208      10   1822.8640269   1808.3177036    0.80%       1   
           329     203      11   1821.5277612   1808.7906394    0.70%       1   
           503     183      12   1821.5115362   1814.4176969    0.39%       2   
           604      74      13   1819.9124340   1817.9289922    0.11%       2   
           672       9      13   1819.9124340   1819.7313148    0.01%       2   
NOTE: Optimal within relative gap.                                              
NOTE: Objective = 1819.91243.