The Linear Programming Solver

Getting Started: LP Solver

The following example illustrates how you can use the OPTMODEL procedure to solve linear programs. Suppose you want to solve the following problem:

\[ \begin{array}{rlllllcc} \mbox{max} & x_1 & + & x_2 & + & x_3 & & \\ \mbox{ subject to } & 3 x_1 & + & 2x_2 & - & x_3 & \leq & 1 \\ & -2x_1 & - & 3 x_2 & + & 2x_3 & \leq & 1 \\ & & & x_1, & x_2, & x_3 & \geq & 0 \\ \end{array} \]

You can use the following statements to call the OPTMODEL procedure for solving linear programs: 

proc optmodel;
   var x{i in 1..3} >= 0;
   max f =    x[1] +   x[2] +   x[3];
   con c1:  3*x[1] + 2*x[2] -   x[3] <= 1;
   con c2: -2*x[1] - 3*x[2] + 2*x[3] <= 1;
   solve with lp / algorithm = ps presolver = none logfreq = 1;
   print x;
quit;

The optimal solution and the optimal objective value are displayed in Figure 7.1.

Figure 7.1: Solution Summary

The OPTMODEL Procedure

Problem Summary
Objective Sense Maximization
Objective Function f
Objective Type Linear
   
Number of Variables 3
Bounded Above 0
Bounded Below 3
Bounded Below and Above 0
Free 0
Fixed 0
   
Number of Constraints 2
Linear LE (<=) 2
Linear EQ (=) 0
Linear GE (>=) 0
Linear Range 0
   
Constraint Coefficients 6

Performance Information
Execution Mode Single-Machine
Number of Threads 1

Solution Summary
Solver LP
Algorithm Primal Simplex
Objective Function f
Solution Status Optimal
Objective Value 8
   
Primal Infeasibility 0
Dual Infeasibility 0
Bound Infeasibility 0
   
Iterations 4
Presolve Time 0.00
Solution Time 0.03

[1] x
1 0
2 3
3 5 




            

            


            
The iteration log displaying problem statistics, progress of the solution, and the optimal objective value is shown in Figure 7.2. 
            

            
  

            


Figure 7.2: Log

               

                  

                     
                     

                        
                        
                        

                           
                           

                              
                              





NOTE: Problem generation will use 4 threads.                                    
                                       
                                    



NOTE: The problem has 3 variables (0 free, 0 fixed).                            
                                       
                                    



NOTE: The problem has 2 linear constraints (2 LE, 0 EQ, 0 GE, 0 range).         
                                       
                                    



NOTE: The problem has 6 linear constraint coefficients.                         
                                       
                                    



NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).      
                                       
                                    



NOTE: The LP presolver value NONE is applied.                                   
                                       
                                    



NOTE: The LP solver is called.                                                  
                                       
                                    



NOTE: The Primal Simplex algorithm is used.                                     
                                       
                                    



                           Objective                Entering      Leaving       
                                       
                                    



      Phase Iteration        Value         Time     Variable      Variable      
                                       
                                    



       P 1          1    0.000000E+00         0                                 
                                       
                                    



       P 2          2    0.000000E+00         0       x[3]             c2 (S)   
                                       
                                    



       P 2          3    5.000000E-01         0       x[2]             c1 (S)   
                                       
                                    



       P 2          4    8.000000E+00         0                                 
                                       
                                    



NOTE: Optimal.                                                                  
                                       
                                    



NOTE: Objective = 8.                                                            
                                       
                                    



NOTE: The Primal Simplex solve time is 0.03 seconds.