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Example 2.2: Altering Arc Data

This example examines the effect of changing some of the arc costs. The backorder penalty costs are increased by 20 percent. The sales profit of 25-inch TVs sent to the shops in May is increased by 30 units. The backorder penalty costs of 25-inch TVs manufactured in May for April consumption is decreased by 30 units. The production cost of 19-inch and 25-inch TVs made in May are decreased by 5 units and 20 units, respectively. How does the optimal solution of the network after these arc cost alterations compare with the optimum of the original network?

These SAS statements produce the new NODEDATA= and ARCDATA= data sets:

  
    title2 'Minimum Cost Flow problem- Altered Arc Data'; 
    data arc2; 
       set arc1; 
       oldcost=_cost_; 
       oldfc=_fcost_; 
       oldflow=_flow_; 
       if key_id='backorder' 
          then _cost_=_cost_*1.2; 
          else if _tail_='f2_may_2' then _cost_=_cost_-30; 
       if key_id='production' & mth_made='May' then 
          if diagonal=19 then _cost_=_cost_-5; 
                         else _cost_=_cost_-20; 
       run; 
  
    proc intpoint 
       bytes=100000 
       printlevel2=2 
       nodedata=node0 
       arcdata=arc2 
       conout=arc3; 
       run; 
  
    proc print data=arc3; 
       var _tail_ _head_ _capac_ _lo_ _supply_ _demand_ _name_ 
           _cost_ _flow_ _fcost_ oldcost oldflow oldfc 
           diagonal factory key_id mth_made; 
           /* to get this variable order */ 
               sum oldfc _fcost_; 
       run;
 

The following notes appear on the SAS log:

  
    NOTE: Number of nodes= 20 . 
    NOTE: Number of supply nodes= 4 . 
    NOTE: Number of demand nodes= 4 . 
    NOTE: Total supply= 4350 , total demand= 4150 . 
    NOTE: Number of arcs= 64 . 
    NOTE: The following messages relate to the equivalent Linear 
          Programming problem solved by the Interior Point algorithm. 
    NOTE: Number of <= constraints= 0 . 
    NOTE: Number of == constraints= 21 . 
    NOTE: Number of >= constraints= 0 . 
    NOTE: Number of constraint coefficients= 136 . 
    NOTE: Number of variables= 68 . 
    NOTE: After preprocessing, number of <= constraints= 0. 
    NOTE: After preprocessing, number of == constraints= 20. 
    NOTE: After preprocessing, number of >= constraints= 0. 
    NOTE: The preprocessor eliminated 1 constraints from the 
          problem. 
    NOTE: The preprocessor eliminated 9 constraint coefficients 
          from the problem. 
    NOTE: 0 columns, 0 rows and 0 coefficients were added to 
          the problem to handle unrestricted variables, 
          variables that are split, and constraint slack or 
          surplus variables. 
    NOTE: There are 48 nonzero elements in A * A transpose. 
    NOTE: Of the 20 rows and columns, 11 are sparse. 
    NOTE: There are 40 nonzero superdiagonal elements in the 
          sparse rows of the factored A * A transpose. This 
          includes fill-in. 
    NOTE: There are 49 operations of the form 
          u[i,j]=u[i,j]-u[q,j]*u[q,i]/u[q,q] to factorize the 
          sparse rows of A * A transpose. 
 Iter Complem_aff Complem-ity  Duality_gap Tot_infeasb  Tot_infeasc Tot_infeasd 
    0   -1.000000   169822194     0.834344       52835        25664       38174 
    1    37113367    17651592     0.912723 1650.135100   801.539460           0 
    2     1990318      739344     0.369751           0 1.363097E-12           0 
    3      358794      186448     0.125234           0            0           0 
    4      116081       45454     0.033002           0            0           0 
    5       26298       16467     0.012077           0            0           0 
    6 8963.676317 3893.499547     0.002875           0            0           0 
    7 1615.258886  718.590270     0.000532           0            0           0 
    8  178.225920   47.380105  0.000035062           0            0           0 
    9    6.698353    0.020987 1.5531175E-8           0            0           0 
    NOTE: The Primal-Dual Predictor-Corrector Interior Point algorithm 
          performed 9 iterations. 
    NOTE: Objective = -1285086.46. 
    NOTE: The data set WORK.ARC3 has 64 observations and 17 
          variables. 
    NOTE: There were 64 observations read from the data set 
          WORK.ARC2. 
    NOTE: There were 8 observations read from the data set 
          WORK.NODE0.
 

The solution is displayed in Output 2.2.1.