Why is there not a feasible solution for a MIP?Infeasibility in mathematical optimization modelsHow to get...

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Why is there not a feasible solution for a MIP?


Infeasibility in mathematical optimization modelsHow to get bounds on ILP optimal solution qualityUsing CPLEX “solution pool” to count feasible pointsFormulation of a constraint in a MIP for an element in different SetsQuerying attributes of LP relaxation at MIP-optimality in GurobiAre there examples of spatially explicit MIP problems?Are valid inequalities worth the effort given modern solvers?How does a warm start work in LP/MIP?Generating all extreme raysDivisibility constraints in integer programmingWhat do solvers like Gurobi and CPLEX do when they run into hard instances of MIP













3












$begingroup$


Is there a way to see why a solver (OR-Tools, CPLEX, Gurobi) cannot find a feasible solution when solving a MIP?



By that I mean, is there a possibility to show at which constraint and exact indices the solver stoped?



Example:




  • $x_i$ a binary variable


  • $a_j$ a parameter


  • $ i in I:|I| =3$


  • constraint: $sum limits_i x_i > a_j forall j$



no solution found because for index $j=2$ not able to fulfill this constraint:



---> $a_2= 100$ and $sum limits_i x_i =3$










share|improve this question











$endgroup$














  • $begingroup$
    Note that my answer relates to infeasibility - the other case of not finding an optimal solution would be an unbounded problem where it typically should be easy to spot the reason.
    $endgroup$
    – CMichael
    14 hours ago






  • 3




    $begingroup$
    Related question: or.stackexchange.com/q/1215/196
    $endgroup$
    – Dipayan Banerjee
    13 hours ago






  • 1




    $begingroup$
    So the magic word that I was missing is infeasibility.
    $endgroup$
    – Georgios
    13 hours ago












  • $begingroup$
    That is an amazingly succinct summary Georgios!
    $endgroup$
    – CMichael
    13 hours ago










  • $begingroup$
    @Georgios I edited your question to ask about infeasibility instead of optimality -- if my edits are not OK, feel free to roll them back or make further edits.
    $endgroup$
    – LarrySnyder610
    51 mins ago
















3












$begingroup$


Is there a way to see why a solver (OR-Tools, CPLEX, Gurobi) cannot find a feasible solution when solving a MIP?



By that I mean, is there a possibility to show at which constraint and exact indices the solver stoped?



Example:




  • $x_i$ a binary variable


  • $a_j$ a parameter


  • $ i in I:|I| =3$


  • constraint: $sum limits_i x_i > a_j forall j$



no solution found because for index $j=2$ not able to fulfill this constraint:



---> $a_2= 100$ and $sum limits_i x_i =3$










share|improve this question











$endgroup$














  • $begingroup$
    Note that my answer relates to infeasibility - the other case of not finding an optimal solution would be an unbounded problem where it typically should be easy to spot the reason.
    $endgroup$
    – CMichael
    14 hours ago






  • 3




    $begingroup$
    Related question: or.stackexchange.com/q/1215/196
    $endgroup$
    – Dipayan Banerjee
    13 hours ago






  • 1




    $begingroup$
    So the magic word that I was missing is infeasibility.
    $endgroup$
    – Georgios
    13 hours ago












  • $begingroup$
    That is an amazingly succinct summary Georgios!
    $endgroup$
    – CMichael
    13 hours ago










  • $begingroup$
    @Georgios I edited your question to ask about infeasibility instead of optimality -- if my edits are not OK, feel free to roll them back or make further edits.
    $endgroup$
    – LarrySnyder610
    51 mins ago














3












3








3


1



$begingroup$


Is there a way to see why a solver (OR-Tools, CPLEX, Gurobi) cannot find a feasible solution when solving a MIP?



By that I mean, is there a possibility to show at which constraint and exact indices the solver stoped?



Example:




  • $x_i$ a binary variable


  • $a_j$ a parameter


  • $ i in I:|I| =3$


  • constraint: $sum limits_i x_i > a_j forall j$



no solution found because for index $j=2$ not able to fulfill this constraint:



---> $a_2= 100$ and $sum limits_i x_i =3$










share|improve this question











$endgroup$




Is there a way to see why a solver (OR-Tools, CPLEX, Gurobi) cannot find a feasible solution when solving a MIP?



By that I mean, is there a possibility to show at which constraint and exact indices the solver stoped?



Example:




  • $x_i$ a binary variable


  • $a_j$ a parameter


  • $ i in I:|I| =3$


  • constraint: $sum limits_i x_i > a_j forall j$



no solution found because for index $j=2$ not able to fulfill this constraint:



---> $a_2= 100$ and $sum limits_i x_i =3$







mixed-integer-programming integer-programming cplex gurobi feasible-points






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 53 mins ago









LarrySnyder610

6,36715 silver badges67 bronze badges




6,36715 silver badges67 bronze badges










asked 14 hours ago









GeorgiosGeorgios

29812 bronze badges




29812 bronze badges















  • $begingroup$
    Note that my answer relates to infeasibility - the other case of not finding an optimal solution would be an unbounded problem where it typically should be easy to spot the reason.
    $endgroup$
    – CMichael
    14 hours ago






  • 3




    $begingroup$
    Related question: or.stackexchange.com/q/1215/196
    $endgroup$
    – Dipayan Banerjee
    13 hours ago






  • 1




    $begingroup$
    So the magic word that I was missing is infeasibility.
    $endgroup$
    – Georgios
    13 hours ago












  • $begingroup$
    That is an amazingly succinct summary Georgios!
    $endgroup$
    – CMichael
    13 hours ago










  • $begingroup$
    @Georgios I edited your question to ask about infeasibility instead of optimality -- if my edits are not OK, feel free to roll them back or make further edits.
    $endgroup$
    – LarrySnyder610
    51 mins ago


















  • $begingroup$
    Note that my answer relates to infeasibility - the other case of not finding an optimal solution would be an unbounded problem where it typically should be easy to spot the reason.
    $endgroup$
    – CMichael
    14 hours ago






  • 3




    $begingroup$
    Related question: or.stackexchange.com/q/1215/196
    $endgroup$
    – Dipayan Banerjee
    13 hours ago






  • 1




    $begingroup$
    So the magic word that I was missing is infeasibility.
    $endgroup$
    – Georgios
    13 hours ago












  • $begingroup$
    That is an amazingly succinct summary Georgios!
    $endgroup$
    – CMichael
    13 hours ago










  • $begingroup$
    @Georgios I edited your question to ask about infeasibility instead of optimality -- if my edits are not OK, feel free to roll them back or make further edits.
    $endgroup$
    – LarrySnyder610
    51 mins ago
















$begingroup$
Note that my answer relates to infeasibility - the other case of not finding an optimal solution would be an unbounded problem where it typically should be easy to spot the reason.
$endgroup$
– CMichael
14 hours ago




$begingroup$
Note that my answer relates to infeasibility - the other case of not finding an optimal solution would be an unbounded problem where it typically should be easy to spot the reason.
$endgroup$
– CMichael
14 hours ago




3




3




$begingroup$
Related question: or.stackexchange.com/q/1215/196
$endgroup$
– Dipayan Banerjee
13 hours ago




$begingroup$
Related question: or.stackexchange.com/q/1215/196
$endgroup$
– Dipayan Banerjee
13 hours ago




1




1




$begingroup$
So the magic word that I was missing is infeasibility.
$endgroup$
– Georgios
13 hours ago






$begingroup$
So the magic word that I was missing is infeasibility.
$endgroup$
– Georgios
13 hours ago














$begingroup$
That is an amazingly succinct summary Georgios!
$endgroup$
– CMichael
13 hours ago




$begingroup$
That is an amazingly succinct summary Georgios!
$endgroup$
– CMichael
13 hours ago












$begingroup$
@Georgios I edited your question to ask about infeasibility instead of optimality -- if my edits are not OK, feel free to roll them back or make further edits.
$endgroup$
– LarrySnyder610
51 mins ago




$begingroup$
@Georgios I edited your question to ask about infeasibility instead of optimality -- if my edits are not OK, feel free to roll them back or make further edits.
$endgroup$
– LarrySnyder610
51 mins ago










3 Answers
3






active

oldest

votes


















8














$begingroup$

Yes - such a question can be answered by looking at the irreducible inconsistent subsystem (IIS).



From the Gurobi documentation:




An IIS is a subset of the constraints and variable bounds with the following properties:
the subsystem represented by the IIS is infeasible, and
if any of the constraints or bounds of the IIS is removed, the subsystem becomes feasible.







share|improve this answer









$endgroup$























    6














    $begingroup$

    There exist various reasons why a solver could not find the optimal solution. You must always check why a solver terminated. Typical reasons are:




    1. optimal solution was found

    2. termination criteria was reached, e.g. time limit or a limit on the optimality gap

    3. solver proved that problem is infeasible


    Popular solvers such as cplex/gurobi can report their status through a getStatus function. When a solver terminates due to some termination criteria, you can end up in any of the following situations:




    1. Some solution was found. The optimality gap gives you insight into the quality of this solution. However, there might exist better solutions, i.e. it is unknown whether this solution is optimal or not.

    2. No solution was found. There might however exist feasible solutions. This status is usually indicated as 'undefined/unknown', as it is unknown whether the solution space is empty or not.


    Frequently, the solver is quickly able to determine feasibility of a problem. This can already happen in the pre-solve status. Modern solvers can search for a Minimal irreducible inconsistent subsystem. This is a subset of constraints which collectively render your problem infeasible. Deleting any constraint of this set would render the subproblem defined by these constraints feasible. Note that there may exist multiple causes of infeasibility in your model, i.e. multiple different IISs.



    If you want to know more about this subject, I recommend the following papers:
    - Finding the minimum weight IIS cover of an infeasible system of linear inequalities by Parker and Ryan, 1996
    - Minimal Infeasible Subsystems and Benders cuts by Fischetti, Salvagnin, Zanette, 2008



    Finding a minimum IIS, i.e. the smallest IIS is an NP-hard problem. Therefore, many solvers use heuristics to find a minimal, but not necessarily minimum IIS. I my experience, trying to find an IIS using CPLEX or Gurobi is a game of hit and miss: finding an IIS might take quite some time, especially in large models. Also, the IIS returned can be big, so interpreting the source of infeasibility might not be trivial. In practice I often use the following approach:




    1. Compute a feasible solution to your problem using for instance a simple heuristic.

    2. Fix all variables in your model to their corresponding values in the solution found in the previous step.

    3. Solve the model and search for an IIS. Gurobi: Model.computeIIS(). Cplex: Cplex.getIIS.

    4. Since all variables are fixed, the IIS returned by the solvers is typically very small.






    share|improve this answer









    $endgroup$















    • $begingroup$
      CPLEX now uses by default conflicts rather than IIS. See e.g. ibm.com/support/knowledgecenter/SSSA5P_12.9.0/…. You would use Cplex.refineConflict to get it.
      $endgroup$
      – Xavier Nodet
      12 hours ago



















    4














    $begingroup$

    Another possibility not mentioned in the other answers is that an optimal solution exists, but the solver is not able to find it, or perhaps conform its optimality, due to numerical difficulties, which might in turn be due to poor scaling or ill-conditioning of the original problem. Double precision floating point computation can be a cruel mistress.






    share|improve this answer









    $endgroup$


















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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      8














      $begingroup$

      Yes - such a question can be answered by looking at the irreducible inconsistent subsystem (IIS).



      From the Gurobi documentation:




      An IIS is a subset of the constraints and variable bounds with the following properties:
      the subsystem represented by the IIS is infeasible, and
      if any of the constraints or bounds of the IIS is removed, the subsystem becomes feasible.







      share|improve this answer









      $endgroup$




















        8














        $begingroup$

        Yes - such a question can be answered by looking at the irreducible inconsistent subsystem (IIS).



        From the Gurobi documentation:




        An IIS is a subset of the constraints and variable bounds with the following properties:
        the subsystem represented by the IIS is infeasible, and
        if any of the constraints or bounds of the IIS is removed, the subsystem becomes feasible.







        share|improve this answer









        $endgroup$


















          8














          8










          8







          $begingroup$

          Yes - such a question can be answered by looking at the irreducible inconsistent subsystem (IIS).



          From the Gurobi documentation:




          An IIS is a subset of the constraints and variable bounds with the following properties:
          the subsystem represented by the IIS is infeasible, and
          if any of the constraints or bounds of the IIS is removed, the subsystem becomes feasible.







          share|improve this answer









          $endgroup$



          Yes - such a question can be answered by looking at the irreducible inconsistent subsystem (IIS).



          From the Gurobi documentation:




          An IIS is a subset of the constraints and variable bounds with the following properties:
          the subsystem represented by the IIS is infeasible, and
          if any of the constraints or bounds of the IIS is removed, the subsystem becomes feasible.








          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered 14 hours ago









          CMichaelCMichael

          8761 silver badge12 bronze badges




          8761 silver badge12 bronze badges


























              6














              $begingroup$

              There exist various reasons why a solver could not find the optimal solution. You must always check why a solver terminated. Typical reasons are:




              1. optimal solution was found

              2. termination criteria was reached, e.g. time limit or a limit on the optimality gap

              3. solver proved that problem is infeasible


              Popular solvers such as cplex/gurobi can report their status through a getStatus function. When a solver terminates due to some termination criteria, you can end up in any of the following situations:




              1. Some solution was found. The optimality gap gives you insight into the quality of this solution. However, there might exist better solutions, i.e. it is unknown whether this solution is optimal or not.

              2. No solution was found. There might however exist feasible solutions. This status is usually indicated as 'undefined/unknown', as it is unknown whether the solution space is empty or not.


              Frequently, the solver is quickly able to determine feasibility of a problem. This can already happen in the pre-solve status. Modern solvers can search for a Minimal irreducible inconsistent subsystem. This is a subset of constraints which collectively render your problem infeasible. Deleting any constraint of this set would render the subproblem defined by these constraints feasible. Note that there may exist multiple causes of infeasibility in your model, i.e. multiple different IISs.



              If you want to know more about this subject, I recommend the following papers:
              - Finding the minimum weight IIS cover of an infeasible system of linear inequalities by Parker and Ryan, 1996
              - Minimal Infeasible Subsystems and Benders cuts by Fischetti, Salvagnin, Zanette, 2008



              Finding a minimum IIS, i.e. the smallest IIS is an NP-hard problem. Therefore, many solvers use heuristics to find a minimal, but not necessarily minimum IIS. I my experience, trying to find an IIS using CPLEX or Gurobi is a game of hit and miss: finding an IIS might take quite some time, especially in large models. Also, the IIS returned can be big, so interpreting the source of infeasibility might not be trivial. In practice I often use the following approach:




              1. Compute a feasible solution to your problem using for instance a simple heuristic.

              2. Fix all variables in your model to their corresponding values in the solution found in the previous step.

              3. Solve the model and search for an IIS. Gurobi: Model.computeIIS(). Cplex: Cplex.getIIS.

              4. Since all variables are fixed, the IIS returned by the solvers is typically very small.






              share|improve this answer









              $endgroup$















              • $begingroup$
                CPLEX now uses by default conflicts rather than IIS. See e.g. ibm.com/support/knowledgecenter/SSSA5P_12.9.0/…. You would use Cplex.refineConflict to get it.
                $endgroup$
                – Xavier Nodet
                12 hours ago
















              6














              $begingroup$

              There exist various reasons why a solver could not find the optimal solution. You must always check why a solver terminated. Typical reasons are:




              1. optimal solution was found

              2. termination criteria was reached, e.g. time limit or a limit on the optimality gap

              3. solver proved that problem is infeasible


              Popular solvers such as cplex/gurobi can report their status through a getStatus function. When a solver terminates due to some termination criteria, you can end up in any of the following situations:




              1. Some solution was found. The optimality gap gives you insight into the quality of this solution. However, there might exist better solutions, i.e. it is unknown whether this solution is optimal or not.

              2. No solution was found. There might however exist feasible solutions. This status is usually indicated as 'undefined/unknown', as it is unknown whether the solution space is empty or not.


              Frequently, the solver is quickly able to determine feasibility of a problem. This can already happen in the pre-solve status. Modern solvers can search for a Minimal irreducible inconsistent subsystem. This is a subset of constraints which collectively render your problem infeasible. Deleting any constraint of this set would render the subproblem defined by these constraints feasible. Note that there may exist multiple causes of infeasibility in your model, i.e. multiple different IISs.



              If you want to know more about this subject, I recommend the following papers:
              - Finding the minimum weight IIS cover of an infeasible system of linear inequalities by Parker and Ryan, 1996
              - Minimal Infeasible Subsystems and Benders cuts by Fischetti, Salvagnin, Zanette, 2008



              Finding a minimum IIS, i.e. the smallest IIS is an NP-hard problem. Therefore, many solvers use heuristics to find a minimal, but not necessarily minimum IIS. I my experience, trying to find an IIS using CPLEX or Gurobi is a game of hit and miss: finding an IIS might take quite some time, especially in large models. Also, the IIS returned can be big, so interpreting the source of infeasibility might not be trivial. In practice I often use the following approach:




              1. Compute a feasible solution to your problem using for instance a simple heuristic.

              2. Fix all variables in your model to their corresponding values in the solution found in the previous step.

              3. Solve the model and search for an IIS. Gurobi: Model.computeIIS(). Cplex: Cplex.getIIS.

              4. Since all variables are fixed, the IIS returned by the solvers is typically very small.






              share|improve this answer









              $endgroup$















              • $begingroup$
                CPLEX now uses by default conflicts rather than IIS. See e.g. ibm.com/support/knowledgecenter/SSSA5P_12.9.0/…. You would use Cplex.refineConflict to get it.
                $endgroup$
                – Xavier Nodet
                12 hours ago














              6














              6










              6







              $begingroup$

              There exist various reasons why a solver could not find the optimal solution. You must always check why a solver terminated. Typical reasons are:




              1. optimal solution was found

              2. termination criteria was reached, e.g. time limit or a limit on the optimality gap

              3. solver proved that problem is infeasible


              Popular solvers such as cplex/gurobi can report their status through a getStatus function. When a solver terminates due to some termination criteria, you can end up in any of the following situations:




              1. Some solution was found. The optimality gap gives you insight into the quality of this solution. However, there might exist better solutions, i.e. it is unknown whether this solution is optimal or not.

              2. No solution was found. There might however exist feasible solutions. This status is usually indicated as 'undefined/unknown', as it is unknown whether the solution space is empty or not.


              Frequently, the solver is quickly able to determine feasibility of a problem. This can already happen in the pre-solve status. Modern solvers can search for a Minimal irreducible inconsistent subsystem. This is a subset of constraints which collectively render your problem infeasible. Deleting any constraint of this set would render the subproblem defined by these constraints feasible. Note that there may exist multiple causes of infeasibility in your model, i.e. multiple different IISs.



              If you want to know more about this subject, I recommend the following papers:
              - Finding the minimum weight IIS cover of an infeasible system of linear inequalities by Parker and Ryan, 1996
              - Minimal Infeasible Subsystems and Benders cuts by Fischetti, Salvagnin, Zanette, 2008



              Finding a minimum IIS, i.e. the smallest IIS is an NP-hard problem. Therefore, many solvers use heuristics to find a minimal, but not necessarily minimum IIS. I my experience, trying to find an IIS using CPLEX or Gurobi is a game of hit and miss: finding an IIS might take quite some time, especially in large models. Also, the IIS returned can be big, so interpreting the source of infeasibility might not be trivial. In practice I often use the following approach:




              1. Compute a feasible solution to your problem using for instance a simple heuristic.

              2. Fix all variables in your model to their corresponding values in the solution found in the previous step.

              3. Solve the model and search for an IIS. Gurobi: Model.computeIIS(). Cplex: Cplex.getIIS.

              4. Since all variables are fixed, the IIS returned by the solvers is typically very small.






              share|improve this answer









              $endgroup$



              There exist various reasons why a solver could not find the optimal solution. You must always check why a solver terminated. Typical reasons are:




              1. optimal solution was found

              2. termination criteria was reached, e.g. time limit or a limit on the optimality gap

              3. solver proved that problem is infeasible


              Popular solvers such as cplex/gurobi can report their status through a getStatus function. When a solver terminates due to some termination criteria, you can end up in any of the following situations:




              1. Some solution was found. The optimality gap gives you insight into the quality of this solution. However, there might exist better solutions, i.e. it is unknown whether this solution is optimal or not.

              2. No solution was found. There might however exist feasible solutions. This status is usually indicated as 'undefined/unknown', as it is unknown whether the solution space is empty or not.


              Frequently, the solver is quickly able to determine feasibility of a problem. This can already happen in the pre-solve status. Modern solvers can search for a Minimal irreducible inconsistent subsystem. This is a subset of constraints which collectively render your problem infeasible. Deleting any constraint of this set would render the subproblem defined by these constraints feasible. Note that there may exist multiple causes of infeasibility in your model, i.e. multiple different IISs.



              If you want to know more about this subject, I recommend the following papers:
              - Finding the minimum weight IIS cover of an infeasible system of linear inequalities by Parker and Ryan, 1996
              - Minimal Infeasible Subsystems and Benders cuts by Fischetti, Salvagnin, Zanette, 2008



              Finding a minimum IIS, i.e. the smallest IIS is an NP-hard problem. Therefore, many solvers use heuristics to find a minimal, but not necessarily minimum IIS. I my experience, trying to find an IIS using CPLEX or Gurobi is a game of hit and miss: finding an IIS might take quite some time, especially in large models. Also, the IIS returned can be big, so interpreting the source of infeasibility might not be trivial. In practice I often use the following approach:




              1. Compute a feasible solution to your problem using for instance a simple heuristic.

              2. Fix all variables in your model to their corresponding values in the solution found in the previous step.

              3. Solve the model and search for an IIS. Gurobi: Model.computeIIS(). Cplex: Cplex.getIIS.

              4. Since all variables are fixed, the IIS returned by the solvers is typically very small.







              share|improve this answer












              share|improve this answer



              share|improve this answer










              answered 13 hours ago









              Joris KinableJoris Kinable

              4911 silver badge9 bronze badges




              4911 silver badge9 bronze badges















              • $begingroup$
                CPLEX now uses by default conflicts rather than IIS. See e.g. ibm.com/support/knowledgecenter/SSSA5P_12.9.0/…. You would use Cplex.refineConflict to get it.
                $endgroup$
                – Xavier Nodet
                12 hours ago


















              • $begingroup$
                CPLEX now uses by default conflicts rather than IIS. See e.g. ibm.com/support/knowledgecenter/SSSA5P_12.9.0/…. You would use Cplex.refineConflict to get it.
                $endgroup$
                – Xavier Nodet
                12 hours ago
















              $begingroup$
              CPLEX now uses by default conflicts rather than IIS. See e.g. ibm.com/support/knowledgecenter/SSSA5P_12.9.0/…. You would use Cplex.refineConflict to get it.
              $endgroup$
              – Xavier Nodet
              12 hours ago




              $begingroup$
              CPLEX now uses by default conflicts rather than IIS. See e.g. ibm.com/support/knowledgecenter/SSSA5P_12.9.0/…. You would use Cplex.refineConflict to get it.
              $endgroup$
              – Xavier Nodet
              12 hours ago











              4














              $begingroup$

              Another possibility not mentioned in the other answers is that an optimal solution exists, but the solver is not able to find it, or perhaps conform its optimality, due to numerical difficulties, which might in turn be due to poor scaling or ill-conditioning of the original problem. Double precision floating point computation can be a cruel mistress.






              share|improve this answer









              $endgroup$




















                4














                $begingroup$

                Another possibility not mentioned in the other answers is that an optimal solution exists, but the solver is not able to find it, or perhaps conform its optimality, due to numerical difficulties, which might in turn be due to poor scaling or ill-conditioning of the original problem. Double precision floating point computation can be a cruel mistress.






                share|improve this answer









                $endgroup$


















                  4














                  4










                  4







                  $begingroup$

                  Another possibility not mentioned in the other answers is that an optimal solution exists, but the solver is not able to find it, or perhaps conform its optimality, due to numerical difficulties, which might in turn be due to poor scaling or ill-conditioning of the original problem. Double precision floating point computation can be a cruel mistress.






                  share|improve this answer









                  $endgroup$



                  Another possibility not mentioned in the other answers is that an optimal solution exists, but the solver is not able to find it, or perhaps conform its optimality, due to numerical difficulties, which might in turn be due to poor scaling or ill-conditioning of the original problem. Double precision floating point computation can be a cruel mistress.







                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered 12 hours ago









                  Mark L. StoneMark L. Stone

                  3,5671 gold badge7 silver badges29 bronze badges




                  3,5671 gold badge7 silver badges29 bronze badges


































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