Chapter 3 
      Linear Programming:  Sensitivity Analysis 
              and Interpretation of Solution
        Introduction to Sensitivity Analysis
        Graphical Sensitivity Analysis
        Sensitivity Analysis:  Computer Solution
        Limitations of Classical Sensitivity Analysis
                                                              
                                                               
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           Introduction to Sensitivity Analysis
       In the previous chapter we discussed:
         • objective function value
         • values of the decision variables
         • reduced costs
         • slack/surplus
       In this chapter we will discuss:
         • changes in the coefficients of the objective 
           function
         • changes in the right-hand side value of a 
           constraint
                                                              
                                                               
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           Introduction to Sensitivity Analysis
       Sensitivity analysis (or post-optimality 
        analysis) is used to determine how the optimal 
        solution is affected by changes, within 
        specified ranges, in:
         • the objective function coefficients
         • the right-hand side (RHS) values
       Sensitivity analysis is important to a manager 
        who must operate in a dynamic environment 
        with imprecise estimates of the coefficients.  
       Sensitivity analysis allows a manager to ask 
        certain what-if questions about the problem.
                                                              
                                                               
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              Graphical Sensitivity Analysis
       For LP problems with two decision variables, 
        graphical solution methods can be used to 
        perform sensitivity analysis on
         • the objective function coefficients, and
         • the right-hand-side values for the 
           constraints.
                                                              
                                                              
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                         Example 1
       LP Formulation
                      Max     5x1 + 7x2
                      s.t.          x1            
                        <   6
                                   2x  + 3x   
                                 1     2
                        <  19
                                     x1 +   x2  
                        <   8
                                         x1, x2  > 
                         0                                    
                                                              
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