# Sensitivity

Sensitivity or post-optimal analysis is extra information that is provided about the current optimal solution. lp_solve provides a substantial amount of sensitivity information. Several API calls are available to retrieve the sensitivity: get_sensitivity_obj, get_ptr_sensitivity_obj, get_sensitivity_objex, get_ptr_sensitivity_objex, get_sensitivity_rhs, get_ptr_sensitivity_rhs, get_dual_solution, get_ptr_dual_solution, get_var_dualresult. The lp_solve program doesn't show the sensitivity by default. So to see the sensitivity information, use the -S4 option. The best way to explain this is via an example.

    min: +COLONE +3 COLTWO +6.24 COLTHREE +0.1 COLFOUR;
    THISROW: +78.26 COLTWO +2.9 COLFOUR >= 92.3;
    THATROW: +0.24 COLONE +11.31 COLTHREE <= 14.8;
    LASTROW: +12.68 COLONE +0.08 COLTHREE +0.9 COLFOUR >= 4;
    COLONE >= 28.6;
    COLFOUR >= 18.00;
    COLFOUR <= 48.98;

The solution of this model is (with the -S4 option):

    Value of objective function: 31.7828

    Actual values of the variables:
    COLONE               28.6
    COLTWO               0
    COLTHREE             0
    COLFOUR              31.8276

    Actual values of the constraints:
    THISROW              92.3
    THATROW              6.864
    LASTROW              391.293

    Objective function limits:
                                  From            Till          FromValue
    COLONE                              0          1e+030         -1e+030
    COLTWO                       2.698621          1e+030       0.5123946
    COLTHREE                            0          1e+030       0.7016799
    COLFOUR                             0       0.1111679         -1e+030

    Dual values with from - till limits:
                              Dual value          From            Till
    THISROW                   0.03448276             52.2         142.042
    THATROW                            0          -1e+030          1e+030
    LASTROW                            0          -1e+030          1e+030
    COLONE                             1        -1.943598        61.66667
    COLTWO                     0.3013793       -0.6355993       0.5123946
    COLTHREE                        6.24         -4841.16       0.7016799
    COLFOUR                            0          -1e+030          1e+030

First look at 'Objective function limits' (via API obtained by get_sensitivity_obj, get_ptr_sensitivity_obj, get_sensitivity_objex, get_ptr_sensitivity_objex). There is a list of all variables with for each variable 3 values. From, Till and FromValue. From - Till gives the limits where between the objective value may vary so that the solution stays the same. For example, variable COLFOUR has a From value of 0 and a Till value of 0.1111679. The coefficient in the objective function of this variable is 0.1. This means that as this coefficient varies between 0 and 0.1111679, the solution doesn't change. The values for the variables and the constraints will remain unchanged as long as the objective coefficient stays in this range. The objective function value will vary of course and also the sensitivity information of the other variables, but the solution will stay the same if only the objective coefficient value for this variable is changed. When the objective coefficient of variable COLFOUR is above 0.1111679 then the solution will change. The FromValue is only meaningful if the variable has a value of 0 (rejected). This is the value that this variable becomes when the From (minimization) or Till (maximization) value is reached. For example, the variable COLTWO that has an amount of 0 will become 0.5123946 if the coefficient in the objective function of COLTWO reaches 2.698621. Note that you only get information about this variable. There is no information what the values will be of the other variables. In a blending example where the coefficients of the objective function are generally the prices of ingredients this information tells you at what point a price may change to have the same composition and the FromValue says at what value a relected ingredient will be taken when the price lowers till the lower range value.

Another piece of information are the Dual values with the from - till limits. This is provided for both the constraints and the variables. The information is the same for both. For example, constraint THISROW has a dual value of 0.03448276 with a From value of 52.2 and a Till value of 142.042. This means that the Dual value specifies how much the objective function will vary if the constraint value is incremented by one unit. This implies that there is only a non-zero dual value if the constraint is active. Constraint THATROW for example is not active because the constraint is <= 14.8, but its value is only 6.864. However constraint THISROW is >= 92.3 and its value is also 92.3, thus active. If the constraint is changed to 93.3 (+1), then the objective value will be the current value + change * dual value = 31.7828 + 1 * 0.03448276 = 31.81728. However this is only true for the range From - Till, which means that the dual value stays only the same as long as the constraint lies between the From - Till limits. The moment that you are outside of these limits, the dual value will change. The dual value gives a very good indication how much this restriction costs. If the dual value is very high then this constraint is very influential on the objective function and if you are able to change it a bit then the solution will be much better. Also the sign of the dual value has a meaning. A positive value means that as the restriction becomes larger, the objective value will be larger, and as it becomes more negative, the objective value will be smaller. Also note that changes in the restrictions, even between the limits, can cause the solution to change. The from - till limits only say that the cost will remain the same, nothing less, nothing more.

#### Inaccurate sensitivity analysis in a model with integer variables

The sensitivity analysis doesn't take the integer restrictions in account. This is almost impossible since it would ask too much calculation time. In particular the from - till limits on both the objective function and the dual values are trustworthy. They only apply for the current solution without the integer restrictions. Keep this in mind. The dual values are correct.