SensitivityAnalysis

Sensitivities of linear program coefficients

Since R2026a

Description

A SensitivityAnalysis object stores sensitivity analysis information about a linear programming solution. The object properties give the rate of change of the objective function value with respect to various coefficients, as well as the limits on the coefficient changes that preserve the optimal basis. For details, see More About

Creation

Create a SensitivityAnalysis object by specifying sensitivity as the sixth output of the linprog function:

[x,fval,exitflag,output,lambda,sensitivity] = linprog(...)

Properties

Each property is a table containing the five columns shown below.

Column Name	Description
LowerLimit	The minimum value the model parameter can take without altering the current optimal basis
UpperLimit	The maximum value the model parameter can take without altering the current optimal basis
ObjectiveValueAtLowerLimit	The objective function value at the optimal solution if the model parameter was changed to LowerLimit
ObjectiveValueAtUpperLimit	The objective function value at the optimal solution if the model parameter was changed to UpperLimit
ObjectiveValueChangeRate	The rate of change of the objective function value with respect to the model parameter in the range (LowerLimit, UpperLimit)

ObjectiveCoefficient — Sensitivity of objective function coefficient f

Read-only: N-by-5 table

This property is read-only.

Sensitivity of the objective function coefficient f, returned as an N-by-5 table, where N is the number of variables in the problem.

EqualityRHS — Sensitivity of linear equality constraint beq

Read-only: Neq-by-5 table

This property is read-only.

Sensitivity of the linear equality constraint beq, returned as an Neq-by-5 table, where Neq is the number of linear equality constraints in the problem.

InequalityRHS — Sensitivity of linear inequality constraint b

Read-only: Nineq-by-5 table

This property is read-only.

Sensitivity of the linear inequality constraint b, returned as an Nineq-by-5 table, where Nineq is the number of linear inequality constraints in the problem.

InequalityLHS — Sensitivity of linear inequality constraint bl

Read-only: Nineq-by-5 table

This property is read-only.

Sensitivity of the linear inequality constraint bl, returned as an Nineq-by-5 table, where Nineq is the number of linear inequality constraints in the problem.

LowerBound — Sensitivity of lower bound lb

Read-only: N-by-5 table

This property is read-only.

Sensitivity of the lower bound lb, returned as an N-by-5 table, where N is the number of variables in the problem.

Data Types: table

UpperBound — Sensitivity of upper bound ub

Read-only: N-by-5 table

This property is read-only.

Sensitivity of the upper bound ub, returned as an N-by-5 table, where N is the number of variables in the problem.

Examples

Obtain Sensitivities

Create a linear program.

f = [-2,-1];
A = [-1 -1;
    1 -1/2;
    -1 1];
bl = [-5 -Inf -Inf];
b = [Inf 2 3];
lb = [0 0];
ub = [];
Aeq = [];
beq = [];

Solve the linear program, requesting sensitivities.

[x,fval,exitflag,output,lambda,sensitivity] = ...
    linprog(f,A,b,Aeq,beq,lb,ub)

Optimal solution found.

x = 2×1

     7
    10

fval = 
-24

exitflag = 
1

output = struct with fields:
         iterations: 0
    constrviolation: 0
            message: 'Optimal solution found.'
          algorithm: 'dual-simplex-highs'
      firstorderopt: 0

lambda = struct with fields:
           lower: [2×1 double]
           upper: [2×1 double]
           eqlin: [0×1 double]
    ineqlinLower: [3×1 double]
         ineqlin: [3×1 double]

sensitivity = 
  SensitivityAnalysis with properties:

   Variables Sensitivity:
    ObjectiveCoefficient: [2×5 table]
              LowerBound: [2×5 table]
              UpperBound: [2×5 table]

   Constraints Sensitivity:
           InequalityLHS: [3×5 table]
           InequalityRHS: [3×5 table]
             EqualityRHS: [0×5 table]

Examine the sensitivities.

sensitivity.ObjectiveCoefficient

ans=2×5 table
    LowerLimit    UpperLimit    ObjectiveValueAtLowerLimit    ObjectiveValueAtUpperLimit    ObjectiveValueChangeRate
    __________    __________    __________________________    __________________________    ________________________

       -Inf           1                    -Inf                           -3                            7           
       -Inf           1                    -Inf                           -4                           10           

sensitivity.LowerBound

ans=2×5 table
    LowerLimit    UpperLimit    ObjectiveValueAtLowerLimit    ObjectiveValueAtUpperLimit    ObjectiveValueChangeRate
    __________    __________    __________________________    __________________________    ________________________

       -Inf            7                   -24                           -24                           0            
       -Inf           10                   -24                           -24                           0            

sensitivity.InequalityLHS

ans=3×5 table
    LowerLimit    UpperLimit    ObjectiveValueAtLowerLimit    ObjectiveValueAtUpperLimit    ObjectiveValueChangeRate
    __________    __________    __________________________    __________________________    ________________________

       -Inf          -17                   -24                           -24                           0            
       -Inf            2                   -24                           -24                           0            
       -Inf            3                   -24                           -24                           0            

sensitivity.InequalityRHS

ans=3×5 table
    LowerLimit    UpperLimit    ObjectiveValueAtLowerLimit    ObjectiveValueAtUpperLimit    ObjectiveValueChangeRate
    __________    __________    __________________________    __________________________    ________________________

        -17          Inf                   -24                            -24                           0           
       -1.5          Inf                    -3                           -Inf                          -6           
         -2          Inf                    -4                           -Inf                          -4           

For descriptions of the returned sensitivities, see SensitivityAnalysis.

More About

Relationship Between Sensitivity Analysis and Lagrange Multipliers

For linear programming problems, the Lagrange multipliers and sensitivity analysis parameters are closely related. Lagrange multipliers give the rate of change of the objective function value with respect to a change in parameters, but only at a solution, meaning the rate of change holds for an infinitesimal change in parameters. However, the same rate of change in objective can hold for a large range of parameters. The sensitivity analysis object give the range of parameters that have the same rate of change for the objective. The absolute value of the returned ObjectiveValueChangeRate in the sensitivity is the same as the absolute value of the associated Lagrange multiplier (they can differ in sign as explained in the next paragraph). For an example comparing Lagrange multipliers to sensitivities, see Obtain Sensitivities.

Why do the signs of the ObjectiveValueChangeRate and corresponding Lagrange multiplier sometimes differ? Lagrange multipliers give the rate of change in the objective function value with respect to a tightening of the constraint, meaning an increase in a lower bound or inequality left hand side, or decrease in an upper bound or inequality right hand side. These Lagrange multipliers are always nonnegative. In contrast, the ObjectiveValueChangeRate represents the rate of change of the objective function value with respect to a change in parameter, whether that change represents a tightening or loosening of a constraint. So the signs of the two rates agree for the lower bounds and the inequality right hand sides, and are opposite for the upper bounds and inequality right hand sides.

Optimal Basis

For a linear program, the optimal basis can be interpreted as a set of constraints that are active at the solution. Typically, this set is unique and changes only when some problem coefficients change sufficiently. For more information, see Basic and Nonbasic Variables.

In some cases, the optimal basis is not unique and can change with the slightest change in coefficients. In such cases, the reported intervals (LowerLimit,UpperLimit) for some parameters can have a width of 0 or close to 0. This singularity reflects a limitation of "single-basis" sensitivity analysis.

Similarly, for some problems with a nonunique basis, each reported range reflects the interval over which one particular basis remains optimal. Therefore, the reported lower and upper limits can be narrower than the full interval with the reported objective value change rate. In other words, a different optimal basis can have a wider range corresponding to the reported change rate.

Properties for Inactive Constraints

For an inactive variable or constraint upper bound, including cases where b or ub is Inf:

• LowerLimit is the variable or constraint value at the optimal solution.

• UpperLimit is Inf.

• ObjectiveValueAtLowerLimit is the objective value at the optimal solution.

• ObjectiveValueAtUpperLimit is the objective value at the optimal solution.

• ObjectiveValueChangeRate is 0.

For an inactive variable or constraint lower bound, including cases where bl or lb is -Inf:

• LowerLimit is -Inf.

• UpperLimit is the variable or constraint value at the optimal solution.

• ObjectiveValueAtLowerLimit is the objective value at the optimal solution.

• ObjectiveValueAtUpperLimit is the objective value at the optimal solution.

• ObjectiveValueChangeRate is 0.

Version History

Introduced in R2026a