Different solutions may be obtained with different
levels of risk. Several solutions are presented below with
increasing objective values and increasing risk. The constraint
risk is the number used to determine the RHS values,
and the system risk is the probability that the solution
will violate at least one constraint. The system risks were
each determined with a simulation of 1000 random observations.
Any number of solutions could added by selecting different
values
of risk
for the several
constraints.
Solution (Constraint Risk) 
Objective Value 
System Risk 
Chance Constrained (10%) 
94.5 
38% 
Chance Constrained (20%) 
105.2

65%

Chance Constrained (30%) 
112.9

82%

Combined Wait and See 
121.5

94%

Expected Value RHS (50%) 
125.6 
96% 
In this example the RHS values are independent random variables
and we have included chance constraints for each individual
constraint. Since the analyst is probably interested in finding
a solution with a given system risk, it would be useful to
construct a single constraint that assures this result. This
is called a joint chance constraint. Unfortunately,
this constraint is not easy to write. For simple cases, it
may be possible to write the constraint, but the resultant
model is not a linear program. In fact, the feasible region
of the resultant model is probably not convex, making the model
difficult to solve.
Even if we use only individual chance constraints, there is
no general guidance on how to set the risks of constraint violation.
The chance constraints do give the analyst a method for explicitly
recognizing uncertainty. 