

Stochastic
Programming 


No Recourse 


With the no recourse situation,
we must fix the decision variables before the random variables
are realized. On this page we investigate some simple strategies
for finding solutions. We will finally conclude that these
strategies are not very satisfactory. The following pages suggest
better, but more difficult, solution methods.
One strategy is to replace the random variables by their expected
values and solve the resultant deterministic model. We might
ask how well this expected value solution serves as
the solution
to
the stochastic
problem. The expected value solution to our example problem
is shown below. Since the expected values of the random variables
are 0, the RHS values are just the constants originally proposed.




To evaluate this solution in a stochastic setting,
we create a different model below. The LP model is the same
as
before,
but now we
allow
the RHS
values
to vary
randomly
but do not solve the LP for each sample. For each sample, the
solution is fixed at the deterministic solution obtained with
the expected RHS. The objective value does not change because
the decision vector is fixed. The only question is what is
the probability that this solution will be feasible?
We have placed logical expressions
in G15:G19 to indicate when the constraints are feasible.
In the function form below, cell G33 contains a logical
expression
that is TRUE only when all constraints are
all satisfied. The LP solution on the worksheet is optimal
when the RHS takes the expected value, so this solution is
feasible. 



When we simulate the model for this
case we do not solve the model for every sample. The word "None"
in cell G23 assures this. The results are shown below. The objective
value shows no variability since the decision variables remain
the same for all samples. The proportion of feasible solutions
is, however, only 6%. The probability that this fixed solution
is feasible is very small. 



These results should
not be surprising. With the expected RHS values, four of the
five constraints are tight in the deterministic optimal solution.
Since the RHS values are Normally distributed, there is a 50%
that a tight constraint will be violated when the random RHS
is realized. Even if the single loose constraint is never violated,
the probability that all the tight constraints are satisfied
is 0.5 raised to the fourth power or 0.0625. The simulated result
is quite close to this value.
One might ask, how should the problem with no
recourse be solved? The answer to this question is the provence
of stochastic programming. 
Combining Wait and See Solutions 

One suggestion often made for
this kind of problem is to solve the problem as if we could
wait
and see the random realizations and then combine the waitandsee
decisions in some way. We do this for the example by creating
a new math programming model identical to the wait
and see model considered earlier, but create a simulation form that records
the values of the decision variables as well as the objective.
The form is below. We now have 11 functions defined. The first
is the objective function and the remaining ten are the values
of the decision variables. All eleven functions have the same
feasibility equation. A solution is recorded only when the LP
has a feasible solution.
We simulate the random variables, solve the LP
for each sample point and record the observed function values.
The results from 100 observations is below. Row 36 shows that
six of the decision variables have nonzero values in some of
the observations.
To continue, we impose the averagedecisionvalue
solution shown in row 36 on the LP model.
To evaluate this
solution, we simulate this model with the
solution fixed as above. Of course, since the solution is fixed
there
is no variability in the objective function. The results indicate
that the solution is feasible for 11% of the simulated RHS
vectors.


We have investigated two solutions to the
norecourse problem, the expected value solution and the mean
waitandsee
solution. The two solutions are shown in the table. When uncertainty
affects the RHS values of the constraint coefficients, there
is always a chance that some constraints will be violated by
a fixed solution. Choosing a solution involves a tradeoff
between
the risk of infeasibility and the expected value of the objective
function. Chance Constraints, described on the next
page, gain some control of the feasibility probability, but
again we will be left with a variety of solutions to choose
between.
Solution 
Objective Value 
Feasibility Probability 
Expected Value RHS 
125.6 
6% 
Combined Wait and See 
120.1 
11% 

