
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 function model below the LP model. We create
the model by calling the Function command of the Random
Variables addin. Five random variables are defined, each
with a Normal distribution. Note that the algorithm field is
blank. We use the Include Address option.
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 variables are fixed at the values
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 place logical expressions
in G15:G19 to indicate when the constraints are feasible.
Cell G31 holds the address of the function value (the objective
function of the LP), and cell G33 holds the address of the
logical expression that returns TRUE if the solution is feasible
and False if it is not. The LP solution on the worksheet is
optimal when the RHS values are the expected values, so this
solution is feasible. With the Include Address option,
cells G32 and G34 are initially blank. They are filled with
the appropriate transfer expression when the simulation begins. 



When we simulate the model for this
case we do not solve the model for every sample. 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 5%. 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
simulation shows that only 5% if the simulated RHS vectors
result in feasible solutions.
One might ask, how should the problem with no
recourse be solved? The answer to this question is the provence
of stochastic programming. 
Combining waitandsee
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 waitandsee
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. None of the solutions were infeasible as indicted
by cell G37. This is not surprising since the waitandsee
solutions observe the RHS vector before finding the solution.
Row 39 shows the mean values for the 100 observations.
The first entry shows the mean objective value of 125.97. The
remainder of the entries on row 39 show the average values
of the decision variables. Variables X1, X2, X4, X5, X8, and
X10 are nonzero for at least some of the observations. 



It is interesting to compare these
results against the results for 1000 observations. Row 39 shows
that the same set of nonzero variables except for X6. This variable
was nonzero for a single selection of random variables. 



The first 20 simulated results are
shown below to illustrate the variability of the waitandsee
solutions. 


The Average Solution


To continue, we impose
the average of the waitandsee
solution 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 of 1000 simulated RHS vectors indicate
that the solution is feasible for only 6.5% of the observations. 



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. The combined
waitandsee
solution has a slightly higher objective value
because it is not feasible for the expected RHS vector. Neither
fixed solution has a very high chance of being feasible when
there is randomness in the RHS vector.
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 alternative solutions.
Solution 
Objective Value 
Feasibility Probability 
Expected Value RHS 
125.61 
5% 
Combined waitandsee

125.97 
6.5% 


