The example on this page is from
the field of stochastic programming. Some
of the
parameters
of
a
situation
are originally not known with certainty, however, probability
distributions for their values are given. Certain decisions
must be made before the random variables are realized. After
they are realized, other decisions may be made. The latter
are called the recourse decisions, and this type of
problem is called decision making with recourse. We
use the Jensen LP/IP addin to solve the example, thus providing
an illustration of the Lshaped method.
This example was borrowed from An Optimization Primer,
An Introduction to Linear, Nonlinear, Large Scale, Stochastic
Programming and Variational Analysis, by Roger JB Wets,
January 11, 2005 (unpublished manuscript). A separate section
of this site considers Stochastic
Programming in detail.
Consider a product
mix problem. A furniture maker
makes
four
products:
P1, P2,
P3 and P4. Two manufacturing resources are required: carpentry
and finishing. The requirements
measured in hours per unit are known and shown in the table
below along with the profit per unit of product.
Product Parameters 
P1 
P2 
P3 
P4 
Carpentry Hours per Unit 
4 
9 
7 
10 
Finishing Hours per Unit 
3 
1 
3 
4 
Profit per Unit 
15 
25 
21 
31 
Our problem is to select the product mix to maximize total
profit, but the availability of the resources are not known.
Rather we have four equally likely estimates of the hours
available for each resource.
Resource Distribution 
1 
2 
3 
4 
Carpentry Hours Available 
4800 
5500 
6050 
6150 
Finishing Hours Available 
3936 
3984 
4016 
4064 
Probability 
0.25 
0.25 
0.25 
0.25 
The mix chosen will require some number of hours for the resources.
Depending on the amount that is available we will pay for extra
hours to obtain the necessary number of hours.
This is called a decision making with recourse. We must chose
the mix before the uncertain quantities are known. Once the
uncertainty is realized we must make additional decisions,
called the recourse decisions, to adjust for the new conditions.
We use the notation below to describe the model with accuracy.
For the example the possible values of the random variables
are given in the table above. The probabilities associated
with these values are 0.25.
