Stochastic Programming - Index of

10 STOCHASTIC PROGRAMMING
1.3 An Illustrative Example
Let us consider the following problem, idealized for the purpose of easy
presentation. From two raw materials, raw1 andraw2, we may simultaneously
produce two different goods, prod1 andprod2 (as may happen for example in
a refinery). The output of products per unit of the raw materials as well
as the unit costs of the raw materials c = (c raw1 ,c raw2 ) T (yielding the
production cost γ), the demands for the products h =(h prod1 ,h prod2 ) T and
the production capacity ˆb, i.e. the maximal total amount of raw materials that
can be processed, are given in Table 2.
According to this formulation of our production problem, we have to deal
with the following linear program:
Table 2
Productivities π(raw i,prodj).
Products
Raws prod1 prod2 c ˆb
raw1 2 3 2 1
raw2 6 3 3 1
relation ≥ ≥ = ≤
h 180 162 γ 100
min(2x raw1 +3x raw2 )
s.t. x raw1 + x raw2 ≤ 100,
2x raw1 +6x raw2 ≥ 180,
3x raw1 +3x raw2 ≥ 162,
x raw1 ≥ 0,
x raw2 ≥ 0.
⎫
⎪⎬
⎪⎭
(3.1)
Due to the simplicity of the example problem, we can give a graphical
representation of the set of feasible production plans (Figure 2).
Given the cost function γ(x) = 2x raw1 +3x raw2 we easily conclude
(Figure 3) that
ˆx raw1 =36, ˆx raw2 =18,γ(ˆx) = 126 (3.2)
is the unique optimal solution to our problem.
Our production problem is properly described by (3.1) and solved by (3.2)
provided the productivities, the unit costs, the demands and the capacity
(Table 2) are fixed data and known to us prior to making our decision on the
production plan. However, this is obviously not always a realistic assumption.
It may happen that at least some of the data—productivities and demands for