Stochastic Programming - Index of

BASIC CONCEPTS 7
1.2 Preliminaries
Many practical decision problems—in particular, rather complex ones—can
be modelled as linear programs
⎫
min{c 1 x 1 + c 2 x 2 + ···+ c n x n }
subject to
a 11 x 1 + a 12 x 2 + ···+ a 1n x n = b 1
⎪⎬
a 21 x 1 + a 22 x 2 + ···+ a 2n x n = b 2
(2.1)
.
.
a m1 x 1 + a m2 x 2 + ···+ a mn x n = b m ⎪⎭
x 1 ,x 2 , ···,x n ≥ 0.
Using matrix–vector notation, the shorthand formulation of problem (2.1)
wouldreadas
⎫
min c T x ⎬
s.t. Ax= b
(2.2)
⎭
x ≥ 0.
Typical applications may be found in the areas of industrial production,
transportation, agriculture, energy, ecology, engineering, and many others. In
problem (2.1) the coefficients c j (e.g. factor prices), a ij (e.g. productivities)
and b i (e.g. demands or capacities) are assumed to have fixed known real values
and we are left with the task of finding an optimal combination of the values for
the decision variables x j (e.g. factor inputs, activity levels or energy flows) that
have to satisfy the given constraints. Obviously, model (2.1) can only provide
a reasonable representation of a real life problem when the functions involved
(e.g. cost functions or production functions) are fairly linear in the decision
variables. If this condition is substantially violated—for example, because of
increasing marginal costs or decreasing marginal returns of production—we
should use a more general form to model our problem:
min g 0 (x)
s.t. g i (x) ≤ 0, i=1, ···,m
x ∈ X ⊂ IR n .
⎫
⎬
⎭
(2.3)
The form presented in (2.3) is known as a mathematical programming problem.
Here it is understood that the set X as well as the functions g i :IR n → IR,i=
0, ···,m, are given by the modelling process.
Depending on the properties of the problem defining functions g i and the
set X, program (2.3) is called
(a) linear, ifthesetX is convex polyhedral and the functions g i ,i=0, ···,m,
are linear;