As it has been explained in section 2.3, there are several approaches in dealing with randomness in stochastic problems. Here, we are considering the scenario-based approach that is pertinent to our problem.
A scenario-based approach is one of the approaches in dealing with randomness or uncertainty. In stochastic programming models, the scenarios are generated to represent the uncertainty in a sensible way while taking into account: the goal of the model and its structure, the available information and the availability of computer software [11, 68, 80].
The scenario-based approach assumes that there are a finite number of decisions that nature can make as the outcomes of randomness [11]. Each of the possible decisions or realizations is called a scenario. Scenarios deal with uncertain aspects of the random variables or parameters that are relevant to the need of the concerned problem [80]. Thus, the future uncertainty in the considered problem is usually described by a set of alternative scenarios. Some examples of scenarios are: the demand for a product is low, medium, or high; the weather is dry or wet; and the market price will go up or down. These are some examples with finite number of future realizations for stochastic modelling. The scenario-based approach can be used in both discrete and continuous random variables provided that there are finite number of realizations. However, even if the nature acts in a continuous manner, often a discrete approximation is mostly used in scenario-based approach [11, 66].
In the scenario-based approach, a scenario tree can be generated which will incorporate all possible realizations of discrete random variables or parameters into the model [80]. For the scenario tree, the number of scenarios as well as the progression of the scenarios from
[11, 68, 80].
To explain the scenario-based approach, we consider a two-stage linear stochastic model with discrete realizations of a random variable. Here, two-stage is based on the stages of decisions taken in solving the stochastic model. The decisions that must be taken before the random experiment, denoted by x, are called first-stage decisions. The period during which they are taken is called the first stage. Decisions that must be taken after the random experiment, denoted by y, are called second-stage decisions and its corresponding period is the second stage. Suppose the result of the random experiment is s ∈ S where S is the sample space of the random experiment, the sequence of decisions and events can be represented diagrammatically as x −→ ξ(s) −→ y(s, x). Thus the second-stage decisions are functions of the outcome of the random experiment and also the first-stage decision [17, 40]. An elementary detailed example for a two-stage stochastic problem is the news- vendor problem found in [11, 17, 68]. We now consider in the next paragraph the general two-stage linear stochastic model that can be transformed into scenario-based approach in dealing with discrete random variables.
Generally, a two-stage stochastic linear program with recourse function can be written as follows [11, 17, 40, 68]: Min x c Tx + E ξQ(x, ξ) (5.1) subject to Ax = b, (5.2) x ≥ 0, (5.3)
where Ax = b is the first stage constraints and Q(x, ξ) is the optimal value of the second stage problem (an extended real valued function or recourse function) given as
Q(x, ξ) = Min
y q
subject to Gx + W y = h, (5.5)
y ≥ 0. (5.6)
where G and W are called technology and recourse coefficient matrices for decision variables, x and y respectively. h is a right hand real value that limits x, y, G and W values. Here x and y are vectors of first and second stage decision variables respectively.
The second stage problem, (5.4) - (5.6), depends on the data ξ := (q, h, G, W ) and some or all elements of which can be random. So ξ is a random vector and Eξ denotes mathematical
expectation with respect to the probability distribution of ξ. This probability distribution is supposed to be known. The two-stage stochastic models where the random variables are fully known or realized, are solved as a “wait-and-see” solution method. On the other hand, when the stochastic models are solved before the realization of random variables, it is a “here-and-now” solution method. In this context, usually the random parameters are estimated using the historical data under probability distributions or density functions [11, 17, 39, 68]. The decisions to be made in “here-and-now” are for single-stage stochastic models [39]. In general, the random parameters or variables for stochastic models can be either in the constraints or in the objective function, or in both [11, 17, 39, 68].
We now consider equations (5.1) - (5.6) to have the discrete distribution in random data with a finite number of |S| possible realizations. These possible realizations, ξs:= (qs, hs, Gs, Ws),
s ∈ S, are called scenarios with corresponding probabilities Ps for its occurrence Pr(ξs) =
Ps. The other interpretation would be that the random vector ξs = ξ(s) depends on the
scenario s, which takes on S different values. In this case, EξQ(x, ξ) = |S| P s=1 PsQ(x, ξs), |S| P s=1
Ps = 1. This consideration is only for a single attribute. For several attributes, Pst or
can also be treated accordingly.
Under scenario-based approach, the model (5.1) - (5.6) can now be written in the form:
Min x,y1,...,ys cTx + S X s=1 PsqsTys (5.7) subject to Ax = b, (5.8) Gsx + Wsys = hs, ∀s, (5.9) x ≥ 0, ys≥ 0, ∀s. (5.10)
Problem (5.7) - (5.10) is the two-stage stochastic problem formulated as one large linear programming problem under scenario-based approach. The constraints (5.8) are known as the first stage constraints and (5.9) are the second stage constraints. Such a stochastic decision model is known as the extensivef orm of the stochastic program since it explicitly describes the second stage decision variables for all scenarios [11].
We would like to point out that the objective function in equation (5.7) is similar to our problem stated in equation (5.22); which is also a scenario-based problem. In our problem the constraints are not stochastic. Examples of scenario-based stochastic problems that are solved numerically can be found in [11, 17, 19, 31, 66, 80].