Características del cine como medio publicitario

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 T_{x + 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].