MAKING STOCHASTIC PROGRAMING MODELS - FILLING THE BLANKS - (in MS-EXCEL/MS-WORD/CSV)
Jesus Velasquez-Bermudez
Decision-Making Artificial Intelligence Entrepreneur & Researcher - Chief Scientific Officer
Papers related:
1.??????????STOCHASTIC PROGRAMMING
?1.1.??????STRUCTURED MATHEMATICAL MODELING (SMM)
?Structured Mathematical Modeling (SMM) is defined, by the author, as a fundamental step in the process of socialization of the mathematical modeling, it is a necessity to ensure that the benefits arising from the applied mathematics extend to as many people as possible. This cannot be achieved, while the mathematical modeling is not within the reach of most professionals in engineering, economics and management sciences.
The main barriers must be overcome is the dependence of mathematical models from the Mathematical Programming technologies used to implement the models. The alternative is to normalize the formulation in such a way to ensure their portability between technological platforms. This standardization would allow professionals interested in the mathematical modeling the possibility of formulating their own models without to know in depth the syntax of a computer language; this fact would expand the number of mathematical modelers and diminish the level of expert computer knowledge required to formulate mathematical models.
?The power of computers, coupled with the power of Large Scale Optimization Methodology (LSOM), coupled with the power of the basic solvers change radically the environment of the mathematical modeler compared with the environment of the modelers of the past; then Stochastic Programming (SP) models should be common to math modelers and end-users of modern Mathematical Programming (MP).
This leads to that the state-of-the-art of applied optimization solutions should migrate, massively, from deterministic optimization models to stochastic models. Stochastic optimization is there for many years, the first work, that the author knows, is the related with Modern Portfolio Theory (MPT), or mean-variance analysis, that was introduced by Economist Harry Markowitz in 1952 (for which he was awarded with the Nobel Prize in Economics).
Then, SMM must include as part of its services the modeling of Multi-Stage Stochastic Programming (MS-SP) that implies to handle random processes over the decision trees and solve problems with different types of objective functions, for example: i) expected value; ii) MiniMax or Maximin and iii) maximum regret; additionally, SMM must include several alternative to risk management; for example, Conditional-Value-at-Risk constraints (CVaR).
?Therefore, it is necessary to normalize the process of conversion of deterministic (core) model into a stochastic model; this process may be automatic, in the sense that the user must only configure the conversion process and SMM generates the stochastic model from the deterministic formulation. Therefore, it is a valid conclusion that to use deterministic models, when uncertainty is an essential part of the decision-making process, is a matter of the past. More information about this topic can be read in Velasquez (2019d, 2019e).
This section provides a brief introduction to the problem of stochastic optimization, concepts that are necessary to understand the implementation of stochastic models using OPTEX. The section can be complemented with the paper:
?For the advanced reader, it is not necessary to read this section.
?1.2.??????STOCHASTIC DECISION-MAKING
?According to Powell (2017) the problem of making decisions uncertainty has motivated applications in several disciplines: spanning business, science, engineering, economics and finance, health and transportation, supply chain design and operations, … ; the decisions may be binary, discrete, continuous. Even richer are the different ways that uncertainty arises that creates a virtually unlimited range of problems. This diversity has generated the evolution of different mathematical modeling styles and solution approaches.
?Powell brings to the stochastic optimization communities a canonical modeling framework that covers all the different perspectives that have evolved over time. From this foundation, Powell reduce the rich array of solution approaches into a relatively small number of fundamental strategies, he is not “replacing these fields but rather building on them, somewhat like standing on the shoulders of giants”.
?The type of stochastic optimization models described by Powell are decision trees, stochastic search, optimal stopping, optimal control, Markov decision processes, approximate/adaptive/neuro dynamic programming, reinforcement learning, online computation, model predictive control, stochastic programming, robust optimization, ranking and selection, simulation optimization and multiarmed bandit.
?1.3.??????STOCHASTIC PROGRAMMING
?In the field of mathematical optimization, Stochastic Programming (SP) is a framework for modeling optimization problems that involve uncertainty. Below is presented the SP based on scenarios, also known as non-anticipative stochastic programming. SP considers multiple scenarios, associating each of them with a probability of occurrence, and the model determines the "best" decision before the occurrence of the scenario (non-anticipative decisions). This alternative requires a process to generate the scenarios, they may be exogenous (calculated outside of optimization model using a special stochastic model whose results are read as optimization parameters), or endogenous (calculated within the model optimization). It should be note, that in stochastic optimization there is no the "best" solution, in contrast to the deterministic optimization, already a fundamental part of the analysis is the management of risk associated with decisions taken; optimize just the expected value of the objective function, may lead to decision highly risks (risk-prone) with greater risk (volatility) than other decisions that do not seek this goal.
?Then, SP is methodology to solve stochastic optimization models using mathematical programing combined with random scenario generation. For this it is necessary to define a process that considers:
1.???Decision Tree, it may be generated using "split" variables with non-anticipative constraints, that is easiest way to express the decision tree.?
?2.???Stochastic Process, the uncertainty dimensions must be defined by the users, considering which is the more convenient stochastic process. An easy way is to link random variables to parameters and/or sets of the core model, including indexes to handle each dimension of uncertainty. Currently the dimensions of uncertainty are included directly in the formulation of the problem implying that change the uncertainty dimensions involves changing the code of the program; this limits the correct use of the models, since in many cases it uses the model that is available and not the model that really requires the problem that is solving.
3.???Risk Management:?The biggest advantage of stochastic models is the inclusion of risk measures in the stochastic model. Nowadays, the risk measure most used is the CVaR (Conditional-Value-at-Risk, Rockafellar and Uryasev).
?4.???Solution Process: the solution of the stochastic model can be accomplished through direct solution of equivalent deterministic problem (the random variables are fixed during the optimization process) or a “real” stochastic model (the random variables change during the optimization process) using LSOM. Sampling methods that may be included in the SMM algorithms.
1.4.??????DECISION PROCESS MODELING
?The decision-making process is represented by a decision-tree where: i) the branches are associated to probabilities of occurrence of events and ii) the nodes represent the decision-moments having as reference the events that have occurred and considering the conditional probability of the events that can occur. This process is based on hypothesis: "I do not know what will happen, but I know what can happen".
?SP involves several steps in the mathematical modeling:
1.???Stochastic Modeling:
2.???Risk Management: Selection of the objective function and the measure of risk and its constraints to calculate it.
3.???Optimization Methodology: Select the optimization methodology to solve the problem
SP not only allows to solve the mathematical it facilitates the probabilistic synthesis of the solution, since all the variables involved in the problem can be characterized probabilistically, based on empirical probability distribution functions that can be subject to hypothesis testing or any other statistical analysis. SP is the way to include risk management in the decision-making process.
?1.5.??????MULTI-STAGE DECISION PROCESS
?1.5.1.??STATIC DECISIONS – TWO STAGE STOCHASTIC PROGRAMMING
?Consider a two-stage decision-making process (SP-2S):
1.???The first phase, corresponding to the initial moment, the beginning of the period 1 (here and now), the decision, and subsequent to the taking of the decision, during the period 1, occurs a realization of the stochastic process w, and
2.???In the second stage, happens a new decision process over x, at the beginning of period/stage 2, when it is known the random process occurred during stage 1. The above process can be represented by a two-stage decision tree.
The application of Benders Theory (Benders 1962) to SP-2S was first done by Van Slyke and Wets (1969) and is known as the L-Shaped method (LS), which is related with two stage/period stochastic problems SP-2S.
?The LS model is equal to the model used to study the Benders Decomposition (BD) in a previous chapter (Velasquez, 2019a); but, LS include the probability qi of each scenario i. All the theory presented to BD is valid to solve LS:
where N represents the number of scenarios.
1.5.2.??DYNAMIC DECISIONS – MULTISTAGE STOCHASTIC PROGRAMMING
?Consider a decision-making process of several stages (SP-MS), which can be interpreted as a concatenation of multiple trees of two stages. At the beginning of the first stage, t=1, the decision x1 is made during stage 1, then occurs a realization of the stochastic process, w1; subsequent to x1 a new decision, x2,?must be make x2 and wait for carrying out the scenario w2;?subsequently decision x3 must be make and so on until to reach the last stage, t=T
?The above process can be represented by a multi-stage decision tree; each branch corresponds to a scenario h and each node n to a decision-making moment. The algebraic formulation of this problem is
1.6.??????UNCERTAINTY DIMENSIONS?
A fundamental aspect of the stochastic optimization models is related to the dimensions of uncertainty, it is the parameters and sets that are considered as random variables in the process of generation of scenarios. When the mathematical model is strong linked to a computer program, change the dimensions of uncertainty implies the reprograming of the computer program code.
?1.7.??????SCENARIO GENERATION
?According to Murphy (2013), in order to make stochastic optimization problems tractable, in general for continuous o nonenumerable stochastic process, it is necessary to make an approximation of the possible future evolution of the universe. The key step is to make a discretization. The idea of discretization is to approximate the probability distribution of the future with a finite number of scenarios, in each of which the universe moves into some specified state. When taken together, the (discrete/empirical) distribution of the unknown variable or variables across these scenarios should reflect the hypothesized (continuous) distribution of the variable(s) being modelled.
?The figure, extracted from Glasserman, P. (2003), shows the distribution of a continuous stochastic variable represented using 10, 100 and 1000 scenarios. The black line shows the normal distribution being approximated. As the number of scenarios improves, the approximation more closely approximates the hypothesized one. There are many sampling techniques to improve the Monte-Carlo scenario generation, but they are not an optimization methodology, they may be associated with experiments design (Chung and Spall, 2015); but nevertheless, the modeler must be conscious that they are fundamental part of the model.
The problem is most interesting when it requires analysis of risks, or strength, of the decisions which it is required to analyze the behavior of the decisions in one of the tails of the distribution, which is of very low probability, but high socio/economic impact.
1.8.??????RISK MANAGEMENT
The risk is inherent in all decisions under uncertainty and therefore your measurement should be taken as a reference for the "strength" of the decisions that are taken. The introduction of a risk constraint, or a risk objective, gives rationality/robustness to the SP problems. In this section only will be present the introduction of Conditional-Value-At-Risk (CVaR) risk constraint in SP.?
?1.8.1.??RISK MODELING??????
?Nowadays, the risk measure more known may be the Value at Risk (VaR) that corresponds to the upper limit of a confidence interval for the losses associated with a portfolio of investments to a certain level of probability. This concept is not new, since it was treated by Edgeworth in 1888, but the practical modern developments date back to 1994 when the J.P. Morgan firm launched its RiskMetricsTM product, date from which use has become a standard of the modern financial engineering.
?Although VaR is a simple and easy-to-understand risk measure, has not desirable mathematical properties (Artzner et al. 1999) such as the no-sub-additivity, which implies that the VaR corresponding to two instruments cannot be greater than the individual sum of the VaR associated to each of them. Additionally, VaR is difficult to introduce in stochastic optimization models based on the concept of scenarios because of their non-convex character (Mausser and Rosen 1991).
?For the above reasons, the concept of Conditional Value-At-Risk (CVaR or mean excess loss) is introduced, like the concept of VaR, and with more attractive mathematical properties (Palmquist et al. 1999). The introduction of VaR constraints on optimization problems has been studied widely (Anderson et al. 2001, Rockafeller and Uryasev, 2000, Uryasev 2000). It has been demonstrated that the appropriate form to consider VaR constraints is using the Conditional Value-at-Risk (CVaR) that is the expected loss exceeding Value-at-Risk, CVaR is also known as Mean Excess, Mean Shortfall, or Tail VaR.
?For an empirical sample, such as those provided by a stochastic optimization model, the equation that relate the VaR and the CVaR depends on the direction of optimization, since the risk is associated with one of the two tails of the empirical distribution of the objective function; for minimization of costs the risk is associated with the upper tail; for maximization the lower tail is the objective of risk management, for this CVaR is associated with deficit of revenue. The figure shows the relations for the minimization case.
The equation for the case of minimization of costs is
?CVaRb(x) = VaRb(x) + (1-b)-1 ∑i=1,N?qi x Max[ 0, Fi(x) - VaRb(x) ]
?where x represents the vector of decisions, Fi(x) the costs associate to the scenario i, b?the probability of VaR(x) can be exceeded and qi the probability of scenario i and CVaRb(x) the associate to VaR(x) ; CVaRb(x) and VaRb(x)?are new variables includes in SP: model, they will be called VaR and CVaR.
To formulate a linear problem, the definition of CVaR must be replaced by the following equations:
where x is the upper bound for CVaRb(x) and wi represents the excess of cost with respect to VaRb(x) if is taken the decision x and occurs the random condition i. The previous constrains must be join to the techno-economical constraints of SP:, and Fi(x) must be replace by f(y) + ct,iT xt,i.
It is important to note that this problem may hasn’t feasible solution due to the constraint CVaR ≤ x.?where x is the upper bound for CVaRb(x) and wi represents the excess of cost with respect to VaRb(x) if is taken the decision x and occurs the random condition i. The previous constrains must be join to the techno-economical constraints of SP:, and Fi(x) must be replace by f(y) + ct,iT xt,i. It is important to note that this problem may hasn’t feasible solution due to the constraint CVaR ≤ x
This may be the most rational approach from the point of view of risk management, since it introduces a measure of risk, quantifying its value and introduces constraints on maximum risk that the decision maker wishes to assume.
?1.8.2.??DECISION-MAKER CRITERIUM??????
Unlike deterministic optimization, SP must face subjectivity problems associated with attitudes that have makers against the risk. In this aspect there is a unique solution which can be considered as universally optimal, and part of the decision-making process is related to the handling by the decision-maker of these concepts.
?Unfortunately, the decision maker must choose between two faced criteria; on the one hand, he/she wants to: i)?minimize the expected cost (or maximize expected income), on the other hand, ii) wants to minimize the risk assumed. The true value of SP models is the construction of a Pareto curve expected value versus risk to help the decision-maker.
There are several measures of the risk associated with a decision; for the case of the cost exist: i) variance (or the standard deviation), ii) maximum cost, iii) maximum regret, iv) Value-At-Risk (VaR) and v) Conditional-Value-At-Risk (CVaR). The position of the decision-maker is directly associated with the objective function of the mathematical models. Four alternatives are considered for the objective function: i) Expect Value, ii) Mean-Variance, iii) Maximin (or Minimax), and i)?Maximum Regret (MR).
?2.??????????OPTEX: ADVANCED CONCEPTS
To read this section, it is suitable that the reader previously read:
?2.1.??????STRUCTURED MATHEMATICAL MODELING (SMM)??
?Structured Mathematical Modeling (SMM) is defined, by the author, as a fundamental step in the process of socialization of the mathematical modeling, it is a necessity to ensure that the benefits arising from the applied mathematics extend to as many people as possible.
?The main barriers must be overcome is the dependence of mathematical models from the Mathematical Programming technologies used to implement the models. The alternative is to normalize the formulation in such a way to ensure their portability between technological platforms.
?The standardization process must define:
1.???SMM Basic: the part of the mathematical modeling process that is included in the standard. This implies: i) regulate by a common agreement made by the representatives of all the mathematical modeling-related communities, and ii) that it should be “mandatory” for the industry and
2.???SMM Advanced: the part of the process that is covered by the optimization companies, as a way of differentiation of products and services offered; it is not binding, but it is convenient for humanity.
?SMM standardizes the management of entities and relationships centered about its database algebraic language that must allow management of linear and non-linear equations.
?Sort the elements that are part of a mathematical model around the concepts of RDB involves the need to structure the process of mathematical modeling in a way to store all elements in the tables of the SMM; this implies organize the mathematical model from an "universal" point of view of a relational information system; then, it is possible to affirm that the information system that supports SMM mathematical modeling is the first step towards normalization of the algebraic formulation and the use of mathematical models.
?The implementation of the SMM can be arranged by stages (levels), at least two must be considered:
1.???SMM Basic: includes entities related with: i) integrated basic models and ii) IDIS data model. Reference: Making MATH Models as LEGO Models
2.???SMM Advanced: includes entities related with stochastic programming and multi-problem models.
The implementation of Stochastic Programming in OPTEX is presented in this document.
2.2.??????CONCEPTUALIZATION?
?The information defined in this section is intended to define problems and mathematical models in accordance with the methodologies chosen for its solution. OPTEX orientation towards handling large size problems, it is important to consider the definitions that are typical of OPTEX. In the setup process, the user must define:
This menu gives access to configuration problems and models and their integration within a decision support system. Additionally, includes the definition of decision trees for non-anticipative stochastic optimization.
Problems or models can be simple problems, in a direct relationship problem-model, or may be associated with cycles of solution depending on the methodology chosen by the modeler to address the solution. For each model OPTEX creates a computer program in an algebraic/computer language (GAMS, IBM OPL, AIMMS, C, … ).
?2.3.??????PROBLEMS (TABLE: PROBLEMA)
?2.3.1.??DEFINITION OF PROBLEMS
?A problem is associated to a set of constraints that define it and a set of variables over which it has control. The configuration of a problem involves the definition of:
?????????- BECO??Benders Theory Coordinator
?????????- BESU??Benders Theory Sub-problem (primary)
?????????- RLCO??Relaxation Lagrangian Coordinator
?????????- RLSU??Relaxation Lagrangian Sub-problem
?????????- IN???????Integrated
A primary problem corresponds to the last level of the hierarchy in a multilevel scheme.
The user must set the restrictions and the variables that define the problem; to do this the user must access the related tables:
When a problem is integrated (IN) only must specify the restrictions, since the variables are automatically defined as appearing in the restrictions.
?2.4.??????MODELS (TABLE: MODELO)
?2.4.1.??DEFINITION
?The model corresponds to the fundamental mathematical that OPTEX takes to solve problems of mathematical programming. To define a model several aspects should be considered:
The process of configuring a model involves the definition of:
In addition, the user must define the problems that make up the model, so the user must access the table that allows to manage the relationship between the model and the mathematical elements that compose it.
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2.4.2.??PROBLEMS OF THE MODEL
It defines a model based on the set of problems that comprise it. The process of configuring a model involves the definition of:
Model: name or the model code.
Problem: name or code of the problem
??3.??????????STOCHASTIC OPTIMIZATION MODELS
?To developed stochastic programming models, the general conception of OPTEX is to transform “automatically” a deterministic optimization model to a stochastic non-anticipative model, using the same equations in both cases.
?The conversion process of a deterministic model to one of stochastic programming only involves fill fields in tables (filling the blanks), since the conversion of all the mathematical elements and structures of information is performed automatically by OPTEX.
?The number of dimensions of uncertainty is defined by the user, who can adjust them to the convenience and representativeness of each type of study realized with the mathematical model.
?3.1.??????GENERAL CONCEPTS
?This expands the definition of the model to contemplate additional concepts that make easier to work in an environment with deterministic models and stochastic models, whose proper implementation depends on the specific problem that the user wants to resolve and the availability of computing capacity that is available.
?This shall be taken into account the following concepts/definitions, which should be added to a model deterministic to convert it into stochastic:
o????Dimension of uncertainty
o????Composite random scenario
o????Random parameters
o????Non-anticipative variables
o????Non-anticipative restrictions
For more information on the technologies and methodologies of stochastic, suggests the interested user to consult:
?It should be noted that the modeler does not require to know this information to formulate models of stochastic optimization in OPTEX.
?3.2.??????RANDOM SCENARIOS
?The non-anticipative stochastic optimization is also known as optimization by random scenarios, which are usually associated with one of the indexes that characterized the existence of all the algebraic elements that make part of the model (parameters, variables, constraints). There are so many scenarios as terminal points as has the tree of scenarios.
?Below, two alternatives for the configuration of scenarios trees:
?In any case it should be considered as a fundamental aspect of the process that is the correlation between random variables considered in the model. Correlation corresponds to a time-space process that links random variables that occurred in different places and at different times of the planning period; and it may be a determining factor of the results and the risks that are taken as a result of the decisions process
To organize the conceptualization of multidimensional random scenario generation, must be consider the following definitions:
o??Results of an exogenous synthetic generation (Monte Carlo simulation) of the random values; or
o??Function of probability distribution parameters/set so that the generation of random variables are part of the stochastic model load process. In this case the random variables may be read or may be calculated during the optimization process.
?3.3.??????DECISION PROCESS (DECISION TREE)
?Non-anticipative stochastic optimization models require the definition of a tree that represents the dynamics and variability of the stochastic process associated with the model and relates it to the process of decision-making that is being analyzed. Below is briefly present the implied mathematical process.
?Consider a two-step decision-making process. In the first phase, corresponding to the initial moment, the beginning of the period 1 (here and now), taking the decision y, subsequent to the taking of the decision, during the period 1, occurs a realization of the stochastic process w, and subsequent to this process must take a new decision x, at the beginning of period 2. Decision y is unique and independent of the stochastic process whereas decision x is dependent, a posteriori of stochastic process, and simulate so many decisions x as random scenarios. The uniqueness of y makes the process non-anticipative, this means that the user doesn’t know that will happen, but the user know that can happen, since it assumes referred to the probability of occurrence of the scenario, and the values of the random parameters on each node.
Now consider a decision-making process of multiples stages, which can be interpreted as a concatenation of multiple trees of two stages. At the beginning of the first stage, identified as t = 1, the x1 decision variable is taken; after, during the period associated with the duration of the stage 1, occurs a realization of the stochastic process w1; subsequent to this process occurs the user must take a new decision x2 and wait for carrying out w2 to later decide to x3 and so on until the user reach the last stage (t=T), last period of the planning horizon.
The above process can be represented by a multi-stage decision tree. In the tree branches and nodes, each branch corresponds to a stage h and each node n to a state of decision-making.
To describe the topology of the decision-making process is required to define the nodes, the periods associated with each node (can be more than one) and branches (random conditions) that pass through the node. In addition, it should know in each node the value of the parameters representing the stochastic process and the probability of reaching the node conditioned in the node that precedes it.
To ensure the condition of non-anticipative, it is necessary to define the control variables on which applies the restriction that guarantees that the decision taken on each node is the same for all non-anticipative variables, for all branches, that they pass through the node. Below it is explained with more mathematical detail this restriction.
To understand the non-anticipative constraints, that there are so many parallel branches as random scenarios ("variable split") and that in each node is imposed a restriction on equality, which ensures that all the decisions that are taken on a node, for the branches that pass through it, are equal in their numeric value.
?To view it mathematically, we define the vector of decision variables of xt,h as the decision which was taken in stage t if stage h?occurs (transition by the branch of the tree h). To ensure the non-anticipative is required to introduce a restriction that even decisions that are taken in the node n, which means to make the non-anticipative restriction as
?zn = xt,h
where e(n) represents the set of branches (scenarios) associated with the node n, B(n) to the set of branches (scenarios) associated with the node n ("scenario bundle") and zn an auxiliary variable that is known as the information state variables ("information state vectors", Higle et al. 2002).
3.4.??????DEFINITION OF STOCHASTIC DECISION TREES
To represent this, in MMIS the user must define the following information in the master table of decision trees:
A?????Decision tree defined by the user in detail based on a single dimension of uncertainty
D?????Decision tree defined based on multiple dimensions of uncertainty that are automatically combined by OPTEX.
?Additionally, the user must set the following aspects:
?3.4.1.??TOPOLOGY
?From the master table of decision trees, through access of subordinate tables must be specified the tree nodes, the nodes relationships with branches (random conditions) and non-anticipative variables.
?It must be defined when defining the nodes of the tree:
?For each node the user must set the branches (basic scenarios, random conditions) passing through the node.
Code Node: code or name associated with the node
Branch of the Tree: code or name associated with the composed random stage passing through the node.
?3.4.2.??NON-ANTICIPATIVE VARIABLES
?For the decision tree the user must define non-anticipative variables that should be considered in each node, for A-type decision trees. D-type scenarios, the non-anticipative variables will be used to establish a model of stochastic optimization of two stages, where the non-anticipative variables correspond to deterministic variable of the first stage.
If non-anticipative variables are not defined, the generated model will be stochastic parallel type, which can incorporate constraints to manage the risk and different types of objective functions of stochastic optimization models
?3.4.3.??MULTIDIMENSIONAL STOCHASTIC PARAMETERS
To convert a deterministic model into a stochastic model is essential to define the parameters that will be considered as random and associate them to a dimension of uncertainty. For this the user must set the table that relates the decision trees with the parameters:
3.5.??????EXAMPLES OF NON-ANTICIPATIVE STOCHASTIC OPTIMIZATION
?Two cases are used to show the process to fill the tables (templates) to convert the deterministic model in a stochastic programming model; the cases are: i) electric dispatch system and ii) design of resilient supply chains.?
?3.5.1.??ELECTRICITY DISPATCH PROBLEM
?It is common to find models of electrical systems that consider fixed the dimensions of uncertainty, in most cases is limited to the hydro-climatic variables) (mainly to water resources). This management of uncertainty may be valid in case of planning/programming in the short/medium term;
?The problem with the dimensions of uncertainty is that these may change depending on: i) the use of the mathematical model, or ii) the dynamics of the system over time. For example, in an electrical system there are multiple dimensions of uncertainty, now consider various circumstances in which the uncertainty dimensions may be characterized:
However, for example, if model that only has the water resource as the unique uncertainty dimension is used to analyze the expansion of a power system in the long run, it would be ignoring important random variables such as: i) price of fuels, ii) demand for electricity, iii) economic growth, iv) infrastructure cost, and v) decisions of the generators (in free markets), ... . Ignore all these dimensions of uncertainty will lead to high financial risks.
The example shows the form to include three random variables in a model of a power system: i) hydro-climatic variables, ii) demand for electricity and iii) fuel prices. This process begin after the deterministic model is ready.
?Below is the form of construction decision tree, for it is initially used an example of reference which only considers a dimension of uncertainty; for example: the hydrological contributions, to a system of hydroelectric generation.
?The characteristics of decision tree in reference are:
?The above is shown loaded in OPTEX information system in the next window
?Now consider the addition of two dimensions of uncertainty: the cost of fuel and electricity demand. For each dimension, consider three possible scenarios with the same probability: high, medium and low. Similarly, to the hydrology, each random condition is represented by a set of series of temporary data, monthly, representing the demand on each node of consumption and price for each thermal power plant. Combining the different scenarios, 4 of hydrology, 3 of demand and 3 of prices, are configured 36 scenarios (4 × 3 × 3).
?To consider the dimensions of uncertainty they should be incorporated into the table of indexes associated with the dimensions of uncertainty.
?Now consider three basic parameters which will be modelled as random variables, each of them associated with a dimension of different uncertainty, implying that it is considered that there is any correlation between these three types of random variables. This implies to characterize them as follows:
?This implies that when the referenced parameters are part of a stochastic model will be displayed in all the equations where appear as CCBt,k,hp, DMAt,si,hh and HAMt,r,hd
?The expansion of the dimension of uncertainty implies that all the equations parameters in which there are these parameters should be considered as random, OPTEX manages this process automatically. This is because a tree associated with the scenario of uncertainty that the user wants to analyze, which must characterize as D, so that it is identified by OPTEX.
When the scenarios considered for a dimension of uncertainty have not the same probability is necessary to define in the master table of scenarios of the dimension of uncertainty, the probability associated with each dimension, thus the master table will have a generic structure such as that below:
?where ddd represents the symbol used to identify the dimension of uncertainty using a relational field
?The scenarios considered for each dimension will be referred through the set _DIM_ddd, this is
?ddd→_DIM_ddd
?The probability associated to each scenario/element is linked to the parameter _PRO_dddddd, where ddd represents the dimension of uncertainty.
?Three tables, one for each dimension of uncertainty are required to describe the random conditions
Tables in which the data is read must include the field COD_ddd to storage the code of the scenario. Previous tables shall be recorded with data from studies of synthetic simulation of exogenous scenarios provided.
?The composed scenario, represented by the set _CAL, is the result of the combination of all dimensions of uncertainty to which belong the random model parameters.
?h→__CAL = {S01, S02, … S36}
?The number of total scenarios shall be equal to the multiplication of standards of every one of the master sets, this is
?_NO(_CAL) = _NO(_DIM_hp) × _NO(_DIM_hh) × _NO(_DIM_hd)
?To establish the relationships between the dimensions of uncertainty requires the following sets
The probability of a set of scenarios of the decision tree scenario is calculated based on the multiplication of probabilities associated to each one of its elements
?PRO_CALh =
_PRO_ hphp→CAL(h) × _PRO_hhhd→CAL(h) × _PRO_dhhd→CAL(h)
?3.5.2.??DESIGN OF RESILIENT SUPPLY CHAIN
?This case presents the process to be followed to implement a stochastic programming model for designing resilient supply chains.
?To consider extreme events that may affect the normal operation of the supply chain are two random parameters are included in the model: i) demand and ii) extreme events. Given that two independent stochastic processes are assumed, each random parameter forms a dimension of uncertainty.
?Below the process the automatic conversion implies:
?1.???To include the indexes related with the uncertainty dimensions: i) ev for catastrophic events and ii) hh for demand
2.???To define a decision tree: DEM
3.???To specify the non-anticipative variables: OPS and OPV
?4.???To specify the parameters with the uncertainty dimensions: DEMM and EFUN
5.???To include in the tables the fields associated to the uncertainty dimensions. Below, table DDMES related with the demand
6.???To link the model with the decision tree
?7.???To link define the risk constraint (CVaR).
Below is presented two screenshot of a GAMS stochastic model generated by OPTEX. The first shows the declaration of non-anticipative constraints.
?The second shows the non-anticipative constraints.
Below, it is presented the results of the stochastic variable (DPZ) that includes the fields related with the uncertainty dimensions.