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Drought conditions at a given location evolve randomly through time and are typically characterized by severity and duration. Researchers interested in modeling the economic effects of drought on agriculture or other water users often capture the stochastic nature of drought and its conditions via multiyear, stochastic economic models. Three major sources of uncertainty in application of a multiyear discrete stochastic model to evaluate user preparedness and response to drought are: (1) the assumption of independence of yearly weather conditions, (2) linguistic vagueness in the definition of drought itself, and (3) the duration of drought. One means of addressing these uncertainties is to re-cast drought as a stochastic, multiyear process using a “fuzzy” semi-Markov process. In this paper, we review “crisp” versus “fuzzy” representations of drought and show how fuzzy semi-Markov processes can aid researchers in developing more robust multiyear, discrete stochastic models.

where

To evaluate the impacts of drought conditions and associated preparedness and response plans, a clear definition of drought must be provided. One definition of drought is the case in which irrigation water supplied is less than irrigation water demanded, due to inadequate rainfall, snow pack, or other weather conditions. As the difference between irrigation water demanded and supplied increases, severity of drought intensifies along a continuous gradient. While the characterization of drought varies across studies, the following definition of drought pro- vided by Yevjevich [

where _{i} is the ith severity state. A Bernoulli variable y_{i} plays a significant role in estimation of the holding time (i.e., duration) probability mass function of drought in later section of semi-Markov chains.

When economic models represent drought in binary terms (i.e., a water allocation either qualifies or does not qualify as drought), this overly-simplified or deceivingly-crisp (as opposed to fuzzy) measurement of drought severity can cause inefficient resource allocation. Unlike this crisp set (in which an element is either a member of the set or not), fuzzy sets allow elements to be included through a degree of membership, as expressed by a membership function, thus relaxing the binary state assumption [

Definition 2. Let

water district. Let

a set of ordered pairs_{i} in W in the mth transition.

where the fuzzy-state grade of membership,

and the powers of the fuzzy transition matrix,

where

A few empirical studies have been conducted to compare efficiencies between the use of classic Markov chains based on conventional crisp set theory and fuzzy Markov chains in the context of stochastic programming models. Mousavi et al. [

where S^{n} represents the state at the nth transition, T_{n} is the time of the nth transition and “t” is arrival time. Equation (3.5) expresses the probability of transitioning to state j at arrival time t, given the system has been in state i after n transitions. Unfortunately no solution has been reported for Equation (3.5) in the literature; therefore, an alternative approach for solving Equation (3.5) has been employed by Cancelliere and Salas [

where _{ij} in the ith drought severity, before moving to the jth

drought severity. This is estimated by

drought before moving to the jth drought, which can be easily observed from historical data.

Mirakbari and Ganji [

where

We now present a numerical example of multiyear water supply forecasts under the assumption that a pair of two random variables {S_{n}, T_{n}} follows a homogeneous fuzzy semi-Markov process (i.e., Equation (3.5) through (3.12)).7 The multiyear forecast associated with the severity and duration of droughts can be incorporated into a multiyear discrete stochastic programming model as shown in Equation (3.12).

Due to the lagging and long-term effects of drought on vegetation and soil moisture, or on cropping choices due to agronomic constraints (e.g., rotations), the resilience of drought is equal to or longer than drought duration [

where π is a stationary distribution.1^{0} The probabilities in Equation (4.4) represent P_{k} (k = 1, 2, 3) in Equation (1.3). In a conventional discrete stochastic programming model, the probability of severe drought (SEE) in the

first year, as well as in the second year, or any other year, is

The possibility mass function of the holding-time spent in each weather condition in a fuzzy semi-Markov

Duration | ||||
---|---|---|---|---|

Weather | Model | d = 1 | d = 2 | d = 3 |

N | FsM | P_{1}(d = 1) = 0.3333 | P_{1}(d = 2) = 0.2222 | P_{1}(d = 3) = 0.1481 |

DSPM | 0.3333 | 0.1111 | 0.0370 | |

MM | FsM | P_{2}(d = 1) = 0.3000 | P_{2}(d = 2) = 0.2222 | P_{2}(d = 3) = 0.1481 |

DSPM | 0.3333 | 0.1111 | 0.0370 | |

SEE | FsM | P_{3}(d = 1) = 0.2500 | P_{3}(d = 2) = 0.1875 | P_{3}(d = 3) = 0.1406 |

DSPM | 0.3333 | 0.1111 | 0.0370 |

N = Normal; MM = Mild/Moderate drought; SEE = Severe/Extreme/Exceptional drought. FsM = Fuzzy semi-Markov; DSPM = Discrete Stochastic Programming Model.

process is estimated with Equation (3.11) and results are presented in

Implementation of adequate measures to control or mitigate drought consequences is a major challenge for irrigators and other water users. Our numerical example, while stylized, demonstrates how economists can use fuzzy semi-Markov processes to incorporate uncertainty about both severity of drought (which necessitates fuzzy sets) and duration of a multiyear drought (which necessitates semi-Markov processes) in stochastic modeling by using a fuzzy semi-Markov process. Such model specification may improve representation of the economic effects of drought severity and duration on water users and the efficacy of alternative mitigation actions.

The views expressed are those of the authors and should not be attributed to USDA or Economic Research Service.

C. S.Kim,Richard M.Adams,Dannele E.Peck, (2016) Multiyear Discrete Stochastic Programming with a Fuzzy Semi-Markov Process. Applied Mathematics,07,482-495. doi: 10.4236/am.2016.76044