Multiyear Discrete Stochastic Programming with a Fuzzy Semi-Markov Process ()
Received 20 October 2016; accepted 21 March 2016; published 24 March 2016
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1. Introduction
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2000 2002 2004 2006 2008 2010 2012 2014
Figure 1. Severity and duration of drought in California (Source: The National Drought Miti- gation Center―USDA).
where
.
2. Linguistic Vagueness and Fuzzy Logic
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 [14] has been widely used [6] [15] [16] :
, (2.1)
where 
is a constant water allotment threshold (such as 40 acre-inches Peck and Adams [3] used for water allotment under normal weather condition) and Si is the ith severity state. A Bernoulli variable yi 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 [6] [17] . Introduced by Zadeh [18] , fuzzy logic and fuzzy set theory have since been widely adopted to deal with linguistic vagueness in various mathematical optimization models, including fuzzy dynamic programming [19] [20] , and optimal fuzzy control [21] .
(2.2)
3. Fuzzy Markov and Semi-Markov Processes
Definition 2. Let 
be a finite state-space of possible water allotments from an irrigation
water district. Let 
be random variables taking values in W. Now, let a fuzzy subset of W be defined as
a set of ordered pairs
, where 
is the grade of membership of Si in W in the mth transition.
, (3.2)
where the fuzzy-state grade of membership, ![]()
, can be calculated as follows:
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, (3.3)
and the powers of the fuzzy transition matrix, ![]()
, are defined by:
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, (3.4)
where ![]()
and ![]()
is an identity matrix at the initial time period.
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. [20] compared a conventional stochastic dynamic programming model (which employed classic Markov chains) and a fuzzy-state stochastic dynamic programming model (which employed fuzzy Markov chains) in the operation of a multipurpose reservoir in Iran. Results from their study show that fuzzy- state stochastic dynamic programming outperforms conventional stochastic dynamic programming in achieving the flood control objective and in overall performance of the reservoir. Chandramouli and Nanduri [27] also compared a conventional stochastic dynamic model and a fuzzy-state stochastic dynamic model in the operation of a multipurpose reservoir in India. They found that the fuzzy-state stochastic dynamic programming model out performed the conventional stochastic dynamic programming model.
3.1. Fuzzy Semi-Markov Process
3.1.1. Homogeneous Fuzzy Semi-Markov Process
where Sn represents the state at the nth transition, Tn 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 [15] and Mirakbari and Ganji [6] .
![]()
(3.9)
where ![]()
represents the probability of the duration dij in the ith drought severity, before moving to the jth
drought severity. This is estimated by ![]()
where ![]()
is the mean holding-time (i.e., duration) in the ith
drought before moving to the jth drought, which can be easily observed from historical data.
Mirakbari and Ganji [6] compared performances between the classic semi-Markov chains and the fuzzy semi- Markov chains by using profust reliability theory to a rangeland system in India.6 Their results indicate that the reliability of rangeland system decisions increased by 22 percent when the fuzzy semi-Markov process is used over the classic semi-Markov process.
3.1.2. Non-Homogeneous Fuzzy Semi-Markov Process
where ![]()
is the transition probability of the embedded fuzzy Markov chain in the process.
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(3.17)
4. Numerical Example
We now present a numerical example of multiyear water supply forecasts under the assumption that a pair of two random variables {Sn, Tn} 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).
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(4.3)
Comparison
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 [38] .9 Therefore, economic impacts of individual years of drought may not necessarily be independent [3] . To compare multiyear drought probabilities between a conventional multiyear discrete stochastic program (which typically assumes the previous year’s state of nature does not influence the probability of future states of nature) and a multiyear homogeneous fuzzy semi-Markov process, the steady-state probabilities of the fuzzy transition matrix in Equation (4.2) are estimated as follows:
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(4.4)
where π is a stationary distribution.10 The probabilities in Equation (4.4) represent Pk (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![]()
, such that the probability of two consecutive years of severe drought is![]()
, while the probability of three consecutive years of severe drought is![]()
.
The possibility mass function of the holding-time spent in each weather condition in a fuzzy semi-Markov
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Table 2. Results of a numerical example.
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 Table 2. In contrast to a conventional discrete stochastic programming model, the possibility of two consecutive years of SEE drought condition is 0.1875, which is 69 percent higher than the probability assumed in a conventional multiyear discrete stochastic programming approach. As the duration of consecutive years of severe drought increases to three years, the possibility of holding time in a fuzzy semi-Markov process is 0.1406, which is 280 percent higher than the probability assumed in a conventional multiyear discrete stochastic programming approach, 0.0370.
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.
5. Summary
Acknowledgements
The views expressed are those of the authors and should not be attributed to USDA or Economic Research Service.
NOTES
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1Both possibilistic and flexible fuzzy programming can be presented in interval fuzzy programming [12] .
2However, we use the terms possibility and probability interchangeably in this paper.
3The max-min operations of fuzzy sets are as follows. For fuzzy subsets A and B of a crisp set W ≠ ϕ, the intersection of A and B is defined as:
, 
Similarly, the union of A and B is defined as follows:
, 
4The set of all subsets of W is called the power set of W.
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5Markov kernel is also called a transition probability function that maps from a measureable space to another space.
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6Profust reliability theory consists of two parts, the fuzzy part which considers vagueness in rangeland system failure and the probabilistic part which incorporates randomness of rangeland failure [6] [17] .
7Multiyear water supply forecast is a forecast in dij = 1, dij = 2, dij = 3, etc. for all drought severities i = 1, 2, ×××, n.
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8See Avrachenkov and Sanchez [25] , Gildeh and Dadgar [34] , Li and He [35] , Mousavi et al. [20] , and Sanchez [36] for fuzzy transition matrix, and Meenakshi and Kaliraja [37] for interval transition matrix.
9Resilience is a measure of the recovery time of the system [38] .
10The stationary distribution must satisfy![]()
, where 1 is a column vector with all entries equals to one.