Fuzzy random chance-constrained programming - Semantic Scholar

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 5, OCTOBER 2001

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Fuzzy Random Chance-Constrained Programming Baoding Liu, Senior Member, IEEE

Abstract—By fuzzy random programming, we mean the optimization theory dealing with fuzzy random decision problems. This paper presents a new concept of chance of fuzzy random events and then constructs a general framework of fuzzy random chance-constrained programming (CCP). We also design a spectrum of fuzzy random simulations for computing uncertain functions arising in the area of fuzzy random programming. To speed up the process of handling uncertain functions, we train a neural network to approximate uncertain functions based on the training data generated by fuzzy random simulation. Finally, we integrate fuzzy random simulation, neural network, and genetic algorithm to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples. Index Terms—Fuzzy programming, genetic algorithm, hybrid intelligent algorithm, neural network, simulation, stochastic programming.

I. INTRODUCTION

R

EAL-LIFE decision problems are usually made in uncertain (random, fuzzy, fuzzy random, etc.) environments. The first type of stochastic programming is the expected value model (EVM), which optimizes the expected objective functions subject to some expected constraints. The second, chance-constrained programming (CCP), was pioneered by Charnes and Cooper [3] as a means of handling uncertainty by specifying a confidence level at which it is desired that the stochastic constraint holds. Sometimes a complex stochastic decision system undertakes multiple tasks, called events, and the decision-maker wishes to maximize the chance functions of satisfying these events. To model this type of problem, Liu [8] initiated the third type of stochastic programming, called dependent-chance programming (DCP). Fuzzy programming has been used in different ways in the past to deal with fuzzy optimization problems. Following the idea of stochastic CCP, in a fuzzy decision system, we assume that the fuzzy constraints will hold with a possibility level; then, we may construct fuzzy CCP models. Several papers have considered fuzzy linear programming or fuzzy multiobjective linear programming problems and proposed a series of ideas of translating the original chance constraints into crisp equivalents (for example, Luhandjula [17], [18] and Yazenin [25], [26]). However, with the development of more effective computer and intelligent algorithms, many new complex optimization problems Manuscript received March 7, 2000; revised February 19, 2001. This work was supported by the National Natural Science Foundation of China under Grant 69804006 and by the Sino-French Joint Laboratory for Research in Computer Science, Control and Applied Mathematics (LIAMA). The author is with the Uncertain Systems Laboratory, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). Publisher Item Identifier S 1063-6706(01)06650-4.

can be processed by digital computers. Thus, we may construct a spectrum of fuzzy CCP models without linearity assumption [9]–[11]. Following the idea of stochastic DCP, Liu [14] provided a theory of fuzzy DCP. Traditionally, mathematical programming models produce crisp decision vectors such that some objectives achieve the optimal values. However, for practical purposes, we should provide a fuzzy decision rather than a crisp one. Bouchon-Meunier et al. [1] surveyed various approaches to maximizing a numerical function over a fuzzy set. Buckley and Hayashi [2] presented a fuzzy genetic algorithm (GA) for maximizing a real-valued function by selecting an optimal fuzzy set. More generally, Liu and Iwamura [15] provided a spectrum of CCP with fuzzy decisions, and Liu [12] constructed the framework of DCP with fuzzy decisions. Recently, Liu [13] laid a foundation for optimization theory in uncertain environments, called uncertain programming. Numerous intelligent algorithms and applications of uncertain programming are also documented. In many cases, fuzziness and randomness simultaneously appear in decision systems. Fuzzy random variables are mathematical descriptions for fuzzy stochastic phenomena and are defined in several ways. Kwakernaak [6], [7] first introduced the notion of fuzzy random variable. This concept was then developed by Puri and Ralescu [23] and Klement et al. [4]. For detailed expositions, the readers may consult Negoita and Ralescu [22] and Kruse and Meyer [5]. For optimization problems with fuzzy random information, we need fuzzy random programming to model them. Some fuzzy random linear programming with single objective has been discussed by several researchers, for example, Wang and Qiao [24], Luhandjula [19], and Luhandjula and Gupta [20]. This paper presents a new concept of chance of a fuzzy random event and then constructs fuzzy random CCP. To compute uncertain functions, we design a spectrum of fuzzy random simulations. However, fuzzy random simulations are clearly a time-consuming process. To speed up the process of handling uncertain functions, we train a neural network (NN) to approximate uncertain functions based on the training data generated by fuzzy random simulation. Lastly, we integrate fuzzy random simulation, NN, and GA to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples. II. FUZZY RANDOM VARIABLES Roughly speaking, a fuzzy random variable is a measurable function from a probability space to a collection of fuzzy variables. In other words, a fuzzy random variable is a random variable taking fuzzy values.

1063–6706/01$10.00 © 2001 IEEE

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Let be a collection of fuzzy variables. For our purpose, we use the following definition of fuzzy random variable. be a probability space. A Definition 1 [16]: Let such that for fuzzy random variable is a function any Borel subset of (1) is the memberis a measurable function of , where . ship function of fuzzy variable be elements of and For example, let be fuzzy sets in . Then the function if if Fig. 1.

.. . if

Remark 1: If consists of a single event, then the fuzzy random variable degenerates to a fuzzy variable. If is a collection of real numbers (rather than fuzzy sets), then the fuzzy random variable degenerates to a random variable. Remark 2: If a fuzzy random variable degenerates to a random variable, then (1) becomes the characteristic function for any Borel subset of the random event of . III. FUZZY RANDOM ARITHMETIC Let and be two fuzzy random variables defined on and , rethe probability spaces is a fuzzy random variable on spectively. Then, , defined by

Now, let us consider the chance of a fuzzy random event. Recall that the probability of the random event [where is a random vector] is defined by

and the possibility of the fuzzy event is defined by

The chances are both a real number. However, for a fuzzy random event, the chance is a function rather than a number. Generally, we have the following definition of chance of fuzzy random event. be an -dimensional Definition 3: Let be real-valued funcfuzzy random vector and . Then the chance of fuzzy random event tions, characterized by (3)

is a fuzzy random variable on , defined by is a function We have

More generally, we have the following definition of operations on fuzzy random variables. be a real-valued function Definition 2: Let over the -dimensional Euclidean space and be fuzzy random , respectively. variables defined on is a fuzzy random variable on Then, , defined by (2) for all

0; j = 1; 2; . . . ; mg.

IV. CHANCE OF FUZZY RANDOM EVENT

is clearly a fuzzy random variable. If is a random variable and is a fuzzy set, then is a fuzzy random variable, defined by

Similarly, the product

f

Chance Ch f ( )

.

from [0,1] to [0,1] such that for any

.

Pos Remark 3: The chance represents “the fuzzy random at the probability .” event holds with possibility is represented by Remark 4: Sometimes, the chance for the sake of clarity. Ch is a noninRemark 5: It is obvious that the function creasing function of (see Fig. 1).

LIU: FUZZY RANDOM CHANCE-CONSTRAINED PROGRAMMING

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Remark 6: If the fuzzy random vector degenerates to a takes value of either zero random vector, then the chance or one. That is

degenerates to a ) is exactly the

Pos

V. CHANCE CONSTRAINTS Assume that is a decision vector, is a fuzzy random vector, is the return function, and are constraint func. It is obvious that the following “fuzzy tions, random programming” subject to:

(4)

is not defined mathematically because it has different interpretations. Hence, the form (4) does not have a mathematical meaning. Because the fuzzy random constraints are not well defined, it is naturally desired that the fuzzy random constraints hold with possibility at probability , where and are specified confidence levels. Then, we have a chance constraint as follows: Ch

(5)

denotes the chance of the fuzzy random event in where Ch . This type of chance constraint is called a joint chance constraint. Remark 8: If the fuzzy random vector degenerates to a , then the chance constraint (5) derandom vector and generates to

which is a standard stochastic chance constraint. Remark 9: If the fuzzy random vector degenerates to a , then the chance constraint (5) degenfuzzy vector and erates to Pos which is a standard fuzzy chance constraint. Sometimes, the following separate chance constraint is employed: Ch

and then employ

(7)

Ch

if otherwise Remark 7: If the fuzzy random vector (with fuzzy vector, then the chance possibility of the event. That is

levels and the following chance constraints:

(6)

and are confidence levels for . where is a function from Because the chance Ch [0, 1] to [0, 1], we may also give two sequences of confidence

VI. OPTIMISTIC AND PESSIMISTIC VALUES Now, we consider the objective function . No matter what the types of fuzzy random parameter and functional form are, for each given decision is a fuzzy random vari, we define two critable. To measure the return function ical values: optimistic value and pessimistic value. Definition 4: For any given decision Ch

(8)

-optimistic value to the return function is called the . will reach upThis means that the return function -optimistic value with possibility at wards of the probability . Definition 5: For any given decision Ch

(9)

-pessimistic value to the return function is called the . will be below the This means that the return function -pessimistic value with possibility at probability . Remark 10: If the fuzzy random vector degenerates to a , then the -optimistic value is random vector and , and the -pes. This simistic value is coincides with the case of stochastic programming. Remark 11: If the fuzzy random vector degenerates to a , then the -optimistic value is fuzzy vector and Pos , and the -pesPos . This simistic value is coincides with the case of fuzzy programming. Remark 12: It is clear that both optimistic and pessimistic values are extreme cases. In fact, we can also define a critical value according to the Hurwicz criterion (10) where is a given number between zero and one. Note that when , it is the optimistic value; when , it is the pessimistic value. VII. MAXIMAX CHANCE-CONSTRAINED PROGRAMMING If we want to maximize the optimistic value to the fuzzy random return function subject to some chance constraints, then we have the following fuzzy random maximax CCP model: subject to: Ch Ch

(11)

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where

and

are specified confidence levels for . The model (11) is called a maximax model because it is equivalent to

subject to: Ch Ch

(12)

is the -optimistic value to the return function , which is more clearly a maximax form. Remark 13: If the fuzzy random vector degenerates to a random vector, then the model (11) degenerates to

where

subject to:

(13)

which is a standard stochastic CCP [13]. Remark 14: If the fuzzy random vector degenerates to a fuzzy vector, then the model (11) degenerates to subject to: Pos Pos

(14)

which is a standard fuzzy CCP [13]. In practice, we may have multiple objectives. Thus, we have to employ the following fuzzy random maximax chance-constrained multiobjective programming (CCMOP): subject to: Ch Ch

(15)

which is equivalent to

subject to: Ch Ch

(16)

subject to: Ch

Ch

preemptive priority factor, which expresses the relative , for all ; importance of various goals weighting factor corresponding to positive deviation for goal with priority assigned; weighting factor corresponding to negative deviation for goal with priority assigned; -optimistic positive deviation from the target of goal ; -optimistic negative deviation from the target of goal ; target value according to goal ; number of priorities. Remark 15: If the fuzzy random vector degenerates to the and deterministic case, then the chances Ch are always possibility 1 at probability Ch and , and 1 provided that Ch Ch become

This coincides with the deterministic goal programming. Remark 16: In deterministic goal programming, at most one and positive deviation is positive. of negative deviation However, for a fuzzy random CCGP, it is possible that both and are positive. VIII. MINIMAX CCP If we want to maximize the pessimistic value to the fuzzy random return function subject to some chance constraints, then we have the following fuzzy random minimax CCP model:

subject to: Ch Ch

are the -optimistic values to the return where , respectively. functions If the priority structure and target levels are set by the decision-maker, then we may formulate a fuzzy random decision system as a chance-constrained goal programming (CCGP)

Ch

where

(17)

(18)

are specified confidence levels for and is the -pessimistic value . to the return function If there are multiple objectives, then we have the following fuzzy random minimax CCMOP:

where

and

subject to: Ch Ch

(19)

are the -pessimistic values to the return where , respectively. functions We can also formulate a fuzzy random decision system as a fuzzy random minimax CCGP according to the priority structure and target levels set by the decision-maker, as shown in the equation at the bottom of the next page, where


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preemptive priority factor, which expresses the rel, for ative importance of various goals all ; weighting factor corresponding to positive deviation for goal with priority assigned; weighting factor corresponding to negative deviation for goal with priority assigned; -pessimistic positive deviation from the target of goal ; -pessimistic negative deviation from the target of goal ; target value according to goal ; number of priorities. Remark 17: If the fuzzy random vector degenerates to the deterministic case, then the two chances Ch and Ch are always possibility 1 at and , and probability 1 provided that Ch Ch imply that

This coincides with the deterministic goal programming. IX. FUZZY RANDOM SIMULATIONS To solve fuzzy random CCP models, we have to check chance constraints and find critical values. In this section, we design some fuzzy random simulations to deal with them. is a fuzzy random Case I: Suppose that are real-valued functions, . For vector and any given and , we design a fuzzy random simulation to check if the chance constraint (20)

Ch holds. Note that the value of Ch

First, we sample ability measure . We define

from

according to the prob-

if Pos otherwise , which are a sequence of random varifor ables (not fuzzy random variables), and the expected values for all provided that meets (21). By the strong law of large numbers, we obtain a. s. . Note that the sum is just the number as satisfying Pos of for . Let be the integer part of . Then can be taken the value of Ch th largest element in the sequence as the with Pos for , where the values may be obtained by fuzzy simulations (Liu [13]). Thus the chance constraint (20) holds if and is only if the value of Ch larger than or equal to the specified confidence level . In other words, the chance constraint (20) holds if and only if there are elements in that are larger than or at least equal to . from according to the Step 1) Generate probability measure . Pos Step 2) Compute the possibility for , respectively, by fuzzy simulation. as the integer part of . Step 3) Set elements in Step 4) If there are at least that are larger than or equal to , return YES, else return NO. Case II: For any given decision and confidence levels and , we sometimes need to find the -optimistic value to . That is, we should design a fuzzy the return function random simulation to find the maximal value such that (22)

Ch

must be achieved at the equality case

holds. It is obvious that the achieved at the equality case

Pos

(21)

subject to: Ch Ch Ch

Pos

-optimistic value

must be (23)

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We sample and define measure

from

according to the probability

if Pos otherwise , which are a sequence of random variables for for all pro(not fuzzy random variables), and vided that meets (23). By the strong law of large numbers, we obtain Fig. 2.

a. s. as number of the value sequence

. Note that the sum is just the satisfying Pos for . Let be the integer part of . Then can be taken as the th largest element in the with Pos

for , which may be obtained by fuzzy simulations. from according to the Step 1) Generate probability measure . such that Step 2) Find the largest values Pos for , respectively, by fuzzy simulation. as the integer part of . Step 3) Set . Step 4) Return the th largest element in Case III: For any given decision and confidence levels and , sometimes we need to find the -pessimistic value . That is, we should design a fuzzy to the return function random simulation to find the minimal value such that (24)

Ch holds. It is obvious that the achieved at the equality case

-pessimistic value

Pos We sample and define measure

must be

(25) from

according to the probability

A hybrid intelligent algorithm.

as number of the value sequence

. Note that the sum is just the satisfying Pos for . Let be the integer part of . Then can be taken as the th smallest element in the with Pos

for , which may be obtained by fuzzy simulations. from according to the Step 1) Generate probability measure . such that Step 2) Find the smallest values Pos for , respectively, by fuzzy simulation. as the integer part of . Step 3) Set th smallest element in Step 4) Return the . X. HYBRID INTELLIGENT ALGORITHM Numerous intelligent algorithms have been developed to solve problems. A natural idea is to integrate these algorithms to produce more powerful and effective hybrid intelligent algorithms. Medsker [21] introduced many good ideas to design hybrid intelligent systems. To solve fuzzy random programming models, we will integrate fuzzy random simulation, NN, and GA to produce a hybrid intelligent algorithm (see Fig. 2). The difference between deterministic programming and fuzzy random programming is that there are uncertain functions in the latter Ch Ch Ch

if Pos otherwise , which are a sequence of random variables for for all pro(not fuzzy random variables), and vided that meets (25). By the strong law of large numbers, we obtain a. s.

(26)

where and are given confidence levels. Although fuzzy random simulations are able to compute the uncertain functions, we need to train a neural network to approximate them because the fuzzy random simulations are a time-consuming process. Then we embed the trained neural network into GA to produce a hybrid intelligent algorithm as follows. Step 1) Generate training data for uncertain function approximations by fuzzy random simulations.


Step 2) Train a neural network to approximate the uncertain functions according to the generated training data. Step 3) Initialize pop size chromosomes in which the trained neural network may be employed to check the feasibility. Step 4) Update the chromosomes by crossover and mutation operations in which the feasibility of offspring may be checked by the trained neural network. Step 5) Calculate the objective values for all chromosomes by the trained neural network. Step 6) Compute the fitness of each chromosome according to the objective values. Step 7) Select the chromosomes by spinning the roulette wheel. Step 8) Repeat the fourth to seventh steps for a given number of cycles. Step 9) Report the best chromosome as the satisfactory solution.

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hybrid intelligent algorithm (5000 cycles in simulation, 2000 training data in NN, 1000 generations in GA) shows that the satisfactory solution is

whose objective value is 13.39. Moreover, we have Ch Ch Example 2: We consider CCP models with both random and fuzzy coefficients subject to: Ch Ch where

are

XI. NUMERICAL EXPERIMENTS Here, we give some numerical examples performed on a personal computer with the following parameters: the population is 0.3, the probability size is 30, the probability of crossover is , and the parameter in the rank-based of mutation evaluation function is 0.05. Example 1: Let us consider the following fuzzy random maximax CCP models:

normally distributed variables are fuzzy numbers with

membership functions

respectively. This is a degenerate fuzzy random CCP model. We first produce 2000 input–output data for the uncertain functions

subject to: Ch Ch

Ch Ch

where are fuzzy random variables taking values of triangular fuzzy number with with with with where represents the normally distributed variable with mean and variance . To solve this model, we produce 2000 input–output data for the uncertain functions

by fuzzy random simulations. According to the generated data, we train a feedforward NN (three input neurons, six hidden neurons, two output neurons) to approximate the uncertain function . After that, the trained NN is embedded into a GA to produce a hybrid intelligent algorithm. A run of the hybrid intelligent algorithm (5000 cycles in simulation, 2000 training data in NN, 300 generations in GA) shows that the satisfactory solution is

whose objective value is 7.43. Furthermore, we have Ch Ch

Ch Ch by fuzzy random simulations. Based on the training data, we train a feedforward NN (four input neurons, six hidden neurons, two output neurons) to approximate the uncertain function . After that, the trained NN is embedded into a GA to produce a hybrid intelligent algorithm. A run of the

Example 3: Here is a fuzzy random minimin CCGP model lexmin subject to: Ch Ch Ch

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where as follows:

are fuzzy random variables defined

with with with with with with with with with where denotes the exponentially distributed variable represents the uniformly distributed with mean and . variable on the interval To solve this CCGP model, we produce 2000 input–output data for uncertain functions

Ch

Ch

Ch

by fuzzy random simulations. Then, we train a feedforward NN (three input neurons, ten hidden neurons, three output neurons) to approximate the uncertain function according to the input–output data. After that, the trained NN is embedded into a GA to search for the satisfactory solution. A run of the hybrid intelligent algorithm (5000 cycles in simulation, 2000 training data in NN, 3000 generations in GA) shows that the satisfactory solution is

which can satisfy the first two goals, but the positive deviation of the third goal is 2.10. XII. CONCLUSION This paper presents a new definition of chance of fuzzy random event and provided a spectrum of fuzzy random CCP models. To solve these models, we integrate fuzzy random simulation, NN, and GA to produce a hybrid intelligent algorithm whose effectiveness is illustrated by some numerical examples.

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Baoding Liu (M’99–SM’00) received the M.S. and the Ph.D. degrees from the Institute of Systems Science, Chinese Academy of Sciences, Beijing, China, in 1989 and 1993, respectively. Since 1998, he has been a full Professor with the Department of Mathematical Sciences, Tsinghua University, Beijing, China. He is the author of several books, including Uncertain Programming (New York: Wiley, 1999), and Decision Criteria and Optimal Inventory Processes (Boston, MA: Kluwer, 1999). He is an Editorial Board Member of Information. Current research interests include stochastic programming, fuzzy programming, fuzzy random programming, random fuzzy programming, intelligent systems, and their applications in various uncertain decision systems. Dr. Liu is an Associate Editor of IEEE TRANSACTIONS ON FUZZY SYSTEMS.