Scientific Programming

Volume 2016 (2016), Article ID 1747425, 7 pages

http://dx.doi.org/10.1155/2016/1747425

## Dependent-Chance Goal Programming for Water Resources Management under Uncertainty

^{1}School of Science, Hebei University of Engineering, Handan 056038, China^{2}School of Economics and Management, Hebei University of Engineering, Handan 056038, China

Received 20 April 2016; Revised 4 July 2016; Accepted 25 July 2016

Academic Editor: Dan Ralescu

Copyright © 2016 Haiying Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Without sufficient data, consulting experts is a good way to quantify unknown parameters in water resources management which will result in human uncertainty. The aim of this paper is to introduce a new tool-uncertainty theory to deal with such uncertainty which is treated as uncertain variable with uncertainty distribution. And a dependent-chance goal programming (DCGP) model is provided for water resources management under such circumstance. In the model uncertain measure is used to measure possibility that an event will occur which is maximized by minimizing the deviation (positive or negative deviation) from target of objective event under a given priority structure. In the end, the developed model is applied to a numerical example to illustrate the effectiveness of the model. The result obtained contributes to the desired water-allocation schemes for decision-markers.

#### 1. Introduction

The water resources management aims to enhance the effective use and guarantee sustainable development of water resources, when requirements about water quality and quantity of users are satisfied. Generally, there are several essential principles for water resources management: maximize the benefits in the development and utilization of water resources; emphasize joint operation of surface water and groundwater; lay equal stress on development and protection of water resources; strike a balance between water quality and water quantity. However, some natural and human factors (extreme climate events, emergency, population growth, etc.) bring severe challenges for efficient management of water resources, such as severe water shortage and water pollution. Obviously, these challenges will intensify the uncertainty of water resources management. However, in earlier study, researchers seldom consider uncertainty which already exists in water resources management. For example, Howes [1] applied linear programming to obtain doable water resource investments for water development projects. Revelle et al. [2] minimized the cost of the treatment plants in water quality management by using linear programming approach. Howard and Shamir [3] established a deterministic linear programming model to study interrelated land and water resource management problem. These approaches above did not take indeterminate factors into consideration.

However, in reality, uncertainty often influences decision-making in water resources allocation, utilization, scheduling, and protection. Later, researchers noted and treated uncertainty as randomness, fuzziness, or both of them. As a result, numerous mathematical methods have been proposed based on stochastic programming to tackle uncertainty treated as randomness. For example, Dupačová et al. [4] utilized stochastic programming to offer alternative decisions for decision-makers in water management. Feiring et al. [5] proposed a stochastic programming model for agricultural irrigation where rainfall was assumed to be a random variable. Huang and Loucks [6] proposed an inexact two-stage stochastic programming model to deal with uncertainty where total water available and net benefit were unknown in water resources management. Kracman et al. [7] developed a multistage stochastic optimization model to handle uncertainty in the inflows into the Highland Lakes of the Highland Lakes system. Zhou et al. [8] proposed a factorial multistage stochastic programming approach for water resources management under uncertainty where some parameters of water allocation are indeterminate. Besides, some researchers used fuzzy programming to handle uncertainty presented as fuzzy set. For example, Jairaj and Vedula [9] formulated a fuzzy mathematical programming model based on fuzzy set theory for a multireservoir system. Sahoo et al. [10] used fuzzy optimization and liner programming to optimize the allocation of land and water resources. Zhang et al. [11] proposed an interactive inexact fuzzy bounded programming approach for agricultural water quality management. Sometimes, multiple uncertainties also exist in water resources management problems. Recent years, some methods are addressed through combining different mathematical programming method to deal multiple uncertainties. Guo et al. [12] developed a fuzzy stochastic two-stage programming approach for water resources management under randomness and fuzziness which integrated the fuzzy robust programming, chance-constrained programming, and two-stage stochastic programming. Li et al. [13] proposed fuzzy-boundary interval-stochastic programming method to tackle uncertainties expressed as probability distributions and fuzzy-boundary intervals. Zeng et al. [14] developed a two-stage credibility-constrained programming with Hurwicz criterion approach to tackle uncertainties presented as probability distributions and fuzzy sets.

These existing methods are effective to deal with objective uncertainty based on probability theory and fuzzy set theory. However, human uncertainty also exists in water resources management. Without sufficient data, consulting experts is another choice to quantify unknown parameter. In this case, human uncertainty always occurs. It is a fact that probability theory is a useful tool when a large number of sample data exists. So it is less effective to tackle such problems based on probability theory if there is lack of sample data. Besides, some researchers used fuzzy set theory to deal with human uncertainty. But it does not emphasize the law of excluded middle and the law of contradiction. In contrast, uncertainty theory [15] follows these principles which was founded in 2007 and became a branch of mathematics based on the normality, duality, and countable subadditivity. It is noteworthy that uncertainty theory is a new mathematical tool rather than a general term for all theory to handle uncertainty. Nowadays, it has been applied to different fields to tackle with uncertain phenomenon. For example, Liu proposed uncertain programming [16], uncertain multiobjective programming and uncertain goal programming [17], and uncertain multilevel programming [18] to deal with decision processing including belief degree successively.

Compared with probability theory and fuzzy set theory, uncertainty theory is effective in handling human uncertainty. So based on uncertainty theory, this paper considers the optimal allocation of water supply systems in uncertain environment. And uncertainty is expressed as uncertain variable with uncertainty distribution. Then the uncertain measure is used to measure possibility that an event will occur. The objective function of this model is to minimize the deviation from target of objective event under a given priority structure. Thus, DCDP model is formulated.

The rest of this paper is organized as follows. Section 2 introduces some basic concepts of uncertainty theory used throughout this paper. Section 3 presents a dependent-chance goal programming model for the water supply system under uncertain environment. Section 4 introduces algorithm to solve the model. Section 5 gives a numerical example to illustrate the effectiveness of the model. Finally, a conclusion is drawn in Section 6.

#### 2. Preliminaries

In this section, we will introduce some useful definitions about uncertain measure, uncertain distribution, and so on.

Let be a nonempty set, and be -algebra over . Each element is called an event. A number indicates the possibility that will occur. Uncertain measure is introduced as a set function satisfying the following axioms [15]:

*Axiom 1 (normality axiom). * for the universal set .

*Axiom 2 (duality axiom). * for any event .

*Axiom 3 (subadditivity axiom). *For every countable sequence of events , we have

The triplet is called an uncertainty space. In addition, the product uncertain measure [19] was defined as follows.

*Axiom 4 (product axiom). *Let be uncertainty spaces for . The product uncertain measure is an uncertain measure satisfying

The concept of uncertain variable was introduced [15] as a measurable function from an uncertainty space to the set of real numbers. In order to describe an uncertain variable, uncertainty distribution is defined [15] as Furthermore, the inverse uncertain distribution of was defined [20]. It plays a crucial role in operations of uncertain variable.

An uncertain variable is called normal if it has a normal uncertainty distribution [15] denoted by , where and are real numbers with .

An uncertain variable is called lognormal if it has a normal uncertainty distribution [15] denoted by , where and are real numbers with .

#### 3. DCGP Model under Uncertainty Environments

Usually, optimal allocation of water supply system is a multiobjective programming problem. However, these goals are often contradictory and achievable at the cost of other goals. In this case, it is a preferred solution to create a priority structure. Then the lower-priority goals are considered after achieving the higher-priority goals. Under such circumstances, based on probability theory, dependent-chance goal programming (DCGP) [21] was firstly proposed to deal with the complex system which contains multiple resources and multiple users. Later the former one was improved and applicable for general stochastic decision systems [22], fuzzy decision systems [23], and fuzzy random systems [24]. Specifically, DCGP is a good tool to deal with water supply-allocation problem, production process [25], capital budgeting [26], and so on.

However, it is reasonable to accept the existence of the system (or phenomenon) which is neither random nor fuzzy. The phenomenon that there is lack of sufficient historical data is a case. In order to cope with such decision systems, uncertainty theory is used to formulate DCGP as follows: where some parameters are shown as follows: is the priority factor which represents the relative importance of each goal; is number of priorities; is the weighting factor of positive deviation for goal under priority assigned; is the weighting factor of negative deviation for goal under priority assigned; is the number of goal constraints; is given target value of goal .

The objective event is expressed as , where is decision vector and is uncertain vector. is uncertain measure of event occurrence. Let be the chance function of the objective event where is the dependant constraints of objective event ().

On the basis of chance functions and goal values, the deviations (positive or negative deviations) can be defined.

Let be the positive deviation from the target of goal , denoted as

Let be the negative deviation from the target of goal , denoted as

The constraints , are the uncertain environment.

#### 4. Hybrid Intelligent Algorithm

The hybrid intelligent algorithm which integrates uncertain simulation, neural network, and genetic algorithm is introduced to solve the model. The algorithm combines the global searching ability of genetic algorithm and the good local searching ability of neural network and improves training effect of neural network by using uncertain simulation.

To be more specific, the aim of uncertain simulation is to generate large amounts of input-output data. And neural network attempts to seek suitable weighting vector to approximate the uncertain functions by training input-output data (general network structure is shown in Figure 1). Finally, the optimum solution is obtained by applying genetic algorithm.