This paper is organized as follows. Section 2 proposes the **uncertain** **variable** method (UVM) to generate uncer- tainty weights from both consistent and inconsistent un- certainty comparison matrices. In Section 3, the defini- tion of consistency for uncertainty comparison and the theorem for checking consistency are expressed, and then we develop a simple yet pragmatic approach that can be used to test whether an uncertainty comparison matrix is consistent or not without solving any mathema- tical program. Moreover an algorithm for identification and modification of inconsistent bounds is discussed. Se- ction 4 provides three numerical examples including a hierarchical (AHP) decision problem to show the sim- plicity and practicality of the proposed methods. The paper is concluded in Section 5.

The paper is organized as follows. In Section 2, un- certainty theory is first introduced in simple word and some basic concepts and properties are given. Then, conventional-AHP ranking method is introduced. Section 3 is the main part of this paper, uncertainty comparison matrix and uncertainty weights are investigated. Section 4 presents two numerical studies to show the applications of the proposed methods. Section 5 discusses the exten- sion of **uncertain** **variable** method to interval comparison matrices and fuzzy matrices, which are transformed into uncertainty comparison matrices using linear uncertainty distribution and zigzag uncertainty distribution respec- tively. Section 6 concludes this paper with a brief sum- mary.

renewal process was introduced by Liu [10] and the canonical process was designed by Liu [11] in 2009. The most important and useful **uncertain** process is canonical process which is the counterpart of Brown motion. Different from Brownian motion, the canonical process is a Lipschitz continuous **uncertain** process with stationary and independent increments and each increment is a normal **uncertain** **variable**. Based on this process, **uncertain** calculus was initialized by Liu [11] in 2009 to deal with differentiation and integration of functions of **uncertain** processes. Following that, **uncertain** differential equation, a type of differential equation driven by canonical process, was defined by Liu [10]. Soon afterwards, Chen and Liu [3] proved an existence and uniqueness theorem of solution for **uncertain** differential equation under Lipschitz condition and linear growth condition. In many cases mathematical finance models are expressed in terms of stochastic differential equation [16, 20]. For **uncertain** financial market, the **uncertain** differential equation also plays an indispensible role. Following the argument that stock price follows geometric canonical process, Liu [9] proposed a basic stock model for **uncertain** financial market. After that, Xu and Peng [17] studied barrier options pricing in **uncertain** financial market, Peng and Yao [18] proposed another stock model to describe the stock price in long-run, Yu[22,23] studied expected payoff of trading strategies with jumps for **uncertain** financial markets and a stock model with jumps for **uncertain** markets.

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tribution of the **uncertain** **variable** – the polynomial basis is orthogonal with a weight function that corresponds up to a constant to the probability density function of the uncer- tain **variable**. Common random (**uncertain**) variables (nor- mal, uniform, exponential, beta) have corresponding classi- cal orthogonal polynomials (Hermite, Legendre, Laguerre, Jacobi) (Eldred et al., 2008). Empirically determined distri- butions, such as those obtained from wind conditions, do not have corresponding classical orthogonal polynomials. For the distributions obtained from the wind conditions, we need to numerically generate custom orthogonal polynomials in order to preserve the optimal convergence property of the polynomial chaos expansion (Oladyshkin and Nowak, 2012). Details about the numerical generation of orthogonal poly- nomials can be found in Gautschi (2004) and an example of the generation of orthogonal polynomials for wind dis- tributions in Padrón (2017). In addition to the optimal con- vergence properties, the use of orthogonal polynomials al- lows us to analytically compute statistics from the polyno- mial chaos expansion (Sect. 4.1.1).

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In order to provide a quantitative measurement of the degree of uncertainty in relation to an **uncertain** **variable**, Liu [3] proposed the definition of **uncertain** entropy re- sulting from information deficiency. Dai and Chen [17] investigated the properties of entropy of function of un- certain variables. The principle of maximum entropy for **uncertain** variables are introduced by Chen and Dai [18]. Besides, there are many literature concerning the defini- tion of entropy of **uncertain** variables, such as Chen [19], Dai [20], etc.

With the rapid development of transportation technology, public passenger traffic system construction investment is getting more and more expensive. Determining the development trend and make the optimal allocation of transportation investment becomes more and more important to the government. To describe the relationship and development trends of the integrated passenger transportation system under **uncertain** environment, an **uncertain** multi-agent model is proposed. Then, **uncertain** **variable** simulation technique is introduced to estimate the uncertainties in the model. A simulation program based on swarm platform is developed to implement the proposed model. Finally, a numerical experiment is conducted to illustrate that this technique is effective and can be used in decision support.

In the process of practical logistics network optimization, **uncertain** factors often appear in competitive logistics distribution center location problem because of lacking of or even without historical data. This paper investi- gated a useful model to handle competitive logistics distribution center location problem with **uncertain** custom- ers demands and **uncertain** setup costs. The mathematical model of this problem was established by **uncertain** programming based on the expected value criterion. In order to solve this model, we took advantage of the properties of **uncertain** **variable**. Then the expected value model was transformed into its crisp equivalent model, and we used mathematical software Lingo to find its optimal solution. At last, a numerical example was pre- sented to illustrate the effectiveness of the proposed model.

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The uncertainty is the mathematical tool to model imprecise quantities of the entities. Uncertainty theory was founded by B. Liu [3] in 2007. The first fundamental concept in uncertainty theory is **uncertain** measure that is used to measure the belief degree of an **uncertain** event. The concepts of membership function and uncertainty distribution are two basic tools to describe **uncertain** sets, where membership function is intuitionistic for us but frangible for arithmetic operations, and uncertainty distribution is hard-to- understand for us but easy-to-use for arithmetic operations. Fortunately, an uncertainty distribution may be uniquely determined by a membership function. The concept of **uncertain** **variable** (neither random **variable** nor fuzzy **variable**) in order to describe im- precise quantities in human systems. There are two type of **uncertain** variables, linear **uncertain** **variable** and zigzag **uncertain** **variable** defined by B. Liu [4].

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In practical decision making process, people often have to deal with indeterminacy data. To handle this situation, many optimization models involved uncertainty are widely discussed. Specifically, in a supply chain, costs of transportation and inventory are important factors in optimizing profits. While, the market demands are often unknown, especially when a new situation arises. In this paper, based on uncertainty theory, a new type of two-stage programming, named **uncertain** programming with recourse (UPR) is first put forward. Then, by employing the expected value of **uncertain** **variable**, an equivalent classic programming of UPR is built. Finally, by regarding the market demands as **uncertain** variables, UPR model is used to solve the integrating transportation and inventory problem under uncertainty.

A number is a mathematical object used to count, label, and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels, for ordering, and for codes. In common use, the word number can mean the abstract object, the symbol, or the word for the number [12]. Now the definition of number involve the uncertainty representation and granular forms which come from **uncertain** sets representation such as: fuzzy set, rough set, grey set vague set ,…..etc

Abstract: We present the results of a series of laboratory economic experiments designed to study compliance behavior of polluting firms when information on the penalty is **uncertain**. The experiments consist of a regulatory environment in which university students face emission standards and an enforcement mechanism composed of audit probabilities and penalties (conditional on detection of a violation). We examine how uncertainty on the penalty affects the compliance decision and the extent of violation under two enforcement levels: one in which the regulator induces perfect compliance and another one in which it does not. Our results suggest that in the first case, **uncertain** penalties increase the extent of the violations of those firms with higher marginal benefits. When enforcement is not sufficient to induce compliance, the **uncertain** penalties do not have any statistically significant effect on compliance behavior. Overall, the results suggest that a cost-effective design of emission standards should consider including public and complete information on the penalties for

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In this paper, I propose a mechanism that explains crucial features of the development of financial crises. In an economy where the financial intermediary sector plays a vital role in allocating investment capital to long-term investment opportunities, I show that asymmetric information between the financial intermediary sector and investors brings rise to two distinct economic states, normal times and booms. Profit maximizing behavior of the intermediary sector to intermediate large quantities of capital to promote growth, but also potentially to take large, inefficient risks at the expense of investors leads to an endogenous concentration of uncertainty in booms. Amidst an **uncertain** boom, investors’ beliefs exhibit fragility; subsequent arrival of negative public information results in an abrupt loss of confidence in the quality of intermediary assets. A crisis breaks out. Investors force early liquidation on intermediary assets and flee to safe assets, in a flight-to-quality episode. In contrast, during normal times, investors’ beliefs are resilient to a negative information signal regarding the quality of intermediary investments. Normal times are quiet.

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OALibJ | DOI:10.4236/oalib.1102366 2 February 2016 | Volume 3 | e2366 **uncertain** renewal reward process which interarrival times and rewards were both regarded as **uncertain** variables and gave the an elementary renewal reward theorem. At present, there is a lack of strict proof for the elementary theorem. Therefore, the paper will give its strict proof with two lemmas by some techniques.

naries required to develop the paper. In Section 3, an **uncertain** embedding theorem is proved and an **uncertain** single/multiple objective optimization method using the embedding theorem is established. In Section 4, we de- velop an **uncertain** linear bi-objective R & D project portfolio selection model. The objectives are 1) maximi- zation of project benefit and 2) minimization of project risk. The risk is defined as the maximum loss that the decision maker may face in the worst case. This is con- sidered as the projected maximum loss in case of failure of the project. Constraints on budget and resources are also considered. Using the embedding theorem estab- lished in Section 3, we convert the bi-objective **uncertain** optimization problem into a tetra-objective crisp optimi- zation problem which is further transformed into a de- terministic convex optimization model by global criteria approach. In Section 5 of this paper a real case study is provided to illustrate our method. The optimization software LINGO is used for the simulation. Finally in Section 6 some concluding remarks are presented.

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M-V portfolio optimization problem with **uncertain** exit-time is considered by Guo and Hu (2005). Zhang and Li (2012) studied a multi-period M-V portfolio selection model with **uncertain** exit-time where the asset returns are serially correlated. Wu and Li (2011) investigated a multi-period M-V model in regime switching markets when exit-time is **uncertain**. Wu et al. (2014) considered a multi-period M-V model with state-dependent exit probability and regime switching. Using mean-field formulation, Yi et al. (2014) studied a multi-period M-V model with an **uncertain** exit-time. Yi et al. (2008) considered an asset-liability management model when the investment horizon is **uncertain**. Blanchet-Scalliet et al. (2008) studied an optimal investment problem when the **uncertain** time-horizon depends on the asset returns. Lv et al. (2016) studied a M-V portfolio optimization problem where market parameters and exit-time are random and market is incomplete.

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In order to cope with the uncertainty of the UFM- CFP, first it is converted to a deterministic form, and then it is solved. For the conversion pur- pose, two main criteria of expected value and critical value of the **uncertain** variables can be considered (see [7, 8]). According to these cri- teria, three different deterministic forms of the UFMCFP as expected value model, expected value and chance-constrained model, and chance- constrained model can be obtained (see [7, 8]). In this paper the chance-constrained model is used to obtain the crisp form of the UFMCFP. Using the chance-constrained model, the UFMCFP is converted to a deterministic form considering the following issues,

The principle of **uncertain** future: the probability of a future event contains an (hidden) uncertainty. The first consequence of the principle: the real values of high probabilities are lower than the preliminarily determined ones; conversely, the real values of low probabilities can be higher than the preliminarily determined ones. The first consequence provides an uniform solution of the underweighting of high and the overweighting of low probabilities, of the Allais paradox, risk aversion, loss aversion, the equity premium puzzle, the “fourfold pattern” paradox, etc. The second consequence: the present probability system of a future event is incomplete. The second consequence provides a solution of the incompleteness of systems of preferences, of ambiguity aversion, of the Ellsberg paradox, etc.

Determining the departure interval is the basic of establishing the operation plan. Based on the existing researches, and consider all kinds of **uncertain** factors in operation process overall, to establish a bilevel programming model which is based on the game of passengers and operation companies. And use the differential evolution bacterial foraging algorithm to solve this model. Set the passengers satisfaction as upper target, and the operational efficiency of enterprise as the lower target. At last examples show that this model can effectively optimize the departure interval, and it has certain feasibility and effectiveness. Keywords: Bus Departure Interval, **Uncertain** Programming, Bilevel Programming Model, Bacterial

Investment decision is a traditional multi-attribute decision making (MADM) prob- lem since it has many uncertainty factors and incomplete information such as in- vestment value, cost, sales, etc. D numbers theory is a useful tool to deal with uncer- tainty factors and incomplete information. In this paper, interval number and D numbers theory are revealed in the **uncertain** factor and incomplete information of investment decision. The weights of **uncertain** factors are calculated using entropy weight method. Thus, a new MADM model for investment decision based on D numbers theory is proposed. Numerical example is used to illustrate the efficiency of the proposed method.

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B. The desription of the problem Suppose there are the evaluation m evaluation projects A ( , a a 1 2 , , a m ) and n evaluation indicators C ( , c c 1 2 , , c n ) , the evaluation indicator’s evaluation values is the matrix X [ x ij m n ] u , x ij denotes the evaluation value of the jth indicator of the ith evaluation project. Suppose x ij is shown by **uncertain** linguistic : [ , ] ij ij ij a b x x x , , , ij ij a b x x S S is the given lingustic scale, ij a x and ij b x are respectively the lower bound and the upper bound of x ij . a ij and b ij are nature number level, and a ij d b ij . The weight of the indicators are unknown. Through the analysis of the matrix X [ x ij m n ] u , the rank of the projects is confirmed finally. C .The **uncertain** linguistic evaluation model based on the TOPSIS method TOPSIS is used to confirm the order of the evaluation objects in virtue of the ideal solution and the negative ideal solution of the multiattribute problems[21]. The ideal solution is a best solution that is assumed(marked as V + ). Each of it’s indicator value is the best value of the optional schemes. The negative solution is another worst solution that is assumed(marked as V - ). Each of it’s indicator value is the worst value of the optional project. V + andV - are compared with each project in the original project set. The distance information of them is used to be the standard to confirm the order of the m projects in X. The ideal solution of **uncertain** lingusitic decision matrix X [ x ij m n ] u is 1 2 ( , , , n ) V v v v (8)