# Dynamic Planning of Mobile Service Teams' Mission Subject to Orders Uncertainty Constraints
## 3 Preliminaries  
### An Ordered Fuzzy Numbers Framework  
The routing and scheduling problems developed to date have limited use due to the data uncertainty observed in practice. The values describing parameters such as transport time or loading/ unloading times depend on the human factor, which means they cannot be determined precisely. It is difficult to account for data uncertainty by using fuzzy variables due to the imperfections of the classical fuzzy numbers algebra [26]. Equations which describe the relationships between fuzzy variables (variables with fuzzy values) using algebraic operations (in particular, addition and multiplication) do not meet the conditions of the Ring (among others if the condition \(\forall_{A\in\mathcal{F}}\ A+0=A\) is met, then condition \(\forall_{A\in\mathcal{F}}\ \exists_{B\in\mathcal{F}}\ A+B=0\) is not met). In addition, algebraic operations based on standard fuzzy numbers follow [PERSON]'s extension principle. In practice, this means that no matter what algebraic operations are used, the support of the fuzzy number, which is the result of these operations, expands. Consequently, it is impossible to solve algebraic equations with fuzzy variables. In particular, this means that for any fuzzy numbers \(a,\ b,\ c\), the following implication \((a+b=c)\Rightarrow\left[(c-b=a)\wedge(c-a=b)\right]\) does not hold. This makes it impossible to solve a simple equation \(A+X=C\). This fact significantly hinders the use of approaches based on declarative models, in which most of the relationships between decision variables are described as linear/nonlinear equations and/or algebraic inequalities. We address these issues by proposing the formalism of ordered fuzzy numbers (OFNs) algebra [39]:  
**Definition 1**. : _An OFN is a pair of continuous real functions:_  
\[\hat{A}=(f_{A},\ g_{A}),\ where:\ f_{A},\ g_{A}:\ [0,1]\rightarrow\mathbb{R}. \tag{1}\]  
The functions \(f_{A}\) and \(g_{A}\) are called the up part and the down part of the OFN \(\hat{A}\), respectively. The values of these continuous functions are limited ranges, which can be defined as the following bounded intervals: \(UP_{A}=(l_{A0},l_{A1})\) and \(DOWN_{A}=(p_{A1},p_{A0})\). Assuming that \(f_{A}\) is increasing and \(g_{A}\) is decreasing as well as that \(f_{A}\leq g_{A}\), the membership function \(\mu_{A}\) of the OFN \(\hat{A}\) is as shown in Figure 1a,b:  
\[\mu_{A}(x)=\left\{ \begin{array}{ll}f_{A}^{-1}(x)&\text{when }x\in UP_{A}\\ g_{A}^{-1}(x)&\text{when }x\in DOWN_{A}\\ 1&\text{when }x\in[l_{A1},p_{A1}]\\ 0&\text{otherwise}\end{array} \right. \tag{2}\]  
A property called the orientation (direction) is defined for an OFN. There are two types of orientation: positive, when \(\hat{A}=(f_{A},\ g_{A})\) the direction is consistent with the direction of the \(OX\) axis, and negative, when \(\hat{A}=(g_{A},\ f_{A})\), the direction is opposite to the direction of the \(OX\) axis. Assuming that the values of all fuzzy variables may have a different orientation, the definitions of the algebraic operations used are as follows:  
**Definition 2**. : _Let \(\hat{A}=(f_{A},\ g_{A})\) and \(\hat{B}=(f_{B},\ g_{B})\) be OFNs. \(\hat{A}\) is a number equal to \(\hat{B}\) (\(\hat{A}=\hat{B}\)), \(\hat{A}\) is a number greater than \(\hat{B}\) or equal to or greater than \(\hat{B}\) (\(\hat{A}>\hat{B}\); \(\hat{A}\geq\hat{B}\)), \(\hat{A}\) is less than \(\hat{B}\) or equal to or less than \(\hat{B}\) (\(\hat{A}<\hat{B}\), \(\hat{A}\leq\hat{B}\)) if: \(\forall_{x\in[0,1]}\ f_{A}(x)*f_{B}(x)\wedge g_{A}(x)*g_{B}(x)\), where the symbol \(*\) stands for: \(=\), \(>\), \(\geq\), \(<\), or \(\leq\)._  
**Definition 3**. : _Let \(\hat{A}=(f_{A},\ g_{A})\), \(\hat{B}=(f_{B},\ g_{B})\), and \(\hat{C}=(f_{C},\ g_{C})\) be OFNs.