New fuzzy approach to facility location problem for extreme environment

Abstract

For the facility location problem under extreme environment a two-stage fuzzy approach is developed. On the first stage, the fuzzy multi-attribute group decision making (MAGDM) model for evaluation of the selection ranking index of a candidate service site is created. For this purpose the triangular fuzzy Choquet averaging (TFCA) operator is constructed. Interaction attributes, influencing the service centers’ selection process, are defined. Interaction indexes between attributes and importance values of attributes are taken into account in the construction of the 2-order additive triangular fuzzy valued fuzzy measure (TFVFM). On the second stage, based on the TFCA operator, a new objective function - selection ranking index of candidate sites is constructed. We consider also two classical objective functions - total cost for opening of service centers and number of agents needed to operate the opened service centers. A new objective function together with latter ones creates the multi-objective fuzzy facility location set covering problem. A Pareto front for this problem is constructed. A simulation example of emergency service facility location planning for a city is considered. The example deals with the problem of planning fire station locations for serving emergency situations in specific demand points – critical infrastructure objects.

Keywords

Multi-objective facility location problem associated probabilities of a fuzzy measure fuzzy numbers choquet integral multi-attribute group decision making

1 Introduction

Timely servicing from emergency service centers to the affected geographical areas (demand points, critical infrastructure objects) is a key task of the emergency management system. Scientific research in this area focuses on distribution networks decision-making problems, which are known as a Facility Location Problem (FLP) ([4, 9] and others).

Location planning for service centers in extreme environment is a complex decision that involves consideration of multiple interaction attributes like maximum customer coverage, minimum service costs, least impacts on geographical points’ residents and environment, and conformance to freight regulations of these points. FLP’s models have to support the generation of optimal locations of service centers in complex and uncertain situations. There are several publications about application of fuzzy methods in the FLP under uncertainty environment. However, all of them have a common approach. They represent experts evaluations of modelling parameters and develop solving methods for facility location problems called in this case fuzzy facility location problem (FFLP) ([14, 20] and others). In this work we consider a two-stage model of FFLP based on the fuzzy aggregation operator’s approach [6] for the optimal selection of facility location centers in extreme environment. As a result, dependence and interactions between attributes hold in MAGDM models. In 1974, Sugeno [21] introduced the concept of non-additive measure (fuzzy measure), which is characterized by monotonicity instead of additivity property. It is the most effective tool to model interaction phenomena [11 , 19] and deal with such decision problems [10, 12]. At the first stage: triangular fuzzy valued fuzzy measure and its associated probabilities on a finite set are developed; Choquet integral (Choquet, 1954) with respect to triangular fuzzy valued fuzzy measure (TFVFM) is defined (Section 3). In Section 4: Experts’ evaluations on candidate service sites with respect to attributes and interaction indexes between attributes and important values of attributes in triangular fuzzy numbers are presented; interactions between attributes and important values of attributes are taken into account in the construction of the 2-order additive TFVFM as an uncertainty pole of experts evaluations. Calculation scheme of selection ranking index of a candidate site by the TFCA operator is constructed in Section 5. The fuzzy multi-attribute decision making model for the evaluation of the selection ranking index of a service site is also created in Section 5. At the second stage, based on the TFCA operator, a new objective function - total selection ranking index of candidate sites is constructed (Section 6). This function together with two classical objective functions, minimization of total cost for opening of service centers and minimization of number of agents needed to operate the opened service centers, creates the fuzzy multi-objective facility location set covering problem (Section 6). A simulation model of emergency service facility location planning for a city is considered in Section 7. The example deals with the problem of planning fire station locations for serving emergency situations in specific demand points – critical infrastructure objects. The main results of the work are presented in the conclusive section.

2 On the lattice of triangular fuzzy numbers and fuzzy probability averaging operators

The fuzzy numbers (FNs) [7] have been studied by many authors. They can be represented in a more complete way as an imprecision variable of the incomplete expert information. It can consider the maximum and minimum and the possibility that the interval values may occur.

Definition 1. [7]. $\tilde{a} (x) : R^{1} \to [0; 1]$ is called the FN which can be considered as a generalization of the interval number: $\tilde{a} (x) = {\begin{matrix} 1, i f x \in [a'_{2}; a''_{2}] \\ \frac{x - a_{1}}{a'_{2} - a_{1}}, i f x \in [a_{1}, a'_{2}] \\ \frac{a_{3} - x}{a_{3} - a''_{2}}, i f x \in [a''_{2}, a_{3}] \\ 0, o t h e r w i s e \end{matrix}$ where $a_{1} \leq a'_{2} \leq a''_{2} \leq a_{3} \in R^{1}$ .

Below, we review arithmetic operations for triangular FNs (TFNs) for which in $\tilde{a} (x)$ $a'_{2} = a''_{2}$ . Let $\tilde{a}$ and $\tilde{b}$ be two TFNs, where $\tilde{a} = (a_{1}, a_{2}, a_{3})$ and $\tilde{b} = (b_{1}, b_{2}, b_{3})$ . Then

$\tilde{a} + \tilde{b} = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3})$ ;

$\tilde{a} - \tilde{b} = (a_{1} - b_{3}, a_{2} - b_{2}, a_{3} - b_{1})$ ;

$\tilde{a} \times k = ({ka}_{1}, {ka}_{2}, {ka}_{3}), k > 0$ ;

${\tilde{a}}^{k} = (a_{1}^{k}, a_{2}^{k}, a_{3}^{k}), k > 0, a_{i} > 0$ ;

$\tilde{a} \cdot \tilde{b} = (a_{1} b_{1}, a_{2} b_{2}, a_{3} b_{3}), a_{i} > 0, b_{i} > 0$ ;

${\tilde{b}}^{- 1} = {\frac{1}{b_{3}}, \frac{1}{b_{2}}, \frac{1}{b_{1}}}, b_{i} > 0$ .

We define: $\begin{matrix} \tilde{a} \cdot {log}_{2} \tilde{a} = a_{2} \cdot {log}_{2} (a_{2}) if a_{2} > 0, \\ \tilde{a} \cdot {log}_{2} \tilde{a} = 0 if a_{2} \leq 0 \end{matrix}$ .

We say that $\tilde{a} > \tilde{b} if a_{2} > b_{2}$ or if a₂ = b₂ then if $\frac{a_{1} + a_{3}}{2} > \frac{b_{1} + b_{3}}{2}$ , otherwise $\tilde{a} = \tilde{b}$ .

The set of all TFNs is denoted by Ψ and nonnegative TFNs (a_i ≥ 0) by $Ψ_{0}^{+}$ . Note that on the lattice Ψ 1_Ψ = (1, 1, 1) and 0_Ψ = (0, 0, 0). The latest notion of inequality induces the total ordering ≥_t on the lattice Ψ and we shall say that $\tilde{a} \geq_{t} \tilde{b}$ iff $\tilde{a} > \tilde{b}$ or $\tilde{a} = \tilde{b}$ . However there are subsets of zero and unit elements on the lattice Ψ: I_Ψ = {(α, 1, β) , (α, 1, β) = _t (1, 1, 1)} and O_Ψ = {(α, 0, β) , (α, 0, β) = _t (0, 0, 0)}. We define the operations of max and min based on the total ordering ≥_t. We say that $max_{t} {\tilde{a}; \tilde{b}} = \tilde{a}$ and $min_{t} {\tilde{a}; \tilde{b}} = \tilde{b}$ iff $\tilde{a} \geq_{t} \tilde{b}$ . A possibility distribution [7] on the Ψis defined as $Pos (\tilde{a} \leq τ) = {\begin{matrix} 1, τ > a_{2}; \\ \frac{τ - a_{1}}{a_{2} - a_{1}}, a_{1} \leq τ \leq a_{2} . \\ 0, τ < a_{1} . \end{matrix}$

Analogously to known probability averaging (PA) operator, based on Def. 1, we present the equivalent form of the fuzzy PA (FPA) operator as an extension of the PA operator on Ψ. If a_i ∈ Ψ ; i = 1, . . . , n, then ${FPA}_{P} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) = \sum_{i = 1}^{n} p_{i} {\tilde{a}}_{i}$

It is easy to prove that if arguments in the FPA present real numbers – ${\tilde{a}}_{i} \equiv (a_{i}, a_{i}, a_{i})$ - then FPA coincides with PA.

As a result, we have required that an aggregation operator satisfies three conditions: monotonicity, boundedness (therefore idempotency) and symmetricity. Such operators are called averaging (mean) aggregation operators. The objective properties required from aggregation operators are presented in [1 , 24] and others. One of the main aims of this work is to construct new extensions of the FPA operators when the probability measure will be changed by the monotone measure [5]. As usual, in FFLPs there are no existing statistical data for definitions of model’s input parameters probability distributions. It is not possible to use probability analysis in aggregation process of the optimization models and objective functions for the solution of a FFLP problem. Therefore, we use experts’ evaluation in a fuzzy probability approach in such cases.

Definition 2. Triangular fuzzy valued probability measure (TFVPM) on a finite set S = {s₁, . . . , s_n} is a mapping $\tilde{P} : 2^{S} \to Ψ$ , ${\tilde{p}}_{i} \equiv \tilde{P} ({s_{i}})$ $\bar{0} \leq_{t} {\tilde{p}}_{i} \leq_{t} \bar{1}$ , ${\tilde{p}}_{i} \in Ψ$ , $\sum_{i = 1}^{n} {\tilde{p}}_{i} =_{t} \bar{1}$ , $\bar{0} \in O_{Ψ}, \bar{1}, \in I_{Ψ}$ and for ∀A ⊆ S $\tilde{P} (A) = \sum_{s_{i} \in A} {\tilde{p}}_{i}$ (1)

Definition 3. Let ${\tilde{a}}_{i} \in Ψ; i = 1, . . ., n$ and $\tilde{P}$ be a TFVPM on S. Then triangular fuzzy probability averaging (TFPA) operator of a fuzzy variable $\tilde{a} = ({\tilde{a}}_{1}, . . ., {\tilde{a}}_{n})$ , ${\tilde{a}}_{i} \equiv \tilde{a} (s_{i}), i = 1, . . ., n$ is called ${TFPA}_{\tilde{P}} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) = \sum_{i = 1}^{n} {\tilde{p}}_{i} {\tilde{a}}_{i},$ (2) where ${\tilde{p}}_{i} \equiv \tilde{P} ({\tilde{a}}_{i})$ .

3 Triangular fuzzy valued fuzzy measure and Choquet averaging aggregations

We introduce the definition of a fuzzy measure[5 , 21] adapted to the case of a finite referential:

Definition 4. [5]. Let S ={ s₁, s₂, . . . , s_n } be a finite set and g be a set function g : 2^S → [0, 1]. We say g is a fuzzy measure (FM) on S if it satisfies

g (Ø) = 0 ; g (S) =1 ;

∀ A, B ⊆ S if A ⊆ B, then g (A) ≤ g (B).

A FM is a normalized and monotone set function. It can be considered as an extension of the probability concept, where additivity is replaced by the weaker condition of monotonicity.

Definition 5. Let S ={ s₁, s₂, . . . , s_n } be a finite set and $\tilde{g}$ be a triangular fuzzy valued set function $\tilde{g} : 2^{S} \to Ψ$ . We say $\tilde{g}$ is a triangular fuzzy valued fuzzy measure (TFVFM) on S if it satisfies

$g (\emptyset) = \bar{0}; g (S) = \bar{1}; \bar{0} \in O_{ψ}, \bar{1} \in I_{ψ}$

$\forall A, B \subseteq S if A \subseteq B, then \bar{g} (A) \leq \bar{g} (B)$ (3)

We consider some aspects of a finite Choquet Integral in aggregations.

Definition 6. [3]. Assume S ={ s₁, s₂, . . . , s_n } is a set of states of nature on which we have a fuzzy measure g and a variable of expert evaluations $a : S \Rightarrow R_{0}^{+}$ such that a (s_i) ≡ a_i ≥ 0, i = 1, 2, . . . , n. Then the aggregation ${CA}_{g} (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} p_{j} a_{i (j)},$ (4) where $\begin{matrix} p_{j} = g ({s_{i (1)}, . . . . ., s_{i (j)}}) - g ({s_{i (1)}, . . . . ., s_{i (j - 1)}}), \\ g ({s_{i (0)}}) \equiv 0, \end{matrix}$ is called a Finite Choquet Averaging (CA) operator. Further, i (·) is an index function such that a_i(j) is the j-th largest of the ${a_{i}}_{i = 1}^{n}$ .

There exist many works on extensions of the Choquet averaging operator for fuzzy arguments. In general, the extensions are constructed on combination of the Zadeh’s extension principle and a Mobius transform of the fuzzy measure ([16, 17, 19 and others). For our reasoning rbased on the Def. 1 rwe present an extension of the CA operator on the lattice Ψ. We consider an extension of the CA operator for a TFVFM:

Definition 7. Assume S ={ s₁, s₂, . . . , s_n } is a set of states of nature on which we have a TFVFM - $\tilde{g}$ and a fuzzy variable of expert evaluations $\tilde{a} : S \Rightarrow Ψ_{0}^{+}$ such that $\tilde{a} (s_{i}) \equiv {\tilde{a}}_{i} \in Ψ_{0}^{+}, i = 1, 2, . . ., n$ . Then the aggregation ${TFCA}_{\tilde{g}} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, . . ., {\tilde{a}}_{n}) = \sum_{j = 1}^{n} {\tilde{p}}_{j} {\tilde{a}}_{i (j)},$ (5) where $\begin{matrix} {\tilde{p}}_{j} = \tilde{g} ({s_{i (1)}, . . . . ., s_{i (j)}}) - \tilde{g} ({s_{i (1)}, . . . . ., s_{i (j - 1)}}), \\ \tilde{g} ({s_{i (0)}}) \equiv 0_{Ψ}, \end{matrix}$ is called a Finite Triangular Fuzzy Choquet Averaging (TFCA) operator. In the proceeding i (·) is index function such that ${\tilde{a}}_{i (j)}$ is the j-th largest of the ${{\tilde{a}}_{i}}_{i = 1}^{n}$ in the sense of the total ordering ≥_t.

It is easy to prove that the TFCA operator is an averaging aggregation operator. If in (5) a TFVFM is a TFVFM then the TFCA value coincides with the TFPA value and if in (5) a TFVFM is a FM and arguments represent real nonnegative numbers – ${\tilde{a}}_{i} \equiv (a_{i}, a_{i}, a_{i}), a_{i} \geq 0$ , then the TFCA value coincides with the CA value. Other analytical properties of the TFCA operator are studied (omitted here).

In general, the possible orderings of the elements of S are given by the permutations of a set with n elements, which form the group S_n. Now we consider a definition of associated probabilities induced by a fuzzy measure g on S.

Definition 8. [2]: The probability functions P_σ defined by $\begin{matrix} P_{σ} (S_{σ (1)}) = g ({S_{σ (1)}}), \dots, \\ P_{σ} (S_{σ (i)}) = g ({S_{σ (1)}, \dots, S_{σ (i)}}) \\ - g ({S_{σ (1)}, \dots, S_{σ (i - 1)}}), \dots \\ P_{σ} (S_{σ (m)}) = 1 - g ({S_{σ (1)}, \dots, S_{σ (i - 1)}}), \\ g ({S_{σ (0)}}) \equiv 0 \end{matrix}$ (6) for each σ = (σ (1) , σ (2) , . . . , σ (n)) ∈ S_n, are called the associated probabilities and the Associated Probability Class (APC)-{P_σ} _{σ∈S_n} of a fuzzy measure g.

Analogous results may be received for the TFVFM - $\tilde{g}$ .

Definition 9. The TFVPM ${\tilde{P}}_{σ}$ on S defined by $\begin{matrix} {\tilde{P}}_{σ} (s_{σ (1)}) = \tilde{g} ({s_{σ (1)}}), . . . ., \\ {\tilde{P}}_{σ} (s_{σ (i)}) = \tilde{g} ({s_{σ (1)}, . . . ., s_{σ (i)}}) \\ - \tilde{g} ({s_{σ (1)}, . . . ., s_{σ (i - 1)}}), . . . ., \\ {\tilde{P}}_{σ} (s_{σ (m)}) = 1_{Ψ} - \tilde{g} ({s_{σ (1)}, . . . ., s_{σ (n - 1)}}), \\ \tilde{g} ({s_{σ (0)}}) \equiv 0_{Ψ} \end{matrix}$ (7) for each σ = (σ (1) , σ (2) , . . . , σ (n)) ∈ S_n, are called the associated triangular fuzzy valued probabilities and the Associated Fuzzy Probability Class (AFPC)- ${{\tilde{P}}_{σ}}_{σ \in S_{n}}$ of a TFVFM - $\tilde{g}$ .

4 Identification of associated fuzzy probabilities in fuzzy MAGDM by attributes interaction and important parameters

The fuzzy measure can represent flexibly a certain kind of an interaction among the decision attributes and can vary from redundancy (negative interaction) to synergy (positive interaction) [13].

Definition 10. [13]. Let g be a set function (not necessarily a fuzzy measure) on S = {s₁, . . . , s_n} . The Mobius representation of g is a set function m_g : 2^S → R defined by $m_{g} (A) = \sum_{B \subset A} (- 1)^{| A ∖ B |} g (B) \forall A \subset S,$ (8) where |A| is the cardinality of the set A, R = (- ∞ , + ∞) .

Any fuzzy measure (any set function) g can be uniquely expressed in terms of its Mobius representation by $g (A) = \sum_{B \subset A} m_{g} (B) \forall A \subset S .$ (9)

Definition 11. [13]. Let g be a fuzzy measure on S. g is said be k, k ∈ {1, 2, . . . , n}, -order additive if its Mobius representation satisfies m_g (A) =0 for all A ⊂ Ssuch that |A| > k and there exists at least one subset A ⊂ S such that m_g (A) ≠0.

Analogously from Def. 5, a fuzzy measure will be defined by $\sum_{l = 1}^{k} (C_{n}^{l})$ coefficients. In our future applications as a fuzzy measure we will use the 2-order additive fuzzy measure [13, 22]. In this case, we only need n (n + 1)/2 coefficients to determine a 2– order additive fuzzy measure.

Definition 12. [13]. Let g be a fuzzy measure on attributes set S

The overall importance value of an attribute s_i ∈ S is called its Shapley value, defined by

$\begin{matrix} I_{i} = \sum_{A \subset S ∖ {s_{i}}} [(| S | - | A | - 1)! / (| S |!)] \\ \cdot [g (A \cup {s_{i}}) - g (A)], \end{matrix}$ (10)

The interaction index of two attributes s_i, s_j ∈ S, i ≠ j is defined by

$\begin{matrix} I_{ij} = \sum_{A \subset S ∖ {s_{i}, s_{j}}} [(| S | - | A | - 2)! / (| S |! - 1)] \\ \cdot [g (A \cup {s_{i}, s_{j}}) \\ - g (A \cup {s_{i}}) - g (A \cup {s_{j}}) + g (A)] . \end{matrix}$ (11)

Using mathematical induction it is easy to prove for a 2-order additive FM and ∀σ = {σ (1) , . . . , σ (n)} ∈ S_n, i = 1, . . . , n, $g ({s_{σ (1)}, . . ., s_{σ (i)}}) = \sum_{l = 1}^{i} I_{σ (l)} - (1 / 2) \cdot \sum_{k \in N_{σ (i)}} \sum_{l = 1}^{i} I_{σ (i) l},$ (12) where N_σ(i) denotes an index subset N_σ(i) ={ 1, …, n } ∖ { σ (1) , …, σ (n) }. Using definition of the associated probabilities of a FM (Def. 8) we obtain connections between associated probabilities, importance values and interaction indexes of attributes:

$\begin{matrix} p_{σ} (s_{σ (i)}) = I_{σ (i)} + (1 / 2) \cdot \sum_{l = 1}^{i - 1} I_{σ (i) σ (l)} - (1 / 2) \\ \cdot \sum_{l = i + 1}^{n} I_{σ (i) σ (l)}, \end{matrix}$ (13) where if i = 1, then the second addend is zero, and if i = n, then the third addend is zero. Representation of the associated probability (13) has an interesting interpretation in terms of the representation of interaction between criteria. In (13) in the positive role are involved corresponding interaction indexesof {s_σ(1)} , {s_σ(1), s_σ(2)} , . . . , {s_σ(1), s_σ(2), . . . , s_σ(i-1)}structure, while in negative role are involved relevant interaction indexes of {s_σ(i+1)} , . . . , {s_σ(i+1), . . . , s_σ(n)} structure. Therefore, n × n ! probabilities are constructed with n (n + 1)/2 interaction indexes J = {I_ij} , i ≠ j, I_ij = I_ji and n values of overall importance values (I = {I_i} , i = 1, . . . , n).

Analogously, we have:

Definition 13. Let $\tilde{g}$ be a TFVFM on S.

The overall fuzzy importance value of an attribute s_i ∈ S is called its fuzzy Shapley value (FSV), defined by

$\begin{matrix} {\tilde{I}}_{i} = \sum_{A \subset S ∖ {s_{i}}} [(| S | - | A | - 1)! / (| S |!)] \\ \cdot [\tilde{g} (A \cup {s_{i}}) - \tilde{g} (A)], \end{matrix}$ (14)

The fuzzy interaction index of two attributes s_i, s_j ∈ S is defined by

$\begin{matrix} {\tilde{I}}_{ij} = \sum_{A \subset S ∖ {s_{i}, s_{j}}} [(| S | - | A | - 2)! / (| S |! - 1)] \\ \cdot [\tilde{g} (A \cup {s_{i}, s_{j}}) \\ - \tilde{g} (A \cup {s_{i}}) - \tilde{g} (A \cup {s_{j}}) + \tilde{g} (A)] . \end{matrix}$ (15)

It is easy to verify that $\sum_{i = 1}^{n} {\tilde{I}}_{i} \in I_{Ψ}$ .

For the definition of APC {P_σ } _{σ∈S_n} of the FM g in the real experiment, experts are able to evaluate importance values {I_i} and interaction indexes {I_ij} of attributes. It will more convenient from the point of intellectual activity of experts if experts present their knowledge in fuzzy terms, such as “very low”, “low” and so on (Table 1). Therefore, in the aggregation procedure, to construct TFVFM and its associated TFVPMs is more natural. For the definition of associated TFVPMs by the fuzzy importance values ${{\tilde{I}}_{i}}$ and fuzzy interaction indexes of attributes ${{\tilde{I}}_{ij}}$ , a fuzzy mathematical programming problem is constructed using entropy maximum principle. On the basis of (13) let the associated TFVPM be presented as

${\tilde{p}}_{σ} (s_{σ (i)}) = {\tilde{I}}_{σ (i)} + c {\tilde{I}}^{σ (i)}$ (16) where ${\tilde{I}}^{σ (i)} \equiv (1 / 2) {\sum_{l = 1}^{i - 1} {\tilde{I}}_{σ (i) σ (l)} - \sum_{l = i + 1}^{n} {\tilde{I}}_{σ (i) σ (l)}}, c \geq 0$ (17)c is a constant, which indicates on the normalization value of supports of TFVs in sum (16). Using maximum entropy principle, the definition of the AFPC can be reduced to the definition of the parameter c ≥ 0 by solution of the following fuzzy mathematical programming problem: $\begin{matrix} max_{c \geq 0} {max_{σ \in S_{n}} (- \sum_{i = 1}^{n} {\tilde{p}}_{σ} (s_{σ (i)}) \cdot {log}_{2} ({\tilde{p}}_{σ} (s_{σ (i)}))}, \\ 0_{Ψ} \leq_{t} {\tilde{p}}_{σ} (s_{σ (i)}) \leq_{t} 1_{Ψ}, \\ \sum_{i = 1}^{n} {\tilde{p}}_{σ} (s_{σ (i)}) =_{t} 1_{Ψ} . \end{matrix}$ (18)

Table 1

Fuzzy terms and their fuzzy ratings.

Fuzzy terms	Ratings in TFNs
Very low (VL)	(1,1,3)
Low (L)	(1,3,5)
Medium (M)	(3,5,7)
High (H)	(5,7,9)
Very high (VH)	(7,9,9)
Negative Very Low (NVL)	(–3,–1,–1)
Negative Low (NL)	(–5,–3,–1)
Negative Medium (NM)	(–7,–5,–3)
Negative High (NH)	(–9,–7,–5)
Negative Very high (NVH)	(–9,–9,–7)

If c = 0 in (16), then fuzzy interaction indexes are not taken into account in associated fuzzy probabilities. Associated fuzzy probabilities are depended only on Shapley indexes and the TFVFM is a TFVPM. In such case in calculations we consider probabilistic environment.

5 Scheme of a calculation of candidate site’s selection ranking index by the TFCA operator

At first, we are focusing on a multi-attribute group decision making approach for location planning for selection of service centers under uncertain and extreme environment. We develop a fuzzy multi-attribute decision making approach for the service center location selection problem for which a fuzzy probability aggregation operators’ approachis used.

The formation of expert’s input data for construction of attributes is an important task of the centers’ selection problem. To decide on the location of service centers, it is assumed that a set of candidate sites already exists. This set is denoted by CC = {cc₁, cc₂, . . . , cc_m}, where we can locate service centers and S = {s₁, s₂, . . . , s_n} be the set of all attributes (transformed in benefit attributes) which define CCs selection. For example: 1. s₁ =” post disaster access by public and special transport modes to the candidate site”; 2. s₂ =” post disaster security of the candidate site from accidents, theft and vandalism”; 3. s₃ =” post disaster connectivity of the location with other modes of transport (highways, railways, seaport, airport etc.)”; 4. s₄ =” costs in vehicle resources, required products and etc. for the location of CCs in candidate site”; 5. s₅ = “impact of the candidate site on the environment, such as important objects of Critical Infrastructure and others”; 6. s₆ =” distances of the candidate site from the central locations”; 7. s₇ =” distances of the candidate site from demand points”; 8. s₈ = “availability of raw material and labor resources in the candidate site”; 9. s₉ =” ability to conform to sustainable freight regulations imposed by emergency managers post disaster for e.g. restricted delivery hours, special delivery zones”; 10. s₁₀ =” ability to increase size to accommodate growing demands post disaster” and others.

Let us assume that DP = {dp₁, dp₂, . . . , dp_l} is the set of all demand points (customers). Let $\tilde{W} = {{\tilde{w}}_{1}, {\tilde{w}}_{2}, . . ., {\tilde{w}}_{n}}$ be the fuzzy weights of attributes. For each expert e_k from invited group of experts (emergency service dispatchers and so on) E = {e₁, e₂, . . . , e_t}, let $\tilde{a}'_{ij}^{k}$ be the fuzzy positive rating (see Table 1) of his/her evaluation for each candidate site cc_i , (i = 1, . . . , m), with respect to each attribute s_j, (j = 1, . . . , n). For the expert e_k we construct binary fuzzy relation $\tilde{A}'_{k} = {\tilde{a}'_{ij}^{k}, i = 1, . . ., m; j = 1, . . ., n}$ , elements of which are represented in fuzzy terms (Table 1). Our task is to build aggregation operators’ approach, which for each candidate site cc_i , (i = 1, . . . , m) aggregates presented objective and subjective data into scalar values – site’ s selection ranking index. This aggregation formally can be represented as: $\begin{matrix} δ_{i} \equiv δ {(cc}_{i}) = Agregg (DP, CC, \tilde{g}, [\tilde{A}'_{k}]_{i}, \\ k = 1, . . ., t), i = 1, . . ., n . \end{matrix}$ where $\tilde{g}$ is – a TFVFM, which take into account fuzzy interaction indexes between attributes and fuzzy important values (weights) of attributes in its construction (Sec. 4).

In fuzzy set theory [7, 25] conversion scales are applied to transform the fuzzy terms into triangular fuzzy numbers. In our approach, we apply a rating scale of 1–9 as semantic form of fuzzy terms of the linguistic variables: attribute’s valuation, importance value of an attribute and interaction index of attributes. Table 1 presents the linguistic variables fuzzy terms and their semantic ratings.

The proposed framework of location planning for candidate sites comprises the following steps:

Step 1: Selection of location attributes. Involves the selection of location attributes for evaluating potential locations for candidate sites. These attributes are obtained from literature review, and discussion with experts and members of the city transportation group. We use 10 attributes defined above. It can be seen that attributes s₃ and s₄ belong to the cost category, that is, the lower the value, the more preferable the alternative for the best location. The remaining attributes are benefit type attributes, which means the higher the value, the more preferable the alternative is for selection.

Step 2: Selection of candidate location sites. Involves selection of potential locations for implementing service centers. The decision makers use their knowledge, prior experience with transportation or other conditions of the geographical area of extreme events and the presence of sustainable freight regulations to identify candidate locations for implementing service centers. For example, if certain areas are restricted for delivery by municipal administration, then these areas are barred from being considered as potential locations for implementing urban service centers. Ideally, the potential locations are those that cater for the interest of all city stakeholders, which are city residents, logistics operators, municipal administrations etc.

Step 3: Locations evaluation using fuzzy aggregation approach. The third step involves evaluation of candidate location sites against the selected attributes (step 1) using the technique of fuzzy approach of alternatives selection.

Step 3.1. Assignment of ratings to the attributes with respect to the candidate sites. Let $\tilde{A}'_{k} = {a'_{ij}^{k}, i = 1, . . ., m; j = 1, . . ., n}$ be the performance ratings of each expert e_k (k = 1, 2 , . . , t) for each candidate site cc_i (i = 1, 2 , . . , m) with respect to attributes s_j (j = 1, 2 , . . , n) and ${\tilde{W}}_{k} = {{\tilde{w}}_{1}^{k}, {\tilde{w}}_{2}^{k}, . . ., {\tilde{w}}_{n}^{k}}$ be weights of attributes presented in positive ratings from Table 1.

Step 3.2. Compute aggregated fuzzy ratings for the attributes and the candidate sites. Let the fuzzy ratings of all experts be described by positive ratings (Table 1) ${\tilde{q}}^{k} = (q_{1}^{k}, q_{2}^{k}, q_{3}^{k}), k = 1, 2, . . ., t$ . If the fuzzy performance ratings and importance values of the attributes evaluated by the k-th expert are $a'_{i j}^{k} = (a'_{i j 1}^{k}, a'_{i j 2}^{k}, a'_{i j 3}^{k})$ and ${\tilde{w}}_{j}^{k} = (w_{j 1}^{k}, w_{j 2}^{k}, w_{j 3}^{k}),$ i = 1, . . . , m : j = 1, 2, . . , n, respectively, then the aggregated fuzzy ratings $(\tilde{a}'_{ij})$ of candidate sites with respect to each attribute are given by $\tilde{a}'_{i j} = (a'_{i j 1}, a'_{i j 2}, a'_{i j 3})$ , where

$\begin{array}{l} a'_{i j l} = (\sum_{k = 1}^{t} q_{l}^{k} a'_{i j l}^{k}) / (\sum_{k = 1}^{t} q_{l}^{k}), l = 1, 2, 3; \\ i = 1, ., m; j = 1, ., n . \end{array}$ (19)

The aggregated fuzzy weights of attributes $({\tilde{w}}_{j}), j = 1, . . ., n$ , are calculated as ${\tilde{w}}_{j} = (w_{j 1}, w_{j 2}, w_{j 3})$ , where $w_{jl} = (\sum_{k = 1}^{t} w_{jl}^{k} q_{l}^{k}) / (\sum_{k = 1}^{t} q_{l}^{k}), i = 1, 2, 3; j = 1, . . ., n$ (20)

Step 3.3. Compute the fuzzy decision matrix. The fuzzy decision matrix $\tilde{A}'$ for the candidate sites CC and the attributes S is constructed as follows: $\begin{matrix} \begin{matrix} s_{1} & s_{2} & s_{n} \end{matrix} \\ \begin{matrix} {cc}_{1} \\ {cc}_{2} \\ {cc}_{m} \end{matrix} [\begin{matrix} \tilde{a}'_{11} & \tilde{a}'_{12} & . . . & \tilde{a}'_{1 n} \\ \tilde{a}'_{21} & \tilde{a}'_{22} & . . . & \tilde{a}'_{2 n} \\ . . . & . . . & . . . & . . . \\ \tilde{a}'_{m 1} & \tilde{a}'_{m 2} & . . . & \tilde{a}'_{mn} \end{matrix}] \end{matrix}$ (21)

Step 3.4. Normalize the fuzzy decision matrix. The raw data are normalized using a linear scale transformation to bring the various attributes scales onto a comparable scale. The normalized fuzzy decision matrix $\tilde{A} = {{\tilde{a}}_{ij}}, i = 1, 2, . . ., m; j = 1, 2, . . ., n$ , is given by fuzzy decision matrix $\tilde{A}'$ , where ${\tilde{a}}_{i j} = (\frac{a'_{i j 1}}{a'_{j}^{}}, \frac{a'_{i j 2}}{a'_{j}^{}}, \frac{a'_{i j 3}}{a'_{j}^{}}) o r {\tilde{a}}_{i j} = (\frac{a_{j}^{-}}{a'_{i j 3}}, \frac{a_{j}^{-}}{a'_{i j 2}}, \frac{a_{j}^{-}}{a'_{i j 1}})$ (22) where $a'_{j} = \max_{i} a'_{i j 3}$ (for benefit attributes) and $a_{j}^{-} = min_{i} a'_{ij 1}$ (for cost attributes).

Step 4: Identify constructive TFVFM which takes into account attributes fuzzy importance values and attributes fuzzy interactions. It is usual that attributes of fuzzy importance values ${{\tilde{I}}_{i}}, i = 1, . . ., n$ can be defined as normalized aggregated fuzzy weights of attributes ${{\tilde{w}}_{i}}, i = 1, . . ., n$ and

${\tilde{I}}_{i} = {\tilde{w}}_{i} / (\sum_{j = 1}^{n} {\tilde{w}}_{j}), i = 1, . . ., n$ . Let for each expert e_k (k = 1, 2 , . . , t) attributes interaction indexes are ${\tilde{I} {''}_{ij}^{k}}, \tilde{I} {''}_{ij}^{k} = \tilde{I} {''}_{ji}^{k}, i \neq j, i, j = 1, . . ., n$ . Aggregated interaction indexes are received by normalization of weighted sums of experts evaluations: ${\tilde{I}}_{ij} = \tilde{I}'_{ij} / (max_{l \neq m} | \tilde{I}'_{lm} |), i \neq j$ , where $\begin{matrix} \tilde{I}'_{ij} = (\sum_{k = 1}^{t} \tilde{I} {''}_{ij}^{k} \cdot {\tilde{q}}^{k}) / (\sum_{k = 1}^{t} {\tilde{q}}^{k}), \\ i \neq j, i, j = 1, . . ., n \end{matrix}$ and | (a₁, a₂, a₃) | = (a₁, a₂, a₃) for the positive rating and | (a₁, a₂, a₃) | = (- a₃, - a₂, - a₁) or the negative rating.

According to Section 4 we construct the associated TFVPMs (16)–(18) of a TFVFM.

Step 5: Compute selection ranking index of candidate sites by the TFCA operator.

Our task is to build aggregation operators’ approach, for which each candidate site cc_i , (i = 1, . . . , m), aggregates presented objective and subjective data into scalar values – site’s selection ranking index. This aggregation we define as the TFCA operator’s value: ${\tilde{δ}}_{i} \equiv \tilde{δ} {(cc}_{i}) = TFCA ({\tilde{a}}_{i 1}, {\tilde{a}}_{i 2}, . . ., {\tilde{a}}_{in}),$ (23) where $\tilde{g}$ is – the 2-order additive TFVFM which takes into account fuzzy interactions between attributes and fuzzy important values of attributes in its construction (step 4).

6 Multi-objective optimization model for fuzzy facility location set covering problem

In this Section we are focusing on the multi-objective optimization model [8] for the location set covering problem (LSCP). LSCP was proposed by C. Toregas and C. Revell in 1972, which seeks a solution for locating the least number of facilities to cover all demand points within the service distance. Different aggregation operators applied in fuzzy extensions of LSCP for facility location were described in [6]. In this work, using a new aggregation approach, we construct new fuzzy LSCP model for emergency service facility location planning.

If x = {x₁, x₂, . . . , x_m} is Boolean decision vector, which defines some selection from candidate sites CC = {cc₁, cc₂, . . . , cc_m} for facility location, we can build sites’ selection ranking index as a linear sum of triangular fuzzy values - ${\tilde{δ}}_{j} x_{j}$ : As a result, new fuzzy objective function selection ranking index of candidate sites $\sum_{j = 1}^{m} {\tilde{δ}}_{j} x_{j}$ is constructed. Its maximization selects a group of sites with the maximal selection ranking index from admissible covering selections. A classical facility location set covering problem “tries” to minimize the total cost needed for opening of service centers - $\sum_{j = 1}^{m} C_{j} x_{j}$ , where C_j is a cost necessary for opening of the candidate site -cc_j. We assume that we know in advance the approximate number of people needed to make a candidate site operate as a service center. This number is denoted by M_i, i = 1, . . . , m. Our goal is to locate service facilities centers with minimal number of agents needed to operate the opened service centers - $\sum_{j = 1}^{m} M_{j} x_{j}$ . The problem aims to locate service facilities in minimal travel time from candidate sites. Let experts evaluated movement fuzzy times between demand points and candidate sites

${\tilde{t}}_{ij}, {dp}_{i} \in DP; i = 1, . . ., l; {cc}_{j} \in CC, j = 1, . . ., m$ .In extreme environment for emergency planning a radius of service center is defined based not on distance but on maximum allowed time T for movement, since the rapid help and servicing is crucial for demand points in such situations. Respectively, a set of candidate sites N_i, covering customer dp_i ∈ DP = {dp₁, dp₂, . . . , dp_l}, is defined as $N_{i} = {{cc}_{j}, {cc}_{j} \in CC / Poss ({\tilde{t}}_{ij} \leq T) \geq λ}$ , 0 < λ <1, where λ is a minimal possibilistic level (selected by the administrators) which coveringcondition ${\tilde{t}}_{ij} \leq T$ is true. Then we can state fuzzymulti-objective facility location set covering problem: $max_{t} z_{1} = \sum_{j = 1}^{m} {\tilde{δ}}_{j} x_{j}$ (24) $min z_{2} = \sum_{j = 1}^{m} C_{j} x_{j}$ (25) $min z_{3} = \sum_{j = 1}^{m} M_{j} x_{j}$ (26) $\sum_{s_{j} \in N_{i}} x_{j} \geq 1 (i = 1, 2, . . ., l); x_{j} \in {0, 1}, j = 1, 2, . . ., n$ (27)

For the exact solution of the problem (24)–(27) an ɛ-constraint approach is developed. A Pareto front is constructed.

7 Simulation of emergency service facility location model

We illustrate the effectiveness of the constructed fuzzy optimization model by the numerical example. Let us consider an emergency management administration of Tbilisi (Georgia) city that wishes to locate some fire stations with respect to timely servicing of critical infrastructure objects. Assume that there are 30 demand points (critical infrastructure objects) and 8 candidate facility sites (fire stations) in the urban area. Let us have 4 experts from Emergency Management Agency (EMA) of the city of Tbilisi for evaluation of the travel times and ratings values of candidate sites, interaction indexes and importance values of attributes. The travel times between demand points and candidate sites are evaluated in triangular fuzzy numbers (evaluated in minutes, omitted because the matrix of fuzzy travel times has large dimensions). According to the standards of the EMA, let the principle of location fire stations be that the fire station can reach the area edge within 5 minutes after receiving the dispatched instruction. Therefore, we set covering radius T = 5 minutes. Minimal possibilistic level λ is equal to 0.9.

Based on the semantic form (Table 1) each expert e_k (k = 1, 2 , 3, 4) with the same “high” rating presented the ratings $\tilde{a}'_{ij}^{k}$ for each candidate site cc_i , (i = 1, . . . , 8), with respect to each attribute s_j, (j = 1, . . . , 10) (for example: first expert’s evaluations see in Table 2. Other experts’ evaluations are omitted here). Let also experts evaluated interaction indexes between attributes $\tilde{I} {''}_{ij}^{k}, i, j = 1, . . ., 10; i < j$ and weights ${\tilde{w}}_{j}^{k}$ for each attribute (for example: first expert’s evaluations see in Table 3. Other experts’ evaluations are omitted here).

Table 2
First expert’s evaluations - $\tilde{a}'_{ij}^{1}$

s ₁ s ₂ s ₃ s ₄ s ₅ s ₆ s ₇ s ₈ s ₉ s ₁₀

cc ₁ L VH H VL H H L VL H VL

cc ₂ VH H H H M H VL H L L

cc ₃ L M L M M H H VL H M

cc ₄ H M M VL H M L M VL M

cc ₅ H L L VH L M M VL M M

cc ₆ L H M VH L L M L M L

cc ₇ H L H L H H VL VH L M

cc ₈ M L M H VH L M H VL H

	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇	s ₈	s ₉	s ₁₀
cc ₁	L	VH	H	VL	H	H	L	VL	H	VL
cc ₂	VH	H	H	H	M	H	VL	H	L	L
cc ₃	L	M	L	M	M	H	H	VL	H	M
cc ₄	H	M	M	VL	H	M	L	M	VL	M
cc ₅	H	L	L	VH	L	M	M	VL	M	M
cc ₆	L	H	M	VH	L	L	M	L	M	L
cc ₇	H	L	H	L	H	H	VL	VH	L	M
cc ₈	M	L	M	H	VH	L	M	H	VL	H

Table 3

First expert’s evaluations: fuzzy importance values $\tilde{I} {''}_{ii}^{1} \equiv w_{i}^{1}$ and fuzzy interaction indexes of attributes ${\tilde{I} {''}_{ij}^{1}, i \neq j}, \tilde{I} {''}_{ij}^{1} = \tilde{I} {''}_{ji}^{1}$

VH	H	VH	VL	VL	H	H	VH	H	VL
	H	VH	VL	L	L	VL	M	M	L
		M	M	VL	M	H	H	L	H
			VH	L	M	M	L	L	VL
				M	VL	VL	M	H	VH
					M	VH	L	M	M
						L	H	H	L
							M	VH	H
								H	M
									L

Based on Section 5, the software is developed, in which the steps 1–5, mathematical programing problem (24)–(27) and ɛ-constraint approach are realized.

The system is interactive with respect to optimization of input data and in a certain extent it is remarkable for intellectual work.

Experts evaluated cc-costs C_j and cc-agents M_j (see criteria z₂ and z₃ in the model (24)–(27), respectively). Coefficients in the objective function z₂ are presented in thousand units. As it was mentioned, movement fuzzy times between demand points and candidate sites ${\tilde{t}}_{ij}, i = 1, . . ., 30; j = 1, . . ., 8$ are evaluated by experts. Therefore, the subsets of service demand points N_i, i = 1, . . . , 8, are received.

Using the software, the normalized fuzzy decision matrix is received; matrix of normalized attributes, fuzzy importance values and fuzzy attributes interaction indexes are also received; the mathematical programming problem (18) is solved (c = 0.02102) and the values of the associated TFVPMs are calculated (calculations are omitted); selection ranking indexes of candidate sites are calculated by the TFCA operator (Table 4). At the end, the Fuzzy Multi-objective Emergency Service Facility Location Set Covering Problem (24)–(27) is constructed:

${\begin{matrix} z_{1} = {\tilde{β}}_{1} x_{1} + {\tilde{β}}_{2} x_{2} + {\tilde{β}}_{3} x_{3} + {\tilde{β}}_{4} x_{4} + {\tilde{β}}_{5} x_{5} + {\tilde{β}}_{6} x_{5} + {\tilde{β}}_{7} x_{7} + {\tilde{β}}_{8} x_{8} \Rightarrow max_{t}, \\ z_{2} = 35 x_{1} + 47 x_{2} + 55 x_{3} + 39 x_{4} + 70 x_{5} + 62 x_{6} + 46 x_{7} + 57 x_{8} \Rightarrow min, \\ z_{3} = 27 x_{1} + 19 x_{2} + 31 x_{3} + 18 x_{4} + 23 x_{5} + 29 x_{6} + 25 x_{7} + 20 x_{8} \Rightarrow min, \\ {Bx}^{T} \geq (1, 1, 1, 1, 1, 1, 1, 1)^{T}, \\ x \equiv {x_{1}, . . ., x_{8}}, x_{i} \in {0, 1}, i = 1, . . ., 8, \end{matrix}$ (28) where the matrix of covering constraints B is a concatenation of vectors N_i in which the covering of demand point is presented by “1” and opposite by “0” (omitted here because of large dimensions).

Table 4

Fuzzy selection ranking indexes of candidate sites

${\tilde{β}}_{1}$	${\tilde{β}}_{2}$	${\tilde{β}}_{3}$	${\tilde{β}}_{4}$	${\tilde{β}}_{5}$	${\tilde{β}}_{6}$	${\tilde{β}}_{7}$	${\tilde{β}}_{8}$
0,0845	0,0822	0,1181	0,0868	0,0951	0,0653	0,1127	0,1169
0,4926	0,4903	0,4991	0,5513	0,4772	0,4497	0,5386	0,4777
1,4506	1,4386	1,4275	1,4752	1,4045	1,3892	1,5820	1,3238

After checking all 2⁸ = 258 possible variants of constraints of the problem (28) we receive the following twelve coverings (Table 5):

Table 5

Coverings and Pareto solutions

TFCA operator	Z₂	Z₃		Z₁
cov1	70	162000	(0,2956	1,4474	4,1790)
cov2	75	148000	(0,3131	1,5281	4,4481)
cov3	62	132000	(0,2818	1,5802	4,4958)
cov4	74	182000	(0,3292	1,454	4,1558)
cov5	85	163000	(0,2962	1,4875	4,3987)
cov6	80	174000	(0,2995	1,4265	4,1405)
cov7	76	158000	(0,3469	1,5154	4,3333)
cov8	72	147000	(0,2649	1,5396	4,4465)
cov9	94	198000	(0,3746	1,9987	5,8758)
cov10	104	213000	(0,3577	1,9581	5,8264)
cov11	80	213000	(0,3801	1,9964	5,6422)
cov12	90	228000	(0,3632	1,9558	5,5928)

Cov1: {cc₁, cc₅, cc₈}, Cov2: {cc₂, cc₃, cc₇}, Cov3: {cc₃, cc₄, cc₇},

Cov4: {cc₃, cc₅, cc₈}, Cov5: {cc₃, cc₆, cc₇}, Cov6: {cc₃, cc₆, cc₈},

Cov7: {cc₃, cc₇, cc₈}, Cov8: {cc₄, cc₆, cc₇}, Cov9: {cc₁, cc₂, cc₅, cc₇},

Cov10: {cc₁, cc₅, cc₆, cc₇}, Cov11: {cc₂, cc₄, cc₅, cc₈}, Cov12: {cc₄, cc₅, cc₆, cc₈}.

In the TFCA aggregation example there exist 6 Pareto solutions - Cov1, Cov2, Cov3, Cov6, Cov7, Cov8. It is clear that increasing the service centers selection ranking index of covering in Pareto solutions gives more worse level of the second objective function – the total cost needed for opening of service centers or of the third objective function – number of agents needed for operating the opened service centers. But the decision on the choice of the fire stations as service centers depends on the decision making person’s preferences with respect to risks of administrative actions.

Of course, elimination of the fire is extreme process and the reliability of choice of service centers is envisaged in Pareto solutions. Notwithstanding, if we had to consider problem (28) without TFCA aggregations (interacting attributes), i.e. we would have two-criteria (z₂, z₃) classical FLSC problem, then we would have only one Pareto solution Cov3, having minimal cost and number of agents. This covering also enters into Pareto solution of fuzzy-FLSC problem, where its selection ranking index is rather low. From this comparison the advantage of the new approach is clear.

8 Conclusions

For the facility location problem under extreme conditions a two-stage fuzzy approach is developed. On the first stage, the Fuzzy Multi-Attribute Group Decision Making model for evaluation of the selection ranking index of a candidate service site is evaluated by the TFCA operator. For the construction of the TFCA operator the following steps are undertaken: 1. The Interaction attributes, which act on the service centers’ selection process, are revealed. 2. Experts’ evaluations on candidate service sites with respect to attributes and interaction indexes between attributes and importance values of attributes in triangular fuzzy numbers are evaluated by the experts. 3. Interactions indexes between attributes and important values of attributes which are taken into account in the construction of the 2-order additive triangular fuzzy valued fuzzy measure (TFVFM). On the second stage, based on the TFCA operator, a new criterion is constructed that maximizes selection ranking index of candidate sites. This criterion together with two classical criteria – which minimize the total cost for opening of service centers and number of agents needed for operating the opened service centers, creates the fuzzy multi-objective facility location set covering problem. A simulation example of emergency service facility location planning for a city is considered. The example deals with the problem of planning fire station locations for serving emergency situations in specific demand points – critical infrastructure objects.

Footnotes

Acknowledgments

This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [FR-18-466].

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VH	H	VH	VL	VL	H	H	VH	H	VL
	H	VH	VL	L	L	VL	M	M	L
		M	M	VL	M	H	H	L	H
			VH	L	M	M	L	L	VL
				M	VL	VL	M	H	VH
					M	VH	L	M	M
						L	H	H	L
							M	VH	H
								H	M
									L

VH	H	VH	VL	VL	H	H	VH	H	VL
	H	VH	VL	L	L	VL	M	M	L
		M	M	VL	M	H	H	L	H
			VH	L	M	M	L	L	VL
				M	VL	VL	M	H	VH
					M	VH	L	M	M
						L	H	H	L
							M	VH	H
								H	M
									L

VH	H	VH	VL	VL	H	H	VH	H	VL
	H	VH	VL	L	L	VL	M	M	L
		M	M	VL	M	H	H	L	H
			VH	L	M	M	L	L	VL
				M	VL	VL	M	H	VH
					M	VH	L	M	M
						L	H	H	L
							M	VH	H
								H	M
									L