Modelling of SWOT Analysis Using Fuzzy Integrals

Fuzzy integrals

Fuzzy integrals are interesting tools to summarize all the pieces of information provided by a function in a single value; this value could be a sort of average of the function, in terms of the underlying fuzzy measure. Fuzzy integrals permit the aggregation of information under different assumptions on the independence of the information sources. In particular, to model situations in which sources are independent as well as in situations in which such independence cannot be assured. Many authors have used fuzzy integrals, among are: Yang (2012), Measuring Software Product Quality with ISO Standards Based on Fuzzy Logic Technique. Authors in China have used fuzzy integrals for comprehensive framework for measuring the performance of an organization resource planning (Wei, Liou, Lee, 2008) and other researchers have used fuzzy integrals for handwritten signature verification (Singh, Madasu, Srivastava, Hamandulu, 2013) and many others (Torra, Narukawa, 2004), (Chang, Wu, Lin, 2008), and (Verkeyn, Botteldooren, Baets, 2011).

Fuzzy integrals use the term fuzzy measure which does not require additive. Fuzzy measure can be defined as:

Let X be a finite index set X = {1, ..., n}.

Definition 1: A fuzzy measure μ defined on X is a set function μ : P(X)→[0,1] satisfying the following axioms (Grabisch et al. 2000) and (Sugeno 1974): (), (X)=1, and AB (A) (B).

The P(X) indicates the power set of X, i.e. the set of all subsets of X.

A fuzzy measure on X needs 2ⁿ coefficients to be defined, which are the values of μ for all the different subsets of X. Fuzzy integrals are integrals of a real function with respect to a fuzzy measure, by analogy with Lebesgue integral which is defined with respect to an ordinary (i.e. additive) measure. There are several definitions of fuzzy integrals, among which the most representatives are those of Sugeno fuzzy Integral (Sugeno 1974) and Choquet fuzzy Integral (Choquet 1953).

Choquet fuzzy integral was chosen over Sugeno fuzzy integral for this paper since the Sugeno method is based on min and max, such integral calculation can only determine interval at which the measured values are possibly located, unlike Choquet fuzzy integral, which provides a unique solution.

Definition 2: Let μ be a fuzzy measure on X. The discrete Choquet fuzzy integral of a function f: X→IR⁺ with respect to μ is defined by

$C_{\mu }\left(f\left(x_1\right),\dots ,f\left(x_n\right)\right)=\stackrel{n}{\underset{i=1}{?}}\left(f\left(x_1\right)-f\left(x_{i-1}\right)\right)\mu \left(A_i\right)$(1)

where i indicates that the indices have been permuted so that 0 ≤ f(x₁) ≤ … ≤ f(x_n) ≤ 1. Also A_i = {x_i, ...,x_n}, and f(x0) = 0.

Definition 3: Let λ(-1, ∞) and let X = {x₁, x₂, …, x_n} be a finite set. If (X, P(X)) is a measurable space and if set function g_λ : P(X)→[0,1] satisfies the following conditions, then g_λ is denoted by a Sugeno λ measure and g_λ()=0, g_λ(X)=1; A∩B=, A∪B≠X g_λ(A∩B)=g_λ(A)+g_λ(B)+ λg_λ(A)g_λ(B) that

$?+1=\stackrel{n}{\underset{i=1}{?}}\left(1+?g_{?}\left(x_i\right)\right),?>-1$ (2)

whereg_λ(x_i) is fuzzy measure.

Definition 4: Let set function g: P(X)→[0,1] be a fuzzy measure on measurable space (X,P(X)), and h: X→[0,1] be a measurable function on X. If h(x₁) ≤ h(x₂) ≤ … ≤h(x_n), A_i={x_i, x_i+1, …., x_n} then (Grabisch, Murofushi, Sugeno, 2000) , (Choquet 1953)

$E^{def}=?hd{g}^{def}=h\left(x_1\right)g\left(A_1\right)+\stackrel{n}{\underset{i=2}{?}}\left(h\left(x_i\right)-h\left(x_{i-1}\right)\right)g\left(A_i\right)$(3)

Where E^def denotes the overall function h(x_i) is viewed as the performance of sub characteristic x_i of the organization at a specific time. g(A_i), express the grade of importance for the subset A_i. The fuzzy integral of h(x_i) with respect to g denotes the overall evaluation.

By using equation (3) the overall evaluation for each, Strength, Weakness, Opportunity and Threat is obtained. From these aggregated values, status of an organization with respect to its environment is determined. The organization can use the output for amending a strategy and/or for developing a new strategy based on the numbers obtained from the fuzzy aggregation. The method can also be used to compare different strategies.

Application procedure of fuzzy integrals

The following are the main steps in evaluating strategy and its effectiveness:

• Change the importance values to decimal values between 0 and 1

• Change the performance values to decimal values between 0 and 1

• Calculate for λ for each level

$\lambda +1=\left(1+\lambda g_{\lambda }\left(S1\right)\right)\left(1+\lambda g_{\lambda }\left(S2\right)\right)\left(1+\lambda g_{\lambda }\left(S3\right)\right),\lambda >-1$ (4)

• Calculate the combined effect of sub characteristics using the formula

$g_i(A,B)=g_i\left(A\right)+g_i\left(B\right)+\lambda g_i\left(A\right)g_i\left(B\right)$(5)

and so on until all sub characteristics at this level are analysed

• Calculate evaluation value for higher level according to equation (3).

The result from this analysis is aggregated performance of the strength of the organization; the same procedures are used to determine Weakness, Opportunity and Threats. Based on the result we can evaluate existing strategy and decide whether to keep the strategy or propose a new one.

Application of proposed method for foreign currency exchange office

There are many currency exchange offices in Prague some of these offices buy and sell foreign currencies for a small difference and they make their profit by buying and selling a large amount of foreign currencies per day while others make a better profit from each unit of currency they buy and sell and make significantly less amount of transaction. The currency exchange company, studied here, uses the second method and has more than five offices each making a small amount of transaction a day.

The data shown in the following table (Table 1 ) was gathered from one of these currency exchange offices. This data was used only as an empirical example to clarify the application of the discussed methods. The SWOT sub characteristics were selected and assigned importance and performance value by the staff of the company based on their experience in that office and in comparison of their other exchange offices located in Prague. The following table contains the data after it was transformed in to [0, 1] scale.

As shown in the above table (Table 1 ), strength of this company is the location, customer service and promotion. The importance of a location of an exchange office is evaluated to be 0.6 out of 1and the location of this particular exchange is very good since it is located on the building right next to a traffic light, 0.9 out of 1. They also have a good customer service, which they believe is 0.9 out of 1 and the importance of good customer service for the success of the exchange is evaluated to be 0.4. The importance of promotion is also 0.4 for exchange and they have a Very good promotion.

It is important to note that this experiment was only done for one branch of the exchange company to explain the application of the method.

The step by step procedure to evaluate the SWOT analysis performed for the exchange office is shown below

• λ was calculated for each level

$$\lambda +1=\left(1+\lambda g_i\left(S1\right)\right)\left(1+\lambda g_i\left(S2\right)\right)\left(1+\lambda g_i\left(S3\right)\right),?>-1$$

$$\lambda +1=(1+0.6\lambda )\left)\right(1+0.4\lambda \left)\right(1+0.4\lambda ),\lambda >-1\text{for Strength (S)}$$

• The data was arranged according to h(x1) ≤ h(x2) ≤ … ≤h(xn)

• Combined effect of sub characteristics was calculated using fuzzy measure

$g_{\lambda }(S\mathrm{1},S\mathrm{3})=\mathrm{0.83},g_{\lambda }(S\mathrm{2},S\mathrm{3})=\mathrm{0.69}$

$$g_{\lambda }\left(S\right)=g_{\lambda }\left(S\mathrm{2}\right)+g_{\lambda }(S\mathrm{1},S\mathrm{3})+(-\mathrm{0.693})g_{\lambda }\left(S\mathrm{2}\right)g_{\lambda }(S\mathrm{1},S\mathrm{3})=\mathrm{1}$$

The same procedure is used for W, O and T

• The aggregated value for each characteristics was calculated using equation (3)

$$\begin{array}{c}{E}^{dff}==h\left(x_1\right)g_2(S1,S2,S3)+\left(h\left(x_2\right)-h\left(x_1\right)\right)g_2(S2,S3)+\left(h\left(x_3\right)-h\left(x_2\right)\right)g_2\left(S3\right)\\ =0.9\times \left(1\right)+(0.9-0.9)\times 0.69+(1-0.9)\times 0.4=0.94\end{array}$$

The same procedure is used to find the values for the rest of the characters. The result of the evaluation is shown in the following table:

Based on these results the company over all has good strength but they also have weakness they could improve their weakness more by reserving more money and they could change their offices to a more tourist centre since the combined effect of these two sub characteristics is significant. The opportunity at their disposal is 0.54, these are the factors the company could not control, but in the future they could choose a place in an area where there are more hostels and restaurants in order to increase their success. Finally the threat is that they worry about the country changing the currency to Euro and that makes them cautious to invest more in the business and that many people are using credit cards, unfortunately they cannot do anything about that.