A country decided to purchase a fleet of jet fighters from the U.S. The Pentagon officials offered the characteristic information of four models that may be sold to that country. The Air Force analyst team of that country agreed that six characteristics (attributes) should be considered. They are maximum speed (
${X}_{1}$) ferry range (
${X}_{2}$), maximum payload (
${X}_{3}$), purchasing cost (
${X}_{4}$), reliability (
${X}_{5}$), and maneuverability (
${X}_{6}$). The measurement units for the attributes are mach, miles, pounds, dollars (in millions), high-low scale, and high-low scale, respectively. The decision matrix for the fighter aircraft selection problem, then, is:
5.2.2. The Decision-Making Based on Fuzzy Soft Sets and Ideal Solution
In this subsection, we illustrate the decision process with the following examples. Suppose three groups of air force analyst team make the following goals:
- Team 1:
“Spare no expense to buy a jet fighter, and the jet fighter that is the fastest, most stable and has the best maneuverability".
- Team 2:
“Buy a jet fighter with a budget of 5 million, and a jet fighter that is stable and has the best maneuverability”.
- Team 3:
“Spend the least money to buy the indicators of a relatively good jet fighter”.
Let $A=\{$${e}_{1}=\u2018\mathrm{maximum}\phantom{\rule{4.pt}{0ex}}\mathrm{speed}\u2019$, ${e}_{2}=\u2018\mathrm{ferry}\phantom{\rule{4.pt}{0ex}}\mathrm{range}\u2019$, ${e}_{3}=\u2018\mathrm{maximum}\phantom{\rule{4.pt}{0ex}}\mathrm{payload}\u2019$, ${e}_{4}=\u2018\mathrm{purchasing}$ cost’, ${e}_{5}=\u2018\mathrm{reliability}\u2019$, ${e}_{6}=\u2018\mathrm{maneuverability}\u2019$}. The attributes { ${e}_{1}=\u2018\mathrm{maximum}\phantom{\rule{4.pt}{0ex}}\mathrm{speed}\u2019$, ${e}_{2}=\u2018\mathrm{ferry}\phantom{\rule{4.pt}{0ex}}\mathrm{range}\u2019$, ${e}_{3}=\u2018\mathrm{maximum}\phantom{\rule{4.pt}{0ex}}\mathrm{payload}\u2019$, ${e}_{5}=\u2018\mathrm{reliability}\u2019$, ${e}_{6}=\u2018\mathrm{maneuverability}\u2019$ } are ‘pros’ attributes, and they are all positive descriptions of jet fighters. For the attribute $\{{e}_{4}=\u2018\mathrm{purchasing}\phantom{\rule{4.pt}{0ex}}\mathrm{cos}\mathrm{t}\u2019\}$, of course, the cheaper, the better.
For Team 1, the attribute $\{{e}_{4}=\u2018\mathrm{purchasing}\phantom{\rule{4.pt}{0ex}}\mathrm{cos}\mathrm{t}\u2019\}$ is a factor that doesn’t need to be considered, no matter how expensive it is. The attribute $\{{e}_{1}=\u2018\mathrm{maximum}\phantom{\rule{4.pt}{0ex}}\mathrm{speed}\u2019\}$ is the primary consideration, $\{{e}_{5}=\u2018\mathrm{reliability}\u2019\}$ second, and finally consider $\{{e}_{6}=\u2018\mathrm{maneuverability}\u2019\}$. Other factors are relatively unimportant. Therefore, the degree of importance is: ${e}_{1}>{e}_{5}>{e}_{6}>({e}_{2}={e}_{3})>{e}_{4}$.
For Team 2, the attribute
$\{{e}_{4}=\u2018\mathrm{purchasing}\phantom{\rule{4.pt}{0ex}}\mathrm{cos}\mathrm{t}\u2019\}$ is the primary consideration. This is a user constraint, and, by Equation (
19), we can get the ideal solution of
${e}_{4}$. The attributes
$\{{e}_{5}=\u2018\mathrm{reliability}\u2019\}$ and
$\{{e}_{6}=\u2018\mathrm{maneuverability}\u2019\}$ are relatively important attributes, and
${e}_{5}>{e}_{6}$ . Therefore, the degree of importance is:
${e}_{4}>{e}_{5}>{e}_{6}>({e}_{1}={e}_{2}={e}_{3})$.
For Team 3, the attribute $\{{e}_{4}=\u2018\mathrm{purchasing}\phantom{\rule{4.pt}{0ex}}\mathrm{cos}\mathrm{t}\u2019\}$ is the primary consideration, and the cheaper the better. All other attributes are secondary attributes that are equally important, which is: ${e}_{4}>({e}_{1}={e}_{2}={e}_{3}={e}_{5}={e}_{6})$.
The attributes are normalized by a small number of samples, and a rigorous decision maker needs to analyze each indicator carefully. To determine its membership function through investigation and research (the definition of membership function is subjective, and the optimal membership function is not the problem discussed in this paper), we can obtain the optimal goal of our decision more accurately.
Suppose the membership function of each attribute is formulated as follows.
Let
${\mu}_{1}(x)$ be the membership function of fast jet fighters:
Let
${\mu}_{2}(x)$ be the membership function of ferry range:
Let
${\mu}_{3}(x)$ be the membership function of maximum payload:
Let
${\mu}_{4}(x)$ be the membership function of the expensive jet fighter:
Let
${\mu}_{5}(x)$ be the membership function of reliability:
Let
${\mu}_{6}(x)$ be the membership function of maneuverability:
The decision table for the fighter aircraft selection can be changed to a fuzzy soft set, as Equation (
32):
Example 7. Team 1: Spare no expense to buy a jet fighter, and the jet fighter that is the fastest, most stable and has the best maneuverability.
Without user constraints, the weight of attributes is: ${e}_{1}>{e}_{5}>{e}_{6}>({e}_{2}={e}_{3})>{e}_{4}$.
Let $\omega =\{{\omega}_{1}=\frac{1}{2},{\omega}_{5}=\frac{1}{4},{\omega}_{6}=\frac{1}{8},{\omega}_{2}=\frac{1}{16},{\omega}_{3}=\frac{1}{16},{\omega}_{4}=0\}$, ${\sum}_{i=1}^{6}{\omega}_{i}=1$.
${\Re}_{1}=1,{\Re}_{2}=1,{\Re}_{3}=1,{\Re}_{4}=0,{\Re}_{5}=1,{\Re}_{6}=1$, ${u}_{goal}=\{1,1,1,0,1,1\}$, From the weighted Hamming distance, it is seen that ${A}_{3}$ will be the choice because it is the closest object to ${A}_{goal}$. In addition, ${d}_{{A}_{2}}\succ {d}_{{A}_{4}}\succ {d}_{{A}_{1}}\succ {d}_{{A}_{3}}$.
Example 8. Team 2: Buy a jet fighter with a budget of 5 million, and a jet fighter that is stable and has the best maneuverability.
The prices include user constraints: ${x}^{\ast}=5,then,{\Re}_{4}={\mu}_{4}(5)=0.881.$
The weight of attributes is: ${e}_{4}>({e}_{5}={e}_{6})>({e}_{1}={e}_{2}={e}_{3})$.
Let $\omega =\{{\omega}_{4}=\frac{14}{60},{\omega}_{5}=\frac{11}{60},{\omega}_{6}=\frac{11}{60},{\omega}_{1}=\frac{8}{60},{\omega}_{2}=\frac{8}{60},{\omega}_{3}=\frac{8}{60}\}$, ${\sum}_{i=1}^{6}{\omega}_{i}=1$.
${\Re}_{1}=1,{\Re}_{2}=1,{\Re}_{3}=1,{\Re}_{4}=0.881,{\Re}_{5}=1,{\Re}_{6}=1$, ${u}_{goal}=\{1,1,1,0.881,1,1\}$. From the weighted Hamming distance, it is seen that ${A}_{3}$ will be the choice. In addition, ${d}_{{A}_{2}}\succ {d}_{{A}_{4}}\succ {d}_{{A}_{1}}\succ {d}_{{A}_{3}}$.
Example 9. Team 3: Spend the least money to buy the indicators of a relatively good jet fighter”.
Without user constraints, the weight of attributes is: ${e}_{4}>({e}_{1}={e}_{2}={e}_{3}={e}_{5}={e}_{6})$.
Let $\omega =\{{\omega}_{4}=\frac{5}{15},{\omega}_{5}=\frac{2}{15},{\omega}_{6}=\frac{2}{15},{\omega}_{2}=\frac{2}{15},{\omega}_{3}=\frac{2}{15},{\omega}_{1}=\frac{2}{15}\}$, ${\sum}_{i=1}^{6}{\omega}_{i}=1$.
${\Re}_{1}=1,{\Re}_{2}=1,{\Re}_{3}=1,{\Re}_{4}=0,{\Re}_{5}=1,{\Re}_{6}=1$, ${u}_{goal}=\{1,1,1,0,1,1\}$. From the weighted Hamming distance, it is seen that ${A}_{3}$ will be the choice. In addition, ${d}_{{A}_{2}}\succ {d}_{{A}_{4}}\succ {d}_{{A}_{1}}\succ {d}_{{A}_{3}}$.
Remark 6. The decision-making based on fuzzy soft sets and the ideal solution has some advantages. Firstly, the soft set model can be combined with other mathematical models. When it is combined with fuzzy decision-making, the soft set is a natural multi-attribute decision making model, which holds a wide range of application prospects in decision-making and analysis. Secondly, in the traditional ideal solution algorithm, it can be seen that the normalization process is the process of establishing the membership function $\mu (x)$. There are many commonly used normalization methods, i.e., Equation (29), but few of them can reflect the nature of the problem. The fuzzy soft set already contains the membership function $\mu (x)$, which can be used well. Thirdly, from the analysis of the attributes of the fuzzy soft sets, we can see that the attributes themselves are associated with each other. For example, ‘price’ and other attributes are related to each other, usually because of their high performance and therefore high pricing. Therefore, some attributes have less impact on the decision results because other attributes already contain information about that attribute.