A Hybrid Fuzzy Goal Programming for Smart Phones and Rate Plan Selection

Abstract

Smartphones are essential personal belongings today. However, smartphone buyers (SBs) are often hesitant in deciding what the best smartphone and rate plan to choose offered by telecom providers. Among many new smartphones, it is often difficult for users to choose the most suitable one in consideration of reaching the salient success aspiration level (SSAL) as close as possible, e.g., large battery capacity, and avoiding falling below the survival aspiration level (SAL) as far as possible, e.g., budget limitation, simultaneously. A novel hybrid weighted SSAL-SAL fuzzy goal programming (FGP) is proposed to assist SBs in finding satisfactory smartphones of their preferences with suitable rate plans. To meet different preferences, SBs can easily set different weights for each smartphone selection goal with linguistic terms, such as high, average, and low. In addition, these linguistic terms can easily be transformed into trapezoidal fuzzy numbers. The weights are then attached to goals in the objective function in the weighted SSAL-SAL-FGP model for choosing the most suitable smartphone. Moreover, this study also allows SBs to set the goal satisfaction levels as a preemptive priority for each goal in the FGP to find the most suitable rate plan option for the selected smartphone. The proposed approach allows SBs to select the best smartphones with suitable rate plans considering their priority preferences. In addition, SBs can save costs and precious time in finding their ideal smartphones and rate plan options. Finally, various examples are used to demonstrate the usefulness of the proposed weighted SSAL-SAL-FGP model, which allows SBs to set different weights for each goal with the intuitive method, such as linguistic terms or preemptive priority to select appropriate smartphones with suitable rate plans to match their habits and budgets. Besides, this study distributed an online questionnaire to investigate smartphone owners and found that female smartphone owners consider the battery capacity and the weight of the smartphone more crucial than the other criteria, while male smartphone owners prefer a smartphone with bigger screen size.

Appendices

Appendices

1.1 Appendix A (SSAL-SAL model)

$${\text{Minimize}}\;\sum\limits_{i = 1}^{4} {w_{i} (e_{i}^{{}} + d_{i}^{ + } )}$$

(1)

Subject to

$$\mu_{1} + e_{1} = 1,$$

(2)

$${\mathbf{f}}({\mathbf{x}}) + ({\mathbf{D}}^{{\mathbf{u}}} - {\mathbf{g}})\mu_{1} + d_{1}^{ + } = {\mathbf{D}}^{{\mathbf{u}}} ,$$

(3)

$$\mu_{2} + e_{2} = 1,$$

(4)

$${\mathbf{f}}({\mathbf{x}}) - {\mathbf{g}}\mu_{2} - d_{2}^{ + } = 0,$$

(5)

$$\mu_{3} + e_{3} = 1,$$

(6)

$${\mathbf{e}}({\mathbf{x}}) - ({\mathbf{N}}^{{\mathbf{u}}} - {\mathbf{h}})\mu_{\delta } + d_{3}^{ + } = {\mathbf{h}},$$

(7)

$$\mu_{4} + e_{4} = 1,$$

(8)

$${\mathbf{e}}({\mathbf{x}}) + {\mathbf{h}}\mu_{\delta } - d_{4}^{ + } = {\mathbf{h}},$$

(9)

${\mathbf{x}} \in {\mathbf{F}}$ (${\mathbf{F}}$ is a feasible set)where $\mu_{1}$, $\mu_{2}$, $\mu_{3}$, and $\mu_{4}$ are the fuzzy scores; $e_{1}$, $e_{2}$, $e_{3}$, and $e_{4}$ are the deviational variables that attach to $1 - \mu_{1}$, $1 - \mu_{2}$, $1 - \mu_{3}$, and $1 - \mu_{4}$. ${\mathbf{f}}({\mathbf{x}}) = \left\{ {f_{1} ({\mathbf{x}})} \right.,f_{2} ({\mathbf{x}}), \ldots ,\left. {f_{n} ({\mathbf{x}})} \right\}$ is an n-vector of the object functions, defined as $f_{i} ({\mathbf{x}}) = {\mathbf{c}}^{j} {\mathbf{x}},j = 1,2, \ldots ,n,$ where ${\mathbf{c}}^{j}$ is the n-vector of coefficients for the jth objective, ${\mathbf{g}} = (g_{1} ,g_{2} , \ldots ,g_{n} ) \in R^{n}$ is the vector of the SSAL, and ${\mathbf{h}} = (h_{1} ,h_{2} , \ldots ,h_{n} ) \in R^{n}$ is the vector of the SAL. ${\mathbf{e}}({\mathbf{x}}) = \left\{ {e_{1} ({\mathbf{x}})} \right.ef_{2} ({\mathbf{x}}), \ldots ,\left. {e_{n} ({\mathbf{x}})} \right\}$ is an n-vector of the object functions, defined as $e_{i} ({\mathbf{x}}) = {\mathbf{c}}^{i} {\mathbf{x}},i = 1,2, \ldots ,n,$ where ${\mathbf{c}}^{i}$ is the n-vector of coefficients for the ith objective. ${\mathbf{e}}({\mathbf{x}})$ and ${\mathbf{f}}({\mathbf{x}})$ can be either the same or different functions, depending on real situations or DM needs.$d_{1}^{ + }$ is the deviational variable that is attached to ${\mathbf{D}}^{{\mathbf{u}}} - ({\mathbf{f}}({\mathbf{x}}) + ({\mathbf{D}}^{{\mathbf{u}}} - {\mathbf{g}})\mu_{1} ).$ $d_{2}^{ + }$ is the deviational variable that is attached to ${\mathbf{f}}({\mathbf{x}}) - {\mathbf{g}}\mu_{2}$. $d_{3}^{ + }$ is the deviational variable that is attached to ${\mathbf{h}} - {\mathbf{e}}({\mathbf{x}}) + ({\mathbf{N}}^{{\mathbf{u}}} - {\mathbf{h}})\mu_{\delta }$. $d_{4}^{ + }$ is the deviational variable that is attached to ${\mathbf{e}}({\mathbf{x}}) + {\mathbf{h}}\mu_{\delta } - {\mathbf{h}}$.

1.2 Appendix B

A trapezoidal fuzzy number, $\tilde{A} = (a_{1} ,b_{1} ,c_{1} ,d_{1} ),$ $a_{1} \le b_{1} \le c_{1} \le d_{1} ,$ and its membership functions are expressed as follows:

$$\mu_{{\tilde{A}}} (x) = \left\{ \begin{array}{*{20}l} \, 0,\qquad \quad x > d_{1} . \hfill \\ \, \frac{{(x - a_{1} )}}{{(b_{1} - a_{1} )}},\quad a_{1} \le x \le b_{1} , \hfill \\ \, 1,\qquad \quad b_{1} \le x \le c_{1} , \hfill \\ \, \frac{{(x - d_{1} )}}{{(c_{1} - d_{1} )}},\quad c_{1} \le x \le d_{1} , \hfill \\ 0,\qquad \quad a_{1} \le x \le b_{1} , \hfill \\ \end{array} \right.$$

(10)

When $b_{1} = c_{1} ,$ $\mu_{{\tilde{A}}} (x)$ become triangular fuzzy numbers.

There are q SBs and l goals. The kth SB can define the trapezoidal fuzzy number $\tilde{A}_{ki}$ by linguistic terms for the ith goal according to their preference as Eq. (11)

$$\tilde{A}_{ki} = (a_{ki} ,b_{ki} ,c_{ki} ,d_{ki} ),\;0 \le a_{ki} \le b_{ki} \le c_{ki} \le d_{ki} \le 1,\;k = 1, \ldots ,q,\;i = 1, \ldots ,l.$$

(11)

For the goal 1, the kth DM can define the trapezoidal fuzzy number, $\tilde{A}_{k1}$ for the ith goal according to their preference as Eq. (12).

$$\tilde{A}_{k1} = (a_{k1} ,b_{k1} ,c_{k1} ,d_{k1} ),\quad k = 1, \ldots ,q.$$

(12)

In solving the group decision-making problem, the different trapezoidal fuzzy numbers from m SBs for goal 1 can be merged with Eq. (13). The average $\tilde{A}_{mean1}$ of all $\tilde{A}_{k1}$ is computed by Eq. (13).

$$\tilde{A}_{{{\text{mean}}1}} = \left( {a_{{{\text{mean}}1}} ,\;b_{{{\text{mean}}1}} ,\;c_{{{\text{mean}}1}} ,\;d_{{{\text{mean}}1}} } \right) = \left( {\frac{1}{q}\sum\limits_{k = 1}^{q} {a_{k1} ,} \;\frac{1}{q}\sum\limits_{k = 1}^{q} {b_{k1} ,} \;\frac{1}{q}\sum\limits_{k = 1}^{q} {c_{k1} ,} \;\frac{1}{q}\sum\limits_{k = 1}^{q} {d_{k1} } } \right).$$

(13)

For an SB, assume that the trapezoidal fuzzy numbers W₁, W₂, and W₃ are the weights of the goals G1, G2, and G3, respectively, where ${\text{W}}_{1} = (a_{1} ,b_{1} ,c_{1} ,d_{1} )$, ${\text{W}}_{2} = (a_{2} ,b_{2} ,c_{2} ,d_{2} )$, and ${\text{W}}_{3} = (a_{3} ,b_{3} ,c_{3} ,d_{3} )$. The normalized weights $\overline{\text{W}}_{1}$, $\overline{\text{W}}_{2}$, and $\overline{\text{W}}_{3}$ of W₁, W₂, and W₃, respectively, can be obtained from Eq. (14).

$$\begin{aligned} \overline{\text{W}}_{1} & = {\text{W}}_{1} \emptyset ({\text{W}}_{1} \oplus {\text{W}}_{2} \oplus {\text{W}}_{3} )= (a_{1} ,b_{1} ,c_{1} ,d_{1} )\emptyset (a_{1} + a_{2} + a_{3} ,b_{1} + b_{2} + b_{3} ,c_{1} + c_{2} + c_{3} ,d_{1} + d_{2} + d_{3} ) \\ & = \left( {\frac{{a_{1} }}{{d_{1} + d_{2} + d_{3} }},\frac{{b_{1} }}{{c_{1} + c_{2} + c_{3} }},\frac{{c_{1} }}{{b_{1} + b_{2} + b_{3} }},\frac{{d_{1} }}{{a_{1} + a_{2} + a_{3} }}} \right) \\ \end{aligned}$$

(14)

where $a_{1}$, $b_{1}$, $c_{1}$, and $d_{1}$ are individual trapezoidal fuzzy numbers of W₁; $a_{2}$, $b_{2}$, $c_{2}$, and $d_{2}$ are individual trapezoidal fuzzy numbers of W₂; $a_{3}$, $b_{3}$, $c_{3}$, and $d_{3}$ are individual trapezoidal fuzzy numbers of W₃.

1.3 Appendix C (FGP)

$${\text{Minimize}}\;\sum\limits_{i = 1}^{n} {(p_{i} + n_{i} ) + a_{i}^{ + } + a_{i}^{ - } )}$$

(15)

Subject to

$$r_{i} \le \left( {\frac{{g_{i,\text{max} } - y_{i} }}{{g_{i,\text{max} } - g_{i,\text{min} } }}} \right)b_{i} ,\quad i = 1,2, \ldots ,n,$$

(16)

$$z_{i} ({\mathbf{x}}) - p_{i} + n_{i} = y_{i} \quad i = 1,2, \ldots ,n,$$

(17)

$$r_{i} - a_{i}^{ + } + a_{i}^{ - } = 1,\quad i = 1,2, \ldots ,n,$$

(18)

$$g_{i,\text{min} } \le y_{i} \le g_{i,\text{max} } ,\quad i = 1,2,\ldots,n,$$

(19)

$$p_{i} ,n_{i} ,a_{i}^{ + } ,a_{i}^{ - } ,r_{i} \ge 0,\quad i = 1,2,\ldots,n,$$

(20)

$$b_{i} \in R_{i} ({\mathbf{x}}),\quad i = 1,2, \ldots ,n,$$

(21)

$$1 - r_{i} \le \sum\limits_{j = 1}^{m - 1} {(\mu_{\text{attribute}} (P_{j} )} - \mu_{\text{attribute}} (P_{j + 1} ))y_{i,j} - k,$$

(22)

$$y_{i,j} \ge y_{i,j + 1} ,\quad j = 1,2, \ldots ,m - 1,$$

(23)

where $p_{i}$ and $n_{i}$ are positive and negative deviational variables from the ith target value $y_{i}$; $a_{i}^{ + }$ and $a_{i}^{ - }$ are deviational variables attached to $\left| {r_{i} - 1} \right|$; $r_{i}$ is the value of the utility function; $b_{i}$ is the binary decision variable that is restricted by the resource constraint function $R_{i} ({\mathbf{x}})$. The target value $y_{i} \in [g_{i,\text{min} } ,g_{i,\text{max} } ]$ of $z_{i} ({\mathbf{x}})$ is a continuous variable, where $g_{i,\text{max} }$ and $g_{i,\text{min} }$ are the upper and lower bound values; k is a small positive value. The binary variables, $y_{i,j}$, are used to indicate which alternatives are selected. For example, if $\mathop \varPi \nolimits_{j = 1}^{m - 1} y_{i,j} = 1$, alternative, $P_{j}$, is selected; otherwise $P_{j}$ is not selected. $\mu_{\text{attribute}} (P_{j} )$ represents the average satisfaction level for the attributes of the jth alternative. With Eq. (22), the summation of the satisfaction level for the attributes of the jth alternative will be forced to achieve a level, the higher, the better. Meanwhile, $y_{i,j}$ will be selected in sequence, and hence, $y_{i,j}$ will be selected before $y_{i,j + 1}$ with Eq. (23).

1.4 Appendix D

$${\text{Minimize}}\;\left( {\frac{{s_{1}^{ + } + s_{1}^{ - } }}{1}} \right) + k_{1}^{ - } + \left( {\frac{{d_{2}^{ + } + d_{2}^{ - } }}{1}} \right) + f_{2}^{ - } + \left( {\frac{{d_{3}^{ + } + d_{3}^{ - } }}{1}} \right) + f_{3}^{ - } + \left( {\frac{{s_{4}^{ + } + s_{4}^{ - } }}{1}} \right) + k_{4}^{ - }$$

!G1: Price$s_{1}^{ + } = 1 - n_{1}^{ + } ,$ $s_{1}^{ - } = 1 - n_{1}^{ - } ,$ $0.63x_{1} + 0.94x_{2} + 0.47x_{3} + 0x_{4} + 1x_{5} - n_{1}^{ + } + n_{1}^{ - } = 1$, ($h_{1,\text{max} } = 1$)

$n_{1}^{ + } \le b_{1}$, $n_{1}^{ - } \le (1 - b_{1} )$, $\eta_{1} \le (0.63x_{1} + 0.94x_{2} + 0.47x_{3} + 0x_{4} + 1x_{5} )$, $\eta_{1} + k_{1}^{ - } = 0$, $x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 1$,@BIN ($x_{i}$); @BIN ($b_{1}$); @BIN ($b_{4}$); $i = 1,2, \ldots ,5$

For example, by referring to the price data in Table 5, we have $e_{i} ({\mathbf{x}}) = 0.63x_{1} + 0.94x_{2} + 0.47x_{3}$ $+ 0x_{4} + 1x_{5}$ in price goal (G1). Since G1 is an SAL goal, the equations of G1 will find the alternatives from $e_{i} ({\mathbf{x}})$, which are moving away from $h_{i,\text{max} } = 1$ (the highest price of all).

1.5 Appendix E

$${\text{Minimize}}\;\sum\limits_{j = 1}^{12} {(p_{1j} + n_{1j} ) + } \sum\limits_{j = 1}^{16} {(p_{2j} + n_{2j} ) + } \sum\limits_{j = 1}^{14} {(p_{3j} + n_{3j} ) + } \sum\limits_{j = 1}^{9} {(p_{4j} + n_{4j} )} + \sum\limits_{k = 1}^{4} {(a_{k}^{ + } + a_{k}^{ - } )} + \sum\limits_{j = 1}^{20} {(y_{1j} + y_{2j} + y_{3j} + y_{4j} )}$$

Subject to G5: The price of the smartphone with a two-year contract

$$\begin{aligned} r_{1} = \frac{{0.08(n_{11} - n_{12} )}}{1990} + \frac{{0.03(n_{12} - n_{13} )}}{1910} + \frac{{0.07(n_{13} - n_{14} )}}{90} + \frac{{0.11(n_{14} - n_{15} )}}{1410} + \frac{{0.1(n_{15} - n_{16} )}}{2590} + \frac{{0.11(n_{16} - n_{17} )}}{410} \hfill \\ + \frac{{0.03(n_{17} - n_{18} )}}{1500} + \frac{{0.19(n_{18} - n_{19} )}}{90} + \frac{{0.1(n_{19} - n_{110} )}}{1500} + \frac{{0.01(n_{110} - n_{111} )}}{910} + \frac{{0.14(n_{111} - n_{112} )}}{40} + \frac{{0.14n_{112} }}{1100} \hfill \\ \end{aligned}$$

$$\begin{aligned} r_{1} - a_{1}^{ + } + a_{1}^{ - } = 1,\;z_{1} - p_{11} + n_{11} = 13990,\;z_{1} - p_{12} + n_{12} = 12800,\;z_{1} - p_{13} + n_{13} = 12400, \hfill \\ z_{1} - p_{14} + n_{14} = 1490,\;z_{1} - p_{15} + n_{15} = 9990,\;z_{1} - p_{16} + n_{16} = 9990,\;z_{1} - p_{17} + n_{17} = 8400, \hfill \\ z_{1} - p_{18} + n_{18} = 7990,\;z_{1} - p_{19} + n_{19} = 5400,\;z_{1} - p_{110} + n_{110} = 3990,\;z_{1} - p_{112} + n_{112} = 1990, \hfill \\ \end{aligned}$$

$\begin{aligned} y_{11} \ge y_{17} ,\;y_{11} = y_{12} = y_{13} = y_{14} = y_{15} = y_{16} ,\;y_{17} \ge y_{18} ,y_{18} \ge y_{19} ,y_{19} = y_{110} ,y_{110} \ge y_{111} ,y_{111} \ge y_{112} , \hfill \\ y_{112} \ge y_{113} ,\;y_{113} \ge y_{114} ,\;y_{114} \ge y_{115} ,\;y_{115} = y_{116} \;y_{116} \ge y_{117} ,\;y_{117} \ge y_{118} ,\;y_{118} \ge y_{119} ,\;y_{119} \ge y_{120} , \hfill \\ \end{aligned}$$y_{120} = 0,\;r_{1} = 0.5,\;r_{2} = 0.5,\;r_{3} = 0.2,\;r_{4} = 0.5,$! Assume lambda 1 = 0.5, lambda 2 = 0.5, lambda 3 = 0.2, and lambda 4 = 0.5; @BIN ($y_{ij}$); $i = 1,2, \ldots ,4;$ $j = 1,2, \ldots ,20.$

1.6 Appendix F