Ranking objectives of advertisements on Facebook by a fuzzy TOPSIS method

Abstract

Social networking sites (SNSs) have become a vital medium for companies to place advertisements and setting an objective of advertisements on SNSs is an important issue of planning a business’s market strategy. The purpose of this work is to develop a fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) method for evaluating and selecting objectives of advertisements on Facebook. In the proposed model, the fuzzy weighted ratings are defuzzified by a centroid method to generate distances of each alternative to the positive and negative ideal solutions. A fuzzy weighted normalized distances index is proposed to rank alternatives, and the centroid method is used for defuzzification. Formulas for the defuzzification of fuzzy weighted ratings and the fuzzy weighted normalized distances index are developed. A numerical example of evaluating objectives of advertisements on Facebook is used to demonstrate the feasibility of the proposed method. Example result reveals that the proposed fuzzy weighted normalized distances index is as effective as the crisp closeness coefficient in ranking objectives under the proposed fuzzy TOPSIS method. An experiment demonstrates that the rankings of objectives may be more likely to change as the gap between two linguistic weights that are assigned to fuzzy weighted normalized distances index increases.

Appendices

Appendix 1

Equation (4.11) is derived as follows.

$$ \begin{aligned} P_{ij2} & = \int_{{Y_{ij} }}^{{Z_{ij} }} {x\mu_{{\tilde{v}_{ij} }}^{R} (x)dx} = \int_{{Y_{ij} }}^{{Z_{ij} }} {x\left( {\frac{{ - K_{ij2} - \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - x)} }}{{2L_{ij2} }}} \right)dx} \\ & = \int_{{Y_{ij} }}^{{Z_{ij} }} {\left( {\frac{{ - K_{ij2} }}{{2L_{ij2} }}} \right)xdx} - \frac{1}{{2L_{ij2} }}\int_{{Y_{ij} }}^{{Z_{ij} }} {\left( {\sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - x)} } \right)xdx} \\ \end{aligned} $$

(7.1)

Let

$$ \begin{aligned} & \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - x)} = t \\ & \Rightarrow \frac{d}{dt}x = \frac{d}{dt}\frac{{t^{2} - K_{ij2}^{2} + 4L_{ij2} Z_{ij} }}{{4L_{ij2} }} = \frac{t}{{2L_{ij2} }}dt \\ \end{aligned} $$

(7.2)

$$ {\text{If }}x = Z_{ij} ,\,t = K_{ij2} \,{\text{and}}\,{\text{If}}\,x = Y_{ij} , \, t = \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - Y_{ij} )} = Y_{ij}^{{\prime \prime }} . $$

(7.3)

Apply Eqs. (7.2)–(7.3) to Eq. (7.1) to obtain:

$$ \begin{aligned} & P_{ij2} = \frac{{K_{ij2} (Y_{ij}^{2} - Z_{ij}^{2} )}}{{4L_{ij2} }} - \frac{1}{{2L_{ij2} }}\int_{{Y_{ij}^{''} }}^{{K_{ij2} }} {t\frac{{t^{2} - K_{ij2}^{2} + 4L_{ij2} Z_{ij} }}{{4L_{ij2} }}\frac{t}{{2L_{ij2} }}dt} \\ & \Rightarrow P_{ij2} = \frac{{K_{ij2} (Y_{ij}^{2} - Z_{ij}^{2} )}}{{4L_{ij2} }} - \frac{1}{{16L_{ij2}^{3} }}\left[ {\frac{{K_{ij2}^{5} - Y_{ij}^{{{\prime \prime }5}} }}{5} + \frac{{K_{ij2}^{3} - Y_{ij}^{{{\prime \prime }3}} }}{3}( - K_{ij2}^{2} + 4L_{ij2} Z_{ij} )} \right] \\ & \;{\text{where}}\,Y_{ij}^{{\prime \prime }} = \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - Y_{ij} )} . \\ \end{aligned} $$

(4.11)

Equation (4.12) is derived as follows.

$$ \begin{aligned} P_{ij3} & = \int_{{Q_{ij} }}^{{Y_{ij} }} {\mu_{{\tilde{v}_{ij} }}^{L} (x)dx} = \int_{{Q_{ij} }}^{{Y_{ij} }} {\left( {\frac{{ - K_{ij1} + \sqrt {K_{ij1}^{2} - 4L_{ij1} (Q_{ij} - x)} }}{{2L_{ij1} }}} \right)dx} \\ & = \int_{{Q_{ij} }}^{{Y_{ij} }} {\left( {\frac{{ - K_{ij1} }}{{2L_{ij1} }}} \right)dx} + \frac{1}{{2L_{ij1} }}\int_{{Q_{ij} }}^{{Y_{ij} }} {\left( {\sqrt {K_{ij1}^{2} - 4L_{ij1} (Q_{ij} - x)} } \right)dx} \\ \end{aligned} $$

(7.4)

Let

$$ \begin{aligned} & \sqrt {K_{ij1}^{2} - 4L_{ij1} (Q_{ij} - x)} = t \\ & \Rightarrow \frac{d}{dt}x = \frac{d}{dt}\frac{{t^{2} - K_{ij1}^{2} + 4L_{ij1} Q_{ij} }}{{4L_{ij1} }} = \frac{t}{{2L_{ij1} }}dt \\ \end{aligned} $$

(7.5)

$$ {\text{If }}x = Q_{ij} , \, t = K_{ij1} \,{\text{and}}\,{\text{If}}\, .x = Y_{ij} , \, t = \sqrt {K_{ij1}^{2} - 4L_{ij1} (Q_{ij} - Y_{ij} )} = Y_{ij}^{'} $$

(7.6)

Apply Eqs. (7.5)–(7.6) to Eq. (7.4) to obtain:

$$ \begin{aligned} & P_{ij3} = \frac{{K_{ij1} (Q_{ij} - Y_{ij} )}}{{2L_{ij1} }} + \frac{1}{{2L_{ij1} }}\int_{{K_{ij1} }}^{{Y_{ij}^{'} }} {t\frac{t}{{2L_{ij1} }}dt} \\ & \Rightarrow P_{ij3} = \frac{{K_{ij1} (Q_{ij} - Y_{ij} )}}{{2L_{ij1} }} + \frac{{Y_{ij}^{'3} - K_{ij1}^{3} }}{{12L_{ij1}^{2} }},\,{\text{where}}\,Y_{ij}^{'} = \sqrt {K_{ij1}^{2} - 4L_{ij1} (Q_{ij} - Y_{ij} )} \\ \end{aligned} $$

(4.12)

Equation (4.13) is derived as follows.

$$ \begin{aligned} P_{ij4} & = \int_{{Y_{ij} }}^{{Z_{ij} }} {\mu_{{\tilde{v}_{ij} }}^{R} (x)dx} = \int_{{Y_{ij} }}^{{Z_{ij} }} {\left( {\frac{{ - K_{ij2} - \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - x)} }}{{2L_{ij2} }}} \right)dx} \\ & = \int_{{Y_{ij} }}^{{Z_{ij} }} {\left( {\frac{{ - K_{ij2} }}{{2L_{ij2} }}} \right)dx} - \frac{1}{{2L_{ij2} }}\int_{{Y_{ij} }}^{{Z_{ij} }} {\left( {\sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - x)} } \right)dx} \\ \end{aligned} $$

(7.7)

Let

$$ \begin{aligned} & \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - x)} = t \\ & \Rightarrow \frac{d}{dt}x = \frac{d}{dt}\frac{{t^{2} - K_{ij2}^{2} + 4L_{ij2} Z_{ij} }}{{4L_{ij2} }} = \frac{t}{{2L_{ij2} }}dt \\ \end{aligned} $$

(7.8)

$$ {\text{If }}x = Z_{ij} , \, t = K_{ij2} \,{\text{and}}\,{\text{If }}x = Y_{ij} , \, t = \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - Y_{ij} )} = Y_{ij}^{''} . $$

(7.9)

Apply Eqs. (7.8)–(7.9) to Eq. (7.7) to obtain:

$$ \begin{aligned} & P_{ij4} = \frac{{K_{ij2} (Y_{ij} - Z_{ij} )}}{{2L_{ij2} }} - \frac{1}{{2L_{ij2} }}\int_{{Y_{ij}^{''} }}^{{K_{ij2} }} {t\frac{t}{{2L_{ij2} }}dt} \\ & \Rightarrow P_{ij4} = \frac{{K_{ij2} (Y_{ij} - Z_{ij} )}}{{2L_{ij2} }} - \frac{{K_{ij2}^{3} - Y_{ij}^{''3} }}{{12L_{ij2}^{2} }},\,{\text{where}}\,Y_{ij}^{''} = \sqrt {K_{ij2}^{2} - 4L_{ij2} (Z_{ij} - Y_{ij} )} . \\ \end{aligned} $$

(4.13)

Appendix 2

Equations (23)–(25) are derived by the following procedure.

$$ \begin{aligned} P_{i2} & = \int_{{r_{ib} }}^{{r_{ic} }} {x\mu_{{\tilde{R}_{i} }}^{R} (x)dx} \\ & = \int_{{r_{ib} }}^{{r_{ic} }} {x\frac{{x - r_{ic} }}{{r_{ib} - r_{ic} }}dx} = \frac{{ - 2r_{ib}^{3} + 3r_{ib}^{2} r_{ic} - r_{ic}^{3} }}{{6(r_{ib} - r_{ic} )}}. \\ \end{aligned} $$

(4.23)

$$ \begin{aligned} P_{i3} & = \int_{{r_{ia} }}^{{r_{ib} }} {\mu_{{\tilde{R}_{i} }}^{L} (x)dx} \\ & = \int_{{r_{ia} }}^{{r_{ib} }} {\frac{{x - r_{ia} }}{{r_{ib} - r_{ia} }}dx} = \frac{{r_{ib} - r_{ia} }}{2}. \\ \end{aligned} $$

(4.24)

$$ \begin{aligned} P_{i4} & = \int_{{r_{ib} }}^{{r_{ic} }} {\mu_{{\tilde{R}_{i} }}^{R} (x)dx} \\ & = \int_{{r_{ib} }}^{{r_{ic} }} {\frac{{x - r_{ic} }}{{r_{ib} - r_{ic} }}dx} = \frac{{r_{ic} - r_{ib} }}{2}. \\ \end{aligned} $$

(4.25)

Appendix 3

See Tables 22, 23, 24, 25, 26, 27 and 28.

Table 22 $K_{ij1}$ of membership functions

Table 23 $K_{ij2}$ of membership functions

Table 24 $L_{ij1}$ of membership functions

Table 25 $L_{ij2}$ of membership functions

Table 26 $Q_{ij}$ of membership functions

Table 27 $Y_{ij}$ of membership functions

Table 28 $Z_{ij}$ of membership functions

Appendix 4

Ranking values of Tables 12, 13, 14, 15 and 16

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{I,I}})$	0.1705	0.1164	0.1081	0.1451	0.1302	0.1325	0.1306
$R_{i} ({\text{I,SS}})$	0.3361	0.2493	0.2372	0.2956	0.2699	0.2753	0.2700
$R_{i} ({\text{I,FS}})$	0.5348	0.4087	0.3920	0.4762	0.4376	0.4466	0.4374
$R_{i} ({\text{I,S}})$	0.7336	0.5682	0.5469	0.6568	0.6053	0.6179	0.6047
$R_{i} ({\text{I,VS}})$	0.8992	0.7010	0.6760	0.8073	0.7450	0.7606	0.7441

	A1	A2	A3	A4	A5	A6	A7
$R_{i} (\text{SS} ,I)$	0.2179	0.1290	0.1141	0.1760	0.1532	0.1554	0.1544
$R_{i} ({\text{SS,SS}})$	0.3836	0.2619	0.2432	0.3265	0.2929	0.2982	0.2938
$R_{i} ({\text{SS,FS}})$	0.5823	0.4213	0.3981	0.5071	0.4606	0.4695	0.4612
$R_{i} ({\text{SS,S}})$	0.7810	0.5808	0.5530	0.6877	0.6283	0.6408	0.6285
$R_{i} ({\text{SS,VS}})$	0.9467	0.7137	0.6820	0.8382	0.7680	0.7836	0.7679

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{FS,I}})$	0.2749	0.1441	0.1214	0.2130	0.1808	0.1829	0.1829
$R_{i} ({\text{FS,SS}})$	0.4405	0.2770	0.2504	0.3635	0.3205	0.3257	0.3224
$R_{i} ({\text{FS,FS}})$	0.6393	0.4364	0.4053	0.5441	0.4882	0.4970	0.4897
$R_{i} ({\text{FS,S}})$	0.8380	0.5959	0.5602	0.7247	0.6559	0.6683	0.6570
$R_{i} ({\text{FS,VS}})$	1.0036	0.7288	0.6893	0.8752	0.7956	0.8110	0.7965

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{S,I}})$	0.3318	0.1592	0.1286	0.2500	0.2084	0.2104	0.2115
$R_{i} ({\text{S,SS}})$	0.4975	0.2921	0.2577	0.4005	0.3481	0.3532	0.3509
$R_{i} ({\text{S,FS}})$	0.6962	0.4516	0.4126	0.5811	0.5158	0.5245	0.5182
$R_{i} ({\text{S,S}})$	0.8950	0.6110	0.5675	0.7617	0.6835	0.6958	0.6856
$R_{i} ({\text{S,VS}})$	1.0606	0.7439	0.6965	0.9122	0.8232	0.8385	0.8250

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{VS,I}})$	0.3793	0.1718	0.1346	0.2809	0.2314	0.2333	0.2353
$R_{i} ({\text{VS,SS}})$	0.5449	0.3047	0.2637	0.4314	0.3711	0.3761	0.3747
$R_{i} ({\text{VS,FS}})$	0.7437	0.4642	0.4186	0.6120	0.5388	0.5474	0.5420
$R_{i} ({\text{VS,S}})$	0.9424	0.6236	0.5735	0.7926	0.7065	0.7187	0.7094
$R_{i} ({\text{VS,VS}})$	1.1080	0.7565	0.7026	0.9431	0.8462	0.8614	0.8488

Ranking values of Tables 17, 18, 19, 20 and 21

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{I,I}})$	0.1705	0.1164	0.1081	0.1451	0.1302	0.1325	0.1306
$R_{i} ({\text{SS,I}})$	0.2179	0.1290	0.1141	0.1760	0.1532	0.1554	0.1544
$R_{i} ({\text{FS,I}})$	0.2749	0.1441	0.1214	0.2130	0.1808	0.1829	0.1829
$R_{i} ({\text{S,I}})$	0.3318	0.1592	0.1286	0.2500	0.2084	0.2104	0.2115
$R_{i} ({\text{VS,I}})$	0.3793	0.1718	0.1346	0.2809	0.2314	0.2333	0.2353

	A1	A2	A3	A4	A5	A6	A7
$R_{i} (I,{\text{SS}})$	0.3361	0.2493	0.2372	0.2956	0.2699	0.2753	0.2700
$R_{i} ({\text{SS,SS}})$	0.3836	0.2619	0.2432	0.3265	0.2929	0.2982	0.2938
$R_{i} ({\text{FS,SS}})$	0.4405	0.2770	0.2504	0.3635	0.3205	0.3257	0.3224
$R_{i} ({\text{S,SS}})$	0.4975	0.2921	0.2577	0.4005	0.3481	0.3532	0.3509
$R_{i} ({\text{VS,SS}})$	0.5449	0.3047	0.2637	0.4314	0.3711	0.3761	0.3747

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{I,FS}})$	0.5348	0.4087	0.3920	0.4762	0.4376	0.4466	0.4374
$R_{i} ({\text{SS,FS}})$	0.5823	0.4213	0.3981	0.5071	0.4606	0.4695	0.4612
$R_{i} ({\text{FS,FS}})$	0.6393	0.4364	0.4053	0.5441	0.4882	0.4970	0.4897
$R_{i} ({\text{S,FS}})$	0.6962	0.4516	0.4126	0.5811	0.5158	0.5245	0.5182
$R_{i} ({\text{VS,FS}})$	0.7437	0.4642	0.4186	0.6120	0.5388	0.5474	0.5420

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{I,S}})$	0.7336	0.5682	0.5469	0.6568	0.6053	0.6179	0.6047
$R_{i} ({\text{SS,S}})$	0.7810	0.5808	0.5530	0.6877	0.6283	0.6408	0.6285
$R_{i} ({\text{FS,S}})$	0.8380	0.5959	0.5602	0.7247	0.6559	0.6683	0.6570
$R_{i} ({\text{S,S}})$	0.8950	0.6110	0.5675	0.7617	0.6835	0.6958	0.6856
$R_{i} ({\text{VS,S}})$	0.9424	0.6236	0.5735	0.7926	0.7065	0.7187	0.7094

	A1	A2	A3	A4	A5	A6	A7
$R_{i} ({\text{I,VS}})$	0.8992	0.7010	0.6760	0.8073	0.7450	0.7606	0.7441
$R_{i} ({\text{SS,VS}})$	0.9467	0.7137	0.6820	0.8382	0.7680	0.7836	0.7679
$R_{i} ({\text{FS,VS}})$	1.0036	0.7288	0.6893	0.8752	0.7956	0.8110	0.7965
$R_{i} ({\text{S,VS}})$	1.0606	0.7439	0.6965	0.9122	0.8232	0.8385	0.8250
$R_{i} ({\text{VS,VS}})$	1.1080	0.7565	0.7026	0.9431	0.8462	0.8614	0.8488