Developing A Fuzzy Enhanced Russell Measure For Media Selection

Abstract: Nowadays, companies spend a lot of money on advertising, but in many situations much of their expenditures are wasted. One of the important factors that can prevent this wasting is selecting the right media. This paper proposes a fuzzy enhanced Russell measure data envelopment analysis (FERM-DEA) model for selecting media in the presence of fuzzy data and enables the decision maker to evaluate full efficiency and compare them with each other. A numerical example validates the proposed model.

Keywords: media selection; data envelopment analysis; enhanced Russell measure; ERM; fuzzy data; full efficiency.

1 Introduction

Advertising is a very important instrument in achieving success of a new product or continuous sale growth of existing products (Ahmad, 1984). The companies spend a lot of money for advertising, but advertising activity is not efficient in many situations. One of the most important issues in the efficient advertising activity is selecting right media. The media selection inherently is a multiple criteria decision problem and should consider various criteria for this purpose (Farzipoor Saen, 2011). As Farzipoor Saen (2011) addressed, media selection is a multiple criteria decision analysis problem.

Data envelopment analysis (DEA) is a mathematical programming technique that uses multiple inputs and outputs to calculate the relative efficiency of multiple decision making units (DMUs). DEA has been widely used to compare the efficiency of DMUs such as supplier selection (Farzipoor Saen, 2008, 2009, 2010; Azadi and Farzipoor Saen, 2012), project portfolio selection (Conka et al., 2008), performance evaluation of incubators (Aaboen et al., 2008), welding process selection (Mirhedayatian et al., 2013), container selection (Shabani et al., 2011), etc. One of the main objectives of DEA is to measure the relative efficiency of a DMU by a scalar measure ranging between zero (the worst) and the one (the best). This scalar is measured through a linear programming model. Specifically, to gauge the relative efficiency of concerned DMU, the Charnes-Cooper-Rhodes (CCR) model deals with the ratio of multiple inputs and outputs. This fractional program is solved by transforming it into an equivalent linear program using the Charnes-Cooper transformation (Charnes and Cooper, 1962). The optimal objective value θ* is called the ratio (or radial) efficiency of the DMU. The optimal solution reveals, at the same time, the existence, if any, of excesses in inputs and shortfalls in outputs (called slacks). A DMU with the full ratio efficiency, θ* = 1 and with no slacks in any optimal solution is called CCR-efficient. Therefore, in delineating total efficiency, it is important to observe the ratio efficiency and slacks. Some attempts have been made to unify θ* and slacks into a scalar measure. Some of them are additive model, slacks-based measure (SBM), enhanced Russell measure (ERM), and range adjusted measure (RAM) (Tone, 2001). Also, traditional DEA models such as CCR and Banker-Charnes-Cooper (BCC) models do not deal with imprecise data and assume that all input and output data are exactly known. However, in real world, this assumption is not always true. Because of uncertain circumstance, DEA sometimes faces the situation of imprecise data. The imprecise data can be expressed in interval or fuzzy number (Wang et al., 2005; Esmaeili, 2012). Consequently, in many applications, including media selection problems, it is vital to take into account the existence of ordinal (qualitative) factors when deciding on the performance of a DMU (medium).

The objective of this paper is to propose a fuzzy enhanced Russell measure DEA (FERM-DEA) model for media selection. The proposed model calculates full efficiency of media (DMUs).

This paper proceeds as follows. In Section 2, literature review is presented. Section 3 introduces the model. Case study and concluding remarks are discussed in Sections 4 and 5, respectively.

2 Literature review

Some approaches have been used for media selection problems in the past. Brown and Warshaw (1965) applied linear programming models for media selection. Bass and Lonsdale (1966) examined the characteristics of the use of linear programming for media selection. Brown (1967) examined the principle of incremental analysis (IA) for magazines selection problem. Charnes et al. (1968) proposed a goal programming model for selecting media. Gensch (1970) reviewed various mathematical approaches that have been applied in media selection problems. Steuer and Oliver (1976) presented a multiple objective linear programming (MOLP) model with interval weights for media selection problem. To illustrate the interval weights approach, a multiple objective hierarchical media selection model is presented. Morard et al. (1976) proposed a model for selecting media in the presence of uncertainty. Their model is based upon the mean-variance method. Wiedey and Zimmermann (1978) presented a fuzzy linear programming for media selection. Using balanced scorecard, Lee et al. (2009) examined effects of information technology (IT) knowledge and media selection on operational performance of small firms. Malthouse et al. (2012) proposed an optimisation model for assessing media for marketing campaigns. Aouni et al. (2012) presented a stochastic multi-objective approach for media planning and proposed two different goal programming formulations with satisfaction function based on scenario forecasting and deterministic equivalent. Aggarwal et al. (2012) developed a model which deals with optimal allocation of advertising budget for a product which is advertised through different media in a segmented market under fuzzy predictions of audience impact.

Dwyer and Evans (1981) used a branch and bound approach for selecting a list in direct mailing advertisement. De Kluyver and Baird (1984) report on an application of mean-variance approach to media scheduling problems. The results illustrate the effect of recognising schedule variance in the formulation of media plans, and illustrate that mean-variance analysis is effective in reducing such schedule variance through parametric risk-return and budget-return analysis. Deckro and Murdock (1987) presented a multiple objective integer linear programming (MOILP) approach for selecting media. Azadi et al. (2013) developed an imprecise neutral DEA (INDEA) model for media selection. They incorporated the imprecise data into neutral DEA model. Korhonen et al. (1989) proposed an evolutionary approach for media selection problem. They used the proposed method in a software company in Finland. Dyer et al. (1992) used analytic hierarchy process (AHP) and integer programming techniques in the media selection problem. Kwak et al. (2005) developed a mixed integer goal programming model to determine the number of advertisements in selected media and to allocate the advertising budget to the selected media.

Since the above-mentioned references do not deal with media selection both in full efficiency context and fuzzy data, a technique that can take into account fuzzy data and full efficiency is needed to better model the media selection problem. To the best of knowledge of authors, there is not any reference that calculates full efficiency and considers fuzzy data as well. This paper has two main contributions:

• a fuzzy ERM model is proposed.
• the proposed model considers both fuzzy data and full efficiency concept for media selection problems.

3 Proposed model

DEA is a non-parametric technique for measuring the relative efficiency of a set of DMUs. According to the DEA literature, the full efficiency is attained if and only if an optimal objective value θ* fulfills both of the following conditions (Cooper et al., 2007a):

• 1 θ* = 1
• 2 all slacks are zero.

Weak efficiency is attained if only condition (i) is satisfied. One of the models that has been developed for measuring full efficiency is ERM model. In this paper, we develop a new FERM-DEA model for media selection. Färe and Lovell (1978), for the first time, introduced the Russell measure. Then, Pastor et al. (1999) developed a model entitled ERM.

For a given DMUo, the value of Russell measure can be obtained by the following model:

min $R = \frac{\sum_{i=1}^{m} \frac{\theta_{i0}}{m}}{\sum_{r=1}^{s} \frac{\varphi_{r0}}{s}}$

$s.t \qquad \sum_{j=1}^{n} x_{ij} \lambda_{j} \leq \theta_{i0} x_{i0}, \quad i = 1, ..., m,$

$\sum_{r=1}^{s} y_{rj} \lambda_{j} \leq \phi_{i0} y_{r0}, \quad r = 1, ..., s,$

$\theta_{i} \leq 1, \quad \phi_{i} \geq 1, \quad \forall i, r,$

$\lambda_{j} \geq 0, \qquad j = 1, ..., n.$

$(1)$

where the variables $(\theta_i \text{ and } \phi_r)$ indicate the level of efficiency related to the $i^{\text{th}}$ input (i=1,…,m) and the $r^{\text{th}}$ output (r=1,…,s), respectively. The variables $(\lambda_j \text{ for } j=1,...,n)$ are used for a structural connection among DMUs in the input-output space. $x_{ij}$ is the $i^{\text{th}}$ input of DMU_j and $y_{rj}$ is the $r^{\text{th}}$ output of DMU_j. $y_{ro}$ is the $r^{\text{th}}$ output of DMU_o (the DMU under evaluation) and $x_{io}$ is the $i^{\text{th}}$ input of the DMU_o.

To transform the fractional model (1) into a linear programming model, the approach suggested by Charnes and Cooper (1962) is used. To this end, Cooper et al. (2007b) introduced a new variable as follows:

$\beta = \left(\sum_{r=1}^{s} \frac{\phi_{r0}}{s}\right)^{-1},$

where $0 < \beta \le 1$ and $\beta \left( \sum_{r=1}^{s} \frac{\phi_{r0}}{s} \right) = 1$ , and also following variables changes are defined:

$u_i = \beta \cdot \theta_i, \quad i = 1, \dots m,$

$v_r = \beta \cdot \phi_i, \quad r = 1, \dots s,$
$t_j = \beta \cdot \lambda_j, \quad j = 1, \dots n.$

As a result, the transformed model is given as follows:

min $\sum_{i=1}^{m} \frac{u_i}{m}$

s.t $\sum_{r=1}^{s} v_r = s$ ,
$\sum_{j=1}^{n} x_{ij} t_j \le u_i x_{i0}$ , $i = 1, ..., m$ ,
$\sum_{j=1}^{n} y_{rj} t_j \ge v_r y_{r0}$ , $r = 1, ..., s$ , (2)

$u_i \leq \beta,$ $i = 1,...,m,$
$\beta \leq v_r,$ $r = 1,...,s,$
$0 \leq t_j,$ $j = 1,...,n,$
$0 \leq \beta \leq 1.$

3.1 Notations and definitions

A tilde is placed over a symbol to denote a fuzzy set. A fuzzy number is fuzzy subset of the real line with a normal, convex and upper semi continuous membership function of bounded support. The family of fuzzy numbers will be denoted by E. The membership function $\tilde{u}$ can be expressed as follows:

u_L(x) & a \le x \le b, \\ 1 & b \le x \le c, \\ u_R(x) & c \le x \le d. \\ 0 & otherwise. \end{cases}$$ (3) where $u_L$ : $[a, b] \rightarrow [0, 1]$ and $u_R$ : $[a, b] \rightarrow [0, 1]$ are left and right membership functions of fuzzy number $\tilde{u}$ , respectively. Definition 1: An arbitrary fuzzy number is presented by an ordered pair of functions $(u(\alpha), \overline{u}(\alpha)), 0 \le \alpha \le 1$ , which satisfies the following requirements: - 1 $\underline{u}(\alpha)$ is a bounded left continuous non-decreasing function over [0, 1], with respect to $\alpha$ - 2 $\overline{u}(\alpha)$ is a bounded right continuous non-increasing function over [0, 1], with respect to $\alpha$ - 3 $\underline{u}(\alpha) \le \overline{u}(\alpha), 0 \le \alpha \le 1.$ The trapezoidal fuzzy $\tilde{u} = (u_{c1}, u_{c2}, s, t)$ , with two defuzzifier $u_{c1}$ , $u_{c2}$ and left fuzziness s > 0 and right fuzziness t > 0 is a fuzzy set where the membership function is as below. $$\tilde{u}(x) = \begin{cases} \frac{1}{s}(x - u_{c1} + s) & u_{c1} - s \le x \le u_{c1} \\ 1 & x \in [u_{c1}, u_{c2}] \\ \frac{1}{t}(u_{c2} - x + t) & u_{c2} \le x \le u_{c2} + t \\ 0 & otherwise. \end{cases}$$ $$(4)$$ and parametric form is $$\underline{u}(\alpha) = u_{c1} - s + s\alpha, \quad \overline{u}(\alpha) = u_{c2} + t - t\alpha.$$ Figure 1 Tarpozidoal fuzzy number  The addition and scalar multiplication of fuzzy numbers are defined by extension principle and can be equivalently represented as follows. For arbitrary $\tilde{u} = (\underline{u}(\alpha), \overline{u}(\alpha))$ , $\tilde{v} = (v(\alpha), \overline{v}(\alpha))$ , and k > 0, we have $$\frac{(\underline{u}+\underline{v})(\alpha) = \underline{u}(\alpha) + \underline{v}(\alpha),}{(\underline{u}+\underline{v})(\alpha) = \overline{u}(\alpha) + \overline{v}(\alpha),}$$ $$k.\tilde{u} = \begin{cases} (k\underline{u}(\alpha), k\overline{u}(\alpha)) & \text{if } k \ge 0, \\ (k\overline{u}(\alpha), k\underline{u}(\alpha)) & \text{if } < 0. \end{cases}$$ (5) ### 3.2 Ranking method for transforming linguistic information into fuzzy data In this paper, we use a comparison method based on the area compensation determined by the membership function of two fuzzy numbers. For two fuzzy numbers $\tilde{A} = (a(\alpha), \bar{a}(\alpha)), \tilde{B} = (b(\alpha), \bar{b}(\alpha))$ . Fortemps and Roubens (1996) defined: $$S_L(\tilde{A} \ge \tilde{B}) = \int_{U(A,B)} \left[\underline{a}(\alpha) - \underline{b}(\alpha)\right] d\alpha, \quad S_R(\tilde{A} \ge \tilde{B}) = \int_{V(A,B)} \left[\overline{a}(\alpha) - \overline{b}(\alpha)\right] d\alpha$$ where $$U(\tilde{A}, \tilde{B}) = \{ \alpha \mid 0 \le \alpha \le 1, \quad \underline{a}(\alpha) \ge \underline{a}(\alpha) \},$$ $$V(\tilde{A}, \tilde{B}) = \{ \alpha \mid 0 \le \alpha \le 1, \quad \overline{a}(\alpha) \ge \overline{a}(\alpha) \}.$$ $S_L(\tilde{A} \geq \tilde{B})$ is the area which claims that the left slope of $\tilde{A}$ is greater than the corresponding part of $\tilde{B}$ . $C(\tilde{A} \geq \tilde{B})$ , the degree to which $\tilde{A}$ is larger than $\tilde{B}$ is computed as $$C: E \times E \to R$$ $$C(\tilde{A} \ge \tilde{B}) = \frac{1}{2} \left\{ S_L(\tilde{A} \ge \tilde{B}) + S_R(\tilde{A} \ge \tilde{B}) - S_L(\tilde{B} \ge \tilde{A}) - S_R(\tilde{B} \ge \tilde{A}) \right\}$$ $$(6)$$ and $$\tilde{A} \succ \tilde{B} \quad iff \quad C(\tilde{A} \ge \tilde{B}) > 0,$$ (7) $$\tilde{A} \pm \tilde{B} \quad iff \quad C(\tilde{A} \ge \tilde{B}) \pm 0.$$ (8) **Figure 2** Comparing *A* and *B*  Theorem 1: $C(\tilde{A} \ge \tilde{B}) = F(\tilde{A}) - F(\tilde{B})$ , where $$F(\tilde{A}) = \frac{1}{2} \int_0^1 (\underline{a}(\alpha) + \overline{a}(\alpha)) d\alpha. \tag{9}$$ *Proof:* See Fortemps and Roubens (1996) $\square$ . Therefore, the equation (8) can be rewritten as $$\tilde{A} \pm \tilde{B} \quad iff \quad F(\tilde{A}) \pm F(\tilde{B}).$$ (10) #### 3.3 Fuzzy ERM model For a given DMU₀, value of Russell measure can be obtained from the following model: min $$R = \frac{\sum_{i=1}^{m} \frac{\theta_{i0}}{m}}{\sum_{r=1}^{s} \frac{\phi_{i0}}{s}}$$ $$s.t \qquad \sum_{j=1}^{n} x_{ij} \lambda_{j} \leq \theta_{i0} x_{i0}, \quad i = 1, ..., m,$$ $$\sum_{r=1}^{s} y_{rj} \lambda_{j} \leq \phi_{i0} y_{r0}, \quad r = 1, ..., s,$$ $$\theta_{i} \leq 1, \quad \phi_{i} \geq 1, \quad \forall i, r,$$ $$\lambda_{j} \geq 0, \qquad j = 1, ..., n.$$ $$(11)$$ In this subsection we develop Russell measure by using fuzzy inputs and fuzzy outputs. Assume that we have a set of n DMUs with m fuzzy inputs and s fuzzy outputs. $$\left\{ \left( \tilde{X}_{j} \tilde{Y}_{j} \right) = \left( \tilde{x}_{1j}, \dots, \tilde{x}_{mj}, \tilde{y}_{1j}, \dots, \tilde{y}_{sj} \right), \quad j = 1, \dots, n \right\}$$ where all fuzzy inputs and fuzzy outputs are fuzzy numbers. Then, for given DMU_o, we have min $$R = \frac{\sum_{i=1}^{m} \frac{\theta_{i0}}{m}}{\sum_{r=1}^{s} \frac{\phi_{i0}}{s}}$$ s.t $$\sum_{j=1}^{n} \tilde{x}_{ij} \lambda_{j} \leq \theta_{i0} \tilde{x}_{i0}, \quad i = 1, ..., m,$$ $$\sum_{r=1}^{s} \tilde{y}_{rj} \lambda_{j} \leq \phi_{i0} \tilde{y}_{r0}, \quad r = 1, ..., s,$$ $$\theta_{i} \leq 1, \quad \phi_{i} \geq 1, \quad \forall i, r,$$ $$\lambda_{j} \geq 0, \qquad j = 1, ..., n.$$ $$(12)$$ By using ranking method based on area compensation (10), model (12) can be written as follows: min $$R = \frac{\sum_{i=1}^{m} \frac{\theta_{i0}}{m}}{\sum_{r=1}^{s} \frac{\phi_{i0}}{s}}$$ s.t $$F\left(\sum_{j=1}^{n} \tilde{x}_{ij} \lambda_{j}\right) \leq F\left(\theta_{i0} \tilde{x}_{i0}\right), \quad i = 1, ..., m,$$ $$F\left(\sum_{r=1}^{s} \tilde{y}_{rj} \lambda_{j}\right) \leq F\left(\phi_{i0} \tilde{y}_{r0}\right), \quad r = 1, ..., s,$$ $$\theta_{i} \leq 1, \quad \phi_{i} \geq 1, \qquad \forall i, r,$$ $$\lambda_{i} \geq 0, \qquad j = 1, ..., n.$$ $$(13)$$ Then, using equation (9) we have min $$R = \frac{\sum_{i=1}^{m} \frac{\theta_{i0}}{m}}{\sum_{r=1}^{s} \frac{\phi_{i0}}{s}}$$ s.t $$\sum_{j=1}^{n} \frac{\lambda_{j}}{2} \int_{0}^{1} \left(\underline{x}_{ij}(\alpha) + \overline{x}_{ij}(\alpha)\right) d\alpha \leq \frac{\theta_{i0}}{2} \int_{0}^{1} \left(\underline{x}_{i0}(\alpha) + \overline{x}_{i0}(\alpha)\right) d\alpha \qquad i = 1, ..., m, \quad (14)$$ $$\sum_{j=1}^{n} \frac{\lambda_{j}}{2} \int_{0}^{1} \left(\underline{y}_{rj}(\alpha) + \overline{y}_{rj}(\alpha)\right) d\alpha \geq \frac{\phi_{i0}}{2} \int_{0}^{1} \left(\underline{y}_{r0}(\alpha) + \overline{y}_{r0}(\alpha)\right) d\alpha \qquad r = 1, ..., s,$$ $$\theta_{i} \leq 1, \quad \phi_{i} \geq 1 \qquad \forall i, r,$$ $$\lambda_{i} \geq 0, \qquad j = 1, ..., n.$$ The model (14) is a fractional problem. To transform the non-linear model (14) into a linear programming problem, we introduce a new variable $\beta$ defined by $$\beta = \frac{1}{\sum_{s=1}^{s} \frac{\phi_0}{s}} \tag{15}$$ such that $0 < \beta \le 1$ and $\beta(\sum_{r=1}^{s} \phi_{r0}/s) = 1$ . Then $\beta$ is used. $$u_i = \beta \theta_i,$$ $$v_r = \beta \phi_r,$$ $$t_i = \beta \lambda_j.$$ Now, using the above definition in model (14), we have $$\min \frac{\sum_{i=1}^{m} u_{i}}{m}$$ s.t $$\sum_{r=1}^{s} v_{r} = s,$$ $$\sum_{j=1}^{n} \frac{\lambda_{j}}{2} \beta \int_{0}^{1} (\underline{x}_{ij}(\alpha) + \overline{x}_{ij}(\alpha)) d\alpha \leq \frac{\theta_{i0}}{2} \beta \int_{0}^{1} (\underline{x}_{i0}(\alpha) + \overline{x}_{i0}(\alpha)) d\alpha \quad i = 1, ..., m,$$ $$\sum_{j=1}^{n} \frac{\lambda_{j}}{2} \beta \int_{0}^{1} (\underline{y}_{rj}(\alpha) + \overline{y}_{rj}(\alpha)) d\alpha \geq \frac{\phi_{i0}}{2} \beta \int_{0}^{1} (\underline{y}_{r0}(\alpha) + \overline{y}_{r0}(\alpha)) d\alpha \quad r = 1, ..., s,$$ $$\theta_{i} \leq 1, \quad \phi_{i} \geq 1 \qquad \forall i, r,$$ $$\lambda_{j} \geq 0, \qquad j = 1, ..., n.$$ The transformed model is as follows: $$\min \frac{\sum_{i=1}^{m} u_{i}}{m}$$ s.t $$\sum_{r=1}^{s} v_{r} = s,$$ $$\sum_{j=1}^{n} \frac{t_{j}}{2} \int_{0}^{1} (\underline{x}_{ij}(\alpha) + \overline{y}_{ij}(\alpha)) d\alpha \leq u_{i0} \int_{0}^{1} (\underline{x}_{i0}(\alpha) + \overline{x}_{i0}(\alpha)) d\alpha, \quad i = 1, ..., m,$$ $$\sum_{j=1}^{n} t_{j} \int_{0}^{1} (\underline{y}_{rj}(\alpha) + \overline{y}_{rj}(\alpha)) d\alpha \leq v_{i0} \int_{0}^{1} (\underline{y}_{r0}(\alpha) + \overline{y}_{r0}(\alpha)) d\alpha, \quad r = 1, ..., s,$$ $$u_{i} \leq \beta, \qquad \qquad i = 1, ..., m,$$ $$\beta \leq v_{r}, \qquad \qquad r = 1, ..., s,$$ $$0 \leq t_{j}, \qquad \qquad j = 1, ..., n,$$ $$0 \leq \beta \leq 1.$$ Finally, using Definition (1), we obtain final model as follows: min $$\frac{\sum_{i=1}^{m} u_{i}}{m}$$ s.t $$\sum_{r=1}^{s} v_{r} = s,$$ $$\sum_{j=1}^{n} t_{j} F\left(\tilde{x}_{ij}(\alpha)\right) \leq u_{i0} F\left(\tilde{x}_{i0}(\alpha)\right), \quad i = 1, ..., m,$$ $$\sum_{j=1}^{n} t_{j} F\left(\tilde{y}_{rj}(\alpha)\right) \leq v_{i0} F\left(\tilde{y}_{r0}(\alpha)\right), \quad r = 1, ..., s,$$ $$u_{i} \leq \beta, \qquad \qquad i = 1, ..., m,$$ $$\beta \leq v_{r}, \qquad \qquad r = 1, ..., s,$$ $$0 \leq t_{j}, \qquad \qquad j = 1, ..., n,$$ $$0 \leq \beta \leq 1.$$ (18) In next section, a case study is presented to validate the proposed model. #### 4 Case study In this section, a case study is presented. In this case study the efficiency of 21 Iranian advertising journals are evaluated. The inputs considered are number of circulation (CIRC), advertising cost (ADV cost), and reputation (REP). The outputs utilised in this study are subscription (SUBS), rate-based digression (R.B. DIG) (return), and discounts (DIS). The dataset are given in Table 1. Here, REP, R.B. DIG, and DIS are fuzzy numbers. The linguistic scale and corresponding triangular fuzzy number for reputation are defined in Figure 3. **Table 1** Related attributes for 21 journals | Journals<br>MUs)<br>(D | CIRC | SUBS | REP | ADV cost | R.B. DIG | DIS | |------------------------|--------|-------|----------------|----------|------------------------|----------------------| | 1 | 10,000 | 2,000 | (2 +α, 4 – α) | 300 | (15 + 2α, 21) | (15 + 2α, 30 – 5 α) | | 2 | 12,000 | 3,500 | (3 + α, 5 – α) | 325 | (8 + 5 α, 11.2 – 5 α) | (20 + 5 α, 40 – 5 α) | | 3 | 7,000 | 2,000 | (4 + 2α, 6) | 320 | (12 + 5 α, 16.8 – 3 α) | (15 + 5 α, 30 – 3 α) | | 4 | 8,000 | 700 | (3 + α, 5 – α) | 295 | (8 + 2α, 11.2 – 3 α) | (10 + 2α, 20 – 3 α) | | 5 | 15,000 | 3,000 | (1 + α, 3 – α) | 272 | (15 + 2α, 21 – 3 α) | (15 + 2α, 30 – 3 α) | | 6 | 5,000 | 1,000 | (0, 2 – 2α) | 372 | (9 + 2α, 12.6 – α) | (20 + 5 α, 40 – 5 α) | | 7 | 8,000 | 1,500 | (1 + α, 3 – α) | 245 | (13 + α, 18.2 – 2α) | (15 + α, 30 – 2α) | | 8 | 12,500 | 1,500 | (4 + 2α, 6) | 425 | (10 + 2α, 14 – α) | (10 + 5 α, 20 – 2α) | | 9 | 8,000 | 5,000 | (1 + α, 3 – α) | 380 | (12 + α, 16.8 – 2α) | (20 + 5 α, 40 – 5α) | | 10 | 10,000 | 1,200 | (2 + α, 4 – α) | 270 | (7 + α, 9.80 – 2α) | (15 + 3 α, 30 – 5α) | | 11 | 40,000 | 4,000 | (4 + 2α, 6) | 300 | (9 + 2α, 12.6 – α) | (10 + 5 α, 20 – α) | | 12 | 8,000 | 1,800 | (3 + α, 5 – α) | 360 | (15 + 2α, 21 – 3 α) | (15 + 5 α, 30 – 5α) | | 13 | 10,000 | 2,300 | (0, 2 – 2α) | 222 | (12 + α, 16.8 – α) | (15 + 5 α, 30 – 5α) | | 14 | 12,000 | 3,000 | (2 + α, 4 – α) | 265 | (10 + α, 14 – 2α) | (20 + 5 α, 40 – 10α) | | 15 | 12,000 | 4,000 | (4 + 2α, 6) | 488 | (8 + α, 11.2 – α) | (15 + 2α, 30 – 5α) | | 16 | 20,000 | 3,800 | (1 + α, 3 – α) | 438 | (9 + α, 12.6 – α) | (15 + 3 α, 30 – 5α) | | 17 | 10,000 | 2,400 | (2 + α, 4 – α) | 365 | (12 + α, 16.8 – α) | (20 + 5 α, 40 – 5α) | | 18 | 8,000 | 1,100 | (2 + α, 4 – α) | 300 | (8 + α, 11.2 – α) | (10 + 5 α, 20 – 3α) | | 19 | 5,000 | 900 | (0, 2 – 2α) | 400 | (9 + α, 12.6 – α) | (20 + 5 α, 40 – 5α) | | 20 | 8,000 | 1,300 | (0, 2 – 2α) | 305 | (15 + 2α, 21 – 3α) | (15 + 2α, 30 – 5α) | | 21 | 12,000 | 2,400 | (3 + α, 5 – α) | 330 | (13 + 2α, 18.2 – α) | (15 + 3 α, 30 – 5α) | **Figure 3** Membership function corresponding to linguistic reputation scale  To solve this problem by model (18), we compute the F functions for fuzzy data. For example, DIS for DMU₁ is computed as follows: $$F(15+2\alpha,30-5\alpha) = \frac{1}{2} \int_0^1 (15+2\alpha+30-5\alpha)d\alpha = \frac{1}{2} \int_0^1 (45-3\alpha)d\alpha = 21.75$$ (19) Table 2 depicts the *F* functions for fuzzy data. Table 3 reports the results of efficiency assessment for 21 advertising journals obtained by the model (18). Table 2 Computing F functions for fuzzy data | DMUs | DIS | R.B. DIG | ADV COST | REP | SUBS | CIRC | |------|-------|----------|----------|-----|-------|--------| | 1 | 21.75 | 17.25 | 300 | 3 | 2,000 | 10,000 | | 2 | 30 | 9.6 | 325 | 4 | 3,500 | 12,000 | | 3 | 23 | 14.9 | 320 | 5.5 | 2,000 | 7,000 | | 4 | 14.75 | 9.35 | 295 | 4 | 700 | 8,000 | | 5 | 23.25 | 17.75 | 272 | 2 | 3,000 | 15,000 | | 6 | 30 | 10.55 | 372 | 1 | 1,000 | 5,000 | | 7 | 22.25 | 15.35 | 245 | 2 | 1,500 | 8,000 | | 8 | 15.75 | 12.25 | 425 | 5.5 | 1,500 | 12,500 | | 9 | 30 | 14.15 | 380 | 2 | 5,000 | 8,000 | | 10 | 22 | 8.15 | 270 | 3 | 1,200 | 10,000 | | 11 | 16 | 11.5 | 300 | 5.5 | 4,000 | 40,000 | | 12 | 22.5 | 17.75 | 360 | 4 | 1,800 | 8,000 | | 13 | 22.5 | 14.4 | 222 | 1 | 2,300 | 10,000 | | 14 | 28.75 | 11.75 | 265 | 3 | 3,000 | 12,000 | | 15 | 21.75 | 9.6 | 488 | 5.5 | 4,000 | 12,000 | | 16 | 22 | 10.8 | 438 | 2 | 3,800 | 20,000 | | 17 | 30 | 14.25 | 365 | 3 | 2,400 | 10,000 | | 18 | 15.5 | 9.6 | 300 | 3 | 1,100 | 8,000 | | 19 | 30 | 10.8 | 400 | 1 | 900 | 5,000 | | 20 | 21.75 | 17.75 | 305 | 1 | 1,300 | 8,000 | | 21 | 22 | 15.85 | 330 | 4 | 2,400 | 12,000 | Table 3 shows journals 5, 6, 7, 9, 11, 13, 14, 19, and 20 are identified as efficient and the rest are inefficient. Table 3 Efficiency assessment for 21 advertising journals | DMUs | Efficiency scores | | |------|-------------------|---| | 1 | 0.70 | _ | | 2 | 0.60 | | | 3 | 0.78 | | | 4 | 0.26 | | | 5 | 1 | | | 6 | 1 | | **Table 3** Efficiency assessment for 21 advertising journals (continued) | DMUs | Efficiency scores | | |------|-------------------|--| | 7 | 1 | | | 8 | 0.29 | | | 9 | 1 | | | 10 | 0.40 | | | 11 | 1 | | | 12 | 0.90 | | | 13 | 1 | | | 14 | 1 | | | 15 | 0.44 | | | 16 | 0.50 | | | 17 | 0.63 | | | 18 | 0.36 | | | 19 | 1 | | | 20 | 1 | | | 21 | 0.56 | | #### **5 Concluding remarks** Media selection is one of the major problems in marketing department of organisations. To select media it is necessary to consider several criteria. Except for DEA, all the mentioned techniques in Sections 1 and 2 rely on subjective weights on criteria for selecting suitable media. The subjective weights are not accurate and they depend totally on decision maker tact. 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