Media Selection In The Presence Of Flexible Factors And Imprecise Data

This paper depicts the media selection problem through a Data envelopment analysis (DEA) model, while allowing for the incorporation of both flexible factors and imprecise data. A numerical example demonstrates the application of the proposed method.

Keywords: imprecise data envelopment analysis; media selection; flexible factors

1. Introduction

Media selection problem is a major concern in advertising. The marketing department is responsible for managerial decisions related to media selection. As Dyer et al (1992) addressed, the media selection problem is an excellent illustration of a complex multi-faceted decision. A decision model used to handle multiple criteria needs to reflect both cardinal and ordinal data. The media selection problem may be stated as follows: Given a set of media options, and various data regarding the media and the audiences to be reached, which options should be selected.

Data envelopment analysis (DEA), developed by Charnes et al (1978), provides a non-parametric methodology for evaluating the efficiency of each of a set of comparable decision making units (DMUs), relative to one another. In the standard use of DEA, it is supposed that one can, given a set of available factors, clearly determine which factors are inputs and which are outputs. However, in many problem situations, the input versus output status of certain measures can be deemed as flexible. For example, in a conventional study of efficiency of research by universities, ‘research income’ is treated as both an output and input (Cook and Zhu, 2007). Similar arguments can be made regarding the evaluation of third-party reverse logistics (3PL) providers, such as described in Farzipoor Saen (2010). There, ‘ratings for service-quality experience (EXP) and service-quality credence (CRE)’ are treated as both outputs and inputs (flexible factors). EXP and CRE could serve as either inputs or outputs. From the perspective of decision maker who intends to select the best 3PL

providers, such measures may play the role of proxy for ‘high quality of services’, hence can reasonably be classified as outputs. On the other hand, from the perspective of 3PL provider that intends to supply reverse logistics services, they can be considered as inputs that help the 3PL provider in obtaining more customers.

In media selection context, the volume of supplied information to audiences can be considered as a flexible factor. From the perspective of media planner who intends to select the best medium, such measure may play the role of proxy for ‘high understanding of audiences’, hence it can be classified as output. However, this factor can be regarded as an input, because competitors by virtue of high volume of supplied information to audiences get more information about company.¹

As well, in many applications (especially media selection problems), it is necessary to consider the existence of ordinal (qualitative) factors when rendering a decision on the performance of a DMU (medium). Very often, it is the case that for a factor such as medium prestige, one can, at most, provide a ranking of the DMUs (media) from best to worst relative to this attribute. In this situation, the data for certain influence factors (inputs and outputs) might better be represented as rank positions in an ordinal, rather than numerical sense. In certain circumstances, the information available may permit one to provide a complete rank ordering of the DMUs on such a factor. Therefore, the data may be imprecise. Note that there may exist

^<sup>1Please note that in traditional DEA, the decision maker decides which criteria are inputs and which are outputs. However, in the flexible factor context, the decision maker is wavered. In other words, the decision maker does not know whether this flexible factor is an input or an output. Therefore, there is a need for a model that determines the status of flexible factor for each DMU, separately. After running the model, the decision maker finds out the status of a flexible factor.

three types of ordinal preference information including (1) strong ordinal preference information; (2) weak ordinal preference information; and (3) indifference relationship. Theses types of ordinal preference information will be discussed in more details in Section 3.

Motivated by those points, the objective of this paper is to propose a model for selecting media in the presence of both flexible factors and imprecise data. This paper depicts the media selection process through a DEA model, while allowing for the incorporation of both flexible factors and imprecise data.

This paper proceeds as follows. In Section 2, literature review is presented. Section 3 discusses the proposed method for media selection. Numerical example is discussed in Section 4 and Section 5 discusses concluding remarks.

2. Literature review

In the following subsections, various studies on the media selection, flexible factors and imprecise data are briefly summarized.

2.1. Media selection

To select media, Dyer et al (1992) present two approaches incorporating the analytic hierarchy process (AHP). The recommended approach utilizes AHP in conjunction with integer programming while an alternative approach uses goal programming (GP) coupled with the AHP. Kwak et al (2005) develop a mixed integer GP model to determine the number of advertisements for selected media and to allocate the advertising media budget to selected media categories for digital products. The model formulation follows the preemptive priority format in the derivation of a satisfying solution. To select media, Wiedey and Zimmermann (1978) suggest a fuzzy GP and linear programming model that seems to be suited to map fuzzy and subjective phenomena as well as incorporates several criteria (objective functions) simultaneously. Deckro and Murdock (1987) present multiple objective integer linear programming (MOILP) as an aid to media selection. A MOILP analysis approach recognizes the lack of infinite division in media choices and provides the decision maker with an efficient set of alternatives. The decision maker may then utilize non-quantifiable criteria to make the final media selection or may use the efficient set developed by the MOILP procedure as a screening mechanism. These screened solutions would then be submitted to further analysis to consider interaction effects, non-linearity and so on. Steuer and Oliver (1976) discuss the application of multiple objective linear programming (MOLP) with interval criterion weights to a multiple criterion media selection problem. This technique actually functions as a generalization of regular linear programming in that it can process problems in which more than one objective function is specified. Consequently, users are no longer necessarily limited to a single composite objective function when more than one measure of effectiveness would be more appropriate as is often the case. Another feature of this solution method is a provision whereby the media planner may associate an interval weight with each different objective.

Bass and Lonsdale (1966) examine the behaviour characteristics of the use of linear programming to media selection. They analysed the influence of weighting systems used to adjust audience data and various restraint systems with actual data. Brown and Warshaw (1965) demonstrate how a linear model can be adapted to solve many media mix problems involving non-linear responses to advertising. They presented two models. The first, illustrates the general linear programming method. The second, a slightly more complex model, shows how a non-linear objective function can be treated within the framework of a linear model under certain conditions.

Brown (1967) examines the principle of incremental analysis as applied to media selection problems by illustrating some magazine-selection problems. However, as Shocker (1970) discussed, incremental analysis will not always lead to a global optimum especially when the surface is characterized by several local optima.

Dwyer and Evans (1981) develop a branch and bound algorithm for solving a list selection problem in direct mail advertising. The special structure and heuristic information imbedded in the problem led to relatively efficient computation by providing quick fathoming of nodes in the branch and the bound tree.

Gensch (1968, 1970) identify the relevant mathematical approaches that might be used to construct an advertising media model to select and evaluate the feasible combinations of media vehicles available for a given product message. The relative merits of the various approaches are then discussed.

Korhonen et al (1989) use the evolutionary approach to modelling and solving a media selection problem for a software company in Finland. They did not solve the problem in a classical sense, but their approach aids the management to see relevant features of the problem and find guidelines for future action. In this approach, the decision-maker does not have to specify the model precisely prior to solving the problem. In fact, the model evolves progressively.

Morard et al (1976) propose a model of advertising media selection taking into account the uncertainty of the audiences reached effectively. The model is based upon the mean-variance point of view but the formulation differs from the now classical model used in portfolio theory. The model goal is to minimize the variability of the total desirable audience. De Kluyver (1980) considers formulations for media scheduling problems. By treating the size of the audience obtained as a random variable, the resulting formulations are shown to be of the mean-variance type, and hence akin to Markowitz’s familiar portfolio model. A binomial probability model is used to derive expected values for gross and net audience segments obtained and of average frequency realized. The proposed model minimizes the variance of gross audience obtained subject to a variety of constraints on gross audience, net audience, frequency and the budget allocated. A parametric solution structure is sought tracing the entire efficiency frontier of solutions, thus allowing trade-offs on a risk-return or budget-return basis. De Kluyver and Baird (1984) report an application of a mean-variance approach to media scheduling problems. The findings illustrate the effect of recognizing schedule variance in formulating candidate media plans, and demonstrate that mean-variance analysis is an effective approach to schedule formulation and assessment through parametric risk-return and budgetreturn analysis.

However, all of the abovementioned references do not use DEA for media selection. This paper, for the first time, proposes a new model that is based on DEA and does not demand weights from the media planner as well.

2.2. Flexible factors

Cook et al (2006) present a methodology for dealing with those situations where a factor is flexible. By treating such a factor on the input side as being non-discretionary, the developed model can be used to determine in which status that factor dominates within each DMU. Specifically, the model determines whether in a DMU the factor is behaving predominantly like an input, hence the DMU would benefit from having less of the factor, like an output where more of the factor is desirable, or where it is in equilibrium. They connect these ideas to those involving increasing, decreasing and constant returns to scale. They apply the model to the analysis of a set of university departments as per Beasley (1990, 1995).

Cook and Zhu (2007) present a modification of the standard constant returns to scale DEA model to accommodate flexible factors. Both an individual DMU model and an aggregate model were suggested as methodologies for deriving the most appropriate designations for flexible factors. However, as addressed by Toloo (2009), their method may produce incorrect efficiency scores due to a computational problem as result of introducing a large positive number to the model. Toloo (2009) introduces a revised model that does not need such a large positive number.

The modification to the traditional models of DEA that permits the inclusion of multiple flexible factors in the analysis was developed by Farzipoor Saen (2010). The proposed model was applied for selecting 3PL providers.

But, all of the aforementioned references do not consider the imprecise data. In addition, they do not apply their model in the area of media selection problem. This paper, for the first time, proposes a new application of dual-role factor in media selection problem.

2.3. Imprecise data

Wang et al (2005) develop a new pair of interval DEA models for dealing with imprecise data such as interval data, ordinal preference information, fuzzy data and their mixture. Farzipoor Saen (2008) introduces a new pair of assurance region-imprecise data envelopment analysis (AR-IDEA) model and employs it for supplier selection. Farzipoor Saen (2009b) provides a model for ranking suppliers in the presence of weight restrictions, cardinal and ordinal data, and non-discretionary factors. Farzipoor Saen (2009c) proposes a new pair of non-discretionary factors-imprecise data envelopment analysis (NF-IDEA) models and employed it for supplier selection. Again, Farzipoor Saen (2009a) introduces a new pair of assurance region-nondiscretionary factors-imprecise data envelopment analysis (AR-NF-IDEA) models and employed it for technology selection. Also, he proposes a systematic analysis to provide peer groups for inefficient technology suppliers. Park (2007) shows two distinct strategies to arrive at an upper and lower bound of efficiency that the evaluated DMU can have within the given imprecise data. Matin et al (2007) employ an additive DEA model to evaluate the technical inefficiency of DMUs under imprecise data. The non-linear DEA model is transformed into an equivalent linear one, then the translation invariant property is used and a one-stage approach is introduced in this inefficiency evaluation. Farzipoor Saen (2011) introduces a model for dealing with selecting third-party reverse logistics (3PL) providers in the presence of imprecise data. The proposed model is based on DEA. Lin (2009) considers how to conduct the efficiency measurement and ranking problem of the tutorial system of some higher education institutions in Taiwan. In order to better reflect the actual situations that different tutors consume various amounts of input to produce various amounts of outputs, an imprecise data envelopment analysis (IDEA) model is proposed to measure the efficiencies of tutors under different possibility levels by incorporating the current methods to serve as output factors and additionally considering the input factor.

However, all of the above references do not take into account the dual-role factors. Besides, they do not use their model in the field of media selection problem. This paper, for the first time, proposes a new use of IDEA in media selection problem.

In summary, all of the abovementioned references do not consider both flexible factors and imprecise data. A technique that can deal with both flexible factors and imprecise data is needed to better model such situation. To the best of author’s knowledge, there is not any reference that discusses media selection in the presence of both flexible factors and imprecise data. Author believes that this paper has a significant contribution to an important and very much under-researched topic. The contributions of proposed model are as follows:

• The proposed model does not demand weights from the decision maker.
• The proposed model considers flexible factors.
• The proposed model considers imprecise data.
• The proposed model considers both flexible factors and imprecise data simultaneously.
• The proposed model can be easily computerized, enabling it to serve as a decision making tool to assist decision makers.
• The paper makes a sufficient contribution to the practice of Operations Research. It is the first study, which proposes advanced DEA model for selecting media.

3. Proposed method for media selection

Let $x_{ij}$ $(i=1,2,\ldots,m)$ be m different inputs and $y_{rj}$ $(r=1,2,\ldots,s)$ be s different outputs for DMU $_j$ , $j=1,2,\ldots,n$ . Suppose also that there exist L flexible factors, whose input/output status is being determined. The values assumed by these flexible factors are denoted by $w_{lj}$ for DMU $_j$ $(l=1,\ldots,L)$ . For each flexible factor l, the binary variable $d_l \in \{0,1\}$ is introduced, where $d_l = 1$ designates that factor l is an output, and $d_l = 0$ designates it as an input. Let $\gamma_l$ be the weight for each flexible factor l.

To deal with flexible factors, Toloo (2009) proposed following model.²

$\max \frac{\sum_{r=1}^{S} \mu_{r} y_{ro} + \sum_{l=1}^{L} \delta_{l} w_{lo}}{\sum_{i=1}^{m} v_{i} x_{io} + \sum_{l=1}^{L} \gamma_{l} w_{lo} - \sum_{l=1}^{L} \delta_{l} w_{lo}},$ st. $\frac{\sum_{r=1}^{S} \mu_{r} y_{rj} + 2 \sum_{l=1}^{L} \delta_{l} w_{lj}}{\sum_{i=1}^{m} v_{i} x_{ij} + \sum_{l=1}^{L} \gamma_{l} w_{lj}} \leq 1 \qquad j = 1, 2, \dots, n,$

$0 \leq \delta_{l} \leq d_{l},$

$\delta_{l} \leq \gamma_{l} \leq \delta_{l} + (1 - d_{l}),$

$d_{l} \in \{0, 1\}, \quad 0 \leq \delta_{l}, \quad \gamma_{l} \leq 1 \quad \forall l,$

$0 \leq \mu_{r}, \quad v_{i} \leq 1 \quad \forall r, \quad i. \quad (1)$

where $\mu_r$ is the weight given to output r and $v_i$ is the weight given to input i. DMU $_o$ is the DMU under consideration. DMU $_o$ consumes $x_{io}$ (i = 1, …, m), the amount of input i, to produce $y_{ro}$ (r = 1, …, s), the amount of output r. Notice that $\delta_l$ is the result of change of variables. For more details, please see Cook and Zhu (2007).

At this juncture, the new model that considers both flexible factors and imprecise data is introduced. The model is based on the interval arithmetic.

Let

$\theta_{j} = \frac{\sum_{r=1}^{S} \mu_{r} y_{rj} + 2 \sum_{l=1}^{L} \delta_{l} w_{lj}}{\sum_{i=1}^{m} v_{i} x_{ij} + \sum_{l=1}^{L} \gamma_{l} w_{lj}} \qquad j = 1, ..., n$ (2)

be the efficiency of DMU_j. According to the operation rules on interval data, there is

\begin{split} \theta_{j} &= \frac{\sum\limits_{r=1}^{S} \mu_{r} \left[ y_{rj}^{L}, y_{rj}^{U} \right] + 2\sum\limits_{l=1}^{L} \delta_{l} \left[ w_{lj}^{L}, w_{lj}^{U} \right]}{\sum\limits_{i=1}^{m} v_{i} \left[ x_{ij}^{L}, x_{ij}^{U} \right] + \sum\limits_{l=1}^{L} \gamma_{l} \left[ w_{lj}^{L}, w_{lj}^{U} \right]} \\ &= \frac{\left[ \sum\limits_{r=1}^{S} \mu_{r} y_{rj}^{L}, \sum\limits_{r=1}^{S} \mu_{r} y_{rj}^{U} \right] + 2\left[ \sum\limits_{l=1}^{L} \delta_{l} w_{lj}^{L}, \sum\limits_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]}{\left[ \sum\limits_{i=1}^{m} v_{i} x_{ij}^{L}, \sum\limits_{i=1}^{m} v_{i} x_{ij}^{U} \right] + \left[ \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{L}, \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U} \right]} \\ &= \frac{\left[ \sum\limits_{r=1}^{S} \mu_{r} y_{rj}^{L}, \sum\limits_{l=1}^{S} \mu_{r} y_{rj}^{U} \right]}{\left[ \sum\limits_{l=1}^{m} v_{i} x_{ij}^{L}, \sum\limits_{i=1}^{m} v_{i} x_{ij}^{U} \right] + \left[ \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U}, \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U} \right]} \\ &+ \frac{2\left[ \sum\limits_{l=1}^{L} \delta_{l} w_{lj}^{L}, \sum\limits_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]}{\left[ \sum\limits_{i=1}^{m} v_{i} x_{ij}^{U} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U} \right]} + \left[ \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U}, \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U} \right]} \\ &= \left[ \frac{\sum\limits_{i=1}^{S} \mu_{r} y_{rj}^{L}}{\sum\limits_{i=1}^{m} v_{i} x_{ij}^{U} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U}}, \sum\limits_{i=1}^{m} v_{i} x_{ij}^{U} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U} \right]} \\ &+ \left[ \frac{2\left[ \sum\limits_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]}{\sum\limits_{i=1}^{m} v_{i} x_{ij}^{U} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U}}, \sum\limits_{i=1}^{m} v_{i} x_{ij}^{U} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lj}^{U} \right]}, \right]

j = 1, \dots, n \tag{3}

^<sup>2Please note that, to develop the new model, the fractional form of the formulation proposed by Toloo (2009) has been rewritten.

It is obvious that $\theta_j$ should be an interval number, which is denoted by $\left[\theta_i^L, \theta_i^U\right] \quad (j = 1, \dots, n)$ Let

$\theta_{j} = \left[\theta_{j}^{L}, \theta_{j}^{U}\right] = \frac{\sum_{r=1}^{S} \mu_{r} y_{rj}^{L}}{\sum_{i=1}^{m} v_{i} x_{ij}^{U} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{U}}, \frac{\sum_{r=1}^{S} \mu_{r} y_{rj}^{U}}{\sum_{i=1}^{m} v_{i} x_{ij}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L}} + \frac{2\left[\sum_{l=1}^{L} \delta_{l} w_{lj}^{U}\right]}{\sum_{i=1}^{m} v_{i} x_{ij}^{U} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{U}}, \frac{2\left[\sum_{l=1}^{L} \delta_{l} w_{lj}^{U}\right]}{\sum_{i=1}^{m} v_{i} x_{ij}^{U} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{U}} \subseteq [0, 1],$

j = 1, \dots, n. \tag{4}

Then

$\theta_{j}^{U} = \frac{\sum_{r=1}^{S} \mu_{r} y_{rj}^{U}}{\sum_{i=1}^{m} v_{i} x_{ij}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L}} + \frac{2 \left[ \sum_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]}{\sum_{i=1}^{m} v_{i} x_{ij}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{U}} \leq 1 \quad j = 1, \dots, n$

$\theta_{j}^{L} = \frac{\sum_{r=1}^{S} \mu_{r} y_{rj}^{L}}{\sum_{i=1}^{m} v_{i} x_{ij}^{U} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{U}} + \frac{2 \left[ \sum_{l=1}^{L} \delta_{l} w_{lj}^{L} \right]}{\sum_{i=1}^{m} v_{i} x_{ij}^{U} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{U}} > 0 \quad j = 1, \dots, n.$

$(5)$

To measure the upper and lower bounds of the efficiency of $DMU_o$ , the following pair of fractional programming models for $DMU_o$ is introduced:

$\begin{aligned} \operatorname{Max} \theta_{o}^{U} &= \frac{\sum\limits_{r=1}^{S} \mu_{r} y_{ro}^{U}}{\sum\limits_{i=1}^{m} v_{i} x_{io}^{L} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lo}^{L}} \\ &+ \frac{2 \left[ \sum\limits_{l=1}^{L} \delta_{l} w_{lo}^{U} \right]}{\sum\limits_{i=1}^{m} v_{i} x_{io}^{L} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lo}^{L}} \end{aligned}$

st.

$\theta_{j}^{L} = \frac{\sum_{r=1}^{S} \mu_{r} y_{rj}^{U}}{\sum_{i=1}^{m} v_{i} x_{ij}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L}} + \frac{2 \left[ \sum_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]}{\sum_{i=1}^{m} v_{i} x_{ij}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L}} \leq 1 \quad j = 1, \dots, n.$

$0 \leq \delta_{l} \leq d_{l},$

$\delta_{l} \leq \gamma_{l} \leq \delta_{l} + (1 - d_{l}),$

$d_{l} \in \{0, 1\}, \quad 0 \leq \delta_{l}, \quad \gamma_{l} \leq 1 \quad \forall l,$

$0 \leq \mu_{r}, \quad v_{i} \leq 1 \quad \forall r, \quad i.$ (6)

$\begin{aligned} \mathbf{Max} \, \theta_{j}^{L} &= \frac{\sum\limits_{r=1}^{S} \mu_{r} y_{ro}^{L}}{\sum\limits_{i=1}^{m} v_{i} x_{io}^{U} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lo}^{U}} \\ &+ \frac{2 \left[ \sum\limits_{l=1}^{L} \delta_{l} w_{lo}^{L} \right]}{\sum\limits_{i=1}^{m} v_{i} x_{io}^{U} + \sum\limits_{l=1}^{L} \gamma_{l} w_{lo}^{U}} \end{aligned}$

st.

$\theta_{j}^{L} = \frac{\sum_{r=1}^{S} \mu_{r} y_{rj}^{U}}{\sum_{i=1}^{m} v_{i} x_{ij}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L}}$

$+ \frac{2 \left[ \sum_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]}{\sum_{i=1}^{m} v_{i} x_{ij}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L}} \leq 1 \quad j = 1, \dots, n.$

$0 \leq \delta_{l} \leq d_{l},$

$\delta_{l} \leq \gamma_{l} \leq \delta_{l} + (1 - d_{l}),$

$d_{l} \in \{0, 1\}, \quad 0 \leq \delta_{l}, \quad \gamma_{l} \leq 1 \quad \forall l,$

$0 \leq \mu_{r}, \quad v_{i} \leq 1 \quad \forall r, \quad i.$

$(7)$

Using Charnes—Cooper transformation, the above pair of fractional programming models can be simplified as the following equivalent linear programming models:

$\operatorname{Max} \theta_o^U = \sum_{r=1}^S \mu_r y_{ro}^U + 2 \left[ \sum_{l=1}^L \delta_l w_{lo}^U \right]$

st.

$\sum_{i=1}^{m} v_{i} x_{io}^{L} + \sum_{l=1}^{L} \gamma_{l} w_{lo}^{L} = 1$

$\sum_{r=1}^{S} \mu_{r} y_{rj}^{U} + 2 \left[ \sum_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]$

$- \sum_{i=1}^{m} v_{i} x_{ij}^{L} - \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L} \leq 0 \quad j = 1, \dots, n$

$0 \leq \delta_{l} \leq d_{l},$

$\delta_{l} \leq \gamma_{l} \leq \delta_{l} + (1 - d_{l}),$

$d_{l} \in \{0, 1\}, \quad 0 \leq \delta_{l}, \quad \gamma_{l} \leq 1 \quad \forall l,$

$0 \leq \mu_{r}, \quad v_{i} \leq 1 \quad \forall r, \quad i.$ (8)

$\operatorname{Max} \theta_{j}^{L} = \sum_{r=1}^{S} \mu_{r} y_{ro}^{L} + 2 \left[ \sum_{l=1}^{L} \delta_{l} w_{lo}^{L} \right]$

st

$\sum_{i=1}^{m} v_{i} x_{io}^{U} + \sum_{l=1}^{L} \gamma_{l} w_{lo}^{U} = 1$

$\sum_{r=1}^{S} \mu_{r} y_{rj}^{U} + 2 \left[ \sum_{l=1}^{L} \delta_{l} w_{lj}^{U} \right]$

$- \sum_{i=1}^{m} v_{i} x_{ij}^{L} - \sum_{l=1}^{L} \gamma_{l} w_{lj}^{L} \leqslant 0 \quad j = 1, \dots, n$

$0 \leqslant \delta_{l} \leqslant d_{l},$

$\delta_{l} \leqslant \gamma_{l} \leqslant \delta_{l} + (1 - d_{l}),$

$d_{l} \in \{0, 1\}, \quad 0 \leqslant \delta_{l}, \quad \gamma_{l} \leqslant 1 \quad \forall l,$

$0 \leqslant \mu_{r}, \quad v_{i} \leqslant 1 \quad \forall r, \quad i.$ (9)

where $\theta_o^U$ stands for the best possible relative efficiency achieved by DMU_o when all the DMUs are in the state of best production activity, while $\theta_o^L$ stands for the lower bound of the best possible relative efficiency of DMU_o. They constitute a possible best relative efficiency interval $[\theta_o^L, \theta_o^U]$ .

Note that model (8) determines the production frontier for all the DMUs and model (9) uses the production frontier as a benchmark to measure the lower bound efficiency of each DMU.

In order to judge whether a DMU is DEA efficient or not, the following definition is given.

Definition 1 A DMU, DMU_o, is said to be DEA efficient if its best possible upper bound efficiency $\theta_o^{U^*} = 1$ ; otherwise, it is said to be DEA inefficient if $\theta_o^{U^*} < 1$ .

Therefore, an integrated approach that deals with all aspects of the imprecise data and flexible factors in a direct manner has been introduced.

Now, the method of transforming ordinal preference information into interval data is discussed, so that the interval DEA models presented in this paper can still work properly even in these situations (Wang et al, 2005).

Suppose some input and/or output data for DMUs are given in the form of ordinal preference information. As discussed in Section 1, there may exist three types of ordinal preference information: (1) strong ordinal preference information such as $y_{rj} > y_{rk}$ or $x_{ij} > x_{ik}$ , which can be further expressed as $y_{rj} > y_{rk}$ and $x_{ij} > \eta_i x_{ik}$ , where $\chi_r > 1$ and $\eta_i > 1$ are the parameters on the degree of preference intensity provided by decision maker; (2) weak ordinal preference information such as $y_{rp} > y_{rq}$ or $x_{ip} > x_{iq}$ ; (3) indifference relationship such as $y_{rl} = y_{rt}$ or $x_{il} = x_{il}$ .

Since DEA model has the property of unit-invariance, the use of scale transformation to ordinal preference information does not change the original ordinal relationships and has no effect on the efficiencies of DMUs (For more details on the property of unit-invariance, please see Chen and Sherman, 2004). Therefore, it is possible to conduct a scale transformation to every ordinal input and output index so that its best ordinal datum is less than or equal to unity and then give an interval estimate for each ordinal datum.

At this point, consider the transformation of ordinal preference information about the output $y_{rj}$ $(j=1,\ldots,n)$ for example. The ordinal preference information about input and other output data can be converted in the same way.

For weak ordinal preference information $y_{r1} \geqslant y_{r2} \geqslant \cdots$ $\geqslant y_{rn}$ , there are following ordinal relationships after scale transformation:

1 \geqslant \hat{y}_{r1} \geqslant \hat{y}_{r2} \geqslant \dots \geqslant \hat{y}_{rn} \geqslant \sigma_r, \tag{10}

where $\sigma_r$ is a small positive number reflecting the ratio of the possible minimum of $\{y_{rj}| j=1,\ldots,n\}$ to its possible maximum. It can be approximately estimated by the decision maker. It is referred as the ratio parameter for convenience. The resultant permissible interval for each $\hat{y}_{rj}$ is given by

$\hat{y}_{ri} \in [\sigma_r, 1], \qquad j = 1, \dots, n.$ (11)

For strong ordinal preference information $y_{r1} > y_{r2} > \cdots > y_{rm}$ , there is the following ordinal relationships after scale transformation:

$1 \geqslant \hat{y}_{r1}, \qquad \hat{y}_{rj} \geqslant \chi_r \hat{y}_{r,j+1} \quad (j = 1, \dots, n \geqslant 1)$

and $\hat{y}_{rn} \geqslant \sigma_r$ , (12)

where $\chi_r$ is a preference intensity parameter satisfying $\chi_r > 1$ provided by the decision maker and $\sigma_r$ is the ratio parameter also provided by the decision maker. The resultant permissible interval for each $\hat{y}_{rj}$ can be derived as follows:

$\hat{y}_{rj} \in \left[\sigma_r \chi_r^{n-j}, \chi_r^{1-j}\right], \quad j = 1, \dots, n \text{ with } \sigma_r \leq \chi_r^{1-n}.$ (13)

Finally, for indifference relationship, the permissible intervals are the same as those obtained for weak ordinal preference information.

Through the scale transformation above and the estimation of permissible intervals, all the ordinal preference information is converted into interval data and can thus be incorporated into models (8) and (9).

In the next section, a numerical example is presented.

4. Numerical example

In this section, a demonstration of methodology is given to exemplify how the proposed approach can be employed

Table 1		Related attributes for 20 media
---------	—	---------------------------------	—	—	—

Media (DMU)	Input	Outputs			Flexible factor
	Cost (10 000 Rials)	SA	ATA*	DU (months)	VS**
	x1j	y1j	y2j	y3j	w1j
Brochures	[240, 300]	[5000, 7000]	3	12	12
Catalogues	[525, 750]	[1500, 3000]	7	24	18
Directories	[1175, 1575]	[4500, 5500]	13	24	14
Advertisement in books of specialized fairs	[1375, 2275]	[4500, 5500]	18	12	11
Specialized magazines	[2750, 4950]	[4500, 5500]	17	3	10
Billboards	[3000, 9000]	[50 000, 200 000]	12	1	8
Internet	[1500, 4000]	[9000, 11 000]	11	24	17
Multimedia CD	[2.5, 3.75]	[4000, 6000]	16	24	20
Cheap gifts	[562.5, 900]	[2000, 2500]	2	1	7
Expensive gifts	[360, 540]	[400, 500]	19	36	6
Overalls	[27 000, 31 500]	[20 000, 25 000]	10	12	5
Specialized fairs	[11 000, 16 500]	[5000, 10 000]	14	6	16
Seminar for customers	[15 000, 22 500]	[50, 100]	20	24	19
Plastic sacks	[500, 625]	[12 000, 13 000]	1	1	4
Cloth sacks	[440, 550]	[5000, 6000]	4	3	3
Almanacs	[11 000, 16 500]	[10 000, 12 000]	9	12	15
Tableaus for sales agents	[6000, 9000]	[95 000, 110 000]	15	60	9
Greeting cards	[1225, 1400]	[3000, 4000]	6	1	2
On wall almanacs	[2200, 2475]	[5000, 6000]	8	12	1
Iconic model of plants	[12 000, 13 500]	[450, 550]	5	120	13

^*Ranking such that 20 highest rank, … , 1 lowest rank (y2,134y2,10 … 4y2,14).

for media selection in steel industry. In particular, this example is used to show how ordinal and bounded data, as well as flexible factor, can be combined into the one unified approach provided by the proposed model. This case study uses actual data in Iranian market.

Sepahan Industrial Group Co. (SIG), was established in 1973 for manufacturing steel tubes, pipes and profiles in Iran. Nowadays after over 36 years, SIG has become a major and pioneer manufacturer and exporter of different kinds of steel pipes, tubes and profiles in Iran as well as the middle-east. SIG production plants, with more than 880 employees and annual production capacity of more than 420 000 metric tons at a total area of more than 1 000 000 square meters, are located near the historical city of Isfahan at the central part of Iran. The media planner at SIG intends to select the best media.

The media bounded input considered is cost.3 The outputs utilized in the study are size of audiences (SA), accuracy in targeting of audiences (ATA), and durability of media (DU). The DU is a cardinal output, while SA and ATA are bounded and ordinal data, respectively. Volume of supplied information to audiences (VS) could be considered as a flexible and ordinal factor simultaneously. From the perspective of media planner who intends to select the best media, such measure may play the role of proxy for ‘high understanding of audiences’, hence can reasonably be classified as output. However, this factor can be regarded as an input, because competitors by virtue of high volume of supplied information to audiences get more information about company. ATA and VS are intangible factors that are not usually explicitly included in evaluation model for media. These qualitative variables are measured on an ordinal scale so that, for instance, with respect to VS for DMU8 (multimedia CD) is given the highest rank, and for DMU19 (on wall almanacs), the lowest. Table 1 depicts the media attributes.

Suppose the preference intensity parameter and the ratio parameter about the strong ordinal preference information are given (or estimated) as w¼ 1.12 and s ¼ 0.01, respectively. Using the transformation technique described in previous section, an interval estimate for ATA and VS of each media can be derived, which is shown in Table 2.

Therefore, all the ordinal data are now transformed into interval numbers and can be evaluated using interval DEA models. Table 3 reports the results of efficiency assessments and their inputs/outputs behaviour for 20 media obtained by using model (8).

The second column of Table 3 shows the optimal d and the third column is the efficiency score for each medium. In this case, all of the media treat the VS measure as an input. As discussed earlier, VS is a flexible factor that media

^**Ranking such that 20 highest rank, … , 1 lowest rank (w1,84w1,13 … 4w1,19).

³The measures selected in this paper are not exhaustive by any means, but are some general measures that can be utilized to evaluate media. Media planners must carefully identify appropriate measures to be used in the decision making process.

Table 2 Interval estimate for 20 media after the transformation of ordinal preference information

Media (DMU)	ATA	VS
Brochures	[0.01254, 0.14564]	[0.03479, 0.40388]
Catalogues	[0.01974, 0.22917]	[0.06866, 0.79719]
Directories	[0.03896, 0.45235]	[0.04363, 0.50663]
Advertisement in books of specialized fairs	[0.06866, 0.79719]	[0.03106, 0.36061]
Specialized magazines	[0.0613, 0.71178]	[0.02773, 0.32197]
Billboards	[0.03479, 0.40388]	[0.02211, 0.25668]
Internet	[0.03106, 0.36061]	[0.0613, 0.71178]
Multimedia CD	[0.05474, 0.63552]	[0.08613, 1]
Cheap gifts	[0.0112, 0.13004]	[0.01974, 0.22917]
Expensive gifts	[0.0769, 0.89286]	[0.01762, 0.20462]
Overalls	[0.02773, 0.32197]	[0.01574, 0.1827]
Specialized fairs	[0.04363, 0.50663]	[0.05474, 0.63552]
Seminar for customers	[0.08613, 1]	[0.0769, 0.89286]
Plastic sacks	[0.01, 0.11611]	[0.01405, 0.16312]
Cloth sacks	[0.01405, 0.16312]	[0.01254, 0.14564]
Almanacs	[0.02476, 0.28748]	[0.04887, 0.56743]
Tableaus for sales agents	[0.04887, 0.56743]	[0.02476, 0.28748]
Greeting cards	[0.01762, 0.20462]	[0.0112, 0.13004]
On wall almanacs	[0.02211, 0.25668]	[0.01, 0.11611]
Iconic model of plants	[0.01574, 0.1827]	[0.03896, 0.45235]

Table 3 Results from Model (8)

Media (DMU)	d *	Efficiency score $(\theta_o^{U^*})$
Brochures	0	0.112
Catalogues	0	0.091
Directories	0	0.088
Advertisement in books of	0	0.11
specialized fairs
Specialized magazines	0	0.097
Billboards	0	1
Internet	0	0.135
Multimedia CD	0	1
Cheap gifts	0	0.038
Expensive gifts	0	0.139
Overalls	0	0.026
Specialized fairs	0	0.04
Seminar for customers	0	0.045
Plastic sacks	0	0.197
Cloth sacks	0	0.092
Almanacs	0	0.033
Tableaus for sales agents	0	0.404
Greeting cards	0	0.054
On wall almanacs	0	0.064
Iconic model of plants	0	0.155

planner does not know whether this factor is an input or an output. After running the proposed model, he/she understands that the factor is as input. In other word, competitors by virtue of high volume of supplied information to audiences get more information about company and as a result they can recognize weaknesses of SIG. Therefore, media planner concludes that less VS will be better and more VS will be risky for future of the business

of SIG. On the other hand, because of the fierce competition in steel industry of Iran, this result is rational.

Based on definition 1, billboards and multimedia CD have the possibility to be DEA efficient. The remaining media with relative efficiency scores of less than 1 are considered to be inefficient. Therefore, media planner can choose one or more of these efficient media.

5. Concluding remarks

Traditional DEA models require that every factor be assigned an explicit designation specifying whether it is an input or output. However, in various situations it remains that there are factors whose status is flexible. In this paper, the modification to the traditional models of DEA that permits the inclusion of both flexible factors and imprecise data in the analysis was developed. The proposed model was applied for selecting media.

The problem considered in this study is at an initial stage of investigation and further research can be done based on the results of this paper. Some of them are as follows.

Similar research can be repeated for dealing with the case that some of the media are slightly non-homogeneous. One of the assumptions of all the classical models of DEA is the homogeneity of DMUs (media), whereas this assumption in many real applications cannot be generalized. In other words, some inputs and/or outputs are not common for all the DMUs. Therefore, there is a need for a model that deals with these conditions.

In this study, the proposed model has been applied to a problem related to media selection. However, the same models could be applied, with minor modifications, to other problems related to selection of technologies, international markets, personnel and many other problems.

Acknowledgements—The author wishes to thank the anonymous reviewers for their valuable suggestions and comments.

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