Multiple attribute decision making (MADM) refers to making preference decisions (e.g., evaluation, prioritization, and selection) over the available alternatives that are characterized by multiple, usually conflicting, attributes. The structure of the alternative performance matrix is depicted in

Table 1, where x

_{ij} is the rating of alternative i with respect to criterion j and w

_{j} is the weight of criterion j (in this paper, we consider the case that the rating of alternative i with respect to criterion j is non-negative).

Since each criterion has a different meaning, it cannot be assumed that they all have equal weights, and as a result, finding the appropriate weight for each criterion is one the main points in MADM. Various methods for finding weights can be found in the literature and most of them can be categorized into two groups: subjective and objective weights. Subjective weights are determined only according to the preference decision makers. The AHP method [

1], weighted least squares method [

2] and Delphi method [

3] belong in this category. The objective methods determine weights by solving mathematical models without any consideration of the decision maker’s preferences, for example, the entropy method, multiple objective programming [

4,

5], principal element analysis [

5],

etc. Since in the most real problems, the decision maker’s expertise and judgment should be taken into account, subjective weighting may be preferable, but when obtaining such reliable subjective weights is difficult, the use of objective weights is useful. One of the objective weighting measures which has been proposed by researchers is the Shannon entropy concept [

6]. Entropy concept was used in various scientific fields. The concept of Shannon’s entropy has an important role in information theory and is used to refer to a general measure of uncertainty. In transportation models, entropy is acted as a measure of dispersal of trips between origin and destinations [

7]. In physics, the word entropy has important physical implications as the amount of “disorder” of a system [

7]. Also the entropy associated with an event is a measure of the degree of randomness in the event. Entropy has also been concerned as a measure of fuzziness [

8]. In MADM the greater the value of the entropy corresponding to an special attribute, which imply the smaller attribute’s weight, the less the discriminate power of that attribute in decision making process.

In many real life problems, the data of the decision making processes cannot be measured precisely and there may be some other types of data, for instance interval data and fuzzy data. In other words, the decision maker would prefer to say his/her point of view in these forms rather than a real number because of the uncertainty and the lack of certain data, especially when data are known to lie within bounded variables, or when facing missing data, judgment data,

etc. In MADM it is most probable that we confront such a case, so finding a suitable weight is an important problem. It is logical that when data are imprecise, weights be imprecise too. In this paper we present a method for solving MADM problems by entropy method consisting of interval data. In this method the weight, which is obtained for each alternative, will be an interval number. We apply the Sengupta approach mentioned in [

9] to compare the interval scores we have found.

This paper has been organized as follows: In

Section 2 the MADM problem is presented with interval data. Then Entropy method is extended for the interval data. In the same section we will also show that if all of the alternatives have deterministic data, then the interval entropy weight leads to the usual entropy weight. In

Section 3, by using α-level set, we will obtain interval weight for fuzzy MADM problem in different levels of confidence. We will also use the data of an empirical example for more explanation and showing the validation of the proposed method. The final section will be the conclusion.