A great deal of uncertain information which is difficult to quantify is taken into account by farmers and experts in the enterprise when making decisions. We are interested in the problems of the implementation of a rabbit-breeding farm. One of the first decisions to be taken refers to the design or type of structure for housing the animals, which is determined by the level of environmental control sought to be maintained in its interior. A farmer was consulted, and his answers were incorporated into the analysis, by means of the fuzzy TOPSIS methodology. The main purpose of this paper is to study the problem by means of the fuzzy TOPSIS method as multicriteria decision making, when the information was given in linguistic terms.
In rural scenarios, the quantitative information to assess in decision making is usually very poor, or difficult to obtain. The factors to be taking into account present a similar pattern, but the data show clear differences between areas or countries. In this way, many recent works have demonstrated the advantages of the implementation of qualitative information to work in rural farms [
The rabbit farmer who wishes to start in the production of rabbits must take a series of decisions, including decisions relating to the environmental comfort (temperature, humidity, ventilation, and illumination) of the animals. The climate zone in which the farm is to be located will condition the environmental control systems that need to be installed in the building to ensure adequate comfort for the animals. In Mediterranean climates, the principal parameter that determines the type of building is the level of temperature control sought, with an excess of heat being much more of a problem than an excess of cold. Thus, we can find open buildings, with open sides, or buildings with the sides closed. The environmental control of the latter is greater. Closed buildings, with more technology, suppose a greater investment cost but generally have higher productions. On the other hand, open buildings present a lower incidence of pathologies since they have greater air renewal although the rabbits may need more looking after as the comfort of the animals cannot be guaranteed. Thus, special care should be taken in choosing the type of building and aspects need to be considered which include not only economic criteria for the implementation, but also aspects related to handling, productivity, and animal welfare.
This type of problem presents the characteristics to be resolved by means of tools to help in decision making, more specifically using the multicriteria method. This type of methods may require consulting experts who can enrich the decision with their judgments regarding the problem. The most common drawback of existing linguistic multi-criteria methods, at least for some classes of problems, is the need to translate the decision makers’ knowledge about a decision problem into numbers and functions. There are decision problems in which qualitative judgments prevail over more or less exact quantitative evaluation. For such problems, it is a somewhat natural choice to use models which incorporate qualitative (descriptive, linguistic, and ordinal) variables. Practical problems are often characterized by several no commensurable and conflicting criteria, and there may be no solution which satisfies all the criteria simultaneously [
The technique for order performance by similarity to ideal solution (TOPSIS), one of the well-known classical multi-criteria decision making (MCDM) methods, was first developed in [
Only a limited number of works have been carried out in recent years about decision making using fuzzy numbers in agriculture [
The purpose of this paper is to contribute to the rabbit farming sector as a MCDM problem in a fuzzy environment, to model the problem of the choice of the best type of structure for a rabbit farm, by means of the fuzzy TOPSIS method. The TOPSIS method is used since the decision maker includes both linguistic as well as numerical data in his assessment.
This paper is organized as follows: the following section introduces the methodology in detail. The linguistic variables and the fuzzy sets are described as is the TOPSIS method, to be used later. Section
Any multi-criteria decision problem (MCDP) may be expressed by means of the following five elements, Once that set of criteria and alternatives have been selected, then a measure of the effect produced by each alternative with respect to each criterion is needed, By means of linguistic terms, the decision-maker represents the goodness of an alternative with respect to a criterion; the different values of There is a preference relation Certain information about the criteria in this case is also linguistic. The decision-maker provides us linguistic information about the importance of each criterion.
Since Zadeh [
By a
A linguistic variable is characterized by a quintuple
In the present case, the linguistic variable is identified with a fuzzy set [
A real triangular fuzzy number (TFN)
where
Unless otherwise specified, it is assumed that
Representation of a triangular fuzzy number.
The TOPSIS approach is an MCDM for the arrangement of preference to an ideal solution by similarity, which was developed by Hwang and Yoon [
The fuzzy TOPSIS methods are derived from the generic TOPSIS method with minor differences, with the pertinent adaptation of the operations associated to the linguistic labels.
The fuzzy TOPSIS procedure consists of the following steps
The construction of the decision matrix is very important when we are interested in obtaining the best alternatives. The outcome of the decision matrix can be one of three types: only deterministic values (numerical values); only linguistic values; or mixed values, as in our case. Conventional MCDM methods require only precise values for a finite set of alternatives. However, true multiple-criteria decision making environments consist of both imprecise and precise values. Therefore, if all or some of the criteria of an alternative are uncertain (or imprecise) or if the given criteria have subjective characteristics, the use of a fuzzy set theory is a reasonable means of resolution. Some examples can be seen in [
The structure of the matrix can be expressed in Table
where
Let
Decision matrix.
C1 | C2 | ||||
A1 | |||||
A2 | |||||
In the classical TOPSIS, the normalized performance matrix can be obtained using the following transformation formula:
The weighted normalized value
The positive ideal value set
Since the problem is to optimize, in this case, the benefit criteria will be that for which the best option corresponds to the smallest value of labels being the contrary for the cost criteria.
The separation of each alternative from the PIS
The relative closeness
Rank the best alternatives according to
In the initial stage of obtaining information, the steps taken are as follows.
In Mediterranean climates, the construction design for housing rabbits offers several alternatives according to the environmental comfort sought for the interior. The different types of houses are the alternatives: closed buildings with a totally controlled environment open buildings with a semicontrolled environment, buildings without closed sides, open buildings without a controlled environment: without closed sides and without environmental control mechanisms.
Each type of building has advantages and disadvantages; closed buildings require a greater investment and have higher maintenance costs for the environmental control systems, but production levels are greater.
Implementation Costs: They include all the costs derived from the construction of the civil engineering and the installations needed to carry out the activity. They depend on the type of structure of the building (open or closed) and the level of technology of the installations.
Running Costs: They include all the payments which are included as such in the cash flow (labour, food, electricity, water, repairs to equipment, medicines, etc.)
Production: It is measured as kilos of meat produced weekly per 100 reproductive females
Mortality in Lactation: Percentage of kits dead in the breeding period (from birth to 35 days after birth)
Mortality in Fattening: It is the percentage of dead rabbits from weaning until the moment of slaughtering (approximately 2 kg of live weight and 65 days of age).
The criteria (C1), (C2) are clearly cost criteria. (C4) and (C5) are criteria which when it comes to optimising are considered as cost criteria, since they indicate mortality, and the greater they are then the worse the criterion is. For this reason, in this case, the only criterion which indicates profit is (C3).
The problems arise from the fact that the information will be both quantitative and qualitative; in these conditions, not all decision models are appropriate, for example, the simple average weight, so a good method is the TOPSIS approach, as it is based on normalization and makes the valuations adimensional.
To obtain the data, a questionnaire has been designed at two levels, one in which the questions are asked to obtain the importance the expert gives to each criterion; this is shown in Table
Linguistic variables for the importance weight of each type 1 criterion and for the ratings of the type 2 alternatives.
Type 1 linguistic label | Type 2 linguistic label | ||||
Label | Description | Fuzzy number | Label | Description | Fuzzy number |
vL | Very low | vG | Very good | ||
L | Low | G | Good | ||
mL | Medium low | mG | Medium good | ||
M | Medium | m | Medium | ||
mH | Medium high | mB | Medium bad | ||
H | High | B | Bad | ||
vH | Very high | vB | Very bad |
To quantify the importance of the criteria, the labels used were importance labels {VL, L, mL, M, mH, H, vH} whose value ranged from the lowest to the highest importance. However, for the rating of the alternatives, given that we are faced with an optimization problem, the linguistics labels used in the questionnaire were labels to define goodness {vG, G, mG, m, mB, B, VB}
In this way, for the expert, the criteria (C1), (C4), and (C5) are the most important with a high importance, whilst (C2) and (C3) have a valuation of medium good. These normalized values can be seen in Table
Importance weight of criteria.
Fuzzy numbers | Normalization | ||||
---|---|---|---|---|---|
C1 | High | (0.700,0.900,1.000) | (0.146,0.220,0.323) | ||
C2 | Medium high | (0.500,0.700,0.900) | (0.104,0.171,0.290) | ||
C3 | Medium high | (0.500,0.700,0.900) | (0.104,0.171,0.290) | ||
C4 | High | (0.700,0.900,1.000) | (0.146,0.220,0.323) | ||
C5 | High | (0.700,0.900,1.000) | (0.146,0.220,0.323) | ||
SUM | (3.100,4.100,4.800) | 1.000 |
For the evaluation of the alternatives for each criterion, as has been indicated, the type 2 labels have been used (Table
Ratings of the expert under the various criteria by linguistic labels and numerical information (percentage ± SD).
C1 | C2 | C3 | C4 | C5 | |
---|---|---|---|---|---|
A1 | vB | vB | vB | 12% ± 1% | 5% ± 1% |
A2 | m | B | B | 14% ± 1% | 8% ± 1% |
A3 | m | m | m | 16% ± 1% | 10% ± 1% |
Note in Table
We have supposed that quantitative and qualitative information is available from the expert.
With regard to the choice of criteria, it should be highlighted that with (C3), (C4), and (C5), we have to take into account both the income that a greater or lower rabbit meat production supposes as well as the cost that is generated by the mortality of animals being fattened, in terms of labour and food costs and that later this does not become income. It can be supposed that the mortality criteria ((C4) and (C5)) are negatively correlated with those of production (C3); however, it is possible to have a low production with low mortality because the number of births is less.
Once Steps
Calculation of the normalized alternatives, weighted alternatives, and
C1 | C2 | C3 | C4 | C5 | |
---|---|---|---|---|---|
(0.146,0.220,0.323) | (0.104,0.171,0.290) | (0.104,0.171,0.290) | (0.146,0.220,0.323) | (0.146,0.220,0.323) | |
A1 | (9.000,10.000,10.000) | (9.000,10.000,10.000) | (0.000,0.000,1.000) | (11.000,12.000,13.000) | (4.000,5.000,6.000) |
A2 | (3.000,5.000, 7.000) | (5.000,7.000,9.000) | (0.000,1.000,3.000) | (13.000,14.000,15.000) | (7.000,8.000,9.000) |
A3 | (1.000,3.000, 5.000) | (3.000,5.000,7.000) | (3.000,5.000,7.000) | (15.000,16.000,17.000) | (9.000,10.000,11.000) |
(0.682,0.864, 1.048) | (0.593,0.758, 0.933) | (0.000,0.000,0.333) | (0.421,0.492,0.573) | (0.259,0.364,0.497) | |
(0.227,0.432, 0.7349 | (0.330,0.531,0.839) | (0.000,0.196,1.000) | (0.497,0.573,0.661) | (0.454,0.582,0.745) | |
(0.076,0.259, 0.524) | (0.198,0.379, 0.653) | (0.391,0.981,2.333) | (0.574,0.655,0.749) | (0.583,0.727,0.910) | |
(0.100,0.190, 0.338) | (0.062,0.129, 0.271) | (0.000,0.000,0.097) | (0.061,0.108,0.185) | (0.038,0.080,0.160) | |
(0.033,0.095, 0.237) | (0.034,0.091,0.244) | (0.000,0.033,0.290) | (0.073,0.126,0.213) | (0.066,0.128,0.240) | |
(0.011,0.057, 0.169) | (0.021,0.065, 0.190) | (0.041,0.167,0.677) | (0.084,0.144,0.242) | (0.085,0.160,0.294) | |
(0.011,0.057, 0.169) | (0.021,0.065,0.190) | (0.000,0.000,0.097) | (0.061,0.108,0.185) | (0.038,0.080, 0.160) | |
(0.100,0.190, 0.338) | (0.062,0.129, 0.271) | (0.041,0.167,0.677) | (0.084,0.144,0.242) | (0.085,0.160,0.294) |
Calculation of
A1 | (0.098,0.148,0.188) | |
A2 | (0.040,0.076,0.228) | |
A3 | (0.066,0.189,0.598) | |
A1 | (0.066,0.189,0.598) | |
A2 | (0.085,0.173,0.406) | |
A3 | (0.098,0.148,0.188) |
In this way, the defuzzified values for each alternative would be calculated, for example
Similarly, we would do the same for
Results.
A1 | 0.146 | 0.237 | 0.619 |
A2 | 0.096 | 0.197 | 0.673 |
A3 | 0.237 | 0.146 | 0.381 |
In Table
Scheme of the decision process for the election of the best type of building for a rabbit-breeding farm.
The difference between Table
Representation of the values corresponding to (
Representation of the values corresponding to
The fact of using the TOPSIS methodology rather than other methods means that the data analysed correspond both to numerical values and linguistic labels, and it therefore becomes necessary to use different measurement scales. This is the case, and therefore, the methodology is very suitable because it allows to analyse the values presented in a clear and concise manner.
Other methodologies [
Today many enterprises use decision making tools to help with their decisions. In rural scenarios, where many important decisions must be taken, these tools may be easily implemented and used by governments and/or farmers. A fuzzy model has great potential as a valuable tool in evaluating such decisions owing to the uncertainty and difficulty in finding quantitative information in some aspects involving this sector. In the illustrative example presented, the problem is affected by many factors which may offer only imprecise and uncertain data. Therefore, a methodology based on fuzzy TOPSIS was developed to resolve a problem of rabbit farm management. The example demonstrates the power of this method to identify preferred options from a given combination of quantitative and qualitative information.
So, the expert’s preferred option for the construction design for housing the rabbits in the southeast of Spain is
This work is partially supported by the DGICYT under Project no. TIN2008-06872-C04-04, FEDER funds and Junta de Andalucía (P07-TIC02970), and the Fundación Séneca (Project no. 8824/PPC/08).