Decision making is one of the most complicated administrative processes in management. Over the years, various methods have been designed to simplify the process as well as developing new methods. Since, there are many imprecise concepts all around us that routinely expressed in different terms. In fact, the human brain works with considering various factors and based on inferential thinking and value of sentences that modeling of them with mathematical formulas if not impossible would be a complex task.
Because crisp data are inexpressive to model real life situations Zadeh in 1965, has suggested fuzzy logic that is closer to human thinking and Chen [3] developed the TOPSIS method to fuzzy decision-making situations. The purpose of fuzzy logic as a decision-making technique is to improve decision making process in vague and unclear circumstances. Fuzzy management science, while creating the flexibility in the model, with entering some data such as knowledge, experience and human judgment in the model also offers fully functional responses to it [5] . However, if a decision is not possible for linguistic variables based on fuzzy sets, Interval-valued fuzzy set theory can provide a more detailed modeling. In this paper, interval-valued fuzzy TOPSIS method is proposed to solve MCDM (Multi-Criteria Decision Making) problems, where the weight of the criterias are unequal [ 2 , 6 , 7 , 10 , 11 , 12 ].
As mentioned, this method was developed by Hwang and Yoon (1981) in which the best alternative should have the shortest distance from an ideal solution and the worst alternative is the furthest from an ideal solution [2, 6].
Assume a multi criteria decision making problem has Normalize the decision matrix using the following evolution for each
Multiply the columns of the normalized decision matrix by the connected weights. The weighted and normalized decision matrix is come as:
specify the ideal and negative ideal alternatives respectively as follows:
With using of the two Euclidean distances to calculate the distance of the existing alternatives from ideal and negative ideal alternatives as:
The relevant closeness to the ideal alternatives can be defined as:
According to the relative closeness to the ideal alternatives rank the alternatives the bigger
Since the theory of fuzzy sets by Zadeh can be used in vague and imprecise terms, many studies, have developed TOPSIS method in the interval-fuzzy environment.Because of the complexity of the socio-economic environment in many practical decision problems that option often would arrange shady by decision-makers [8, 9]. An interval-valued fuzzy set
Figure 1 Shows the value of membership at
Here are two interval-valued fuzzy numbers
The Normalized Euclidean distance between
A standard MCDM (Multi-Criteria Decision Making) problem can be briefly demonstrated in a decision matrix that Definition of linguistic variables for the ratings Definition of linguistic variables for the importance of each criterionVery Poor (VP) [(0,0);0;(1,1.5)] Poor (P) [(0,0.5);1;(2.5,3.5)] Moderately Poor (MP) [(0,1.5);3;(4.5,5.5)] Fair (F) [(2.5,3.5),5,(6.5,7.5)] Moderately Good (MG) [(4.5,5.5),7,(8.9.5)] Good (G) [(5.5,7.5),9,(9.5,10)] Very Good (VG) [(8.5,9.5),10,(10,10)] Very low (VL) [(0,0);0;(0.1,0.15)] Low (L) [(0,0.05);0.1;(0.25,0.35)] Medium low (ML) [(0,0.15);0.3;(0.45,0.55)] Medium (M) [(0.25,0.35),0.5,(0.65,0.75)] Medium high (MH) [(0.45,0.55),0.7,(0,8,0.95)] High (H) [(0.55,0.75),0.9,(0.95,1)] Very high (VH) [(0.85,0.95),1,(1,1)]
Suppose that
Here
Therefore, the normalized matrix
Here the suggested technique for building up the TOPSIS to interval-valued fuzzy TOPSIS can be described as follows:
Make the weighted normalized fuzzy decree matrix with notice that each criterias has own importance as: Defined the optimal and negative optimal solution as:
Normalized Euclidean distance can be figured out using Definition 3.2 as follows:
Hence, we can calculate distance from the ideal alternative for each alternative as follows:
As the same way, calculate gap of the negative ideal solution by:
Eqs. (12) and (13) are used to specify the distance from ideal and negative ideal alternatives in interval values.
The involved sepreation can be calculated by:
The latest worths of
Suppose aninvestment corporation plans to allocate its limited resources toinvest on four projectsaccording toimportance and profitability of each project respectively. In thispaper, a model is presented for prioritizing investments in various industrial fields. In this case, committee of company’s decision makers intend to evaluate and ultimately rank the possible company’s investment options. Desired options for investment are given in the following table.
Firstly, criteria and sub-criteria were determined by applying the strategic documents of company. There are three main criteria as: “Industrial efficiency”, “Compliance with company’s strategy” and “The campany’s industrial experience”. The hierarchy of criteria and sub-criteria shows in Table 4.
Desired options for investment
Code | Title |
---|---|
A1 | Project 1 |
A2 | Project 2 |
A3 | Project 3 |
A4 | Project 4 |
The hierarchy of criteria and sub-criteria
C1-Industrial efficiency | C2-Compliance withcompany’sstrategy | C3-Industrial experience | |||
---|---|---|---|---|---|
C11 | Increasing demand for industrial products (P) | C21 | Ability to attract foreign investors (P) | C31 | Receivables in the subordinate (C) |
C12 | Alternative Products (C) | C22 | Entrepreneurship (P) | C32 | Implementation process of industrial projects (P) |
C13 | government intervention in product pricing (P) | C23 | Technology transfer Capacity (P) | ||
C14 | Current value of the industry on the exchange (P) | C24 | Ability to reduce dependency on foreign products (P) | ||
C15 | Average process for the delivery of industrial projects (C) | C25 | Amount of dependency on foreign raw material (C) | ||
C26 | Exportamount (P) |
After weighing the basic criteria by decision makers separately and unaware each other based on the target, then decision matrix is created by specified linguistic variables.
As already mentioned, each linguistic variable has an interval fuzzy value. Table 6. gives these values as. So, the final decision matrix is given in Tables 7 with interval fuzzy numbers.
In this step, decision matrix is normalized by the equation (8) and the results are expressed in Tables 8.
As stated earlier, the weight of each criteria was previously determined by the decision makers (Shannon entropy) as given in Tables 9.
Now we can make the weighted normalized fuzzy decision matrix by using the Eq. (9) given that each criterion has different importance. As in Table 10.
By using Eq. (11,12,13) the distance of each alternative is calculated from the ideal alternative
At this step, the fuzzy relative closeness of each alternative is calculated by using the respective distinctions of each pair and the results are given in Table (11).
In the last step alternatives are listed in Table (12) according to their relative closeness.
Now calculate 10 4.83 5.5 10 4.83 10 9.83 10 10 5.5 9.83 5.5 10
Decision matrix according to linguistic variables
C11 | C12 | C13 | C14 | C15 | C21 | C22 | C23 | C24 | C25 | C26 | C31 | C32 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A1 | MG | P | MP | G | MP | MG | MG | M | M | MP | M | MP | MG |
A2 | VG | VP | P | MG | VP | G | G | G | G | P | MG | P | VG |
A3 | P | M | M | MP | P | M | MP | P | VP | MP | MP | M | M |
A4 | M | MP | MP | MG | P | MG | M | P | MG | MP | MG | MP | M |
Interval fuzzy value of linguistic variables
[(3.83,4.83);6.33;(7.5,8.83)] | VP |
[(4.5,5.5);6.67;(7.67,8.33)] | P |
[(5.17,6.17);7.33;(8.17,9)] | MP |
[(6.17,7.5);8.67;(,9.179.83)] | M |
[(7.17,8.17);9;(9.33,9.83)] | MG |
[(7.5,8.83);9.67;(9.83,10)] | G |
[(8.5,9.5);10;(10,10)] | VG |
Interval valued fuzzy decision matrix
C11 | C12 | C13 | C14 | C15 | |
A1 | [(7.17,8.17);9;(9.33,9.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(5.17,6.17);7.33;(8.17,9)] | [(7.5,8.83);9.67;(9.83,10)] | [(5.17,6.17);7.33;(8.17,9)] |
A2 | [(8.5,9.5);10;(10,10)] | [(3.83,4.83);6.33;(7.5,8.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(7.17,8.17);9;(9.33,9.83)] | [(3.83,4.83);6.33;(7.5,8.83)] |
A3 | [(4.5,5.5);6.67;(7.67,8.33)] | [(6.17,7.5);8.67;(,9.179.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(4.5,5.5);6.67;(7.67,8.33)] |
A4 | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(5.17,6.17);7.33;(8.17,9)] | [(7.17,8.17);9;(9.33,9.83)] | [(4.5,5.5);6.67;(7.67,8.33)] |
C21 | C22 | C23 | C24 | C25 | |
A1 | [(7.17,8.17);9;(9.33,9.83)] | [(7.17,8.17);9;(9.33,9.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] |
A2 | [(8.5,9.5);10;(10,10)] | [(6.17,7.5);8.67;(,9.179.83)] | [(8.5,9.5);10;(10,10)] | [(7.5,8.83);9.67;(9.83,10)] | [(4.5,5.5);6.67;(7.67,8.33)] |
A3 | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(3.83,4.83);6.33;(7.5,8.83)] | [(5.17,6.17);7.33;(8.17,9)] |
A4 | [(7.17,8.17);9;(9.33,9.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(7.17,8.17);9;(9.33,9.83)] | [(5.17,6.17);7.33;(8.17,9)] |
C26 | C31 | C32 | |||
A1 | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(7.17,8.17);9;(9.33,9.83)] | ||
A2 | [(7.17,8.17);9;(9.33,9.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(8.5,9.5);10;(10,10)] | ||
A3 | [(5.17,6.17);7.33;(8.17,9)] | [(6.17,7.5);8.67;(,9.179.83)] | [(6.17,7.5);8.67;(,9.179.83)] | ||
A4 | [(7.17,8.17);9;(9.33,9.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(6.17,7.5);8.67;(,9.179.83)] |
Normalize Decision Matrix
C11 | C12 | C13 | C14 | C15 | |
A1 | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.88,1.07);0.72;(0.58,0.63)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.75,0.88);0.97;(0.98,1)] | [(0.78,0.93);0.66;(0.54,0.59)] |
A2 | [(0.85,0.95);1;(1,1)] | [(1,1.3);0.76;(0.55,0.64)] | [(1,1.22);0.82;(0.62,0.72)] | [(0.72,0.82);0.9;(0.93,0.98)] | [(1,1.3);0.76;(0.55,0.64)] |
A3 | [(0.45,0.55);0.67;(0.77,0.83)] | [(0.64,0.78);0.56;(0.49,0.53)] | [(0.73,0.89);0.63;(0.56,0.6)] | [(0.52,0.62);0.7;(0.82,0.9)] | [(0.88,1.07);0.72;(0.58,0.63)] |
A4 | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.78,0.93);0.66;(0.54,0.59)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.88,1.07);0.72;(0.58,0.63)] |
C21 | C22 | C23 | C24 | C25 | |
A1 | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.73,0.83);0.92;(0.95,0.95)] | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.89,1.06);0.75;(0.61,0.67)] |
A2 | [(0.85,0.95);1;(1,1)] | [(0.63,0.76);0.88;(0.93,1)] | [(0.85,0.95);1;(1,1)] | [(0.75,0.88);0.97;(0.98,1)] | [(1,1.22);0.82;(0.62,0.72)] |
A3 | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.53,0.63);0.75;(0.83,0.92)] | [(0.45,0.55);0.67;(0.77,0.83)] | [(0.38,0.48);0.63;(0.75,0.88)] | [(0.89,1.06);0.75;(0.61,0.67)] |
A4 | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.63,0.76);0.88;(0.93,1)] | [(0.45,0.55);0.67;(0.77,0.83)] | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.89,1.06);0.75;(0.61,0.67)] |
C26 | C31 | C32 | |||
A1 | [(0.63,0.76);0.88;(0.93,1)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.72,0.82);0.9;(0.93,0.98)] | ||
A2 | [(0.73,0.83);0.92;(0.95,0.95)] | [(1,1.22);0.82;(0.62,0.72)] | [(0.85,0.95);1;(1,1)] | ||
A3 | [(0.53,0.63);0.75;(0.83,0.92)] | [(0.73,0.89);0.63;(0.56,0.6)] | [(0.62,0.75);0.73;(0.82,0.9)] | ||
A4 | [(0.73,0.83);0.92;(0.95,0.95)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.62,0.75);0.73;(0.82,0.9)] |
Weight values of criteria
[(0.85,0.95);1;(1,1)] | VH |
[(0.55,0.75);0.9;(0.95,1)] | H |
[(0.45,0.55);0.7;(0.8,0.95)] | MH |
[(0.25,0.35);0.5;(0.65,0.75)] | M |
[(0,0.15);0.3;(0.45,0.55)] | ML |
[(0,0.05);0.1;(0.25,0.35)] | L |
[(0,0);0;(0.1,0.15)] | VL |
Weight of criterias
C11 | VH | C21 | L | C31 | M |
C12 | H | C22 | ML | C32 | ML |
C13 | H | C23 | M | ||
C14 | L | C24 | ML | ||
C15 | MH | C25 | VL | ||
C26 | M |
Weighted normalize fuzzy decision matrix
C11 | C12 | C13 | C14 | C15 | |
A1 | [(0.61,0.78);0.9;(0.93,0.98)] | [(0.48,0.80);0.65;(0.55,0.63)] | [(0.49,0.8);0.68;(0.58,0.67)] | [(0,0.04);0.09;(0.25,0.35)] | [(0.35,0.51);0.46;(0.43,0.56)] |
A2 | [(0.72,0.9);1;(1,1)] | [(0.55,0.98);0.68;(0.52,0.64)] | [(0.55,0.92);0.74;(0.59,0.72)] | [(0,0.04);0.09;(0.23,0.34)] | [(0.45,0.72);0.53;(0.44,0.61)] |
A3 | [(0.38,0.52);0.67;(0.77,0.83)] | [(0.35,0.59);0.50;(0.47,0.53)] | [(0.40,0.67);0.57;(0.53,0.6)] | [(0,0.03);0.07;(0.21,0.32)] | [(0.4,0.6);0.5;(0.46,0.6)] |
A4 | [(0.53,0.71);0.73;(0.82,0.9)] | [(0.43,0.70);0.59;(0.51,0.59)] | [(0.49,0.8);0.68;(0.58,0.67)] | [(0,0.04);0.09;(0.23,0.34)] | [(0.4,0.6);0.5;(0.46,0.6)] |
C21 | C22 | C23 | C24 | C25 | |
A1 | [(0,0.04);0.09;(0.23,0.35)] | [(0,0.12);0.28;(0.43,0.52)] | [(0.16,0.26);0.37;(0.53,0.68)] | [(0,0.11);0.22;(0.37,0.5)] | [(0,0);0;(0.06,0.1)] |
A2 | [(0,0.05);0.1;(0.25,0.35)] | [(0,0.11);0.26;(0.42,0.55)] | [(0.21,0.33);0.5;(0.65,0.75)] | [(0,0.13);0.29;(0.44,0.55)] | [(0,0);0;(0.06,0.11)] |
A3 | [(0,0.04);0.07;(0.21,0.32)] | [(0,0.08);0.23;(0.37,0.51)] | [(0.11,0.19);0.34;(0.5,0.62)] | [(0,0.07);0.19;(0.34,0.48)] | [(0,0);0;(0.06,0.1)] |
A4 | [(0,0.04);0.09;(0.23,0.34)] | [(0,0.11);0.26;(0.42,0.55)] | [(0.11,0.19);0.34;(0.5,0.62)] | [(0,0.12);0.27;(0.42,0.54)] | [(0,0);0;(0.06,0.1)] |
C26 | C31 | C32 | |||
A1 | [(0.16,0.27);0.44;(0.6,0.75)] | [(0.22,0.37);0.38;(0.4,0.5)] | [(0,0.12);0.27;(0.42,0.54)] | ||
A2 | [(0.18,0.29);0.46;(0.62,0.71)] | [(1,0.43);0.41;(0.4,0.54)] | [(0,0.14);0.3;(0.45,0.55)] | ||
A3 | [(0.13,0.22);0.38;(0.54,0.69)] | [(0.18,0.31);0.32;(0.36,0.45)] | [(0,0.11);0.22;(0.37,0.5)] | ||
A4 | [(0.18,0.29);0.46;(0.62,0.71)] | [(0.22,0.37);0.38;(0.4,0.5)] | [(0,0.11);0.22;(0.37,0.5)] |
Distance of alternatives from ideal alternatives
A1 | A2 | ||||||||
---|---|---|---|---|---|---|---|---|---|
C11 | 0.234521 | 0.14 | 0.826035 | 0.890468 | C11 | 0.161658 | 0.057735 | 0.916224 | 0.967815 |
C12 | 0.443471 | 0.31459 | 0.564329 | 0.697472 | C12 | 0.422414 | 0.278328 | 0.587452 | 0.781537 |
C13 | 0.424264 | 0.288039 | 0.588473 | 0.719097 | C13 | 0.382187 | 0.225389 | 0.631981 | 0.798415 |
C14 | 0.892562 | 0.851293 | 0.153406 | 0.209921 | C14 | 0.898332 | 0.853483 | 0.142595 | 0.204369 |
C15 | 0.585947 | 0.491155 | 0.415933 | 0.511631 | C15 | 0.528205 | 0.3879 | 0.475044 | 0.624873 |
C21 | 0.898146 | 0.851293 | 0.142595 | 0.209921 | C21 | 0.889288 | 0.843603 | 0.155456 | 0.212132 |
C22 | 0.783156 | 0.712928 | 0.296254 | 0.312463 | C22 | 0.792465 | 0.716984 | 0.28519 | 0.356931 |
C23 | 0.665833 | 0.591608 | 0.384448 | 0.471487 | C23 | 0.576368 | 0.503786 | 0.48874 | 0.554196 |
C24 | 0.816497 | 0.742092 | 0.248529 | 0.321714 | C24 | 0.778396 | 0.698451 | 0.304248 | 0.366742 |
C25 | 0.791623 | 0.776745 | 0.034641 | 0.057735 | C25 | 0.980408 | 0.964728 | 0.034641 | 0.063509 |
C26 | 0.627163 | 0.549363 | 0.439394 | 0.525674 | C26 | 0.607838 | 0.541541 | 0.457675 | 0.516333 |
C31 | 0.67082 | 0.585064 | 0.342929 | 0.420833 | C31 | 0.485833 | 0.543016 | 0.665357 | 0.463537 |
C32 | 0.789515 | 0.710868 | 0.288271 | 0.355387 | C32 | 0.772981 | 0.690917 | 0.31225 | 0.37063 |
Sum | 8.623517 | 7.605039 | 4.725234 | 5.703802 | Sum | 8.276371 | 7.305862 | 5.456853 | 6.281021 |
A3 | A4 | ||||||||
---|---|---|---|---|---|---|---|---|---|
C11 | 0.426693 | 0.350333 | 0.628808 | 0.685128 | C11 | 0.329747 | 0.235938 | 0.703847 | 0.784644 |
C12 | 0.563738 | 0.461519 | 0.444747 | 0.541295 | C12 | 0.494335 | 0.376917 | 0.514166 | 0.628808 |
C13 | 0.505239 | 0.38893 | 0.505239 | 0.614763 | C13 | 0.423832 | 0.289425 | 0.588473 | 0.719097 |
C14 | 0.91086 | 0.869521 | 0.127802 | 0.189912 | C14 | 0.898332 | 0.853483 | 0.142595 | 0.204369 |
C15 | 0.548209 | 0.43589 | 0.455192 | 0.568624 | C15 | 0.528205 | 0.3879 | 0.475044 | 0.624873 |
C21 | 0.91086 | 0.865814 | 0.127802 | 0.190526 | C21 | 0.898332 | 0.853483 | 0.142595 | 0.204369 |
C22 | 0.814412 | 0.748198 | 0.251529 | 0.326292 | C22 | 0.792465 | 0.716984 | 0.28519 | 0.356931 |
C23 | 0.701831 | 0.641898 | 0.354824 | 0.422729 | C23 | 0.701831 | 0.641898 | 0.354824 | 0.422729 |
C24 | 0.835005 | 0.772744 | 0.22487 | 0.300777 | C24 | 0.789367 | 0.711548 | 0.288271 | 0.355387 |
C25 | 0.980408 | 0.967815 | 0.034641 | 0.057735 | C25 | 0.848528 | 0.967815 | 0.34641 | 0.057735 |
C26 | 0.671541 | 0.602467 | 0.388544 | 0.472193 | C26 | 0.607838 | 0.541541 | 0.457675 | 0.516333 |
C31 | 0.717496 | 0.643169 | 0.296873 | 0.365605 | C31 | 0.671516 | 0.586316 | 0.342929 | 0.420833 |
C32 | 0.817578 | 0.741732 | 0.248529 | 0.321714 | C32 | 0.817578 | 0.741732 | 0.248529 | 0.321714 |
Sum | 9.403869 | 8.490031 | 4.089401 | 5.057293 | Sum | 8.801903 | 7.904981 | 4.890547 | 5.617823 |
The final ranking of Options
RC1 | RC2 | RC* | RANK | |
---|---|---|---|---|
A1 | 0.428572 | 0.353983 | 0.391278 | 2 |
A2 | 0.462286 | 0.602653 | 0.532469 | 1 |
A3 | 0.373306 | 0.30307 | 0.338188 | 4 |
A4 | 0.415433 | 0.357171 | 0.386302 | 3 |
The increasing complexity of socio-economic communities causes the intricacy and ambiguity in the priorities of decision-makers; because decision-making is often done in some circumstances such as lack of information and knowledge, lack of decision-makers consensus, time limits. . . So, in such situation, Decision-making in an interval-valued fuzzy environment would be convenient. The main characteristic of using interval-valued fuzzy environment is that the membership functions would be an interval rather than an exact number. In fuzzy set theory, it is difficult to express a thought or linguistic variables entirely by an integer number in [0, 1]. Thus, expressing degree of certainty by an interval of [0, 1] would be more appropriate. It’s worth paying attention, the use of interval valuation numbers gives an occasion to proficients to define lower and upper bounds values as an interval for matrix elements and weights of criteria.