A compensatory model for computing with words under discrete labels and incomplete information

Knowledge-Based Systems 27 (2012) 29–37

Contents lists available at SciVerse ScienceDirect

Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

A compensatory model for computing with words under discrete labels and incomplete information Ozan Çakır ⇑ _ Department of Industrial Engineering, Gediz University Seyrek Köy, Menemen, Izmir 35665, Turkey

a r t i c l e

i n f o

Article history: Received 15 September 2009 Received in revised form 23 September 2011 Accepted 10 October 2011 Available online 20 October 2011 Keywords: Compensatory programming Computing with words Muti-attribute decision making Discrete linguistic labels Linguistic variables Incomplete weight information Information fusion

a b s t r a c t In this paper, we propose a compensatory model for computing with words under discrete linguistic labels and incomplete weight information. This particular model will be useful in the context of multi-attribute decision making problems characterized by discrete linguistic attribute evaluations and partially-known weight information. This group of multi-attribute decision making problems may be modeled as multiobjective programs by using the concept of satisfactory degree, defined for each decision alternative under study. We derive a compensatory program which can be substituted for such multi-objective models. Further, we prove that the optimal solution of this compensatory program is a Pareto solution to the original multi-objective model. To show the working principles of this approach, we illustrate the procedure on two numerical examples from the published literature. We then analyze a concrete example we developed for illustrating the real-life meanings of several model constructs and managerial connotations of the results obtained by using this new approach. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction and related literature Multi-attribute decision making (MADM) problems are comprised of ranking a set of decision alternatives based on their overall performances according to a set of attributes. While analyzing complex MADM problems, decision makers usually have to deal with the problem of managing uncertain, poor and partially-known (i.e. incomplete) data. Moreover, intangible nature of some attributes under study may add complexity to the problem. For example, the values of attributes such as competitiveness, quality, flexibility, and conformance cannot be expressed in numerical scales. Hence, in many MADM problems attribute values may only be available in terms of discrete linguistic labels, called the linguistic variables. A linguistic variable stands for ‘‘an expression in natural or artificial language’’ [27], describing a collection of values. When linguistic variables are employed to typify qualitative phenomena, one needs to compute with words in natural language for solving the master MADM problem. The current paper is not intended to be an exhaustive review of standard procedures for MADM problems, hence we refer the reader to the books [11,28,12,8] and the references therein. Among the MADM methods, the procedures based on computing with words (CWW) under discrete linguistic label sets are being increasingly

⇑ Tel.: +90 232 3550000x2310. E-mail address: [email protected] 0950-7051/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2011.10.006

applied. Let L = {lk :k = K, . . . , 0, . . . , K} denote a totally ordered set of discrete labels lk where k is a label index. These labels are arranged symmetrically around the neutral label l0 with satisfying the following two conditions. Condition 1. If k 6 h, then lk 6 lh where h is another label index. Condition 2. There exists a negation operator m such that m(lk) = lk. Consider a set of n decision alternatives indexed by i and a set of m attributes indexed by j, which are common for all decision alternatives. We may group the procedures for CWW under discrete linguistic label sets into four categories. 1.1. CWW based on the extension principle In this method [5], a set of fuzzy numbers that comply with the semantics of linguistic label set is defined and arithmetic operations are performed on these numbers. Such operations are carried out using fuzzy arithmetic based on the extension principle [7], yet in this case, it is well-known that the resultant fuzzy scores have large supports. Hence, they may not overlap with original linguistic labels. Therefore, a linguistic approximation operator [5] is employed to project the final fuzzy scores onto the original label set. The procedure can be described as follows: OðF;AgÞ

OðL;AxÞ

Lnm ! F ¼ fF i : F i # Rg ! V ¼ fv i : v i 2 fK; . . . ; Kgg

ð1Þ

30

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37

where O(F, Ag) is a fuzzy aggregation operator, F is the set of resultant fuzzy scores Fi, O(L, Ax) is a linguistic approximation operator and V is the set of all overall values vi (i.e. scores) of the decision alternatives.

1.2. CWW based on the symbolic model In the symbolic model [6], computations are performed on the indices of the linguistic labels. Symbolic aggregation operators [6] are employed to obtain a set of intermediate numerical scores. Then, the intermediate scores are approximated with a linguistic approximation operator that returns a numerical value denoting the index of a particular linguistic label from the original label set. The procedure can be described as follows: nm OðS;AgÞ

L

OðL;AxÞ

! S ¼ fSi : Si 2 ½K; Kg ! V ¼ fv i : v i 2 fK; .. .; Kgg

ð2Þ

where O(S, Ag) is a symbolic aggregation operator, S is the set of intermediate scores Si, O(L, Ax) is the linguistic approximation operator that returns an index k 2 {K, . . . , K} for the overall value of each decision alternative.

1.3. CWW based on the 2-tuple representation model Herrera and Martinez [9,10] argued that the above approximation procedures used to represent the results in the original domain of the linguistic label sets lead to a constant loss of information. They developed the 2-tuple representation model [9] where the linguistic information is expressed as 2-tuples containing a linguistic term and a number. Hence, the linguistic information is managed within a continuous domain instead of discrete labels. To illustrate this, consider an intermediate score Si 2 [K, K] for decision alternative i. Let k = round(Si) and ai = Si  k where round() is the rounding operation that returns the closest label index k to Si, and ai 2 [0.5, 0.5) is the deviation from the closest label index, referred to as the value of the symbolic translation [9]. Consider a symbolic translation function

s : ½K; K ! L  ½0:5; 0:5Þ;

ð3Þ

which converts the score to a 2-tuple containing a label and a symbolic translation value. That is to say we have s(Si) = (lk, ai) where k = round(Si) leading to k 2 {K, . . . , K}, and ai = Si  k hence ai 2 [0.5, 0.5). In this method, the original linguistic information is expressed as a set of 2-tuples (lk, 0), and 2-tuple linguistic aggregation operators [9] are employed to derive the overall values of the decision alternatives. Thus, the resultant overall values are 2-tuples as well. The procedure can be described as follows: s

Oð2T;AgÞ

Lnm ! L ¼ f‘ij : ‘ij ¼ ðlk ; 0Þg ! V ¼ fv i : v i ¼ ðlk ; ai Þg

ð4Þ

where s is the symbolic translation function, L is the 2-tuple representation of the original linguistic information and O(2T, Ag) is a 2-tuple linguistic aggregation operator. Herrera and Martinez [9] suggested that the resultant overall values vi in this method are more useful than those derived by using the extension principle and the symbolic model. Moreover, they provided with an example where these two methods return the same linguistic labels for the overall values of a subset of the decision alternatives. Since the resultant overall values vi in the 2-tuple representation model have a label lk and a unique symbolic translation value ai, the decision maker can differentiate between tied alternatives according to their symbolic translation values.

1.4. CWW directly based on optimization models In this method [24–26], the overall values of the decision alternatives are computed directly from the indices of the linguistic labels by employing optimization models. A linguistic negative (positive) ideal sequence of attribute values and a distance function between the overall values of decision alternatives and this ideal sequence are defined. Finally, the maximization (minimization) problem associated with the collective distance function between the overall values of n decision alternatives and overall value of the negative (positive) ideal sequence is modeled as a multi-objective optimization model. The procedure can be described as follows: OPT

Lnm ! V ¼ fv i : v i 2 ½K; Kg

ð5Þ

where OPT is a particular optimization model. The resultant overall values are derived in the continuum of the interval [K, K] defined by the limits of the linguistic label index. In this paper, we propose a compensatory program for CWW under discrete linguistic labels and incomplete weight information, in the context of MADM problems. The procedure we suggest falls into the realm of the fourth category described above. In a recent paper, Xu [24] introduced the concept of satisfactory degrees for decision alternatives under study, and proposed a three-stage interactive procedure for the solution of multi-objective optimization model associated with such MADM problems. The compensatory program we discuss in this paper can be avowed as an enhancement of this procedure, as we show how to compute the ranking of decision alternatives in a single stage. We also allow the decision maker to control the tradeoff associated with the achievement and balance of resultant satisfactory degrees of the decision alternatives. Thereby, we incorporate a compensatory mechanism to reflect the decision maker’s disposition to achievement and his/her aversion to imbalance. Finally, we prove that the optimal solution of our compensatory program is a Pareto solution to the multi-objective optimization model associated with original MADM problem. All studies we listed in this section suggest different mechanisms both in information processing and computation/number crunching. Hence, several distinct features of these methods emerge, such as aggregation, mathematical operations, approximations, usage of labels and so on. To better see the overview of these methods we provide a comparative analysis in Table 1. Recall that, we limit our attention to CWW approach in this paper. The reader should note that linguistic modeling can not be constrained only to the CWW approach. For an alternative framework and a clear discussion of the notion of label semantics, we suggest the readers an important paper by Lawry [14]. A decision support system which can both work with linguistic labels and Boolean labels can be found in [18], a discussion of group consensus under unbalanced linguistic label sets is provided in [3], a linguistic recommender system is analyzed in [19], a MADM problem where attribute values come from an uncertain linguistic label set is solved with using principles of prospect theory in [15], for a recent analysis of preference relations under linguistic modeling one may refer to [4]. The remaining of the paper is structured as follows. In Section 2, we provide some basic definitions, notation, and give a description of the problem accompanied by a brief overview of Xu’s [24] threestage interactive procedure. Section 3 is devoted to introducing the single-stage compensatory program and proving its efficiency. In Section 4, we illustrate the procedure on two numerical examples from the published literature. To better illustrate the managerial implications and real life connotations of several constructs we employed under the proposed method, we analyze another

31

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37 Table 1 An analysis of procedures for CWW under discrete linguistic label sets. Paper

Aggregation

Approximation

Operations on

Output label domain

Computational feature

Degani and Bortolan [5] Delgado et al. [6] Herrera and Martinez [9] Herrera and Martinez [10] Xu [24]

Fuzzy Symbolic Linguistic Linguistic None

Linguistic Linguistic None None None

Fuzzy numbers Label indices 2-tuples 2-tuples Label indices

Discrete Discrete Continuous Continuous Continuous

Fuzzy arithmetic

Xu [25]

None

None

Label indices

Continuous

Xu and Da [26]

None

None

Label indices

Continuous

Current paper

None

None

Label indices

Continuous

concrete multi-attribute decision making example, in detail. Furthermore, with using the information fashioned by this example, we pointed out two advantages of our compensatory program, namely, the variety of compensatory solutions attainable and assurance of efficiency. We end our discussion with conclusions in Section 5.

Definition 3. Let a+ = (lK)m and a = (lK)m be two sequences of order m. Then, a+ and a are called linguistic positive and negative ideal sequences, respectively. By definition, the positive and negative ideal sequences stand for the arrangement of the best and worst possible attribute values to vectors of order m, respectively. P Definition 4 (Xu [24]). Let v i ðwÞ ¼ m j¼1 wj  aij ; 8i. Then, vi(w) is the overall value (score) of the decision alternative i under a particular weight vector w. Pm By using the normalization constraint j¼1 wj ¼ 1 and Definitions 3 and 4, the overall values of linguistic positive and negative ideal sequences can be derived as follows [24]:

wj  lK ¼ lK

ð6Þ

j¼1

v



ðwÞ ¼

m X

to to

ð8Þ

Let, d(vi(w), v (w)) be the distance between the overall value of alternative i and overall value of the linguistic negative ideal sequence. Similarly, let d(v+(w), v(w)) be the distance between the overall value of the linguistic positive ideal sequence and overall value of the linguistic negative ideal sequence. Definition 5 (Xu [24]). The ratio:

lðv i ðwÞÞ ¼

dðv i ðwÞ; v  ðwÞÞ dðv þ ðwÞ; v  ðwÞÞ

ð9Þ

is called the satisfactory degree of decision alternative i. It is also easy to see that, when the linguistic label set is symmetric (for example, the set L), the satisfactory degree of decision alternative i is equivalent to:

lðv i ðwÞÞ ¼

1  dðv i ðwÞ; v  ðwÞÞ 2K

ð10Þ

We now describe the problem and provide a brief overview of Xu’s [24] three-stage interactive solution procedure. Through the incomplete-weight MADM process, the decision maker’s aim is to find a weight vector such that the satisfactory degree of each decision alternative is maximized. To illustrate the process, first we define the following polytope for notational convenience, by using the set of incomplete weight information I, non-negativity constraints, and the normalization constraint:

( P

w ¼ ðw1 ; . . . ; wm ÞT : wj 2 I; wj P 0;

m X

) wj ¼ 1 :

The multi-objective program for maximizing the satisfactory degree of each decision alternative is given by the following [24,26]:

max flðwÞ  ½lðv 1 ðwÞÞ; . . . ; lðv n ðwÞÞ : s:t: w 2 Pg: ð12Þ

ð7Þ

j¼1

One may observe that, the closer the overall value vi(w) of alternative i to the overall value v+(w) of the linguistic positive ideal sequence, the better the alternative is. Or equivalently, the further away the overall value vi(w) of alternative i from the overall value v(w) of the linguistic negative ideal sequence, the better the alternative is. In this paper, we adopt the latter to measure the performance of a decision alternative. To facilitate this, we now

ð11Þ

j¼1

ðP  1Þ wj  lK ¼ lK :

to



In this section, for completeness, we provide some basic definitions and notation which are useful for problem representation. Let w = (w1, . . . , wm)T be the weight vector of attributes Pm such that wj P 0 and j¼1 wj ¼ 1. Further, let A = (aij)nm be the decision matrix where aij 2 L are the attribute values of alternatives. Hence, ai = (aij)m are n vectors of attribute values for the decision alternatives. Xu [24] used the term point for describing two positive and negative ideal reference sequences of attribute values, which later used to define the satisfactory degree of each decision alternative. Ma et al. [17] referred to each of these artificial alternatives as an ideal solution. In this paper, we find it sensible to use the term ideal sequence to name the positive and negative ideal vectors of attribute values.

m X

to

consider a metric d on the set of all overall values V ¼ V [ fv þ ðwÞ; v  ðwÞg, measuring the distance between a pair of overall values. It is also worthwhile to note that we employ the rectilinear metric throughout this paper. Hence, the distance between the overall values vi(w) = lk and vq(w) = lh of decision alternatives i and q is of the form:

dðv i ðwÞ; v q ðwÞÞ ¼ jk  hj:

2. Basic definitions and problem description

v þ ðwÞ ¼

Multiobjective programming with transformation separate max–min and max-sum equivalents Multiobjective programming with transformation a single objective equivalent Multiobjective programming with transformation a goal program equivalent Multiobjective programming with transformation a compensatory equivalent

At this point, the decision maker may consider two scenarios: Scenario A. He/she can maximize the minimum satisfactory degree and obtain a threshold such that any satisfactory degree is guaranteed to be larger than this value. Scenario B. He/she can maximize the sum of the satisfactory degrees of the decision alternatives, as a whole.

32

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37

The mathematical program associated with Scenario A is as follows:

ðP  2Þ

max fminflðv i ðwÞÞg : s:t: w 2 Pg:

ð13Þ

Letting k = min{l(vi(w))} and employing Zimmermann’s [29] approach, we arrive at Stage 1 problem of Xu [24]:

ðP  3Þ

max fk : s:t:

lðv i ðwÞÞ P k 8i; w 2 Pg:

ð14Þ

Next, we may consider Scenario B. Corresponding mathematical program is given by the following:

ðP  4Þ

max

( n X

)

lðv i ðwÞÞ : s:t: w 2 P :

ð15Þ

i¼1

The decision maker may be able to define a set of lower bounds bi associated with the minimum satisfactory degree of each alternative, during the decision making process. Hence, allowing such bounds and letting ki be n free variables we arrive at Stage 2 problem of Xu [24]: ðP  5Þ

max

( n X

) ki : s:t:

lðv i ðwÞÞ P ki P bi 8i; w 2 P; ki unrestricted :

i¼1

ð16Þ Finally, the three-stage interactive procedure described by Xu [24] for solving (P  1) can be summarized as follows.

3. Single stage compensatory program Observe that there is an inherent tradeoff between the achievement and balance of the satisfactory degrees in the program (P  1). To illustrate this, consider two scenarios introduced in Section 3. For Scenario A, the decision maker solves (P  3) and obtains a lower bound on all satisfactory degrees. In this case, more attention is paid to ensure that each satisfactory degree is larger than this bound. Accordingly, the resultant satisfactory degrees are close to each other and highly balanced, but no attention is paid to maximize them as a whole. Conversely, in Scenario B, the decision maker solves (P  5) and maximizes the satisfactory degrees. Accordingly, some satisfactory degrees assume high resultant values, and no attention is paid to their balance. As a result, there may be considerable differences between the satisfactory degrees for some decision alternatives. It is well-known that such unbalanced and poorly compromised solutions are usually undesirable from a multi-attribute decision making point-of-view. To overcome these hitches, we now describe a single-stage linear compensatory program which simultaneously maximizes the satisfactory degrees and ensures a balance between their resultant values. For this purpose, similarly to the compensation parameter convention [30,16], we employ a control parameter d 2 [0, 1] to achieve a convex combination between the conflicting objectives of maximization and balance. With combining the models (P  2) and (P  4), we obtain the following hybrid model: ( ðP  6Þ

Stage 1. Solve (P  3) and calculate the satisfactory degrees of decision alternatives l(vi(w0)) associated with the optimal weight vector w0. Utilize the lower bounds bi according to this solution, and utilize a counter c = 1. Stage 2. Solve (P  5) and calculate the satisfactory degrees l(vi(wc)) associated with the new optimal weight vector wc. Stage 3. If (P  5) is feasible and the decision maker is satisfied with the result, rank the alternatives with respect to the overall values vi(wc) and stop. If (P  5) is infeasible or the decision maker is not satisfied with the result, then the decision maker should increase the satisfactory degrees of some alternatives, and decrease the satisfactory degrees of some other alternatives, update c, and return to Stage 2.

max d  minflðv i ðwÞÞg þ ð1  dÞ 

n X

)

lðv i ðwÞÞ : s:t: w 2 P; d 2 ½0; 1 :

i¼1

ð17Þ A higher value of the control parameter d will result more attention to be paid for obtaining a balance between the resultant satisfactory degrees and a larger threshold associated with the minimum satisfactory degree. Similarly, a lower value of the control parameter d will result more attention to be paid for simultaneous maximization of the satisfactory degrees, as a whole. Hence, the decision maker has full control of the tradeoff associated with these two conflicting objectives with the aid of a single control parameter. With utilizing the free variables k, ki and lower bounds bi for the satisfactory degrees, formulation (P  6) can be transformed into the following linear program:

ðP  7Þ

max

f ðwÞ  d  k þ ð1  dÞ 

n X

ki

ð18Þ

i¼1

Theorem 6 (Xu [24]). The optimal solution of the model (P  5) is a Pareto solution to the model (P  1). In the above interactive procedure, at each iteration succeeding the first one, the decision maker solves a maximization problem and evaluates the resultant satisfactory degrees through two consecutive stages. When (P  5) is infeasible or the decision maker is not satisfied with the result, it is suggested that the decision maker may tune-up the satisfactory degrees of some alternatives. However, as it stands, this task is neither that simple, nor a procedure is described for it. Observe that, the decision maker can not simply arrange the satisfactory degrees without checking that all the constraints in the formulations (P  3) and (P  5) are not violated. Moreover, the tradeoff between the conflicting objectives of maximizing the satisfactory degrees and ensuring their balance can not be captured by this three-stage model. We now describe a single-stage compensatory program which is equivalent to the multiobjective model (P  1). The compensatory approach we introduce requires solving a single linear program, and it is capable of capturing the achievement-balance tradeoff. In the sequel we also prove that the optimal solution of this new single-stage model is indeed a Pareto solution to (P  1).

s:t:

lðv i ðwÞÞ P ki P k P bi 8i;

ð19Þ

w 2 P; d 2 ½0; 1;

ð20Þ ð21Þ

k; ki unrestricted:

ð22Þ

Unlike Xu’s [24] three-stage model, observe that: (1) the decision maker solves only (P  7) at each iteration, and (2) the fine-tuning of resultant satisfactory degrees solely depend on the control parameter d. Hence, if he/she is not satisfied with the result, violation of the constraints need not be checked at each iteration. Instead, the decision maker simply adjusts the control parameter according to his/her objective (i.e. simultaneous maximization vs. balance) and re-solves (P  7). Now we prove the efficiency of this new method. Theorem 7. The optimal solution of (P  7) is a Pareto solution to (P  1).

Proof. By contradiction. Suppose that w7 is the optimal solution of (P  7), but not a Pareto solution to (P  1). Then, (P  1) has a Par      eto solution w1 such that l v i w1 P l v i w7 ; 8i. It immediately follows that:

33

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37

        9S ¼ i : k0i ¼ l v i w1 > l v i w7 ¼ ki – ;:

ð23Þ

⁄

Let i be the decision alternative such that

lðv i ðwÞÞ ¼ minflðv i ðwÞÞg ¼ k: We consider two cases: P    P (1) If i⁄ R S, we have: f w7 ¼ d  k þ ð1  dÞ  ki þ iRS ki i2S P    P 0 and f w1 ¼ d  k þ ð1  dÞ  i2S ki þ iRS ki . Upon simplification and comparison we obtain,

X 0 X     f 0 w7 ¼ ð1  dÞ  ki < ð1  dÞ  ki ¼ f 0 w1 : i2S

ð24Þ

i2S

P    P (2) If i⁄ 2 S, we have: f w7 ¼ d  k þ ð1  dÞ  ki þ iRS ki i2S P    P 0 and f w1 ¼ d  k0 þ ð1  dÞ  i2S ki þ iRS ki . Upon simplification and comparison we obtain, X 0 X     f 00 w7 ¼ d  k þ ð1  dÞ  ki < d  k0 þ ð1  dÞ  ki ¼ f 00 w1 : i2S

ð25Þ

i2S

Observe that, both cases lead to a contradiction with the optimality of w7 . h 4. Numerical examples In this section, first we illustrate how the proposed hybrid procedure works on two examples from the published literature. Having presented the ‘‘mechanics’’ of our procedure on these two examples, we then analyze a budget allocation example we developed for illustrating the practical implications of this method under multi-attribute decision making environments. The roles and practical inferences for satisfactory degrees, overall values, the influence of control parameter and possible alteration(s) in the final ranking will be discussed. Finally, we comment on some practical advantages of our compensatory mechanism with comparing it to two other analogues that have objective functions derived by means of alternative compensatory operators.

also solved the single-stage model (P  7) with d = 0.1 and d = 0.9. For d = 0.1, more attention is paid for maximizing the satisfactory degrees as a whole. For this case, we again obtain the solution presented in Table 2. The compensatory solution for d = 0.9 is illustrated in Table 3, where more attention is paid for balancing the resultant satisfactory degrees. A comparison between Tables 2 and 3 shows that: for the maximizing solution d = 0.1, except alternative 3, all other alternatives assume higher satisfactory degrees than the results of the balanced solution d = 0.9, as expected. Let D(d) be the difference between the maximum and minimum satisfactory degrees under a particular control parameter. This term can be used as an indicator of the spread of resultant satisfactory degrees. For the maximizing solution d = 0.1, we have D(0.1) = 0.7487  0.6935 = 0.0587. Similarly, for the balanced solution d = 0.9 we have D(0.9) = 0.747  0.6935 = 0.0535. Hence, the spread of resultant satisfactory degrees in the balanced solution is more compact, as desired. The difference between spread values may seem very marginal in this example, but observe that the satisfactory degrees are essentially ratios. Thus, the terms D(0.1) and D(0.9) are differences between two ratios. 4.2. Example 2 Consider the following example ([25], page 721, Example 3) where the performance of three decision alternatives are evaluated according to seven attributes under the discrete linguistic label set introduced in Example 1. The decision matrix and incomplete weight information for this example are given as follows:

0

l3 B A ¼ @ l2 l2

l3

1

l2

l1

l0

l2

l1

l1

l4

l2

l3

l1

C l2 A;

l2

l1

l3

l0

l3

l1

I ¼ f0:2 6 w1 6 0:3; 0:05 6 w2 6 0:1; 0:5 6 w3 6 0:6; 0:2  w2 6 w4 ;

0:05 6 w5  w4 ; w4 6 w6 ;

w6  w4 6 w5  w7 ; 0:4  w6 6 w7 g: 4.1. Example 1 Consider the following example ([24], page 23) which was originally adapted from [2]. This is a MADM problem where the performance of five decision alternatives are evaluated according to three attributes under a discrete linguistic label set. This set is given by: L = {l4: extremely poor, l3: very poor, l2: poor, l1: slightly poor, l0: fair, l1: slightly good, l2: good, l3: very good, l4: extremely good}. The decision matrix A and incomplete weight information I are given as follows: 0 1 l0 l2 l3 B C B l3 l4 l2 C B C B A ¼ B l0 l3 l2 C C; I ¼ f0:25 6 w1 6 0:4; 0:15 6 w2 6 0:3; w2 < w3 g B C @ l4 l1 l2 A l2 l3 l1 Finally, lower bounds for the minimum satisfactory degrees are as follows: b1 = 0.7, b2 = 0.71, b3 = 0.69, b4 = 0.73, b5 = 0.73. The three-stage interactive procedure solution regarding this example is summarized in Table 2. For obtaining the equi-weight compensatory solution of this example, we solved (P  7) with d = 0.5. In a single stage, we were able to obtain satisfactory degrees presented at Table 2, with overall values v1(w) = l1.68; v2(w) = l1.75; v3(w) = l1.52; v4(w) = l1.88; v5(w) = l1.99, and the resultant ranking 5 4 2 1 3. To illustrate how the decision maker has control over the conflicting objectives of simultaneous maximization and balance, we

Lower bounds for the minimum satisfactory degrees are not specified for this example. To illustrate the tradeoff associated with the objectives of maximization and balance, we solved (P  7) for a set of different values of the control parameter. The resultant satisfactory degrees for each alternative are summarized in Table 4. For all different solutions presented at Table 4, we obtained the robust 2 1 3 ranking which agrees with the ranking obtained by using the method presented in [25]. Fig. 1 is a depiction of this resultant satisfactory degrees for the set of values under study. For small values of d, more attention is paid to the maximization objective, hence Alternatives 1 and 2 assume their highest satisfactory degrees. Yet, Alternative 3 attains its lowest satisfactory degrees leaving these solutions poorly compromised. As the value of d

Table 2 Summary of three-stage interactive solution and equi-weight compensatory solution d = 0.5. Alternative

ki

vi(w)

1 2 3 4 5 k = 0.69 w = (0.39, 0.3, 0.31)T Ranking: 5 4 2 1 3

0.71 0.7188 0.69 0.735 0.7487

1.68 1.75 1.52 1.88 1.99

34

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37

Table 3 Summary of compensatory solution d = 0.9. Alternative

ki

vi(w)

1 2 3 4 5 k = 0.6935 w = (0.376, 0.3, 0.324)T Ranking: 5 4 2 1 3

0.7065 0.71 0.6935 0.7315 0.747

1.652 1.68 1.548 1.852 1.976

increases, more attention is paid to the balance, hence the maximum (i.e. Alternative 2) and minimum (i.e. Alternative 3) satisfactory degrees tend to come closer. Fig. 2 is a depiction of the spread of resultant satisfactory degrees for the aforementioned set of d values. For d = 0, the spread of resultant satisfactory degrees attains its maximum value. As the value of d increases, it can be recognized that the spread becomes more compact. 4.3. Example 3: Budget allocation Consider an R&D firm organized as separate project teams. The budget for a planning horizon will be apportioned among such teams according to their past performances under a number of relevant attributes. For example: profoundness of the project proposal, team organization, effectiveness of planning, conformance to project schedule, prospects for partnering and/or chartering, accuracy of cost estimation, resource levelling accomplishments, reliability of reporting, monitoring success, and auditing skills may emerge as significant attributes under this setting. Suppose at the end of each planning horizon, a committee of experts evaluate project teams with considering multiple attributes from the above set, and provide their assessment in the form of a decision matrix. Proximately recognizable is the fact that several attributes utilized are intangible in nature. As such, the appropriate representation for this information is a decision matrix constructed by means of discrete linguistic labels after a structured decision making process by the committee. Budget for the next planning horizon is then allocated to teams with respect to their overall assessment values (i.e. scores) obtained from processing this matrix. The ranking, in this manner, is an indicator of the firm’s priority for funding a specific project team. Noticeably, the composition and weighting of the attribute set may differ among planning horizons to conform to firm’s inclusive goals and strategy. Thereby, the committee is totally flexible to impose weight restrictions during the decision-making process to reflect firm’s perseverance towards the importance of attributes employed. On the other hand, satisfactory degrees attained may serve both as performance targets for the next planning horizon or valuable Table 4 Summary of satisfactory degrees for Example 2. d

l(v1(w))

l(v2(w))

l(v3(w))

D

d=0 d = 0.1 d = 0.2 d = 0.3 d = 0.4 d = 0.5 d = 0.6 d = 0.7 d = 0.8 d = 0.9 d=1

0.598125141 0.598125141 0.584375084 0.584375084 0.571428644 0.571428644 0.571428644 0.571428644 0.575892929 0.575892929 0.575892929

0.881875141 0.881875141 0.875000112 0.875000112 0.868526892 0.868526892 0.868526892 0.868526892 0.862946535 0.862946535 0.862946535

0.421875028 0.421875028 0.439062598 0.439062598 0.452009039 0.452009039 0.452009039 0.452009039 0.452232253 0.452232253 0.452232253

0.460000112 0.460000112 0.435937514 0.435937514 0.416517853 0.416517853 0.416517853 0.416517853 0.410714282 0.410714282 0.410714282

media for internal benchmark processes. By definition, each satisfactory degree will show, as a proportion, the distance associated with the overall values of a project team and the negative ideal versus the distance associated with the overall values of positive and negative ideals. Hence, the higher the satisfactory degree of a project team, the better the team is. Under this convention, the satisfactory degrees show how far the teams are from the worst possible project performance. Thereby, the firm management may utilize preceding planning horizon’s satisfactory degrees as realistic performance targets for an upcoming planning horizon. This is the point where balancing objective of our hybrid model sets in. A high-performing project team should not be enforced to meet relatively high performance standard by assigning its satisfactory degree as a lower bound for the upcoming planning horizon. On the other hand, a low-performing project team should indeed be required to meet a higher, but realistic, performance standard for the upcoming planning horizon. Moreover, this target should preferably be close by targets set for other teams. Under simultaneous maximization objective, there is no such mechanism to control unfair target-setting for high- and also low-performing project teams, whereas balancing objective introduced in our hybrid model is designed to undertake such tasks. Furthermore, when maximization is enforced the overall values of low-performing teams turn out to be lower than those attained under balanced solutions. Yet, such project teams should be supported at least budgetwise to leverage their performance for the upcoming planning horizon. Another feature of the balancing objective we employed is that it curbs out such impractical reductions on the overall values of low-performing alternatives. Let us now illustrate these implications numerically. For the sake of simplicity, let us assume there exist four project teams being evaluated under three attributes. Assume the committee of experts evaluated each project team according to these attributes with using the linguistic label set provided in Example 1, and structured the decision matrix as follows:

1 3 3 1 B3 2 3 C C B A¼B C: @1 3 4 A 0

2 2

3

The firm is also impringing upon attribute weights leading to the incomplete weight information:

I ¼ f0:2 6 w1 6 0:45; 0:25 6 w2 6 0:35; w2 6 w3 g: Suppose the satisfactory degrees at the preceding planning horizon transpired as b1 = 0.6, b2 = 0.7, b3 = 0.5 and b4 = 0.6. These values are set forth as minimum expected satisfactory degrees by the firm management. First we analyze the case where more attention is paid to the maximization of satisfactory degrees as a whole. For this purpose we solve the compensatory program (P  7) with using the above information under d = 0.1. This solution is summarized in Table 5. Under compensatory solution with d = 0.1, except team 1, all teams outperform their targets given by bi values and reach high satisfactory degrees. Yet, the solution is poorly compromised leaving team 1 at a relatively low satisfactory degree with k1 = k = 0.7. This is the baseline of all satisfactory degrees and points out the minimum expected performance during the next planning horizon, intended for the worst performing team of the current planning horizon. The weights of attributes 2 and 3 come out as equal with a little less emphasis is given to attribute 1 under this scheme. The funding priority of the firm will be team 3 verified by the resultant ranking 3 2 4 1. When overall assessment values vi(w) are investigated after normalization, we see that team 3 is receiving 29.41% of the budget for the next planning horizon, followed by

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37

35

Fig. 1. Satisfactory degrees for Example 2.

Fig. 2. Spread of resultant satisfactory degrees.

team 2 with 28.34%, team 4 with 25.13% and team 1 with 17.12% shares, respectively. Nevertheless, there exist some hitches with this solution. First of all, team 3 is clearly the best performing team under this scenario, reaching a satisfactory degree of 0.84375 when its target value was merely 0.5. Team 2 attained a nearby satisfactory degree of 0.83125, yet the target set for team 2 was as high as 0.7. Given that team 3 already has the best performance, expecting a new target of 0.84375 from this team for the upcoming period will result in an imbalance, as far as the expectations from all teams are considered. Especially, when the new target for the worst performing team came out relatively low at a level of 0.7. On the other hand, team 1 is the contributor with the worst performance requiring immediate management support, at modest, by means of a fair budget. Yet again, we see a clear imbalance with team 1 qualifying for a 17.12% Table 5 Summary of compensatory solution d = 0.1. Team

ki

1 0.7 2 0.83125 3 0.84375 4 0.79375 k = 0.7 w = (0.3, 0.35, 0.35)T Ranking: 3 2 4 1

vi(w)

Target

1.6 2.65 2.75 2.35

0.6 0.7 0.5 0.6

of the total budget amid considerable differences with funding of other teams. Lastly, the baseline k = 0.7 for satisfactory degrees is also problematic as the teams could do better under a balancing scheme. This is particularly desired from the viewpoint of firm management, as total performance is always a decisive objective rather than individual team performances. To overcome such hitches let us now analyze the problem under a balancing scheme. For this purpose, we solve the same decisionmaking problem with assigning a high value to the control parameter. Suppose we assign d = 0.7 to better balance the resultant satisfactory degrees and resolve the compensatory program (P  7) with the same information. In this case we obtain the solution summarized in Table 6. Under compensatory solution with d = 0.7, again all teams outperform their targets given by bi values and reach high satisfactory degrees. The weights of attributes 2 and 3 are still equal, but this time more emphasis is given to attribute 1 resulting in a natural alteration in the final ranking. This time, the funding priority of the firm management will be team 2 verified by the resultant ranking 2 3 4 1. Thus, with employing the compensatory program towards a balancing strategy, one also has the flexibility to apply a different ranking scheme completely aligned with comprehensive goals. In this example, since team 3 has the best performance and this well extended beyond its target, firm management turned its funding emphasis to another promising alternative, such as team 2, with a resultant alteration in the final ranking. One question arise at this point is how could the management compute an appropriate value of d with the aim of having significantly different alternative solutions. It is impractical to try to develop such a generic procedure as the breakpoints where ranking change essentially occur are specific to the subject decision problem input. Though, the compensatory program (P  7) could be used in an interactive way to detect such cutoff values on the control parameter. To better understand the hurdle, the reader is referred to the studies [30,16] by the originators of compensatory operators. If we investigate the baseline of all satisfactory degrees, we see that the minimum of the expected performances for the next planning horizon increased to k = 0.7375 under balancing scheme. This really shows that firm management have the opportunity to expect more from the worst performing team with setting k = 0.7375 to ensure a rational target assignment amongst the project teams. This is indeed visible from the spread of resultant satisfactory degrees as we have D(0.1) = 0.14375 and D(0.7) = 0.10312, where we

36

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37

Table 6 Summary of compensatory solution d = 0.7. Team

ki

1 0.7375 2 0.840625 3 0.796875 4 0.784375 k = 0.7375 w = (0.45, 0.275, 0.275)T Ranking: 2 3 4 1

Table 7 Summary of two possible solutions.

vi(w)

Target

1.9 2.727 2.375 2.275

0.6 0.7 0.5 0.6

After the illustration of its managerial inferences, let us now point out two advantages of the hybrid approach we developed with using this example. Note that, one can modify (17) with employing other compensatory operators to create alternative compensatory mechanisms for solving the anticipated budget allocation problem. There exists a collection of compensatory operators well practiced under various domains [1,20–23]. For our purposes in this section, let us consider modified Zimmermann and augmented max–min compensatory operators. If we utilize modified Zimmermann compensatory operator, the objective function of the model (17) is of the form:

max fd  minflðv i ðwÞÞg þ ð1  dÞ  maxflðv i ðwÞÞgg:

ð26Þ

where d 2 [0, 1] is the control parameter. This model can also be transformed to a linear program with bringing in additional free variables, a set of binary variables and adjusting the constraints of (17) correspondingly. It is not our aim here to further work through this construction. Some well-structured models can be found in [21]. On the other hand, in the case we apply augmented max–min compensatory operator, the objective function of the model (17) will look like:

max minflðv i ðwÞÞg þ h 

Solution 2

ki

0.7, 0.83125, 0.84375, 0.79375 1.6, 2.65, 2.75, 2.35 0.3, 0.35, 0.35

0.7375, 0.840625, 0.796875, 0.784375 1.9, 2.727, 2.375, 2.275 0.45, 0.275, 0.275

wj

4.4. Example 4

n X

Solution 1

vi

observe a considerable reduction on the spread to better institute a fair target assignment. When it comes to overall assessment values vi(w), with normalization we have that team 2 is receiving 29.39% of the budget for the next planning horizon, followed by team 3 with 25.60%, team 4 with 24.52% and team 1 with 20.49% shares, respectively. The main objective of firm management is to secure total performance and a 20.49% budget allotment to the worst contributor indicate enhanced management support.

(

Model output

) r i  lðv i ðwÞÞ :

ð27Þ

i¼1

where h > 0 is a sufficiently small number and each ri is a weight indicating the relative importance of the satisfactory degree i such P that ni¼1 ri ¼ 1. It is worthwhile to note that this set of weights have no association with w and solely indicate decision maker’s priorities. h is also a pre-determined number which is usually assigned to 0.01 (see, for example [13,22]), or even to smaller values such as 0.0001 (see, [21]). To ensure a comparison between the solutions of our hybrid model and its two other analogues (26) and (27), we solved the budget allocation example with these three compensatory mechanisms using various values of the parameters d and h. Specifically, we used the set {0, 0.1, 0.2, . . . , 1} for the control parameter d in the hybrid model and in its variant (26) with modified Zimmermann compensatory operator. To better explore the possible solutions between the values used for h in the literature, we used the set {0.0001, 0.001, 0.002, . . . , 0.01} in the model (27) with augmented max–min compensatory operator. We also assumed that the deci-

sion maker is indifferent between the satisfactory degrees, hence set ri = 1/n in our experiments. When the finite possibilities prescribed by the above parameter settings are explored with all three models, two unique solutions were attainable. These are summarized in Table 7. Our hybrid model generates a compensatory solution between the ‘‘min’’ operator and the ‘‘sum’’ operator depending on the value of d. Similarly, (26) equipped with modified Zimmermann compensatory operator produces a compensatory solution between the ‘‘min’’ operator and the ‘‘max’’ operator depending on the value of d. It is also clear that (27) with augmented max–min compensatory operator will provide a compensatory solution between the ‘‘min’’ operator and the ‘‘sum’’ operator. But since h is a very small number, the compensatory solution generated by this approach is very close to the ‘‘min’’ operator. To sum, all three models are well able to produce compensatory solutions. This is illustrated at the first line in Table 8. In compensatory decision making, usually decision makers prefer to see a wider set of compensatory solutions available, just to ensure the freedom of choice and flexibility. Hence, variety of the compensatory solutions generated emerge as an important characteristic of compensatory decision making methods. When we inspect this example with our hybrid approach, we see that it is well performing on solution variety and both possible solutions in Table 7 were attainable. Nonetheless, Modified Zimmermann compensatory operator is also well known to offer a wide set of solutions to choose, and this was the case for this example as well. We were able to attain two possible solutions with using (26). Model (27) with augmented max–min compensatory operator fails on solution variety for this problem as only Solution 2 was reached after our numerous trials which even stretched to those where h used beyond its setting h 2 [0.0001, 0.01]. Line two in Table 8 is devoted to comparison of three models on variety of compensatory solutions produced. A compensatory mechanism is deemed as efficient if it guarantees Pareto optimality of the compensatory solutions generated. As we have noted, augmented max–min compensatory operator is known to guarantee a unique Pareto solution in the neighbourhood of the ‘‘min’’ operator with its proximity depending on h. Yet, model (26) fails to satisfy this property as it is known that modified Zimmermann compensatory operator, by construction, does not guarantee to obtain a Pareto solution. In this example, we see that the two solutions listed in Table 7 are Pareto solutions and modified Zimmermann compensatory operator have detected them. But this is not the case for large problem instances. For a comprehensive example the reader is referred to [21]. On the other hand, by Theorem 7, we know that our hybrid approach guarantees a Table 8 A comparison of three models. Feature Compensatory Solution variety Pareto guarantee

Modified Zimmermann p p 

Augmented max–min p  p

Hybrid approach p p p

O. Çakır / Knowledge-Based Systems 27 (2012) 29–37

Pareto solution at every case. We summarize this discussion at line three of Table 8. In a few words, the solutions produced by our hybrid approach are compensatory, of adequate diversity, and indeed Pareto optimal whereas other mechanisms tested here failed to satisfy all these desirable properties simultaneously. 5. Conclusions In many real life MADM problems, decision makers are usually faced with uncertain and partially-known data due to the complexity of the problem domain, lack of accurate estimates for attribute values, and time restrictions. Moreover, many important attributes may not be quantified appropriately due to their intangible nature. Hence, the information regarding these attributes may be available to the decision maker only in discrete linguistic labels. In this paper, we proposed a procedure for solving such MADM problems by computing with words under discrete linguistic labels and incomplete weight information. In particular, we developed a single-stage linear compensatory model which is equivalent to the multi-objective program associated with such MADM problems. This new compensatory model is capable of capturing the tradeoff associated with the maximization and balance of resultant satisfactory degrees of each decision alternative under study. Further, we proved that the optimal solution of this model is a Pareto solution to the original multi-objective program. We illustrated the procedure on two numerical examples from the published literature. Subsequently, we analyzed an ill-structured multi-attribute decision making example with managerial difficulties as far as the fairness and rationality principles are concerned. We showed that the hybrid procedure we developed produces sensible results with handy managerial implications on such imbalanced cases. For future research, we propose interested researchers to come up with different objective, sometimes referred to as the achievement, functions and derive new alternative compensatory approaches. Such approaches may suggest different mechanisms to control the maximization-balance tradeoff. Also, it may be valuable to show the applications of this particular method to some interesting real-life MADM problems. One may also come up with an experimental study using real industry data to investigate possible useful inferences on the behaviour of the control parameter under given break-even points between the ranks of alternatives. Acknowledgements The author thanks three anonymous referees for their careful reading and fruitful comments which improved the content and presentation of this manuscript. References [1] C. Araz, H. Selim, I. Ozkarahan, A fuzzy multi-objective covering-based vehicle location model for emergency services, Computers & Operations Research 34 (2007) 705–726. [2] N. Bryson, A. Mobolurin, An action learning evaluation procedure for multiple criteria decision making problems, European Journal of Operational Research 96 (1995) 379–386.

37

[3] F.J. Cabrerizo, I.J. Pérez, E. Herrera-Viedma, Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information, Knowledge-Based Systems 23 (2010) 169–181. [4] H. Chen, L. Zhou, B. Han, On compatibility of uncertain additive linguistic preference relations and its application in the group decision making, Knowledge-Based Systems 24 (2011) 816–823. [5] R. Degani, G. Bortolan, The problem of linguistic approximation in clinical decision making, International Journal of Approximate Reasoning 2 (1988) 143–162. [6] M. Delgado, J.L. Verdegay, M.A. Villa, On aggregation operations of linguistic labels, International Journal of Intelligent Systems 8 (1993) 351–370. [7] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [8] J. Figueira, M. Ehrgott, S. Greco, Multiple Criteria Decision Analysis: State of the Art Surveys, Springer-Verlag, New York, 2005. [9] F. Herrera, L. Martinez, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8 (2000) 746–752. [10] F. Herrera, L. Martinez, A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision making, IEEE Transactions on Systems, Man and Cybernetics Part B 31 (2000) 227–234. [11] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making, Springer-Verlag, Berlin, 1981. [12] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making: An Introduction, Sage Publications, Thousand Oaks, 1995. [13] Y.J. Lai, C.L. Hwang, Possibilistic linear programming for managing interest rate risk, Fuzzy Sets and Systems 54 (1993) 135–146. [14] J. Lawry, A framework for linguistic modeling, Artificial Intelligence 155 (2004) 1–39. [15] P. Liu, F. Jin, X. Zhang, Y. Su, M. Wang, Research on the multi-attribute decision-making under risk with interval probability based on prospect theory and the uncertain linguistic variables, Knowledge-Based Systems 24 (2011) 554–561. [16] M.K. Luhandjula, Compensatory operator in fuzzy linear programming with multiple objective, Fuzzy Sets and Systems 8 (1982) 245–252. [17] J. Ma, Z.P. Fan, L.H. Huang, A subjective and objective integrated approach to determine attribute weights, European Journal of Operational Research 112 (1999) 397–404. [18] J. Ma, J. Lu, G. Zhang, Decider: A fuzzy multi-criteria group decision support system, Knowledge-Based Systems 23 (2010) 23–31. [19] C. Porcel, E. Herrera-Viedma, Dealing with incomplete information in a fuzzy linguistic recommender system to disseminate information in university digital libraries, Knowledge-Based Systems 23 (2010) 32–39. [20] H. Selim, I. Ozkarahan, A supply chain distribution network design model: an interactive fuzzy goal programming-based solution approach, International Journal of Advanced Manufacturing Technology 36 (2008) 401–418. [21] F. Tiryaki, Interactive compensatory fuzzy programming for decentralized multi-level linear programming (DMLLP) problems, Fuzzy Sets and Systems 157 (2006) 3072–3090. [22] S.A. Torabi, E. Hassini, An interactive possibilistic programming approach for multiple objective supply chain master planning, Fuzzy Sets and Systems 159 (2008) 193–214. [23] S.A. Torabi, E. Hassini, Multi-site production planning integrating procurement and distribution plans in multi-echelon supply chains: an interactive fuzzy goal programming approach, International Journal of Production Research 47 (2009) 5475–5499. [24] Z. Xu, An interactive procedure for linguistic multiple attribute decision making with incomplete weight information, Fuzzy Optimization and Decision Making 6 (2007) 17–27. [25] Z. Xu, A method for multiple attribute decision making with incomplete weight information in linguistic setting, Knowledge-Based Systems 20 (2007) 719–725. [26] Y. Xu, Q. Da, A method for multiple attribute decision making with incomplete weight information under uncertain linguistic environment, Knowledge Based Systems 21 (2008) 837–841. [27] L.A. Zadeh, The concept of linguistic variable and its application to approximate reasoning: Parts 1–2, Information Sciences 8 (1975) 199–249. 301–357. [28] M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982. [29] H.J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer, Dordrecht, 1990. [30] H.J. Zimmermann, P. Zynso, Latent connectiveness in human decision making, Fuzzy Sets and Systems 4 (1980) 37–51.