Visualizing decision process on spheres based on the
even swap concept
Han-Lin Li
a, Li-Ching Ma
b,⁎
aInstitute of Information Management, National Chiao Tung University, Taiwan, ROC
bDepartment of Information Management, National United University, Taiwan, ROC
Received 24 April 2007; received in revised form 11 October 2007; accepted 21 January 2008 Available online 4 February 2008
Abstract
The Even Swap method, originally outlined by Benjamin Franklin 230 years ago, is a rational way of finding the best alternative by evenly swapping decision criteria. This study develops a Decision Ball model to assist a decision maker in ranking alternatives and visualizing decision process based on the Even Swap concept. By viewing the moving trajectories of alternatives on spheres, a decision maker can specify trade-offs among criteria via the Even Swap process thus ranking alternatives more consistently. © 2008 Elsevier B.V. All rights reserved.
Keywords: Visualization; Preference; Even Swap; Decision balls; Ranking
1. Introduction
In a multi-criteria decision making process, the more the decision criteria, the more the difficulties the decision maker (DM) has to face. Therefore, assistance in making reliable trade-offs among criteria thus ranking alternatives consistently is a critical issue in manage-ment research.
More than 230 years ago, Benjamin Franklin outlined the concept of Even Swaps in a letter (see Appendix A) about choosing between two alternatives. Franklin's fundamental idea is that if every alternative for a given criterion is rated equally, then the criterion can be ignored in making decision. Following this idea, Hammond, Keeney and Raiffa developed a mechanism
for Even Swaps to provide a useful way for making trade-offs with a range of criteria across a range of alternatives [14]. “Even” implies equivalence and “Swap” represents exchange. An even swap increases the value of one criterion while decreasing the value of another criterion by an equivalent amount. By iteratively crossing out equally rated criteria to reduce the number of criteria, the best option can be determined.
The Even Swap method is an algorithm for multi-criteria decision making under certainty. Each alter-native has a scaled ranking of a number of criteria, some positive and the remainder negative. The DM is asked to make a number of indifferent judgments between the original alternative and the modified alternative. These adjustments are made to equalize all alternatives with respect to one of the criteria, thus rendering it irrele-vant in the comparison. By successively applying this principle, as suggested by Benjamin Franklin, and recognizing when one alternative is dominated by
Decision Support Systems 45 (2008) 354–367
www.elsevier.com/locate/dss
⁎ Corresponding author. Tel.: +886 37 381828. E-mail addresses:hlli@cc.nctu.edu.tw(H.-L. Li),
lcma@nuu.edu.tw(L.-C. Ma).
0167-9236/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2008.01.004
another, alternatives can be ruled out until only one remains.
The Even Swap approach is a rational and practical way for finding the most preferred alternative. How-ever, current Even Swap methods have the following inadequacies:
(i) Only the most preferred alternative is found. In an actual decision environment, the DM may also want to know the second or the third preferred alternatives.
(ii) Some trade-offs of criteria values, as specified by the DM, may not be consistent with each other. Current methods have no mechanism to check the consistency of these trade-offs.
(iii) The similarities among alternatives are not taken into account. Actually, the DM does not only want to know what the best option is but also the differences (or similarities) among alternatives. The two main reasons for the above inadequacies in the current Even Swap methods are: First, they do not have a way to display differences (or similarities) among alternatives according to the trade-off values specified by the DM. Such a display can help the DM to see the differences among alternatives with different trade-off values. Second, they may not rank alternatives con-sistently according to even swaps made by the DM.
This study therefore develops a visualization model, the so called Decision Ball model, to assist a decision maker in ranking alternatives and visualizing decision process based on the Even Swap concept. By displaying all alternatives on spheres, the DM can see the differences among alternatives, can calculate the effects of different trade-off values, and can examine the moving trajectories of alternatives to check the consistency of even swaps. Thus, the DM can rank alternatives by viewing the ad-justment outcomes displayed on the spheres.
Several graphic techniques have been developed to support decision-making process: for instance, deduction
graphs to treat decision problems associated with expanding competence sets[19], a hyperbolic tree and a hierarchical list to visualize criminal relationships [23], and Gower Plots to detect inconsistencies in a decision maker's preferences and rank alternatives [11,12]. All these methods, however, used a 2-dimensional plane geo-metry to illustrate multidimensional data. A 2-dimensional plane model cannot depict three points that do not obey the triangular inequality (i.e. the total length of any two edges must be larger than the length of the third edge) neither can it display four points that are not on the same plane. For instance, as illustrated in Fig. 1, consider three points, A, B, C, where the distance between AB, BC, and AC are 3, 1, and 6, respectively, as shown inFig. 1(b). It is impossible to show their relationships by three line segments on a 2-dimensional plane, as shown inFig. 1
(a). If there are four points, A, B, C and D, which are not on the same plane, as shown inFig. 1(c), it is impossible to present these four points on a 2-dimensional plane too.
Multidimensional scaling (MDS) [3,9] and a self-organizing map (SOM) [17] are commonly used tech-niques to map the similarities between points in a high dimensional space into a lower dimensional space (usually Euclidean). For instance, a visualization model, based on a scaling technique known as Sammon map
[22], was proposed to visualize adjacency data[7]; a SOM network was extended to classify decision groups[16]; the fisheye views and fractal views were used to support the visualization of a category map based on SOM[24]. However, there are two restrictions in current multi-dimensional scaling and SOM models limiting their use in visualizing Even Swap process. First, they do not show inconsistencies in even swaps. Inconsistencies in prefer-ences are common phenomena in decision-making. If these inconsistencies are significant, the reliability of decision-making might be reduced. Second, neither method displays the priorities of alternatives, which are essential for decision-making.
This study develops a Decision Ball model, based on the concept of multidimensional scaling techniques, to
Fig. 1. Advantages of a sphere model (a) Display line segments on a 2-D plane (b) Display curves on a sphere (c) Display four points that are not on the same plane.
visualize Even Swap process on a sphere. A sphere model can display more information than a 2-dimensional plane model, and is easier to read than a 3-dimensional cube model. By mapping the alternatives into the points on the surface of a hemisphere, the Even Swap process is illustrated as moving trajectories among related points. The DM can examine these trajectories of points to obtain the information listed below:
(i) Dissimilarities between alternatives. The longer the distance between alternatives on a sphere, the larger the dissimilarity between them.
(ii) The superiority (or dominance) of some alter-natives over others by checking their longitude. Alternatives, which are located on the same longitude, exhibit clear dominance in relation to each other.
(iii) The consistency of even swaps by checking the latitude of alternatives after each even swap. The even swap, which causes the largest latitudinal shift of a given alternative, is the most inconsistent. The proposed approach can be extensively applied in many fields. Possible applications are the selection of promotion plans in Marketing, investment decisions regarding financial products in Finance, evaluation of suppliers in Supply Chain Management, choice of col-leges in Personal Decisions…etc.
This paper is organized as follows: Section 2 briefly reviews the conventional Even Swap method. Section 3 develops a Decision Ball model based on the Even Swap concept to rank and display alternatives forming the main theoretical part of this paper. Therefore, readers only interested in the application of proposed method can skip Section 3. Section 4 uses an example to de-monstrate the whole decision process. Mathematical proofs of propositions and theorems are provided in the Appendices. A prototype Even Swap Decision Ball system has also been developed in this study, accessible from
http://140.113.72.1/~hlli01/index.htmto illustrate the use-fulness of the proposed method.
2. Review of the conventional Even Swap method Consider a set of alternatives A = {A1, A2,…, An} for solving a decision problem, where the decision maker selects m criteria to fulfill, denoted as c1, …, cm. Suppose the decision problem is a discrete problem, in which no combination of alternatives can be selected. The conventional Even Swap method[13,14]begins by creating a consequence table specified by the DM. Such a table contains the consequences that the alternatives
have for the given criteria. The DM can find the best alternative based on the following three steps.
Step 1. Eliminating dominated alternatives. The Even Swap method intends to eliminate as many alter-natives as possible. Since the fewer the alteralter-natives, the fewer the trade-offs the DM has to make. Aiis said to dominate Ajif alternative Aiis better than Aj in some criteria and no worse than Aj in all other criteria. All dominated alternatives are eliminated first.
Step 2. Choosing a target criterion. After eliminating dominated alternatives, the Even Swap method suggests that the DM chooses a target criterion whose values for all alternatives can be adjusted to be the same.
Step 3. Making even swaps. The DM chooses another criterion that can compensate for the changes in the target criterion. Then the DM assesses what changes in this criterion will compensate for the needed change in the target criterion. Finally, the even swaps are made and the target criterion is cancelled out. Steps 1 through Step 3 are applied iteratively until the best alternative is found. Here, an example is given to illustrate the steps of the conventional Even Swap method.
Example 1. This example comes from Harvard Busi-ness Review[14]which describes a business problem: which office to rent. The DM has five major decision criteria to fulfill (Table 1): (c1) sufficient space, (c2) good access to his clients, (c3) good office services, (c4) a short commuting time from home to office, and (c5) low cost. Office size is measured in square feet. The percentage of clients within an hour's drive from the office is used to measure the access to clients. A simple three-letter scale is used to describe the office services provided:“A” indicates full service; “B” means partial service; and “C” implies no service available. The commuting time is the average time in minutes needed to travel to work during rush hour, and cost is measured by monthly rent. Five alternative locations from A1 through A5are considered. The two rightmost columns of Table 1 are the upper and lower bounds of each criterion, as illustrated in the next Section.
Using the Even Swap method, the problem can be solved as follows:
Iteration 1 bStep 1N The DM can eliminates A5 immediately because A2dominates A5. The remain-ing alternatives are A1, A2, A3and A4.bStep 2N The
DM chooses “commuting time” for target criterion. bStep 3N He decides to increase it from 20 to 25 for A3and to decrease it from 45 to 25 for A1so that the commuting time of all four alternatives would be equivalent. He uses 8 percentage points increase in customer access for A3and 150 increases in monthly cost for A1to compensate for the changes in com-muting time for A3and A1, respectively (Table 2(a)).
Iteration 2 bStep 1N The DM can eliminate A1
because A4dominates A1. The remaining alternatives are A2, A3and A4.bStep 2N The DM chooses “office services” as a target criterion. bStep 3N He equates an
increase in service level from C to B for A3with a $100 increase in monthly costs, and equates a decrease in service level from A to B for A4with a $100 decrease per month (Table 2(b)). Clearly, both “time” and “services” criteria are the same for all alternatives and can be eliminated.
Iteration 3bStep 1N There is no dominated alternative. bStep 2N “Office size” is chosen as target criterion. bStep 3N The DM equates an increase in office size from 500 to 700 for A3with a $50 increase in monthly costs, and equates a decrease in office size from 950 to 700 for A4with a $300 decrease per month (Table 2(c)). Iteration 4 bStep 1N Alternative A3 is eliminated because A4 dominates A3. Only alternatives A2and A4 are remaining now (Table 2(d)). bStep 2N The DM chooses“customer access” as a target criterion. bStep 3N He makes an even swap between customer access and monthly cost by increasing 5 percentage points access for A2with an increase of $100 per month. Iteration 5 bStep 1N Alternative A2 is eliminated because A4 dominates A2. Since there is only one alternative remaining, the process can be terminated. Alternative A4is the best option.
The Even Swap method provides a rational process for reaching the best option in making a decision. However,
Table 1
The consequence table of Example 1 (A2≻A5)
Table 2
there still are some inadequacies. Take Example 1 for instance, illustrated as follows:
(i) As illustrated inTable 2, A4is the most preferred alternative. However, it is difficult to say which one of A1, A2, A3or A5is second and which is third.
(ii) The dissimilarities among alternatives are difficult to determine. For example, as illustrated in
Table 1, the DM finds it difficult to tell which are the dissimilarities among A1, A2, A3, A4and A5. (iii) Current Even Swap methods lack a mechanism
for showing up serious inconsistencies in Even Swaps. (The detailed illustrations will be dis-cussed in Section 3).
This study extends the concept of Even Swaps and proposes a model to assist the DM to rank alternatives, illustrate differences among alternatives, and check the inconsistencies of preferences.
3. The proposed decision ball model based on the even swap process
Consider a decision with a set of alternativesA={A1, A2, …, An}. The decision maker has m main decision criteria to fulfill, expressed as c1,…, cm. Denote ci,kas the kth raw criterion value of alternative Ai, expressed as Ai= Ai(ci,1, …, ci,k, …, ci,m). Denote Pck and cPk as the
lower and upper bounds of the raw criterion value of ck respectively. The value ofPck andPckcan be either given
by the decision maker directly or calculated by the minimum and maximum raw criterion value of ck. In Example 1, the value of Pck and cPk are assumed to be
specified by the decision maker, as listed in the two rightmost columns ofTable 1.
Here a sphere model based on the Even Swap process to rank n alternatives is proposed. An important assumption is that the proposed approach is dealing with objectives which can compensate for each other. In addition, the data types are restricted to continuous or ordinal data in this study. First, the following preproces-sing should be performed.
3.1. Data preprocessing
(i) Data transformation. All ordinal data has to be transformed into numerical data in advance. There are several methods to deal with such transforma-tion, such as monotonic transformation[15]. Since data transformation is not addressed by this study, it is assumed ordinal data can be mapped directly
into numerical data by the DM, for simplicity (This transformation has not to be linear).
(ii) All criterion values, cost and benefit, are trans-formed to a scale of 1 to 10 based on min–max normalization.
(iii) Criteria values representing costs, which the DM prefers to be as small as possible, are transformed by subtracting from 11.
The symbols ci,k and Ci,k are used for the kth cri-terion value of alternative Aibefore and after preproces-sing to distinguish between the raw and preprocessed criterion value. Denote PCk and CPk as the lower and
upper bounds of preprocessed criterion value Ck, where
P
Ck¼ 1 andCPk¼ 10. The preprocessed consequence
table of Example 1 is listed inTable 3. Take a benefit criterion value C1,1 and a cost criterion value C1,5 as examples, C1;1¼ 9T c1;1Pc1 Þ= cP 1Pc1 þ 1 ¼ 9T 800 500ð Þ= 1200 500ð Þ þ 1 ¼ 4:86 and C1;5¼ 11 9T c1;5Pc5 = n P c5Pc5 þ 1g ¼ 3:70. In order to rank alternatives, one kind of score function has to be chosen before developing the ranking model. There are two main types of score functions: additive and multiplicative score functions. Instead of using an additive function, the score func-tion of Ai is assumed to be in a multiplicative, non-linear Cobb-Douglas [6] form with constant return to scale in this study because it is a well established and commonly used form, and also a kind of power func-tion. Based on the concept of Brugha[4,5]and Barzilai
[1,2], relative measured weights and scores should be synthesized using a power function. In addition, a multiplicative score function is good at reflecting a reasonable marginal rate of substitution. Denote wkas the weight of criterion k. In order to reduce the com-plexity of the score function, all weights are assumed to be positive.
Table 3
The score function of Aiis expressed below Sið Þ ¼ ww 0Ci;1w1Ci;2w2 N Cwi;mm; ð1Þ
where w0, w1,…, wm≥0 and Pm
k¼1 wk¼ 1.
It is assumed that the values of the weights wkare implicitly in the DM's mind, but he can express them by even swaps.
For the purpose of comparison, two reference alterna-tives are defined: an ideal alternative A44¼ A44ðCP1; P
C2; N ;CPmÞ and a worst alternative A4¼ A4ðCP1; CP2;
N ; CPmÞ. Both alternatives may not be included in the
original alternative set A. Let the score of A⁎⁎be 10. Then, w0= 1 and S⁎= 1.
In order to distinguish between alternatives, the weighted difference δi,j(w) between alternative Aiand Ajis defined as di;jð Þ ¼w Max Ci;1; Cj;1 Min C i;1; Cj;1 " #w1 N Max C i;m; Cj;m Min C i;m; Cj;m " #wm ; ð2Þ where w0, w1,…, wm≥0 and Pm
k¼1 wk¼ 1. Since Max(Ci,k,
Cj,k)≤10 and Min(Ci,k, Cj,k)≥1 for all k, 1≤δi,j(w)≤10 andδi,j(w) =δj,i(w).
The idea of Expression (2) comes from the definition of an additive dissimilarity function, which is commonly defined asdi;jð Þ ¼w P m k¼1 wkjCi;k Cj;kj ¼ Pm k¼1ðwkðMax Ci;k; Cj;k
Min C i;k; Cj;kÞ with P m
k¼1wk¼ 1, where Ci,k
and Cj,kare the kth normalized criterion values of alter-native i and j. The multiplicative dissimilarity function can then be constructed in a similar way as di; jð Þ ¼w
jm k¼1 Max Cð i;k;Cj;kÞ Min Cð i;k;Cj;kÞ wk withP m
k¼1 wk¼ 1. Because all
cri-teria values in multiplicative form have been nor-malized to a[1,10]scale during the preprocessing stage, 1≤δi,j(w)≤10 and δi,j (w) =δj,i (w). For instance, the scores of alternatives with consequences (1, 1) and (2, 2) are 1 and 2, respectively. Based on the multiplicative concept, the score of the later alternative is 2 times that of the former one. From Expression (2), the dissimilarity between these two alternatives is 2. Comparing with the alternatives with consequences (9, 9) and (10, 10), the scores of these two alternatives are 9 and 10, respectively, where the second score is 1.1 times of the first one. From Expression (2), the dissimilarity between these two alternatives is 1.1.
Here Aiand Ajare mapped into the two points Piand Pj(denoted as the mapping points) on the surface of a hemisphere, such that the arc length connecting these two points expresses dissimilarity between Ai and Aj.
Since it is easier to compute the Euclidean distance than to compute the arc length, it is essential to have the following proposition:
Proposition 1. Let Piand Pjbe two points on the surface of a sphere centered at point O(0, 0, 0) with radius r. Denoteθi,jas the angle PiOPj, and denote PwiPj as the
shortest arc length along the great circle that passes through the two points. It is true that arc lengthPwiPjis
monotonically related to the Euclidean distancePPiPj.
The proof of this proposition is given in Appendix B. Referring to the non-metric multidimensional scaling method[9], it is more convenient to use the Euclidean distancePPiPj rather than the arc length to approximate
dissimilarities. Both approximation methods make very little difference to the resulting configuration[8]. There-fore, Euclidean distances are used in this paper for convenience.
Based on A⁎⁎, A⁎and Proposition 1, a hemisphere is generated. It is centered at (0, 0, 0) with radius 10. P⁎⁎ (the mapping point of A⁎⁎) is located at the north pole of this hemisphere with (x⁎⁎, y⁎⁎, z⁎⁎) = (0, 10, 0), while P⁎ (the mapping point of A⁎) is located at the equator with x⁎, y⁎, z⁎= (x⁎, 0, z⁎) where x⁎2+ z⁎2= 102, as depicted in
Fig. 2. It is clear that the distance between P⁎⁎(0, 10, 0) and O(0,0,0) is 10, and the distance between P⁎⁎(0, 10, 0) and P⁎(x⁎, 0, z⁎) is 10
ffiffiffi 2 p
. The Euclidean distance between Pi and Pj, denoted as di,j, is used to represent the logarithm of dissimilarity between Ai and Aj (i.e. ln(δi,j)): the larger the difference, the longer the distance. Furthermore, the alternative with a higher score is de-signed to be closer to the north pole so that alternatives are located on the concentric circles in the order of score from top view.
The relationship between Siand di,⁎⁎is defined as
di;44¼ 10 ffiffiffi 2 p 1 ln ð ÞSi ln ð Þ10 ; ð3Þ where if Si= 1 then di,⁎⁎= 10 ffiffiffi 2 p and if Si= 10 then di,⁎⁎= 0.
To map each Aito a point Pi(xi, yi, zi) on the surface of a hemisphere, the following conditions should be satisfied: ðiÞ di;44¼ 10 ffiffiffi 2 p 1 ln ð ÞSi ln ð Þ10 ; ðiiÞ x2 i þ y2i þ z2i ¼ 100; ðiiiÞ x2 i þ yð i 10Þ2þz2i ¼ di2;44:
The following proposition is deduced.
Proposition 2. The relationship between yi and Si is expressed as yi¼ 10 10 1 ln ð ÞSi ln ð Þ10 2 : ð4Þ
The proof of Proposition 2 is given in Appendix C. By mapping all Ai into the points on a sphere, rela-tionships among alternatives can be examined. These relationships are discussed below.
Consider the following propositions:
Proposition 3. On a hemisphere, suppose there are two alternatives Aiand Ajwith SiNSj. P⁎⁎, Piand Pjare on the same longitude if and only if ln(δi,j) = ln(δj,⁎⁎)− ln(δi,⁎⁎).
The proof of this proposition is given in Appendix D. Given an alternative set A = (A1, A2,…, An) and a weight vector w, a corresponding Decision Ball of A and w is denoted as DB(w, I) = {(xi, yi, zi)| i∊I={1,2,…, n}}, where (xi, yi, zi) is the coordinate of alternative Ai on the Decision Ball and yi≥0.
Proposition 4. Consider a DB(w, I) with two alter-natives Aiand Ajonly, i.e., I = {i, j}. If Ai≻Aj, then Pi and Pjare on the same longitude.
The proof of this proposition is given in Appendix E. Proposition 5. For a DB(w, I) for I= {i, j}. If Si(w)NSj(w) and Piand Pjare on the same longitude, then Ai≻Aj.
The proof of this proposition is given in Appendix F. The following theorem is then deduced:
Theorem 1. For a DB(w, I), I = {i, j}, given Aiand Aj where Si(w)NSj(w), if and only if Ai≻Aj, then Piand Pj are on the same longitude of the ball connecting P⁎⁎, Pi and Pj.
Denote DS(p) = {Ai1, Ai2,…, Aip} as a dominant set
composed of p alternatives with dominant relationships Ai1≻Ai2≻…≻Aip.
Proposition 6. Consider a dominant set DS(k) = {Ai1,
Ai2,…, Aik}. Let DB(w, I), I = {1, 2, …, k} be the
corresponding Decision Ball for the alternatives A1, A2, …, Ak, where A1≻A2≻…≻Ak. Connecting the mapping points P⁎⁎, P1, P2, …, Pk forms a longitude on the surface of this Decision Ball. That implies axi+ czi= 0 for i = 1, 2,…, k, where a and c are constants.
The proof of Proposition 6 is given in Appendix G. A Decision Ball DB(w, I) = {(xi, yi, zi)| i∊I={1,2,…, n}} is obtained by solving the model below.
Model 1. (A Decision Ball model with MDS concept) Min xi; yi;zi f g Obj¼ Xn i¼1 Xn jNi dˆi; jdi; j 2 s:t: dˆi; jV dˆp;qe; 8di; jb dp;q; 1V i; j; p; q V n; ð5Þ d2i; j¼ xi xj 2þ y i yj 2þ z i zj 2; 8i; j; ð6Þ yi¼ 10 10 1 ln ð ÞSi ln ð Þ10 2 ; 8i; ð7Þ x2i þ y2i þ z2i ¼ 100;8i; ð8Þ xizj¼ xjzi; 8AidAj; ð9Þ 10 V xiV 10; 0 V yiV 10; 10 V ziV 10; 8i; ð10Þ ε is a tolerable error.
The objective of Model 1 is to minimize the sum of squared differences between di,jand dˆi,j. Eq. (5) is the monotonic transformation from ln(δi,j) to dˆi,j based on the concept of non-metric MDS [9,18]: the higher the dissimilarity, the longer the distance. Because 1≤δi,j≤ 10 for all i, j,δi,jbδp,qimplies ln(δi,j)bln(δp,q) for all i, j, p, q. That is, if δi,jbδp,q, dˆi,j is smaller than dˆp,q; therefore, the distance between Aiand Ajis shorter than the distance between Apand Aq. The ε in Eq. (5) is a computational precision which can be normally set as 10− 6. Eq. (7) is from Proposition 2. All alternatives are graphed on the surface of the ball described by Eq. (8). Eq. (9) is obtained from Proposition 6. In Eq. (10), all alternatives are located on the upper hemisphere.
The number of variables used in Model 1 is n(n−1)+ 3n, where 3n is the number of decision variables used for xi, yi, ziand n(n−1) is the number of variables used for di,jand dˆi,j. The maximal number of constraints used in Model 1 is n(n−1)+6n in which Eqs. (5) and (6) account for n(n− 1) constraints and Eqs. (7)–(10) contain no more than 6n constraints. Model 1 is a non-linear model, which can be solved by some commercialized optimization software, such as Global Solver of Lingo 9.0[20], to obtain an optimum solution.
Fig. 3. Moving trajectory of concurrent points.
Let A′ be the alternative converted from Ai i by the DM through making even swaps. Aiand A′ are calledi concurrent alternatives. Piand Pi′, which are mapping points of Aiand A′, are called concurrent points.i Remark 1. Given two alternatives Aiand Aj, suppose the DM can stably make even swaps based on the score function in ( 1 ), then Pjcan be converted into another concurrent point Pj′ such that P⁎, Piand Pj′ are on the same longitude.
Fig. 3is used to interpret Remark 1. Here Si≥Sjbut Aidoes not dominate Aj. Via Even Swap process, Ajis converted to Aj′ where Ai≻Aj′. From Theorem 1, Pjis moved to a concurrent point Pj′ where P⁎, Piand Pj′ are on the same longitude. Aiis said to be consistently even swapped into Ai′ ifjSiS VSiijVe, where ε is a tolerable error. Theorem 2. Given Aiwith its concurrent alternative Ai′, and Piwith its concurrent point Pi′, Aiis consistently even swapped into Ai′ if and only if Piand Pi′ are on the same latitude.
The proof of this theorem is given in Appendix H. Current Even Swap methods lack a mechanism to advise the DM when there are serious inconsistencies among even swaps. For instance, as illustrated inTable 2(a) and (d), based on the score function in Eq. (1), the weights of criteria can be calculated as follows (all criterion values have been transformed in the data preprocessing stage as listed inTable 3):
(i) For A1 in Table 2(a) (mapped to A1 column of
Table 3),3:256:25w4w43:71w5w5¼ 1; then ww45¼ 2:
(ii) For A3 in Table 2(a) (mapped to A3 column of
Table 3),8:027:3w2w26:257w4w4 ¼ 1; then ww24¼ 1:2:
(iii) For A2 in Table 2(d) (mapped to A2 column of
Table 3),8:658:2w2w26:43:6w5w5¼ 1; then ww25¼ 10:76:
From (i) and (ii),w2
w5¼ 2:4, which is quiet different
from the result in (iii). These inconsistencies among even swaps, based on the same Cobb–Douglas score function, are not checked by the conventional Even Swap methods.
This study proposes Theorem 2 to check the con-sistency of Even Swap process made by the DM. For instance, as illustrated inFig. 3, Pj is consistently even swapped into Pj′, however, Pj″ is not even swapped from Pjconsistently. The more inconsistent a swap the DM has made, the bigger differences in score before and after even swap. That is, the difference between coordinate yjand yj′ is bigger.
Both Theorem 1 and Theorem 2 are utilized in this study to develop an algorithm to visualize the Even Swap process via a Decision Ball. By examining the moving trajectories of related points on a Decision Ball, the DM can rank the alternatives more consistently.
4. Decision process
This section uses the previous example (Example 1) to illustrate the process of ranking the alternatives using the proposed method. First, the DM sets initial weights for each criterion. If the DM cannot specify initial weights, equal weights are assumed at the beginning. These weights are iteratively adjusted when new preference information from the DM is acquired. The DM is assisted by a decision support system (DSS) composed of data, models and graphic interfaces. The process is summarized as follows:
Step 1 (Initialization). The DSS asks the DM to input a consequence table, to select criteria with cost features, to quantify the non-numerical criteria, and to specify the initial weights w(J ) for J = 0. J is used to record the number of iterations, and J = 0 indicates initial settings. A dominant set is initialized as DS(J ) =ϕ, for J=0. Step 2 (Displaying an initial Decision Ball). Set J = 0. Based on w(J), the DSS computes Si(w) and δi,j(w)
in Eqs. (1) and (2), respectively. A Decision Ball DB(w, I) is displayed to the DM after solving Model 1.
Fig. 4. Iteration 1: initial sphere.
Step 3 (Choosing the next alternative for even swap). The alternative Ai∉ DS(J) with the highest score is chosen as the next swap alternative by the DSS. The process stops if all alternatives are in DS(J) or the DM ceases to make further even swaps.
Step 4 (Making even swaps). The DM makes even swaps between Aiand alternatives in DS(J). Aiis changed to a concurrent alternative Ai′ such that Ai′ dominates or is dominated by an alternative in DS(J).
Step 5 (Updating weights and displaying a resulting sphere). For each even swap, the system computes the related weights by solving the following linear program:
Model 2. (Updating weights)
Min wp;wq f g a s:t: jwp ln Ci;p ln C V i; p þ wq ln Ci;q ln C V i;q
jVa; for an even swap Cp; Cq
in Ai; Xm k¼1 wk ¼ 1; wkz0; 8k: ð11Þ
The weights of unadjusted criteria are kept the same as those in the previous step.
Variables Ci,k and Ci,k′ are the value of criterion k of Ai before and after the even swap respectively. The resulting sphere based on the new weights is displayed. Then J is incremented, i.e. J = J + 1.
Step 6 (Updating the dominant set). Aiis added into DS(J). Reiterate Steps 3–6.
Take Example 1 to illustrate the whole decision process. It is important to note that the decision maker still deals with raw criterion values. After the decision maker inputs these values, the system automatically transforms the raw criterion values into preprocessed criterion values. In addition, the even swaps made here are different from those made in the original example described in Section 2 because all dominated alternatives are eliminated in the original example; however, the proposed approach tries to rank all alternatives so that all alternatives have to be kept and compared.
Iteration 1. At Step 1, the DM inputs his consequence table, upper and lower bound values of each criterion (Table 1), where c4, c5are criteria with a cost feature. Suppose the DM inputs the initial weights w(1) = (w1, w2, w3, w4, w5) = (0.2, 0.2, 0.2, 0.2, 0.2). The DSS asks the DM to answer some questions.
bDSSN Consider the qualitative criterion c3. Please quantify the values of service level A, B and C. bDMN 4, 2, 1. (The preprocessed values are 10, 4, 1, respectively, using min–max normalization).
Fig. 5. Iteration 2 (a) Adjusting A2with respecting to A4(b) Resulting sphere.
Fig. 6. Iteration 3 (a) Adjusting A1with respecting to A2(b) Resulting sphere.
The transformed consequence table after preprocessing is listed inTable 3. At Step 2, based on the initial weights, the dissimilarities between alternatives and scores of alternatives are calculated. An initial sphere (Fig. 4) is displayed to the DM. Here A4has the highest score. DS(1) = {A4}.
Iteration 2. A2is chosen as the swapped alternative with A4since A2is the next best alternative.
bDSSN Examining the table values inFig. 5(a). Choose a target criterion of A2from {c1, c2, c3, c5}, and adjust its value. The adjusted value should be the same as the target criterion of A4.
bDMN c5and 1900.
In the same way, the DM makes a 150 increase in c1to compensate for the increase of c5from 1700 to 1900 (A2is changed to a concurrent point A2′, and A4≻A′). Model 2 is then formulated as the following program:2
Min w1;w5 f g a s:t: jw1ðln 3:57ð Þ ln 5:5ð ÞÞ þ w5ðln 6:4ð Þ ln 2:8ð ÞÞjV a; w2¼ w3¼ w4¼ 0:2; X5 k¼1 wk¼ 1; wkz0; 8k ¼ 1; N ; 5:
Solving the above program yields w(2) = (0.26, 0.2, 0.2, 0.2, 0.14). The resulting sphere is shown inFig. 5(b). At Step 6, the DSS sets DS(2) = {A4, A2′}.
Fig. 7. Iteration 4 (a) Adjusting A5with respecting to A1(b) Resulting sphere.
Fig. 8. Iteration 5 (a) Adjusting A3with respecting to A5(b) Resulting sphere.
Iteration 3. Alternative A1is chosen as the swap alternative. Suppose the DM equates a decrease in c3from A to B with a 25 increase in c2, and equates an increase in c5from 1850 to 1900 with a 50 increase in c1. The corresponding changes are depicted inFig. 6(a) and (b). DS(3) = {A4, A′ , A2 1′}. The top three options have been found. The DM can then choose to terminate or continue to the next iteration.
Iteration 4. A5is chosen as a swap alternative. Suppose the DM equates an increase in c3from C to B with a 200 increase in c5, and equates an increase in c4from 30 to 45 with a 100 increase in c1, as listed inFig. 7(a).Fig. 7(b) shows the resulting sphere.
Iteration 5. Suppose the DM wants to continue the process. A3 is chosen as a swap alternative. Suppose the consequence table and corresponding Decision Ball after even swaps are as shown inFig. 8(a) and (b), where DS(5) = {A4, A2′, A1′, A′, A5 3′}. The process is then terminated.
The consistencies among even swaps can be checked by the moving trajectory of concurrent points. The even swap, which causes the largest latitudinal shift of a given alternative, is the most inconsistent. For instance, the moving trajectory of A3is shown inFig. 9, where 3Jstands for concurrent point P3after the Jth iteration. The most inconsistent even swaps the DM has made are at Iteration 2 and 5 because 32and 35are furthest away from the latitude formed by all 3Jbased on Theorem 2. Here the scores of points 3132, 33, 34and 35are 3.48, 3.01, 3.49, 3.40 and 4.00, respectively. The DM can revise these inconsistencies by re-iterating the even swap process at Iteration 2 or 5. For instance, if the DM chooses to re-iterate the even swap process at Iteration 2 (as listed inFig. 5a) and equates an increase in c5from
Fig. 9. The moving trajectories of A3after even swaps.
1700 to 1900 for A2with a 250 increase in c1, the score of 32is changed from 3.01 to 3.25. It is worth noticing that, since the sphere can be rotated to present different views, the relative longitude positions of concurrent points might be different from those inFigs. 4–8. In addition, the position of the concurrent point in the first iteration can be ignored because the initial weights may be given arbitrarily.
This problem was solved by Global Solver of Lingo 9.0[20]on a Pentium 4 personal computer. The running time was less than five seconds for each iteration.
5. Concluding remarks
The Even Swap is a rational and straightforward method which provides a mechanism for making trades so that a DM can make the best choice. Based on the concept of Even Swaps, this study proposes a graphic method to help the DM rank and visualize alternatives. Rather than revealing the best option only in a conven-tional Even Swap method, the proposed approach can fully rank all alternatives. In addition, the DM can find the similarities among alternatives, can iteratively adjust preferences, and see the corresponding changes on the Decision Ball.
The proposed approach meets most of the require-ments of a useful decision model, known as decision calculus[21]. First, it is simple because it is easy for a DM to understand. Second, it is robust because the method is logically correct for finding a rational solution. Third, it is easy to control, adapt, and complete. Finally, since the DM can adjust inputs and visualize outputs via the Decision Balls, the proposed approach facilitates convenient communication between the DM and the DSS.
One restriction of this approach is the running time that may considerably increase when the number of alternatives becomes large because the time complexity of Model 1 is n2. In future study, how to linearize this non-linear model to deal with large size problems can be
addressed. Nevertheless, because the Even Swap method is good for small size problems or the final stage of decisions, the proposed approach is especially help in the case of alternatives fewer than 10.
Appendix A. A letter from Benjamin Franklin to Joseph Priestly
In the affair of so much importance to you, wherein you ask my advice, I cannot, for want of sufficient premises, advise you what to determine, but if you please I will tell you how. When those difficult cases occur, they are difficult, chiefly because while we have them under consideration, all the reasons pro and con are not present to the mind at the same time; but sometimes one set present themselves, and at other times another, the first being out of sight. Hence the various purposes or inclinations that alternatively prevail, and the uncertainty that perplexes us. To get over this, my way is to divide half a sheet of paper by a line into two columns; writing over the one Pro, and over the other Con. Then, during three or four days consideration, I put down under the different heads short hints of the different motives, that at different times occur to me, for or against the measure. When I have thus got them all together in one view, I endeavor to estimate their respective weights; and where I find two, one on each side, that seem equal, I strike them both out. If I find a reason pro equal to some two reasons con, I strike
out the three. If I judge some two reasons con, equal to three reasons pro, I strike out the five; and thus proceeding I find at length where the balance lies; and if, after a day or two of further consideration, nothing new that is of importance occurs on either side, I come to a determination accordingly. And, though the weight of the reasons cannot be taken with the precision of algebraic quantities, yet when each is thus considered, separately and comparatively, and the whole lies before me, I think I can judge better, and am less liable to make a rash step, and in fact I have found great advantage from this kind of equation, and what might be called moral or prudential algebra. Wishing sincerely that you may determine for the best, I am ever, my dear friend, yours most affectionately. (London, Sept 19, l772)
From:“Letter to Joseph Priestly”, Benjamin Franklin Sampler[10].
Appendix B
Proof of Proposition 1.PPiPj¼ 2r sinh2i;j;
w PiPj¼ rhi;j¼ 2r sin1 P PiPj 2r . Since 0 V P PiPjV 2r, we have 0 V P PiPj 2r V 1.
That is, 0V sin1
P PiPj 2rV p 2[15]. Because sin 1 PPiPj 2r is mono-tonically related toPPiPj 2r while 0V P PiPj 2r V1; w PiPj, is mono-tonically related toPPiPj. □ Appendix C
Proof of Proposition 2. The variable di,⁎⁎ represents the Euclidean distance between Ai located at (xi, yi, zi) and A⁎⁎located at the north pole (0, 10, 0).
di;442 ¼ xð i 0Þ2þ yð i 10Þ2þ zð i 0Þ2 ¼ x2 i þ y 2 i þ z 2 i 20yiþ 100 ¼ 102 20y iþ 100 ¼ 200 20yi:
From Eq. (3), d2i;44¼ 200 1 lnð ÞSi
lnð Þ10 2 ¼ 200 20yi, we can obtain yi¼ 10 10 1 lnlnð ÞSi 10 ð Þ 2 . That is, if Si= 1 then yi= 0, and if Si= 10 then yi= 10. □ Appendix D
Proof of Proposition 3. If P⁎⁎, Pi and Pj are on the same longitude with SiNSj, then
w
PiPj¼ wP44Pj wP44Pi.
That is, the value ofPw44Piþ wPiPj wP44Pjis minimal for
known Siand Sj. By referring to Proposition 1, the value of d⁎⁎,i (w) + di,j(w)−d⁎⁎, j(w) is minimal. Since di,jis used to represent ln(δi,j), it implies ln(δi,⁎⁎) + ln(δi,j)− ln(δ⁎⁎,j) is minimal. ln di;44þ ln d i;j ln d 44;j ¼X m k¼1 wkln Cð Þ ln Ck i;kþ Xm k¼1 wkð ln Max C i;k; Cj;k ln Min C i;k; Cj;kÞ Xm k¼1 wklnð Þ ln CCPk j;k ¼Xm k¼1 wk ln Max Ci;k; Cj;k ln C i;k þXm k¼1 wkln C j;k ln Min C i;k; Cj;k:
Since 1≤ci,k≤10 for all i, the minimum value of ln(δi,⁎⁎) + ln(δi,j)− ln(δ⁎⁎,j) is 0. That implies ln(δi,j) = ln(δj,⁎⁎)− ln(δi,⁎⁎). On the other hand, if ln(δi,j) = ln(δj,⁎⁎)−ln(δi,⁎⁎), it implies d⁎⁎,i(w) + di,j(w)−d⁎⁎, j(w) is minimal, which means P⁎⁎, Piand Pjare located on the same arc along the great circle. That is, P⁎⁎, Piand Pj are on the same
longitude. □
Appendix E
Proof of Proposition 4. Ai≻Aj implies Ci,k≥Cj,k, for all k. From Eq. (3),
ln di;j
¼ Xm
k¼1
wkln Max C i;k; Cj;k ln Min C i;k; Cj;k
! ¼X m k¼1 wk ln Ci;k ln C j;k ¼X m k¼1 wk ðlnð10Þ ln Cj;k ln 10 ð Þ ln C i;kÞ ¼ ln d j;44 ln d i;44:
From Proposition 3, P⁎⁎, Piand Pjare on the same longitude.
Appendix F
Proof of Proposition 5. Since Si(w)NSj(w) and Pi, Pj are on the same longitude, ln(δi,i) = ln(δj,⁎⁎)−ln(δi,⁎⁎). lndj;44 ln d i;44¼P m k¼1wk ln 10ð Þ ln Cj; k ln 10 ð Þ þ ln C i; kÞ ¼ P m k¼1 wk ln Ci; k ln C j; k ¼ ln d i; j¼ Pm k¼1wk ln Max Ci;k;Cj;k ln Min C i;k;Cj;k ; which implies Ci,k≥Cj,kfor all k. That is Ai≻Aj.
Appendix G
Proof of Proposition 6. The proof is similar to Pro-positions 4 and 5. In addition, all points mapped at the same longitude of a sphere must be located at the same
cutting plane of a sphere, i.e. axi+ byi+ czi+ d = 0, where a, b, c and d are constants. Because the cutting plane has to pass through the origin (0, 0, 0) and the north pole (0, 10, 0), b = d = 0. That is, all points located at the same longitude of a sphere must satisfy equality axi+ czi= 0. Appendix H
Proof of Theorem 2. (i) If Ai is consistently even swapped into A′, then Si i=Si′, which means yi= yi′ (Proposition 2). Therefore, Pi and Pi′ are on the same latitude. (ii) If Piand Pi′ are on the same latitude, then yi= yi′ which implies Si= Si′. A′ therefore is consistentlyi even swapped from Ai.
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Han-Lin Li is a Chair Professor of National Chiao Tung University, Taiwan. He received his PhD degree from University of Pennsylvania, USA. His articles have appeared in Decision Support Systems, Operations Research, Fuzzy Sets and Systems, Journal of the Operational Research Society, European Journal of Operational Research, Journal of Global Optimization, Computers and Operational Research, and many other publications.
Li-Ching Ma is an Associate Professor in the Department of Information Management at National United University, Taiwan. She received her PhD degree in Information Management from National Chiao Tung University, Taiwan. Her research interests include decision-making, visualization, and optimization.
