( )/( )
kj j kj j j
r = f −f f −f− . (A10)
Step 2: Calculate the group utility mean and maximum regret. The values can be computed using
1 n
k j kj
j
S w r
=
=
∑
(the average synthesized gap for all criteria) and Qk =max {j kjr j=1,2,..., }n (the maximum gap in k criterion for priority improvement) respectively.Step 3: Calculate the index value using Eq. (A11).
* * * *
( )/( ) (1 )( )/( )
k k k
R =v S S− S−−S + −v Q Q− Q−−Q , (A11) where k = 1, 2,…, m, S* = mini Si or S* = 0 (when all criteria have been achieved to the aspira-tion level) and S−=maxi iS or S−= (the worst situation); 1 Q*=mini iQ or setting Q =* 0 and Q−=maxi iQ or setting Q−= , and v is presented as the weight of the strategy of the 1 maximum group utility (priority improvement). Conversely, 1– v is the weight of individual regret. Therefore, we can rewrite Rk=vSk+ −(1 )v Qk, when S = , * 0 S−= , 1 Q = and * 0 Q−= .1
Downloaded by [National Chiao Tung University ] at 15:01 28 April 2014
APPENDIx III.
Extension of De Novo programming to changeable spaces for an MOP problem Multi-objective programming (MOP) problems can mathematically be represented as
1 2
maxf ( ), ( ),..., ( )x f x fk x ⇒ Objective space s t. . ≤
≥0 Ax b
x ⇒ Decision space, ′ ≤ ′ =
≥ 0
p Ax p b B x
where p denotes a vector of unit prices of resources. The superscript
'
denotes transpose.Traditional MOP problems find the objective space subject to the decision space, which assumes that the decision space cannot be changed. We developed new “changeable spaces”
method to relax these assumptions for MOP problems based on De Novo programming (Zeleny 1986, 1990).
mincx . . ( )i i
s t f x ≥ f∗, 1,2,...,i= k (or setting f to be an aspiration level)i*
≥0 x ,
where vector c p A denotes unit prices of decision variables, and = ′ fi∗ denotes the ideal point of objective i (we also can set f to be an aspiration level).i*
[Example]
Graph Example max f1 …… profit max f2 …… quality
Reshaping the feasible set to include the missing “Good” alternative Given design with natural quality – profit trade-offs as follows:
Good unavailable options max
max Feasible Space
a
b
c Profit
Quality Ideal Point Trade-off
Fig. A2. Graphic example of De Novo programming
Downloaded by [National Chiao Tung University ] at 15:01 28 April 2014
x and 1 x , with each of them consuming five different resources (unit market prices 2 of resources are given).
According to De Novo programming, the Maximum levels of two products can be cal-culated by mathematical programming:
Profit: max ( , ) 400f x x1 1 2 = x1+300x2
The data are summarized as follows:
Unit price
$ Resources
(Raw material) Technological coefficients
(Resource Requirement) No. of units (Resource portfolio)
x1 x2
30 Nylon 4 0 20
40 Velvet 2 6 24
9.5 Silver thread 12 4 60
20 Silk 0 3 10.5
10 Golden thread 4 4 26
– The costs of the given resources portfolio:
(30 20) (40 24) (9.5 60) (20 10.5) (10 26) $2600× + × + × + × + × = – Unit costs of producing one unit of each of the two products:
1:(30 4) (40 2) (9.5 12) (20 0) (10 4) $354
Downloaded by [National Chiao Tung University ] at 15:01 28 April 2014
– Maximum f x x in total quality index2 1 2( , )
where vector p denotes vector unit price of each resource; vector ′c p A denotes “product = ′ unit cost”, B denotes budget.
De Novo programming: – Cost of the newly designed system:
Answer: (30 16.12) (40 23.3) (9.5 58.52) (20 7.62) (10 26.28) $2386.74× + × + × + × + × =
The data are summarized as follows:
Unit price
$ Resources
(Raw material) Technological coefficients
(Resource Requirement) No. of units (Resource portfolio)
Downloaded by [National Chiao Tung University ] at 15:01 28 April 2014
Toward a MCDM New Era – Professor Tzeng’s Roadmap Philosophy
Taking True Responsibility, Creating Added Value, and
Making Contribution through MCDM Knowledge to Global Society
concept Graphical Representation Approach
Value Fig. A3. Extension of changeable decision space and aspiration level
James J. H. LIOU received his Ph.D. degree from the Department of Mechanical and Aerospace Engineer-ing of University of Missouri-Columbia, USA in 1996. After workEngineer-ing for airline industry for 8 years, he became an assistant professor in the Department of Air Transportation at the Kainan University, Taiwan.
He turned out to be an associate professor in 2008. Currently, he is an associate professor in the Depart-ment of Industrial Engineering and ManageDepart-ment at National Taipei University of Technology. Dr Liou’s primary research interest is data mining, including feature selection, clustering, ensemble methods, and decision support system. Recently, he became interested in applying data mining algorithms to solve some business problems in customer targeting, e-commerce, and safety science. He has publications in numerous journals, including Journal of Air Transport Management, Expert Systems with Applications, International Journal of Production Research, Applied Soft computing, Information Sciences and Euro-pean Journal of Operational Research etc.
Downloaded by [National Chiao Tung University ] at 15:01 28 April 2014
Gwo-Hshiung TZENG was born in 1943 in Taiwan. In 1967, he received the Bachelor’s degree in busi-ness management from the Tatung Institute of Technology (now Tatung University) , Taiwan; in 1971, he received the Master’s degree in urban planning from Chung Hsing University (Now Taipei University), Taiwan; and in 1977, he received the Ph.D. degree course in management science from Osaka University, Osaka, Japan.
He was an Associate Professor at Chiao Tung University, Taiwan, from 1977 to 1981, a Research Associate at Argonne National Laboratory from July 1981 to January 1982, a Visiting Professor in the Department of Civil Engineering at the University of Maryland, College Park, from August 1989 to August 1990, a Visiting Professor in the Department of Engineering and Economic System, Energy Modeling Forum at Stanford University, from August 1997 to August 1998, a professor at Chaio Tung University from 1981 to 2003, and a Chair Professor at Chiao Tung University. He got National Distinguished Chair Professor (Highest Honor offered by the Ministry of Education Affairs, Taiwan) and Distinguished Re-search Fellow (Highest Honor Offered by NSC, Taiwan) in 2000. His current reRe-search interests include statistics, multivariate analysis, network, routing and scheduling, multiple criteria decision making, fuzzy theory, hierarchical structure analysis for applying to technology management, energy, environ-ment, transportation systems, transportation investenviron-ment, logistics, location, urban planning, tourism, technology management, electronic commerce, global supply chain, etc. He has got a Highly Cited Paper (March 13, 2009) ESI “Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS” published in the “European Journal of Operational Research” on July 16th, 156(2), 445–455, in 2004 it has been recently identified by Thomson Reuters’ Essential Science IndicatorsSM to be one of the most cited papers in the field of Economics.
He has got the MCDM Edgeworth-Pareto Award from the International Society on Multiple Criteria Decision Making (June 2009), has got the Pinnacle of Achievement Award 2005 of the world, and had got the National Distinguished Chair Professor and Award (highest honor offered) of the Ministry of Educa-tion Affairs of Taiwan and three times of distinguished research award and two times of distinguished research fellow (highest honor offered) of National Science Council of Taiwan. Fellow IEEE Member (from September 30, 2002). He organized a Taiwan affiliate chapter of the International Association of Energy Economics in 1984 and he was the Chairman of the Tenth international Conference on Multiple Criteria Decision Making, July 19–24, 1992, in Taipei; the Co-Chairman of the 36th International Confer-ence on Computers and Industrial Engineering, June 20–23, 2006, Taipei, Taiwan; the Chairman of the International Summer School on Multiple Criteria Decision Making 2006, July 2–14, Kainan University, Taiwan. He is a member of IEEE, IAEE, ISMCDM, World Transport, the Operations Research Society of Japan, the Society of Instrument and Control Engineers Society of Japan, the City Planning Institute of Japan, the Behaviormetric Society of Japan, the Japan Society for Fuzzy Theory and Systems; and partici-pating many Society of Taiwan. He is editors-in-Chief of International Journal of Information Systems for Logistics and Management, and so on.
Downloaded by [National Chiao Tung University ] at 15:01 28 April 2014