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Flow

1. Download the pmflow file.
2. Make it executable
   chmod +x filename

3. Example: (save as input.txt): Copy
   
   
   flowname: {test}; vertices: {A,B,C,D,E,F,G}; idxin: {0, 1, 2, 3, 3, 4, 4, 4}; idxout: {1, 3, 4, 4, 1, 3, 3, 0}; flowvalues: {-1, 0, -2, 4, 7, 2, 1, -2}; aggmethod: {average}; #flownorm: {2.5}; Xrange: {2,10};

Option: -a average

Showing edgelist: (test): names: {A,B,C,D,E,F,G} (size: 7) idxin: {1,0,3,4,3} (size: 5) idxout: {0,4,1,2,4} df: {1,2,3.5,2,0.333333} Showing components: 1 -> {0,1,2,3,4} isolated_vertices: {5,6} Potential (project 'test'): Comp_1 (node -> X):: ------------------- D(3) -> 10 A(0) -> 7.07527 E(4) -> 6.12903 B(1) -> 5.95699 C(2) -> 2 inc = 41.85° max inc (5% quantile) = 9.9° (Caklovic) max inc (10% quantile) = 11.75° (Caklovic) aggregation: average base: 2 xrange: (2,10) file: input.txt

Option: -a sum

Showing edgelist: (test): names: {A,B,C,D,E,F,G} (size: 7) idxin: {1,0,3,4,3} (size: 5) idxout: {0,4,1,2,4} df: {1,2,7,2,1} Showing components: 1 -> {0,1,2,3,4} isolated_vertices: {5,6} Potential (project 'test'): Comp_1 (node -> X):: ------------------- D(3) -> 10 E(4) -> 5.04762 A(0) -> 4.66667 B(1) -> 2.7619 C(2) -> 2 inc = 35.86° max inc (5% quantile) = 9.9° (Caklovic) max inc (10% quantile) = 11.75° (Caklovic) aggregation: sum base: 2 xrange: (2,10) file: input.txt

Option: -a undef

Showing edgelist: (test): names: {A,B,C,D,E,F,G} (size: 7) idxin: {1,1,4,3,3,4,4,0} (size: 8) idxout: {0,3,2,4,1,3,3,4} df: {1,0,2,4,7,2,1,2} Showing components: 1 -> {0,1,2,3,4} isolated_vertices: {5,6} Potential (project 'test'): Comp_1 (node -> X):: ------------------- D(3) -> 10 E(4) -> 7.23077 A(0) -> 6.76923 B(1) -> 3.69231 C(2) -> 2 inc = 59.45° max inc (5% quantile) = 9.9° (Caklovic) max inc (10% quantile) = 11.75° (Caklovic) aggregation: multigraph base: 2 xrange: (2,10) file: input.txt

Flow visualization (igraph)

1. Download R-script source file flow.lib.R.
2. Download R-script file flow.plot.R.
3. Open flow.plot.R in R-Studio.
4. For input input.txt you wil get the graph

   

5. You may notice that node size depends upon its potential (option -a average).

Table

1. Download the table file.
2. Make it executable
   chmod +x filename

3. Example: (save as table.csv): Copy

   Tab;Physics;Gym;Drawing;Singing John;1.5;2.5;1;* Mary;*;1;3;1 Ted;2.2;*;2;* Fida;4;3;1;3 Anne;4;3;*;1 Peter;*;5;*;6

Options: -a [sum, average, multigraph]

Another example

Grades;I;II;III A;3;9;6 B;6;7;9 C;9;6;8 weight;1;1;1 flownorm;1;1;1

Duality

Dual;u;v;x;y;flownorm A;5;4;6;5;2 B;5;5;5;4;1 C;0;4;4;3;2 D;0;4;3;2;4 flownorm;2;3;1;5 weight;1;1;1

R-matrix

1. Download the pmrmatrix file.
2. Make it executable
   chmod +x filename

Single DM

1. Copy (save as rmatrix.csv)

   BanaeCosta;A;B;C;D;E A;1;2;3;5;9 B;*;1;2;4;9 C;*;*;1;2;8 D;*;*;*;1;7 E;*;*;*;*;1

2. Command line:
   ./pmrmatrix -f rmatrix.csv -m EVM
   
   
   You will get the output:

   Arguments: 2 - file: rmatrix.csv - method: EVM Testing transitivity ... OK Eigenvalue method: Perron vector: ------------------- 0 -> 0.426245 1 -> 0.280924 2 -> 0.16523 3 -> 0.10074 4 -> 0.0268609 CR = 0.051 lambda (Perron root) = 5.22993 max lambda (Saaty cr=0.1) = 5.4497 max lambda (Alonso, Lamata alpha=0.02) = 5.08996 max lambda (Alonso, Lamata alpha=0.1) = 5.44982 max lambda (Alonso, Lamata alpha=0.5) = 7.2491 Testion COP ... OK COP ratio is violated on (i,j)/(k,l): [((3,4)/(0,3))] COP ratio values (w_i/w_j)/(w_k/w_l): [(3.750423/4.231155)] data source: rmatrix.csv

   
   You may also try:
   ./pmrmatrix -f rmatrix.csv -m GMM
   ./pmrmatrix -f rmatrix.csv -m PM

   Incomplete matrix

   Copy (save as matrix.csv)

   Incomplete;A;B;C;D;E A;1;2;3;5;* B;*;1;2;4;9 C;*;*;1;2;* D;*;*;*;1;7 E;*;*;*;*;1

   Methods: GMM and EVM makes no sense, but
   

   ./pmrmatrix -f rmatrix.csv -m PM
   is legal.

   An example

   Copy (save as extreme.csv)

   Ex;A;B;C;D;E A;1;2;3;5;9 B;2;1;2;4;9 C;3;2;1;2;8 D;5;4;2;1;7 E;9;9;8;7;1

   Using 3 options EVM, GMM, PM you will get three different results:

   	EVM	GMM	PM
   A	0.196265	0.183133	0.2
   B	0.181652	0.161498	0.2
   C	0.162806	0.148919	0.2
   D	0.18307	0.18447	0.2
   E	0.276206	0.32198	0.2

Aggregation of reciprocal matrices

1. Download group.zip
2. Unpack it in the same folder where is rmatrix executable.
3. Command line (group aggregation with GMM):
   ./pmrmatrix -f group -m gGMM

4. Command line (group aggregation with PM):
   ./pmrmatrix -f group -m gPM

Hierarchy

1. Download the pmhier file.
2. Make it executable
   chmod +x filename

Top down hierarchy

1. Copy Example: (save as hier1.txt):

   name: {Hier1} ; base: {1.5} ; levels: {A,B,C} ; # level nodes structure: {A: {a1, a2}, B: {b1, b2, b3}, C: {c1, c2, c3}} ; # initialization weight: {1.1:1, 1.2:1} ; # parents and children children: { 1.1: {3.1, 3.2, 3.3}, 1.2: {2.1, 2.2, 2.3}, 2.1: {3.1, 3.2, 3.3}, 2.2: {3.1, 3.3}, 2.3: {3.1, 3.2, 3.3} } ; #flownorm: {1.1:1,1.2:1}; # preferences by pairs preferences: { 1.1: {(3.1, 3.2): 2, (3.1, 3.3): 4, (3.3, 3.2):6}, 1.2: {(2.1, 2.2):0, (2.2, 2.3):1, (2.1, 2.3):3}, 2.1: {(3.1, 3.2):1, (3.1, 3.3):2}, 2.2: {(3.1, 3.3):1}, 2.3: {(3.1, 3.2): -1, (3.1, 3.3): 1, (3.3, 3.2):0} } ; # sum, average aggregation : {average} ; # top -> bottom (once) htype: {top-down} ;

   
   This hierarchy is an example with a parent node (1.1) which has child nodes in a third level and 'jumps over' the second level. Such feature is not allowed (as I know) in the Expert choice software.
   Command line:

   ./pmhier -f hier1.txt > a.out

   
   You will get the output (in a.out file):

   --------------------------- HIERARCHY (Hier) model: top-down -------------------------- init.weight 1.1 -> 0.5 1.2 -> 0.5 Level 2. B Weights: Comp_1 (node -> W) (sorted) 2.1(0) -> 0.400531 2.2(1) -> 0.349896 2.3(2) -> 0.249572 inc = 21.42° Level 3. C Weights: Comp_1 (node -> W) (sorted) 3.1(0) -> 0.387554 3.3(2) -> 0.358145 3.2(1) -> 0.254301 inc = 40° Aggregation method(s): average base: 1.5 call: ./pmhier -f hier1.txt

2. Restriction of the hierarchy

   A command line
   ./pmhier -f hier1.txt -r 1.2
   restricts the hierarchy down from the 'root 1.2' (a2) and solves the restricted hierarchy.
   The output is:

   --------------------------- HIERARCHY (Hier1) model: top-down Restriction (root = 1.2) -------------------------- init.weight: 1.2 -> 1 Level 2. Weights: Comp_1 (node -> W) (sorted) 2.3(2) -> 0.464813 2.2(1) -> 0.354719 2.1(0) -> 0.180468 inc = 21.42° Level 3. Weights: Comp_1 (node -> W) (sorted) 3.1(0) -> 0.344602 3.2(1) -> 0.339556 3.3(2) -> 0.315842 inc = 47.66° Aggregation method(s): average base: 1.5 call: ./pmhier -f hier1.txt -r 1.2

   Group decision (GDM)

   Top-down aggregation

   A group of flows may be aggregate using a hierarchical model. In example hier1.txt a hierarchy is structered with 3 levels:
   structure: {A: {a1, a2}, B: {b1, b2, b3}, C: {c1, c2, c3}}.
   The first level $A=\{a1, a2\}$ defines a group of decision makers, the second level $B=\{b1,b2,b3\}$ is the set of criteria for evaluation of the alternatives $C=\{c1,c2,c3\}$ in the third level. Such idea is promoted by Saaty, which, in my opinion is not a correct one, because the weights of the second level in the hierarchy are influenced by the preferences of all DMs from the first level.
   Remark. A hidden assumption in such formulation is that all decision makers use the same set of criteria which may not be the case in reality.

   Restrict and aggregate (ABC)

   A command line:

   ./pmhier -f hier1.txt --abc

   (--abc means Aggregation By Criteria) aggregates a hierarchy by criteria in the top level. For each criterion in the top level, an individual flow of the bottom level is calcualted solving a restricted hierarchy (of that criteria).
   The output is:

   --------------------------- HIERARCHY (Hier1) model: ABC --------------------------- init.weight 1.1 -> 0.5 1.2 -> 0.5 flow.norm Aggregation by criteria: Hier1 Comp_1 (node -> W) (sorted) 3.1(0) -> 0.453036 3.3(2) -> 0.346428 3.2(1) -> 0.200536 inc = 40.01° Aggregation method(s): average base: 1.5 call: ./pmhier -f hier1.txt --abc

   
   Remark. A hidden assumption in such formulation is that all decision makers use the same set of criteria which may not be the case in reality.

   Divide and aggregate

   This kind of aggregation assume that each decision maker defines his own hierarchy with his own criteria and subcriteria (or even without them). The only restriction is that hierarchies have the same bottom level (names of the alternatives).
   Example. Download a group data and unpack them in the same folder where is pmhier executable. In the folder named group there are hierarchies of all decision makers and a file named group.info

   #! interface: {members}; # Example of group.info file. A list of members # which is included in analysis is given below. # This is a subset of the set of members. members: {dm1, dm2}; # initialization weight: {dm1:1, dm2:1, dm2.cpy:0} ; # sum, average, undef aggregation : {sum} ; ## group type: dual = exclude self comparisons # uncomment if neccessary # gtype: {dual} ; ## run [onestep|iterate] only if gtype=dual # uncomment if neccessary # method: {iterate};

   A command line
   ./pmhier -f ./group/ -m
   (-m means members) gives as ouput

   ---------------------------------------- Group aggregation (Group_A) ---------------------------------------- Members weights dm1 -> 0.5 dm2 -> 0.5 dm2.cpy -> 0 Weight (dm1+dm2) Comp_1 (node -> W) (sorted) c1 -> 0.594233 c3 -> 0.290978 c2 -> 0.114789 inc = 44.59° Aggregation method(group): sum Aggregation method(s): dm1:undef,dm2:sum base: 0 call: ./pmhier -f group/ -m

   Remark. Please pay attention on the results of ABC aggregation and aggregation of individual hierarchies: they are the same!.

   Group self-ranking

   As above, each DM has his own hierrarchy with one difference: the alternatives are decision makers themselves. There are 2 possibilities of self-evaluation of a group of decision makers: (1) with supervisor and (2) without supervisor.

   (1) In the first case the supervisor gives a priori members weights. The procedure is the same as in Divide and aggregate. The supervisor also has a possibility to exclude each DM from the preferences in his hierarchy. For such possiblity a group type gtype:{dual} is introduced in the file group.info as shown in the textarea above. A full example is linked here.

   (2) The second case is more complicated to understand at this moment. To understand this case you should understand first the idea of self-dual hierarchy (the next chapter). Self-ranking is meaningfull for dual type groups, i.e. for those groups with the set of alternatives equal to the set of group members. The software recognize such case in group.info (option method:{iterate}) and test data for such possibility. A full example is linked here.

Self dual hierarchy

Self-dual hierarchy has the same nodes in the first and in the last level. A procedure starts with some initial weight on the first level, calculates the last level weight, substitute the new weight as new initial weight and so on... The procedure has a fix point which is a robust weight of the alternatives. It is proved that the fix point does not depend upon the initial weight.

1. Example: (save as hier.dual.txt): Copy

   name: {H2.dual} ; # must be > 1 base: {2} ; levels: {A,B,C} ; # top level and bottom level have the same nodes structure: {A: {a1, a2, a3, a4}, B: {b1, b2, b3, b4}, C:{a1, a2, a3, a4}} ; # Initialization. The result doesn't depend on weight: {1.1:1, 1.2:0, 1.3: 0, 1.4: 0}; children: { 1.1: {2.1, 2.2, 2.3, 2.4}, 1.2: {2.1, 2.2, 2.3, 2.4}, 1.3: {2.1, 2.2, 2.3, 2.4}, 1.4: {2.1, 2.2, 2.3, 2.4}, 2.1: {3.1, 3.2, 3.3}, 2.2: {3.1, 3.2, 3.4}, 2.3: {3.2, 3.3, 3.4}, 2.4: {3.1, 3.2, 3.3} } ; preferences: { 1.1: {(2.1, 2.2): 5, (2.1, 2.3): 4, (2.4, 2.2):1}, 1.2: {(2.2, 2.4):1, (2.3, 2.1):-4, (2.2, 2.1):0}, 1.3: {(2.4, 2.2): 0, (2.1, 2.3): 2, (2.2, 2.3):5}, 1.4: {(2.4, 2.2): -2, (2.1, 2.3): 6, (2.4, 2.3):0}, 2.1: {(3.1, 3.2):5, (3.2, 3.3):8 }, 2.2: {(3.1, 3.2):0, (3.2, 3.4):-5 }, 2.3: {(3.3, 3.2):6, (3.4, 3.2): 5 }, 2.4: {(3.2, 3.1):6, (3.3, 3.2): 7, (3.1, 3.3):0 }} ; aggregation: {} ; htype: {dual} ;

2. Command line:
   
   ./pmhier -f hier.dual.txt

3. 
   You will get the output:
   

   --------------------------- HIERARCHY (H2.dual) model: dual ------------------- Fixed point (sorted) a4 -> 0.411932 a1 -> 0.206305 a2 -> 0.194542 a3 -> 0.187221 laststep = 8 error = 2.80157e-12 Aggregation method(s): undef base: 2 call: ./pmhier -f hier.dual.txt

Choice modeling

1. Download the choice file.
2. Make it executable
   chmod +x filename

3. Example: (save as choice_binomial.txt): Copy

   #! interface : {choice}; flowname: {Choice binomial}; vertices: {A,B,C,D}; ratio: {A:B=4:3}; ratio: {B:C=2:1}; ratio: {C:A=3:2};

4. Command line:
   ./pmchoice -f choice_binomial.txt
   Output:

   show input:: A:B => 4:3 B:C => 2:1 C:A => 3:2 Showing edgelist: (Choice binomial): names: {A,B,C,D} (size: 4) idxin: {0,1,2} (size: 3) idxout: {1,2,0} df: {0.415037,1,0.584963} Weight (project 'Choice binomial'): Comp_1 (node -> W):: ------------------- B(1) -> 0.379701 A(0) -> 0.318929 C(2) -> 0.301369 aggregation: sum file: choice_binomial.txt

5. Other examples (play with input data to see the difference):
   choice_1
   choice_2
   choice_puzzle
   choice_test
   Conclude that: (A:B=2:1 & B:C=2:1) is equivalent to A:B:C=4:2:1.

Choce model as a generalization of R-matrix approach

1. ./pmchoice -f ctest.txt
2. ./pmrmatrix -f rtest.csv -m PM

Geometrical interpretation of ratio(s)

In a Future ...

1. Ordinal Potential Method.
2. Missing Data - a proxy approach (partialy done).
3. Machine Learning and Prediction by PM (partialy done).
4. PM Score Matching in Observational Studies (partialy done).
5. Input-Output models (DEA vs. PM) (partialy done)