The tradeoff algorithm performs an evaluation of multi-attribute products from a rank ordering of preferences for attribute level pairs. As a brief example, consider two attributes, each having three levels (cost = $1.75, $1.50, and $1.25 and vitamin content = 80%, 50%, and 30% of minimum daily requirement). The procedure requires that the nine combinations of cost and vitamin content be rank ordered as to preference with 1 being the most preferred and 9 the least preferred combination.
TRADEOFF produces an estimate of the relative utilities of each of the levels of the two attributes considered. Factorial designs may be used to provide more than one set of utility estimates for a given attribute. For each set of attributes, the estimate of the utilities is given by the function:
where:
Uim = the estimated utility value of attribute i, level m.
Rimjn = the rankings for level m of attribute i and level n of attribute j
For multiplicative models, the functional form appears:
The TRADEOFF program is based on an iterative algorithm which calculates the inconsistencies between the rankings and the utilities. The iterative procedure continues as utility values are changed and re-evaluated for consistency.
Program input is done consecutively for each tradeoff matrix (attribute x attribute matrix), with program options selected from the TRADEOFF menu.
Each line of the data file contains the data for each attribute x attribute tradeoff matrix. For example a tradeoff matrix that is 2 attribute levels by 3 attribute levels in size would contain 6 tradeoff pairs, or data points within a 2 row by 3 column matrix.
Data would be entered in rowwise order, from cells 1,1; 1,2; 1,3; 2,1; 2,2; 2,3. The input line would appear:
2 2 3
(8I2)
1 2 1 2 3 4 5 6
where the first line contains three numbers. The first number indicates that two attributes are considered in this analysis. The second number indicates that the first attribute has 2 levels. The third number indicates that the second attribute has three levels.
The second line contains a format statement. In this case, we read 8 integers, each being 2 columns wide. The first two numbers (1 2) are id numbers and the next six are data points for the 2 x 3 matrix.
The third line first identifies the matrix as representing the tradeoff of attributes 1 and 2. This matrix identification is followed by the six preference values for the 2 x 3 data matrix: 1 2 3 4 5 6.
Let's consider another more complicated example: In all analyses, one control parameter line is required and appears in the first line of the data file. The parameter line uses a FREE format to specify the number of attributes and levels. If more than 20 attributes are specified, successive lines are used. For the example problem, the control line includes:
1) The number of attributes (for example, 5)
2) The number of levels of each attribute. For this 5 attribute example, they are 4,5,4,4, and 4 levels.
5 4 5 4 4 4 (2I2,20F3.0) 1 2 20 19 14 6 8 18 15 11 4 7 17 13 10 2 5 16 12 9 1 3 3 4 6 10 13 16 3 8 11 15 2 5 9 14 1 4 7 12 2 5 14 19 20 12 13 17 18 6 7 15 16 3 4 8 10 1 5 9 11 2 3 2 20 18 15 9 10 19 17 11 4 6 16 13 8 3 5 14 12 7 1 2 1 3 16 14 8 6 15 11 7 3 13 10 5 2 12 9 4 1 5 4 2 5 10 14 3 7 11 15 6 8 12 16 1 4 9 13 1 5 8 12 16 4 7 11 15 3 6 10 14 2 5 9 13 1
Each line of the data file contains information for a tradeoff table. In line 1, attributes 1 and 2 are specified as having 4 and 5 levels respectively. This tradeoff table would have 20 cells (4*5).
The first two numbers in the data line specify the attribute numbers (1 and 2) in the tradeoff matrix. The attribute numbers are followed by the ranks from 1 to 20. Tradeoff tables may be entered in any order, as long as attribute numbers are identified in the first two numbers of each data line.
An example of a three attribute model is now considered. The three attributes are defined as having 2, 3,and 3 levels respectively. Suppose that we have collected data for three tradeoff matrices: (1 x 2), (2 x 3), and (1 x 3).
MATRIX 1 MATRIX 2 MATRIX 3
ATTRIBUTE #2 ATTRIBUTE #3 ATTRIBUTE #3
1 2 3 1 2 3 1 2 3
+-------------+ +-------------+ +-------------+
ATT ¦ 1 2 4 ¦ ATT¦ 1 2 4 ¦ ATT.¦ 1 2 4 ¦
#1 ¦ 3 5 6 ¦ #2 ¦ 3 5 8 ¦ #1 ¦ 3 6 5 ¦
+-------------+ ¦ 6 7 9 ¦ +-------------+
+-------------+
The actual data input into TRADEOFF would appear:
3 2 3 3 (2I2,9F2.0) 1 2 1 2 4 3 5 6 2 3 1 2 4 3 5 8 6 7 9 1 3 1 2 4 3 6 5
The first data line indicates 3 attributes, being 2, 3, and 3 levels each. Lines 2, 3, and 4 identify the respective tradeoff matrices. Line 2 is for the tradeoff of attributes 1 and 2, followed by the list of the 6 attribute combinations; Line 3 shows the tradeoff of attributes 2 and 3, followed by the list of the 9 attribute combinations; and line 4 shows the tradeoff for attributes 1 and 3 followed by the list of the 6 attribute combinations.
The maximum size of the tradeoff table is limited to 10 attribute levels by 10 attribute levels. A maximum of 200 attributes may be specified.
Data for multiple, rather than single or aggregate respondent may also be analyzed by TRADEOFF. The multiple respondent data option is specified interactively at the beginning of the program. The data for the first respondent is entered as for a single respondent analysis, with successive respondents data entered and separated by a blank line. A sample data file for 5 attributes with 4,5,4,4, and 4 levels appears:
5 4 5 4 4 4
(2I2,20F3.0)
1 2 20 19 14 6 8 18 15 11 4 7 17 13 10 2 5 16 12 9 1 3
3 4 6 10 13 16 3 8 11 15 2 5 9 14 1 4 7 12
2 5 14 19 20 12 13 17 18 6 7 15 16 3 4 8 10 1 5 9 11 2
3 2 20 18 15 9 10 19 17 11 4 6 16 13 8 3 5 14 12 7 1 2
1 3 16 14 8 6 15 11 7 3 13 10 5 2 12 9 4 1
5 4 2 5 10 14 3 7 11 15 6 8 12 16 1 4 9 13
1 5 8 12 16 4 7 11 15 3 6 10 14 2 5 9 13 1
1 2 20 19 14 6 8 18 15 11 4 7 17 13 10 2 5 16 12 9 1 3
3 4 6 10 13 16 3 8 11 15 2 5 9 14 1 4 7 12
2 5 14 19 20 12 13 17 18 6 7 15 16 3 4 8 10 1 5 9 11 2
3 2 20 18 15 9 10 19 17 11 4 6 16 13 8 3 5 14 12 7 1 2
1 3 16 14 8 6 15 11 7 3 13 10 5 2 12 9 4 1
5 4 2 5 10 14 3 7 11 15 6 8 12 16 1 4 9 13
1 5 8 12 16 4 7 11 15 3 6 10 14 2 5 9 13 1
1 2 20 19 14 6 8 18 15 11 4 7 17 13 10 2 5 16 12 9 1 3
3 4 6 10 13 16 3 8 11 15 2 5 9 14 1 4 7 12
2 5 14 19 20 12 13 17 18 6 7 15 16 3 4 8 10 1 5 9 11 2
3 2 20 18 15 9 10 19 17 11 4 6 16 13 8 3 5 14 12 7 1 2
1 3 16 14 8 6 15 11 7 3 13 10 5 2 12 9 4 1
5 4 2 5 10 14 3 7 11 15 6 8 12 16 1 4 9 13
1 5 8 12 16 4 7 11 15 3 6 10 14 2 5 9 13 1
Options for the TRADEOFF program are selected from the TRADEOFF menu. Options include
(1) Model Type: (Additive or Multiplicative)
(2) Convergence Acceleration parameter (Default=2.5)
A multiplicative constant used in estimating the utility matrix. This constant is increased in proportion to the order of comparisons (Parameter 4).
(3) Utility Increment (Default=.05)
Utility for attribute I, level M is increased by this amount at each iteration of the program. Increasing this value speeds up the program, but if too large, may create instability in convergence.
(4) Comparison Order parameter (Default=2)
This parameter specifies the order of comparisons that are made in computing the badness of fit measure. The default value of 2 indicates that adjacent pairs are tested (1,2; 2,3; 3,4; 4,5; 5,6) for inconsistencies between the tradeoff rank orders and the computed utility values. A value of 3 would indicate that pairs (1,2; 1,3; 2,3; 2,4; 3,4; 3,5; 4,5; 4,6; 5,6) are tested. As the Comparison Order parameter is increased, the Convergence Acceleration parameter be proportionately increased.
(5) Iterations (Default=10)
(6) Proportions Correct to stop (Default=.85)
This is the proportion of rank orders that are consistent with the order of the utilities. When this proportion of rank orders is reached, iterations stop.
(7) Minimum Convergence Rate to stop (Default=.005)
The step size decreases as the model fit improves. This parameter sets a lower bound on the minimal size of improvement for which iterations will continue.
+----------------------------------------------------------------------------+ ¦ ¦ ¦ T R A D E O F F M E N U ¦ ¦ ¦ ¦---------------------------------------------------------------------------¦ ¦ 1 SELECT MODEL TYPE: ¦ ¦ (0=ADDITIVE 1=MULTIPLICATIVE) ¦ ¦ 2 SELECT CONVERGENCE ACCELERATION FACTOR ¦ ¦ (DEFAULT = 2.5, CHANGE IN PROPORTION TO ORDER) ¦ ¦ 3 SELECT DELTA, THE UTILITY INCREMENT ¦ ¦ (DEFAULT=.05, CHANGE TO .1 IF MULTIPLICATIVE) ¦ ¦ 4 SELECT ORDER, THE COMPARISON ORDER OPTION, DEF=2 FOR 2ND ORDER ¦ ¦ ¦ ¦ 5 SELECT ITERATIONS TO TERMINATE ¦ ¦ (DEFAULT = 10) ¦ ¦ 6 SELECT PROPORTION CORRECT OPTION ¦ ¦ (DEFAULT=.85 OF COMPARISONS CORRECT) ¦ ¦ 7 SELECT MINIMUM CONVERGENCE RATE ¦ ¦ (DEFAULT=.005 MINIMUM DECREASE IN BADNESS OF FIT ¦ ¦ 8 SHOW CURRENT PARAMETERS ¦ ¦ 9 CHANGES COMPLETE, CONTINUE PROGRAM ¦ ¦ 10 QUIT PROGRAM ¦ +---------------------------------------------------------------------------+
Output for TRADEOFF is presented as a m x n matrix, where utilities generated for the attribute level pairs are listed. Individual respondent utilities are also output to a user specified file. These utilities may be analyzed in the simulator provided with the program CONJOINT.
Nehls, L., Seaman, B., and Montgomery, D. (1976), "FTROFF: A Basic Program for Trade-off Analysis," Report No. 76-105, Marketing Science Institute.
1. The tradeoff program is based on the program "$FTROFF: A Basic Program for Tradeoff Analysis," Report No. 76-105, Marketing Science Institute.
NONMETRIC TRADEOFF ANALYSIS
BASED ON THE $FTROFF ALGORITHM WRITTEN BY
LYLE NEHLS, BRUCE SEAMAN, AND DAVID B. MONTGOMERY
PC-MDS VERSION
ANALYSIS TITLE: TRADEOFF TEST DATA
DATA IS READ FROM FILE: TRADEOFF.DAT
OUTPUT FILE IS: TRADEOFF.PRN
TITLE: TRADEOFF TEST DATA
INITIAL PREFERENCE MATRIX GENERATED USING
A STANDARDIZED SUM OF THE TRADEOFF SCORES HEURISTIC
AXES: ROWS = ATTRIBUTES / COLUMNS = ATTRIBUTE LEVELS
IMPORTANCE IMPORT.% 1
1 .459 .095 -.249 -.055 .094 .210
2 1.206 .251 -.627 -.373 .000 .579 .421
3 .866 .180 -.447 -.156 .185 .418
4 1.338 .278 .647 .250 -.206 -.691
5 .941 .196 .168 -.196 -.457 .485
TOTAL POSSIBLE COMPARISONS: 1050.
COMPARISONS ACTUALLY MADE: 227.
COMPUTING UTILITY MATRIX....
ITERATION 1: UTILITY SCORE MATRIX
AXES: ROWS = ATTRIBUTES / COLUMNS = ATTRIBUTE LEVELS
IMPORTANCE IMPORT.% 1
1 .325 .068 -.181 -.035 .072 .144
2 1.217 .256 -.640 -.354 -.005 .578 .420
3 .793 .166 -.413 -.131 .164 .380
4 1.481 .311 .698 .263 -.179 -.782
5 .949 .199 .149 -.206 -.446 .503
TOTAL INCONSISTENCIES REMAINING = 36.50
PROPORTION OF COMPARISONS CORRECT = .84 BADNESS OF FIT MEASURE = .10
ITERATION 2: UTILITY SCORE MATRIX
AXES: ROWS = ATTRIBUTES / COLUMNS = ATTRIBUTE LEVELS
IMPORTANCE IMPORT.% 1
1 .351 .073 -.210 -.030 .099 .141
2 1.183 .245 -.612 -.364 -.004 .571 .410
3 .772 .160 -.407 -.123 .165 .365
4 1.572 .325 .740 .280 -.188 -.832
5 .953 .197 .147 -.219 -.440 .512
TOTAL INCONSISTENCIES REMAINING = 29.50
PROPORTION OF COMPARISONS CORRECT = .87 BADNESS OF FIT MEASURE = .08
ITERATION STOPS: PROPORTION OF CORRECT COMPARISONS IS SUFFICIENTLY HIGH
*****IDENTIFICATION KEY FOR PLOTS WITH IDENTIFIED POINTS*****
PT # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
CHAR 1 2 3 4 5 6 7 8 9 A B C D E F
PT # 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
CHAR G H I J K L M N O P Q R S T U
PT # 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
CHAR V W X Y Z + / = * & $ @ % ? <
PT # 46 47 48 49 50
CHAR ( ) " ; @
POINT NUMBERS ABOVE 50 IDENTIFIED AS >, MULTIPLE POINTS IDENTIFIED AS #
ONE NUMBER IDENTIFIES A GIVEN ATTRIBUTE ACROSS ATTRIBUTE LEVELS
+....+....+....+....+....+....+....+....+....+....+....+....+
U 1.50+| +
.| .
T .| .
1.15+| +
I .| .
.| .
L .81+| +
.| 4 .
I .| 2 .
.46+| 5 2+
T .| 3 .
.| 4 .
Y .12+| 5 # 1 +
.0-----------------------1-----------2------------------------.
.| 3 .
-.23+| 1 5 4 +
V .| 2 .
.| 3 5 .
A -.58+| 2 +
.| .
L .| 4 .
-.92+| +
U .| .
.| .
E -1.27+| +
.| .
S .| .
+....+....+....+....+....+....+....+....+....+....+....+....+
.0 .4 .8 1.3 1.7 2.1 2.5 2.9 3.3 3.8 4.2 4.6 5.0
A T T R I B U T E L E V E L S