Permutations based model for business performance

Arthur Fétiveau; Gilles Durrieu; Emmanuel Frénod; Claude-Henri Meledo; Benoît Prat

doi:10.3934/dcdss.2024033

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2025, Volume 18, Issue 7: 1699-1722. Doi: 10.3934/dcdss.2024033

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Permutations based model for business performance

1.
Univ Bretagne Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France
2.
Aldecis, Paris, France

^*Corresponding author: Arthur Fétiveau
^*Corresponding author: Arthur Fétiveau

Received: July 2023

Early access: March 19, 2024

Published online: March 19, 2024

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Abstract

This article is devoted to the development of a machine learning statistical framework to drive company's objectives. To this end, their sales data was used to target efficiently the issues or opportunities by a ranking. We implemented a permutation based model using generalized Mallows models dealing with quantitative values, considering that a ranking is a permutation. The advantage of the generalized version is the possibility to differentiated the cost to move each element in the permutation. In our model, we differentiate the cost of an inversion in the permutation by using the gap value between the two elements. We proposed model parameters estimators and illustrated our estimation procedure on simulated data and a real application.

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Mathematics Subject Classification: Primary: 65C20, 62F10; Secondary: 62P05.

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Figure 1. Permutohedron of order $ 4 $ for the Kendall distance

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Figure 2. Permutohedron of order $ 4 $ for the Kendall distance

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Figure 3. Boxplot of the Kendall distances between $ \hat{\pi} $ and $ \pi $ for $ 5 $ $ \sigma_j $, permutations of $ n = 30 $ items, with the same $ \theta $ and the same $ \pi $ over $ 3000 $ realizations

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Figure 4. Kendall distance between $ \hat{\pi} $ and $ \pi $ for different numbers of $ \sigma_j $, permutations of $ n = 30 $ items, with the same $ \theta = 0.3 $ and the same $ \pi $ over $ 3000 $ realizations

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Figure 5. Boxplot of the Kendall distance between $ \hat{\pi} $ and $ \sigma $ for $ n = 30 $, in the set of parameters $ P_4 $

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Figure 6. Boxplot of the Kendall distance between $ \hat{\pi} $ and $ \sigma $ for $ n = 30 $, in the set of parameters $ P_8 $

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Figure 7. Boxplot of the Kendall distance between $ \hat{\pi} $ and $ \sigma $ for $ n = 30 $, in the set of parameters $ P_9 $

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Table 1. Means $ \pm $ standard deviations of Kendall distances between $ \pi $ and $ \hat{\pi} $ for $ 50 $ $ \sigma_j $, permutations with $ n = 30 $ items, with the same $ \theta $ and the same $ \pi $ over $ 3000 $ realizations

Choice of $ \theta $	BORDA	MLE	QUAD
$ 0.01 $	$ 166.19 \pm 26.269 $	$ 168.03 \pm 26.491 $	$ 168.447 \pm 26.287 $
$ 0.05 $	$ 63.08 \pm 10.843 $	$ 64.61 \pm 11.530 $	$ 65.09 \pm 11.590 $
$ 0.1 $	$ 31.13 \pm 6.050 $	$ 30.65 \pm 6.456 $	$ 31.05 \pm 6.525 $
$ 0.15 $	$ 19.43 \pm 4.298 $	$ 18.10 \pm 4.505 $	$ 18.41 \pm 4.610 $
$ 0.2 $	$ 13.36 \pm 3.410 $	$ 11.55 \pm 3.439 $	$ 11.80 \pm 3.538 $
$ 0.25 $	$ 9.53 \pm 2.769 $	$ 7.77 \pm 2.722 $	$ 8.02 \pm 2.739 $
$ 0.3 $	$ 7.16 \pm 2.351 $	$ 5.30 \pm 2.234 $	$ 5.52 \pm 2.281 $
$ 0.35 $	$ 5.35 \pm 2.054 $	$ 3.70 \pm 1.851 $	$ 3.80 \pm 1.879 $
$ 0.4 $	$ 4.025 \pm 1.785 $	$ 2.54 \pm 1.557 $	$ 2.70 \pm 1.608 $
$ 0.45 $	$ 3.08 \pm 1.569 $	$ 1.79 \pm 1.269 $	$ 1.89 \pm 1.295 $
$ 0.5 $	$ 2.22 \pm 1.367 $	$ 1.18 \pm 1.045 $	$ 1.30 \pm 1.089 $
$ 0.55 $	$ 1.65 \pm 1.189 $	$ 0.79 \pm 0.876 $	$ 0.87 \pm 0.898 $
$ 0.6 $	$ 1.23 \pm 1.060 $	$ 0.54 \pm 0.712 $	$ 0.62 \pm 0.779 $

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Table 2. Mean $ \pm $ standard deviation of the standard deviation of the Kendall distances between $ \sigma $ and $ \hat{\pi} $ for $ 50 $ $ \sigma_j $, permutations of $ n = 30 $ items, with the same $ \theta $ and the same $ \pi $ over $ 3000 $ realizations

Choice of $ \theta $	BORDA	MLE	QUAD
$ 0.01 $	$ 27.40 \pm 2.785 $	$ 27.86 \pm 2.801 $	$ 25.21 \pm 2.571 $
$ 0.05 $	$ 26.17 \pm 2.616 $	$ 26.64 \pm 2.646 $	$ 25.13 \pm 2.492 $
$ 0.1 $	$ 23.95 \pm 2.449 $	$ 24.38 \pm 2.491 $	$ 23.46 \pm 2.388 $
$ 0.15 $	$ 21.15 \pm 2.180 $	$ 21.55 \pm 2.210 $	$ 20.90 \pm 2.158 $
$ 0.2 $	$ 18.46 \pm 1.929 $	$ 18.79 \pm 1.972 $	$ 18.31 \pm 1.918 $
$ 0.25 $	$ 16.17 \pm 1.686 $	$ 16.42 \pm 1.707 $	$ 16.07 \pm 1.668 $
$ 0.3 $	$ 14.21 \pm 1.509 $	$ 14.41 \pm 1.532 $	$ 14.15 \pm 1.505 $
$ 0.35 $	$ 12.67 \pm 1.331 $	$ 12.83 \pm 1.350 $	$ 12.63 \pm 1.329 $
$ 0.4 $	$ 11.38 \pm 1.196 $	$ 11.51 \pm 1.210 $	$ 11.35 \pm 1.195 $
$ 0.45 $	$ 10.31 \pm 1.116 $	$ 10.42 \pm 1.118 $	$ 10.30 \pm 1.108 $
$ 0.5 $	$ 9.43 \pm 0.999 $	$ 9.52 \pm 1.005 $	$ 9.42 \pm 0.994 $
$ 0.55 $	$ 8.65 \pm 0.920 $	$ 8.72 \pm 0.927 $	$ 8.65 \pm 0.920 $
$ 0.6 $	$ 8.02 \pm 0.847 $	$ 8.07 \pm 0.848 $	$ 8.02 \pm 0.844 $

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Table 3. Mean $ \pm $ standard deviation of the standard deviation of the Kendall distances between $ \sigma $ and $ \hat{\pi} $ for $ 5 $ $ \sigma_j $, permutations of $ n = 30 $ items, with the same $ \theta $ and the same $ \pi $ over $ 3000 $ realizations

Choice of $ \theta $	BORDA	MLE	QUAD
$ 0.01 $	$ 21.43 \pm 7.759 $	$ 27.08 \pm 9.761 $	$ 16.76 \pm 6.101 $
$ 0.05 $	$ 20.07 \pm 7.318 $	$ 25.20 \pm 8.963 $	$ 16.32 \pm 6.013 $
$ 0.1 $	$ 17.42 \pm 6.425 $	$ 22.04 \pm 8.096 $	$ 15.05 \pm 5.618 $
$ 0.15 $	$ 15.23 \pm 5.623 $	$ 19.43 \pm 7.126 $	$ 13.79 \pm 5.096 $
$ 0.2 $	$ 13.00 \pm 4.823 $	$ 16.75 \pm 6.070 $	$ 12.30 \pm 4.554 $
$ 0.25 $	$ 11.38 \pm 4.322 $	$ 14.47 \pm 5.468 $	$ 10.92 \pm 4.170 $
$ 0.3 $	$ 9.89 \pm 3.721 $	$ 12.57 \pm 4.702 $	$ 9.57 \pm 3.674 $
$ 0.35 $	$ 8.86 \pm 3.315 $	$ 11.12 \pm 4.144 $	$ 8.63 \pm 3.226 $
$ 0.4 $	$ 8.00 \pm 2.924 $	$ 9.98 \pm 3.616 $	$ 7.91 \pm 2.909 $
$ 0.45 $	$ 7.29 \pm 2.745 $	$ 9.04 \pm 3.338 $	$ 7.25 \pm 2.647 $
$ 0.5 $	$ 6.63 \pm 2.417 $	$ 8.13 \pm 3.009 $	$ 6.61 \pm 2.403 $
$ 0.55 $	$ 6.21 \pm 2.281 $	$ 7.58 \pm 2.762 $	$ 6.23 \pm 2.175 $
$ 0.6 $	$ 5.77 \pm 2.117 $	$ 7.00 \pm 2.573 $	$ 5.79 \pm 2.011 $

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Table 4. Means $ \pm $ standard deviations of Kendall distances between $ \pi $ and $ \hat{\pi} $ for $ 2 $ to $ 50 $ $ \sigma_j $, permutations of $ n = 30 $ items, with the same $ \theta = 0.3 $ and the same $ \pi $ over $ 3000 $ realizations

Choice of $ J $	BORDA	MLE	QUAD
$ 2 $	$ 51.52 \pm 10.300 $	$ 69.13 \pm 14.736 $	$ 63.59 \pm 12.554 $
$ 3 $	$ 42.51 \pm 8.530 $	$ 43.76 \pm 9.118 $	$ 43.83 \pm 9.074 $
$ 4 $	$ 37.00 \pm 7.409 $	$ 39.49 \pm 8.238 $	$ 38.81 \pm 7.908 $
$ 5 $	$ 32.73 \pm 6.766 $	$ 32.91 \pm 7.222 $	$ 33.10 \pm 7.117 $
$ 6 $	$ 29.97 \pm 6.166 $	$ 30.65 \pm 6.768 $	$ 30.40 \pm 6.661 $
$ 7 $	$ 27.42 \pm 5.642 $	$ 26.66 \pm 5.921 $	$ 26.94 \pm 5.929 $
$ 8 $	$ 25.37 \pm 5.320 $	$ 25.20 \pm 5.752 $	$ 25.27 \pm 5.797 $
$ 9 $	$ 23.77 \pm 5.037 $	$ 22.79 \pm 5.371 $	$ 23.12 \pm 5.307 $
$ 10 $	$ 22.07 \pm 4.728 $	$ 21.45 \pm 5.157 $	$ 21.53 \pm 5.137 $
$ 15 $	$ 17.33 \pm 3.992 $	$ 15.79 \pm 4.178 $	$ 16.25 \pm 4.244 $
$ 20 $	$ 14.29 \pm 3.588 $	$ 12.78 \pm 3.635 $	$ 13.02 \pm 3.648 $
$ 25 $	$ 12.23 \pm 3.255 $	$ 10.46 \pm 3.398 $	$ 10.82 \pm 3.314 $
$ 30 $	$ 10.66 \pm 3.055 $	$ 8.95 \pm 2.976 $	$ 9.15 \pm 3.044 $
$ 40 $	$ 8.57 \pm 2.655 $	$ 6.82 \pm 2.504 $	$ 7.05 \pm 2.570 $
$ 50 $	$ 7.07 \pm 2.431 $	$ 5.29 \pm 2.204 $	$ 5.49 \pm 2.242 $

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Table 5. Mean $ \pm $ standard deviations of the standard deviation of the Kendall distances between $ \sigma $ and $ \hat{\pi} $ for $ 2 $ to $ 50 $ $ \sigma_j $, permutations of $ n = 30 $ items, with the same $ \theta $ and the same $ \pi $ over $ 3000 $ realizations

Choice of $ J $	BORDA	MLE	QUAD
$ 2 $	$ 2.40 \pm 1.845 $	$ 50.24 \pm 8.921 $	$ 0.25 \pm 0.250 $
$ 3 $	$ 7.01 \pm 3.675 $	$ 11.74 \pm 6.181 $	$ 7.55 \pm 4.217 $
$ 4 $	$ 8.81 \pm 3.824 $	$ 13.38 \pm 5.848 $	$ 5.09 \pm 3.924 $
$ 5 $	$ 9.92 \pm 3.670 $	$ 12.63 \pm 4.659 $	$ 9.66 \pm 3.640 $
$ 6 $	$ 10.73 \pm 3.554 $	$ 13.64 \pm 4.446 $	$ 8.89 \pm 3.733 $
$ 7 $	$ 11.25 \pm 3.440 $	$ 13.10 \pm 3.899 $	$ 10.93 \pm 3.208 $
$ 8 $	$ 11.74 \pm 3.224 $	$ 13.47 \pm 3.670 $	$ 10.56 \pm 3.262 $
$ 9 $	$ 12.16 \pm 3.221 $	$ 13.57 \pm 3.581 $	$ 11.77 \pm 2.949 $
$ 10 $	$ 12.22 \pm 3.083 $	$ 13.57 \pm 3.408 $	$ 11.38 \pm 3.116 $
$ 15 $	$ 13.16 \pm 2.575 $	$ 13.95 \pm 2.724 $	$ 12.83 \pm 2.448 $
$ 20 $	$ 13.58 \pm 2.233 $	$ 14.16 \pm 2.331 $	$ 13.28 \pm 2.204 $
$ 25 $	$ 13.77 \pm 2.054 $	$ 14.23 \pm 2.127 $	$ 13.57 \pm 2.016 $
$ 30 $	$ 13.84 \pm 1.867 $	$ 14.21 \pm 1.930 $	$ 13.69 \pm 1.855 $
$ 40 $	$ 14.07 \pm 1.618 $	$ 14.33 \pm 1.658 $	$ 13.97 \pm 1.609 $
$ 50 $	$ 14.22 \pm 1.482 $	$ 14.41 \pm 1.498 $	$ 14.14 \pm 1.480 $

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Table 6. Mean $ \pm $ standard deviations of Kendall distances between $ \pi $ and $ \hat{\pi} $ for a set of parameters $ \theta $ : $ P_1 $ : $ 50 $ parameters at $ 0.05 $; $ P_2 $ : $ 50 $ parameters at $ 0.6 $; $ P_3 $ : $ 2 $ parameters at $ 0.6 $; $ P_4 $ : $ 2 $ parameters at $ 0.6 $ and $ 5 $ parameters at $ 0.05 $; $ P_5 $ : $ 2 $ parameters at $ 0.6 $ and $ 48 $ parameters at $ 0.05 $; $ P_6 $ : $ 25 $ parameters at $ 0.6 $; $ P_7 $ : $ 25 $ parameters at $ 0.6 $ and $ 25 $ parameters at $ 0.05 $. The permutations are of $ n = 30 $ items

Choice of $ \theta $	BORDA	MLE	QUAD
$ P_1 $	$ 63.08 \pm 10.843 $	$ 64.61 \pm 11.530 $	$ 65.09 \pm 11.590 $
$ P_2 $	$ 1.23 \pm 1.060 $	$ 0.54 \pm 0.712 $	$ 0.62 \pm 0.779 $
$ P_3 $	$ 22.83 \pm 5.811 $	$ 32.67 \pm 8.107 $	$ 30.90 \pm 7.436 $
$ P_4 $	$ 25.20 \pm 5.661 $	$ 27.38 \pm 6.716 $	$ 29.76 \pm 6.395 $
$ P_5 $	$ 26.94 \pm 5.456 $	$ 22.74 \pm 5.631 $	$ 33.13 \pm 6.380 $
$ P_6 $	$ 3.32 \pm 1.625 $	$ 2.04 \pm 1.374 $	$ 2.25 \pm 1.437 $
$ P_7 $	$ 3.51 \pm 1.664 $	$ 2.06 \pm 1.372 $	$ 2.70 \pm 1.564 $

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Table 7. Mean $ \pm $ standard deviations of the standard deviation of the Kendall distances between $ \sigma $ and $ \hat{\pi} $ for a set of parameters $ \theta $ : $ P_1 $ : $ 50 $ parameters at $ 0.05 $; $ P_2 $ : $ 50 $ parameters at $ 0.6 $; $ P_3 $ : $ 2 $ parameters at $ 0.6 $; $ P_4 $ : $ 2 $ parameters at $ 0.6 $ and $ 5 $ parameters at $ 0.05 $; $ P_5 $ : $ 2 $ parameters at $ 0.6 $ and $ 48 $ parameters at $ 0.05 $; $ P_6 $ : $ 25 $ parameters at $ 0.6 $; $ P_7 $ : $ 25 $ parameters at $ 0.6 $ and $ 25 $ parameters at $ 0.05 $. The permutations are of $ n = 30 $ items

Parameters $ \theta $	BORDA	MLE	QUAD
$ P_1 $	$ 26.17 \pm 2.616 $	$ 26.64 \pm 2.646 $	$ 25.13 \pm 2.492 $
$ P_2 $	$ 8.02 \pm 0.847 $	$ 8.07 \pm 0.848 $	$ 8.02 \pm 0.844 $
$ P_3 $	$ 1.73 \pm 1.344 $	$ 25.12 \pm 5.053 $	$ 0.25 \pm 0.25 $
$ P_4 $	$ 20.36 \pm 6.735 $	$ 20.97 \pm 6.689 $	$ 19.53 \pm 6.405 $
$ P_5 $	$ 25.99 \pm 2.639 $	$ 26.20 \pm 2.657 $	$ 25.45 \pm 2.570 $
$ P_6 $	$ 7.76 \pm 1.187 $	$ 7.91 \pm 1.209 $	$ 7.70 \pm 1.177 $
$ P_7 $	$ 19.62 \pm 2.511 $	$ 19.66 \pm 2.507 $	$ 19.58 \pm 2.505 $

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Table 8. Mean $ \pm $ standard deviations of the standard deviation of the Kendall distances between $ \sigma $ and $ \hat{\pi} $ for a set of parameters $ \theta $ : $ P_8 $ : $ 8 $ different parameters, $ \theta_1 = 0.1 $, $ \theta_2 = 0.15 $, $ \theta_3 = 0.2 $, $ \theta_4 = 0.25 $, $ \theta_5 = 0.3 $, $ \theta_6 = 0.35 $, $ \theta_7 = 0.4 $ and $ \theta_8 = 0.6 $; $ P_9 $ : $ 1 $ parameters at $ 0.6 $ and $ 6 $ parameters at $ 0.1 $. The permutations are of $ n = 30 $ items

Choice of $ \theta $	BORDA	MLE	QUAD
$ P_8 $	$ 23.95 \pm 5.075 $	$ 21.78 \pm 5.368 $	$ 24.74 \pm 5.531 $
$ P_9 $	$ 38.01 \pm 7.309 $	$ 32.66 \pm 7.921 $	$ 45.98 \pm 8.356 $

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Table 9. Kendall distance between $ \pi $ and $ \hat{\pi} $ when we rank only the $ 40 $ evaluated items

Month	MM + MLE	MM + QUAD	MINMAX + MLE	MINMAX + QUAD
$ 1 $	$ 160 $	$ 181 $	$ 189 $	$ 183 $
$ 2 $	$ 91 $	$ 147 $	$ 121 $	$ 137 $
$ 3 $	$ 128 $	$ 135 $	$ 168 $	$ 168 $
$ 4 $	$ 137 $	$ 173 $	$ 182 $	$ 198 $
Month	MED + MLE	MED + QUAD	MEAN + MLE	MEAN + QUAD
$ 1 $	$ 135 $	$ 136 $	$ 128 $	$ 133 $
$ 2 $	$ 106 $	$ 106 $	$ 97 $	$ 97 $
$ 3 $	$ 125 $	$ 132 $	$ 123 $	$ 125 $
$ 4 $	$ 132 $	$ 132 $	$ 135 $	$ 134 $

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Table 10. Kendall distance between $ \hat{\pi} $ ranked over every items and $ \hat{\pi} $ ranked over the $ 40 $ evaluated items, on the $ 40 $ evaluated items

Month	MM + MLE	MM + QUAD	MEAN + MLE	MEAN + QUAD
$ 1 $	$ 0 $	$ 40 $	$ 0 $	$ 7 $
$ 2 $	$ 0 $	$ 111 $	$ 0 $	$ 0 $
$ 3 $	$ 0 $	$ 82 $	$ 0 $	$ 0 $
$ 4 $	$ 0 $	$ 108 $	$ 0 $	$ 13 $

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