Distributionally Preference Robust Optimization in Multi-Attribute Decision Making

TL;DR

This paper is the first attempt to use distributionally robust optimization methods for PRO and applies the proposed DPRO model in car manufacturing and facility location planning and shows how the random samples may be extracted by multinomial logit method and conjoint analysis/machine learning.

Abstract

Utility preference robust optimization (PRO) has recently been proposed to deal with optimal decision making problems where the decision maker's (DM) preference over gains and losses is ambiguous. In this paper, we take a step further to investigate the case that the DM's preference is not only ambiguous but also potentially inconsistent or even displaying some kind of randomness. We propose a distributionally preference robust optimization (DPRO) approach where the DM's preference is represented by a random utility function and the ambiguity is described by a set of probability distributions of the random utility. An obvious advantage of the new DPRO model is that it no longer concerns the DM's preference inconsistency. In the case when the random utility functions are of piecewise linear structure, we propose two statistical methods for constructing the ambiguity set: an ellipsoidal method and a bootstrap method both of which are fundamentally based on the idea of confidence region with the sample mean of the random parameters, and demonstrate how the resulting DPRO can be solved by a cutting surface algorithm and an MISOCP respectively. We also show how the DPRO models with general random utility functions may be approximated by those with piecewise linear random utility functions. Finally, we apply the proposed DPRO model in car manufacturing and facility location planning and show how the random samples may be extracted by multinomial logit method and conjoint analysis/machine learning. The paper is the first attempt to use distributionally robust optimization methods for PRO.

Figures (9)

• Figure 1: PLUF $u(x;v)=u_1(x_1;v_1)+u_2(x_2;v_2)$ with $x=(x_1,x_2)^T$, $v=(v_1^T,v_2^T)^T$, $v_1=(v_{1,1},v_{1,2},\cdots,v_{1,7})^T$, $v_2=(v_{2,1},v_{2,2},\cdots,v_{2,6})^T$.
• Figure 2: $K=1000$, $N=50$. (a) convex hull $\mathfrak{W}^*_{1-\alpha}$. (b)$100(1-\alpha)\%$ bootstrap confidence region $\mathfrak{C}^*_{1-\alpha}$. (c)$\mathfrak{C}^*_{1-\alpha}$ v.s. ellipsoid $\left\{v_{<3-1>}\in \mathbb{ R}^2_+: \|S^{-1/2}_{<3-1>}(v_{<3-1>}-\bar{V}_{<3-1>})\|^2\leq \gamma \right\}$.
• Figure 3: $N=20$, $N=50$, $N=100$, $K=10,000$. The sets of coloured points represent the set $\mathfrak{W}^*_{1-\alpha}$ with different values of $\alpha$.
• Figure 4: $N=20$, $N=50$, $N=100$, $K=10,000$. The sets of coloured points represent the set $\mathfrak{C}^*_{1-\alpha}$ with different values of $\alpha$.
• Figure 5: (a) The optimal value of DUPRO with ambiguity set by bootstrap; (b)-(d) The worst expected utility functions when $\alpha = 0.15$, $\alpha=0.30$ and $\alpha=0.55$.

Theorems & Definitions (10)

• proposition 1
• proposition 2
• proposition 3
• proposition 4
• proposition 5
• proposition 6
• proposition 7
• theorem 1: \ref{['DUPRO']} with ellipsoidal ambiguity (\ref{['eq:set-fV']})
• lemma 1
• theorem 2: \ref{['DUPRO']} with bootstrap ambiguity \ref{['set:fP_hat B']}