Multi-objective stochastic linear programming with recourse and flexible decision making

TL;DR

This paper introduces a novel set-valued optimization framework for multi-objective stochastic linear programs with recourse, embedding a manager’s preference for flexibility directly into the first-stage decision via a polyhedral convex set objective. By formulating the problem with an upper image $\mathcal{P}$ and a set-valued objective $F(x)$, the approach yields a whole spectrum of Pareto-efficient outcomes while ensuring maximal second-stage flexibility, operationalized through forward (DP1) and backward (DP2) decision procedures. It develops deterministic surrogates—the wait-and-see problem and the expected-value problem—and analyzes their relationships to the original recourse problem, including conditions under which their upper images contain or equal $\mathcal{P}$. The framework is demonstrated on a multi-objective newsvendor with sustainability and a risk-management example with transaction costs, highlighting practical managerial guidelines and outlining extensions to broader settings and risk measures.

Abstract

Optimal inventory leads to stochastic optimization problems where deterministic delivery decisions have to be made in advance of stochastic demand realizations. Similarly, risk deposits have to be given before the random outcomes of investments are known. In this paper, multi-criteria versions of such stochastic recourse problems are studied. In addition to traditional concepts like Pareto-optimality, a decision maker for the multi-criteria problem may have a preference for greater flexibility in the second stage decision. This idea leads to a first stage optimization problem with a set-valued objective instead of a mere multi-criteria one. Under linearity assumptions, this problem becomes a polyhedral convex set optimization problem instead of a multi-objective linear program. Solution concepts for multi-objective/set-valued recourse problems are given as well as deterministic surrogates of the stochastic problem such as its deterministic equivalent, the so-called wait-and-see problem and the expected-value problem for the multi-objective case. Managerial decision making guidelines are obtained based on set optimization methods and the preference-for-flexibility approach: choose the deterministic first stage variable such that a maximum of flexibility is combined with a guarantee for Pareto minimal objective values. Two major examples illustrate the findings, a multi-objective newsvendor problem with an additional health/sustainability objective and a risk compensation problem where the availability of more than one asset for risk compensation, e.g., several currencies, leads to multiple objectives.

Figures (6)

• Figure 1: A decision maker could be interested in the Pareto efficient expected outcome $\geneuro 250$ in $100$ minutes working time, i.e., in the point $y=(250,100)$. There are several options to realize this outcome. One option is to purchase $100$ pieces of JP and no piece of BT, i.e., $x=(100,0)$. Another option is purchasing $100$ copies of JP and $100$ copies of BT, i.e., $x=(100,100)$. This ambiguity has consequences for the second-stage decision process whenever the newsvendor changes her preference. The grey sets in the background illustrate all possible outcomes (and outcomes worse than these) for all possible first-stage decisions $x$. The vertices and the edges between adjacent vertices represent the efficient outcomes. The colored sets display all possible second-stage outcomes (and outcomes worse than these) after fixing $x$. We see that the efficient first-stage decision $x=(100,100)$ provides more flexibility in the second stage than the first-stage decision $x=(100,0)$. Moreover, $x=(100,100)$ is the most flexible among all first-stage decisions which enables the outcome of $\geneuro 250$ in $100$ minutes. Note that the $y_1$-axis is inverted as we maximize the gain while in the standard form of \ref{['spr1']} we would minimize the loss.
• Figure 2: Solution of the recourse problem from Example \ref{['ex:1']}. The gray set in the background is the upper image $\mathcal{P} = \mathcal{P}^\star$. The colored sets in the front represent the objective values $F(x^1)$, $F(x^2)$ and $F(x^3)$. Every vertex of $\mathcal{P}$ is contained in one of the sets $F(x^1)$, $F(x^2)$ and $F(x^3)$ (infimum attainment). The three displayed sets $F(x^i)$ are not properly contained in some other set $F(x)$, $x \in \mathbb{R}^n$ (minimality). Thus $\bar{X} = \Set{x^1, x^2 , x^3}$ is a solution to the polyhedral convex set optimization problem \ref{['sprs']}. The last picture shows a comparison of the three solution elements. $F(x^i)$ represents the second-stage flexibility obtained for the first-stage decision $x^i$.
• Figure 3: Upper image of the wait-and-see problem (red, rear) compared to the upper image of the recourse problem (gray, front). Left: based on data from Monday and Tuesday only. Right: based on data from Monday to Wednesday.
• Figure 4: Upper image of the expected value problem (blue) compared to the upper image of the recourse problem (gray). Left: based on data from Monday and Tuesday only. Right: based on data from Monday to Wednesday, where we see that inclusion \ref{['eq:52']} can is violated.
• Figure 5: The upper images of \ref{['eevx']} are displayed in the first three pictures. The gray set in the background is the upper image of the recourse problem. We see that not all possible outcomes of the recourse problem can be obtained by this solution of \ref{['ev']}, not even if we take the convex hull over the upper images of \ref{['eevx']}. For instance, the maximal gain is not reached.

Theorems & Definitions (18)

• Remark 3.1
• Definition 3.2
• Proposition 3.3
• Example 3.4
• Remark 3.5
• Proposition 4.1
• Proposition 5.1
• Proposition 5.2
• Definition 5.3
• Remark 5.4