Using integer programming for balancing return and risk in problems with individual chance constraints

Abstract

In this paper, we study probabilistically constrained problems involving individual chance constraints, random univariate right-hand sides, and risk tolerances defined as decision variables which affect part of the objective function. Built on the concept of efficient points, we formulate the problems as mixed-integer programs by using binary variables to determine an optimal risk tolerance for each chance constraint. We develop two benchmark approaches, both of which solve chance-constrained programs with fixed risk values in a bisection algorithm or by enumeration. We specify our approaches for a minimum cost flow problem and a network capacity design problem, both of which involve chance constraints for bounding the risk of demand shortages. We test instances with diverse size and complexity of the two network problems, and demonstrate the computational efficacy as well as give managerial insights.

Introduction

Chance-constrained programming (CCP) techniques are often used to evaluate quality of service (QoS) or system reliability for decision making under uncertainty. A generic CCP formulation is in the form of

$$\min \{c^{T}x:x\in X,\mathbb{P}(\mathit{Tx}-\xi \ge 0)\ge 1-ϵ\},$$

where $x\in {\mathbb{R}}^n$ is a decision vector in a deterministic feasible region $X\subseteq {\mathbb{R}}_n^{+}$, and $c\in {\mathbb{R}}^n$ represents a deterministic cost vector. Additionally, solution x needs to satisfy a chance constraint in which $T\in {\mathbb{R}}^{m\times n}$ is a technology matrix and $\xi \in {\mathbb{R}}^m$ is a right-hand side vector. Both T and ξ can be random parameters, following some underlying probability distributions. The minimum probability for satisfying the chance constraint is $1-ϵ$, where ϵ explicitly indicates a value of risk tolerance.

The study of CCP problems dates back to Charnes et al. [3] and Miller and Wagner [13]. The problems are generally intractable unless specific assumptions imposed (see [15]). Various approaches have emerged for handling general CCP problems, using convex or scenario approximations (e.g., [14]) as well as equivalent binary integer programs by assuming a finite support of the uncertainty (see, e.g., [12], [7]). In particular, when only vector ξ is random, the literature defines the concept of non-dominated points, or the so-called p-efficient points (where p refers to $1-ϵ$ in this paper) for optimizing the CCP model (1). The following definition of p-efficient point is due to Prékopa [16] and is used by Dentcheva et al. [4] for solving joint CCP problems with the right-hand side vector ξ being random.

Definition 1

Let $p\in [0,1]$. A point $v\in {\mathbb{R}}^m$ is called a p-efficient point of the probability distribution F, if $F\left(v\right)\ge p$ and there is no $w\le v$, $w\ne v$ such that $F\left(w\right)\ge p$.

For joint chance constraints with random technology matrix T, p-efficient points are used by Ruszczyński [19] for solving equivalent integer programming reformulations, where 0–1 knapsack cover inequalities are employed for improving the computation. In this paper, we use the results of efficient points to analyze two model (1) variants that involve adjustable risk tolerance ϵ being treated as a decision variable.

We consider a new class of CCP formulations by letting risk tolerance ϵ be a decision variable and taking into account a new type of cost that is associated with variable ϵ. We consider

$$[\mathbf{CCRV}-\mathbf{S}]:\min \{c^{T}x+g(ϵ):x\in X,\mathbb{P}(\mathit{Tx}-\xi \ge 0)\ge 1-ϵ,1-ϵ\in [1-\overline{ϵ},1\left]\right\},$$

where variable ϵ denotes the maximum probability of violating $\mathit{Tx}-\xi \ge 0$, and is upper bounded by parameter $\overline{ϵ}<1$. The objective function trades off between the original operational cost $c^{T}x$ and the cost $g\left(ϵ\right):[0,\overline{ϵ}]\to \mathbb{R}$ that is a monotonically increasing function in ϵ. That is, if the chance constraint bears with higher risk ϵ, then we pay higher cost $g\left(ϵ\right)$, which reflects the risk-and-return tradeoff.

In this paper, we use discretely distributed ξ, and denote $\Omega$ as a finite set of scenarios containing realizations ${\xi }^1,\dots ,{\xi }^{\left|\Omega \right|}$, with p^s being the probability of realizing ${\xi }^s,\forall s\in \Omega$. This assumption is given without loss of generality, since if ξ follows some continuous distribution, one can use sampling (see [11]) to generate a finite number of scenarios to form the set $\Omega$. Moreover, we focus on individual chance constraints (i.e., m=1) with the random right-hand side ξ following a univariate distribution. In such a case, there is a unique (1−ϵ)-efficient point coinciding the (1−ϵ)-quantile of the distribution function of ξ (see Definition 1), and thus we can narrow the set of feasible solutions of the risk variable ϵ to a finite number of threshold values derived from possible realizations of ξ.

In a parallel study by Lejeune and Shen [10], the authors consider joint chance constraints (i.e., $m>1$) with random technology matrix T, and develop approaches based on sufficient points instead of efficient points for balancing the tradeoff between risk and return. A point k is called p-sufficient if $F\left(k\right)\ge p$ and is p-insufficient if $F\left(k\right)<p$. This definition is more relaxed compared with Definition 1, so that any p-efficient point is also p-sufficient, but not vice versa. Moreover, one can derive all p-sufficient realizations using marginal probability distributions of random parameters in a joint chance constraint when $m>1$ and then use the Boolean modeling framework to define the satisfiability of that constraint. The method is used to handle joint chance constraints with random right-hand sides [9], and used by Lejeune [8] to elicit the exhaustive list of p-efficient points through solutions of a mixed-integer linear program. When $m>1$, the Boolean modeling method requires much fewer binary variables to reformulate a joint chance constraint as compared to using p-efficient points that may require an exponential number of binary variables depending on the sample size $\left|\Omega \right|$. When m=1, we cannot take the advantage of using marginal probability functions to derive all p-sufficient realizations and the corresponding reformulation while limiting the number of binary variables used. In this paper, we employ $(1-ϵ)$-efficient point to reformulate individual chance constraints in generic CCRV-S formulations and for each fixed ϵ, we only have one $(1-ϵ)$-efficient point.

For solving CCRV-S, we first replace ϵ with binary variables that are specially ordered set of type one (SOS1), and transform CCRV-S into a mixed-integer programming (MIP) formulation. We develop two benchmark approaches which vary the values of ϵ in a bisection algorithm and by enumeration. In each benchmark approach, we optimize the corresponding CCP model (1) for each fixed choice of ϵ. We also develop a polynomial-time algorithm for solving CCRV-S when $X={\mathbb{R}}_{+}^n$. Furthermore, we study a variant of CCRV-S with multiple individual chance constraints, each of which contains a univariate random right-hand side and a risk tolerance variable. The summation of all risk tolerances is bounded by a given budget, and each risk tolerance variable affects the objective through a monotonically increasing cost function.

We apply our approaches to two types of network-flow problems under demand uncertainty, respectively involving single- or multi-commodity flows. We formulate single or multiple individual chance constraints to restrict the risk of node- and commodity-wise demand shortages, and trade off between the cost affected by the value of risk tolerance and the cost of network flow/capacity design. We demonstrate the computational efficacy of our approaches by testing a set of diverse instances of the two problems, and also reveal managerial insights of risk management in network-related applications.

For applications where the value of ϵ is unlikely to be fixed before making decision x, CCRV-S provides a framework for balancing return and risk. A CCRV-S variant in which cost and risk are two components of a scalarized objective function is due to Evers [6], motivated by a chemical engineering problem in the presence of uncertainty in a compound production process. The literature also performs a Pareto analysis of the “efficient frontier” of CCP problems by changing ϵ in model (1) and computing optimal objective values for fixed values of ϵ. Rengarajan and Morton [17] and Rengarajan et al. [18] use such an analysis to trade off between the cost of building a network and the probability of an adversarial event causing failure of the network. The CCRV-S variants with multiple chance constraints are commonly seen in stochastic network flow applications [20], [21]. We show an example to demonstrate problems for applying CCP models with adjustable risk variables

Example 1 Chance-constrained optimal power flow (OPF)

Generation re-dispatch is a routine operation performed periodically (e.g., every 15 min) for adjusting outputs to meet the loads for power in transmission systems. The related OPF problems typically minimize a (convex) cost of generation, subject to flow balance constraints, transmission capacity constraints, ramping constraints, etc. With high renewable penetration in future smart grids, a dispatcher faces both fluctuations of wind farm outputs and uncertain loads, leading to situations where power flow ratings could be significantly exceeded, and thus this calls to re-examine the use of standard cost-based OPF.

To achieve system stability, one can consider a chance-constrained OPF problem by formulating multiple chance constraints to restrict the risk of (i) overloads exceeding thermal limits, (ii) violating synchronization conditions [2], [5] and (iii) renewable outputs being under-utilized. The three chance constraints may or may not agree on the same set of economic dispatch decisions. (To satisfy (iii) we intend to use more outputs from some wind farm having high generation rates which may lead to violations of (i) or (ii).) Reliability levels change in the chance constraints may affect yield quantity, solution quality, and the sensitivity of results at different scales. Such a multi-objective nature motivates us to make an appropriate selection of reliability targets associated with the multiple chance constraints, according to some pre-existing relations of the reliability values. For instance, the risk associated with loss of synchrony should be much smaller than the risk of exceeding thermal limits depending on underlying physics and engineering restrictions in transmission systems.

CCRV-S is also applicable to analyzing risk management problems with hidden objectives (e.g., reputation damage) associated with reliability $1-ϵ$. For instance, risk faced by a company and risk faced by a manager in the company may need separate consideration as the manager׳s reputation might suffer from a highly visible mistake. In CCRV-S, function $g\left(ϵ\right)$ is used to capture the cost of reputation associated with a reliability level $1-ϵ$. It can also represent a company׳s long-term reputation or brand name. This is reflected by the objective of CCRV-S which combines a short-term profit $c^{T}x$ goal with a long-term brand name objective $g\left(ϵ\right)$.

Structure of the paper: The rest of the paper is organized as follows. 2 Optimizing risk tolerance of a single chance constraint, 3 Optimizing risk in multiple individual chance constraints consider CCRV-S and its variant with multiple individual chance constraints, respectively. For both problems, we reformulate equivalent MIP models and develop benchmark algorithms for solving the reformulations. In Section 4, we specify our models and approaches in 2 Optimizing risk tolerance of a single chance constraint, 3 Optimizing risk in multiple individual chance constraints, respectively, for a minimum cost flow problem and a multi-commodity network capacity design problem. In Section 5, we demonstrate computational results and insights by testing instances of the network-flow problems in Section 4. In Section 6, we summarize the paper and describe future research directions.