Satisficing Awakens: Models to Mitigate Uncertainty Patrick Jaillet Department of Electrical Engineering and Computer Science, Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected] Sanjay Dominik Jena Singapore-MIT Alliance for Research and Technology, Singapore, 138602, [email protected] Tsan Sheng Ng Department of Industrial & Systems Engineering, National University of Singapore, Singapore 117576, [email protected] Melvyn Sim NUS Business School, National University of Singapore, Singapore 119245, [email protected] Satisficing, as an approach to decision-making under uncertainty, aims at achieving solutions that satisfy the problem's constraints as well as possible. Mathematical optimization problems that are related to this form of decision-making include the P-model of Charnes and Cooper (1963), where satisficing is the objective, as well as chance-constrained and robust optimization problems, where satisficing is articulated in the constraints. In this paper, we first propose the R-model, where satisficing is the objective, and where the problem consists in finding the most \robust" solution, feasible in the problem's constraints when uncertain outcomes arise over a maximally sized uncertainty set. We then study the key features of satisficing decision making that are associated with these problems and provide the complete functional characterization of a satisficing decision criterion. As a consequence, we are able to provide the most general framework of a satisficing model, termed the S-model, which seeks to maximize a satisficing decision criterion in its objective, and the corresponding satisficing-constrained optimization problem that generalizes robust optimization and chance-constrained optimization problems. Next, we focus on a tractable probabilistic S-model, termed the T-model whose objective is a lower bound of the P-model. We show that when probability densities of the uncertainties are log-concave, the T-model can admit a tractable concave objective function. In the case of discrete probability distributions, the T-model is a linear mixed integer program of moderate dimensions. We also show how the T-model can be extended to multi-stage decision-making and present the conditions under which the problem is computationally tractable. Our computational experiments on a stochastic maximum coverage problem strongly suggest that the T-model solutions can be highly effective, thus allaying misconceptions of having to pay a high price for the satisficing models in terms of solution conservativeness. History : January 28, 2016 1 P. Jaillet, S. D. Jena, T. S. Ng, and M. Sim: Satisficing Awakens: Models to Mitigate Uncertainty 2 Article submitted; 1. Introduction Parametric uncertainty in mathematical programming problems is ubiquitous in many real-world problems. When uncertain parameters are revealed, the solution obtained may become infeasi- ble and the actual objective value attained may be significantly worse than the \optimal" value achieved based on solving optimization models assuming deterministic parameter values (Ben-Tal et al. 2004). A common approach to deal with uncertainty in an optimization problem is to incor- porate attitudes of risks, when probability distributions are available, or ambiguity, whenever they are not. Such models have been extensively studied in stochastic programming (see, e.g., Pr´ekopa 1995, Birge and Louveaux 1997, Pr´ekopa 2003), robust optimization (see, e.g., Ben-Tal and Nemirovski 1999, Bertsimas et al. 2011) and distributionally robust optimization (see, e.g., Delage and Ye 2010, Goh and Sim 2010, Wiesemann et al. 2014). The concept of satisficing, a portmanteau of the terms `satisfy' and ‘suffice’ first introduced by Simon (1959), addresses uncertainty with the aims of achieving feasibility in an uncertain envi- ronment. In many real world problems under uncertainty, the goal is not necessarily to maximize or minimize objective functions such as profits or costs. Instead, decision-makers may be more interested in obtaining solutions that can “satisfice” the constraints of the problem, in some sense, as well as possible. For instance, it is reasonable in a project management problem with uncertain activity completion times to ensure that the project can be completed on schedule and within the allocated budget (see, e.g., Goh and Hall 2013). Satisficing as an objective in decision making has mostly been explored from the economic perspective (see, e.g. G¨uth2010, St¨uttgenet al. 2012). Charnes and Cooper (1963) were the first to incorporate the principal idea of satisficing in the mathematical framework of success probability maximization, which has been termed the P-model. In simplified form, the P-model can be stated as: max ln (A(z~)x ≥ b(z~)) P (1) s:t: x 2 X ; where x are decision variables of dimension N defined on a feasible set X and z~ is a K dimensional random vector that influences the entries of the function maps A : RK 7! RM×N and b : RK 7! RM . We refer to the problem's constraints as the set of randomly perturbed linear constraints, A(z)x ≥ b(z), where z is a random outcome of z~ (under the probability measure P). A satisficing decision criterion evaluates how well a solution x would remain feasible in the problem's constraints under uncertainty. In this regard, the P-model is an optimization problem that maximizes a log- N probability satisficing decision criterion, νP : R 7! R [ {−∞}, given by νP (x) = lnP (A(z~)x ≥ b(z~)) : (2) P. Jaillet, S. D. Jena, T. S. Ng, and M. Sim: Satisficing Awakens: Models to Mitigate Uncertainty Article submitted; 3 Note that the decision criterion is based on a log-probability function, instead of directly a probabil- ity function. Using the convention ln 0 = −∞, if x is always infeasible in the problem's constraints, then νP (x) = −∞. Moreover, if the probability function is log-concave in x, which may arise in useful instances, then the objective function of Problem (1) would be concave. The P-model is closely related to the more popular chance-constrained optimization problem pioneered by Charnes and Cooper (1959, 1963) where the satisficing decision criterion (2) is imposed in the constraints. The following gives a simplified definition of a chance-constrained optimization model: min c0x s:t: ln P (A(z~)x ≥ b(z~)) ≥ ∆ (3) x 2 X : In contrast to the P-model, the objective in (3) is a deterministic cost function, where c 2 RN defines the objective function coefficients, and the satisficing decision criterion of the P-model is now subject to a lower bound parameter, ∆ 2 R. In other words, the satisficing decision criterion constraint above enforces how well the problem's constraints A(z)x ≥ b(z) must be satisfied under uncertainty, and among feasible solutions x 2 X that fulfill this decision criterion constraint, select one that is cost-minimizing. We refer readers to Pr´ekopa (2003) and Henrion (2004) for an excellent introduction to the models, algorithms and theory of chance-constrained programming. Chance-constrained programming has also found wide-ranging applications in important finance and engineering problems, for instance in renewable energy planning under uncertain loads and intermittent renewable supplies (see e.g., Van et al. 2014, Bremer et al. 2015). The P-model and the chance-constrained optimization problem are known to be generally intractable, because evaluating the probability typically demands high dimensional integration and is a computationally excruciating procedure. It is also noteworthy that the log-probability satisfic- ing decision criterion (2) is concave for the special case when A(·) is a constant (see, e.g. Pr´ekopa 2003, Theorem 2.5) and the vector b(z~) is affinely dependent onz ~1;:::; z~K , which are independently distributed random variables with log-concave density functions. However, notwithstanding the convexity of the resultant optimization problems in (1) and (3), the evaluation of the log-probability satisficing decision criterion (2) can itself still be computationally challenging even when M = 1 and the random variables are iid uniformly distributed (Nemirovski and Shapiro 2006). Hence, more restrictive conditions are required to obtain computationally scalable results. For more general distributions, one may use sample average approximation approaches, such as Monte Carlo meth- ods (see, e.g. Shapiro, A. 2003) to approximate the objective function. Unfortunately, even small problems with relatively simple structure can require hundreds of samples (Shapiro and Homem- de-Mello 2000, Pagnoncelli et al. 2009) to achieve a desired level of accuracy. When the problem size is large or the variability of the uncertain parameters is high, the sample size required is likely P. Jaillet, S. D. Jena, T. S. Ng, and M. Sim: Satisficing Awakens: Models to Mitigate Uncertainty 4 Article submitted; to become prohibitively large. Furthermore, a sample-based optimization model for solving these problems requires a large number of binary variables in its formulation, making it computationally expensive. Robust optimization is a more tractable alternative to chance-constrained optimization and has become an important approach in addressing practical optimization problems with data uncer- tainty. A robust optimization problem can generally be formulated as follows, min c0x s:t: A(z)x ≥ b(z) 8z 2 U(Γ) (4) x 2 X : In the above, the vector z denotes a realization of z~ from an uncertainty set, U(Γ) ⊆ W, with W ⊆ K R being the support of z~, and where Γ 2 R+ is a user specified parameter that relates to the level of uncertainty that must be tolerated. In particular, U(·) is designed such that U(α1) ⊆ U(α2) ⊆ W for all 0 ≤ α1 ≤ α2. The parameter Γ is sometimes referred to as a \budget of uncertainty", and is a simple and powerful tool that allow decision-makers to adjust and customize the robust optimization models according to their degree of aversion to uncertainty. Such an approach can provide some flexibility in preventing linear programming solutions from being overly conservative (see, e.g., Ben-Tal and Nemirovski 1999, Bertsimas et al.