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Connections between Mathematical and Behavioral Decision-Making Models
Bearden, J. N., & Connolly, T. Satisficing in Sequential Search (under review, Organizational Behavior & Human Decision Processes)
We present empirical and theoretical results from two multi-attribute sequential search tasks. In both, the DM sequentially encounters options composed of two attributes and must pay to learn the value of the attributes. In the continuous version of the task the DM learns precise numerical values of the attributes when she pays to view them. In the threshold version the DM learns only whether the value of an attribute is above or below a threshold that she sets herself. Results from the continuous condition reveal that DMs tend to terminate their searches too early relative to the optimal policy. The pattern is reversed in the threshold condition, with DMs searching for too long. Maximum likelihood comparisons of two different stochastic decision models show that DMs under both information conditions perform in ways consistent with the optimal policies. We relate our results to Herbert Simon’s notion of satisficing, and conclude that satisficing can
be a very effective and efficient search policy. Read Paper.
Bearden, J. N., Rapoport, A., & Murphy, R. O. Sequential Observation and Selection with Rank-Dependent Payoffs: An Experimental Test. (under review, Management Science)
We report the results of two experiments with different parameter sets designed to study sequential observation and selection decision problems in which payoffs increase monotonically in the quality of the selected alternative. In this class of search problems, applicants are interviewed one at a time, decision makers only learn the applicant’s rank relative to the applicants that have been interviewed and rejected, the decision to stop terminates the search, and payoffs decrease in the overall (and unknown) absolute
rank of the selected applicant. Compared to the optimal policy, which we computed numerically, the subjects terminated their search too early in both experiments. Results from a probability estimation task conducted in Experiment 2 show that subjects overestimate the absolute quality of applicants given their relative rank. We competitively tested three behavioral decision rules and found that a multi-threshold rule, which has the same form as the optimal policy but is parameterized differently, best accounts for the data. The alternative parameterization that results in stopping too early can be explained by the estimation results from Experiment 2. Read Paper.
Bearden, J. N., & Murphy, R. O. On Generalized Secretary Problems. (under review, Theory and Decision)
This paper is composed of two related parts. In the first, we present a
dynamic programming procedure for finding optimal policies for a class of sequential
search problems that includes the well-known “secretary problem." In the second,we propose a stochastic model of choice behavior for this class of problems and test
the model with two extant data sets. We conclude that the previously reported bias for decision makers to terminate their search too early can, in part, be accounted
for by a stochastic component of their search policies. Read Paper.
Bearden, J. N., Murphy, R. O., & Rapoport, A. A Multi-Attribute Extension of the Secretary Problem: Theory and Experiments. (in press) Journal of Mathematical Psychology.
We present a generalization of a class of sequential search problems with ordinal ranks (“secretary" problems) in which applicants are characterized by multiple attributes that are evaluated independently. We then present a procedure for numerically computing the optimal search policy and test it in two experiments with incentive-compatible payoffs. With payoffs dependent on the absolute ranks of the attributes, we test the optimal search model with both symmetric (Experiment 1) and asymmetric (Experiment 2) search problems. In both experiments we find that, relative to the optimal search policy, subjects stop the search too early. Our results show that this bias is largely driven by a propensity to stop prematurely on applicants of intermediate (relative) quality. Read Paper.
Lim, C. Bearden, J. N., & Smith, J. C. Sequential Search with Multi-Attribute Options (in press). Decision Analysis.
We describe a search problem in which a decision maker (DM) must select several sequentially-encountered options. Each option is described by multiple attributes, and the value of an option is given by a function of its attribute values. However, the attribute values are not known with certainty, and can only be ascertained in a predefined order, at some fixed cost. During the search the DM can choose to select an option, purchase information about an attribute value, reject (permanently) the current option and continue the search, or terminate the search and accept a status quo outcome. We introduce a Threshold Policy for this search process, and provide a class of option value functions for which the policy is optimal. We then furnish a dynamic programming procedure for prescribing an optimal policy for this problem. Finally, we derive analytic solutions to some special cases of the problem, and present a case study that demonstrates a possible use of the proposed approach. Read Paper.
Bearden, J. N. Skip the Square Root of n: A New Secretary Problem (under review, Journal of Mathematical Psychology)
We present an extension of the secretary problem in which the DM sequentially
observes up to n applicants whose actual values X1,X2,…,Xn are drawn i.i.d. from a uniform distribution on [0,1]. The DM must select exactly one applicant, cannot recall released applicants, and receives a payoff of xt, the realization of Xt, for selecting the tth applicant. For each encountered applicant, the DM only learns whether the applicant is the best so far. We prove that the optimal policy dictates skipping the first n1/2-1 applicants, and then selecting the next encountered
applicant with rank 1. Read Paper.
Bearden, J. N., Rapoport, A., & Murphy, R. O. Assigning Patients to Hospitals in a Time of Disaster: An Experimental Test of Sequential Selection and Assignment (under review, Journal of Behavioral Decision Making)
We study a class of sequential selection and assignment problems, where a decision maker (DM) must sequentially assign applicants to positions in an attempt to minimize total expected cost. In modeling this class of problems, we assume that the DM is only informed of the rank of an applicant relative to the applicants that she previously observed and assigned. We report the results of three experiments designed to study sequential assignment behavior. In comparing the aggregate results from all three experiments to the optimal decision policy that we present and discuss, we identify a systematic bias to assign early applicants to overly costly positions, but with experience subjects approach, though not reach, optimal assignments. Read Paper.
Bearden, J. N., & Connolly, T. Optimal Satisficing. (under review, Management Science)
Herbert Simon introduced the notion of satisficing to explain how boundedly rational agents might approach diffcult sequential decision problems. His satisficing decision makers were offered as an alternative to optimizers, who have impressive computational capacities which allow them to maximize. There is no reason, however, why satisficers can not do their task optimally. In this paper, we present a simplified sequential search problem for a satisficing decision maker, and show how to compute its optimal satisficing search policies. Our findings demonstrate that satisficing, when done properly, can be a quite effective search policy. Read Paper.
Bearden, J. N., & Rapoport, A. Operations Research in Experimental Psychology (in press). Tutorials In Operations Research: Emerging Theory, Methods, and Applications.
This chapter reviews some of the uses of operations research methods in experimental
psychology. We begin by describing some basic methodological issues that arise in the
study of human decision making. Next, we describe in more detail research in experimental psychology that has used methods common to operations research - such as dynamic programming - to understand sequential observation and selection behavior. We then suggest some ways in which experimental psychology and operations research can each provide the other with new research questions. Read Paper.
Casey, M. and Sen, S. The Scenario Generation Algorithm for Multistage Stochastic Linear Programming.
A multistage stochastic linear program (MSLP) is a model of sequential stochastic optimization where the objective and constraints are linear. When any of the random variables used in the MSLP are continuous, the problem is infinite dimensional. In order to numerically tackle such a problem we usually replace it with a finite dimensional approximation. Even when all the random variables have finite support, the problem is often computationally intractable and must be approximated by a problem of smaller dimension. One of the primary challenges in the field of stochastic programming deals with discovering effective ways to evaluate the importance of scenarios, and to use that information to trim the scenario tree in such a way that the solution to the smaller optimization problem is not much different than the problem stated with the original tree. The Scenario Generation (SG) algorithm proposed in this paper is a finite element method that addresses this problem for the class of MSLP with random right-hand sides. Read Paper.
Modeling and exploiting decision-making weaknesses in enemy behavior
Andreas, A. and Smith, J.C., Mathematical Programming Algorithms for Two-Path Routing Problems with Reliability Constraints. (submitted to) Operations Research.
Most traditional routing problems assume perfect operability of all arcs and nodes. However, when independent arc failure probabilities exist, a secondary objective must be present to retain some measure of expected functionality, introducing nonlinear, nonconvex constraints. We examine the Robust Two-Path Problem, which seeks to establish two paths between a source and destination node wherein at least one path must remain fully operable with some threshold probability. We consider the case where both paths must be arc-disjoint and the case where arcs can be shared between the paths. We begin by proving the NP-hardness of these problems, and then examine various strategies for solving the resulting nonlinear integer program, including pruning, coefficient tightening, lifting, and branch-and-bound partitioning schemes. We discuss the advantages and disadvantages of these methods, and conclude with computational results. Read Paper.
Lim, C., and Smith, J.C., Algorithms for Discrete and Continuous Multicommodity Flow Network Interdiction Problems. (revision submitted to) IIE Transactions.
We consider a network interdiction problem on a multicommodity flow network, in which an attacker disables a set of network arcs in order to minimize the maximum profit that can be obtained from shipping commodities across the network. The attacker is assumed to have some budget for destroying (or “interdicting") arcs, and each arc is associated with a positive interdiction expense. In this paper, we examine problems in which interdiction must be discrete (i.e., each arc must either be left alone or completely destroyed), and in which interdiction can be continuous (the capacities of arcs may be partially reduced). For the discrete problem, we describe a linearized model for optimizing network interdiction that is similar to previous studies
in the field, and compare it to a penalty model that does not require linearization constraints. For the continuous case, we prescribe an optimal partitioning algorithm along with a heuristic procedure for estimating the optimal objective function value. We demonstrate on a set of randomly generated test data that our penalty model for the discrete interdiction problem significantly reduces computational time when compared to that consumed by the linearization model. Read Paper.
Genc, T., Reynolds, S. and Sen, S. Dynamic Oligopolistic Games Under Uncertainty: A Stochastic Programming Approach.
This paper studies several stochastic programming formulations of dynamic oligopolistic games under uncertainty. We argue that one of the models, namely Games with Probabilistic Scenarios (GPS), provides an appropriate formulation. For such games, we show that symmetric players earn greater expected profits as demand volatility increases. This result suggests that even in an increasingly volatile market, players may have an incentive to participate in the market. The key to our approach is the so-called scenario formulation of stochastic programming. In addition to several modeling insights, we also discuss the application of GPS to the electricity market in Ontario, Canada. The examples presented in this paper illustrate that this approach can address dynamic games that are clearly out of reach for dynamic programming, a common approach in the literature on dynamic games. Read Paper.
Application study on sequential search problems
Bearden, J. N., & Connolly, T. Optimal Satisficing. (under review, Management Science)
Herbert Simon introduced the notion of satisficing to explain how boundedly rational agents might approach diffcult sequential decision problems. His satisficing decision makers were offered as an alternative to optimizers, who have impressive computational capacities which allow them to maximize. There is no reason, however, why satisficers can not do their task optimally. In this paper, we present a simplified sequential search problem for a satisficing decision maker, and show how to compute its optimal satisficing search policies. Our findings demonstrate that satisficing, when done properly, can be a quite effective search policy. Read Paper.
Bearden, J. N., & Connolly, T. Satisficing in Sequential Search (under review, Organizational Behavior & Human Decision Processes)
We present empirical and theoretical results from two multi-attribute sequential search tasks. In both, the DM sequentially encounters options composed of two attributes and must pay to learn the value of the attributes. In the continuous version of the task the DM learns precise numerical values of the attributes when she pays to view them. In the threshold version the DM learns only whether the value of an attribute is above or below a threshold that she sets herself. Results from the continuous condition reveal that DMs tend to terminate their searches too early relative to the optimal policy. The pattern is reversed in the threshold condition, with DMs searching for too long. Maximum likelihood comparisons of two different stochastic decision models show that DMs under both information conditions perform in ways consistent with the optimal policies. We relate our results to Herbert Simon’s notion of satisficing, and conclude that satisficing can
be a very effective and efficient search policy. Read Paper.
Bearden, J. N., & Murphy, R. O. On Generalized Secretary Problems. (under review, Theory and Decision)
This paper is composed of two related parts. In the first, we present a
dynamic programming procedure for finding optimal policies for a class of sequential
search problems that includes the well-known “secretary problem." In the second,we propose a stochastic model of choice behavior for this class of problems and test
the model with two extant data sets. We conclude that the previously reported bias for decision makers to terminate their search too early can, in part, be accounted
for by a stochastic component of their search policies. Read Paper.
Bearden, J. N., Murphy, R. O., & Rapoport, A. A Multi-Attribute Extension of the Secretary Problem: Theory and Experiments. (in press) Journal of Mathematical Psychology.
We present a generalization of a class of sequential search problems with ordinal ranks (“secretary" problems) in which applicants are characterized by multiple attributes that are evaluated independently. We then present a procedure for numerically computing the optimal search policy and test it in two experiments with incentive-compatible payoffs. With payoffs dependent on the absolute ranks of the attributes, we test the optimal search model with both symmetric (Experiment 1) and asymmetric (Experiment 2) search problems. In both experiments we find that, relative to the optimal search policy, subjects stop the search too early. Our results show that this bias is largely driven by a propensity to stop prematurely on applicants of intermediate (relative) quality. Read Paper.
Bearden, J. N., & Rapoport, A. Operations Research in Experimental Psychology (in press). Tutorials In Operations Research: Emerging Theory, Methods, and Applications.
This chapter reviews some of the uses of operations research methods in experimental
psychology. We begin by describing some basic methodological issues that arise in the
study of human decision making. Next, we describe in more detail research in experimental psychology that has used methods common to operations research - such as dynamic programming - to understand sequential observation and selection behavior. We then suggest some ways in which experimental psychology and operations research can each provide the other with new research questions. Read Paper.
Bearden, J. N. Skip the Square Root of n: A New Secretary Problem (under review, Journal of Mathematical Psychology)
We present an extension of the secretary problem in which the DM sequentially observes up to n applicants whose actual values X1,X2,…,Xn are drawn i.i.d. from a uniform distribution on [0,1]. The DM must select exactly one applicant, cannot recall released applicants, and receives a payoff of xt, the realization of Xt, for selecting the tth applicant. For each encountered applicant, the DM only learns whether the applicant is the best so far. We prove that the optimal policy dictates skipping the first n1/2-1 applicants, and then selecting the next encountered applicant with rank 1. Read Paper.
Lim, C. Bearden, J. N., & Smith, J. C. Sequential Search with Multi-Attribute Options (in press). Decision Analysis.
We describe a search problem in which a decision maker (DM) must select several sequentially-encountered options. Each option is described by multiple attributes, and the value of an option is given by a function of its attribute values. However, the attribute values are not known with certainty, and can only be ascertained in a predefined order, at some fixed cost. During the search the DM can choose to select an option, purchase information about an attribute value, reject (permanently) the current option and continue the search, or terminate the search and accept a status quo outcome. We introduce a Threshold Policy for this search process, and provide a class of option value functions for which the policy is optimal. We then furnish a dynamic programming procedure for prescribing an optimal policy for this problem. Finally, we derive analytic solutions to some special cases of the problem, and present a case study that demonstrates a possible use of the proposed approach. Read Paper.
Software models for human decision-making
Xu, Y. and Sen, S. A Distributed Computing Architecture for Simulation and Optimization.
We design a generic framework to integrate distributed simulation and optimization models. Many problems require the integration of these two types of models. For example, stochastic programming can use simulation models as a scenario generator for optimization models; in some other cases, simulation models need optimization models to help determine system parameters. The framework is shown to be able to provide various services to help the integration of simulation and optimization models. We illustrate our implementation with a product-mix example. The example integrates a discrete event simulation of a product-mix problem with a linear programming (optimization) model of such a system. The simulation updates the parameters in the optimization model, which as a result will generate a new production plan. Read Paper.
Zhao, X., Venkateswaran, J, & Son, Y. Modeling Human Operator Decision-Making in Manufacturing Systems Using BDI Agent Paradigm, Industrial Engineering Research Conference, 2005
Human operators are imperative to the functioning and success of a manufacturing system. Yet there is a noticeable lack of research in modeling of human operators, especially decision-making aspects. In this paper, the complex task of modeling of human operator decision-making in a manufacturing system is addressed using Belief-Desire-Intention (BDI) agent paradigm. The roles, responsibilities and services of the operator are mapped on to mental models of beliefs, desires and intentions. The dynamic evolution of the mental models is also highlighted. The functioning of the human operator model is illustrated by integrating the model with a shop floor control system. Read Paper.
Ryu, K., Son, Y, & Jung, M. Modeling and Specifications of Dynamic Agents in Fractal Manufacturing Systems, Computers in Industry, 52(2), 161-182.
In order to respond to a rapidly changing manufacturing environment and market, manufacturing systems must be flexible, adaptable, and reusable. The fractal manufacturing system (FrMS) is one of the new manufacturing paradigms that address the need for these characteristics. The FrMS is comprised of a number of “basic components”, each of which consists of five functional modules: 1) an observer, 2) an analyzer, 3) an organizer, 4) a resolver, and 5) a reporter. Each of these modules, using agent technology, autonomously cooperates and negotiates with others while processing its own jobs. The resulting architecture has a high degree of self-similarity, one of the main characteristics of a fractal. Despite the many conceptual advantages of the FrMS, it has not been successfully elaborated and implemented to date because of the difficulties involved in doing so. In this paper, the static functions and dynamic activities of each agent are modeled using the unified modeling language (UML). Then, relationships among agents, working mechanisms of each agent, and several fractal-specific characteristics (self-similarity, self-organization, and goal-orientation) are modeled using the UML. Then, a method for dealing with several types of information such as products, orders, and resources in the FrMS is presented. Finally, a preliminary prototype for the FrMS using AgletsÔ is presented. Read Paper.
Venkateswaran, J, & Son, Y. Hybrid System Dynamic--Discrete Event Simulation-based Architecture for Hierarchical Production Planning, International Journal of Production Research, 43(20), 4397-4429.
Multi-plant production planning problem deals with the determination of type and quantity of products to produce at the plants over multiple time periods. Hierarchical production planning provides a formal bridge between long-term plans and short-term schedules. A hybrid simulation-based hierarchical production planning architecture consisting of system dynamics (SD) components for the enterprise level planning and discrete event simulation (DES) components for the shop-level scheduling is presented. The architecture consists of the Optimizer, Performance Monitor and Simulator modules at each decision level. The Optimizers select the optimal set of control parameters based on the estimated behaviour of the system. The enterprise-level simulator (SD model) and shoplevel simulator (DES model) interact with each other to evaluate the plan. Feedback control loops are employed at each level to monitor the performance and update the control parameters. Functional and process models of the proposed architecture are specified using IDEF. The internal mechanisms of the modules are also described. The modules are interfaced using High Level Architecture (HLA). Experimental results from a multi-product multi-facility manufacturing enterprise demonstrate the potential of the proposed approach. Read Paper.