Large-Scale Solution Approaches for Healthcare and Supply Chain Scheduling

This research proposes novel solution techniques for two real world problems. We first consider a patient scheduling problem in a proton therapy facility with deterministic patient arrivals. In order to assess the impacts of several operational constraints, we propose single and multi-criteria linear programming models. In addition, we ensure that the strategic patient mix restrictions predetermined by the decision makers are also enforced within the planning horizon. We study the mathematical structures of the single criteria model with strict patient mix restrictions and derive analytical equations for the optimal solutions under several operational restrictions. These efforts lead to a set of rule of thumbs that can be utilized to assess the impacts of several input parameters and patient mix levels on the capacity utilization without solving optimization problems. The necessary and sufficient conditions to analytically generate exact efficient frontiers of the bicriteria problem without any additional side constraint are also explored. In a follow up study, we investigate the solution techniques for the same patient scheduling problem with stochastic patient arrivals. We propose two Markov Decision Process (MDP) models that are capable of tackling the stochasticity. The second problem of interest is a variant of the parallel machine scheduling problem. We propose constraint programming (CP) and logic-based Benders decomposition algorithms in order to make the best decisions for scheduling nonidentical jobs with time windows and sequence dependent setup times on dissimilar parallel machines in a fixed planning horizon. This problem is formulated with (i) maximizing total profit and (ii) minimizing makespan objectives. We conduct several sensitivity analysis to test the quality and robustness of the solutions on a real life case study

Subproblem Separation in Logic-Based Benders\u27 Decomposition for the Vehicle Routing Problem with Local Congestion

Subproblem separation is a common strategy for the acceleration of the logic-based Benders\u27 decomposition (LBBD). However, it has only been applied to problems with an inherently separable subproblem structure. This paper proposes a new method to separate the subproblem using the connected components algorithm. The subproblem separation is applied to the vehicle routing problem with local congestion (VRPLC). Accordingly, new Benders\u27 cuts are derived for the new subproblem formulation. The computational experiments evaluate the effectiveness of subproblem separation for different methods applying new cuts. It is shown that subproblem separation significantly benefits the LBBD scheme

Combining Optimisation and Simulation Using Logic-Based Benders Decomposition

Operations research practitioners frequently want to model complicated functions that are are difficult to encode in their underlying optimisation framework. A common approach is to solve an approximate model, and to use a simulation to evaluate the true objective value of one or more solutions. We propose a new approach to integrating simulation into the optimisation model itself. The idea is to run the simulation at each incumbent solution to the master problem. The simulation data is then used to guide the trajectory of the optimisation model itself using logic-based Benders cuts. We test the approach on a class of stochastic resource allocation problems with monotonic performance measures. We derive strong, novel Benders cuts that are provably valid for all problems of the given form. We consider two concrete examples: a nursing home shift scheduling problem, and an airport check in counter allocation problem. While previous papers on these applications could only approximately solve realistic instances, we are able to solve them exactly within a reasonable amount of time. Moreover, while those papers account for the inherent variance of the problem by including estimates of the underlying random variables as model parameters, we are able to compute sample average approximations to optimality with up to 100 scenarios.Comment: 30 page

Improved cut generation algorithms for the acceleration of Logic-Based Benders Decomposition

Logic-based Benders' decomposition (LBBD) is a solution method that integrates mixed-integer programming (MIP) and constraint programming (CP). LBBD solution scheme is a finite iterative algorithm, the central element of which are Benders' cuts. It is crucial for the convergence of the algorithm to strengthen the generated Benders' cuts. This thesis aims to improve cut generation algorithms for the acceleration of LBBD. The cut generation algorithms are the steps of the LBBD solution scheme that can include cut-strengthening techniques and subproblem separation. As the initial step, this thesis provides an extensive computational evaluation of cut-strengthening techniques in LBBD. The evaluation also includes subproblem separation and its influence on the effectiveness cut-strengthening techniques and the overall acceleration of LBBD. The computational experiments solve cumulative facility scheduling, single-facility scheduling, and vehicle routing problems, which are representative of LBBD problems and are routinely solved with benchmark datasets available. The results of this study indicate that cut-strengthening techniques can benefit from variable sorting. Another observation from this study is that cut-strengthening techniques and subproblem separation can be used interchangeably. Three heuristics based on variable sorting are proposed to improve efficiency of cut-strengthening techniques. The main features of the proposed heuristics are simplicity and no additional computational cost. The computational results show that improved cut-strengthening techniques lead to reduction in overall solution time of the LBBD solution scheme. A novel way of separating the subproblem is developed in this thesis. The subproblem separation is based on the connected components algorithm and can be applied to subproblems that do not separate naturally. The computational results solving vehicle routing problem with local congestion show that substantial acceleration is achieved by subproblem separation. This thesis shows that improvements to cut generation algorithm can significantly accelerate LBBD. The proposed improvements can be implemented as a part of general LBBD framework

Leveraging Decision Diagrams to Solve Two-stage Stochastic Programs with Binary Recourse and Logical Linking Constraints

Two-stage stochastic programs with binary recourse are challenging to solve and efficient solution methods for such problems have been limited. In this work, we generalize an existing binary decision diagram-based (BDD-based) approach of Lozano and Smith (Math. Program., 2018) to solve a special class of two-stage stochastic programs with binary recourse. In this setting, the first-stage decisions impact the second-stage constraints. Our modified problem extends the second-stage problem to a more general setting where logical expressions of the first-stage solutions enforce constraints in the second stage. We also propose a complementary problem and solution method which can be used for many of the same applications. In the complementary problem we have second-stage costs impacted by expressions of the first-stage decisions. In both settings, we convexify the second-stage problems using BDDs and parametrize either the arc costs or capacities of these BDDs with first-stage solutions depending on the problem. We further extend this work by incorporating conditional value-at-risk and we propose, to our knowledge, the first decomposition method for two-stage stochastic programs with binary recourse and a risk measure. We apply these methods to a novel stochastic dominating set problem and present numerical results to demonstrate the effectiveness of the proposed methods