Troels Martin Range

A column generation approach for solving the patient admission scheduling problem

This paper addresses the Patient Admission Scheduling (PAS) problem. The PAS problem entails assigning elective patients to beds, while satisfying a number of hard constraints and as many soft constraints as is possible, and arises at all planning levels for hospital management. There exist a few, different variants of this problem. In this paper we consider one such variant and propose an optimization-based heuristic building on branch-and-bound, column generation, and dynamic constraint aggregation to solve it. We achieve tighter lower bounds than previously reported in the literature and, in addition, we are able to produce new best known solutions for five out of twelve instances from a publicly available repository.

A Column Generation-Based Heuristic for Rostering with Work Patterns

This paper addresses the Ground Crew Rostering Problem with Work Patterns, an important manpower planning problem arising in the ground operations of airline companies. We present a cutting stock-based integer programming formulation of the problem and describe a powerful heuristic decomposition approach, which utilizes column generation and variable fixing, to construct efficient rosters for a six-month time horizon. The time horizon is divided into smaller blocks, where overlaps between the blocks ensure continuity. The proposed methodology is able to circumvent one step of the conventional roster construction process by generating rosters directly based on the estimated workload. We demonstrate that this approach has the additional advantage of being able to easily incorporate robustness in the roster. Computational results on real-life instances confirm the efficiency of the approach.

An adaptive large neighborhood search procedure applied to the dynamic patient admission scheduling problem

OBJECTIVE The aim of this paper is to provide an improved method for solving the so-called dynamic patient admission scheduling (DPAS) problem. This is a complex scheduling problem that involves assigning a set of patients to hospital beds over a given time horizon in such a way that several quality measures reflecting patient comfort and treatment efficiency are maximized. Consideration must be given to uncertainty in the length of stays of patients as well as the possibility of emergency patients. METHOD We develop an adaptive large neighborhood search (ALNS) procedure to solve the problem. This procedure utilizes a Simulated Annealing framework. RESULTS We thoroughly test the performance of the proposed ALNS approach on a set of 450 publicly available problem instances. A comparison with the current state-of-the-art indicates that the proposed methodology provides solutions that are of comparable quality for small and medium sized instances (up to 1000 patients); the two approaches provide solutions that differ in quality by approximately 1% on average. The ALNS procedure does, however, provide solutions in a much shorter time frame. On larger instances (between 1000-4000 patients) the improvement in solution quality by the ALNS procedure is substantial, approximately 3-14% on average, and as much as 22% on a single instance. The time taken to find such results is, however, in the worst case, a factor 12 longer on average than the time limit which is granted to the current state-of-the-art. CONCLUSION The proposed ALNS procedure is an efficient and flexible method for solving the DPAS problem.

A shortest-path-based approach for the stochastic knapsack problem with non-decreasing expected overfilling costs

Abstract The knapsack problem (KP) is concerned with the selection of a subset of multiple items with known positive values and weights such that the total value of selected items is maximized and their total weight does not exceed capacity. Item values, item weights, and capacity are known in the deterministic case. We consider the stochastic KP (SKP) with stochastic item weights. For this variant of the SKP we combine the chance constrained KP (CCKP) and the SKP with simple recourse (SRKP). The chance constraint allows for a violation of capacity, but the probability of a violation beyond an imposed limit is constrained. The violation of the capacity constraint is also included in the objective function in terms of a penalty function as in the SRKP. Penalty is an increasing function of the expected number of units of violation with proportionality as a special case. We formulate the SKP as a network problem and demonstrate that it can be solved by a label-setting dynamic programming approach for the shortest path problem with resource constraints (SPPRC). We develop a dominance criterion for an elimination of states in the dynamic programming approach using only the deterministic value of items along with mean and variance of the stochastic weight of items corresponding to the associated paths in the underlying network. It is shown that a lower bound for the impact of potential extensions of paths is available as an additional means to limit the number of states provided the penalty cost of expected overtime is convex. Our findings are documented in terms of a computational study.

First-order dominance: stronger characterization and a bivariate checking algorithm

How to determine whether one distribution first-order dominates another is a fundamental problem that has many applications in economics, finance, probability theory, and statistics. Nevertheless, little is known about how to efficiently check first-order dominance for finite multivariate distributions. Utilizing that this problem can be formulated as a transportation problem with a special structure, we provide a stronger characterization of multivariate first-order dominance and develop a linear time complexity checking algorithm for the bivariate case. We illustrate the use of the checking algorithm when numerically assessing first-order dominance among continuous bivariate distributions.

Solving the selective multi-category parallel-servicing problem

In this paper, we present a new scheduling problem and describe a shortest path-based heuristic as well as a dynamic programming-based exact optimization algorithm to solve it. The selective multi-category parallel-servicing problem arises when a set of jobs has to be scheduled on a server (machine) with limited capacity. Each job requests service in a prespecified time window and belongs to a certain category. Jobs may be serviced partially, incurring a penalty; however, only jobs of the same category can be processed simultaneously. One must identify the best subset of jobs to process in each time interval of a given planning horizon, while respecting the server capacity and scheduling requirements. We compare the proposed solution methods with a Mixed Integer Linear Programming (MILP) formulation and show that the dynamic programming approach is faster when the number of categories is large, whereas the MILP can be solved faster when the number of categories is small.

Dynamic Job Assignment: A Column Generation Approach with an Application to Surgery Allocation

We consider the assignment of jobs to agents in a stochastic and dynamic setting. Focus is on a dynamic scenario with due dates and service levels reflecting the completion of jobs within certain deadlines. Due dates and other relevant characteristics for currently uncompleted jobs generated in the past are known, but the consumption of resources needed for their completion is stochastic. Distributions for the generation of future jobs as well as their characteristics are known. Capacity is limited, and an arriving job that cannot be assigned to an agent within its due date must be outsourced. Outsourcing is accompanied by a cost. We develop an optimization model based on column generation for the assignment of known and future jobs to agents such that the expected cost of outsourcing is minimum. The model is an extension of a generalized assignment problem and provides an allocation of known as well as tentative future jobs to agents. The model is embedded in a rolling horizon framework and subjected to a series of computational tests. The results indicate that taking stochastic information about future job arrivals into account in the assignment of jobs to agents implies an improved performance. The model is highly relevant in the context of patient scheduling in an operating theater. For this reason patient scheduling constitutes the storyline in the development of the model.

Decomposing Bivariate Dominance for Social Welfare Comparisons

The lower orthant dominance relation is frequently used for multidimensional social welfare comparisons. Recently it has been shown that bivariate dominance can be characterized in terms of elementary mass transfer operations. We provide an algorithm which explicitly decomposes the mass transfers into welfare differences and inequality differences.

A Shortest-Path-Based Approach for the Stochastic Knapsack Problem with Non-Decreasing Expected Overfilling Costs

The knapsack problem (KP) is concerned with the selection of a subset of multiple items with known positive values and weights such that the total value of selected items is maximized and their total weight does not exceed capacity. Item values, item weights, and capacity are known in the deterministic case. We consider the stochastic KP (SKP) with stochastic item weights. We combine the formulation of the SKP with a probabilistic capacity constraint (CKP) and the SKP with simple recourse (SRKP). Capacity is made an integral part of the constraint set in terms of a chance constraint as in the CKP. The chance constraint allows for a violation of capacity, but the probability of a violation beyond an imposed limit is constrained. The capacity constraint is also included in the objective function in terms of a penalty function as in the SRKP. Penalty is an increasing function of the expected number of units of violation with proportionality as a special case. We formulate the SKP as a network problem and demonstrate that it can be solved by a label-setting dynamic programming approach for the Shortest Path Problem with Resource Constraint (SPPRC). We develop a dominance criterion for an elimination of states in the dynamic programming approach using only the deterministic value of items along with mean and variance of the stochastic weight of items corresponding to the associated paths in the underlying network. It is shown that a lower bound for the impact of potential extensions of paths is available as an additional means to limit the number of states provided the penalty cost of expected overtime is convex. Our findings are documented in terms of a computational study.

A Benders decomposition-based matheuristic for the Cardinality Constrained Shift Design Problem

The Shift Design Problem is an important optimization problem which arises when scheduling personnel in industries that require continuous operation. Based on the forecast, required staffing levels for a set of time periods, a set of shift types that best covers the demand must be determined. A shift type is a consecutive sequence of time periods that adheres to legal and union rules and can be assigned to an employee on any day. In this paper we introduce the Cardinality Constrained Shift Design Problem; a variant of the Shift Design Problem in which the number of permitted shift types is bounded by an upper limit. We present an Integer Programming model for this problem and show that its structure lends itself very naturally to Benders decomposition. Due to convergence issues with a conventional implementation, we propose a matheuristic based on Benders decomposition for solving the problem. Furthermore, we argue that an important step in this approach is finding dual alternative optimal solutions to the Benders subproblems and describe an approach to obtain a diverse set of these. Numerical tests show that the described methodology significantly outperforms a commercial Mixed Integer Programming solver on instances with 1241 different shift types and remains competitive for larger cases with 2145 shift types. On all classes of problems the heuristic is able to quickly find good solutions.