A constraint programming-based solution approach for medical resident scheduling problems

Persistent calls come from within the graduate medical education community and from external sources for regulating the resident duty hours in order to meet the obligations about the quality of resident education, the well-being of residents themselves, and the quality of patient care services. The report of the Accreditation Council for Graduate Medical Education (ACGME) proposes common program requirements for resident hours. In this paper, we first develop a mixed-integer programming model for scheduling residents’ duty hours considering the on-call night, day-off, rest period, and total work-hour ACGME regulations as well as the demand coverage requirements of the residency program. Subsequently, we propose a column generation model that consists of a master problem and an auxiliary problem. The master problem finds a configuration of individual schedules that minimizes the sum of deviations from the desired service levels for the day and night periods. The formulation of this problem is possible by representing the feasible schedules using column variables, whereas the auxiliary problem finds the whole set of feasible schedules using constraint programming. The proposed approach has been tested on a series of problems using real data obtained from a hospital. The results indicate that high-quality schedules can be obtained within a few seconds.     

A two-stage stochastic programming model for the parallel machine scheduling problem with machine capacity

This paper proposes a two-stage stochastic programming model for the parallel machine scheduling problem where the objective is to determine the machines' capacities that maximize the expected net profit of on-time jobs when the due dates are uncertain. The stochastic model decomposes the problem into two stages: The first (FS) determines the optimal capacities of the machines whereas the second (SS) computes an estimate of the expected profit of the on-time jobs for given machines' capacities. For a given sample of due dates, SS reduces to the deterministic parallel weighted number of on-time jobs problem which can be solved using the efficient branch and bound of M’Hallah and Bulfin [16]. FS is tackled using a sample average approximation (SAA) sampling approach which iteratively solves the problem for a number of random samples of due dates. SAA converges to the optimum in the expected sense as the sample size increases. In this implementation, SAA applies a ranking and selection procedure to obtain a good estimate of the expected profit with a reduced number of random samples. Extensive computational experiments show the efficacy of the stochastic model.     

Solving single-product economic lot-sizing problem with non-increasing setup cost,constant capacity and convex inventory cost in O(N log N) time

This paper considers an economic lot sizing model with constant capacity, non-increasing setup cost, and convex inventory cost function. Algorithms with computational time of O(N×TD_N)have been developed for solving the model, where N is the number of planning periods and TD_N is the total demand. This study partially characterizes the optimal planning structure of the model. A new efficient algorithm with computational time of O(N log N) has also been developed based on the partial optimal structure. Moreover, computational study demonstrates that the new algorithm is efficient.     

Selection of a dynamic supply portfolio in make-to-order environment withrisks

The problem of a multi-period supplier selection and order allocation in make-to-order environment in the presence of supply chain disruption and delay risks is considered. Given a set of customer orders for finished products, the decision maker needs to decide from which supplier and when to purchase product-specific parts required for each customer order to meet customer requested due date at a low cost and to mitigate the impact of supply chain risks. The selection of suppliers and the allocation of orders over time is based on price and quality of purchased parts and reliability of supplies. For selection of dynamic supply portfolio a mixed integer programming approach is proposed to incorporate risk that uses conditional value-at-risk via scenario analysis. In the scenario analysis, the low-probability and high-impact supply disruptions are combined with the high probability and low impact supply delays. The proposed approach is capable of optimizing the dynamic supply portfolio by calculating value-at-risk of cost per part and minimizing expected worst-case cost per part simultaneously. Numerical examples are presented and some computational results are reported.     

A computational study of parametric tabu search for 0–1 mixed integer programs

We present a computational study of parametric tabu search for solving 0–1 mixed integer programming (MIP) problems, a generic heuristic for general MIP problems proposed by Glover [Glover F. Parametric tabu-search for mixed integer programs. Computers and Operations Research 2006; 33: 2449–94.]. This approach solves a series of linear programming problems by incorporating branching inequalities as weighted terms in the objective function. New strategies are proposed for uncovering feasible and high-quality solutions and extensive computational tests are performed on instances from the literature.     

Exact algorithms for a generalization of the order acceptance and scheduling problem in a single-machine environment

This paper studies a generalization of the order acceptance and scheduling problem in a single-machine environment where a pool consisting of firm planned orders as well as potential orders is available from which an over-demanded company can select. The capacity available for processing the accepted orders is limited and each order is characterized by a known processing time, delivery date, revenue and a weight representing a penalty per unit-time delay beyond the delivery date. We prove that the existence of a constant-factor approximation algorithm for this problem is unlikely. We propose two linear formulations that are solved using an IP solver and we devise two exact branch-and-bound procedures able to solve instances with up to 50 jobs within reasonable CPU times. We compare the efficiency and quality of the results obtained using the different solution approaches.     

A revised particle swarm optimization based discrete Lagrange multipliers method for nonlinear programming problems

In this paper, a new algorithm for solving constrained nonlinear programming problems is presented. The basis of our proposed algorithm is none other than the necessary and sufficient conditions that one deals within a discrete constrained local optimum in the context of the discrete Lagrange multipliers theory. We adopt a revised particle swarm optimization algorithm and extend it toward solving nonlinear programming problems with continuous decision variables. To measure the merits of our algorithm, we provide numerical experiments for several renowned benchmark problems and compare the outcome against the best results reported in the literature. The empirical assessments demonstrate that our algorithm is efficient and robust.     

Combined planning and scheduling in a divergent production system with co-production: A case study in the lumber industry

Many research initiatives carried out in production management consider process planning and operations scheduling as two separate and sequential functions. However, in certain contexts, the two functions must be better integrated. This is the case in divergent production systems with co-production (i.e. production of different products at the same time from a single product input) when alternative production processes are available. This paper studies such a context and focuses on the case of drying and finishing operations in a softwood lumber facility. The situation is addressed using a single model that simultaneously performs process planning and scheduling. We evaluate two alternative formulations. The first one is based on mixed integer programming (MIP) and the second on constraint programming (CP). We also propose a search procedure to improve the performance of the CP approach. Both approaches are compared with respect to their capacity to generate good solutions in short computation time.     

Some algebraic methods for solving multiobjective polynomial integer programs

Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the past few years. However, the polynomial case has not been studied in detail due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem to the problem of solving one or more systems of polynomial equations and we use certain Gröbner bases to solve these systems. Different transformations give different methodologies that are theoretically stated and compared by some computational tests via the algorithms that they induce.     

Shortest path problem with uncertain arc lengths

Uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths. With help from the operational law of uncertainty theory, this paper gives the uncertainty distribution of the shortest path length. Also, it investigates solutions to the α-shortest path and the most shortest path in an uncertain network. It points out that there exists an equivalence relation between the α-shortest path in an uncertain network and the shortest path in a corresponding deterministic network, which leads to an effective algorithm to find the α-shortest path and the most shortest path. Roughly speaking, this algorithm can be broken down into two parts: constructing a deterministic network and then invoking the Dijkstra algorithm.