Longest Expected Processing Time Research Articles

Robust allocation of operating rooms: A cutting plane approach to handle lognormal case durations

The problem of allocating operating rooms (OR) to surgical cases is a challenging task, involving both combinatorial aspects and uncertainty handling. We formulate this problem as a parallel machines scheduling problem, in which job durations follow a lognormal distribution, and a fixed assignment of jobs to machines must be computed. We propose a cutting-plane approach to solve the robust counterpart of this optimization problem. To this end, we develop an algorithm based on fixed-point iterations that identifies worst-case scenarios and generates cut inequalities. The main result of this article uses Hilbert’s projective geometry to prove the convergence of this procedure under mild conditions. We also propose two exact solution methods for a similar problem, but with a polyhedral uncertainty set, for which only approximation approaches were known. Our model can be extended to balance the load over several planning periods in a rolling horizon. We present extensive numerical experiments for instances based on real data from a major hospital in Berlin. In particular, we find that: (i) our approach performs well compared to a previous model that ignored the distribution of case durations; (ii) compared to an alternative stochastic programming approach, robust optimization yields solutions that are more robust against uncertainty, at a small price in terms of average cost; (iii) the longest expected processing time first (LEPT) heuristic performs well and efficiently protects against extreme scenarios, but only if a good prediction model for the durations is available. Finally, we draw a number of managerial implications from these observations.

Comparing concepts for shop floor control of information-processing services in a job shop setting: a case from the financial services sector

Controlling information-based service processes in the short term is a major challenge. Within these processes, information is collected, generated and transformed while customers are directly involved. Typically, such processes are characterised by a job shop layout that increases complexity relating to shop floor control. In contrast to manufacturing, such settings have rarely been addressed in the literature to date. This is surprising, as information-processing services exist in every industry. Thus, evidence about the impact of concepts for shop floor control in such pure service environments is needed. As a case study, we use one of the most intense information-processing industries – the financial services sector. A simulation model is built using process mining as a novel approach to gather a major part of the relevant data from business information systems. We identify nine concepts for shop floor control and simulate the usage of these in six scenarios. The results show that the concept that prioritises customer orders with the longest expected processing time is superior in cycle time over all six scenarios. The application of this concept leads to a reduction of the average cycle time of almost 50% in the case study.

Decision support for hospital bed management using adaptable individual length of stay estimations and shared resources

BackgroundElective patient admission and assignment planning is an important task of the strategic and operational management of a hospital and early on became a central topic of clinical operations research. The management of hospital beds is an important subtask. Various approaches have been proposed, involving the computation of efficient assignments with regard to the patients’ condition, the necessity of the treatment, and the patients’ preferences. However, these approaches are mostly based on static, unadaptable estimates of the length of stay and, thus, do not take into account the uncertainty of the patient’s recovery. Furthermore, the effect of aggregated bed capacities have not been investigated in this context. Computer supported bed management, combining an adaptable length of stay estimation with the treatment of shared resources (aggregated bed capacities) has not yet been sufficiently investigated. The aim of our work is: 1) to define a cost function for patient admission taking into account adaptable length of stay estimations and aggregated resources, 2) to define a mathematical program formally modeling the assignment problem and an architecture for decision support, 3) to investigate four algorithmic methodologies addressing the assignment problem and one base-line approach, and 4) to evaluate these methodologies w.r.t. cost outcome, performance, and dismissal ratio.MethodsThe expected free ward capacity is calculated based on individual length of stay estimates, introducing Bernoulli distributed random variables for the ward occupation states and approximating the probability densities. The assignment problem is represented as a binary integer program. Four strategies for solving the problem are applied and compared: an exact approach, using the mixed integer programming solver SCIP; and three heuristic strategies, namely the longest expected processing time, the shortest expected processing time, and random choice. A baseline approach serves to compare these optimization strategies with a simple model of the status quo. All the approaches are evaluated by a realistic discrete event simulation: the outcomes are the ratio of successful assignments and dismissals, the computation time, and the model’s cost factors.ResultsA discrete event simulation of 226,000 cases shows a reduction of the dismissal rate compared to the baseline by more than 30 percentage points (from a mean dismissal ratio of 74.7% to 40.06% comparing the status quo with the optimization strategies). Each of the optimization strategies leads to an improved assignment. The exact approach has only a marginal advantage over the heuristic strategies in the model’s cost factors (≤3%). Moreover,this marginal advantage was only achieved at the price of a computational time fifty times that of the heuristic models (an average computing time of 141 s using the exact method, vs. 2.6 s for the heuristic strategy).ConclusionsIn terms of its performance and the quality of its solution, the heuristic strategy RAND is the preferred method for bed assignment in the case of shared resources. Future research is needed to investigate whether an equally marked improvement can be achieved in a large scale clinical application study, ideally one comprising all the departments involved in admission and assignment planning.

Cost minimization and workload balancing in printed circuit board assembly

We consider an operation assignment problem arising from a Printed Circuit (PC) board assembly process. The research was inspired by applications at Hewlett-Packard Company where hundreds of types of PC boards require the insertion of thousands of types of components. The components can be inserted manually or by automated insertion machines. The machines can only hold a limited number of different component types. We investigate how to assign the boards and components to the machines and manual process so as to minimize cost while at the same time balancing machine workloads. We first present a Binary Integer Program (BIP) formulation of the problem. We then develop optimality results that allow us to reduce significantly the size of the BIP. Using the improved BIP formulation, and upper bounds generated using a Cost Minimizing Workload Balancing (CMWB) heuristic that we develop, we show how branch-and-bound can be used to find optimal solutions to small and medium-sized problems in reasonable time. We also show that the CMWB heuristic finds solutions in seconds of CPU time that are within a few percent of optimal. In addition, the CMWB heuristic outperforms the heuristic that has been used at Hewlett-Packard, as well as the longest expected processing time heuristic. Although this paper specifically addresses a problem of partially-automated PC board assembly, the results apply to a more general set of problems, including job and tool assignment in flexible manufacturing systems, and general operation assignment problems.

Optimal Scheduling of Jobs with Exponential Service Times on Identical Parallel Processors

The scheduling of jobs with stochastically independent, exponentially distributed service times on identical parallel processors is considered. General sufficient conditions for optimality in expectation of priority policies for certain cost functions are given, including cases of the weighted flow time. The priority policies under consideration may be more general than the longest expected processing time (LEPT) or the shortest expected processing time (SEPT) policy. We deal with a fixed number of processors as well as certain more general resource constraints. Finally, precedence relations between jobs given by strict interval orders are admitted and an optimality result for LEPT is stated for this situation.

Minimizing expected makespan in a two-machine stochastic open shop with Poisson arrival

Longest Expected processing time (LEPT) policy is a machine loading rule where out of all the jobs waiting to be processed by a machine, with their processing times following given probability distributions, the one with the largest expected processing time is chosen first. Using a method based on Markov process and dynamic programming, we show that a LEPT policy will minimize the expected makespan for a two-machine stochastic open shop with Poisson arrival for jobs. Processing time of any job at any machine is exponential. We assume that all jobs are identical but the two machines are not.

Minimizing expected makespan in stochastic open shops

Suppose that two machines are available to process n tasks. Each task has to be processed on both machines; the order in which this happens is immaterial. Task j has to be processed on machine 1 (2) for random time Xj (Yj) with distribution Fj (Gj). This kind of model is usually called an open shop. The time that it takes to process all tasks is normally called the makespan. Every time a machine finishes processing a task the decision-maker has to decide which task to process next. Assuming that Xj and Yj have the same exponential distribution we show that the optimal policy instructs the decision-maker, whenever a machine is freed, to start processing the task with longest expected processing time among the tasks still to be processed on both machines. If all tasks have been processed at least once, it does not matter what the decision-maker does, as long as he keeps the machines busy. We then consider the case of n identical tasks and two machines with different speeds. The time it takes machine 1 (2) to process a task has distribution F (G). Both distributions F and G are assumed to be new better than used (NBU) and we show that the decision-maker stochastically minimizes the makespan when he always gives priority to those tasks which have not yet received processing on either machine.

Minimizing expected makespan in stochastic open shops

Suppose that two machines are available to process n tasks. Each task has to be processed on both machines; the order in which this happens is immaterial. Task j has to be processed on machine 1 (2) for random time Xj (Yj ) with distribution Fj (Gj ). This kind of model is usually called an open shop. The time that it takes to process all tasks is normally called the makespan. Every time a machine finishes processing a task the decision-maker has to decide which task to process next. Assuming that Xj and Yj have the same exponential distribution we show that the optimal policy instructs the decision-maker, whenever a machine is freed, to start processing the task with longest expected processing time among the tasks still to be processed on both machines. If all tasks have been processed at least once, it does not matter what the decision-maker does, as long as he keeps the machines busy. We then consider the case of n identical tasks and two machines with different speeds. The time it takes machine 1 (2) to process a task has distribution F (G). Both distributions F and G are assumed to be new better than used (NBU) and we show that the decision-maker stochastically minimizes the makespan when he always gives priority to those tasks which have not yet received processing on either machine.

Scheduling of stochastic tasks on two parallel processors

AbstractWe consider the problem of scheduling n tasks on two identical parallel processors. We show both in the case when the processing times for the n tasks are independent exponential random variables, and when they are independent hyperexponentials which are mixtures of two fixed exponentials, that the policy of performing tasks with longest expected processing time (LEPT) first minimizes the expected makespan, and that in the hyperexponential case the policy of performing tasks with shortest expected processing time (SEPT) first minimizes the expected flow time. The approach is simpler than the dynamic programming approach recently employed by Bruno and Downey.