Four-Day Workweek Scheduling with Two or Three Consecutive Days Off

A four-day workweek days-off scheduling problem is considered. Out of the three days off per week for each employee, either two or three days must be consecutive. An optimization algorithm is presented which starts by utilizing the problem's special structure to determine the minimum workforce size. Subsequently, workers are assigned to different days-off work patterns in order to minimize either the total number or the total cost of the workforce. Different procedures must be followed in assigning days-off patterns, depending on the characteristics of labor demands. In some cases, optimum days-off assignments are determined without requiring mathematical programming. In other cases, a workforce size constraint is added to the integer programming model, greatly improving computational performance.     

A dynamic production planning and scheduling algorithm for two products processed on one line

This paper presents a dynamic production planning and scheduling algorithm for two products processed on one line over a fixed time horizon. Production rates are assumed fixed, and restrictions are placed or inventory levels and production run lengths. The resulting problem is a nonlinear binary program, which is solved using an implicit enumeration strategy. The algorithm focuses on the run changeover period while developing tighter bounds on the length of the upcoming run to improve computational efficiency. About 99% pf 297 randomly generated problems with varying demand patterns are solved in less than 15 seconds of CPU time on a CDC Cyber 172 Computer. A mixed integer programming formulation of the generalized multi-product case under no-backlogging of demand is also given.     

A mixed integer programming model for advanced planning and scheduling (APS)

This paper presents a mixed integer programming (MIP) model which succeeds in a system integration of the production planning and shop floor scheduling problems. The proposed advanced planning and scheduling (APS) model explicitly considers capacity constraints, operation sequences, lead times and due dates in a multi-order environment. The objective of the model is to seek the minimum cost of both production idle time and tardiness or earliness penalty of an order. The output of the model is operation schedules with order starting time and finish time. Numerical result shows that the suggested APS model can favorably produce optimal schedules.     

不确定彩排时长下节目调度的鲁棒优化

实际节目彩排调度中,节目的表演时长受内外因素影响,具有不确定性。为了合理调度所有节目,控制演员的空闲时间,使得演员的总等待成本最小,采用了鲁棒优化方法进行研究。首先,建立了节目彩排调度的确定型模型;进一步,考虑节目表演时长的不确定性,采用有界区间描述节目表演时长并考虑决策者风险偏好,在确定型模型的基础上构建区间型两阶段鲁棒优化模型;接着,将鲁棒优化模型转化为0-1混合线性规划模型;最后,采用Matlab进行数值实验,结果表明决策者越偏好规避风险,演员的总等待成本越大。     

The implementor/adversary algorithm for the cyclic and robust scheduling problem in health-care

A general problem in health-care consists in allocating some scarce medical resource, such as operating rooms or medical staff, to medical specialties in order to keep the queue of patients as short as possible. A major difficulty stems from the fact that such an allocation must be established several months in advance, and the exact number of patients for each specialty is an uncertain parameter. Another problem arises for cyclic schedules, where the allocation is defined over a short period, e.g. a week, and then repeated during the time horizon. However, the demand typically varies from week to week: even if we know in advance the exact demand for each week, the weekly schedule cannot be adapted accordingly. We model both the uncertain and the cyclic allocation problem as adjustable robust scheduling problems. We develop a row and column generation algorithm to solve this problem and show that it corresponds to the implementor/adversary algorithm for robust optimization recently introduced by Bienstock for portfolio selection. We apply our general model to compute master surgery schedules for a real-life instance from a large hospital in Oslo.     

小批量随机块坐标下降算法

针对机器学习中广泛存在的一类问题：结构化随机优化问题（其中“结构化”是指问题的可行域具有块状结构,且目标函数的非光滑正则化部分在变量块之间是可分离的）,我们研究了小批量随机块坐标下降算法（mSBD）。按照求解非复合问题和复合问题分别给出了基本的mSBD和它的变体,对于非复合问题,分析了算法在没有一致有界梯度方差假设情况下的收敛性质。而对于复合问题,在不需要通常的Lipschitz梯度连续性假设条件下得到了算法的收敛性。最后通过数值实验验证了mSBD的有效性。     

A note on “A mixed integer programming model for advanced planning and scheduling (APS)”

In a recent paper, Chen and Ji [Chen, K., Ji, P., 2007. A mixed integer programming model for advanced planning and scheduling (APS). European Journal of Operational Research 181, 515–522] develop a mixed integer programming model for advanced planning and scheduling problem that considers capacity constraints and precedence relations between the operations. The orders require processing of several operations on eligible machines. The model presented in the above paper works for the case where each operation can be processed on only one machine. However, machine eligibility means that only a subset of machines are capable of processing a job and this subset may include more than one machine. We provide a general model for advanced planning and scheduling problems with machine eligibility. Our model can be used for problems where there are alternative machines that an operation can be assigned to.     

Bicriteria hierarchical optimization of two-machine flow shop scheduling problem with time-dependent deteriorating jobs

We study a two-machine flowshop scheduling problem with time-dependent deteriorating jobs, i.e. the processing times of jobs are an increasing function of their starting time. The objective is to minimize the total completion time subject to minimum makespan. We propose a mixed integer programming model, and develop two pairwise interchange algorithms and a branch-and-bound procedure to solve the problem while using several dominance conditions to limit the size of the search tree. Several polynomial-time solvable special cases are discussed. Finally, numerical studies are performed to examine the effectiveness and the efficiency of the proposed algorithms.     

Coal bank scheduling using a mathematical programming model

Coals are extracted from mines and upgraded on the surface for customers. The upgraded coals must be aged at least four weeks in a bank before being supplied to customers. Different sizes of banks are required for different lengths of time at different points in time. Bigger banks increase the floor space capacity and reduce handling costs. The proper location of banks reduces the total space requirement and bank movement after building. In this paper, we address the bank size and location problem and solve it by using a mathematical programming approach.     

Retail store scheduling for profit

In spite of its tremendous economic significance, the problem of sales staff schedule optimization for retail stores has received relatively scant attention. Current approaches typically attempt to minimize payroll costs by closely fitting a staffing curve derived from exogenous sales forecasts, oblivious to the ability of additional staff to (sometimes) positively impact sales. In contrast, this paper frames the retail scheduling problem in terms of operating profit maximization, explicitly recognizing the dual role of sales employees as sources of revenues as well as generators of operating costs. We introduce a flexible stochastic model of retail store sales, estimated from store-specific historical data, that can account for the impact of all known sales drivers, including the number of scheduled staff, and provide an accurate sales forecast at a high intra-day resolution. We also present solution techniques based on mixed-integer (MIP) and constraint programming (CP) to efficiently solve the complex mixed integer non-linear scheduling (MINLP) problem with a profit-maximization objective. The proposed approach allows solving full weekly schedules to optimality, or near-optimality with a very small gap. On a case-study with a medium-sized retail chain, this integrated forecasting–scheduling methodology yields significant projected net profit increases on the order of 2–3% compared to baseline schedules.     

Simultaneously scheduling multiple turns for steel color-coating production

This paper investigates a large-scale scheduling problem in the iron and steel industry, called color-coating production scheduling (CCPS). The problem is to generate multiple production turns for the galvanized coils that dynamically arrive from upstream lines within a given scheduling horizon, and at the same time determine the sequence of these turns so that the productivity and product quality are maximized while the production cost and the number of generated turns are minimized. We formulate this problem as a mixed integer nonlinear program and propose a tabu search heuristic to obtain satisfactory solutions. Results on real production instances show that the presented model and heuristic are more effective and efficient with comparison to manual scheduling. A practical scheduling system for CCPS combining the model and heuristic has been developed and successfully implemented in a major iron and steel enterprise in China.     

A heuristic algorithm for the hospital health examination scheduling problem

This study considers the problem of health examination scheduling. Depending on their gender, age, and special requirements, health examinees select one of the health examination packages offered by a health examination center. The health examination center must schedule all the examinees, working to minimize examinee/doctor waiting time and respect time and resource constraints, while also taking other limitations, such as the sequence and continuity of the examination procedures, into consideration. The Binary integer programming (BIP) model is one popular way to solve this health examination scheduling problem. However, as the number of examinees and health examination procedures increase, solving BIP models becomes more and more difficult, if not impossible. This study proposes health examination scheduling algorithm (HESA), a heuristic algorithm designed to solve the health examination scheduling problem efficiently and effectively. HESA has two primary objectives: minimizing examinee waiting time and minimizing doctor waiting time. To minimize examinee waiting time, HESA schedules the various parts of each examinee’s checkup for times when the examinee is available, taking the sequence of the examination procedures and the availability of the resources required into account. To minimize doctor waiting time, HESA focuses on doctors instead of examinees, assigning waiting examinees to a doctor as soon as one becomes available. Both complexity analysis and computational analyses have shown that HESA is very efficient in solving the health examination scheduling problem. In addition to the theoretical results, the results of HESA’s application to the concrete health examination scheduling problems of two large hospitals in Taiwan are also reported.     

Workforce planning at USPS mail processing and distribution centers using stochastic optimization

Service organizations that operate outside the normal 8-hour day and face wide fluctuations in demand constantly struggle to optimize the size and composition of their workforce. Recent research has shown that improved personnel scheduling methods that take demand uncertainty into account can lead to significant reductions in labor costs. This paper addresses a staff planning and scheduling problem that arises at United States Postal Service (USPS) mail processing & distribution centers (P&DCs) and develops a two-stage stochastic integer program with recourse for the analysis. In the first stage, before the demand is known, the number of full-time and part-time employees is determined for the permanent workforce. In the second stage, the demand is revealed and workers are assigned to specific shifts during the week. When necessary, overtime and casual labor are used to satisfy demand. This paper consists of two parts: (1) the analysis of the demand distribution in light of historical data, and (2) the development and analysis of the stochastic integer programming model. Using weekly demand for a three-year period, we first investigate the possibility that there exists an end-of-month effect, i.e., the week at the end of month has larger volume than the other weeks. We show that the data fail to indicate that this is the case. In the computational phase of the work, three scenarios are considered: high, medium, and low demand. The stochastic optimization problem that results is a large-scale integer program that embodies the full set of contractual agreements and labor rules governing the design of the workforce at a P&DC. The usefulness of the model is evaluated by solving a series of instances constructed from data provided by the Dallas facility. The results indicate that significant savings are likely when the recourse problem is used to help structure the workforce. This work was supported in part by the National Science Foundation under grants DMI-0218701 and DMI-0217927.     

Tropical optimization problems in time-constrained project scheduling

We consider a project that consists of activities to be performed in parallel under various temporal constraints, which include start-start, start-finish and finish-start precedence relationships, release times, deadlines and due dates. Scheduling problems are formulated to find optimal schedules for the project with respect to different objective functions to be minimized, such as the project makespan, the maximum deviation from the due dates, the maximum flow-time and the maximum deviation of finish times. We represent these problems as optimization problems in terms of tropical mathematics, and then solve them by applying direct solution methods of tropical optimization. As a result, new direct solutions of the scheduling problems are obtained in a compact vector form, which is ready for further analysis and practical implementation. The solutions are illustrated by simple numerical examples.     

A general model for cyclic machine scheduling problems

A general framework for modeling and solving cyclic scheduling problems is presented. The objective is to minimize the cycle time. The model covers different cyclic versions of the job-shop problem found in the literature, robotic cell problems, the single hoist scheduling problem and tool transportation between the machines.It is shown that all these problems can be formulated as mixed integer linear programs which have a common structure. Small instances are solved with CPLEX. For larger instances tabu search procedures have been developed. The main ideas of these methods are indicated.     

多种产品生产系统的生产计划问题

研究一个为满足确定性需求而进行产品生产的系统.系统中有一台用于生产的机器, 生产$n$种不同的产品.每一种产品都有确定的日需求. 我们将能够满足所有产品需求的生产计划称之为可行计划.主要想通过数学模型, 来建立一套判定可行计划存在性的理论. 在确保存在可行计划的前提下, 设计了一种找寻出具体可行计划的计算方法. 并且, 进一步可以通过0-1规划来优化系统的效率,称之为最小化产能占用率.     

Robust discrete optimization and network flows

We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. In particular, when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation. When only the cost coefficients are subject to uncertainty and the problem is a 0–1 discrete optimization problem on n variables, then we solve the robust counterpart by solving at most n+1 instances of the original problem. Thus, the robust counterpart of a polynomially solvable 0–1 discrete optimization problem remains polynomially solvable. In particular, robust matching, spanning tree, shortest path, matroid intersection, etc. are polynomially solvable. We also show that the robust counterpart of an NP-hard -approximable 0–1 discrete optimization problem, remains -approximable. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network. The research of the author was partially supported by the Singapore-MIT alliance.The research of the author is supported by a graduate scholarship from the National University of Singapore.Mathematics Subject Classification (2000): 90C10, 90C15     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

Patient mix optimization in tactical cardiothoracic surgery planning: a case study

Cardiothoracic surgery planning involves different resourcessuch as operating theatre (OT) time, medium care beds, intensivecare beds and nursing staff. Within cardiothoracic surgery differentcategories of patients can be distinguished with respect totheir requirements of resources. The mix of patients is, therefore,an important aspect of decision making for the hospital to managethe use of these resources. A master OT schedule is used atthe tactical level of planning for deriving the weekly OT plan.It defines for each day of a week the number of OT hours availableand the number of patients operated from each patient category.We develop a model for this tactical level planning problem,the core of which is a mixed integer linear program. The modelis used to evaluate scenarios for surgery planning at tacticalas well as strategic levels, demonstrating the potential ofinteger programming for providing recommendations for change.