Tour Scheduling and Task Assignment of a Heterogeneous Work Force: A Heuristic Approach

The dual problem of work tour scheduling and task assignment involving workers who differ in their times of availability and task qualifications is examined in this paper. The problem is presented in the context of a fast food restaurant, but applies equally well to a diverse set of service operations. Developing a week-long labor schedule is a nontrivial problem, in terms of complexity and importance, which a manager spends as much as a full workday solving. The primary scheduling objective (the manager's concern) is the minimization of overstaffing in the face of significant hourly and daily fluctuations in minimum staffing requirements. The secondary objective (the workers’ concern) is the minimization of the sum of the squared differences between the number of work hours scheduled and the number targeted for each employee. Contributing to scheduling complexity are constraints on the structure of work tours, including minimum and maximum shift lengths and a maximum number of workdays. A goal programming formulation of a representative problem is shown to be too large, for all practical purposes, to be solved optimally. Existing heuristic procedures related to this research possess inherent limitations which render them inadequate for our purposes. Subsequently, we propose and demonstrate a computerized heuristic procedure capable of producing a labor schedule requiring at most minor refinement by a manager.     

一种求解时变条件下有宵禁限制最短路的算法

在组合优化过程中,往往需要获得从起点到终点之间的最短路.由于道路、天气、交通条件等因素的影响,使得网络具有很强的时变特性.同时,对于网络中的节点往往有宵禁的限制.对时变条件下有宵禁限制并有到达时间限制的最短路进行了研究,建立了软、硬宵禁限制下的数学模型,给出并证明了时变条件下获得有宵禁限制最短路的最优条件,并设计了求解的多项式算法,通过此算法可以获得时变条件下有宵禁限制的最短路.同时,算法和模型还考虑了不同的起点出发时间,使路径决策者可以根据自身的情况,选择合适的出发时间和路径.最后给出了一个应用算例,分析了宵禁对于获得的最短路的影响.     

Reducibility of stochastic networks

This paper deals with the problem of reducing a stochastic network to one equivalent activity. The problem was motivated by the question of determining or approximating the probability distribution function of the duration of the longest or shortest path in a stochastic network. We define a particular reduction and use it to characterize the reducibility of such a network. The network can be reduced to one equivalent activity if the network does not have a special graph which we call the ‘interdictive graph’, or IG for short. If an IG is embedded in the network, the network is irreducible. In this case, its reduction becomes possible by duplicating some of the arcs in the irreducible network. The concept of duplicating an arc is introduced, then it is used to identify the arcs which can be duplicated. The reduction procedure is stated and illustrative examples are provided. An upper bound on the number of duplications is established.     

ALLOCATING AND SCHEDULING MOBILE DIAGNOSTIC IMAGING EQUIPMENT AMONG HOSPITALS

The allocation and weekly scheduling of mobile magnetic resonance imaging (MRI) units leased to a group of hospitals that share the equipment can be a complex problem. Similar problems occur in other domains where expensive equipment or facilities such as video conference facilities, aircraft, and supercomputers are leased. The crux of the problem was determining the number of days and which days of the week various types of equipment types should be leased to hospitals, so as to maximize the rental revenues and satisfy client preferences for days of the week and equipment types. We found rental revenues were a decreasing function of the number of days allocated to a hospital. We considered two sub-problems linked by a set of variables to model the problem. We show that one of these subproblems is a minimum cost network flow problem and the other is an integer multi-commodity transportation problem. We developed a procedure for solving the latter problem by exploiting earlier results for specialized networks. We conducted a computational study to evaluate the performance of this procedure and showed that it generally provides near-optimal integer solutions. We describe the development and implementation of a spreadsheet-based decision support system based on this model. This system was successfully implemented by a small firm with no expertise or prior experience using models.     

Distributionally robust maximum probability shortest path problem

In this study, we discuss and develop a distributionally robust joint chance-constrained optimization model and apply it for the shortest path problem under resource uncertainty. In sch a case, robust chance constraints are approximated by constraints that can be reformulated using convex programming. Since the issue we are discussing here is of the multi-resource type, the resource related to cost is deterministic; however, we consider a robust set for other resources where covariance and mean are known. Thus, the chance-constrained problem can be expressed in terms of a cone constraint. In addition, since our problem is joint chance-constrained optimization, we can use Bonferroni approximation to divide the problem into L separate problems in order to build convex approximations of distributionally robust joint chance constraints. Finally, numerical results are presented to illustrate the rigidity of the bounds and the value of the distributionally robust approach.

     

Graph Algorithms with Small Communication Costs

We study minimizing communication cost in parallel algorithm design, by minimizing the number of communication phases in coarse-grained parallel computers. There have been several recent papers dealing with parallel algorithms of small communication cost under different models. Most of these results are for computational geometry problems. For these problems it has been possible to decompose tasks into appropriate subproblems in a communication-efficient way. It appears to be somewhat more difficult to design parallel algorithms with small communication phases for graph theory problems. In this paper we focus on the design of deterministic algorithms with a small number of communication phases for the list ranking problem and the shortest path problem.     

Robust Resource Allocation Decisions in Resource‐Constrained Projects*

The well‐known deterministic resource‐constrained project scheduling problem involves the determination of a predictive schedule (baseline schedule or pre‐schedule) of the project activities that satisfies the finish–start precedence relations and the renewable resource constraints under the objective of minimizing the project duration. This baseline schedule serves as a baseline for the execution of the project. During execution, however, the project can be subject to several types of disruptions that may disturb the baseline schedule. Management must then rely on a reactive scheduling procedure for revising or reoptimizing the baseline schedule. The objective of our research is to develop procedures for allocating resources to the activities of a given baseline schedule in order to maximize its stability in the presence of activity duration variability. We propose three integer programming–based heuristics and one constructive procedure for resource allocation. We derive lower bounds for schedule stability and report on computational results obtained on a set of benchmark problems.     

Flows with unit path capacities and related packing and covering problems

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log?m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an O(?m)O(\sqrt{m}) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.     

Some graph optimization problems with weights satisfying linear constraints

In this paper, we study several graph optimization problems in which the weights of vertices or edges are variables determined by several linear constraints, including maximum matching problem under linear constraints (max-MLC), minimum perfect matching problem under linear constraints (min-PMLC), shortest path problem under linear constraints (SPLC) and vertex cover problem under linear constraints (VCLC). The objective of these problems is to decide the weights that are feasible to the linear constraints, and find the optimal solutions of corresponding graph optimization problems among all feasible choices of weights. We find that these problems are NP-hard and are hard to be approximated in general. These findings suggest us to explore various special cases of them. In particular, we show that when the number of constraints is a fixed constant, all these problems are polynomially solvable. Moreover, if the total number of distinct weights is a fixed constant, then max-MLC, min-PMLC and SPLC are polynomially solvable, and VCLC has a 2-approximation algorithm. In addition, we propose approximation algorithms for various cases of max-MLC.

     

Technology adoption and training practices as a constrained shortest path problem

Technology adoption is not a new venue for research. Much work of decision modeling, diffusion of new technology and statistical analysis of survey data has been done. Some studies focus on finding the optimal forms of technology to adopt within a complementarity framework, but there is no mention of finding an optimal path from a firm's current state to its optimal state. This represents a significant gap in the literature. The paper applies a constrained shortest path problem to training and technology adoption decisions by firms. Given the current set of training and technology adoption the method solves for what technology/practice should be adopted or removed from the complete set of combinations and in what order so as to maximize performance subject to budget constraints. To the authors' knowledge, this is the first application of the constrained shortest path problem to technology adoption decisions.     

资源受限下平行工序顺序对优化的0-1规划模型

资源受限是工程项目时刻都可能面对的挑战。由于资源限制,需要将原项目计划中相互之间无优先关系的平行工序调整为顺序工序。平行工序顺序化可导致项目工期延迟,因此需考虑如何使项目工期延迟最小。该平行工序顺序优化问题是项目调度问题,也是排列组合问题,通常难度很大,包括一些NP-hard问题。本文主要研究该问题的一类典型子问题——平行工序顺序对优化,即如何将项目中某2n个平行工序调整为n个顺序工序对,并且对项目工期的影响最小。该问题的总方案数可达到（2n）!/n!。本文借助工序网络（如CPM网络）,运用简单的时间参数量化了平行工序顺序化对项目工期的影响,进而降低问题的求解难度,建立了纯0-1规划模型。实验验证了该模型的求解效率,求解100个平行工序规模的问题平均耗时0.2605秒,而求解500个平行工序规模的问题平均耗时10.66秒。     

Balancing assembly line with skilled and unskilled workers

In this paper, we present the process of rebalancing the line at motorcycle-assembly plant. The company found it necessary to rebalance its line, since it needs to increase production in the spring and summer months. The main characteristics of the problem are as follows: (i) the company hires temporary staff, who need more time to carry out their tasks than permanent workers; (ii) there must always be at least one skilled employee working alongside an unskilled one; and (iii) different task groups are incompatible with each other (clean-hands tasks and dirty-hands tasks). The goal is to minimise the number of temporary workers required, given a cycle time and the team of workers on staff. The problem is modelled as a binary linear program (BLP) and solved optimally by means of the ILOG CPLEX 9.0 optimiser. The solution provided, namely 12 permanent workers (skilled) and two temporary workers (unskilled), is an improvement on the solution implemented by the business, which involved 12 permanent workers and four temporary workers.     

Reliable Ad Hoc Routing Based on Mobility Prediction

Reliability is a very important issue in Mobile Ad hoc NETworks (MANETs). Shortest paths are usually used to route packets in MANETs. However, a shortest path may fail quickly, because some of the wireless links along a shortest path may be broken shortly after the path is established due to mobility of mobile nodes. Rediscovering routes may result in substantial data loss and message exchange overhead. In this paper, we study reliable ad hoc routing in the urban environment. Specifically, we formulate and study two optimization problems. In the minimum Cost Duration-bounded Path (CDP) routing problem, we seek a minimum cost source to destination path with duration no less than a given threshold. In the maximum Duration Cost-bounded Path (DCP) routing problem, we seek a maximum duration source to destination path with cost no greater than a given threshold. We use a waypoint graph to model the working area of a MANET and present an offline algorithm to compute a duration prediction table for the given waypoint graph. An entry in the duration prediction table contains the guaranteed worst-case duration of the corresponding wireless link. We then present an efficient algorithm which computes a minimum cost duration-bounded path, using the information provided in the duration prediction table. We also present a heuristic algorithm for the DCP routing problem. In addition, we show that the proposed prediction and routing schemes can be easily applied for designing reliable ad hoc routing protocols. Simulation results show that our mobility prediction based routing algorithms lead to higher network throughput and longer average path duration, compared with the shortest path routing. This research was supported in part by ARO grant W911NF-04-1-0385 and NSF grant CCF-0431167. The information reported here does not reflect the position or the policy of the federal government.     

Direct Observation of Road Construction Worker Behavior

Many construction workers are paid with public dollars and typically earn large salaries within the manual labor workforce. The present study used a momentary time sampling procedure to directly observe the proportion of road construction workers engaged in work-related behavior and compared the observation to repeated observations of one worksite. Both observations suggested that a small number of highway workers may be performing work tasks at any time, suggesting a need for behavior analysts to devise more effective work-related contingencies to ensure taxpayers receive more return on their investment.     

Optimizing the number of stations in assembly lines under learning for limited production

This paper considers finding the optimal number of stations for an assembly line producing a limited quantity of a new product under learning conditions. This type of production characterizes heavy products (such as airplanes and communication systems) and science-based industries (e.g. laser cutting devices and special equipment for hospitals). These products are manufactured in assembly line fashion in small batches of a few hundred. Since the products are assumed to be totally new to the workers, the learning phenomenon is significant (i.e. task times decrease from cycle to cycle as experience is gained). Therefore, standard balancing methods are no longer applicable. Determining the number of stations has a large effect on the production rate while the actual assignment of tasks to stations helps to fine tune the cycle time. Thus, the number of stations can be regarded as a strategic decision variable . This paper discusses two ways for determining the optimal number of stations, namely as a cost minimization problem and as a profit maximization problem.     

子网络视角下项目群合同项目工期延误诊断模型研究

项目群工期延误诊断是项目群进度目标控制的一项重要任务。总时差可用于判断项目群中某项工作延误对项目群总工期的延误程度,但没有解决某项工作延误对其自身合同项目和后续合同项目工期延误程度的判断问题。本文根据多项目管理和项目群管理理论,通过引入项目群子网络,研究并构建了基于子网络的项目群结构。在此基础上,运用关键路径法（CPM）,系统地研究了因子网络中合同项目某工作工期延误对自身子网络以及项目群中后续子网络工期的影响,提出了子网络后主链定理以及前主链总时差定理,从而实现子网络视角下项目群合同项目工期延误的诊断分析。结合算例进行了具体阐述与应用。最后就如何应用人工智能算法实现项目群进度及其影响因素进行实时监控提出研究思路。本文研究成果为子网络承包商的工期延误责任划分以及索赔提供依据。     

A Model for Resource Constrained Production and Inventory Management

We present a general model for multi-item production and inventory management problems that include a resource restriction. The decision variables in the model can take on a variety of interpretations, but will typically represent cycle times, production batch sizes, number of production runs, or order quantities for each item. We consider environments where item demand rates are approximately constant and performing an activity such as producing a batch of a product or placing an order results in the consumption of a scarceresource that is shared among the items. Some examples of shared resources include limited machine capacity, a restriction on the amount of money that can be tied up in stock, orlimited storage capacity. We focus on the case where the decision variables must be integer valued or selected from a discrete set of choices, such as when an integer number of production runs is desired for each item, or in order quantity problems where the items come in pack sizes containing more than one unit and, therefore, the order quantities must be an integer multiple of the pack sizes. We develop a heuristic and a branch and bound algorithm for solving the problem. The branch and bound algorithm includes reoptimization procedures and the heuristic to improve its performance. Computational testing indicates that the algorithms are effective for solving the general model.     

Aircraft routing and crew pairing: Updated algorithms at Air France

Aircraft routing and crew pairing problems aim at building the sequences of flight legs operated respectively by airplanes and by crews of an airline. Given their impact on airlines operating costs, both have been extensively studied for decades. Our goal is to provide reliable and easy to maintain frameworks for both problems at Air France. We propose simple approaches to deal with Air France current setting. For routing, we introduce an exact compact IP formulation that can be solved to optimality by current MIP solvers in at most a few minutes even on Air France largest instances. Regarding crew pairing, we provide a methodology to model the column generation pricing subproblem within a new resource constrained shortest path framework recently introduced by the first author. This new framework, which can be used as a black-box, leverages on bounds to discard partial solutions and speed-up the resolution. The resulting approach enables to solve to optimality Air France largest instances. Recent literature has focused on integrating aircraft routing and crew pairing problems. As a side result, we are able to solve to near optimality large industrial instances of the integrated problem by combining the aforementioned algorithms within a simple cut generating method.     

Cellular Bucket Brigades on U‐Lines with Discrete Work Stations

It is challenging to maximize and maintain productivity of a U‐line with discrete stations under the impact of variability. This is because maximizing productivity requires assigning workers to suitable tasks and maintaining productivity requires sufficient flexibility in task assignment to absorb the impact of variability. To achieve this goal, we propose an operating protocol to coordinate workers on the U‐line. Under the protocol the system can be configured such that its productivity is maximized. Workers are allowed to dynamically share work so that the system can effectively absorb the impact of variability. Analysis based on a deterministic model shows that the system always converges to a fixed point or a period‐2 orbit. We identify a sufficient condition for the system to converge to the fixed point. Increasing the number of stations improves productivity only under certain circumstances. The improvement is most significant when the number of stations in each stage increases from one to two, but further dividing the U‐line into more stations has diminishing return. Simulations based on random work velocities suggest that our approach significantly outperforms an optimized, static work allocation policy if variability in velocity is large.     

EDUCATION

The power of an integer programming approach—and more particularly a 0,1 programming approach—to resource allocation problems is not widely appreciated. Recent developments in mathematical programming and in problem formulation enable decision makers to deal explicitly with the special conditions which characterize real world problems in capital budgeting, scheduling, and facilities planning. In this paper, somewhat tutorial in nature, we seek to demonstrate formulation and solution of an equipment selection resource allocation problem with several special conditions. The problem can be solved on any reasonably equipped computer system. Sensitivity studies are also performed and discussed.