A decomposed branch-and-price procedure for integrating demand planning in personnel staffing problems

The personnel staffing problem calculates the required workforce size and is determined by constructing a baseline personnel roster that assigns personnel members to duties in order to cover certain staffing requirements. In this research, we incorporate the planning of the duty demand in the staff scheduling problem in order to lower the staffing costs. More specifically, the demand originates from a project scheduling problem with discrete time/resource trade-offs, which embodies additional flexibility as activities can be executed in different modes. In order to tackle this integrated problem, we propose a decomposed branch-and-price procedure. A tight lower and upper bound are calculated using a problem formulation that models the project scheduling constraints and the time-related resource scheduling constraints implicitly in the decision variables. Based upon these bounds, the strategic problem is decomposed into multiple tactical subproblems with a fixed workforce size and an optimal solution is searched for each subproblem via branch-and-price. Fixing the workforce size in a subproblem facilitates the definition of resource capacity cuts, which limit the set of eligible project schedules, decreasing the size of the branching tree. In addition, in order to find the optimal integer solution, we propose a specific search strategy based upon the lower bound and dedicated rules to branch upon the workload generated by a project schedule. The computational results show that applying the proposed search space decomposition and the inclusion of resource capacity cuts lead to a well-performing procedure outperforming different other heuristic and exact methodologies.     

The new Fundamental Tree Algorithm for production scheduling of open pit mines

The problem of annual production scheduling in surface mining consists of determining an optimal sequence of extracting the mineralized material from the ground. The main objective of the optimization process is usually to maximize the total Net Present Value of the operation. Production scheduling is typically a mixed integer programming (MIP) type problem. However, the large number of integer variables required in formulating the problem makes it impossible to solve. To overcome this obstacle, a new algorithm termed “Fundamental Tree Algorithm” is developed based on linear programming to aggregate blocks of material and decrease the number of integer variables and the number of constraints required within the MIP formulation. This paper proposes the new Fundamental Tree Algorithm in optimizing production scheduling in surface mining. A case study on a large copper deposit summarized in the paper shows substantial economic benefit of the proposed algorithm compared to existing methods.     

Project selection,scheduling and resource allocation with time dependent returns

In this paper we formulate and analyze the joint problem of project selection and task scheduling. We study the situation where a manager has many alternative projects to pursue such as developing new product platforms or technologies, incremental product upgrades, or continuing education of human resources. Project return is assumed to be a known function of project completion time. Resources are limited and renewable. The objective is to maximize present worth of profit. A general mathematical formulation that can address several versions of the problem is presented. An implicit enumeration procedure is then developed and tested to provide good solutions based on project ordering and a prioritization rule for resource allocation. The algorithm uses an imbedded module for solving the resource-constrained project scheduling problem at each stage. The importance of integrating the impact of resource constraints into the selection of projects is demonstrated.     

Multi-mode resource constrained multi-project scheduling and resource portfolio problem

This paper introduces a multi-project problem environment which involves multiple projects with assigned due dates; activities that have alternative resource usage modes; a resource dedication policy that does not allow sharing of resources among projects throughout the planning horizon; and a total budget. Three issues arise when investigating this multi-project environment. First, the total budget should be distributed among different resource types to determine the general resource capacities, which correspond to the total amount for each renewable resource to be dedicated to the projects. With the general resource capacities at hand, the next issue is to determine the amounts of resources to be dedicated to the individual projects. The dedication of resources reduces the scheduling of the projects’ activities to a multi-mode resource constrained project scheduling problem (MRCPSP) for each individual project. Finally, the last issue is the efficient solution of the resulting MRCPSPs. In this paper, this multi-project environment is modeled in an integrated fashion and designated as the resource portfolio problem. A two-phase and a monolithic genetic algorithm are proposed as two solution approaches, each of which employs a new improvement move designated as the combinatorial auction for resource portfolio and the combinatorial auction for resource dedication. A computational study using test problems demonstrated the effectiveness of the solution approach proposed.     

Using Benders Decomposition to Implicitly Model Tour Scheduling

This paper introduces a new integrated model for the combined days-off and shift scheduling problem (the tour scheduling problem). This model generalizes the forward and backward constraints, previously introduced by Bechtold and Jacobs for the shift scheduling problem, to the tour scheduling problem. This results in a general and compact formulation that can handle several types of scheduling flexibility. We also provide a new proof of the correctness of forward and backward constraints based on Benders decomposition. The latter approach is interesting in itself because it can be used to solve the problem when extraordinary overlap of break windows or start-time bands is present. A discussion of model size for a set of hypothetical test problems is presented to show the merits of the new formulation.     

Tabu search for a class of scheduling problems

Scheduling problems are often modeled as resourceconstrained problems in which critical resource assignments to tasks are known and the best assignment of resource time must be made subject to these constraints. Generalization toresource scheduling, where resource assignments are chosen concurrently with times results is a problem which is much more difficult. A simplified model of the general resource scheduling model is possible, however, in which tasks must be assigned a singleprimary resource, subject to constraints resulting from preassignment ofsecondary, or auxiliary, resources. This paper describes extensions and enhancements of tabu search for the special case of the resource scheduling problem described above. The class of problems is further restricted to those where it is reasonable to enumerate both feasible time and primary resource assignments. Potential applications include shift oriented production and manpower scheduling problems as well as course scheduling where classrooms (instructors) are primary and instructors (rooms) and students are secondary resources. The underlying model is a type of quadratic multiple choice problem which we call multiple choice quadratic vertex packing (MCQVP). Results for strategic oscillation and biased candidate sampling strategies are shown for reasonably sized real and randomly generated, synthetic, problem instances. The strategies are compared with other variations using consistent measures of solution time and quality developed for this study.     

Real-world flexible resource profile scheduling with multiple criteria: learning scalarization functions for MIP and heuristic approaches

This article addresses a scheduling problem for a chemical research laboratory. Activities with potentially variable, non-rectangular resource allocation profiles must be scheduled on discrete renewable resources. A mixed-integer programming (MIP) formulation for the problem includes maximum time lags, custom resource allocation constraints, and multiple nonstandard objectives. We present a list scheduling heuristic that mimics the human decision maker and thus provides reference solutions. These solutions are the basis for an automated learning-based determination of coefficients for the convex combination of objectives used by the MIP and a dedicated variable neighborhood search (VNS) approach. The development of the VNS also involves the design of new neighborhood structures that prove particularly effective for the custom objectives under consideration. Relative improvements of up to 60% are achievable for isolated objectives, as demonstrated by the final computational study based on a broad spectrum of randomly generated instances of different sizes and real-world data from the company’s live system.     

Integrated production and material handling scheduling using mathematical programming and constraint programming

In this article, we propose an integrated formulation of the combined production and material handling scheduling problems. Traditionally, scheduling problems consider the production machines as the only constraining resource. This is however no longer true as material handling vehicles are becoming more and more valuable resources requiring important investments. Their operations should be optimized and above all synchronized with machine operations. In the problem addressed in this paper, a job shop context is considered. Machines and vehicles are both considered as constraining resources. The integrated scheduling problem is formulated as a mathematical programming model and as a constraint programming model which are compared for optimally solving a series of test problems. A commercial software (ILOG OPLStudio) was used for modeling and testing both models.     

智能制造环境下的备件生产与运输协同调度问题研究

智能制造和即时配送环境下的备件生产与运输协同调度问题是目前国内研究的一大热点,这是因为备件供应链响应速度已成为当前备件制造企业赢得客户的关键因素。为了提高客户满意度,尽可能缩短从客户下达定制化生产订单到订单配送完成的时间,本文建立了以所有客户总等待时间最短为目标的混合整数规划模型和集合覆盖模型,推导了最优解性质,并设计改进的分支定价算法求得最优解。通过将小规模算例结果与CPLEX进行对比,验证了模型和算法的有效性。多组算例测试结果表明,所提出的模型和算法可以有效提升智能制造环境下的备件供应链运作效率。     

Tool allocation and machining conditions optimization for CNC machines

In the literature, there exist many variations of machining economics problem in terms of modelling approaches and solution methodologies. However most of the existing studies focus on the single machining operation which is seldom in practice. On the other hand, tool management approaches at the system level fail to relate the tooling issues to the machining conditions, and ignore the tool availability and tool wear restrictions. A new solution methodology is developed to determine the optimum machining conditions and tool allocation simultaneously to minimize the production cost of a multiple operation case where there can be alternative tools for each operation. As a result, we can both improve the solution by exploiting the interaction between these two decisions, and also prevent any infeasibility that might occur for the tool allocation problem due to tool contention among the operations for a limited number of tool types by considering the tool availability and tool life limitations.     

The resource-constrained modulo scheduling problem: an experimental study

In this paper, we focus on the resource-constrained modulo scheduling problem (RCMSP), a general periodic scheduling problem, abstracted from the problem solved by compilers when optimizing inner loops at instruction level for VLIW parallel processors. Heuristic solving scheme have been proposed since many years to solve this problem, among which the decomposed software pipeling method. In this method, a cyclic scheduling problem ignoring resource constraints is first considered and a so-called legal retiming of the operations is issued. Second, a standard acyclic problem, taking this retiming as input, is solved through list scheduling techniques. In this paper, we propose a novel hybrid approach, which uses the decomposed software pipeling method to obtain a good retiming. Then the obtained retiming is used to build an integer linear programming formulation of reduced size, which allows to solve it exactly. Experimental results show that a lot more problems are solved with this new approach. The gap to the optimal solution is less than 1 % on most of the tested problem instances and the method appears to be competitive with a recently proposed constraint programming algorithm (Bonfietti et al., Lect. Notes Comput. Sci. 6876:130–144, 2011).     

Modeling lotsizing and scheduling problems with sequence dependent setups

Several production environments require simultaneous planing of sizing and scheduling of sequences of production lots. Integration of sequencing decisions in lotsizing and scheduling problems has received an increased attention from the research community due to its inherent applicability to real world problems. A two-dimensional classification framework is proposed to survey and classify the main modeling approaches to integrate sequencing decisions in discrete time lotsizing and scheduling models. The Asymmetric Traveling Salesman Problem can be an important source of ideas to develop more efficient models and methods to this problem. Following this research line, we also present a new formulation for the problem using commodity flow based subtour elimination constraints. Computational experiments are conducted to assess the performance of the various models, in terms of running times and upper bounds, when solving real-word size instances.     

Production scheduling with alternative process plans

This paper deals with a scheduling problem with alternative process plans that was motivated by a production of wire harnesses where certain parts can be processed manually or automatically by different types of machines. Only a subset of all the given activities will form the solution, so the decision whether the activity will appear in the final schedule has to be made during the scheduling process. The problem considered is an extension of the resource constrained project scheduling problem (RCPSP) with unary resources, positive and negative time-lags and sequence dependent setup times. We extend classic RCPSP problem by a definition of alternative branchings, represented by the Petri nets formalism allowing one to define alternatives and parallelism within one data structure. For this representation of the problem, an integer linear programming model is formulated and the reduction of the problem, using time symmetry mapping, is shown. Finally, a heuristic algorithm based on priority schedule construction with an unscheduling step is proposed for the nested version of the problem and it is used to solve the case study of the wire harnesses production.     

Resource-constrained job scheduling with recyclable resources

In resource-constrained project scheduling problems, resources are typically classified as being either renewable, non-renewable, or doubly-constrained. A new resource classification, recyclable, is introduced. Notation and a generalized problem formulation are developed for resource-constrained job scheduling problems where resources are recyclable. This foundation is then used for studying the single-machine scheduling problem with tooling constraints. For a given set of jobs, the problem is to find the job sequence, tool type quantities, and tool recycling schedule such that the sum of job completion times and quantity of tools allocated are both minimized. Two solution approaches are developed, and examples are used to compare and contrast the approaches. The results indicate that the ‘traditional’ job scheduling approach (i.e. schedule jobs first, then tools) can lead to sub-optimal solutions. Furthermore, by scheduling jobs and tools simultaneously, it may be possible to attain a given level of performance with fewer tools.     

A hybrid method for large-scale short-term scheduling of make-and-pack production processes

Due to the ongoing trend towards increased product variety, fast-moving consumer goods such as food and beverages, pharmaceuticals, and chemicals are typically manufactured through so-called make-and-pack processes. These processes consist of a make stage, a pack stage, and intermediate storage facilities that decouple these two stages. In operations scheduling, complex technological constraints must be considered, e.g., non-identical parallel processing units, sequence-dependent changeovers, batch splitting, no-wait restrictions, material transfer times, minimum storage times, and finite storage capacity. The short-term scheduling problem is to compute a production schedule such that a given demand for products is fulfilled, all technological constraints are met, and the production makespan is minimised. A production schedule typically comprises 500–1500 operations. Due to the problem size and complexity of the technological constraints, the performance of known mixed-integer linear programming (MILP) formulations and heuristic approaches is often insufficient.     

A two pass heuristic algorithm for scheduling ‘blocked out’ units in continuous process industry

This paper addresses the problem of scheduling cascaded ‘blocked out’ continuous processing units separated by finite capacity storage tanks. The raw materials for the product lines arrive simultaneously on the input side of the first unit. But every unit can process only one product line at a time, thus giving rise to the possibility of spillage of raw material due to limited storage capacity. The need to process multiple product lines and the added constraint of multiple intermediate upliftment dates aggravate the problem. This problem is quite common in petrochemical industry. The paper provides a MINLP (Mixed Integer Non-Linear Programming) formulation of the problem. However, for any realistic scheduling horizon, the size of the problem is too large to be solved by standard packages. We have proposed a depth first branch and bound algorithm, guided by heuristics, to help planners in tackling the problem. The suggested algorithm could output near optimal solutions for scheduling horizons of 30 time periods when applied to real life situations involving 3 units and 3 product lines. Preliminary version of the paper appeared in the proceedings of MISTA, 2005.     

Multiple orders per job batch scheduling with incompatible jobs

The multiple orders per job (MOJ) scheduling problem is presented for the batch-processing environment such as that exemplified by diffusion ovens. A mixed-integer programming formulation is presented for the incompatible job family case wherein only jobs that belong to the same family may be grouped together in a production batch. This optimization formulation is tested through an extensive experimental design with the objective of minimizing total weighted tardiness (maximizing on-time delivery performance). Optimal solutions are achievable for this initial set of 6-to-12 order problems, but it is noted that the optimization model takes an unreasonable amount of computation time, which suggests the need for heuristic development to support the analysis of larger, more practical MOJ batch scheduling problems. A number of simple heuristic approaches are investigated in an attempt to find near-optimal solutions in a reasonable amount of computation time. It is seen that a combination of the heuristics produces near-optimal solutions for small order problems. Further testing proves that these heuristic combinations are the best for large order problems as well.     

An Integer Programming Approach to the Vehicle Scheduling Problem

This paper describes an integer programming formulation of the vehicle scheduling problem and illustrates how such a formulation can be extended to incorporate restrictions on work load, coverage and service that occur in real world vehicle scheduling problems. The integer programme is solved using the Revised Simplex method, additional constraints being introduced to retain integrality during convergence. The feasible region of this integer programme is initially restricted so that only routes constructed through sets of radially contiguous locations are considered. The effect of relaxing these over-constraints is explored. The method is demonstrated on fifteen problems ranging in size from 21 to 100 locations and the results generally show an improvement on previously published results. This is particularly true of the larger problems. This method compares favourably with other methods in computational efficiency.     

A composite branch and cut algorithm for optimal shift scheduling with multiple breaks and break windows

This paper presents a new branch and cut algorithm for optimal shift scheduling with multiple breaks and break windows. The proposed algorithm is based on an implicit formulation of the problem requiring significantly smaller number of variables than the set covering formulation of Dantzig. The new algorithm, adding cuts, developing upper bounds for the variables, and employing an efficient rounding heuristic, was tested successfully with 90 test problems involving between 2160 and 32??928 shift variations and five demand profiles. Our results show that large shift scheduling problems can be solved optimally with the proposed branch and cut algorithm and the implicit formulation.