An MP/CP-Based Hybrid Approach to Optimization of the Resource-Constrained Scheduling Problems

Constrained scheduling problems are common in manufacturing, project management, transportation, supply chain management, software engineering, computer networks etc. Multiple binary and integer decision variables representing the allocation of resources to activities and numerous specific constraints on these variables are typical components of the constraint scheduling problem modeling. With their increased computational complexity, the models are more demanding, particularly when methods of operations research (mathematical programming, network programming, dynamic programming) are used. By contrast, most resource-constrained scheduling problems can be easily modeled as instances of the constraint satisfaction problems (CSPs) and solved using constraint programming (CP) or others methods. In the CP-based environment the problem definition is separated from the methods and algorithms used to solve the problem. Therefore, a constraint-based approach to resource-constrained scheduling problems that combines an OR-based approach for problem solving and a CP-based approach for problem modeling is proposed. To evaluate the applicability and efficiency of this approach and its implementation framework, illustrative examples of resource-constrained scheduling problems are implemented separately for different environments.

Resource-constrained scheduling problems are commonly found in various areas, such as project management, manufacturing, transportation, software engineering, computer networks, and supply chain management. Its problem models involve a large number of constraints and discrete decision variables, including binary and integer. In effect, the representation of resource allocation, for instance, is often expressed using binary or integer decision variables to form several constraints according to the respective scheduling problem. It significantly increases the number of decision variables and constraints as the problem scales; such kind of traditional approaches based on operations research is insufficient. Therefore, a hybrid approach to decision support for resource-constrained scheduling problems which combines operation research (OR) and constraint logic programming (CLP) is proposed. Unlike OR-based approaches, declarative CLP provides a natural representation of different types of constraints. This approach provides: (a) decision support through the answers to the general and specific questions, (b) specification of the problem based on a set of facts and constraints, (c) reduction to the combinatorial solution space. To evaluate efficiency and applicability of the proposed hybrid approach and implementation platform, implementation examples of job-shop scheduling problem are presented separately for the three environments, i.e., Mathematical Programming (MP), CLP, and hybrid implementation platform.

Contributors

Handbook of Constraint Programming

In this paper a constraint programming (CP) approach for calculating release dates for lots within a supply chain environment is investigated. The lot start times are verified by a simulation model using different dispatching rules focusing on tardiness. To test the presented CP approach a simple fab model is constructed. The fab model consists of parallel batch machines as well as work centers and single machines. The investigated objectives are tardiness, earliness and cycle time. Due to the high complexity decomposition methods for the CP approach are tested. The results from the CP method are lot start dates which are verified by a downstream simulation run. The results show that the presented CP approach could outperform the simulation for all objectives. The content of this paper could be used as a first investigation for new scheduling methods within a supply chain management.

We describe a modeling framework that integrates mathematical programming (MP), constraint programming (CP) and heuristic methods. It is extendible to other solution methods as well. The problem structure is mirrored in the model structure, and the solver exploits this structure in a principled way to combine methods effectively. The approach generalizes and extends past research on the integration of MP and CP. Six modeling examples are given. In particular, it is shown that a recent integration scheme for CP and MP based on Benders decomposition is a special case of the framework described here.

Many applications involve the concepts of scheduling, such as communications, packet routing, production planning [Zhai et al., 2006], classroom arrangement [Mathaisel & Comm, 1991], aircrew scheduling [Chang, 2002], nurse scheduling [Ohki et al., 2006], food industrial [Simeonov & Simeonovova, 2002], control system [Fleming & Fonseca, 1993], resource-constrained scheduling problem [Chen, 2007] and grid computing. There are many different types of scheduling problems such as real-time, job-shop, permutation flow-shop, project scheduling and other scheduling problems have been studied intensively. However, in this work, the studied grid task scheduling problem is much more complex than above stated classic task scheduling problems. Restated, a grid application is regarded as a task scheduling problem involving tasks with inter-communication and distributed homogeneous or heterogeneous resources, and can be represented by a task interaction graph (TIG). Grid is a service for sharing computing power and data storage capacity over the Internet. The grid systems outperform simple communication between computers and aims ultimately to turn the global network of computers into one vast computational resource. Grid computing can be adopted in many applications, such as high-performance applications, large-output applications, data-intensive applications and community-centric applications. These applications major concern to efficiently schedule tasks over the available multi-processor environment provided by the grid. A grid is a collaborative environment in which one or more tasks can be submitted without knowing where the resources are or even who owns the resources [Foster et al., 2001]. The efficiency and effectiveness of grid resource management greatly depend on the scheduling algorithm [Lee et al., 2007]. Generally, in the grid environment, these resources are different over time, and such changes will affect the performance of the tasks running on the grid. In grid computing, tasks are assigned among grid system [Salman, 2002]. The purpose of task scheduling in grid is to find optimal task-processor assignment and hence minimize application completion time (total cost). Most scheduling problems in these applications are categorized into the class of NP-complete problems. This implies that it would take amount of computation time to obtain an optimal solution, especially for a large-scale scheduling problem. A variety of approaches have been applied to solve scheduling problems, such as

In this study, the U-type assembly line balancing problem (UALBP) with assignment restrictions (AR-UALBP) is considered. Linked and incompatible distance and station restrictions are taken into account. New mathematical programming (MP) and constraint programming (CP) models are proposed. The objective function of the models is minimizing the cycle time for a given number of workstations. In addition to this objective, line efficiency and CPU time are also considered as other performance measurements. Numerical experiments on some problems from the literature are performed to evaluate the effectiveness of the proposed models. The results are compared with the results of the CP model of the straight ALBP with assignment restrictions and also the results from MP and CP models are compared with each other. The numerical results indicate that the proposed CP and MP models are more effective in obtaining better and optimal results when solving the AR-UALBP.

A guided local search with iterative ejections of bottleneck operations for the job shop scheduling problem

In the new dynamic economic environment where supply chains increasingly face constant change and instability, the better supply chain planning and management enabled by advanced data management systems provides enhanced value proposition for customers resulting in improved profitability for firms along the supply chain. However, achieving such high quality level of supply chain visibility is not an easy task requiring technological capability, organizational willingness and data quality management intensively demanding attention from both managers and scholars. Therefore, the chapter begins with an overview of the role of information systems in supply chain management followed by a discussion regarding the role of information quality in successful supply chain interactions. Data quality management in terms of strategy and data governance is then reviewed. Finally, data quality tools complementing strategy dimension of data quality management are studied.

This paper is aimed at researchers and practitioners in Operational Research who are interested in the new field of Constraint Programming/Constraint Logic Programming. Due to differing terminology and problem representation they might have found it difficult to access the field. The paper focuses on discrete optimisation problems. The first part lists frequently used terms in Constraint Programming (CP), contrasting them with their counterparts in Mathematical Programming (MP). The second part explains some of the most important concepts and techniques in more detail by comparing the CP and MP implementations of a small example problem, the ‘Change Problem’. It includes an overview of the respective results. In conclusion a more generalised comparison of CP and MP techniques is given.

This work is concerned with approximating constraint satisfaction problems (CSPs) with an additional global cardinality constraints. For example, \maxcut is a boolean CSP where the input is a graph $G = (V,E)$ and the goal is to find a cut $S \cup \bar S = V$ that maximizes the numberof crossing edges, $|E(S,\bar S)|$. The \maxbisection problem is a variant of \maxcut with an additional global constraint that each side of the cut has exactly half the vertices, i.e., $|S| = |V|/2$. Several other natural optimization problems like \minbisection and approximating Graph Expansion can be formulated as CSPs with global constraints. In this work, we formulate a general approach towards approximating CSPs with global constraints using SDP hierarchies. To demonstrate the approach we present the following results: Using the Lasserre hierarchy, we present an algorithm that runs in time $O(n^{poly(1/\epsilon)})$ that given an instance of \maxbisection with value $1-\epsilon$, finds a bisection with value $1-O(\sqrt{\epsilon})$. This approximation is near-optimal (up to constant factors in $O()$) under the Unique Games Conjecture. By a computer-assisted proof, we show that the same algorithm also achieves a 0.85-approximation for \maxbisection, improving on the previous bound of 0.70 (note that it is \uniquegames hard to approximate better than a 0.878 factor). The same algorithm also yields a 0.92-approximation for \maxtwosat with cardinality constraints. For every CSP with a global cardinality constraints, we present a generic conversion from integrality gap instances for the Lasserre hierarchy to a {\it dictatorship test} whose soundness is at most integrality gap. Dictatorship testing gadgets are central to hardness results for CSPs, and a generic conversion of the above nature lies at the core of the tight Unique Games based hardness result for CSPs. \cite{Raghavendra08}

In a facility with front room and back room operations, it is useful to switch workers between the rooms in order to cope with changing customer demand. Assuming stochastic customer arrival and service times, we seek a policy for switching workers such that the expected customer waiting time is minimized while the expected back room staffing is sufficient to perform all work. Three novel constraint programming models and a shaving algorithm are presented. Experimental results show that the best constraint programming model, using shaving, is able to find and prove optimal solutions for almost all problem instances within a reasonable run-time, but that an existing heuristic algorithm performs better in terms of solution quality over time. A hybrid method combining the heuristic and the best constraint programming method is shown to perform better than either of these approaches separately. This is the first work of which we are aware that solves a queueing control problem with constraint programming.

We investigate the following resource constrained scheduling problem. Given m identical processors, s renouvable, but limited resources, n independent tasks each of unit length, where each task needs one processor and a task dependent amount of every resource in order to be processed. In addition each task possesses an integer ready-time, which means it cannot be processed before its ready-time. The optimization problem is to assign all tasks to discrete times in ℕ, minimizing the latest completion time C max of any task subject to the processor, resource and ready-time constraints.The problem is NP-hard even under much simpler assumptions. The best previously known polynomial-time approximation is due to Röck and G. Schmidt, who gave in 1983 in the case of zero ready-times an O(m)-factor algorithm. For the analysis of their algorithm, the assumption of identical (zero-) ready-times is essential.The main contribution of this paper is to break the O(m) barrier substantially (even for non zero ready-times) and to show the first constant factor approximations, which are best possible, unless P=N P.Furtheron we can prove for the apparently simpler problem with zero ready-times the N P-completeness of the problem of deciding whether there exists an integral schedule of size C, even if a fractional schedule of size C and an integral schedule of size C+1 are known. The proof is based on an interesting reduction to the chromatic index problem.KeywordsSchedule ProblemIndependent TaskFractional ScheduleUnrelated Parallel MachineChromatic IndexThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

One of the difficulties in high-level synthesis and compiler optimization is obtaining a good schedule without knowing the exact computation time of the tasks involved. The uncertain computation times of these tasks normally occur when conditional instructions are employed and/or inputs of the tasks influence the computation time. The relationship between these tasks can be represented as a data-flow graph where each node models the task associated with a probabilistic computation time. A set of edges represents the dependencies between tasks. In this research, we study scheduling and optimization algorithms taking into account the probabilistic execution times. Two novel algorithms, called probabilistic retiming and probabilistic rotation scheduling, are developed for solving the underlying nonresource and resource constrained scheduling problems, respectively. Experimental results show that probabilistic retiming consistently produces a graph with a smaller longest path computation time for a given confidence level, as compared with the traditional retiming algorithm that assumes a fixed worst-case and average-case computation times. Furthermore, when considering the resource constraints and probabilistic environments, probabilistic rotation scheduling gives a schedule whose length is guaranteed to satisfy a given probability requirement. This schedule is better than schedules produced by other algorithms that consider worst-case and average-case scenarios.