Optimal timing of a sequence of tasks with general completion costs

Scheduling a sequence of tasks––in the acceptation of finding the execution times––is not a trivial problem when the optimization criterion is irregular as for instance in earliness–tardiness problems. This paper presents an efficient dynamic programming algorithm to solve the problem with general cost functions depending on the end time of the tasks, idle time costs and variable durations also depending on the execution time of the tasks. The algorithm is also valid when the precedence graph is a tree and it can be adapted to determine the possible execution windows for each task not exceeding a maximum fixed cost.     

Optimal engineering design via Benders’ decomposition

The optimal engineering design problem consists in minimizing the expected total cost of an infrastructure or equipment, including construction and expected repair costs, the latter depending on the failure probabilities of each failure mode. The solution becomes complex because the evaluation of failure probabilities using First-Order Reliability Methods (FORM) involves one optimization problem per failure mode. This paper formulates the optimal engineering design problem as a bi-level problem, i.e., an optimization problem constrained by a collection of other interrelated optimization problems. The structure of this bi-level problem is advantageously exploited using Benders’ decomposition to develop and report an efficient algorithm to solve it. An advantage of the proposed approach is that the design optimization and the reliability calculations are decoupled, resulting in a structurally simple algorithm that exhibits high computational efficiency. Bi-level problems are non-convex by nature and Benders algorithm is intended for convex optimization. However, possible non-convexities can be detected and tackled using simple heuristics. Its practical interest is illustrated through a realistic but simple case study, a breakwater design example with two failure modes: overtopping and armor instability.     

Precedence theorems and dynamic programming for the single-machine weighted tardiness problem

We tackle precedence-constrained sequencing on a single machine in order to minimize total weighted tardiness. Classic dynamic programming (DP) methods for this problem are limited in performance due to excessive memory requirements, particularly when the precedence network is not sufficiently dense. Over the last decades, a number of precedence theorems have been proposed, which distinguish dominant precedence constraints for a job pool that is initially without precedence relation. In this paper, we connect and extend the findings of the foregoing two strands of literature. We develop a framework for applying the precedence theorems to the precedence-constrained problem to tighten the search space, and we propose an exact DP algorithm that utilizes a new efficient memory management technique. Our procedure outperforms the state-of-the-art algorithm for instances with medium to high network density. We also empirically verify the computational gain of using different sets of precedence theorems.     

Iterated local search for workforce scheduling and routing problems

The integration of scheduling workers to perform tasks with the traditional vehicle routing problem gives rise to the workforce scheduling and routing problems (WSRP). In the WSRP, a number of service technicians with different skills, and tasks at different locations with pre-defined time windows and skill requirements are given. It is required to find an assignment and ordering of technicians to tasks, where each task is performed within its time window by a technician with the required skill, for which the total cost of the routing is minimized. This paper describes an iterated local search (ILS) algorithm for the WSRP. The performance of the proposed algorithm is evaluated on benchmark instances against an off-the-shelf optimizer and an existing adaptive large neighbourhood search algorithm. The proposed ILS algorithm is also applied to solve the skill vehicle routing problem, which can be viewed as a special case of the WSRP. The computational results indicate that the proposed algorithm can produce high-quality solutions in short computation times.     

Balancing assembly lines with tabu search

Balancing assembly lines is a crucial task for manufacturing companies in order to improve productivity and minimize production costs. Despite some progress in exact methods to solve large scale problems, softwares implementing simple heuristics are still the most commonly used tools in industry. Some metaheuristics have also been proposed and shown to improve on classical heuristics but, to our knowledge, no computational experiments have been performed on real industrial applications to clearly assess their performance as well as their flexibility. Here we present a new tabu search algorithm and discuss its differences with respect to those in the literature. We then evaluate its performance on the Type I assembly line balancing problem. Finally, we test our algorithm on a real industrial data set involving 162 tasks, 264 precedence constraints, and where the assembly is carried out on a sequential line with workstations located on both sides of the conveyor, with two possible conveyor heights and no re-positioning of the product. We discuss the flexibility of the metaheuristic and its ability to solve real industrial cases.     

Scheduling tasks and communications on a virtual distributed system

A set of tasks has to be scheduled on identical parallel processors subject to precedence constraints and small communication delays. A polynomial algorithm is known to exist if task duplication is allowed and the number of available processors is not limited. However the problem of communications scheduling is not taken into account. In this paper, we prove that this algorithm also never saturates communication channels and always delivers messages on time, if slightly stronger constraints are imposed on the tasks.     

Robust balancing of straight assembly lines with interval task times

This paper addresses the balancing problem for straight assembly lines where task times are not known exactly but given by intervals of their possible values. The objective is to assign the tasks to workstations minimizing the number of workstations while respecting precedence and cycle-time constraints. An adaptable robust optimization model is proposed to hedge against the worst-case scenario for task times. To find the optimal solution(s), a breadth-first search procedure is developed and evaluated on benchmark instances. The results obtained are analysed and some practical recommendations are given.     

On the Hierarchical Chinese Postman Problem with linear ordered classes

The Hierarchical Chinese Postman Problem (HCPP) is a Chinese Postman Problem with the arcs partitioned into priority classes ordered by a precedence relation. The problem under the sum criterion is polynomially solvable if the ordering is linear and each class is connected. For a known HCPP algorithm we give an O(n) improvement (n the number of nodes) leading to O(kn⁴) with k the number of classes. The same complexity appears to hold for the lexicographic criterion which minimises the costs of the first priority class, then the costs of the second class, etc. The notions of servicing and traversing related to arcs, allow for more real life models of arc routing problems. We show how to incorporate these notions in known algorithms, without increasing the complexity.     

An iterative algorithm for scheduling UET tasks with due dates and release times

A new polynomial-time iterative algorithm is presented for the scheduling problem with a unit execution time task system, parallel identical processors, precedence constraints, release times, and the criterion of maximum lateness. For the maximum lateness and makespan problems the algorithm allows to achieve the performance guarantees previously known only for the problems without release times.     

Scheduling tasks on a flexible manufacturing machine to minimize tool change delays

This paper considers the problem of scheduling a given set of precedence constraint tasks on a flexible machine equipped with a tool magazine where each task requires exactly one of the tools during its execution. Changing from one tool to another requires a certain amount of time that depends on the pair of tools being exchanged. We present a new algorithmic approach for general task precedence relations when it is desired to sequence the tasks in such a way that the total time required for tool changes is minimized. The proposed algorithm is of polynomial time complexity in case of task precedences of limited width w, i.e. for precedence relations where each subset of independent tasks has not more than w elements. Since the task precedences width w could be arbitrary, we describe two heuristic algorithms and empirically evaluate their effectiveness in finding schedules with minimum total time required for tool changes.     

Heuristics for multimode scheduling problems with dedicated resources

We study scheduling problems with multiple modes and dedicated resources arising in production and project management, which constitute a special class of the general multimode resource-constrained project scheduling problem. A task may require simultaneously a set of discrete, renewable resources to be processed and the processing can be performed in different modes, that is with different resource sets, processing times, or costs. Precedence constraints can exist among tasks. The total budget that can be allocated to the project can be limited. The problem consists of identifying a mode for each task and a starting time for its processing respecting precedence, resource, and budget constraints. A graph model and an iterative solution scheme are presented. Specific heuristic algorithms for the cases with and without budget constraints are given and computational results are discussed.     

A Precedence Graph Algorithm for the Shop Scheduling Problem

This paper is concerned with the development of a precedence graph algorithm for solving certain combinatorial problems. This algorithm is applied mainly to job-shop scheduling problems; however, the extension of its applicability can be demonstrated by considering project scheduling, travelling salesman and explosion problems. The algorithm employs linear graphs to construct the quantified precedence matrix, a powerful criterion to resolve the conflict between the tied operations, and the use of a quasi-Boolean procedure to evaluate the obtained sequence.Considerable experimentation is conducted to evaluate the performance of the algorithm. Significant results pertaining to the quality of solution, the computation time and the number of iterations and conflicts encountered in obtaining a solution are given.     

Task scheduling with and without communication delays: A unified approach

The problem of scheduling directed acyclic task graphs on an unbounded number of processors is considered. We present a single algorithm which is applicable to several special cases, thus effecting a unified approach to task scheduling independent of the task graph. We start by considering multi-stage dags and present an algorithm that computes a schedule in O(Nq log q) time, where N is the number of stages, and q is the maximum number of edges between any two stages of the graph. We show that the schedule produced by the algorithm is optimal when: (i) all communication delays are zero or, (ii) the precedence graph is an in-tree or an out-tree and communication times are small or, (iii) the task graph is densely connected and communication costs and processing costs are unity. For multi-stage dags with small communication times we show that the makespan of the schedule generated by our algorithm is less than twice that of the optimal. We also bound the makespan for the case when communication times are arbitrary. We then show how the algorithm may be applied to schedule arbitrary dags and derive the performance bounds for this case. Finally, we present the results of tests we carried out with randomly generated task graphs. These seem to indicate that, on the average, the algorithm performs substantially better than theoretical worst case predictions.     

An efficient genetic algorithm for the traveling salesman problem with precedence constraints

The traveling salesman problem with precedence constraints (TSPPC) is one of the most difficult combinatorial optimization problems. In this paper, an efficient genetic algorithm (GA) to solve the TSPPC is presented. The key concept of the proposed GA is a topological sort (TS), which is defined as an ordering of vertices in a directed graph. Also, a new crossover operation is developed for the proposed GA. The results of numerical experiments show that the proposed GA produces an optimal solution and shows superior performance compared to the traditional algorithms.     

A variable neighbourhood search algorithm for the constrained task allocation problem

A variable neighbourhood search algorithm that employs new neighbourhoods is proposed for solving a task allocation problem whose main characteristics are: (i) each task requires a certain amount of resources and each processor has a capacity constraint which limits the total resource of the tasks that are assigned to it; (ii) the cost of solution includes fixed costs when using processors, task assignment costs, and communication costs between tasks assigned to different processors. A computational study shows that the algorithm performs well in terms of time and solution quality relative to other local search procedures that have been proposed.     

Optimal spare ordering policy for preventive replacement under cost effectiveness criterion

This paper presents a spare ordering policy for preventive replacement with age-dependent minimal repair and salvage value consideration. The spare unit for replacement is available only by order and the lead-time for delivering the spare due to regular or expedited ordering follows general distributions. To analyze the ordering policy, the failure process is modelled by a non-homogeneous Poisson process. By introducing the costs due to ordering, repairs, replacements and downtime, as well as the salvage value of an un-failed system, the expected cost effectiveness in the long run are derived as a criterion of optimality. It is shown, under certain conditions, there exists a finite and unique optimum ordering time which maximizes the expected cost effectiveness. Finally, numerical examples are given for illustration.     

A discrete time optimal control model with uncertainty for dynamic machine allocation problem and its application to manufacturing and construction industries

This paper proposed a discrete time optimal control model in which machine failure time is modeled assuming a Weibull distribution and machine productivity is regarded as a fuzzy variable for dealing with a dynamic machine allocation problem (DMAP) in manufacturing and construction industries. The aim is to maximize total production or construction throughput when uncertainties such as machine breakdowns are taken into account. A failure probability-work time equation is presented to describe the relationship between machine failure probability and mean time to work. To transform the uncertain optimal control model into a deterministic one, the expected value model (EVM) was introduced for forming an equivalent crisp model. The fuzzy variables in the model are also defuzzified by using an expected value operator with an optimistic–pessimistic index. Then a number of lemmas and theorems are presented and proved to formulate the theoretical algorithm so that the crisp model of the DMAP can be solved. Three actual construction and production projects are used as practical application examples. The theoretical algorithm results for the three project examples are compared with a particle swarm optimization approach and a genetic algorithm method, which demonstrates the practicality and efficiency of our optimization method.     

广义优先关系约束下Max-npv项目调度问题及其遗传算法

以往Max-npv项目调度问题的研究都假定活动之间的关系为单一结束-开始类型,现实中活动之间关系复杂多变,因此,将广义优先关系引入Max-npv项目调度问题中,构建了广义优先关系约束下的Max-npv项目调度模型。针对该优化模型设计了一种双层遗传算法,外层遗传算法负责任务执行模式的优化,内层遗传算法负责任务调度的优化。在内层遗传算法中,采用任务开始时间之差作为新的编码方式,大大简化了交叉变异算子,针对网络图中的环状结构设计了修复算子,确保了编码的有效性。通过一个算例对算法进行了测试,实验结果验证了算法的有效性。     

Shortest-route formulation of mixed-model assembly line balancing problem

A shortest-route formulation of the mixed-model assembly line balancing problem is presented. Common tasks across models are assumed to exist and these tasks are performed in the same stations. The formulation is based on an algorithm which solves the single-model version of the problem. The mixed-model system is transformed into a single-model system with a combined precedence diagram. The model is capable of considering any constraint that can be expressed as a function of task assignments.