Dynamic Programming Solution Algorithms Research Articles

On scheduling of step-improving jobs to minimize the total weighted completion time

Kim et al. (2022) studied single-machine scheduling of step-improving jobs with a common discount factor to minimize the total weighted completion time. This study proposes a pseudo-polynomial algorithm to solve the problem. We then consider the general problem in which the discount factors are job dependent. It is proven that there is an optimal solution in which the early normal jobs are sequenced in the WSPT order and the late discounted jobs are sequenced in the WSPT order. Based on two WSPT lists, a dynamic programming solution algorithm is proposed. The run time is pseudo-polynomial when the distance of any job is bounded by a constant, where the distance of a job is the absolute difference of the two positions of this job in two WSPT-ordered job lists.

Minimizing total late work on a single machine with generalized due-dates

We study single machine scheduling problems with generalized due-dates. The scheduling measure is minimum total late work. We show that unlike the classical version (assuming job-specific due-dates), this problem has a polynomial time solution. Then, the problem is extended to allow job rejection. First, an upper bound on the total permitted rejection cost is assumed. Then we study the setting whereby the rejection cost is part of the objective function, which becomes minimizing the sum of total late work and rejection cost. We prove that both versions are NP-hard, and introduce pseudo-polynomial dynamic programming solution algorithms. We then consider a setting in which the machine is not available for some period (e.g., due to maintenance). Again, a pseudo-polynomial dynamic programming is introduced for the (NP-hard) problem of minimizing total late work with generalized due-dates and unavailability period. These dynamic programming algorithms are tested numerically, and proved to perform well on problems of various input parameters. Then, the extension to the weighted case, i.e., the problem of minimizing total weighted late work with generalized due-dates is proved to be NP-hard. Finally, we study a slightly different setting, in which the given due-dates are assigned to jobs, but there is no restriction on their order, i.e., the j-th due-date is not necessarily assigned to the j-th job in the sequence. We show that this problem (known as scheduling assignable due-dates) to minimize total late work is NP-hard as well.

Extensions of dynamic programming for multi-stage combinatorial optimization

We propose a dynamic programming framework for an exact multi-stage (lexicographic) combinatorial optimization. Unlike conventional algorithms of dynamic programming that return one optimal solution, two dynamic programming algorithms proposed in this paper are coping with the whole set of optimal solutions or with its essential part. We describe the set of elements for optimization by a labeled directed acyclic graph, which in some sense, is similar to the structure of subproblems of the considered problem. For a given cost function (objective), the first algorithm constructs a subgraph of this graph that describes the whole set of optimal elements or its essential part. This algorithm can be used for multi-stage optimization of elements relative to a sequence of cost functions. The second algorithm counts elements before the optimization and after each optimization step. The considered labeled directed acyclic graph is a kind of circuit. This circuit builds the set of elements for optimization from one-element sets attached to input nodes. It uses the operation of union of sets attached to unifying nodes and functional operations attached to functional nodes. The algorithms for optimization and counting elements are defined for an arbitrary circuit without repetitions in which each element is generated exactly one time. For a considered problem with a known conventional dynamic programming solution algorithm, usually, it is easy to construct a corresponding circuit without repetitions. Once the circuit and cost functions are defined, our framework provides the correctness proofs and detailed time complexity analysis for the proposed algorithms. To make this approach more intuitive, we consider an illustrative example related to the maximum subarray problem. We tested our approach on the following nine combinatorial optimization problems: matrix chain multiplication, global sequence alignment, optimal paths in directed graphs, binary search trees, optimal bitonic tour, segmented least squares, convex polygon triangulation, one-dimensional clustering, and line breaking (text justification). We consider the last three problems in detail: construct a circuit without repetitions, describe at least two cost functions, evaluate a number of operations and time required by the algorithms, discuss an example, and consider the results of computer experiments with randomly generated instances of the problem.

Regular scheduling measures on proportionate flowshop with job rejection

In this article, we study the method of job rejection in the setting of proportionate flowshop, and focus on minimizing regular performance measures, subject to the constraint that the total rejection cost cannot exceed a given upper bound. In particular, we study total completion time, maximum tardiness, total tardiness, and total weighted number of tardy jobs. All the addressed problems are NP-hard as their single machine counterpart are known to be NP-hard. To the best of our knowledge, there are no detailed solutions in scheduling literature to the first two problems, whereas the last two problems were never addressed to date. For each problem, we provide a pseudo-polynomial dynamic programming solution algorithm, and furthermore, we enhance the reported running time of the first two problems. Our extensive numerical study validates the efficiency of the provided solutions.

Single-machine batch scheduling problem with job rejection and resource dependent processing times

This paper addresses single-machine batch scheduling with job rejection and convex resource allocation. A job is either rejected, in which case a rejection penalty will be incurred, or accepted and processed on the machine. The accepted jobs are combined to form batches containing contiguously scheduled jobs. For each batch, a batch-dependent machine setup time, which is a function of the number of batches processed previously, is required before the first job in the batch is processed. Both the setup times and job processing times are controllable by allocating a continuously divisible nonrenewable resource to the jobs. The objective is to determine an accepted job sequence, a rejected job set, a partition of the accepted job sequence into batches, and resource allocation that jointly minimize a cost function based on the total delivery dates of the accepted jobs, and the job holding, resource consumption, and rejection penalties. An dynamic programming solution algorithm with running time O (n6) is developed for the problem. It is also shown that the case of the problem with a common setup time can be solved in O (n5) time.

A bicriterion approach to common flow allowances due window assignment and scheduling with controllable processing times

AbstractWe investigate a single‐machine scheduling problem for which both the job processing times and due windows are decision variables to be determined by the decision maker. The job processing times are controllable as a linear or convex function of the amount of a common continuously divisible resource allocated to the jobs, where the resource allocated to the jobs can be used in discrete or continuous quantities. We use the common flow allowances due window assignment method to assign due windows to the jobs. We consider two performance criteria: (i) the total weighted number of early and tardy jobs plus the weighted due window assignment cost, and (ii) the resource consumption cost. For each resource consumption function, the objective is to minimize the first criterion, while keeping the value of the second criterion no greater than a given limit. We analyze the computational complexity, devise pseudo‐polynomial dynamic programming solution algorithms, and provide fully polynomial‐time approximation schemes and an enhanced volume algorithm to find high‐quality solutions quickly for the considered problems. We conduct extensive numerical studies to assess the performance of the algorithms. The computational results show that the proposed algorithms are very efficient in finding optimal or near‐optimal solutions. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 41–63, 2017

Joint optimization of high-speed train timetables and speed profiles: A unified modeling approach using space-time-speed grid networks

This paper considers a high-speed rail corridor that requires high fidelity scheduling of train speed for a large number of trains with both tight power supply and temporal capacity constraints. This research aims to systematically integrate problems of macroscopic train timetabling and microscopic train trajectory calculations. We develop a unified modeling framework using three-dimensional space-time-speed grid networks to characterize both second-by-second train trajectory and segment-based timetables at different space and time resolutions. The discretized time lattices can approximately track the train position, speed, and acceleration solution through properly defined spacing and modeling time intervals. Within a Lagrangian relaxation-based solution framework, we propose a dynamic programming solution algorithm to find the speed/acceleration profile solutions with dualized train headway and power supply constraints. The proposed numerically tractable approach can better handle the non-linearity in solving the differential equations of motion, and systematically describe the complex connections between two problems that have been traditionally handled in a sequential way. We further use a real-world case study in the Beijing-Shanghai high-speed rail corridor to demonstrate the effectiveness and computational efficiency of our proposed methods and algorithms.

Finding optimal solutions for vehicle routing problem with pickup and delivery services with time windows: A dynamic programming approach based on state–space–time network representations

Optimization of on-demand transportation systems and ride-sharing services involves solving a class of complex vehicle routing problems with pickup and delivery with time windows (VRPPDTW). This paper first proposes a new time-discretized multi-commodity network flow model for the VRPPDTW based on the integration of vehicles’ carrying states within space–time transportation networks, so as to allow a joint optimization of passenger-to-vehicle assignment and turn-by-turn routing in congested transportation networks. Our three-dimensional state–space–time network construct is able to comprehensively enumerate possible transportation states at any given time along vehicle space–time paths, and further allows a forward dynamic programming solution algorithm to solve the single vehicle VRPPDTW problem. By utilizing a Lagrangian relaxation approach, the primal multi-vehicle routing problem is decomposed to a sequence of single vehicle routing sub-problems, with Lagrangian multipliers for individual passengers’ requests being updated by sub-gradient-based algorithms. We further discuss a number of search space reduction strategies and test our algorithms, implemented through a specialized program in C++, on medium-scale and large-scale transportation networks, namely the Chicago sketch and Phoenix regional networks.

Minimizing hierarchical routing error

In hierarchical routing schemes, nodes are grouped into clusters at multiple levels, and a given node sees only a summarized view of the entire network. Hierarchical routing introduces error, which is the difference between the hierarchical path length and the optimal path length using flat routing. Since in practice the routing table size at each node is limited, we formulate the constrained optimization problems of finding a hierarchy structure that minimizes either the worst case or average case routing error. We prove results characterizing solutions of these problems, and present dynamic programming solution algorithms and computational results.

Scheduling to minimize the total compression and late costs

We consider a single-machine scheduling model in which the job processing times are controllable variables with linear costs. The objective is to minimize the sum of the cost incurred in compressing job processing times and the cost associated with the number of late jobs. The problem is shown to be NP-hard even when the due dates of all jobs are identical. We present a dynamic programming solution algorithm and a fully polynomial approximation scheme for the problem. Several efficient heuristics are proposed for solving the problem. Computational experiments demonstrate that the heuristics are capable of producing near-optimal solutions quickly. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 67–82, 1998