Pseudo-polynomial Dynamic Programming Algorithm Research Articles

Scheduling with a due-window for acceptable lead-times

Due-dates are often determined during sales negotiations in two stages: (i) in the pre-sale stage, the customer provides a time interval (due-window) of his acceptable due-dates, (ii) in the second stage, the parties agree on the delivery penalties. Thus, the contract reflects penalties of both parts of the sales negotiations: earliness/tardiness penalties of the due-dates (as a function of the deviation from the agreed upon due-window), and earliness/tardiness penalties of the actual delivery times (as a function of the deviation from the due-dates). We model this setting of a two-stage negotiation on a single machine, and reduce the problem to a well-known setting of minimizing the weighted earliness/tardiness with a given (fixed) due-window. We adopt (and correct) a pseudo-polynomial dynamic programming algorithm for this NP-hard problem. The algorithm is extended to a setting of parallel identical machines, verifying that this case remains NP-hard in the ordinary sense. Moreover, an efficient greedy heuristic and a tight lower bound are introduced and tested. Extremely small optimality gaps are obtained in our numerical tests.

Single-machine batch scheduling with job processing time compatibility

We consider scheduling problems with job processing time compatibility on a single unbounded batch machine. Each job's processing time is characterized by an interval. Any number of jobs can be processed in a batch under the condition that the processing time intervals of the jobs in the same batch have a nonempty intersection. The processing time of a batch is given by the left endpoint of the intersection of the processing time intervals of the jobs in the batch. For the makespan minimization problem with individual job release dates, we design a pseudo-polynomial dynamic programming algorithm for the case where the number of distinct release dates is fixed. We also present a class of online algorithms that are 2-competitive and a polynomial-time approximation scheme for the case where the number of release dates is arbitrary. For the problem to minimize the weighted number of tardy jobs under a common due date, we show that it is binary NP-hard and provide a polynomial-time algorithm when the jobs have a common weight.

Integrating two-agent scheduling and order acceptance problems to maximise total revenue by bounding each agent penalty function

This paper integrates the two–agent scheduling problem and the order acceptance and scheduling problem, to form a single problem for further investigation. The idea originates from real life situations in which manufacturers deal with different customers, each having different requests. It is, therefore, assumed in the new problem that two customer (agent) types exist while each has their specific penalty functions. The penalty function of the first agent is the total lateness while that of the second is the number of tardy orders. The objective is maximising the total revenue of accepted orders by bounding each agent's penalty function. For this problem, a pseudo–polynomial dynamic programming algorithm is developed. It was shown that the algorithm is capable of optimally solving all the problem instances up to 50 orders in size. The algorithm is found to be capable of solving 87.15% of the instances with 90 orders.

Two-agent scheduling with agent specific batches on an unbounded serial batching machine

We study a scheduling problem, in which jobs of two agents are performed in agent specific batches on the same serial unbounded batching machine. On this machine, jobs of the same batch complete simultaneously, and the batch processing time is equal to the total processing time of its jobs plus a setup time. Each agent aims at minimizing a function which depends only on the completion times of its jobs. The problem is to find a schedule that minimizes the objective function of one agent, subject to the objective function of the other agent does not exceed a given threshold. The problem appears in optimizing product consolidation operations of a cross-docking distribution center. Polynomial and pseudo-polynomial dynamic programming algorithms are derived for settings with various combinations of the objective functions.

Pseudo-polynomial dynamic programming for an integrated due date assignment, resource allocation, production, and distribution scheduling model in supply chain scheduling

In this study, we consider an integrated due date assignment, production, and batch delivery scheduling problem with controllable processing times for multiple customers in a supply chain. The objective is to minimize the sum of the weighted number of tardy jobs as well as the due date assignment, resource allocation, and batch delivery costs. This model can also be applied when some parts of the jobs are outsourced. The problem is NP-hard. We propose a pseudo-polynomial dynamic programming algorithm to solve this problem, which shows that the problem is ordinary NP-hard. We performed computational tests to evaluate the proposed method.

Single Machine Scheduling with an Availability Constraint and Rejection

This paper considers single machine scheduling with an availability constraint and rejection. It is assumed that the machine is not available for processing during a given time interval. A job is either rejected, in which case a rejection penalty has to be paid, or accepted and processed on the machine. The objective is to minimize the sum of the weighted total completion time of the accepted jobs and the total rejection penalty of the rejected jobs. For this NP-hard problem, we present a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS).

Parallel machines scheduling with deteriorating jobs and availability constraints

We consider parallel machines scheduling problem in which the actual processing time of the job is a proportional function of its starting time and each machine is not available in a specified time period. We consider the non-resumable case. The objective is to minimize the weighted sum of completion times. We show that the general case of the problem is inapproximable unless P = NP and present a pseudo-polynomial dynamic programming algorithm. We also present a fully polynomial-time approximation scheme for the special case of the problem where only one machine is not available in a specified time period.

Minimum weighted number of tardy jobs on an m-machine flow-shop with a critical machine

Abstract We study a flow-shop problem, where each of the jobs is limited to no more than two operations. One of these operations is common for all the jobs, and is performed on the same (”critical”) machine. Reflecting many applications, jobs are assumed to be processed in blocks on the critical machine. All the jobs share a common due-date, and the objective is minimum weighted number of tardy jobs. We prove that the problem is NP-hard. Then we formulate the problem as an integer program, and introduce a pseudo-polynomial dynamic programming algorithm, proving that the problem is NP-hard in the ordinary sense. We also propose an efficient heuristic, which is shown numerically to produce very close-to-optimal schedules. Finally, we show that the special case of identical weights is polynomially solvable.

Scheduling deteriorating jobs under disruption

This paper considers machine scheduling problems with deteriorating jobs under disruption. There are two types of disruption. In the first one we cannot know the disruption in advance and can only start to revise the schedule until the disruption actually occurs. In the second one we can know a disruption is going to occur at some point in the future and can revise the schedule in advance to respond disruption before it occurs. We want to create a new schedule that takes into account both the original objective function and a measure of deviation from the original schedule. The objective is weighted sum of total completion time and total tardiness. We discuss single machine problems and two parallel machine problems. In each case, either a polynomial algorithm or pseudo-polynomial dynamic programming algorithm be provided, respectively.

A job-shop problem with one additional resource type

We consider a job-shop scheduling problem with n jobs and the constraint that at most p<n jobs can be processed simultaneously. This model arises in several manufacturing processes, where each operation has to be assisted by one human operator and there are p (versatile) operators. The problem is binary NP-hard even with n=3 and p=2. When the number of jobs is fixed, we give a pseudopolynomial dynamic programming algorithm and a fully polynomial time approximation scheme (FPTAS). We also propose an enumeration scheme based on a generalized disjunctive graph, and a dynamic programming-based heuristic algorithm. The results of an extensive computational study for the case with n=3 and p=2 are presented.

Optimization Problems and Algorithms in Double-layered Food Packing Systems

In this paper, we discuss lexicographic bi-criteria combinatorial optimization problems arising in two types of double-layered food packing systems, which we call upright and diagonal types. Such food packing systems are known as so-called automatic combination weighers. The first and second layers in a double-layered food packing system consist of n weighing hoppers and n booster hoppers, respectively. Some amount of food such as a green pepper and a handful of potato chips is thrown into each hopper, and it is called an item. The food packing system performs an operation of choosing a subset I' from the set I of the current 2n items to produce a package of the food. Then, the resulting empty hoppers are supplied with next items, and the set I is updated. By repeating the packing operation, a large number of packages are produced one by one. The boosters are just hoppers without weighing function, but the weights of items in the boosters can be known since each type of double-layered food packing systems has its own constructional feature such that they receive next items from the weighing hoppers. The primary objective of lexicographic bi-criteria food packing problems is to minimize the the total weight of chosen items for each package, making the total weight no less than a specified target weight T. The second objective is to maximize the total priority of chosen items for each package so that items with longer durations in hoppers are preferably chosen. The priority of an item is given as its duration in hopper. In this paper, we prove that the lexicographic bi-criteria food packing problems can be solved in O(nT) time by dynamic programming if all input data are integral. We also show the executive efficiency of the pseudo-polynomial dynamic programming algorithms by conducting numerical experiments.

Two-machine flow shop scheduling with common due window to minimize weighted number of early and tardy jobs

This article studies two due window scheduling problems to minimize the weighted number of early and tardy jobs in a two-machine flow shop, where the window size is externally determined. These new scheduling models have many practical applications in real life. However, results on these problems have rarely appeared in the literature because of a lack of structural and optimality properties for solving them. In this article, we derive several dominance properties and theorems, including elimination rules and sequencing rules based on Johnsos order, lower bounds on the penalty, and upper bounds on the window location, which help to significantly trim the search space for the problems. We further show that the problems are NP-hard in the ordinary sense only. We finally develop efficient pseudopolynomial dynamic programming algorithms for solving the problems. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009

Exact methods for single-item capacitated lot sizing problem with alternative machines and piece-wise linear production costs

In this paper, we study a special case of the capacitated lot sizing problem (CLSP), where alternative machines are used for the production of a single-item. The production cost on each machine is assumed to be piece-wise linear with discontinuous steps (step-wise costs). The over-produced finished products can be stored in an unlimited storage space to satisfy future demand. The aim is to achieve optimal production planning without backlogging. This problem can be seen as an integration of production and transportation activities in a multi-plant supply chain structure, where finished goods are sent directly from the plants to the distribution center using capacitated vehicles. For this problem, which we show to be NP-hard, we propose an exact pseudo-polynomial dynamic programming algorithm which makes it NP-hard in the ordinary sense. We also give three mixed integer linear programming (MILP) formulations that we have found in the literature for the simplest case of the CLSP. These formulations are adapted to the multi-machine case with a step-wise cost structure, to which some valid inequalities have been added to improve their efficiency. We then compare the computational time of the dynamic program to that of one MILP which we selected among MILP formulations based on its lower computational time and its lower and upper bound quality.

The coordination of transportation and batching scheduling

We study a coordinated scheduling problem of production and transportation in which each job is transported to a single batching machine for further processing. There are m vehicles that transport jobs from the holding area to the batching machine. Each vehicle can transport only one job at a time. The batching machine can process a batch of jobs simultaneously where there is an upper limit on the batch size. Each batch to be processed occurs a processing cost. The problem is to find a joint schedule of production and transportation such that the sum of the total completion time and the total processing cost is optimized. For a special case of the problem where the job assignment to the vehicles is predetermined, we provide a polynomial time algorithm. For the general problem, we prove that it is NP-hard (in the ordinary sense) and present a pseudo-polynomial time algorithm. A fully polynomial time approximation scheme for the general problem is obtained by converting an especially designed pseudo-polynomial dynamic programming algorithm.

Scheduling a maintenance activity to minimize total weighted completion-time

We study a single machine scheduling problem. The processor needs to go through a maintenance activity, which has to be completed prior to a given deadline. The objective function is minimum total weighted completion time. The problem is proved to be NP-hard, and an introduction of a pseudo-polynomial dynamic programming algorithm indicates that it is NP-hard in the ordinary sense. We also present an efficient heuristic, which is shown numerically to perform well.

Faster Combinatorial Algorithms for Determinant and Pfaffian

Computation of a determinant is a very classical problem. The related concept is a Pfaffian of a matrix defined for skew-symmetric matrices. The classical algorithm for computing the determinant is Gaussian elimination. It needs O(n 3) additions, subtractions, multiplications and divisions. The algorithms of Mahajan and Vinay compute determinant and Pfaffian in a completely non-classical and combinatorial way, by reducing these problems to summation of paths in some acyclic graphs. The attractive feature of these algorithms is that they are division-free. We present a novel algebraic view of these algorithms: a relation to a pseudo-polynomial dynamic-programming algorithm for the knapsack problem. The main phase of Mahajan-Vinay algorithm can be interpreted as a computation of an algebraic version of the knapsack problem, which is an alternative to the graph-theoretic approach used in the original algorithm. Our main results show how to implement Mahajan-Vinay algorithms without divisions, in time $\tilde{O}(n^{3.03})$using the fast matrix multiplication algorithm.

Scheduling parallel dedicated machines with the speeding‐up resource

AbstractWe consider a problem of scheduling jobs on m parallel machines. The machines are dedicated, i.e., for each job the processing machine is known in advance. We mainly concentrate on the model in which at any time there is one unit of an additional resource. Any job may be assigned the resource and this reduces its processing time. A job that is given the resource uses it at each time of its processing. No two jobs are allowed to use the resource simultaneously. The objective is to minimize the makespan. We prove that the two‐machine problem is NP‐hard in the ordinary sense, describe a pseudopolynomial dynamic programming algorithm and convert it into an FPTAS. For the problem with an arbitrary number of machines we present an algorithm with a worst‐case ratio close to 3/2, and close to 3, if a job can be given several units of the resource. For the problem with a fixed number of machines we give a PTAS. Virtually all algorithms rely on a certain variant of the linear knapsack problem (maximization, minimization, multiple‐choice, bicriteria). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

Integrated Routing and Scheduling of Hazmat Trucks with Stops En Route

We consider an integrated routing and scheduling problem in hazardous materials transportation where accident rates, population exposure, and link durations on the network vary with time of day. We minimize risk (accident probability multiplied by exposure) subject to a constraint on the total duration of the trip. We allow for stopping at the nodes of the network. We consider four versions of this problem with increasingly more realistic constraints on driving and waiting periods, and propose pseudopolynomial dynamic programming algorithms for each version. We use a realistic example network to experiment with our algorithms and provide examples of the solutions they generate. The computational effort required for the algorithms is reasonable, making them good candidates for implementation in a decision-support system. Many of the routes generated by our models do not exhibit the circuitous behavior common in risk-minimizing routes. The en route stops allow us to take full advantage of the time-varying nature of accident probabilities and exposure and result in the generation of routes that are associated with much lower levels of risk than those where no waiting is allowed.

A fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date

We develop a fully polynomial-time approximation scheme (FPTAS) for minimizing the weighted total tardiness on a single machine, provided that all due dates are equal. The FPTAS is obtained by converting an especially designed pseudopolynomial dynamic programming algorithm.

Simultaneous minimization of total completion time and total deviation of job completion times

This paper addresses a single-machine scheduling problem with the objective of minimizing a linear combination of total job completion times and total deviation of job completion times from a common due-date. The due-date is assumed to be non-restrictive, i.e., sufficiently large to have no impact on the optimal sequence. When the weights are job-independent, the problem is shown to have a polynomial time solution, and the optimal schedule is fully characterized as a function of the different parameters. When job-dependent weights are assumed, the problem is known to be NP-hard. We introduce a pseudo-polynomial dynamic programming algorithm, indicating that this case is NP-hard in the ordinary sense. The algorithm is shown experimentally to perform extremely well when tested on high-multiplicity instances with up to 1000 jobs.