Gur Mosheiov

Scheduling problems with a learning effect

Abstract In many realistic settings, the production facility (a machine, a worker) improves continuously as a result of repeating the same or similar activities; hence, the later a given product is scheduled in the sequence, the shorter its production time. This “learning effect” is investigated in the context of various scheduling problems. It is shown in several examples that although the optimal schedule may be very different from that of the classical version of the problem, and the computational effort becomes significantly greater, polynomial-time solutions still exist. In particular, we introduce polynomial solutions for the single-machine makespan minimization problem, and two for multi-criteria single-machine problems and the minimum flow-time problem on parallel identical machines.

Scheduling with general job-dependent learning curves

Abstract Several recent papers focused on the effect of learning on the optimal solution of scheduling problems. We extend the setting studied so far to the case of job-dependent learning curves, that is, we allow the learning in the production process of some jobs to be faster than that of others. Our learning curve approach, which assumes learning takes place as a function of repetition of the production process, is otherwise completely general, and is not based upon any particular model of learning acquisition. We show that in the new, possibly more realistic setting, the problems of makespan and total flow-time minimization on a single machine, a due-date assignment problem and total flow-time minimization on unrelated parallel machines remain polynomially solvable.

Scheduling jobs under simple linear deterioration

Abstract We consider simple linear deterioration of processing times, i.e. Pi = αit, where αi is a job dependent deterioration rate and t > 0 is the jobs starting time. In this environment, we study most classical n-job, non-preemptive, single machine scheduling models, i.e. most commonly used performance measures are considered: makespan, flow-time, total tardiness, number of tardy jobs, etc. We show that all these models remain polynomially solvable.

Parallel machine scheduling with a learning effect

The phenomenon of ‘learning’ has been extensively studied in many different areas of Operational Research. However, the ‘learning effect’ of the producer/processor has rarely been studied in the general context of production scheduling, and has never been investigated in multi-machine scheduling settings. We focus in this paper on flow-time minimization on parallel identical machines. We show that this problem has a polynomial time solution, although the computational effort required is much larger than the effort required for solving the classical version of the problem.

V-shaped policies for scheduling deteriorating jobs

A set of N jobs has to be processed on a single machine. Jobs have the same basic processing time, but the actual processing time of each job grows linearly with its starting time. A (possibly) different rate of growth is associated with each job. We show that the optimal sequence to minimize flow time is V-shaped: Jobs are arranged in descending order of growth rate if they are placed before the minimal growth rate job, and in ascending order if placed after it. Efficient (0(N log N)) asymptotically optimal heuristics are developed. Their average performance is shown to be extremely good: The average relative error over a set of 20-job problems is on the order of 10−5.

The Travelling Salesman Problem with pick-up and delivery

Demand points for pick-up and delivery are located in the plane. A single vehicle of a given capacity performs both pick-ups and deliveries. The objective is to find the shortest feasible tour. A real life application is described. We compare this problem to the classic Travelling Salesman Problem. Based on their similarity we develop a simple heuristic consisting of pick-up and delivery along the travelling salesman tour. Asymptotic behavior and worst case analysis are provided. We then introduce an alternative solution method which is an extension of the well known Cheapest Insertion heuristic. Performance of both heuristics is tested on various-size simulated problems.

Complexity analysis of job-shop scheduling with deteriorating jobs

This paper addresses job-shop scheduling problems with deteriorating jobs, i.e. jobs whose processing times are an increasing function of their starting time. A simple linear deterioration is assumed and our objective is makespan minimization. We provide a complete analysis of the complexity of flow-shops, open-shops and job-shop problems. We introduce a polynomial-time algorithm for the two-machine flow-shop, and prove NP-hardness when an arbitrary number of machines (three and above) is assumed. Similarly, we introduce a polynomial-time algorithm for the two-machine open-shop, and prove NP-hardness when an arbitrary number of machines (three and above) is assumed. Finally, we prove NP-hardness of the job-shop problem even for two machines.

Multi-Machine Scheduling With Linear Deterioration

AbstractWe study multi-machine makespan minimization of deteriorating jobs, i.e. jobs whose processing times are increasing functions of their starting times. We assume simple linear deterioration, and parallel identical machines. The problem is proven to be NP-hard even for two machines. A heuristic and lower bound are introduced and tested numerically. The heuristic is shown to be asymptotically optimal under very general conditions. We also study a non-conventional performance measure relevant to settings with deteriorating jobs, namely, minimum total load on all machines.

Vehicle routing with pick-up and delivery: tour-partitioning heuristics

Abstract Many applications of the classical vehicle routing problem involve pick-up and delivery services between the depot and peripheral locations (warehouses, stores, stations). This paper studies an important version of the vehicle routing problem with pick-up and delivery (the so-called delivery and backhaul problem): delivery in our case refers to transportation of goods from the depot to customers, and pick-up (backhaul) refers to shipment from customers to the depot. The objective is to find a set of vehicle routes that service customers such that vehicle capacity is not violated and the total distance traveled is minimized. Tour partitioning heuristics for solving the capacitated vehicle routing problem are based on breaking a basic tour into disjoint segments served by different vehicles. This idea is adapted for solving the delivery and backhaul problem. Two heuristics that focus on efficient utilization of vehicles’ capacities are introduced, analyzed and tested numerically.

Scheduling problems with two competing agents to minimize minmax and minsum earliness measures

A relatively new class of scheduling problems consists of multiple agents who compete on the use of a common processor. We focus in this paper on a two-agent setting. Each of the agents has a set of jobs to be processed on the same processor, and each of the agents wants to minimize a measure which depends on the completion times of its own jobs. The goal is to schedule the jobs such that the combined schedule performs well with respect to the measures of both agents. We consider measures of minmax and minsum earliness. Specifically, we focus on minimizing maximum earliness cost or total (weighted) earliness cost of one agent, subject to an upper bound on the maximum earliness cost of the other agent. We introduce a polynomial-time solution for the minmax problem, and prove NP-hardness for the weighted minsum case. The unweighted minsum problem is shown to have a polynomial-time solution.