Minimizing task completion time with the execution set method

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Abstract

We consider a partial ordered set (POSET) of assembling operations, with known execution durations, that must be accomplished. The assembling operations can be executed on an acyclic network with an identical set of robots on each conveyer (arc). The number of depots (nodes) is a known integer. Between each pair of depots we can locate only one conveyer. We seek an arrangement of the network and a plan that divides the task operations among the conveyers, minimizing the overall task completion time. We use linear programming optimization, subject to reasonably general rules for distributing the operation-fragments among the conveyers.

Introduction

A manufacturing system of electronic products was erected as an acyclic network of identical conveyers located between depots. Only one conveyer can be installed between a given pair of depots. The transportation of assemblies, sub-assemblies and items can be performed only via the network conveyers. Unlimited types of assemblies, sub-assemblies, and items can be transported on the conveyer simultaneously without slowing the processing speed. However, a standard set of robots is installed lengthwise on each conveyer, executing the assembly one by one. The conveyer speed is regulated in synchronization with the robot motions. The depot functions as a collector, a buffer, and a distributor of the sub-assemblies and items. The time required for collecting or distributing assemblies, sub-assemblies, and items is negligible. Quality inspection determines that distributing items is unallowable before the completion of the batch, i.e., until the whole assembly operation is accomplished. Anticipating significant robot setup durations, the network was originally planned so that each conveyer is allocated for each assembly operation, and the layout of the network was determined by the operations’ precedence priorities. Technical improvements have reduced the robot setups to a negligible duration. Thus, dividing the task operations into fragments that can be executed in parallel on different conveyers is now a reasonable option for seeking a shorter completion time of the assembly task. The objective is to attain the minimal overall task completion time by redistributing the workload among the conveyer layout. However, (1) the start and end of each operation’s workload fragment execution must be set on a chain of conveyers with the corresponding assembly operation’s distributor and collecting depots, and (2) increment of the depots is unallowable.

The Program Evaluation and Review Technique (PERT) developed by Malcolm et al. (1959) and the Critical Path Method (CPM) developed by Kelley and Walker (1959) considered a Directed Acyclic Graph (DAG) with a Partial Ordered Set (POSET) of activities. The CPM enables us to shorten the project completion time by speeding the performance of the project activities which are located on the critical path. This action requires additional budget. Some time-cost tradeoff optimization procedures assume the existence of redistributable resources within the project from the outset (i.e., some of the project activities are planned initially to be accomplished in a shorter time than their normal duration), while others assume an increase in the project budget. In both cases, the performance of some project activities is accelerated, thus, the project is not accomplished in the most economic manner (e.g., Kelley and Walker, 1959, Kelley, 1961, Phillips and Dessouki, 1977, Tufeci, 1982, Monde et al., 1990, Golenko-Ginzburg, 1993, Karamburowski, 1993, Wu and Li, 1994, Laslo, 2003).

An alternative approach for shortening overall task completion time is to execute the task operations in parallel by dividing the task operations, if allowable, into small fragments which can be redistributed within the DAG. The absence of speeded performance enables us to accomplish the task with minimal budget. Parallel processing is a promising approach to meet the computational requirements of a large number of current and emerging applications. Multiprocessor scheduling has been an active research area and, therefore, many different assumptions and terminology are independently suggested (see a survey by Yu-Kwong and Ishfaq, 1999). Early static scheduling research made simplifying assumptions about the architecture of the parallel program and the parallel machine, such as uniform node weights, zero edge weights, and the availability of an unlimited number of processors. However, even with some of these assumptions, the scheduling problem has been proven to be NP-complete, except for a few restricted cases (Garey and Johnson, 1979).

Parallel processing may have a significant impact in manufacturing systems as well. Luh and Lin (1985) claim that it is possible to achieve minimum production time and increase productivity through the use of parallel operations in parts fabrication and the assembly, computation and control of industrial robots.

Keren et al. (2004) studied the problem of distributing the overall workload among the arcs of an n-nodes DAG, with the goal of minimizing the value of the critical path from node N1 (source node) to node Nn (sink node), where the precedence priorities of the task operations are not taken into consideration. While solving this problem by linear programming requires a number of constraints that explodes with the number of nodes with-in the DAG, it was shown that using Dilworth theorem (Cook et al. (1998)) enables us to get an efficient algorithm for the problem of distributing the cumulative spread-out workload within any DAG. The optimal solution to the problem is to equally distribute the overall workload over the maximum cut (max cut) in the graph which separates the nodes N1 and Nn. Since the max cut in any n-node DAG is less than or equal to the max cut in the complete n-node DAG, ⌊n2/4⌋, one can quickly compute a lower bound of the solution to the problem. In practice, these precedence priorities are unavoidable, and continuation of this work, where effective solution is not yet available, is required.

A paper by Chen et al. (1997) considers two improvements over the traditional activity network by including two types of time constraints. The first type is the time window constraint (Solomon, 1987), which assumes that a task operation is to be executed only in a specified time interval. The second one is the time schedule constraint (Chen et al., 1997), which specifies that a task operation can only be executed at one of an ordered schedule of beginning times. Yang and Chen (2000) assumed that time can be treated as repeating cycles where each cycle consists of two categories: (1) some pairs of rest and work windows and (2) a leading number specifying a maximal number of times each pair should iterate. In this context, task operations can be executed within a time window, and cannot be executed outside the time window. In these models, the time windows are defined by a calendar, while we intend to introduce execution sets which are defined by the precedence priorities, i.e., the occurrences of the beginning and the end of these execution sets are variables.

The execution set method presented here delivers significant benefits in many network model problems where the execution set concept is applicable. The solution of the introduced problem is applicable in additional areas, e.g., computing tasks by a network of identical processors and also project management.

Section snippets

The model

A DAG G, activity-on-arc (AOA) network with one starting event (source node) N1 and one terminal event (sink node) Nn denotes the initial network (e.g., see Fig. 1). The integer n > 1 represents a predetermined number of graph nodes, denoting the depots. The m graph arcs Ai,j, 1  i < j  n, denote the location of the identical conveyers between the supplying depot (functioning as a distributor), node Ni, and the delivering depot (functioning as a collector), node Nj. Since G is acyclic it is assumed

The linear programming formulation

  • Decision variables: we introduce two types of decision variables:

    • (1)

      For any task operation Oi,j and arc Ak,l  ES(Oi,j) pair, we introduce a non-negative variable ti,j,k,l which expresses the Oi,j fragment of workload for execution on arc Ak,l.

    • (2)

      The occurrence time Tk, is a variable denoting the time in which all the task operations with the ending event (node) Nk are completely accomplished.

  • Objective: minimizing the overall task completion time, i.e., Min Z = Tn.

  • Constraints: there is a set of

The extended layouts of the DAG

The extension of the DAG’s layout by adding arcs (conveyers), between existing nodes (depots), provides additional opportunities for shortening the overall task completion time.

If a supplemental arc does not provide additional precedence constraints, such arc supplementation may provide improved results for the overall task completion time, but will never worsen it, since it is allowable not to use any arc as a target for the redistributed workload. The supplementation of arcs in a chain, i.e.,

Implementing the method in project management

Implementing the execution set method in project management is considered to be more economic than the time-cost tradeoffs method, since each activity is performed in its normal time and at its respective normal cost, which is the minimal possible cost. The shortening of the overall completion time with the CPM time-cost tradeoffs (Kelley and Walker, 1959, Kelley, 1961) is based on speeded performances that require additional budget. The optimization results introduced in Table 2 demonstrate

Summary

Implementing the execution set method provides a new concept for minimizing the overall task completion time of a directed acyclic graph in some applications and to shorten it in other applications.

For a partial ordered set of assembly operations, the redistribution of the initial directed acyclic graph, and then its extension of adding more identical arcs in a chain, are both linear programming problems, and can be easily solved. The simplicity of the linear programming formulation makes our

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