Analysis of Heuristics for Stochastic Programming: Results for Hierarchical Scheduling Problems

Abstract

PREFACEThis paper -the second in a series by an international group of researcherscontinues recent trends in IIASA research involving studies of hierarchical systems and optimization of stochastic systems. It is a sequel to RR-84-4.In the earlier paper, the authors observed that practical hierarchical planning involves a top-down temporal sequence of decisions at an increasing level of detail and with increasingly accurate information. In this paper they analyze certain two-level problems of machine shop design and scheduling of this form. Emphasis is on proving the asymptotic optimality of approximate and heuristic procedures as the number of tasks in the system becomes large and the random task processing times become small relative to the planning horizon. Such results tend to reinforce the long-held views of practical persons faced with difficult decisions -in sufficiently complex environments suitable rules of thumb can be highly efficient.All the authors are active in the development of computer software for planning and operations management i11 various environments, so that in a very real sense this paper describes theoretical research stemming from practice. M.A.H. DEMPSTER MATHEMATICS OF OPERATIONS RESEARCH Vol. 8, No. 4. November 1983Printed in U.S.A. ANALYSIS OF HEURISTICS FOR STOCHASTIC PROGRAMMING: RESULTS FOR HIERARCHICAL SCHEDULING PROBLEMS*Certain multistage decision problems that arise frequently in opera1ions management planning and control allow a natural formulalion as multistage stochastic programs. In job shop scheduling, for example, Ihe firs! stage could correspond Io the acquisition of resources subject lo probabilisiic information about the jobs 10 be processed, and Ihe second stage lo Ihe aclual allocation of the resources to the jobs given determinisiic information about their processing requirements. For two simple versions of this two-stage hierarchica l scheduling problem, we describe heuristic solulion methods and show that their performance is asymptotically optimal both in expeclaiion and in probability. Introduction.Certain multistage decision problems that arise frequently in operations management planning and control allow a natural formulation as multistage stochastic programming problems. In the context of job shop scheduling, for example, at least two decision stages can usually be recognized. At the aggregate level, one has to decide upon the acquisition of resources ; precise information about what will be required of them, however, is either unavailable because it results from · unknown future developments, or intentionally suppressed to facilitate the decision making. Subsequently, at the detailed level, one has to decide upon the actual allocation of the resources over time, when all the relevant information is at hand.Problems of this type occur in other settings as well, such as the design of distribution and vehicle routing systems [3]. They always involve a sequence of decisions over time, at an increasing level of detail and with increasing information becoming ...

Computational complexity of stochastic programming problems

“…Problems of this type appear in the literature under the name of hierarchical planning problems (see e.g. [2], [17]). …”

Stochastic Integer Programming

“…Usually the solution methods consist of solving the problems at the different levels separately and glue them together. Dempster et al [7,8] gave the first mathematically rigorous analysis of such a hierarchical planning system. They presented the result on the hierarchical scheduling problem exposed in Section 4.…”

Stochastic Integer Programming by Dynamic Programming

“…The two-stage scheduling problem studied in this section was first formulated in (DEMPSTER et al (1983)). At the aggregate level, one has to decide on the number X of identical parallel machines that are to be acquired, while knowing the cost c of a single machine, the number n of jobs that are to be processed, and the probability distribution of the vector w = (wi.…”

“…Previous work on this problem concerned the design and analysis of a two-stage heuristic (DEMPSTER et al ( 1983)). This heuristic sets the number of machines equal to the value of X that minimizes the lower bound VLB(X) = cX+nµ,/ X on EV* (X, w) and assigns the jobs to the machines by a list scheduling rule.…”

“…· Given the formidable difficulty of stochastic integer programming, most research in the area has so far concentrated on the design and analysis of approximation algorithms. This approach is exemplified in (DEMPSTER et al (1981(DEMPSTER et al ( ), (1983; FRENK et al (1984); MARCHETII-SPACCAMELA et al (1984)), where simple heuristics are proposed for a variety of two-stage production and distribution planning problems. Probabilistic analyses of the heuristics then provide exact statements about the quality of the approximations, such as some form of asymptotic optimality.…”