Multiproject Scheduling with Limited Resources: A Zero-One Programming Approach

Abstract

A zero-one (0-1) linear programming formulation of multiproject and job-shop scheduling problems is presented that is more general and computationally tractable than other known formulations. It can accomodate a wide range of iwal-world situations including multiple resource c'.nstraints, due dates, job splitting, resource substitutabiilty, and concurrency and nonconcurrency of job performance requirements. Three possible objective functions are discussed: minimizing total throughput time for all projects: minimizing the time by which all projects are completed (i.e., minimizing makespan); and minimizing total lateness or lateness penalty for all projects. however, another look at the problem from a mathematical programming point of view seems in order, especially in light of recent developments in 0-I programming [7,9,18,25]. INTRODUCTIONThe -;chcduling problens cunsidered here deal with det,,idnng when a job should be processed, given limited e'.ailabilitius of rcources, e.g., rmen, equiptwnL, and facilities. The words .ob adI project will be used Lhvuugliout to denote the two levels of work aggregation being considered. A project consOi.tr of a set of jobs.In Luthei literaturc describing acht-d4ling research, the following equivalent descriptors way be found: Equations are developed to ensure that a schedule meets the foliowing constraints when they are imposed:

Integrated scheduling, assembly area- and part-assignment for large-scale, make-to-order assemblies

“…First, the binary decision variable xj}t = 1, if operation j is started at time instant t, and 0, otherwise (cf. Pritsker et al [34]). That is, X^Q = 1 denotes that Operation j starts at time instant t = 0, ends at time instant t = pj and is processed during periods t = 1,..., pj.…”

Constraint-Propagation-Based Cutting Planes: An Application to the Resource-Constrained Project Scheduling Problem

“…The first one, presented in §3.1, follows the disjunctive graph approach by Balas (1970). The second one, in §3.2, was presumably given first in Pritsker et al (1969).…”

“…The most frequently encountered integer linear formulation of the RCPSP (Christofides et al 1987, Möhring et al 2003, Pritsker et al 1969) is based on time-indexed Boolean variables y jt where y jt = 1 if and only if activity j starts at time t, for each activity j ∈ V and for each time period t = 0 T . Given these variables, the RCPSP can be formulated as follows:…”

On project scheduling with irregular starting time costs

“…Pritsker et al [29] were likely the ÿrst to use variables x jt (j ∈ V; t ∈ I ), where x jt = 1 if job j is started at time t and x jt = 0, otherwise. The problem now reads as follows.…”

“…Instead of using (9), they model temporal constraints in the way originally suggested by Pritsker et al [29]:…”