Theory and application of reciprocal transformation of “path problem” and “time float problem”

Abstract

The concept of analytic geometry, i.e., the reciprocal transformation of geometry and algebra, hints a prospect for the reciprocal transformation of the "path problem" and the "time float problem". A reciprocal transformation can be used to solve a complex problem in one field by translating it into a simpler one in another field. In this case, owing to the generalized concept of length, various types of non-path problems such as the optimum allocation problem and equipment replacement problem can be represented as "path problems". A "length network", which is generalized in nature, is translated into a "time network" by changing the meanings of arcs and lengths. Furthermore, "path problems" can be represented as "time float problems" by discovering the relationships of paths in the length network and time floats in the time network. Base on the relationships, "time float problems" also can be represented as "path problems". The relationships are keys to updating the mutual correspondence of path problems and time float problems. The relationships mirror the uniform qualities of networks in various disciplines and fields. We apply such relationships to solve optimum allocation problems, equipment replacement problems, and path problems with required lengths and to analyze anomalies in projects under generalized precedence relations. These applications test the effectiveness of our proposed approach to theoretical and applied researches in various disciplines and fields.

New Quantization Approach for the Anomaly: The Increase in Time Float following Consumption

“…However, according to (9) and (10) and the computations of ℎ and , ℎ and may be changed following the duration prolongation of activity if there are paths from the activity to nodes (ℎ) or ( ). Figures 1-3 show the changed 6 of (6, 9) ∈ following the duration prolongation of activity 5, and the reason is that there is a path = (9) → (4) → (3) → (5) → (6) from the start node (9) of activity 5 to node (6) and the path is a part of the longest path from the beginning node (0) to node (6). Therefore, (20) will lead to erroneous results in the cases dissatisfying the above condition.…”

“…Wiest [5] first discovered some unusual characteristics of activity network under GPRs, which brought the special characteristics of GPRs to the front line of project management [3,[6][7][8][9][10][11][12][13]. In particular, besides Wiest [5], some other authors also focus on anomalies of time floats under GPRs [3,6,7,9,11,12]. Elmaghraby and Kamburowski [3] found another two anomalies in which (1) the reduction (increase) in project completion time is a consequence of prolonging (shortening) the duration of a critical activity (time float is 0) and (2) shortening the duration of an activity may result in an infeasibility of the project.…”

“…This study analyzes the objective of make-span minimization for RCPSP. The anomalies of time floats under GPRs may appear in the case of changing activity durations [3,6,7,12], particularly the duration prolongations of activities. Therefore, the anomalies may affect RCPSP-GPRs involving changeable activity durations, such as the multimode RCPSP (MRCPSP) and RCPSP with activity splitting.…”