有限資源專案排程問題(The Resource Constrained Project Scheduling Problem, RCPSP)比一般專案排程問題複雜之處在於增加了資源使用量的限制。至今的相關研究大多探討資源以單期配置或以整個規劃時程配置的方式，鮮少研究探討資源配置為時間區間配置。然而，一周若干勞動小時的工時制度或每月編排固定預算的情形等，都是專案管理者經常面臨資源配置為時間區間配置的例子。因此，資源以時間區間配置的專案排程問題絕對是值得研究的方向。
　　本研究建立一套資源配置以時間區間配置之RCPSP混合整數線性規劃模式，排程目標為專案完成時間極小化。此模式放寬了活動工期、資源量及專案完成時間為整數的假設，並且有以下兩項別於其他相關研究的特性：(一)活動排程的分佈不受制於時間區間，亦即活動開始與完成時間可座落於同一時間區間、跨時間區間或涵蓋一個以上的時間區間。(二)活動於各時間區間之資源使用量以該活動於各時間區間所佔工期天數對其總工期之比例而定。
　　在資源以時間區間配置的情形下，若排程目標非專案完成時間極小化，亦可以本研究模式為基礎，加以修正與調整，將其應用於各類型的排程目標。另外，本研究模式亦可推廣至資源以時間區間配置，單一工作站環境下之物料需求規劃(Material Requirements Planning, MRP)問題。由測試結果得本研究模式於中度複雜的參數條件下，可在合理的運算時間內解決一般甚至中大型規模的問題。

　　The Resource Constrained Project Scheduling Problem (RCPSP) is more complicated than the general project scheduling problem owing to the fact that the scheduling of activities of a project must meet the resource constraints. Most researches on RCPSP focus on the renewable resource or nonrenewable resource so far and there are few researches focusing on “partially renewable resource”. However, the project manager often faces this kind of resource like the available labor hours of a week or the budget of a month. Therefore, the project scheduling under partially renewable resource constraints is much worthy of being studied.
　　A mixed-integer linear program (MILP) for the project scheduling under partially renewable resource constraints is presented and the objective is makespan minimization. The integer assumptions on the activity durations, resources and the completion time of project are relaxed. There are two characteristics in this model: (1) The scheduling of activities is not involved by the time intervals, i.e. an activity can start and complete in the same time interval, across time intervals or including time intervals. (2) The resource usage amount of an activity in every time interval is determined by the proportion of its covering periods in every time interval to its duration.
　　This model also allows for the use of a wider variety of objectives than makespan minimization after moderately adjusted under partially renewable resource constraints. Furthermore, this model can be used to solve Material Requirements Planning (MRP) in a job shop environment under partially renewable resource constraints. Using this model, the general or even large size problems can be solve in reasonable computational time under the problem parameters of moderately complicated levels.

1. Ahn T. and Erenguc S.S. (1998). “The resource constrained project scheduling problem with multiple crashable modes: A heuristic procedure.” European Journal of Operational Research, 107(2), pp.250-259.
2. Bandellomi M., Tucci M. and Rinaldi R. (1994). “Optimal resource leveling using non-serial dynamic programming.” European Journal of Operational Research, 78, pp.167-177.
3. Böttcher J., Drexl A., Kolisch R. and Salewski F. (1999). “Project scheduling under partially renewable resource constraints.” Management Science, 45(4), pp.543-559.
4. Brinkmann K. and Neumann K. (1996). “Heuristic procedures for resource constrained project scheduling with minimal and maximal time lags: the resource leveling and minimum project-duration problem.” Journal of Decision Systems, 5, pp.129-155.
5. Davis K.R., Stam A. and Grzybowski R.A. (1992). “Resource constrained project scheduling with multiple objectives: A decision support approach.” Computer & Operation Research, 19(7), pp.657-669.
6. De P., Dunne J., Ghosh J.B. and Wells C.E. (1995). “The discrete time-cost tradeoff problem for project networks.” European Journal of Operational Research, 81, pp.225-238.
7. Demeulemeester E. (1995). “Minimizing resource availability costs in time-limited project networks.” Management Science, 41(10), pp.1590-1598.
8. Elmaghraby S.E. (1977). “Activity network: project planning and control by network models.” Wiley, New York.
9. Elmaghraby S.E. and Herroelen W.S. (1990). “The scheduling of activities to maximize the net present value of projects.” European Journal of Operational Research, 49, pp.35-49.
10. Etgar R., Shtub A. and LeBlanc L.J. (1995). “Scheduling projects to maximize net present value-the case of time-dependent, contingent cash flows.” European Journal of Operational Research, 96, pp.90-96.
11. Icmeli O. and Rom W.O. (1996). “Solving the resource constrained project scheduling problem with Optimization Subroutine Library.” Pergamom, Computers and Operations Research, 23(8), pp.801-817.
12. Icmeli-Tukel O. and Rom W.O. (1997). “Ensuring quality in resource constrained project scheduling.” European Journal of Operational Research, 103, pp.483-496.
13. Kolisch R. and Padman R. (2001). “An integrated survey of deterministic project scheduling.” Omega, The International Journal of Management Science, 29, pp.249-272.
14. Kolisch, R. and Sprecher A. (1996). “PSPLIB - A project scheduling problem library.” European Journal of Operational Research, 96, pp.205-216.
15. Rom W.O., Icmeli-Tukel O. and Muscatello J.R. (2002). “MRP in a job shop environment using a resource constrained project scheduling model.” Omega, The International Journal of Management Science, 30, pp.275-286.
16. Skutella M. (1998). “Approximation algorithm for the discrete time-cost tradeoff problem.” Mathematics of Operations Research, 23(4), pp.909-929.
17. Slowinski R. (1989). “Multiobjective project scheduling under multiple-category resource constraints.” Advances in project scheduling, pp.151-167, Elsevier, Amsterdam.
18. Sprecher A. and Drexl A. (1998). “Multi-mode resource-constrained project scheduling by a simple, general and powerful sequencing algorithm.” European Journal of Operational Research, 107(2), pp.431-450.
19. Talbot F.B. (1982). “Resource-constrained project scheduling with time-resource tradeoffs: the nonpreemptive case.” Management Science, 28(10), pp.1197-1210.