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MILP format of the MRCPSP

For a restoration project with a set of restoration tasks, tasks 2 to −1 represent actual restoration tasks. Task 1 and task n are dummy tasks, representing the start and the end of all restoration tasks. The precedence relationships between tasks are represented in the form of a finish-to-start project network. Each task can be processed in one of several different modes ∈ {1, 2, ..., }. For every task , there is a duration and demand of for the type r resource when executing at mode s for every time step. For the dummy start and end tasks, there is only a single mode s = 1. The duration is , and ; the resource demands are and , ∀. The set of renewable resource types is denoted by and the set of non-renewable resource types is denoted by . The resource availability of type is represented by at time step ∀ = 1, ..., , and can be either constant values or nonuniform values over time.

The mixed integer linear programming (MILP) for the multi-mode resource-constrained project scheduling problem (MRCPSP) is presented as follows, based on (Klein 2000; Cheng et al. 2015).

Find

, , , ,

so that the completion time (CT) of all restoration tasks is minimal.

subjected to

,

, ,

, ,

,

where is the binary decision variable to determine whether task finishes with mode at time step ; is the set of restoration tasks ; is the total number of tasks; is an upper bound for the total restoration duration, which can be determined from pre-processing; represents the set of tasks which precedes task ; ,2,...,; and are the earliest finishing time and the latest finishing time of task , respectively.