2.3.4 Project planning
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1 .. Project planning project consists of a set of m activities with their duration: activity i has duration d i, i =,..., m. estimate Some pairs of activities are subject to a precedence constraint: i j indicates that j can start only after the end of i. xample ctivities:,,,, Precedences:,,,,. maldi Fondamenti di R.O. Politecnico di Milano
2 project can be represented by a directed graph G = (N, ) : arc activity arc cost = activity duration To account for the precedence constraints, the arcs must be positioned so that: i j there exists a directed path where the arc associated to i preceeds the arc associated to j i j. maldi Fondamenti di R.O. Politecnico di Milano
3 xample ctivities:,,,, Precedences:,,,, ummy activities with duration = N..: The graph is acyclic (without circuits) otherwise we would have a logical incoherence: i,..., jk ki (with topological ordering of the nodes : i < j (i, j) ).. maldi Fondamenti di R.O. Politecnico di Milano
4 ! When simplifying the graph, pay attention not to introduce unwanted precedence constraints mantaining only this dummy activity also implies besides,,,, Node v event corresponding to the end of all the activities (i, v) δ - (v) and hence to the possibile beginning of all those (v, j) δ + (v) δ - (v) v δ + (v). maldi Fondamenti di R.O. Politecnico di Milano
5 Introducing nodes and/or artificial arcs we obtain a graph G that contains a unique initial node s corresponding to the event beginning of the project, contains a unique final node t corresponding to the event end of the project, does not contain multiple arcs (with same endpoints). s t or s t. maldi Fondamenti di R.O. Politecnico di Milano 5
6 Planning a project amounts to schedule and coordinate the activities so as to minimize the overall duration (the time needed to complete all activities). Since any path from s to t represents a sequence of activities that must be executed in the specified order, its cost provides a lower bound on the minimum overall duration of the project. Minimum overall duration = cost of a maximum path from the initial node s to the final node t. maldi Fondamenti di R.O. Politecnico di Milano
7 ritical-path method (PM) For each node (event) h N we determine: the earliest time Tmin[h] at which the event associated to node h can occur ( Tmin[n] corresponds to the minimum overall duration of the project ) the latest time Tmax[h] at which the event associated to node h can occur without delaying the completion of the project beyond the minimum project duration For each activity (i, j), the slack is Tmax[ j] Tmin[i] - d i. maldi Fondamenti di R.O. Politecnico di Milano 7
8 xample onsider the following project ctiv. F G H I uration Predecessors -,, F Precedence constraints:,,,,, F, G, G, H, F I etermine the overall minimum duration of the project and the slack for each activity. maldi Fondamenti di R.O. Politecnico di Milano
9 ctivities:,,,,, F, G, H, I Precedence constraints:,,,,, F, G, G, H, F I Graphical model: 5 F 7 ummy activities H I G. maldi Fondamenti di R.O. Politecnico di Milano 9 ( duration )
10 xample 5 F 7 H G I ummy activities ( duration ) [, ] 5 [, ] [ 5, ] 7 [, ] [, ] [, ] [ Tmin[i], Tmax[i] ] [, ] [, ]. maldi Fondamenti di R.O. Politecnico di Milano
11 [, ] [, ] [ 5, ] 5 7 [, 9 ] [ [,, ] [[, ]] [ Tmin[i], Tmax[i] ] [, ] [, ] [, ] 5 [, ] [ 5, ] [, 9 ] 7 [, ] [, ] [, ]. maldi Fondamenti di R.O. Politecnico di Milano [, ] [ Tmin[i], Tmax[i] ] Maximum cost path
12 ritical-path Method (PM) input output G = (N, ) with n = N, duration d ij associated to each arc (i, j) Tmin[i] e Tmax[i] per i =,, n GIN order the nodes topologically; Tmin[] := ; FOR h:= TO n O Tmin[h] := MX{ Tmin[i] + d ih : (i,h) δ - (h) }; N-FOR Tmax[n] := Tmin[n]; // minimum project duration FOR h:=n OWNTO O Tmax[h] := MIN{ Tmax[j] d hj : (h,j) δ + (h) }; N-FOR N. maldi Fondamenti di R.O. Politecnico di Milano
13 n activity (i, j) with zero slack is critical N..: Tmin[i] = Tmax[i] and Tmin[j] = Tmax[j] do not suffice to have Tmax[j] - Tmin[i] - d ij =! [, ] 5 [, ] [ 5, ] [, ] (, ) is not critical since - - = [, 9 ] 7 [, ]. maldi Fondamenti di R.O. Politecnico di Milano [, ] [, ] [ Tmin[i], Tmax[i] ] The critical activities are in red Slack of (, 7) = = Slack of (, ) = - - =
14 path from to is critical if all its arcs are critical (one such path always exists). [, ] 5 [, ] [ 5, ] [, 9 ] 7 [, ] [, ] [ Tmin[i], Tmax[i] ] [, ] [, ] critical path with minimum project duration =. maldi Fondamenti di R.O. Politecnico di Milano
15 Gantt diagrams Provide temporal representations of the project iagram at earliest : each activity (i, j) starts at time Tmin[i] (i,j) d ij Tmin[i] Tmax[j] H F I G F G H I. maldi Fondamenti di R.O. Politecnico di Milano 5
16 iagram at latest : each activity (i, j) ends at time Tmax[j] (i,j) d ij Tmin[i] Tmax[j] H F I F 5 9 G G H I. maldi Fondamenti di R.O. Politecnico di Milano
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