2.3.5 Project planning

Transcription

1 efinitions:..5 Project planning project consists of a set of m activities with their duration: activity i has duration d i, i = 1,..., 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 Foundations of Operations Research Politecnico di Milano 1

2 Model project can be represented by a directed graph G = (N, ) : arc activity arc length = 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 Foundations of Operations Research Politecnico di Milano

3 Therefore we have: Node v event corresponding to the end of all the activities (i, v) - (v) and hence to the possibile beginning of all those (v, j) + (v). - (v) v + (v). maldi Foundations of Operations Research Politecnico di Milano

4 xample ctivities:,,,, Precedences:,,,, ummy activities with duration = Property The graph G is acyclic (it is a G). y contradiction: if i1 1,..., jk ki there would be a logical contradiction.. maldi Foundations of Operations Research Politecnico di Milano

5 The above graph G can be simplified by contracting some arcs:! When simplifying the graph, pay attention not to introduce unwanted precedence constraints. If this dummy activity is maintained, unwanted is implied, besides,,,,.. maldi Foundations of Operations Research Politecnico di Milano 5

6 rtifical nodes and/or artificial arcs are introduced so that graph G 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 Foundations of Operations Research Politecnico di Milano

7 Problem Given a project defined by its activities (with their durations) and precedence constraints, schedule the activities so as to minimize the overall project duration (the time needed to complete all activities). s t Property Minimum overall project duration = length of a longest path from s to t. Since any s-t path represents a sequence of activities that must be executed in the specified order, its length provides a lower bound on the minimum overall project duration.. maldi Foundations of Operations Research Politecnico di Milano 7

8 ritical path method (PM) etermines a schedule (a plan for executing the activities specifying the order and the alotted time) that minimizes the overall project duration, the slack of each activity, i.e., the amount of time by which its execution can be delayed, without affecting the overall minimum duration of the project. Initialization: onstruct the graph G representing the project and sort the nodes topologically.. maldi Foundations of Operations Research Politecnico di Milano 8

9 Phase I: onsider the nodes by increasing indices and, for each node h N, determine: the earliest time Tmin[h] at which the event associated to node h can occur ( Tmin[n] corresponds to the minimum overall project duration ). Phase II: onsider the nodes by decreasing indices and, for each node h N, determine: 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. Phase III: For each activity (i, j), compute the slack ij = Tmax[j] Tmin[i] d ij.. maldi Foundations of Operations Research Politecnico di Milano 9

10 xample onsider the following project: ct. F G H I uration 1 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 Foundations of Operations Research Politecnico di Milano 1

11 ctivities:,,,,, F, G, H, I Precedence constraints:,,,,, F, G, G, H, F I Graphical model: 1 5 F Precedence 7 H I G 1 8 ummy activities ( duration ). maldi Foundations of Operations Research Politecnico di Milano 11

12 xample 1 5 F 7 H G I 8 ummy activities ( duration ) Phase I: [, ] 1 5 [, ] [ 5, ] 7 [ 8, ] 1 8 [ 1, ] [ 11, ] [ Tmin[i], Tmax[i] ] [, ] [, ]. maldi Foundations of Operations Research Politecnico di Milano 1

13 Phase II: [, ] [, ] [ 5, ] 1 5 [, ] 7 [ 8, 9 ] 1 [, ] [ [ 1, 1, 1 ] 8 [[ 11, 11 ]] [ Tmin[i], Tmax[i] ]. maldi Foundations of Operations Research Politecnico di Milano 1

14 [, ] 1 5 [, ] [ 5, ] [, ] [ 8, 9 ] 7 1 [, ] [ 1, 1 ] 8 [ 11, 11 ] [ Tmin[i], Tmax[i] ] Longest path. maldi Foundations of Operations Research Politecnico di Milano 1

15 Pseudocode of critical path method (PM) Input Output G = (N, ) with n = N, duration d ij associated to each arc (i, j) Tmin[i] and Tmax[i] for i =1,, n GIN Order the nodes topologically; Tmin[1] := ; FOR h:= TO n O Tmin[h] := MX{ Tmin[i] + d ih : (i,h) - (h) }; N-FOR Tmax[n] := Tmin[n]; /* minimum project duration */ FOR h:=n 1 OWNTO 1 O Tmax[h] := MIN{ Tmax[j] d hj : (h,j) + (h) }; N-FOR N O(n+m) O(n+m) O(n+m) Overall complexity: O(m). maldi Foundations of Operations Research Politecnico di Milano 15

16 efinition: n activity (i, j) with zero slack ( ij = ) is critical. [, ] 1 5 [, ] [ 5, ] [ 8, 9 ] 7 1 [ 1, 1 ] 8 [ 11, 11 ] critical activities [ Tmin[i], Tmax[i] ] [, ] [, ] (, ) is not crtitical since = =1 7 = = 1 8 = = 1 Observation: Tmin[i] = Tmax[i] and Tmin[j] = Tmax[j] do not suffice to have: ij = Tmax[j] - Tmin[i] - d ij =!. maldi Foundations of Operations Research Politecnico di Milano 1

17 efinition: critical path is an s t path only composed of critical activities (one such path always exists). [, ] 1 5 [, ] [ 5, ] [, ] 7 [ 8, 9 ] 1 [, ] 8 [ 1, 1 ] [ 11, 11 ] [ Tmin[i], Tmax[i] ] critical path with minimum project duration =11. maldi Foundations of Operations Research Politecnico di Milano 17

18 Gantt charts Introduced in the 191s by Henry Gantt ( ). Provide temporal representations of the project. Gantt chart at earliest : each activity (i, j) starts at time Tmin[i] (i,j) d ij Tmin[i] Tmax[j] H 1 F I G F G H I maldi Foundations of Operations Research Politecnico di Milano 18

19 Gantt chart at latest : each activity (i, j) ends at time Tmax[j] (i,j) d ij Tmin[i] Tmax[j] H 1 F I F G G H I maldi Foundations of Operations Research Politecnico di Milano 19