DMP87 - Scheduling, Timetabling and Routing
Weekly Notes
Lecture 11, Fall 2008

Lecture March 3

We treated the following:

• Iterated Greedy for $Fm\:\vert\:prmu\:\vert\:C_{max}$ (Slides or sections 3, 3.4 excluded, of the paper by Ruiz and Stützle)

• Tabu Search for $Fm\:\vert\:prmu\:\vert\:C_{max}$ (Slides or sections 1-3 of Grabowski and Wodecki)

• Job shop, definition, disjunctive graph representation, and shifting bottleneck heuristic (textbook pages 84 through 93).

The next time:

• Tabu search for $Jm\:\vert\:\:\vert\:C_{max}$ (slides or photocopies available from the Documents section of the Black Board, pages 194-197).

• Job shop generalizations (photocopies available from the Documents section of the Black Board, pages 203-208).

• Resource Constrained Project Scheduling, Definitions (Slides from Lecture 7).

• Discussion of the Assignments below.

Assignment

Formulate the following problems in the form of RCPSP. [The goal of this assignment is gaining confidence with the RCPSP models and start dealing with timetabling.]

Project Management in Construction Industry

A contractor has to complete $n$ activities. The duration of activity $j$ is $p_j$ and it requires a crew of size $W_j$. The activities are not subject to precedence constraints. The contractor has $W$ workers at his disposal and his objective is to complete all $n$ activities in minimum time.

Exam Scheduling

Exams in a college may have different duration. The exams have to be held in a gym with $W$ seats. The enrollment in course $j$ is $W_j$ and all $W_j$ students have to take the exam at the same time. The goal is to develop a timetable that schedules all $n$ exams in minimum time. Consider both the cases in which each student has to attend a single exam as well as the situation in which a student can attend more than an exam.

Course scheduling

In a basic high-school timetabling problem we are given $m$ classes $c_1,\ldots,c_m$, $h$ teachers $a_1,\ldots,a_h$ and $T$ teaching periods $t_1,\ldots,t_T$. Furthermore, we have lectures $i=l_1,\ldots,l_n$. Associated with each lecture is a unique teacher and a unique class. A teacher $a_j$ may be available only in certain teaching periods. The corresponding timetabling problem is to assign the lectures to the teaching periods such that

• each class has at most one lecture in any time period
• each teacher has at most one lecture in any time period,
• each teacher has only to teach in time periods where he is available.

Audit-staff scheduling

A set of jobs $J_1,\ldots, J_g$ are to be processed by auditors $A_1,\ldots,A_m$. Job $J_l$ consists of $n_l$ tasks ($l=1,\ldots,g$). There may be precedence constraints $i_1 \longrightarrow i_2$ between tasks $i_1,i_2$ of the same job. Associated with each job $J_l$ is a release time $r_l$, a due date $d_l$ and a weight $w_l$.

Each task must be processed by exactly one auditor. If task $i$ is processed by auditor $A_k$, then its processing time is $p_{ik}$. Auditor $A_k$ is available during disjoint time intervals $[s_k^\nu,l_k^\nu]$ ( $\nu=1,\ldots,m$) with $l^\nu_k<s^\nu_k$ for $\nu = 1,\ldots,m_k-1$. Furthermore, the total working time of $A_k$ is bounded from below by $H^-_k$ and from above by $H^+_k$ with $H^-_k \leq H^+_k$ ($k=1,\ldots,m$).

We have to find an assignment $\alpha(i)$ for each task $i=1,\ldots, n := \sum_{l=1}^g n_l$ to an auditor $A_{\alpha(i)}$ such that

• each task is processed without preemption in a time window of the assigned auditor

• the total workload of $A_k$ is bounded by $H^-_k$ and $H_k^k$ for $k=1,\ldots,m$.

• the precedence constraints are satisfied,

• all tasks of $J_l$ do not start before time $r_l$, and

• the total weighted tardiness $\sum^g_{l=1} w_lT_l$ is minimized.

(Hint: this case extends the definition of RCPSP by multi-mode case, that is, the possibility that the processing time of an activity $i$ in mode $m$ is given by $p_{im}$ and the per periods usage of renewable resource $k$ is given by $r_{ikm}$. One has to assign a mode to each activity and schedule the activities in the assinged modes).

PS: Funny(?): Finding the longest path, Daniel Barrett.