Flow formulations for curriculum-based course timetabling

Abstract

In this paper we present two mixed-integer programming formulations for the curriculum based course timetabling problem (CTT). We show that the formulations contain underlying network structures by dividing the CTT into two separate models and then connect the two models using flow formulation techniques. The first mixed-integer programming formulation is based on an underlying minimum cost flow problem, which decreases the number of integer variables significantly and improves the performance compared to an intuitive mixed-integer programming formulation. The second formulation is based on a multi-commodity flow problem which in general is NP-hard, however, we prove that it suffices to solve the linear programming relaxation of the model. The formulations show competitiveness with other approaches based on mixed-integer programming from the literature and improve the currently best known lower bound on one data instance in the benchmark data set from the second international timetabling competition. Regarding upper bounds, the formulation based on the minimum cost flow problem performs better on average than other mixed integer programming approaches for the CTT.

Notation

In this section we provide an overview of all the notation that we used throughout the paper. We divided the notation into five lists; abbreviations, functions, sets, parameters, variables and flor graph notations. In each list we have ordered the notation alphabetically. For each notation, we have provided a short description and a reference to the page number where the notation is first used.

Abbreviations

$\mathbf {A}$ :

Abbreviation of the Availability $\mathbf {(\mathbf {A})}$ constraint

BMPR10:

The results reported by Burke et al. (2010)

$\mathbf {C}$ :

Abbreviation of the Conflicts $\mathbf {(\mathbf {C})}$ constraint

CCRT13:

The results reported by Cacchiani et al. (2013)

CTT:

Abbreviation of Curriculum-based Course Timetabling

$\mathcal {F}_{LP}$ :

The LP-relaxation of model (63)–(67)

HB11:

The results reported by Hao and Benlic (2011)

$\mathbf {IL}$ :

Abbreviation of the Isolated Lectures $\mathbf {(\mathbf {IL})}$ constraint

ITC2007:

The second international timetabling competition (ITC2007) held in 2007

$\mathbf {L}$ :

Abbreviation of the Lectures $\mathbf {(\mathbf {L})}$ constraint

MIN:

The results obtained by the minimum cost flow based formulation

$\mathbf {MWD}$ :

Abbreviation of the Minimum Working Days $\mathbf {(\mathbf {MWD})}$ constraint

MULT:

The results obtained by the multi-commodity flow based formulation

$\mathbf {RC}$ :

Abbreviation of the Room Capacity $\mathbf {(\mathbf {RC})}$ constraint

$\mathbf {RO}$ :

Abbreviation of the Room Occupancy $\mathbf {(\mathbf {RO})}$ constraint

$\mathbf {RStab}$ :

Abbreviation of the Room Stability $\mathbf {(\mathbf {RStab})}$ constraint

Functions

$(\cdot )^+$ :

A function returning the the input if it is a non-negative value; zero is returned otherwise

$o(\cdot )$ :

The objective value of the solution given as input

Sets

A :

The set of feasible period-room solutions in the multi-commodity flow formulation

$\mathcal {C}$ :

The set of courses

$\mathcal {C}_l$ :

The set of courses that is taught by lecturer $l\in \mathcal {L}$

$\mathcal {C}_q$ :

The set of courses that belongs to curriculum $q\in \mathcal {Q}$

$\mathcal {C}_\gamma$ :

The set of courses that belongs to clique $\gamma \in \Gamma$

$\mathcal {D}$ :

The set of days

$\Gamma$ :

The set of cliques in the course conflict graph

$\mathcal {L}$ :

The set of lecturers

$\mathcal {P}$ :

The set of periods

$\mathcal {P}_d$ :

The set of periods in $\mathcal {P}$ that belongs to day $d\in \mathcal {D}$

$\varPhi _p$ :

The set of periods that are adjacent to the period $p\in \mathcal {P}$

$\mathcal {Q}$ :

The set of curricula

$\mathcal {R}$ :

The set of rooms

$\mathbb {Z}^+$ :

The set of non-negative integers, i.e., $\mathbb {Z}^+=\left\{ x\in \mathbb {Z}\,:\,x\ge 0\right\}$

$\emptyset$ :

The empty set

Parameters

$C_r$ :

The capacity of room $r\in \mathcal {R}$

$F_{c,p}$ :

Binary value indicating whether or not course $c\in \mathcal {C}$ can be assigned to period$p\in \mathcal {P}$

$L_c$ :

The number of lectures to schedule for course $c\in \mathcal {C}$

$M_c$ :

The desired number minimum working days for course $c\in \mathcal {C}$

$S_c$ :

The number of students attending course $c\in \mathcal {C}$

$\mathcal {T}$ :

The maximum number of periods for any day, i.e., $\mathcal {T}=\max _{d\in \mathcal {D}}\left\{ |\mathcal {P}_d|\right\}$

$W^{\mathbf {IL}}$ :

The cost of violating the Isolated Lectures $\mathbf {(\mathbf {IL})}$ constraint

$W^{\mathbf {MWD}}$ :

The cost of violating the Minimum Working Days $\mathbf {(\mathbf {MWD})}$ constraint

$W^{\mathbf {RC}}$ :

The cost of violating the Room Capacity $\mathbf {(\mathbf {RC})}$ constraint

$W^{\mathbf {RStab}}$ :

The cost of violating the Room Stability $\mathbf {(\mathbf {RStab})}$ constraint

Variables

$f_{c,p,r}$ :

The amount of flow on the arc $(c,p)\rightarrow (r,p)$ in the minimum cost flow graph. A Non-negative continuous variable indicating by how big a fraction course $c\in \mathcal {C}$ is assigned to period $p\in \mathcal {P}$ and room $r\in \mathcal {R}$ in the combined model

$f^u_{c,p}$ :

The amount of flow on the arc $(u)\rightarrow (c,p)$ in the minimum cost flow graph

$f^v_{r,p}$ :

The amount of flow on the arc $(r,p)\rightarrow (v)$ in the minimum cost flow graph

$s_{q,p}$ :

Binary variable indicating whether or not curriculum $q\in \mathcal {Q}$ has an isolated lecture in period $p\in \mathcal {P}$

$t_{c,d}$ :

Binary variable indicating whether or not course $c\in \mathcal {C}$ is assigned to at least one period from the set $\mathcal {P}_d$ for day $d\in \mathcal {D}$

$w_{c}$ :

Non-negative integer variable indicating by how many days the Minimum Working Days $\mathbf {(\mathbf {MWD})}$ constraint is violated for course $c\in \mathcal {C}$

$x_{c,p}$ :

Binary variable indicating whether or not course $c\in \mathcal {C}$ has a lecture assigned in period $p\in \mathcal {P}$

$x_{c,p,r}$ :

Binary variable indicating whether or not course $c\in \mathcal {C}$ is assigned to period $p\in \mathcal {P}$ and room $r\in \mathcal {R}$

$y_{c,r}$ :

Integer variable indicating the number of times that course $c\in \mathcal {C}$ is assigned to room $r\in \mathcal {R}$

$z_{c,r}$ :

Binary variable indicating whether or not course $c\in \mathcal {C}$ has at least one lecture assigned in room $r\in \mathcal {R}$

Flow graph notations

$(\cdot )$ :

A node in a graph, e.g., a node corresponding to a course and period pair is denoted (c, p). The source node and sink node of the minimum cost flow graph are denoted (u) and (v)

$(\cdot )\rightarrow (\cdot )$ :

An arc in a graph

$(\cdot )\rightarrow \cdots \rightarrow (\cdot )$ :

A path in a graph