Reservations without slack Reservations with slack Timetabling with one Op. Timetabling w. Operators Outline Educational Timetabling DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Reservations without slack 2. Reservations with slack Lecture 18 Reservations and 3. Timetabling with one Operator Educational Timetabling 4. Timetabling with Operators 5. Educational Timetabling Marco Chiarandini Introduction School Timetabling 2 Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Outline Educational Timetabling Educational Timetabling Timetabling 1. Reservations without slack Educational Timetabling 2. Reservations with slack School/Class timetabling University timetabling 3. Timetabling with one Operator Personnel/Employee timetabling Crew scheduling 4. Timetabling with Operators Crew rostering Transport Timetabling 5. Educational Timetabling Introduction Sports Timetabling School Timetabling Communication Timetabling 3 5
Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Reservations without slack Timetabling w. Operators Timetabling w. Operators Polynomially solvable cases Educational Timetabling Educational Timetabling Interval Scheduling 1. p j = 1 Given: Solve an assignment problem at each time slot m parallel machines (resources) n activities 2. w j = 1 , M j = M , Obj. minimize resources used r j starting times (integers), Corresponds to coloring interval graphs with minimal number of d j termination (integers), colors w j or w ij weight, M j eligibility Optimal greedy algorithm (First Fit): order r 1 ≤ r 2 ≤ . . . ≤ r n without slack p j = d j − r j Step 1 assign resource 1 to activity 1 Task: Maximize weight of assigned activities Step 2 for j from 2 to n do Assume k resources have been used. Examples: Hotel room reservation, Car rental Assign activity j to the resource with minimum feasible value from { 1 , . . . , k + 1 } 6 7 Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Educational Timetabling Educational Timetabling 8 9
Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Outline Educational Timetabling Educational Timetabling 3. w j = 1 , M j = M , Obj. maximize activities assigned 1. Reservations without slack Corresponds to coloring max # of vertices in interval graphs with k colors 2. Reservations with slack Optimal k -coloring of interval graphs: order r 1 ≤ r 2 ≤ . . . ≤ r n 3. Timetabling with one Operator J = ∅ , j = 1 Step 1 if a resource is available at time r j then assign activity j 4. Timetabling with Operators to that resource; include j in J ; go to Step 3 5. Educational Timetabling Step 2 Else, select j ∗ such that C j ∗ = max j ∈ J C j Introduction if C j = r j + p j > C j ∗ go to Step 3 School Timetabling else remove j ∗ from J , assign j in J Step 3 if j = n STOP else j = j + 1 go to Step 1 10 11 Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Reservations with Slack Heuristics Educational Timetabling Educational Timetabling Most constrained variable, least constraining value heuristic | M j | indicates how much constrained an activity is Given: ν it : # activities that can be assigned to i in [ t − 1 , t ] m parallel machines (resources) � � w j Select activity j with smallest I j = f p j , | M j | n activities Select resource i with smallest g ( ν i,t +1 , . . . , ν i,t + p j ) (or discard j if no place free for j ) r j starting times (integers), d j termination (integers), Examples for f and g : w j or w ij weight, � w j � = | M j | M j eligibility f , | M j | p j w j /p j with slack p j ≤ d j − r j g ( ν i,t +1 , . . . , ν i,t + p j ) = max( ν i,t +1 , . . . , ν i,t + p j ) Task: Maximize weight of assigned activities p j ν i,t + l � g ( ν i,t +1 , . . . , ν i,t + p j ) = p j l =1 12 13
Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Outline Timetabling with one Operator Educational Timetabling Educational Timetabling There is only one type of operator that processes all the activities 1. Reservations without slack Example: A contractor has to complete n activities. 2. Reservations with slack The duration of activity j is p j 3. Timetabling with one Operator Each activity requires a crew of size W j . The activities are not subject to precedence constraints. 4. Timetabling with Operators The contractor has W workers at his disposal His objective is to complete all n activities in minimum time. 5. Educational Timetabling Introduction School Timetabling RCPSP Model If p j all the same ➜ Bin Packing Problem (still NP-hard) 14 15 Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Educational Timetabling Educational Timetabling Heuristics for Bin Packing Example: Exam scheduling Exams in a college with same 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 Construction Heuristics minimum time. Best Fit Decreasing (BFD) Each student has to attend a single exam. C max ( FFD ) ≤ 11 9 C max ( OPT ) + 6 First Fit Decreasing (FFD) 9 Local Search: [Alvim and Aloise and Glover and Ribeiro, 1999] Bin Packing model Step 1: remove one bin and redistribute items by BFD In the more general (and realistic) case it is a RCPSP Step 2: if infeasible, re-make feasible by redistributing items for pairs of bins, such that their total weights becomes equal (number partitioning problem) 16 17
Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Outline Educational Timetabling Educational Timetabling [Levine and Ducatelle, 2004] 1. Reservations without slack 2. Reservations with slack 3. Timetabling with one Operator 4. Timetabling with Operators 5. Educational Timetabling Introduction School Timetabling 18 19 Reservations without slack Reservations without slack Reservations with slack Reservations with slack Timetabling with one Op. Timetabling with one Op. Timetabling w. Operators Timetabling w. Operators Timetabling with Operators Educational Timetabling Educational Timetabling There are several operators and activities can be done by an Mapping to Graph-Vertex Coloring operator only if he is available Two activities that share an operator cannot be scheduled at the activities ➜ vertices same time Examples: if 2 activities require the same operators ➜ edges aircraft repairs time slots ➜ colors scheduling of meetings (people ➜ operators; resources ➜ rooms) feasibility problem (if # time slots is fixed) exam scheduling (students may attend more than one exam ➜ optimization problem operators) If p j = 1 ➜ Graph-Vertex Coloring (still NP-hard) 20 21
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