One Problem, Two Structures, Six Solvers and Ten Years of Personnel Scheduling Louis-Martin Rousseau

One Problem, Two Structures, Six Solvers, and Ten Years of Personnel Scheduling

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!

Louis-‐Mar@n Rousseau CIRRELT -‐ École Polytechnique de Montréal !

ShiI Scheduling: Once upon a @me...• It all started in 1950’s...

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Personnel scheduling: nowadays

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Clopening happens when one works

both a closing and the following opening shift

Outline• GOAL: illustrate that the right structures for a problem can be useful whatever the solu:on technology !

• Using formal languages structures in: – CP, SAT and MIP – Hybrid CP-‐MIP Branch and Bound – Dynamic Programming

• with Large Neighborhood Search • with Column Genera@on

!• Discuss two applica@ons using CP techniques

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!

The mul@-‐ac@vity ShiI Scheduling Problem

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A very extensive surrey of the mono-activity version in: !• Ernst, A. T., Jiang, H., Krishnamoorthy, M., & Sier, D. (2004). Staff

scheduling and rostering: A review of applications, methods and models. European journal of operational research, 153(1), 3-27.

• Ac@vi@es (what employees can do) • Time Periods (15 minutes) • ShiIs

• Star@ng period • End period • Work segments • Rest segments !

• Assign shiIs to employees that meet – work regula@ons – their exper@se – their availabili@es !

• Op@mize – Labor costs – divergence from desired number of employees

Problem descrip@on

Work activities:

Shift 1:

Shift 2:

Rest activities:

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A typical example1. If an employee is working, he must work between 3 hours and 8 hours 2. If a working employee covers at least 6 hours of work ac@vi@es, he must have two 15

minute-‐breaks and a lunch break of 1 hour. 3. If a working employee covers less than 6 hours of work ac@vi@es, he must have a 15

minute break, but no lunch. 4. If performed, the dura@on of any ac@vity a ∈ W is at least 1 hour 5. A break (or lunch) is necessary between two different work ac@vi@es. 6. Work ac@vi@es must be inserted between breaks, lunch and rest stretches. 7. Rest shiIs have to be assigned at the beginning and at the end of the day. 8. Ac@vi@es a ∈ Ft are not allowed to be performed at period t ∈ T.

Minimize labor costs as well as over and under covering costs

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Constraints based on Regular & Context-‐free Grammar languages

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• Pesant, G. (2004). A regular language membership constraint for finite sequences of variables. In Principles and Practice of Constraint Programming–CP 2004 (pp. 482-495). Springer Berlin Heidelberg.

• Beldiceanu, N., Carlsson, M., & Petit, T. (2004). Deriving filtering algorithms from constraint checkers. In Principles and Practice of Constraint Programming–CP 2004 (pp. 107-122). Springer Berlin Heidelberg.

• Sellmann, M. (2006). The theory of grammar constraints. In Principles and Practice of Constraint Programming-CP 2006 (pp. 530-544). Springer Berlin Heidelberg.

• Quimper, C. G., & Walsh, T. (2006). Global grammar constraints. In Principles and Practice of Constraint Programming-CP 2006 (pp. 751-755). Springer Berlin Heidelberg.

• Quimper, C.G., & Walsh, T. (2007). Decomposing global grammar constraints. In Principles and Practice of Constraint Programming–CP 2007 (pp. 590-604). Springer Berlin Heidelberg.

• Kadioglu, S., & Sellmann, M. (2010). Grammar constraints. Constraints, 15(1), 117-144. • Katsirelos, G., Narodytska, N., Walsh, T., (2011), The weighted grammar constraint, Annals of OR, 184:1,179-207 • Gange, G., Stuckey, P. J., & Van Hentenryck, P. (2013). Explaining Propagators for Edge-Valued Decision

Diagrams. In Principles and Practice of Constraint Programming (pp. 340-355).

A regular language constraint• Regular language membership constraint (CP)

– X = <x1,..., xn>, be a finite sequence of variables – Π denote a DFA or NFA – L(Π) be the regular language recognized by Π – aabbaa and ccc are words of L(Π), acab is not.

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Building the correct DFA (or NFA)

Sequence regula@ons Sequence length

Λ

Filtering Regular in CP

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N4 N5 N63

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A simple automaton for rostering

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initial domains!X1 ∈ {a,b,c,o}!X2 ∈ {b,o}!X3 ∈ {a,c,o}!X4 ∈ {a,b,o}!X5 ∈ {a}!

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Forward pass Backward Pass

filtered domains!X1 ∈ {a,b,c,o}!X2 ∈ {o}!X3 ∈ {a,c,o}!X4 ∈ {a,o}!X5 ∈ {a}

Decomposing the Regular Constraint• It is not mandatory to introduce a new propagator, we can simply introduce new variables

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Let Si be the state of the system at transition i:

Initially S1 ∈ {1} S{2..6} ∈ {1,2,3,4,5,6,7}

Every transition in the DFA is given !as a tuple t = < OriginState, Value, DestinationState >

TableConstraint(Si, Xi, Si+1, Π), i : 1..5

Domain consistency of the table constraint will filter S such that!S1 ∈ {1}!S2 ∈ {1,2,3,4}!S3 ∈ {1,5,6,7}!S4 ∈ {1,2,4,5,7}!S5 ∈ {1,2,5,7}!S6 ∈ {2}

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On our example1. If an employee is working, he must work between 3 hours and 8 hours 2. If a working employee covers at least 6 hours of work ac@vi@es, he must have two

15 minute-‐breaks and a lunch break of 1 hour. 3. If a working employee covers less than 6 hours of work ac@vi@es, he must have a 15

minute break, but no lunch. 4. If performed, the dura@on of any ac@vity a ∈ W is at least 1 hour 5. A break (or lunch) is necessary between two different work ac@vi@es. 6. Work ac@vi@es must be inserted between breaks, lunch and rest stretches. 7. Rest shiIs have to be assigned at the beginning and at the end of the day. 8. Ac@vi@es a ∈ Ft are not allowed to be performed at period t ∈ T.

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A Regular language for our example

Replacing sequencing rules (4 to 8)

• A context free grammar is defined by – Σ : alphabet or ac@vi@es (also called terminals) – N: a set of characters or work structures called non terminals – S ∈ N: the star@ng character – P: a set of produc@ons in the form A -‐> w where:

• A is a non terminal • w is a sequence of terminal and non terminal

!

• Example: !!

A sequence of length 3 can only be : aab or abb

The Grammar Global Constraint

CYK Algorithm

S � AB

A � AA

B � BB

A � a

B � b

S

a b a b a b

S B A SA

B

B

A A AB B B

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A

FILTERING

dom(Xi)

The Grammar Graph

�

S

��

� �

BAA

A B

bbaa

�

B

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• The grammar graph (Γ) encapsulates all feasible solutions • A tree in Γ represents one solu:on

Solu@on aab

Solu@on abb

(Γ) is a DNNF

(Γ) is hypergraph and a solution (tree)

is an hyperpath

• We can enrich the grammar descrip@on to include in the descrip@on of a produc@on • Valid set of periods (P) • Valid interval for the span of sub sequence (S) • Cost of choosing a produc@on

!

!

!

• These can be easily incorporated in the CYK algorithm. • Here is the enriched grammar that captures all the rules (1..8)

Modeling SSP with a Grammar Constraint

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14 C.-G. QUIMPER AND L.-M. ROUSSEAU

6. Experimental Results

6.1. Problem Description

We consider the following shift scheduling problem with n employees and m activities

over a period of time discretized into T = 96 time slots of 15 minutes each. During eachtime slot, an employee can either rest, be on a break, lunch, or work on a given activity.

The schedule of an employee must respect the following rules.

1. A break has a duration of 15 minutes.

2. A lunch has a duration of 1 hour.

3. A part-time employee can take one break during his shift.

4. A full-time employee is entitled to one break, one lunch, and another break in this

order.

5. When an employee starts working on an activity, he works for at least one hour on the

same activity.

6. Lunches and breaks are scheduled between two periods of work.

7. An employee must have a break or a lunch before changing activities.

8. Periods of rest are to be scheduled at the beginning or the end of the day.

9. A part-time employee must work a minimum of 3 hours and fewer than 6 hours a day.

10. A full-time employee must work a minimum of 6 hours and a maximum of 8 hours a

day.

11. At specific times of the day, when the business is closed, every employee must either

rest, lunch, or be on a break.

As an example, a full time employee could rest between midnight and 9am, then work

on activity 1 from 9am to 10:30am, break, work on activity 2 from 10:45am to 12:00pm,

lunch, work on activity 3 from 1:00pm to 2:30pm, break, work on activity 4 from 2:45pm

to 5:00pm, and rest until midnight.

We model the rules using the following context-free grammar. Identifiers between de-

limiters ⟨. . .⟩ are terminal symbols.

R → ⟨rest⟩R | ⟨rest⟩ Ai → ⟨activityi⟩Bi Ai | ⟨activityi⟩

Bi

W[4,∞) → Ai for 1 ≤ i ≤ m P → W ⟨break⟩WF → PLP S → RP[13,24]R | RF[30,38]RL → ⟨lunch⟩ ⟨lunch⟩ ⟨lunch⟩ ⟨lunch⟩


for every leaf nodeN(A, i, 1) do1

weight(N(A, i, 1)) ← min{cost(S[i] \ {sr[i]} ∪ {a}, i) | A → a ∈ G}33

for every inner node N(A, i, j) in post-order do4

weight(N(A, i, j)) ← min{weight(N(B, x, y)) + weight(N(C, w, z)) |66

⟨N(B, x, y), N(C, w, z)⟩ ∈7

Children(N(A, i, j))}8

return weight(N(S, 1, T ))9

Algorithm 2: Compute the cost of the best schedule obtained by replacing sr ∈ S by

another sequence.

The local search repeatedly chooses a sequence sr from S and replaces it by the sequenceN that minimizes the cost of the overall schedule. This process continues until it reaches

a local minimum. Notice that the grammar graph only needs to be constructed once. Only

the weights need to be recomputed at each iteration.

4.2. Enhancing Context-Free Languages

Following [29], one can enrich a context-free language by imposing some conditions on

the applicability of a productionA → BC. For instance, one can restrict A to represent a

sequence of a specific length. The non-terminals B and C might also have to meet some

requirements about the length of the sequences they represent. Similarly, a production can

be restrained to be effective only at particular times of the day. We denote a constrained

literal A with APS where P ⊆ N is the set of positions where A can occur in the sequence

and S ⊆ N is the set of valid spans for the subsequence A. If the symbol S or P are

omitted, we assume that the set of valid positions and valid spans is N. A weight w can

also be associated with a production. Each time a production is used in the schedule of

an employee, the weight of the production is added to the weight of the parsing tree. We

indicate the weight w of a production on the arrow as follows: Aw→ BC. One can omit to

mention the weight whenever it is null.

For instance, the production A[2,4]8.9→ BC{2,5,7}

[1,3] can be applied only if the sequence

obtained has a length between 2 and 4 characters, the span of the literalC is between 1 and

3, and the subsequence C appears at positions 2, 5, or 7. The weight associated with this

production is 8.9 and the literal B is unconstrained.

These new restrictions on the productions are useful to model situations where a worker

must work a certain amount of time before changing activities or can have a lunch break

only at lunch time. Weights on production can be used to encode preferences or to model

the case when extra salary must be given to an employee doing some over-time.

Implementing these constraints only require minor modifications to the algorithm. On

line 16 in Algorithm 1, when processing the production APa

Sa

w→ BPb

SbCPc

Sc, we add the

extra-conditions that i ∈ Pa, i ∈ Pb, and i + k ∈ Pc as well as j ∈ Sa, k ∈ Sb, and

j − k ∈ Sc.


for every leaf nodeN(A, i, 1) do1

weight(N(A, i, 1)) ← min{cost(S[i] \ {sr[i]} ∪ {a}, i) | A → a ∈ G}33

for every inner node N(A, i, j) in post-order do4

weight(N(A, i, j)) ← min{weight(N(B, x, y)) + weight(N(C, w, z)) |66

⟨N(B, x, y), N(C, w, z)⟩ ∈7

Children(N(A, i, j))}8

return weight(N(S, 1, T ))9

Algorithm 2: Compute the cost of the best schedule obtained by replacing sr ∈ S by

another sequence.

The local search repeatedly chooses a sequence sr from S and replaces it by the sequenceN that minimizes the cost of the overall schedule. This process continues until it reaches

a local minimum. Notice that the grammar graph only needs to be constructed once. Only

the weights need to be recomputed at each iteration.

4.2. Enhancing Context-Free Languages

Following [29], one can enrich a context-free language by imposing some conditions on

the applicability of a productionA → BC. For instance, one can restrict A to represent a

sequence of a specific length. The non-terminals B and C might also have to meet some

requirements about the length of the sequences they represent. Similarly, a production can

be restrained to be effective only at particular times of the day. We denote a constrained

literal A with APS where P ⊆ N is the set of positions where A can occur in the sequence

and S ⊆ N is the set of valid spans for the subsequence A. If the symbol S or P are

omitted, we assume that the set of valid positions and valid spans is N. A weight w can

also be associated with a production. Each time a production is used in the schedule of

an employee, the weight of the production is added to the weight of the parsing tree. We

indicate the weight w of a production on the arrow as follows: Aw→ BC. One can omit to

mention the weight whenever it is null.

For instance, the production A[2,4]8.9→ BC{2,5,7}

[1,3] can be applied only if the sequence

obtained has a length between 2 and 4 characters, the span of the literalC is between 1 and

3, and the subsequence C appears at positions 2, 5, or 7. The weight associated with this

production is 8.9 and the literal B is unconstrained.

These new restrictions on the productions are useful to model situations where a worker

must work a certain amount of time before changing activities or can have a lunch break

only at lunch time. Weights on production can be used to encode preferences or to model

the case when extra salary must be given to an employee doing some over-time.

Implementing these constraints only require minor modifications to the algorithm. On

line 16 in Algorithm 1, when processing the production APa

Sa

w→ BPb

SbCPc

Sc, we add the

extra-conditions that i ∈ Pa, i ∈ Pb, and i + k ∈ Pc as well as j ∈ Sa, k ∈ Sb, and

j − k ∈ Sc.

24, 40]12, 23]

• Filtering algorithm by Meinolf Sellmann in 2006, Quimper & Walsh 2006 • Decomposi@on by Quimper & Walsh 2007 • Incremental filtering algorithm by Kadogliu & Sellmann 2010. !

• A boolean representa@on of a parsing tree: – A leaf is true if its corresponding assignment is true

– If a AND node is true, both its children must be true

– If a OR node is true, one its children must be true

– if a node is true (OR/AND) one of its parent must be true

– The root is true: N(S,1,3) = 1

!

• This model can be used in CP or SAT.

Filtering the Grammar Constraint

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SAT vs CP, Monoli@c vs Decomposi@ons

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|A| # m GAC SAT SAT Mono Decompsol time bt opt sol time bt opt sol bt sol bt

1 2 4 26.0 2666 507215p

26.0 1998 443546p26.75 28072 26.25 625683

1 3 6 37.25 1128199 36.75 1953562 37.0 34788 37.0 47715771 4 6 38.0 256 84999

p38.0 287 91151

p- 15539 38.0 56488

1 5 5 24.0 153 67376p

24.0 60 40008p

24.0 40163 24.0 79144131 6 6 33.0 98 48638

p33.0 70 40361

p- 11537 33.0 33405

1 7 8 49.5 236066 49.5 682715 49.0 27635 49.0 26637211 8 3 20.5 80 36348

p20.5 44 25502

p21.0 24343 20.5 635589

1 10 9 54.0 202699 54.25 507749 - 9365 - 5194462 1 5 25.0 453 146234 25.0 301 103918

p25.0 1180 25.0 3828461

2 2 10 58.75 313644 59.0 151076 58.0 14887 58.0 21166022 3 6 38.25 236850 38.25 214203 41.0 1419 41.0 2142012 4 11 71.25 230777 69.75 239519 - 9983 - 7749422 5 4 23.75 2945 644496

p23.75 1876 522780

p26.5 25573 26.25 117105

2 6 5 26.75 4831 777572p27.25 2162816 26.75 10681 26.75 1054531

2 8 5 31.5 244837 31.75 391858 32.0 218 31.5 37718312 9 3 19.0 2283 701474

p19.0 1227 481395

p19.25 20473 19.0 45516

2 10 8 55.0 372870 55.0 333520 - 9968 - 909857Table 1. Benchmark problems solved by MiniSat+. GAC SAT: results from MiniSat+ with allCNF clauses; SAT: results from MiniSat+ with all CNF clauses but clause (4); Mono: resultsfrom a CP solver using the monolithic propagator; Decomp: results from a CP solver using thedecomposition; |A|: number of activities; #: problem number; m: number of employees; sol:number of worked hours (boldfonted if best solution found amongst the different methods); time(s): CPU time in seconds. Times are omitted when the search is suspended by a lack of memory;bt: number of backtracks (boldfonted if least backtracks amongst methods that prove optimality);opt: solution was proved optimal. ILog solver did not prove any problems optimal within onehour of computation.

Even though Algorithm 1 can produce a graph with up to O(n3|G|) nodes, wenoticed that in practice many nodes are never created. The size of the resulting DAGis much smaller in practice than the theoretical bound of O(n3|G|). For instance, thegrammar G for problems with one activity can be written in Chomsky normal form in15 productions. The upper bound on the number of and-nodes in the DAG is 15 963

2 =6635520 nodes whilst there were 71796 nodes on average with these instances.

We also tried modelled the schedule of an employee using an automaton. Due tothe constraints on the number of hours a full-time and a part-time employee must work,many states in the automaton needed to be duplicated resulting in an automaton withseveral thousands of states. Moreover, patterns such as those produced by the non-terminals P andW cannot be reused in an automaton without further duplicating states.The DAG based on the regular language ended up much larger than the one producedby the context-free grammar.

Using Lazy-‐Clause Genera@on

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• ShiI constraints are described with a context-‐free grammar, which is then converted into an MDD.

• Coverage constraints are implemented with a BDD decomposi@on. • Solved with LCG with explana@ons over MDD with costs

21

Experimental Results - Shift Scheduling

Table : Comparison of different methods on shift scheduling problems.

Inst. dec

mdd

mdd ev-mdd ev-mdd

I

time fails time fails time fails time fails1,2,4 14.51 39700 3.49 21888 0.20 607 0.17 6351,3,6 11.25 40675 19.00 76348 0.87 4045 0.91 41561,4,6 2.62 7582 0.69 3518 0.11 350 0.27 10771,5,5 0.41 1585 0.52 3955 0.07 239 0.06 2381,6,6 0.40 1412 0.21 1161 0.08 249 0.11 4131,7,8 4.03 13149 2.43 12046 0.73 3838 0.83 42791,8,3 0.85 5002 0.39 3606 0.06 219 0.07 2621,10,9 17.55 44222 19.77 68688 1.23 5046 1.31 74192,1,5 0.14 691 0.24 1490 0.02 78 0.01 452,2,10 — — 131.29 286747 43.62 99583 49.05 1009582,3,6 188.77 187760 144.99 289568 2.39 6443 5.94 136952,4,11 — — 391.59 918438 42.38 111567 92.89 2205682,5,4 25.85 59635 12.18 50340 0.65 1545 0.48 15412,6,5 83.78 104911 30.27 80046 6.18 12100 7.63 160742,8,5 90.28 153331 34.69 110917 4.99 15507 10.02 265652,9,3 6.10 20472 9.17 42105 0.86 1898 0.47 15932,10,8 349.88 303227 95.61 168720 8.85 26331 17.22 37356Total — — 896.53 2139581 113.29 289645 187.44 436874Mean — — 52.74 125857.71 6.66 17037.94 11.03 25698.47Geom. — — 7.92 31465.82 0.86 2667.65 1.03 3428.35

Results for COST-MDD decomposition are omitted; they were awful.Peter J. Stuckey EV-MDDs September 25, 2013 38 / 41

State-‐of-‐the-‐art

for 1-‐2 ac:vi:es

22

!

Solving SSP with Regular and Context-‐free Grammar in Mixed Integer Programming

• Côté, M. C., Gendron, B., Quimper, C. G., & Rousseau, L. M. (2011). Formal languages for integer programming modeling of shift scheduling problems. Constraints, 16(1), 54-76. !

• Côté, M. C., Gendron, B., & Rousseau, L. M. (2011). Grammar-based integer programming models for multiactivity shift scheduling. Management Science, 57(1), 151-163.

Compact assignment formula@on

A Regular language for our example

Replacing sequencing rules (4 to 8)

Deriving a network flow problem

Building a network flow problem

Network flow formula@on

Linking constraints

Derived automa:cally from the automaton

• The model should find a tree in the Γ graph • The lineariza@on of the CFG constraint is simply:

Linearizing a context-‐free grammar

27

�

v�c(u)

av =�

v�p(u)

av ⇤u ⇥ Nor

�

v�c(R)

av = 1

av � 0, ⇤v ⇥ Nand

REGULAR

Results on our examples

28

Table 1: Results for the classical MIP model

LP MIPId LP Value LP Time |C| |V | MIP Value MIP Time

1 138,8952 36,53 29871 4040 172,6670 2142,172 162,9396 104,73 56743 5104 > >3 168,8223 89,06 56743 5104 > >4 131,6560 32,95 45983 4704 152,2240 3610,005 143,7443 63,61 40447 4360 171,9930 3610,016 129,0947 36,65 40447 4360 137,5180 3616,577 148,4681 38,26 45887 4608 > >8 147,2002 90,49 56743 5104 > >9 142,4836 27,15 36175 4156 182,5370 3607,7610 145,9563 36,89 45983 4704 149,1810 3611,72

Table 2: Results for the MIP regular model


1 138,8952 15,95 1491 1856 172,6670 1,032 162,9396 13,32 2719 3976 164,1370 40,093 168,8223 15,70 2719 3976 169,0120 64,644 131,6560 19,15 2183 3144 133,3830 46,395 144,4182 30,81 1915 2728 145,4640 14,036 133,0766 12,34 1915 2728 135,2180 3,287 149,2739 26,55 2183 3144 150,6810 5,998 147,2002 12,81 2719 3976 148,0470 131,779 142,4836 11,84 1759 2416 182,5370 16,1410 146,2410 23,66 2183 3144 147,5030 20,22

22

ASSIGNMENT

Table 3: Results for the partial MIP grammar model


1 138,8952 34,63 1683 2060 172,6665 1,162 162,9396 31,67 3040 4285 163,9210 66,263 168,8223 41,62 3040 4285 170,5663 46,744 131,6560 42,89 2459 3408 133,1414 98,005 144,4182 52,52 2143 2956 145,2850 21,836 133,0766 57,56 2143 2956 135,1286 1,567 149,2739 44,03 2456 3405 150,7675 53,828 147,2002 47,63 3040 4285 148,0467 263,439 142,4836 27,46 1957 2614 182,4833 12,9110 146,2410 37,19 2458 3408 147,6853 19,28

Table 4: Results for the complete MIP grammar model


1 > 3612,35 2727 6356 172,6665 7,422 > 3610,68 35419 190480 > >3 > 3615,64 35419 190480 > >4 > 3614,92 19163 84672 133,0630 1850,385 > 3609,11 10387 41464 145,8830 322,576 > 3618,18 10387 41464 134,8178 130,217 > 3618,83 19163 84672 151,2082 1662,758 > 3611,39 35419 190480 > >9 > 3614,71 5503 21580 182,5370 1015,1010 > 3610,10 19163 84672 147,0870 1313,28

23

GRAMMAR

Removing Symmetries • In the original model of Dantzig the variables actually were defined to capture the number of employees working on each shiI !

• Which removes symmetries in the case where employees are iden:cal !

• Can we ask the same ques@on using formal language based MIP ?

29

NO YES

�

v�c(u)

av =�

v�p(u)

av ⇤u ⇥ Nor

�

v�c(R)

av = 1

av � 0, ⇤v ⇥ Nand

Implicit Grammar in MIP• Every node in the three now generates general integers:

– av = number of @me node and is selected by an employee

– NS = total number of employees needed

Ns

This model was shown to have the Linear Relaxation as mono-activity models of !!• Set Covering formulation of Dantzig!• Implicit models of Aykin, Bechtold &

Jacob, and Rekik.!•

• From 1 to 10 ac@vi@es • number of shiI increases by a factor of 10,000 • number of contraints increases by a factor of 10.

– Grammar based model: • number of variables is less than doubled • number of constraint increases by about 50%

• All problem solved to op:mality in less than 10 minutes

Results on the full mul@-‐ac@vity problem

31

Cote, Gendron, and Rousseau: Grammar-Based Integer Programing Models for Multi-Activity Shift Scheduling

Article submitted to Management Science; manuscript no. MS-0001-1922.65 23

Table 4 Multi-activity problems with the Implicit Grammar model

NbAct NbShifts |C| |V | Time Nopt(10)

2 13404928 70143 18202 423,90 93 67752783 73529 20144 337,14 94 214010944 76915 22092 212,07 95 522350575 80301 24046 182,02 106 1082991744 83687 26002 179,51 107 2006203423 87073 27970 317,31 108 3422303488 90459 29899 274,66 109 5481658719 93845 31821 296,37 1010 8354684800 97231 33799 518,02 10

tion approach for these problems. Their method does not succeed in finding optimal solution for

the single-activity and multi-activity problems. As for our modeling approach, the multi-activity

problems can easily be handled by adding a small group of productions to the one-activity problem

grammar and results show that they can rapidly be solved on almost all the problems available.

Note that the growth of constraints and variables when augmenting the number of work-activities

is much slower than the increase in the number of allowed shifts.

6. Conclusions

In this paper, we present an IP model for optimal shift scheduling problems that di�ers from

the other models in the literature. Our modeling approach uses context-free grammars to model

problem restrictions in the generation of shifts and converts it into an IP model. This model succeed

in finding the optimal explicit solution and leads to the same integrality gap as Dantzig (1954) set

covering model and other well-known models in the literature for the same problem (see Bechtolds

and Jacobs (1990), Aykin (1996) and Rekik et al. (2004)).

The experimentations show that solving times for our model is comparable with solving times

for Aykin (1996) model on classical shift scheduling problems. Moreover, we tested our model in

comparison with an extension of Aykin (1996) suggested by Rekik et al. (2004) on a problem

allowing a large set of shifts. We also show that our model can be solved to optimality e⇥ciently

on the latter problem when up to ten work-activities have labor requirements. To our knowledge,

no implicit model in the literature can solve the multi-activity problems that e⇥ciently.

!

An Hybrid MIP and CP Approach

32

• Salvagnin, D., & Walsh, T. (2012). A hybrid MIP/CP approach for multi-activity shift scheduling. In Principles and Practice of Constraint Programming (pp. 633-646). Springer Berlin Heidelberg.

ShiI Scheduling Problems: Personalized Cases

• Par@culari@es: – List of available employees

– Each employee has individual caracteris@cs

• Availability within the planning horizon • Mul@-‐ac@vity case: skills to perform only a subset of the ac@vi@es

• Problem: • Assign one shiI to each employee;

• while taking into account the constraints on the composi@on of shiIs;

• minimizing the overall cost.

!

• No longer possible to use implicit models… at least not solely !

33

Orbital Shrinking• Given a model contains symmetries, all variables which are symmetric can be replace by a single variable which is equal to their sum. !

!

!

!

!

!

!

!

!

• Thus if X1, X2, X3 are totally equal (the represent iden@cal employee), we replace them by a single variable

34

variable orbits

cons

trai

nt o

rbits

take averages!within orbits

Y 31 =

X

i:1..3

Xi

Orbital Shrinking• This trick only works for convex op@miza@on ! (not MIP in general) • Grammars and Regulars allow to define a convex space !

!

!

!

!

!

!

!

• In that sense, the Implicit Grammar Model was a case of Orbital Shrinking. !

• Now if the employees are not all iden@cal, can we s@ll use this trick ?!?35

Exact Reformulation

convex optimization

assignment

a

b b b

a

a a a

a a

regular/grammar constraint only

An Hybrid MIP/CP approach

36

MIP solver on the shrunken model

Hybrid MIP/CP decomposition

CP solver to check if a shrunken solution can be disaggregated into a feasible solution of the original problem

inner node

pruned by bound

infeasible integer solution

feasible integer solution

A Mutliphase Hybrid MIP/CP Approach

37

Multi-phase approach1.

solve aggregated modeland collect solutions

2.apply MIP heuristic

to collected solutions

3.final run with

real hybrid decomposition

✓ no callbacks!✓ dual reductions on!✓ very fast

could stop here!if gap is small!

✓ callbacks!✓ CP check!✓ dual reductions off

LNS + Grammar DP

Implicit Grammar MIP formula@on

Integrated CP/MIP Branch and Bound

Experimental Results

38

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10

# solved

cpx-reg hybrid grammar

average time

1

10

100

1000

10000

1 2 3 4 5 6 7 8 9 10

cpx-reg hybrid grammar

gap

0.1%

1.0%

10.0%

100.0%

1 2 3 4 5 6 7 8 9 10

cpx-reg hybrid

!

Dynamic Programming 1 Large Neighborhood Searh

39

• Quimper, C. G., & Rousseau, L. M. (2010). A large neighbourhood search approach to the multi-activity shift scheduling problem. Journal of Heuristics, 16(3), 373-392.

A Simple Problem• Lets look at a smaller example:

– There are no more than two consecu@ve work period. – The day starts and finishes with a work period. – Overstaffing and understaffing costs 1 unit

40

Local Search

41

1

0

1

0

0

1

1

0

Marginal costs

Best replacement shiI using Regular

42

Schedule regula@ons Schedule length

1

0

1

1

0 1

1

1

Marginal costs

Best replacement shiI using Grammar

OS14

OX13

OW12OX

12 OW32

OW11 OB

21 OW21 OB

31 OW31 OW

41

Ow11 Ob

21 Ow21 Ob

31 Ow31 Ow

41

AS�XW,114 AS�XW,2

14

AX�WB,113

AX�WB,112 AW�WW,1

12 AW�WW,132

AW�w,111 AB�b,1

21 AW�w,121 AB�b,1

31 AW�w,131 AW�w,1

41

�2

0

0 �3 �20 �1

�2 0 �3 0 �2 �1

�2 0 �3 �2 �1

�2 �5 �3

�5 �3�2

�5

�5

�6

�6

�5

w w wb

43

Marginal costs

1 10 01

1 1 1

1

00

0

1

1 1011

2

2

1

1

1

1

2

1

1

3

Local Search

44

1

0

0 1

A simple large neighborhood search• Generate a random solu@on

• Repeat

– Select an employee and remove its schedule

– Compute the over-‐costs and under-‐costs for every ac@vity

– Find the best replacing solu@on by:

• Solving shortest path in the automaton layered graph (DAG)

• Finding the lightest tree in an AND/OR Graph

– Insert the new schedule in the solu@on

• Un@l a local minimum is reached

• Restart

45

Experimental Results 1 ac@vity, compared with MIP

46

Experimental Results 1 to 10 ac@vi@es

47

Ind runs

Ind runs

1 min runs

1 min runs

Times in milliseconds to construct one shiI

48

!

Dynamic Programming 2 Column Genera@on

• Demassey, S., Pesant, G., & Rousseau, L. M. (2006). A cost-regular based hybrid column generation approach. Constraints, 11(4), 315-333. !

• Côté, M. C., Gendron, B., & Rousseau, L. M. (2013). Grammar-based column generation for personalized multi-activity shift scheduling. INFORMS Journal on Computing, 25(3), 461-474.

Column Genera@on Framework

49

Select among a subset of possible shiIs, those that together fit best the demand

(using a LP set covering formula@on)

Find the shiIs that we haven’t seen that would improve the current solute

and so on…

NEW

SHIFTS

DUAL VALUES How ?

Grammar-‐based Column Genera@on

Set covering model:

Restricted master problem:

Set of ac@vi@es

Set of periods in the planning horizon

Set of allowed shiIs for employee e

Set of employees

Set of allowed shiIs for employee e at column genera@on itera@on k

Integrality constraints relaxed50

dual values

1

0

1

1

0 1

1

1

Grammar-‐based Column Genera@on approach: Solving the subproblem

OS14

OX13

OW12OX

12 OW32

OW11 OB

21 OW21 OB

31 OW31 OW

41

Ow11 Ob

21 Ow21 Ob

31 Ow31 Ow

41

AS�XW,114 AS�XW,2

14

AX�WB,113

AX�WB,112 AW�WW,1

12 AW�WW,132

AW�w,111 AB�b,1

21 AW�w,121 AB�b,1

31 AW�w,131 AW�w,1

41

�2 0 �3 �1�20

51

dual values

116

Tableau 6.2 Comparison with Demassey et al. (2006) approach

Root Node Branch and Price

NbA Time(s) NbIts NbCols NbS(0.01%) Time(s) NbS(1%)

Grammar-based CG

1 0.1 10 257 5 62 102 0.2 16 601 6 100 93 0.6 20 975 6 2074 84 1.1 25 1321 5 2096 95 3.0 46 2321 0 > 2h 106 4.0 47 2457 9 915 107 6.6 47 3117 5 2426 98 8.3 62 3459 7 2163 109 9.6 62 3626 5 1886 710 9.9 43 3405 6 3754 8

Demassey et al. (2006) CG

1 0.4 19 889 8 144 �2 3.7 48 2340 8 394 �3 2.0 52 2550 4 1592 �4 12.5 103 5063 0 > 2h �5 6.2 86 4288 0 > 2h �6 13.8 130 6493 0 > 2h �7 18.4 137 6839 0 > 2h �8 25.4 155 7736 0 > 2h �9 25.9 155 7741 0 > 2h �10 42.0 179 8974 0 > 2h �

Computa@onal Results

52

Regular-based CG

Table 5: Multi-activity problems with the Implicit Grammar model

NbAct NbShifts |C| |V | Time(s) Nopt(10)

2 13404928 18068 69893 409.07 103 67752783 19945 73152 205.38 94 214010944 21822 76417 300.47 105 522350575 23699 79688 146.16 106 1082991744 25576 82961 213.79 107 2006203423 27453 86246 230.88 108 3422303488 29330 89492 257.06 109 5481658719 31207 92731 289.08 1010 8354684800 33084 96026 516.74 10

approach, the multi-activity problems can easily be handled with a few more productions

than in the one-activity case, and results show that they can rapidly be solved on almost all

available instances. Note that the growth in the number of constraints and variables when

increasing the number of work activities is much slower than the increase in the number of

feasible shifts.

Comparison with existing IP formulations based on formal languages. Table 6

presents the times reported by Cote et al. (2009) to solve the one and two work-activities

instances with two employee-based explicit formulations, the IP Regular model, based on a

finite automaton, and the IP Grammar model, based on a context-free grammar. In both

cases, 0-1 assignment variables for each employee are used, instead of the general integer

variables used in model Q. These experiments were run on a 2.4 GHz Dual AMD Opteron

Processor 250 with 3 GB of RAM, using the MIP solver CPLEX 10.0 and a time limit of

3600 seconds. In this experiment, we also used CPLEX 10.0 to solve our implicit

grammar models to make a more fair comparison. The table reports the time in

seconds to obtain an integer solution with a relative IP gap smaller or equal than 1%. The

symbol “>” represents an instance for which that gap could not be reached within the time

limit. The Implicit Grammar column shows the time needed by the implicit grammar

model to reach the 1% relative IP gap limit.

Table 6 illustrates that the two employee-based explicit formulations su⌃er from scal-

23

Implicit Grammar MIP 2+ activities

State-‐of-‐the-‐art for Iden:cal Employess

State-‐of-‐the-‐art for personalized

employees

Used week form of branching

!

Retail Scheduling for Profit… when the objec@ve is non linear

• Chapados, N., Joliveau, M., L’Ecuyer, P., & Rousseau, L. M. Retail Store Scheduling for Profit. European Journal of Operations Research, Forthcoming.

Mo@va@on• How useful are salespersons in retail…

54

Fundamental Accoun@ng Equa@on

ProfitSales x

MarginSalaries

Goal: maximize this

Sales Decomposi@on per Period

Sales Traffic Conversion Rate Average Basket

Nb of shoppers entering the store

Fraction of shoppers that become buyers Amount bought,

knowing that something was bought

Salary Decomposi@on

Salary Hours Worked

Hourly Wage

Commission

• Given the sets :of @me periods T, of employee E and ac@vi@es A • inputs

– Rnt the revenues generated if n people are working at @me t. – Cta the payroll cost of an employee working at @me t

• variables – xte ∈ A specifies the ac@vity performed by employee e at @me t. – yta ∈ {1..|E|} specifies the number of employees performing a at @me t

A Constraint Programming model

Retail Store Workforce Scheduling 5Revenue Increase (%)

Sun

Mon

Tue

Wed

Thu

Fri

Sat

09:00 09:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:30 19:00 19:30 20:00 20:30

−4

−2

0

2

4

6

8

10

Fig. 2. Average revenue increase by day-of-week and time-of-day across all stores covered by thecase study.

maxX

t�T

Rtywt�

X

a�A

Cat y

at (4)

s.t. Regular(�, xe(all t�T )) ⇤e ⇥ E (5)

GCC(x(all e�E)t , A, y(all a�A)

t ) ⇤t ⇥ T (6)yat ⇥ {0, . . . , |E|} ⇤a ⇥ A, t ⇥ T (7)

xet ⇥ A ⇤e ⇥ E, t ⇥ T (8)

The model is based on two sets of variables: xet specifies the activity (among those

in A) performed by employee e at time t, and yat represents the number of employeesperforming each activity a at time t. The objective function (4) maximizes the revenueforecasted at each time period given the chosen number of working employees (thoseforecasted values are stored in table R) minus the salary expenses associated to eachtype of activity (provided in table C). The values in table R are obtained from thestatistical forecasting model (cf. eq. (2)) for working employees; we can assume thatthe revenue associated with other activities is zero. Constaints (5) enforce that each em-ployee’s schedules follows a certain number of regulations, which are captured in � (asproposed in [4] and [10]), constaints (6) link the x and y variables, while constraints (7)–(8) define the domain of all variables. We plan to implement this straighforward modelin the coming weeks in order to present detailed results during the conference.

6 Conclusion

We have proposed a new methodology to forecast to expected revenue function asso-ciated to the presence of a given number of employees in a retail store. The use of aConstraint Program, as opposed to an Integer Program for instance, allows to easilyincorporate this non-linear function into a shift scheduling model. The overall approachwill thus allow to build schedules where employees are considered as a source a revenuerather than a sole expense.

• Given the sets :of @me periods T, of employee E and ac@vi@es A • inputs

– Rnt the revenues generated if n people are working at @me t – Cta the payroll cost of an employee working at @me t

• variables – xaet ∈ {0,1} 1 if ac@vity a is performed by employee e at @me t. – yakt ∈ {0,1} 1 if exactly k employees are performing a at @me t

MIP model

Scheduling over a day

60

Table 7: Estimated relative profit increase �p and relative gap � to the best upper bound, for thesolutions obtained by solving the proposed CP model after 30 seconds and the MIP and CP models after300 seconds of CPU time.

CP after 30 sec. CP and MIP after 300 sec.Instances description �p (%) � (%) �p (%) � (%)

Store #Days #Periods #Empl. CP CP MIP CP MIP CP

1 408 33.8 6 3.4 0.3 3.5 3.5 0.2 0.22 326 32.4 5 2.8 0.2 2.6 2.6 0.1 0.13 372 40.6 5 2.6 0.2 2.6 2.6 0.1 0.14 377 36.7 5 2.9 0.2 3.0 3.0 0.1 0.15 309 40.4 5 2.6 0.2 2.7 2.7 0.1 0.16 204 32.0 6 4.0 0.2 3.4 4.2 0.8 0.07 399 37.5 5 2.7 0.2 2.8 2.8 0.1 0.18 384 41.7 6 3.0 0.3 2.0 3.2 1.4 0.29 241 33.0 5 2.9 0.2 3.0 3.1 0.1 0.010 401 33.4 4 1.3 0.0 1.3 1.3 0.0 0.011 369 42.0 5 2.1 0.2 2.2 2.2 0.2 0.212 381 40.3 5 2.2 0.1 2.2 2.2 0.1 0.013 366 41.4 5 2.2 0.1 2.1 2.3 0.2 0.114 368 42.6 8 2.9 0.5 2.8 3.2 0.6 0.215 184 42.1 9 2.7 0.8 2.4 3.2 1.1 0.3

Average 339.3 38.0 5.6 2.7 0.2 2.6 2.8 0.3 0.1

and

y

wd,t

� 2 8d 2 D, t 2 T

d

, (51)

respectively. The constraints (38) and (47) guarantee that there is at least one full-timeemployee working at any time. Constraints (39) and (48) ensure that an employee cannotwork more than five shifts (days) over a week. The days o↵ are managed di↵erently inthe MIP and CP; the MIP requires the introduction of new variables. For the MIP, con-straints (40) and (41) are added to link the new variables ve

d

and z

e to the variables se,d,tq,a,q

0 ,respectively, so the constraints (42) can guarantee that an employee has two consecutivedays o↵ during the week or one weekend day o↵. Constraints (43) and (44) must alsobe added to define the domain of variables ve

d

and z

e. For CP, only the constraints (49)must be added, as they directly ensure that an employee has two consecutive days o↵ ina week or one weekend day o↵.

5.3. Scheduling over a day

By taking the set D as a single element, the proposed models can be used to constructwork schedules over a day. In this context, the weekly regulations are ignored.

Table 7 compares the expected profit increases provided by solving our models, overthe scheduling solutions used by the retailer, over a single day. These numbers areaverages over the 5089 days of our data. The table gives two types of informations:(i) the estimated relative increase �

p

in expected profit with the schedules constructedby MIP after 300 seconds of CPU time, and by CP after 30 seconds and 300 seconds ofCPU time, compared with the schedules used by the retailer; and (ii) the relative gap� between those MIP and CP solutions and the best upper bound extracted from theMIP solution after 300 seconds. As the solutions provided by solving the MIP with PLA

30

Scheduling over a week

61

Table 8: Evolution of the relative gap � for the solutions identified by CP when building schedules overa week with two hours of CPU time, in comparison with the best upper bound obtained after 24 hours.

Instances description Relative gap bound � (in %)

Store #Weeks #Periods #Empl. after 30 min after 60 min after 120 min

1 37 236 6 2.2 2.0 1.62 23 224 5 1.9 1.7 1.53 21 282 5 3.8 3.6 3.44 34 256 5 2.5 2.3 2.15 17 280 5 4.4 4.0 3.66 15 232 6 2.8 2.5 2.47 38 262 5 3.1 2.7 2.58 38 288 6 3.4 3.3 3.19 14 230 5 2.0 1.8 1.610 36 234 4 2.0 1.9 1.711 22 294 5 3.4 3.3 3.212 23 282 5 2.9 2.7 2.613 22 288 5 3.0 2.8 2.714 18 296 8 1.9 1.7 1.715 12 292 9 8.3 7.2 6.9

Average 24.7 265.1 5.6 3.2 2.9 2.7

generate less profit than the solutions of the full MIP while using the same amount ofCPU time to construct feasible schedules, we always used the full MIP (no PLA) in thissection. For many instances, the MIP failed to return a feasible solution after 30 seconds;for this reason we do not report the MIP results for 30 seconds. The solutions returnedby either MIP or CP after 300 seconds are very accurate. They are nearly optimal witha respective average gap � of only 0.3% and 0.1%, and correspond to estimated profitincreases �

p

of 2.6% and 2.8%, respectively. CP seems to provide a slightly larger profitgrowth than MIP, and also provides good solutions already after 30 seconds. On the otherhand, CP does not provide an upper bound on the optimal value during the resolution.In practice, the CP approach seems to be the most e�cient method overall.

5.4. Scheduling over a week

For the scheduling problem over a week, CP was able to construct feasible weeklyschedules consistently in less than two hours of CPU time, whereas for the MIP, even thecombination with PLA with a single linear piece (MIP-PLA

1

) failed to provide feasiblesolutions for over 30% of the weeks. Table 8 shows the evolution of the gap � betweenthe solutions identified by solving the CP model after 30 minutes, 60 minutes, and 120minutes of CPU time, and the best upper bound extracted after 24 hours of CPU timewith the MIP model. Given that the weekly schedules used historically in the stores donot satisfy the new rules imposed by the retailer, we cannot make a fair comparison withour solutions. This is why we only compare with an upper bound. The quality of theschedules constructed by CP is demonstrated by the small average relative gap (less than� ⇡ 3% after 60 minutes) between the value of these solutions and the correspondingupper bounds (which are likely to be larger than the optimal values).

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An Applica@on to Hydro-‐Québec Call-‐Center Scheduling

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• Pelleau M, Rousseau L-M, L’Écuyer P, Zegal W, Delorme L. (2014), Scheduling of Agents from Forecasted Future Call Arrivals at Hydro-Québec’s Call Centers, In Principles and Practice of Constraint Programming. Springer Berlin Heidelberg.

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a. Impact Based Search

Objective Value with HQ’s tool

Obj

ectiv

eVa

lue

with

CP

b. Phone First

Objective Value with HQ’s tool

M. Pelleau, L-M. Rousseau, P. L’Ecuyer, W. (1. Université de Montréal2. Polytechnique Montréal3. Institut de recherche d’Hydro-Québec)Zegal, L. Delorme Scheduling of Agents May 7, 2014 15 / 17

10 %

20 %

30 %

Improvement in agent staffing levels met

QUESTIONS ?

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Education

One Problem, Two Structures, Six Solvers and Ten Years of Personnel Scheduling Louis-Martin Rousseau