MIP Relaxation and Large Neighborhood Search for a Multi-Mode - - PowerPoint PPT Presentation

MIP Relaxation and Large Neighborhood Search for a Multi-Mode Resource-Constrained Multi-Project Scheduling Problem

Christian artigues Emmanuel H´ ebrard

LAAS-CNRS, Univ of Toulouse, France

August 29, MISTA 2013, Ghent, Belgium

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 1 / 10

the MMRCMPSP (1/2) P set of projects Aij activity j of project i, i ∈ P, j ∈ Ji Ei set of precedence constraints inside project i ∈ P G set of global (renewable) resources ck capacity of global resource k ∈ G Lρ

i set of local renewable resources for project i ∈ P

Lν

i set of local non-renewable resources for project i ∈ P

cik capacity of resource k ∈ Lρ

i ∪ Lν i

Mij set of modes for activity Aij, i ∈ P, j ∈ Ji dijm duration of activity Aij in mode m rijm demand of activity Aij for global resource k in mode m rρ

ijm demand of activity Aij for renewable local resource k in mode m

rν

ijm demand of activity Aij for non-renewable local res. k in mode m

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 2 / 10

the MMRCMPSP (2/2)

MMRCMPSP Lex(

• i∈P

ci, max

i∈P ci)

ci ≥ sij + dijmij ∀i ∈ P, ∀j ∈ Ji

• Aij∈A(t)

r ρ

ijmijk ≤ ck

∀t ∈ {0, . . . , T}, ∀k ∈ G

• Aij∈A(t)

r ρ

ijmijk ≤ cik

∀t ∈ {0, . . . , T}, ∀i ∈ P, ∀k ∈ Lρ

i

• i∈P,j∈Ji

r ν

ijmijk ≤ cik

∀i ∈ P, ∀k ∈ Lν

i

sij ≥ siq + diqmiq ∀i ∈ P, ∀(q, j) ∈ Ei sij ≥ 0 ∀i ∈ P, ∀j ∈ Ji mij ∈ Mij ∀i ∈ P, ∀j ∈ Ji T: scheduling horizon (upper bound of the makespan for an optimal

i∈P ci)

A(t): set of activities in progress at time t

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 3 / 10

Our approach Some facts Need for a relatively good method within a short time. MMRCMPSP is made of

◮ a multiple knapsack subproblem (mode assignment/non renewable res.

const)

◮ a RCPS subproblem

Mode assignements implicitly induces highly disconnected RCPSPs = ⇒ extremely difficult to solve the problem in an integrated way So we opted for use off-the-shelf solver for initial solution / relaxation computations, constraint handling and (tree-search based) local search MKP solving via integer (0–1) programming RCPSP solving via constraint programming with global scheduling constraints

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 4 / 10

The algorithm

Preliminary definitions S: schedule (start times) M: mode assignment Problem RCMPSP(S, M): solve the RCMPSP with all modes fixed to M and find a solution improving f (S) Problem MMRCMPSP(A, S, M): solve the MMRCMPSP with all modes fixed to M except for activities in A and find a solution improving f (S)

Algorithm MIP&LNS

• 1. Compute an initial solution for MMRCMPSP → M, S

(CP)

• 2. Compute an initial mode assignment → M

(MIP)

• 3. While not time over do

4. Solve RCMPSP(S, M) → S (CP) 5. Select a set A of critical activities according to (M, S) 6. Solve MMRCMPSP(A, S, M)→ M, S (CP)

• 7. end while
• 8. return (M, S)
• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 5 / 10

Compute an initial mode assignment → M

MIP relaxation

MIP α

• i∈P

˜ ci + β ˜ C ˜ C ≥

• i∈P,j∈Ji

rρ

ijmkxijm/ck

∀k ∈ G ˜ ci ≥

• i∈P,j∈Ji

rρ

ijmkxijm/cik

∀i ∈ P, ∀k ∈ Lρ

i

˜ ci ≥ sij + pijmxijm ∀i ∈ P

• i∈P,j∈Ji

rν

ijmkxijmk ≤ cik

∀i ∈ P, ∀k ∈ Lν

i

sij ≥ siq + diqmiq ∀i ∈ P, ∀(q, j) ∈ Ei sij ≥ 0 xijm ∈ {0, 1} ∀i ∈ P, ∀j ∈ Ji, ∀m ∈ Mij

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 6 / 10

Select a set A of critical activities according to (M, S)

Usage Rate

demand time

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 7 / 10

Select a set A of critical activities according to (M, S)

Usage Rate

profile area demand time Usage = profile area / (time * capacity) Usage(A) = same thing over activity A’s time window

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 7 / 10

Select a set A of critical activities according to (M, S)

Resource Efficiency

most economical mode greediest mode current mode

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 8 / 10

Select a set A of critical activities according to (M, S)

Resource Efficiency

most economical mode greediest mode current mode Efficiency(A) =

A′s req in greediest mode−A′s req in current mode A′s req in greediest mode−A′s req in most economical mode

(averaged over all renewable resources)

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 8 / 10

Select a set A of critical activities according to (M, S)

Resource Efficiency

most economical mode greediest mode current mode Efficiency(A) =

A′s req in greediest mode−A′s req in current mode A′s req in greediest mode−A′s req in most economical mode

(averaged over all renewable resources) Arbitrarily partition the schedule in n time slices and select the activities which Span the current slice have the highest distance between their inefficiency and resource usage (Ex: high resource usage during activity time window and low efficiency = ⇒ we have to change activity mode)

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 8 / 10

Results

• CP : IBM CPOptimizer 12.5 tree search (RCMSP: multipoint search, fail

limit set to 100000000.0/nb activities. MMRCMSP: restart search, fail limit set to 100000000.0/(nb activities*slice nb)

• MIP : IBM CPLEX 12.5 (single thread, default parameters, time limit

set to 10s) Results on set A and B Instance Objective A-1.txt 100023 A-2.txt 200041 A-3.txt 50 A-4.txt 6500042 A-5.txt 16300106 A-6.txt 16300087 A-7.txt 64300194 A-8.txt 30500146 A-9.txt 23400116 A-10.txt 108700304 Instance Objective B-1.txt 43600129 B-2.txt 52600156 B-3.txt 66300207 B-4.txt 141600281 B-5.txt 103600249 B-6.txt 109700217 B-7.txt 90700230 B-8.txt 360800524 B-9.txt 552400747 B-10.txt 405300429

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 9 / 10

Register to the ROADEF/EURO 2014 Challenge (SNCF) !

Trains don’t vanish!

http://roadef.challenge.org Increasing traffic, reduced capacities around stations: how to solve this ? Aim: schedule activities of each train between its arrival and its departure, while minimizing Uncovered departures, Conflicts on tracks between trains and Miscellaneous industrial performance costs Integrated “railway system” problem on a local perimeter (Parking on yards, Maintenance

• perations on dedicated facilities, Potential

conflicts between trains on tracks, Assignment

• f platforms to arrivals/departures,

Multiple-unit trains assembling/disassembling)

• C. Artigues, E. H´

ebrard (LAAS-CNRS) MIP&LNS for the MMRCMPSP MISTA 2013 10 / 10