Resource Constrained Job Scheduling Dissertantenseminar Matthias - - PowerPoint PPT Presentation

Resource Constrained Job Scheduling

Dissertantenseminar Matthias Horn1 G¨ unther R. Raidl1

1 Institute of Logic and Computation, TU Wien, Vienna, Austria,

{horn|raidl}@ac.tuwien.ac.at

Mar 27, 2019

Decision Diagrams (DDs)

◮ well known in computer science for decades

◮ logic circuit design, formal verification, . . .

◮ get popular in combinatorial optimization in the last decade

◮ graphical representation of solutions

• f a combinatorial optimization problem (COP)

◮ weighted directed acyclic multi-graph

with one root node r and one target node t

◮ each r-t path corresponds to a solution of the COP ◮ length of a path coincides with the solution’s objective value

◮ state of the art results could be obtained on several problems

Decision Diagrams (DDs)

r t Exact DD π0 ∈ {1, 2, 3} π1

. . .

πn 1 2 3

Exact DDs

◮ represent precisely the set of feasible solutions of a COP ◮ longest path: corresponds to optimal solution ◮ tend to be exponential in size ⇒ approximate exact DD

Decision Diagrams (DDs)

r t Exact DD π0 ∈ {1, 2, 3} π1

. . .

πn 1 2 3 r t Restricted DD π0 π1

. . .

πn 1 2 3

Restricted DDs

◮ represent subset of feasible solutions of a COP ◮ by removing nodes and edges ◮ length of longest path: corresponds to a primal bound

Decision Diagrams (DDs)

r t Exact DD π0 ∈ {1, 2, 3} π1

. . .

πn 1 2 3 r t Relaxed DD π0 π1

. . .

πn 1 2 3 r t Restricted DD π0 π1

. . .

πn 1 2 3

Relaxed DDs

◮ represent superset of feasible solutions of a COP ◮ by merging nodes ◮ length of longest path: corresponds to an upper bound ◮ discrete relaxation of solution space

Relaxed DDs

◮ discrete relaxation of solution space ◮ usage

◮ to obtain dual bounds ◮ as constraint store in constraint propagation ◮ derivation of cuts in mixed integer programming (MIP) ◮ branch-and-bound: branching on merged nodes ◮ . . .

◮ excellent results on e.g.

◮ set covering (Bergman et al., 2011) ◮ independent set (Bergman et al., 2014) ◮ time dependent traveling salesman (Cire and Hoeve, 2013) ◮ time dependent sequential ordering (Kinable et al. 2017)

DDs and Dynamic Programming

◮ dynamic programming (DP)

◮ controls xi, current state si ◮ transitions: si+1 = φi(si, xi),

i = 1, . . . , n

◮ objective function: f(x) = n

i=1 ci(si, xi)

◮ can be solved recursively

gi(xi) = min

xi∈Xi(si) {ci(si, xi) + gi+1(φi(si, xi))} ,

i = 1, . . . , n

◮ exact DDs are strongly related to DP

◮ J. N. Hooker, Decision Diagrams and Dynamic Programming, 2013. ◮ each DP state is associated with a node in the DD ◮ root node s0, target node sn+1 ◮ arc (si, φi(si, xi)) with cost ci(si, xi) for each control xi ∈ Xi ◮ create a DD based on a DP formulation without solving it ◮ provides recursive formulations of the COP

DDs - Construction Methods

◮ Top-Down Construction (TDC)

◮ compile relaxed DD layer by layer ◮ layer width is limited ◮ if current layer gets too large ⇒ merge nodes

◮ Incremental Refinement (IR)

◮ start with relaxed DD of width one ◮ iteratively refine by splitting nodes and filtering arcs

◮ A∗-based Construction (A∗C)

◮ construct a relaxed DD by a modified A∗ algorithm ◮ the size of the open list is limited by parameter φ ◮ if φ would be exceeded, nodes are merged ◮ for PC-JSOCMSR: obtained smaller DDs with stronger bounds in

shorter time

Resource Constrained Job Scheduling (RCJS)

◮ jobs J = {1, . . . , n} ◮ machines M = {1, . . . , l} ◮ one renewable shared resource ◮ each job j ∈ J has

◮ an assigned machine mj ∈ M ◮ a release time rj, a due time dj and a processing time pj ◮ weight wj ◮ cumulative resource requirement gj ◮ set of preceding jobs Γj

◮ total amount of resource consumed by concurrently executed jobs

is limited by G

◮ objective: minimize the total weighted tardiness

Solution Representation

◮ represented by permutation π of jobs ◮ assuming that π satisfies the precedence constraints ◮ a feasible schedule S(π) from π can be obtained by

◮ assigning start time si for job i in order of π ◮ such that all constraints are satisfied

◮ objective function: f(S(π)) = n i=1 wi max(0, si + pj − dj)

time horizon T = [T min, . . . , T max]

T min = min

r∈J rj and

T max = max

r∈J rj +

• j∈J

pj

Motivation

◮ mining supply chains

◮ transfer minerals from mining sites to ports by rail or road ◮ rail wagons and trucks are limited and have to be shared ◮ materials have to arrive at the ports by specific times, otherwise

demurrage costs must be paid

Earlier Work on RCJS

Singh, G., Ernst, A.T., (2011)

◮ integer linear programming (ILP) formulation ◮ Lagrangian relaxation (LR) based heuristic ◮ a simulated annealing (SA) based approaches ◮ genetic algorithm (GA)

Singh, G., Ernst, A.T., (2012)

◮ Lagrangian Particle Swarm Optimization

Thirduvady, D., Singh, G., Ernst, A.T., (2014)

◮ combining column generation (CG) and LR with ◮ ant colony optimization (ACO)

Thirduvady, D., Singh, G., Ernst, A.T., (2016)

◮ parallelization of ACO and SA

Further related works on RCJS with hard deadlines,. . .

Mixed Integer Programming Formulation

Model

min

• j∈J

T max

• t=T min+1

cjt

• zjt − zjt−1
• zjt = 0

∀j ∈ J, ∀t = T min, . . . , rj + pj − 1

• j∈Jm
• zjt+pj − zjt
• ≤ 1

∀t ∈ T , ∀m ∈ M zjt ≥ zjt−1 ∀j ∈ J, ∀T min + 1 . . . , T max zjT max = 1 ∀j ∈ J zkt ≤ zjt−pk ∀j → k, ∀t ∈ T

• j∈J

gj

• zjt+pj − zjt
• ≤ G

∀t ∈ T

Variables

◮ Decision variables zjt: one if job j is completed at time t or earlier ◮ Costs: cjt = wj max (t − dj, 0), ∀t ∈ T , ∀j ∈ J

Lagrangian Relaxation

Sup-problems for each machine m ∈ M:

Lm(λ) = min

z

• j∈Jm
• t>0

 cjt +

min{pj,t}

• i=1

λt−igj   (zjt − zj,t−1) subject to . . .

Lagrangian function:

L(λ) =

• m∈M

Lm(λ) − G

• t∈T

λt

States and Transitions

DD for RCJS: directed acyclic multigraph M = (V, A) Each node u ∈ V corresponds to a state (P(u), ˆ P(u), t(u), η(u))

◮ set P(u) ⊆ J of jobs that can be scheduled immediately ◮ set ˆ

P(u) ⊆ J of jobs that are already scheduled

◮ vector t(u) = (tr(u))m∈M of earliest times for each machine m ◮ vector η(u) = (ηt(u))t∈T representing the resource consumption at

time t Initial (root) state: r = (J, ∅, (0, . . . , 0), (0, . . . , 0)) An arc (v, w) ∈ A represents a transition from (P(v), ˆ P(v), t(v), η(v)) to (P(u), ˆ P(u), t(u), η(u)) by scheduling a job j ∈ P(v) at its earliest possible time considering t(v)

v u j | cj

States and Transitions

DD for RCJS: directed acyclic multigraph M = (V, A) Each node u ∈ V corresponds to a state (P(u), ˆ P(u), t(u), η(u))

◮ set P(u) ⊆ J of jobs that can be scheduled immediately ◮ set ˆ

P(u) ⊆ J of jobs that are already scheduled

◮ vector t(u) = (tr(u))m∈M of earliest times for each machine m ◮ vector η(u) = (ηt(u))t∈T representing the resource consumption

at time t Initial (root) state: r = (J, ∅, (0, . . . , 0), (0, . . . , 0)) An arc (v, w) ∈ A represents a transition from (P(v), ˆ P(v), t(v), η(v)) to (P(u), ˆ P(u), t(u), η(u)) by scheduling a job j ∈ P(v) at its earliest possible time considering t(v)

v u j | cj

Basic lower bound on the total weighted tardiness (1)

Preprocessing

As long as there exists two jobs j, k ∈ J s.t. j ≪ k and rk < rj + pj ⇒ set rk := rj + pj

Observation

After preprocessing it frequently happens that there are few jobs j with dj < rj.

Naive lower bound

T LB

≪ =

• j∈J

wj max(0, rj + pj − dj)

Basic lower bound on the total weighted tardiness (2)

Idea: sort jobs according to their due times in increasing order and

◮ sum up the total processing time for each job j ∈ J,

T LB1

j

= min

j′∈Jqj

wj max  0,

• j′∈Jqj |dj′≤dj

pj′ − dj  

◮ and the the average resource consumption

T LB2

j

= min

j′∈J wj max

 0,

• j′∈J|dj′≤dj

pj′gj′ G − dj   Lower bound on the total tardiness: T LB

due =

• j∈J

max(T LB1

j

, T LB2

j

)

Basic lower bound on the total weighted tardiness (3)

Baptiste, P., and Pape, C.L., (2005): relax non-preemption assumption of jobs, two steps

◮ (1) compute vector Cm = (C[1], . . . , C[nm]) for each m ∈ M

◮ where C[i] is a lower bound for the i-th smallest completion time in

any schedule

◮ shortest remaining processing time (SRPT):

each time a job becomes available or is completed, a job with the shortest remaining processing time among the available and uncompleted jobs is scheduled

◮ (2) solve assignment problems to get T LB C

=

m∈M T LB Cm

◮ between jobs in Jm and the elements in vector Cm ◮ optimal solution can be computed by the Hungarian algorithm

in cubic time

◮ compute fast lower bound on the assignment problem

see: Baptiste, P., and Pape, C.L., (2005)

First Strike – A∗ algorithm for RCJS

◮ perform A∗ algorithm

◮ on the defined state graph for RCJS ◮ using lower bound max{T LB

≪ , T LB due, T LB C } to guide the search

◮ perform beam search based diving to find primal solutions

◮ (expected) poor performance due to

◮ rather weak lower bounds ◮ high dimensional states ◮ high memory consumption

◮ possible improvements:

◮ using relaxed DDs to compute (hopefully) stronger lower bounds

Multivalued DDs for Sequencing Problems

Cire, A., van Hoeve, W.J., (2013)

Filtering

◮ try to identify infeasible arcs ◮ NP-hard problem ◮ associate states to each node of the DD

States for node u of DD M

◮ All↓ u ⊆ J, jobs scheduled on all paths from r to node u

All↓

v =

• a=(u,v)∈in(v)

(All↓

u ∪ {val(a)}) ◮ Some↓ u ⊆ J, jobs scheduled on some paths from r to node u

Some↓

v =

• a=(u,v)∈in(v)

(Some↓

u ∪ {val(a)})

Multivalued DDs for Sequencing Problems

Cire, A., van Hoeve, W.J., (2013)

Lemma 1

An arc a = (u, v) is infeasible if any of the following conditions hold: val(a) ∈ All↓

u,

|Some↓

u| = l(a)

and val(a) ∈ Some↓

u.

Lemma 2

An arc a = (u, v) is infeasible if any of the following conditions hold: val(a) ∈ All↑

v,

|Some↑

v| = n − l(a)

and val(a) ∈ Some↑

v,

|Some↓

u ∪ {val(a)} ∪ Some↑ v| < n.

Relaxed DD for Sequencing Problems

Cire, A., van Hoeve, W.J., (2013)

Incremental Refinement (IR)

◮ sort jobs according to some priority J∗ = {j∗ 1, . . . , j∗ n} ◮ jobs with high priority

◮ play a greater role in feasibility and optimality ◮ should be assigned to exactly one position in all orderings ◮ for weighted total tardiness: decreasing due times

Theorem 1

Let W > 0. There exists a relaxed MDD M where at least ⌊log2 W⌋ jobs are assigned to exactly one position in all orderings identified by M.

Lemma 7

A job j is assigned to exactly one position in all orderings identified by M if and only if j ∈ Some↓

u \ All↓ u for all nodes u ∈ M.

Relaxed DD for RCJS

State Merger for RCJS

Merging two states u and v: u ⊕ v = (P(u) ∪ P(v), ˆ P(u) ∪ ˆ P(v), (min(tm(u), tm(v))m∈M, (min(ηt(u), ηt(v))t∈T )

Compilation of relaxed DD M

◮ apply IR algorithm (Cire, A., van Hoeve, W.J., (2013)) ◮ sort jobs according to decreasing due times

Relaxed DD for RCJS

State Merger for RCJS

Merging two states u and v: u ⊕ v = (P(u) ∪ P(v), ˆ P(u) ∪ ˆ P(v), cancel out each other (min(tm(u), tm(v))m∈M, (min(ηt(u), ηt(v))t∈T )

Compilation of relaxed DD M

◮ apply IR algorithm (Cire, A., van Hoeve, W.J., (2013)) ◮ sort jobs according to decreasing due times

(too) many jobs with late due times

Results

in most cases: T sp(M) ≈ 0

Relaxed DD for RCJS

State Merger for RCJS

Merging two states u and v: u ⊕ v = (P(u) ∪ P(v), ˆ P(u) ∪ ˆ P(v), cancel out each other (min(tm(u), tm(v))m∈M, (min(ηt(u), ηt(v))t∈T )

Compilation of relaxed DD M

◮ apply IR algorithm (Cire, A., van Hoeve, W.J., (2013)) ◮ sort jobs according to decreasing wj max(0, rj + pj − dj) values

Results

in most cases: T sp(M) ≥ T LB

≪

Decomposition Based Approach for RCJS

Idea: compile relaxed DD Mm for each single machine m ∈ M

States

◮ set P(u) ⊆ J of jobs that can be scheduled immediately ◮ set ˆ

P(u) ⊆ J of jobs that are already scheduled

◮ time marker t of earliest available time

State Merger

u ⊕ v = (P(u) ∪ P(v), ˆ P(u) ∪ ˆ P(v), min(t(u), t(v)))

Lower Bound

T LB

SM =

• m∈M

T sp(Mm)

Decomposition Based Approach for RCJS

Advantages and Disadvantages

◮ on average 10 jobs are assigned to one machine

⇒ size of single relaxed DD is independent of the problem size

◮ in most cases paths have no job repetition ◮ no “cancel out” effect regarding time marker t ◮ shared resource is ignored completely

Improvement

Use Cm to strengthen states of Mm

Empirical Results (no exhaustive tests on the cluster done yet)

◮ usually T LB SM is substantial stronger than T LB ≪ , T LB due, or T LB C ◮ however weaker compared to the LP relaxation of the MIP

A∗ revisited

Idea: use T LB

SM to guide the A∗ search

Current Status

◮ compute T LB SM for each encountered node during the A∗ search ◮ A∗ is able to find the proven optimal solution at least for the

smallest instances (with three machines)

◮ for larger instances: too slow

Future Improvements

◮ create just one relaxed DD for each machine at the beginning ◮ A∗ will keep for each A∗-node references to the corresponding

relaxed DD-nodes

◮ provides constant look up time ◮ use T sp(Mm) as tie breaking criterion

Future work

◮ try to incorporate shared resource ◮ derive solution from shortest paths obtained from Mm, m ∈ M ◮ consider alternative cost structure ◮ ACO-based approach, attach pheromones to arcs of relaxed DD? ◮ find another problem

Thank you for your attention