Multi-Product Lot-Sizing and Scheduling
on Unrelated Parallel Machines
Mikhail Y. Kovalyov (presenter), Belarusian State University
Alexandre Dolgui, Ecole des Mines de S.Etienne
Anton V. Eremeev, Institute of Mathematics, SB RAS, Omsk
1.
2.
3.
4.
5.
6.
7.
Problem formulation.
Computational complexity.
Motivation.
Related studies.
Triangle inequality case.
Given number of products.
Perspectives for future research.
1
1. Problem formulation.
R|slij,β|γ,
βє{cntn,dscr}, γє{Cmax,Lmax} – 4 versions
n products, m machines
pli – per unit processing requirement for product i on machine l,
Di ≤Xi≤Bi,
Xi – total production of i,
q0li≤xli,
xli – production quantity of i on l,
- variables
Ci≤di,
Ci – completion time for i,
Mi – set of eligible machines for i,
Nl – set of eligible products for machine l, nl=|Nl|, nmax=max{nl}.
Machine m
…
…
Machine l
…
Machine 1
sl0i
i
slij
…
j
sljk
k
time
2
2. Computational complexity.
TSP
Hamiltonian Path of Minimum Weight
βє{cntn,dscr}, γє{Cmax,Lmax}
R1|slij,β|γ
R1|slij,β|γ - NPO-complete (no constant factor poly. approximation)
(TSP - Orponen and Mannila, 1987)
∆TSP
R1|∆slij,β|γ
R1|∆slij,β|γ is 220/219-non-approximable
(∆TSP - Papadimitrou & Vempala, 2006)
3
3. Motivation.
Medium-range production scheduling applications in:
• textile industry (Silva,Magalhaes 2006; Taner,Hodgson,King,
Schultz 2007);
• metal production in foundries (dos Santos-Meza,dos Santos,
Arenales 2002; de Araujo,Arenales,Clark 2008);
• multi-product chemical plants (Bitran,Gilbert 1990; Lin,Floudas,
Modi,Juhasz 2002; Shaik,Floudas,Kallrath,Pitz 2007)
slij – cleaning operations;
Non- ∆slij – some chemicals have cleaning effect;
dscr – production of granules in bags or packets;
cntn – continuous production of granules;
Cmax – latest plant completion time minimization;
Lmax – equitable customer treatment w.r.t. due date satisfaction
(if product=customer order).
4
4. Related studies.
Monma, Potts, OR 1989: identical machines, ∆ case, each product i
consists of Di distinct items having their own processing times and
due dates, preemptions allowed (which makes the problem
continuous), Length=O(mn2+Σ Di).
Results: D.P. algorithms for single machine case (no pmtn can be
assumed), NP-hardness for two machine case.
OurLength=O(mn2).
Brucker, Kovalyov, Shafransky, Werner, AOR 1998: discrete case,
Bi =Di , sequence independent setup times.
Results: NP-hardness proofs, polynomial special cases, D.P.
algorithms, (1+ε)-approximation algorithms.
5
5. Triangle inequality case.
A
slij+sljk≥slik ,
l,i,j,k
Property 1. There exists an optimal solution for R|Δslij,β|γ,
βє{cntn,dscr}, γє{Cmax,Lmax}, in which each product is produced
in at most one lot on each machine.
A schedule is fully specified by:
for each machine: a set of products to be manufactured, their
sequence and the corresponding lot sizes.
6
5. Triangle inequality case.
Allocation 0-1 matrix Y={yli} :
yli=1, if product i is manufactured on machine l, yli=0 otherwise.
Feasible Y : {l | yli=1} є Mi & Σyli≥1, i.
#feasibleY=O(2mn ).
max
A
Associated with Y:
P(Y,l) – set of product permutations consistent with Y for each machine l.
Matrix of lot sizes X={xli}
A
X consistent with Y: q0liyli ≤xli, lєMi, Di ≤∑xli≤Bi,
i
Schedule =(Y, π1,…,πm,X)
#(m+1)-tuples (Y, π1,…,πm) = O(2mnmax(nmax!)m).
7
5. Triangle inequality case.
Two-stage solution procedure:
1. Enumeration of Y, and given Y, m-tuples of permutations
(π1,…,πm), πlєP(Y,l), l=1,…,m.
2. Given (m+1)-tuple (Y, π1,…,πm), solve lot-sizing subproblem LP with O(mn) variables:
Minimize Cmax (Lmax), subject to
t(πl,l)+∑iєN_l plixli≤ Cmax, (…- di^l_k≤Lmax), l=1,…,m,
Di ≤∑lєM_ixli≤Bi, i=1,…,n,
q0liyli≤xli≤Diyli, l=1,…,m, i=1,…,n.
Variables: Cmax (Lmax) and {xli}. Rational if β=cntn, integer if
β=dscr.
8
5. Triangle inequality case.
Statement 1. Problem R|Δslij,β|γ, βє{cntn,dscr}, γє{Cmax,Lmax},
can be solved in O(τβ2mnmax(nmax!)m) time, where τβ – solution time
for LP (β=cntn) or ILP (β=dscr) with O(mn) variables and
O(m+n) constraints.
If minimum lot processing times q0li pli and setup times satisfy
certain inequalities (similar to ∆), the two-stage procedure can be
modified to be used for the case, in which ∆ for slij is violated.
9
5. Triangle inequality case.
DP1 for R|Δslij,β|Cmax (modified from Held and Karp 1962, for ∆TSP):
T(l,S,i) – minimum total setup time on machine l for processing
products of set SєNl provided that iєS is processed last.
Initialization: T(l,S,i)=sl0i for S={i}, iєNl, l=1,…,m.
Recursion: T(l,S,i)=minjєS\{i}{T(l,S\{i},j)+slji}
Minimum total setup times T*(l,Y) can be computed in
O(m(nmax)2 2nmax) time.
Statement 2. Problem R|Δslij,β|Cmax is solved by DP1 in
O(τβ2mnmax+m(nmax)2 2nmax) time (reduced by a factor of (nmax!)m
comparing with an enumeration of all permutations).
10
6. Given number of products.
Statement 3. R|slij=s,β|γ, βє{cntn,dscr}, γє{Cmax,Lmax}, is NPhard, even if n=2, q0li=0, pli=pl, Di=D, Bi=∞, di=d, and any two
pli differ by at most a factor of 2.
Proof: Bounded Partition: Given 2k+1 positive integer numbers
e1,...,e2k and E, which satisfy ∑el=2E and E/(k+1)< el<E/(k-1),
l=1,...,2k, is there a subset XєK:={1,...,2k} such that ∑lєX el=E?
Calculate A=∏er. Instance of R|slij=s,β|γ, βє{cntn,dscr},γє{Cmax,Lmax}:
n=2, m=2k, slij=A, q0li=0, pli=A/el, Di=E, Bi=∞, di=2A.
Bounded Partition has solution
Cmax≤2A (Lmax≤0).
(slij=A
each machine l can process at most el units of the same
product within the remaining A available time units
X=MachineSet-For-Product-1, ∑iєX el≥E, ∑iєK\X el≥E).
11
6. Given number of products.
DP2 for R|slij,dscr|Cmax :
(assigns product lots to machines 1,…,m in this order)
Cl(z1,…,zn,j,t) – minimum Cmax value for a partial schedule, in
which zi units of product i, i=1,…,n, are processed on machines
1,…,l, product jєNl is processed last on machine l, and the last
unit of this product completes at time t, t≤nmax(pmaxBmax+smax).
Initialization:C0(z1,…,zn,j,t)=0 for (z1,…,zn,j,t)=(0,…,0), and
C0(z1,…,zn,j,t)=∞ for (z1,…,zn,j,t)≠(0,…,0).
Recursion:Cl(z1,…,zn,j,t)=min
miniєN_{l-1}U{0},t {Cl-1(z1,…,zn,i,t)}, if (j,t)=(0,0),
miniє(N_lU{0})\{j},δє{q^o_{lj},…,z_j}{max{t,Cl(z1,…,
zj-δ,…,zn,i,t-(slij+δplj)}, if (j,t) ≠(0,0).
12
6. Given number of products.
Statement 4. Problem R|slij,dscr|Cmax is solved by DP2 in
O(m(nmax)3(Bmax+1)n+1(smax+pmaxBmax)) time, which is
pseudopolynomial for a given n and linear in m.
Reduced running time:
In the case
because
slij=Δslij
each product has at most
one lot on each machine
slij=sli
order of lots on the same
machine is immaterial
13
6. Given number of products.
DP3 for R|slij,dscr|Lmax :
(modification of DP2 for Cmax)
Difference:
t≤nmax(pmaxBmax+smax)+dmax.
Statement 5. Problem R|slij,dscr|Lmax is solved by DP3 in
O(m(nmax)3(Bmax+1)n+1(smax+pmaxBmax+dmax)) time.
14
7. Perspectives for future research.
LP and ILP models for commercial solvers
Heuristics, metaheuristics
Efficient enumeration techniques
Subexponential algorithms
PTASes, FPTASes
15