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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. Problem formulation.2. Computational complexity.3. Motivation.4. Related studies. 5. Triangle inequality case.6. Given number of products.7. Perspectives for future research.

1. Problem formulation.R|slij,β|γ, βє{cntn,dscr}, γє{Cmax,Lmax} – 4 versions

sl0i i slij j sljk k

Machine 1

Machine l

Machine m

…

…

…

…

time

n products, m machines

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pli – per unit processing requirement for product i on machine l,

Di ≤Xi≤Bi, Xi – total production of i, q0

li≤xli, xli – production quantity of i on l,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}.

- variables

dem

and

min

lot s

izeseq. and machine dep.

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2. Computational complexity.

TSP Hamiltonian Path of Minimum Weight R1|slij,β|γ βє{cntn,dscr}, γє{Cmax,Lmax}

R1|slij,β|γ - NPO-complete (no constant factor poly. approximation)

∆TSP R1|∆slij,β|γ R1|∆slij,β|γ is 220/219-non-approximable

(∆TSP - Papadimitrou & Vempala, 2006)

(TSP - Orponen and Mannila, 1987)

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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).

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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.

4. Related studies.

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5. Triangle inequality case.

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.

slij+sljk≥slik , l,i,j,k

A

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Matrix of lot sizes X={xli}

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).

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(2mnmax).

Associated with Y:

P(Y,l) – set of product permutations consistent with Y for each machine l.

A

A

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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.

5. Triangle inequality case.

total set

up time

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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.

min lot size

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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.

5. Triangle inequality case.

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).

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Statement 3. R|slij=s,β|γ, βє{cntn,dscr}, γє{Cmax,Lmax}, is NP-hard, even if n=2, q0

li=0, pli=pl, Di=D, Bi=∞, di=d, and any two pli differ by at most a factor of 2.

6. Given number of products.

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, q0

li=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=Machine-Set-For-Product-1, ∑iєX el≥E, ∑iєK\X el≥E).

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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), andC0(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).

no product is assigned

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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.

unusual for batch scheduling problems

Reduced running time:

In the case because

slij=Δslijeach product has at most one lot on each machine

slij=sli order of lots on the same machine is immaterial

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6. Given number of products.

Statement 5. Problem R|slij,dscr|Lmax is solved by DP3 in O(m(nmax)3(Bmax+1)n+1(smax+pmaxBmax+dmax)) time.

DP3 for R|slij,dscr|Lmax :(modification of DP2 for Cmax)

Difference:t≤nmax(pmaxBmax+smax)+dmax.

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7. Perspectives for future research.

LP and ILP models for commercial solvers

Heuristics, metaheuristics

Efficient enumeration techniques

Subexponential algorithms

PTASes, FPTASes