markov mills, reliable rolls and monte carlo mines ... · introduction markov model reliability...

IntroductionMarkov Model

Reliability ModelSimulationProspects

Markov Mills, Reliable Rolls and Monte CarloMines: Minimizing the Operating Costs of

Grinding Mills

Matthias Rupp, Wolfgang Mergenthaler, Bernhard Mauersberg,Jens Feller

Frankfurt Consulting Engineers GmbHwww.frankfurt-consulting.de

International Conference of Numerical Analysis and AppliedMathematics 2005, Rhodes, Greece

M. Rupp, W. Mergenthaler, B. Mauersberg, J. Feller Minimizing the Operating Costs of Grinding Mills



Outline

Introduction

Markov Model

Reliability Model

Monte Carlo Simulation

Prospects

Background, problem definition

Equilibrium distribution

Mill availability

Ordering strategy

Comparison, future steps




BackgroundProblem DefinitionSummary

Background

Mining equipment:

I Ore extraction → crushing and grinding → further processing.

I Grinding mills (two rolls rotating in opposite directions).

I Rolls are very large and expensive.

I One year to manufacture one, but they last shorter.

Goal:

I Minimize overall costs.

I Keep the mills running.

I Keep costs for rolls low.





Problem Definition: Mill Failures

I A mill fails iff one or both rolls fail.

I A mill can be repaired by replacing both rolls.

I Downtime costs cd(t) = cdown ∗#defect(t).

I µ, σ for roll lifetime, µ̂, σ̂ for mill lifetime.





Problem Definition: Roll Orders

I Roll pairs can be ordered anytime.

I They arrive after a replacement time has passed.

I λ for roll replacement time, λ̃ for roll pair replacement time.

I Costs qcroll immediately, (1− q)croll on delivery.

I Prefinance costs cf (t) = #ordered(t) ∗ croll ∗ q ∗ r .





Problem Definition: Inventory

I Rolls can stay in an inventory.

I Inventory costs ci (t) = #inv(t) ∗ croll ∗ p,where p is the inventory cost factor.





Problem Definition: Costs

I Downtime costs cd(t) = cdown ∗#defect(t).

I Prefinance costs cf (t) = #ordered(t) ∗ croll ∗ q ∗ r .

I Inventory costs ci (t) = #inv(t) ∗ croll ∗ p.

I Total cost c(t) = cd(t) + ci (t) + cf (t).





Problem Definition: Summary

I Grinding mills with two rolls each.

I Rolls are ordered and replaced in pairs.

I Goal is to minimize costs (downtime and ordering related).




ModelEquilibrium DistributionSummary

Markov Model: Ordering Strategy

Simple ordering strategy:

I Fixed size inventory for k roll pairs.I A new roll pair is ordered if

I a roll pair is taken out of the inventory.I a mill breaks and the inventory is empty.

I A mill is repaired as soon as a roll pair is available.





Markov Model: States & Transitions

I Idea: Keep track ofI number of roll pairs in inventory andI number of defect mills

with the states.

I Exponential distribution with rates µ̂′ = 1/µ̂ and λ̃′ = 1/λ̃.

��

��

��

��

��

��

· · · · · · · ·0 1 N N+1 N+2 N+k

R R R R R R R

I I I I I I I

(k+N)λ̃′ (k+N−1)λ̃′ (k+1)λ̃′ kλ̃′ (k−1)λ̃′ (k−2)λ̃′ 1λ̃′

1µ̂′ 2µ̂′ Nµ̂′ Nµ̂′ Nµ̂′ Nµ̂′ Nµ̂′





Markov Model: Equilibrium Distribution

I Goal: Calculate distribution π of states for t →∞.

I Use P[entering state i ] = P[leaving state i ] to obtain

π0(k + N)λ̃ = π1µ̂

πi−1(k + N − i + 1)λ̃ + πi+1(i + 1)µ̂ = πi (k + N − i)λ̃ + πi i µ̂

πN+i−1(k − i + 1)λ̃ + πN+i+1Nµ̂ = πN+i (k − i)λ̃ + πN+iNµ̂

πN+k−1λ̃ = πN+kNµ̂

��

��

��

��

��

��

· · · · · · · ·0 1 N N+1 N+2 N+k

R R R R R R R

I I I I I I I

(k+N)λ̃′ (k+N−1)λ̃′ (k+1)λ̃′ kλ̃′ (k−1)λ̃′ (k−2)λ̃′ 1λ̃′

1µ̂′ 2µ̂′ Nµ̂′ Nµ̂′ Nµ̂′ Nµ̂′ Nµ̂′





Markov Model: Equilibrium Distribution

I Solving the recurrence yields

πi = π0

(k + N

i

)(λ̃′

µ̂′

)i

, 0 ≤ i < N

πN+i = π0

(k + N

N

)(N

i

)(λ̃′

µ̂′

)N+i

N−i i !, 0 ≤ i ≤ k

I With #defect = max(N − i , 0), #inv = 2max(i − N, 0) and#ordered = N + k − i , the total cost c can be computed forgiven k,N.





Markov Model: Summary

I Simple ordering strategy.

I Exponential distributions.

I Equilibrium distribution gives total cost.




Ordering StrategyMill AvailabilitySummary

Reliability Model: Ordering Strategy

I Time ordering so that roll pairs arrive”just in time“,

i.e. before the next mill failure.

I Order earlier to increase chances that roll pairs arrive in time.

I By ordering at time µ̂− ασ̂ − λ̃ after a mill failure, rolls arrivein expectation ασ̂ days before the next mill failure.

I This works only for a lookahead of one, i.e. µ̂ ≥ λ̃ + ασ̂.

I Maintain a reserve of size k.

I Restrict mills to rolls ordered for it and to the reserve.

I Deterministic replacement times.

I Normally distributed mill failure times.





Reliability Model: Mill Availability

For the mill availability µ̂µ̂+m , we need the mean time to repair m:

m = s

∫ ∞

0d F̂ (u)

∫ ∞

u−w(w + v − u) d G̃ (v)

= s

∫ µ̂−ασ̂

0(µ̂− ασ̂ − u) d F̂ (u)

= s(µ̂− ασ̂)F̂ (µ̂− ασ̂)− s

∫ µ̂−ασ̂

0u d F̂ (u) ≤ s(µ̂− ασ̂)Φ(−α)

with w := µ̂− ασ̂ − λ̃, F̂ the lifetime distribution of a mill, G̃ thedeterministic replacement time distribution for a pair of rolls, ands the steady state probability that there is no roll pair in inventory.





Reliability Model: Mill Availability

I Goal: Estimate s = P[no roll pair in inventory].

I Model the request for reserve roll pairs as a Poisson processwith demand rate b := N

µ̂ Φ(−α).

⇒ s = 1− e−bλ̃∑k−1

i=0(bλ̃)i

i! .





Reliability Model: Costs

I k − bλ̃ demands for roll pairs from the reserve.

I A fraction of 1− s can be satisfied.

⇒ k − bλ̃(1− s) roll pairs in inventory on average.

I Nασ̂/µ̂ roll pairs due to arrivals before mill failures.

⇒ #inv = 2(k − N

µ̂ (Φ(−α)λ̃(1− s)− ασ̂)).

I N/µ̂ mill failures per day.

I #ordered/λ̃ rolls arrive ⇒ #ordered = 2Nλ̃/µ̂.





Reliability Model: Summary

I Advanced ordering strategy, but only for lookahead one.

I Normally distributed mill lifetime, deterministic replacementtime.

I Upper bound for total cost by estimating mill availability.




SimulationSummary

Monte Carlo Simulation: Ordering Strategy

I Same as reliability model, but for arbitrary lookahead iby ordering at time i µ̂−

√iασ̂ − λ̃.

I For given µ̂, σ̂, λ̃, determine the lookahead i asmin{i ∈ N|i µ̂−

√iασ̂ − λ̃ ≥ 0}.




SimulationSummary

Monte Carlo Simulation: Summary

I Advanced ordering strategy for arbitrary lookahead.

I No restrictions between the mills.

I Arbitrary distributions.




ComparisonFuture StepsSummary

Prospects: Qualitative Comparison

Trade-off between accuracy and suitability for analysis:

Model Accuracy limitations Analysis limitations

Markov model Simple ordering strategy Exact costsExponential distributions

Reliability model Lookahead one Upper boundNormal mill lifetimesDet. replacement times

Simulation No limitations No analysis





Prospects: Quantitative Comparison

Key figures for reliability model and simulation:

Figure Reliability Simulation

µ̂/(µ̂ + m) 1.99 1.99#inv 0.26 0.44#ordered 1.63 1.08

For N = 2, k = 0, α = 1.5, µ̂ = 475, σ̂ = 47.5, λ̃ = 365,croll = 5 · 105, cdown = 105, p = 0.15, q = 0.2 and r = 0.08.





Prospects: Future Steps

I Expand models to better fit the real situation.I Other ordering strategies.I Different distributions, e.g. lognormal or Weibull.I More realistic cost model.I Roll renewal.

I Extend to other mill components.





Summary

I Application from the mining industry.

I Minimization of operating costs of grinding mills.

I Three models (Markov, reliability, simulation).


markov mills, reliable rolls and monte carlo mines ... · introduction markov model reliability...

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