delivery lead time and flexible capacity setting for repair shops with homogenous customers n.c....
TRANSCRIPT
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Delivery Lead Time andFlexible Capacity Setting forRepair Shops with Homogenous Customers
N.C. Buyukkaramikli1,2
J.W.M. Bertrand1
H.P.G. van Ooijen1
1- TU/e IE&IS 2- EURANDOM
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OUTLINE
• Introduction & Motivation (give some spoilers)
• Literature Review
• Model & Assumptions
• Setting the Scene for Flexibility
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INTRODUCTION & MOTIVATION
• After Sales Services become more important (Cohen et. al, HBR 2006)
• For Capital Goods maintenance• Corrective
• Area of Interest: Capital Goods which are commoditized to some extent:
Forklifts
Trucks
Construction Eq.
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INTRODUCTION & MOTIVATION
• Commoditized Capital Goods Environment• Numerous users• Rental suppliers available• Maintenance
− Hiring a substitute machine during repair
One of the biggest Forklift Supplier & Service Provider in the Benelux Area that has numerous customers (Hypothetically at )
Repair Shop & Rental Store are nearby
Upon a failure a substitute forklift from the rental store can be hired for a fixed amount of time.
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INTRODUCTION & MOTIVATION
RESEARCH QUESTIONS
Given the availability of exogenous rental suppliers: 1. How should the repair shop capacity & hiring duration
decisions be given?Integrated vs. Non-integrated systems
2. What is the role of Lead Time Performance Requirements in the coordination of these decisions?
3. How can one make use of capacity flexibility in this environment?
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LITERATURE REVIEW
• Surveys on Maintenance:• Pierskalla and Voelker (1976), Sherif and Smith (1982), Cho and
Parlar (1990), Dekker(1996), Wang (2002)
• Flexible Capacity Management in Machine Interference Problem:• Crabill(1974), Winston(1977,1978), Allbright (1980)
• Capacity Flexibility Management in Repairable-Item Inventory models:• Gross et al. (1983,1987), Scudder (1985), De Haas (1995)
• Lead Time Management• Duenyas and Hopp (1995), Spearman and Zhang (1999),
Elmaghraby and Keskinocak (2004)
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MODEL & ASSUMPTIONS
Repair Shop
.......
m/c
m/c
m/c
m/c
.......
Exogenous Rental Supplier for substitute m/c
m/c
Resupply Time
subs.m/csubs.m/c L units of time
m/c
.......
m/c
.......
m/c
.......
m/c
m/c m/c
......
subs.m/c
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MODEL & ASSUMPTIONS
Instantaneous Shipment from/to the Repair shop & the Rental Store
Failures ~Poisson (λ) (w.l.o.g λ = 1 failure per week.)
Each failure a random service time at the repair shop
Repair Shop ~ a single Server Queue
Capacity of the Repair shop= Service Rate (interpreted as the weekly working hours)
We pay h$ during L units of time to the rental supplier, (non-refundable)
If (resupply time) > L we loose B$ per unit time until the repaired machine is returned (B>h)
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MODEL & ASSUMPTIONS
Repair Shop’s Total Costs per unit time:
RSTC(µ) = K + cp µ.K: Capacity unrelated costs
cp : Wage factor
Repair Shop: cost -plus (C+) strategy for determining price per repair
p(µ) = RSTC(µ)/λ + α .
µ Sojourn time distribution (density) function , Fµ(.), (f µ(.))
Given µ and L, total cost during downtime cycle TCDT (µ, L)(B > h)
,x L
TCDT L p hL B x L f x dx
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MODEL & ASSUMPTIONS
INTEGRATED DECISION MAKING:Assumptions:
Minimize TCDT (µ, L) when all info. is available(K, cp, h, B, λ, α, Fµ(.), fµ(.))
(1)
Special Case: Jointly Convex when M/M/1Fµ ~ Exponential(µ-λ)
Is Integrated Decision Making Realistic? Confidentiality concerns of the Repair Shop?Reluctant to give repair time distribution…
Laws of Confidentiality Walls of Confidentiality
,min ,L TCDT L
*
1 ln
p
hh
B
c
*ln( )
1 ln
p
hBL
hh
B
c
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MODEL & ASSUMPTIONS
DECOMPOSED DECISION MAKING:
Customer SideInformation available:
h, BDecision to be Given:
L
Repair Shop SideInformation available:
cp, K, α, Fμ(.)Decision to be Given:
μ
Lead Time Performance with Li & γ=h/BP(S>Li )=γ
Min RSTC(µ) s.t.
P(S>Li)=γ
p(µ*(Li )), HR(Li)=hazard rate @ Li
µ*(Li )
Approximate
From HR(Li)
x L
B x L f x dx
New Li
Wa
ll of
Co
nfid
en
tiality
i.1
i.2
i.3
i.4
i i+1
Start from here
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MODEL & ASSUMPTIONS
DECOMPOSED DECISION MAKING:
Lead Time Performance Constraint reduces TCDT(L) to a single variable function
For general service times exponential tail asymptotic (Glynn and Whitt (1994), Abate et al (1995)).
Total area can be derived from the hazard rate at L with µ*(L).
L* (integrated solution)can be reached with an arbitrary precision.
Further savings? Capacity Flexibility
γ=h/Bp(µ*(L))
hL+ * ( )Lx L
B x L f x dx
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Research Question 2Setting the Scene for Capacity Flexibility
Hire Immediately-Send Periodically
• Each failed machine is sent to the repair shop only in equidistant points in time. (Period of length D)
• However a substitute machine is hired immediately (until next period + L)
• Time until next period ~ Uniform(0,D)
• Repair Shop D[X]/M/1,X~Poisson(λD) (Buyukkaramikli et al. (2009))
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Research Question 2Setting the Scene for Capacity Flexibility
Negative Effects
• Additional Hiring Time(hD/2)
• Burstiness in the arrival pattern.
T=0
T=3
T=5
ρ=1.1, λ=1, L:P(S<1) R:P(S<20)
For small values of D, the performance can be better
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Setting the Scene for Capacity Flexibility
Positive Effects
Recall that RSTC(µ) = K + cp µ
1. Savings in the fixed component due to economies of scale in transportation.
1 2
(1-e-D) /D
1/(1+β1D)
% Savings in K
D
4 failures in a period4 trucks
4 failures in a period1 truck
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Setting the Scene for Capacity Flexibility
Positive Effects
2. Certainty in arrival times : Once all the repairs are completed idle (for sure!) at least until the next period.
− Opportunity for capacity flexibility… − Agreement (with the union or individuals) on the Max. number of
working hours per week (µ), payment for actual hours worked (λ)
− Would cp be the same? (D=0) Compensating differentials?
21
pc
D
pc before after
D
Β2=0.1Β2=0.25
Β2=0.5
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Decomposition Method?
The Decomposed Method can be applied mutadis mutandis in this scheme, by updating the cost formulations:
RSTC(µ,D) = K/(1+β1D) +
p(µ,D) = RSTC(µ,D)/λ + α
21
pc
D
,, , ,2 D
x L
DTCDT D L p D h hL B x L f x dx
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DECOMPOSED DECISION MAKING:
Customer SideInformation available:
h, B,DDecision to be Given:L to minimize TCDT
Repair Shop SideInformation available:
cp, K, α, Fμ,D(.),DDecision to be Given:µ to minimize RSTC
Lead Time Performance with Li & γ=h/BP(S>Li )=γ
Min RSTC(µ) s.t.
P(S>Li)=γ
p(µ*(Li |D),D), HR(Li)=hazard rate @ Li
µ*(Li |D)
Approximate
From HR(Li)
,D
x L
B x L f x dx
New Li
Wa
ll of
Co
nfid
en
tiality
D=0D=0.5D=1D=1.5D=2D=2.5D=3D=3.5D=4D=4.5D=5i.1
i.2
i.3
i.4
i i+1
Start from here
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CONCLUSIONS
1. Maintenance Operations of a Commoditized Capital Goods Environment
1. Hiring a Substitute Machine Alternative
2. Decision Making Framework 1. Integrated vs. Decomposed
3. Setting the Scene for Strategic Capacity Flexibility1. Periodic Customer Admissions
4. Applying Labor Economics Concepts
to OM models