delivery lead time and flexible capacity setting for repair shops with homogenous customers n.c....
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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
OUTLINE
• Introduction & Motivation (give some spoilers)
• Literature Review
• Model & Assumptions
• Setting the Scene for Flexibility
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.
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.
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?
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)
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
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)
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
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
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
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
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))
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
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
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
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
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
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