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