Delivery Lead Time and Flexible Capacity Setting for Repair Shops with Homogenous Customers N.C. Buyukkaramikli 1,2 J.W.M. Bertrand 1 H.P.G. van Ooijen 1 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 Exogenous Rental Supplier for substitute m/c m/c Resupply Time subs.m/c L units of time 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 Shops Total Costs per unit time: RSTC(µ) = K + c p µ. K: Capacity unrelated costs c p : 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, F µ (.), (f µ (.)) Given µ and L, total cost during downtime cycle TCDT (µ, L) (B > h)

MODEL & ASSUMPTIONS INTEGRATED DECISION MAKING: Assumptions: Minimize TCDT (µ, L) when all info. is available( K, c p, h, B, λ, α, F µ (.), f µ (.) ) (1) Special Case: Jointly Convex when M/M/1 F µ ~ 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

MODEL & ASSUMPTIONS DECOMPOSED DECISION MAKING: Customer Side Information available: h, B Decision to be Given: L Repair Shop Side Information available: c p, K, α, F μ (.) Decision to be Given: μ Lead Time Performance with L i & γ=h/B P(S>L i )=γ Min RSTC(µ) s.t. P(S>L i )=γ p(µ * (L i )), HR(L i )= hazard L i µ * (L i ) Approximate From HR(L i ) New L i Wall of Confidentiality 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/B p(µ * (L)) hL+

Research Question 2 Setting 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 2 Setting 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 + c p µ 1.Savings in the fixed component due to economies of scale in transportation. 1 2 (1-e -D ) /D 1/(1+β 1 D) % Savings in K D 4 failures in a period 4 trucks 4 failures in a period 1 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 c p be the same? ( D=0 ) Compensating differentials? before after D Β 2 =0.1 Β 2 =0.25 Β 2 =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+β 1 D) + p(µ,D) = RSTC(µ,D)/λ + α

DECOMPOSED DECISION MAKING: Customer Side Information available: h, B,D Decision to be Given: L to minimize TCDT Repair Shop Side Information available: c p, K, α, F μ,D (.),D Decision to be Given: µ to minimize RSTC Lead Time Performance with L i & γ=h/B P(S>L i )=γ Min RSTC(µ) s.t. P(S>L i )=γ p(µ * (L i |D),D), HR(L i )= hazard L i µ * (L i |D) Approximate From HR(L i ) New L i Wall of Confidentiality D=0D=0.5D=1D=1.5D=2D=2.5D=3D=3.5D=4D=4.5D=5 i.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 Flexibility 1.Periodic Customer Admissions 4.Applying Labor Economics Concepts to OM models